Wavelets and renormalization

0 -

i:


A j' ,j" ,j'"

j;

)0

distinct

fl

( ",P.;", ~(f;)),1. dx (~.(X)~(f".,»), + ~A' [4> X

X

i i i i

+ (4

2

(fl ~(f;)),

j', j"~tinct #~j"
dx

dx' (
dx

dx' (
+ 2!

G)

X

dx

2 j' ,j" ,j'"

i i

dx'

j;

distinct (

)3!

(

#j'~,j(4)


0

0

(
X (
))0(
+

2

4

j' , ..

,j(~istinct #j'~,j(6)
0

1.13. THE NEED FOR PHASE CELL ANALYSIS

99

(1.13.2)

The structure of each term can be represented by a graph in the following manner. Associate a line segment to each covariance that includes at least one integration variable over A. Let each such integration variable be represented by a point joining all lines whose covariances include that variable. The resulting graphs are Feynman graphs, and the above expansion is represented as follows:

(fl

(h) exp (-,\

I:

CJL,., ~(fj)U

,,(X)4 : dX)

X

,j"~tinct #~j" j#~,j(4) II

(

3! j'

(

(fj)) 0

xlI

~

(h) )

>eX

(h)) 0

- , \ j', ...

,j(~istinct

~A'HQ~(fj))H §

+

2

+4

(fl

)0=

0

I I -fJ2

+ 4

j' , ...

+

Gr

L

2 !,

J , .. ,j(4) distinct

,j(~istinct (#j'~,j(6)

(fj) )

X

0

r

(1.13.3) If both endpoints of a line represent integration variables, that line is an internal leg. The points representing integration variables are vertices. If only one endpoint of a line is a vertex - i.e., if the covariance involves a test function - that line is an external leg .

No vertex is ever connected to itself by a leg, as the Wick ordering of the interaction rules that out. Now in dimension d = 2, all of the integrals represented by these Feynman graphs are convergent in the K. = 00 limit . Indeed, all higher-order graphs represent convergent integrals in the ultraviolet limit, and this observation is the perturbative way to understand why Wick ordering is the only renormalization required in two space-time dimensions. In dimension d = 3, some of these graphs represent divergent integrals, but only a mass renormalization and a couple of energy renormalizations are needed to cancel the ultraviolet divergences. Indimension d =4, the divergences are so severe

100

CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS

that every possible ultraviolet renormalization of the physical parameters is believed to yield a trivial (Gaussian) Euclidean field theory. For a Euclidean field theory as simple as a self-interacting scalar field, dimension d = 3 seems to be the most challenging case available. We devote the remainder of our discussion to the scalar quartic interaction in three space-time dimensions - the ~ field theory. In this case it is easy enough to check that among the second-order Feynman graphs in (1.13.3),

(i are convergent in the

K,

=

00

X

)2

Fig. 1.13.1:

limit, while

Fig. 1.13.2:

are divergent. (The second and third graphs in Fig. 1.13.1 are actually convergent in arbitrary dimension, while the first is divergent for d = 4.) If we write the third-order terms of the perturbation series, the Feynman graphs are:

iii

iii @ ii @ i

@

il -fJ- I ill II

g

Iii

X

ii

'KX

~

I

ill

::><:::::>c::::

X

i

<>( X

X Fig. 1.13.3:

Iii A

Iii ~ (i iii £)

X

r,

Fig. 1.13.4:

The graphs in Fig. 1.13.4 converge in the K, = 00 limit, while the graphs in Fig. 1.13.3 are divergent. Obviously, the number of topologically different Feynman graphs escalates with the order, so the ultraviolet renormalization problem may appear formidable

1.13. T H E NEED FOR PHASE CELL ANALYSIS

101

just from a combinatorial and algebraic point of view. However, the problem can be simplified by the notion of primitive graph. A primitive graph is a connected divergent graph which becomes convergent if any one of its internal legs is replaced by two external legs. The second graph in Fig. 1.13.2 and the first graph in Fig. 1.13.3 are both primitive; there are no others in the entire perturbation series. Moreover, every connected divergent graph other than the third-order primitive graph is divergent only because it contains at least one second-order primitive graph as a subgraph. For convenience, we denote the basic quantities associated with these two primitive graphs by:

Note that for an arbitrary set of points XI,. . . ,X J ,

+ 4a2 j' , j l ' distinct

(

@ x j ) o j+j1.jtr

+

7

(1.134

while

It is a fundamental algebraic fact that the power series in X of the free expectation

is precisely the series obtained from the power series in X of

(fi j=1

m(ji) exp (-A

L

: @ , ( x ) ~: dx

by discarding all graphs containing primitive graphs as subgraphs. The algebra is a variation on the Linked Cluster Theorem, which we shall not pursue here, but will discuss in Chap. 3 while introducing polymer expansions. The point here is that a simple, direct modification of the interaction generates a perturbation series all of whose terms are finite because the divergent Feynman graphs are automatically omitted.

102

CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS

On the other hand, this alteration is no renormalization - specifically, the new term NII.,,,(iP) in the interaction is not a mass renormalization. Actually, £A~~ poses no problem because it has no iP-dependence; it is just an energy renormalization. To implement a mass renormalization, define the parameter shift by

8m~,,,(x)

dx'

= 48>.2 [

(iP,,(x)iP,,(x'))~

(1.13.8)

and consider the problem of replacing Nil., " (iP) with the local term MII.,,,(iP)

= [ dx8m~,,,(x)

: iP,,(X)2 : .

(1.13.9)

On the other hand, (1.13.6) indicates that : NII.,,,(iP) : +£,e, 48>.2 [

dx [ dx'

(1.13 .10)

(iP,,(x)iP,,(x'))~ .

(1.13.11)

Moreover, it is the difference 48>.2 [

dx [

dx'

(iP,,(x)iP,,(x'))~

x : iP,,(x)(iP,,(x) - iP,,(x')) :

(1.13.12)

that contains no divergences as K, -t 00, so the additional energy renormalization £A~~ is required in this formal, perturbative scheme. Thus (1.13.13) is the actual renormalization of the interaction, while the renormalization cancellations are implemented term-by-term in the formal perturbation series by writing

The random variable (1.13.12) controls the difference between the algebraic cancellation of divergences and the renormalization cancellation. Note that

[<XX>

-2

+

], (1.13 .15)

which can be shown to be finite in the K, = 00 limit - i.e., the divergences in the diagrams cancel - so this random variable is square-integrable with respect to the free measure dJ.Lo .

1.13. THE NEED FOR PHASE CELL ANALYSIS

103

It has already been emphasized that the power series in the coupling constant ).. cannot converge, so term-by-term control of the series does not establish rigorous control of the model. In §1.11 the key importance of an ultraviolet stability bound (1.11.29) to a renormalized interaction was emphasized. In our case, the required bound is

Unfortunately, the underlying K, = 00 measure is actually singular with respect to the free measure dflO, so the exponential cannot be dflo-integrable. Thus, the Glimm-JaffeNelson ultraviolet expansion, which involves U(dflo)-estimation, cannot be applied here. The Guerra-Rosen-Simon results, while independent of dimension, are useless for the same reason. On the other hand, the Seiler-Simon estimation used to derive A-uniform bounds on normalized expectations can be adapted to this model because it depends on ultraviolet stability as an input bound and exploits OS positivity. The greatest difficulty lies in establishing (1.13.16), and what is significantly different about this problem in dimension d > 2 is that the ultraviolet limit and infinite-volume limit are not so neatly separated. Phase space must be decomposed into phase cells, subject to the constraints of the uncertainty principle.

References 1. T . Balaban , "Ultraviolet Stability in Field Theory: The c/>~ Model ," in Scaling and Self-Similarity in Physics, J . Frohlich, ed., Birkhiiuser, Boston, 1983. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical c/>~ Model," Commun. Math. Phys. 88 (1983),263-293. 3. G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicolo, E. Oliveri, E. Presutti, and E. Scacciatelli, "On the Ultraviolet Stability in the Euclidean Scalar Field Theories," Commun. Math. Phys. 11 (1980),95-130 . 4. A. Bovier and G . Felder, "Skeleton Inequalities and the Asymptotic Nature of Perturbation Theory for <J>4- t heories in Two and Three Dimensions," Commun. Math. Phys . 93 (1984), 259-275. 5. D. Brydges, J . Frohlich, and A. Sokal, "A New Proof of the Existence and NonTriviality of the Continuum c/>~ and c/>~ and Quantum Field Theories," Commun. Math. Phys. 91 (1983), 141-186. 6. P. Federbush, "Unitary Renormalization of (c/> 4 h+1 ," Commun. Math. Phys . 21 (1971), 261-268. 7. J . Feldman, "The )..c/>~ Field Theory in a Finite Volume," Commun. Math. Phys. 37 (1974) , 93-120. 8. J . Feldman and K. Osterwalder, "The Wightman Axioms and the Mass Gap for Weakly Coupled (c/> 4 h Quantum Field Theories," Ann. Phys. 97 (1976) , 80-135.

104

CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS

9. G. Gallavotti, "Renormalization Theory and Ultraviolet Stability for Scalar Fields via Renormalization Group Methods," Rev. Mod. Phys. 57 (1985),471- 561. 10. J. Glimm, "Boson Fields with the ¢4 Interaction in Three Dimensions," Commun. Math. Phys. 10 (1968) , 1-47. 11 . J. Glimm and A. Jaffe, "Positivity of the ¢~ Hamiltonian," Fortschr. Phys. 21 (1973) , 327- 376. 12. K. Hepp, TMorie de la Renormalisation, Springer-Verlag, New York, 1970. 13. A. Jaffe, "Divergence of Perturbation Theory for Bosons," Commun. Math. Phys. 1 (1965) , 127-149. 14. J . Kogut and K. Wilson, "The Renormalization Group and the c:-Expansion," Phys. Rep. 12 (1974), 75-200. 15. J . Magnen and R. Seneor, "The Infinite Volume Limit of the ¢~ Model," Ann. Inst. H. Poincare 24 (1976) , 95-159. 16. Y. Park, "Convergence of Lattice Approximations and Infinite Volume Limit in the p.¢4 - U¢2 - l.ufJ)S Field Theory," J . Math. Phys. 18 (1977),354-366. 17. V. Rivasseau, From Perlurbative to Constructive Renormalization, Princeton University Press, Princeton, 1991. 18. E. Seiler and B. Simon, "Nelson's Symmetry and All That in the YUkawa2 and ¢~ Field Theories," Ann. Phys. 97 (1976), 470-518. 19. A. Sokal, "An Alternate Constructive Approach to the ¢~ Quantum Field Theory, and a Possible Destructive Approach to ¢:," Ann. Inst . H. Poincare A37 (1982) , 317-398. 20. E. Speer, "Analytic Renormalization ," J . Math. Phys. 9 (1968), 1404-1410. 21. K. Symanzik, "Small Distance Behavior in Field Theory and Power Counting," Commun. Math. Phys. 18 (1970),227- 246.

Chapter 2

Wavelets - Basic Theory and Construction Phase space analysis is any useful decomposition of a function into modes that are well-localized and - at the same time - have a small spread in momentum. The properties of the Fourier transform impose limitations on such a decomposition at the very outset, but the need for phase space analysis in quantum field theory has already been made clear in Chap. 1. It is important in other areas as well - e.g., partial differential equations and signal analysis. One limitation is that a mode cannot be sharply localized in both momentum and position - i.e., no compactly supported function has a compactly supported Fourier transform. Another limitation is that - while exponential localization in both momentum, and position is possible - a small standard deviation in momentum (resp. position) creates a lower bound on the standard deviation in position (resp. momentum) by the uncertainty principle. Further limitations are inevitable if one desires the set of modes (expansion functions) to be coherent in some sense - e.g., related to one another through group operations. Perhaps the oldest type of phase space analysis is the discrete windowed Fourier transform - an analysis based on expansion functions of the form

with 6 and ranging over cubic lattice sites in d-dimensional space with integer coordinates. These functions are coherent in the sense that they are generated by discrete translates in 2d-dimensional phase space with f as the generating function. Naturally, the regularity and decay properties of f are an important issue. In particular, if f is the characteristic function of the unit cube [0, lId, then each expansion function is sharply localized, but discontinuous. Moreover, {f;;) is an orthonormal basis for the Hilbert space L ~ ( R in ~ )this case; indeed, the decomposition is just a cubic partition of @ with Fourier series analysis applied on each cube. The jump discontinuities are the obvious drawback, as they correspond to poor localization in momentum sp_acz This implies that singularities in the function analyzed are not reflected by the (m, n)-dependence of the expansion coefficients. Can a more regular f be found such

106

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

that {f1ii;} is still an orthonormal basis? The answer is yes, but only by sacrificing localization; the trade-off is very strict, as it turns out. It has been shown by Balian [B9] and also by Low [L47] that if {f1ii;} is an orthonormal basis, then the Heisenberg uncertainty of f must be infinite! This means that no orthonormal basis of this type can be much better than the one generated by the characteristic function of the unit cube. The common practice in phase space localization of this type is to sacrifice orthogonality to obtain more regularity for f . The obvious constraint is that the calculation of expansion coefficients for a given function must still be realistic. One of the earliest proposals - independently examined by Gabor [G1] and von Neumann [unpublished] - was to choose f as a Gaussian function peaked at the center of the unit cube - a function with the best possible phase space localization. The price of this analysis was immediately evident in the over-completeness of the set {f1ii;} generated. Expansion coefficients were not unique for an arbitrary function decomposed in this way. Indeed, such an expansion was known to be numerically unstable in the sense that the sum of squares of inner products of these functions with the function to be analyzed could be arbitrarily small relative to the L 2-norm of the given function. On the other hand - since this set is over-complete anyway - one can enhance this overcompleteness by generating a set with phase space translates smaller than unity in the spatial directions or smaller than 211" in the momentum directions. Expansions of this type are numerically stable, and they have been exploited in signal analysis despite the computational inconvenience due to the redundancy (see, e.g., Baastians [B1] and Janssen [J8-J10J) . This decomposition scheme is an example of a frame - a certain generalization of orthonormal basis where the elements are not even necessarily linearly independent. Frames were first studied in the context of nonharmonic analysis by Duffin and Schaeffer [D28] and can be defined for an abstract Hilbert space. The frame property is a generalization of Parseval's identity to a double inequality for the sum of squares of inner products, where the upper and lower bounds are proportional to the square of the norm of the test vector. From a computational point of view, frame expansions are "painless non-orthogonal expansions," and their usefulness to phase space localization in signal analysis was widely advertised by Daubechies, Grossmann, and Meyer [D4]. They even constructed frames generated by a-translates in position space and translates < 211"/ a in momentum space where the generating function f is class Coo with compact support. Even as they established impressive results for the old-fashioned windowed Fourier transform, Daubechies, Grossmann, and Meyer were already engaged in the development of a rather different type of phase space analysis based on scaling - namely, wavelet analysis . The meaning of the word "wavelet" has varying degrees of generality in the literature, but the coherence of a wavelet basis or wavelet frame usually involves scaling operations. Indeed, the orthogonality of discrete scalings is a property that shall be included in our own definition of the word, so there are results on wavelet frames that lie outside the scope of our concerns. For our purposes, a dyadic wavelet is a square-integrable function llJ such that (2.2)

for all non-zero integers r. Note that this definition does not require orthogonality of lattice translates on the same scale or completeness of the basis generated. Negative values of r correspond to dyadic scalings of 9 to large length scales, while positive values of r yield small length scales. Throughout most of this chapter we concentrate on dimension d = 1 (the dimension appropriate to signal analysis in any case) and extend the wavelet constructions to arbitrary dimension in the very last section. This decision is due, in part, to the ease with which the L2-orthogonal constructions extend to the multi-dimensional case. On the other hand, we shall also construct Sobolevorthogonal wavelets in that section. Obtaining well-behaved wavelets of this type is more involved because the coherent structure is rectangular while the Sobolev norm is elliptic. The earliest example of an orthonormal wavelet basis was constructed by Haar [H6] decades before the subject was born. It is a dyadic scale hierarchy - including arbitrarily large and arbitrarily small length scales - of piecewise constant functions whose integrals vanish. On the same scale, the discrete translates are orthogonal because they have disjoint supports, while functions on differing scales are orthogonal because the smaller-scale function is supported on some rectangle over which the larger-scale function is constant. This type of decomposition has the same regularity and decay properties as the windowed Fourier decomposition with the characteristic function the functions are discontinuous, but their localization is sharp. The Haar basis was originally introduced in the context of a general study of orthogonal systems, and at the time, no one seemed to have any curiosity about the existence of similar bases of more regular functions. The special features of the construction may have led some mathematicians to suspect that nothing better than the Haar basis was possible. Since that time, ideas involving scaling transformations have slowly emerged in other areas. In statistical mechanics, for example, Wilson introduced the renormalization group [W16] as a means of isolating the long-distance behavior of the generalized Ising model. The basic transformation averages spin values over a block of sites with the cubic lattice partitioned into such blocks and the new configuration defined by scaling the coarser lattice back to the unit lattice. The Gibbs measure itself is transformed by integrating out the fluctuations and applying the additional transformation induced by the scaling. Iteration of this transformation has the effect of reducing the long-distance decay of correlation functions to behavior at large length scales. Moreover, in his effort to justify a rather drastic approximation aimed at obtaining qualitative results, Wilson assumed the existence of a wavelet basis [W17]. At roughly the same time, Glimm and Jaffe were developing phase cell methods in their effort to construct the pure scalar quantum field theory with quartic selfinteraction in three space-time dimensions [G45, G57]. We have already discussed this model in Chap. 1, where we advertised the need for phase space analysis. While their decomposition of phase space into phase cells was primitive and their expansion method complex, the convergence proof contained several ingenious arguments that involved scaling. In signal analysis, the use of scaling as an analytic tool was first proposed by Grossmann and Morlet [G96], inspired in part by the work of Morlet et al. [M48]. Specifically, they applied the wavelet transform to the analysis of seismic data in geophysics, using a basic signal function f (t). The one-dimensional wavelet transform \EE of an arbitrary

108

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

function E(t) is given by QjE

1 (a, s) = -

m

dt f (a-l(t - s))*E(t),

-,

a

# 0,

(2.3)

with a fixed choice of f (t). Subsequently, Grossmann, Morlet, and Paul studied this integral transform as a square-integrable representation of the affine group [G97, G981. The resolution of the identity (i.e., the inversion formula) is given by

((t) =

1,

1 da lal-512 , c(f) -

1, w

ds (a-'(t

- s ) ) 4 ( a ,s),

(2.4)

The finiteness of the constant c(f) is a condition on the basic signal f ( t ) ,which is regarded as a "wavelet" in this context. Note that if f ( w ) is continuous, then f(0) = 0.-This weak vanishing-moment condition is satisfied by wavelets in virtually every context, independently of how they are defined. Meanwhile, Stromberg constructed a one-dimensional orthonormal basis of wavelets more regular than the Haar wavelets [S85]. Inspired by the construction of an orthonorma1 basis for L2([0,11) due to Franklin [F53] - which is more regular than the Haar basis but has little wavelet coherence - Stromberg derived a piecewise linear, continuous function with exponential decay that generates an orthonormal basis for L2(R) through dyadic scaling and discrete scale-commensurate translation. This achievement attracted little attention at the time, as it occurred in the context of harmonic analysis, and interdisciplinary communication failed in this instance. In mathematical physics at about the same time, Gawedzki and Kupiainen were engaged in establishing a sound mathematical basis for the renormalization group ideas of Wilson. For the transformation they rigorously determined the flow of the iteration in the neighborhood of the Gaussian fixed point for the dipole gas [G16, G251. The type of phase space localization used for the definition of the renormalization group transformation varies with the context, but in the case where the transformation is based on block spin averaging, the constrained minimization technique for integrating out the fluctuations can be adapted as a method for constructing wavelets (see, e.g., [B12, B131). Concurrent to the work of Gawedzki and Kupiainen was the work of Federbush on a class of infrared problems [FlO]. Instead of using the renormalization group formalism, however, he developed a phase cell cluster expansion in much the same spirit as Glimm and Jaffe, except that his phase cells were expansion functions. Not only was his orthonormal basis scale-coherent, but it was a polynomial generalization of the Haar basis! His construction combined the Haar scheme with the homogeneity of monomials to yield a coherent basis of piecewise-polynomial functions with as many vanishing moments as one wished. The vanishing moments directly implied the interscale-orthogonality but were most important to the multi-scale decomposition of long-distance singularities. On the other hand, short-distance number divergences in the cluster expansion of a Euclidean field theory could not be cancelled in this phase cell formalism for a technical reason related to the jump discontinuities in these wavelets, so

109

the rigorous treatment of ultraviolet problems in this formalism had to wait on the discovery of smoother wavelets [BI2. BI3). The Stromberg construction was unknown to the mathematical physics community, but the Stromberg wavelet would have required modification, as it does not have an abundance of vanishing moments. Interest in wavelets finally quickened when Meyer constructed an orthonormal basis of wavelets that were smooth and yet of rapid decrease [M40). Indeed, the Fourier transform of a Meyer wavelet is class e oo with compact support! On the other hand, if a function has compact support in momentum space, it cannot have exponential decay in position space - only the rapid decrease, at best. Any basis of wavelets that are class eN for conveniently chosen N and have exponential decay would be more useful to mathematical physics in general. Subsequent to the Meyer construction, Lemarie found such orthonormal bases - one for each degree N of smoothness [L22) . Mallat and Meyer almost immediately realized that the construction of a variety of wavelets could be organized into a scheme which they dubbed multi-scale resolution analysis [M41 , M42). Any input function for this constructive machine is called a scaling junction, and its regularity and decay properties are shared by the wavelet produced. This point of view was central to the decomposition and reconstruction algorithm of Mallat [M8MIO], and that algorithm, in turn, eventually led to Daubechies' discovery of compactly supported wavelets with a finite but arbitrary degree of smoothness [DI). As in the case of windowed Fourier analysis, there was immediate interest in the advantages gained by sacrificing orthogonality in favor of some weaker property. One idea - already suggested by the constrained minimization technique [BI2) - was to sacrifice intrascale orthogonality while retaining interscale orthogonality. In fact, the variational approach differs from the multi-scale resolution analysis approach, since wavelets that are only interscale-orthogonal are derived first . In the Lemarie case, these wavelets are more explicit than those comprising the orthonormal basis. At the same time, the realization of the dual basis - necessary for the calculation of expansion coefficients - is equally explicit because the interscale orthogonality reduces it to a single-scale derivation (see, e.g., [BI7) and [CIO)) . Chui and Wang subsequently constructed a basis of this type where the wavelets are compactly supported splines [Cll, CI2). This is a computational advantage over Daubechies wavelets, which are not splines. On the other hand, the wavelets dual to the Chui-Wang wavelets do not have compact support. Another industry was the construction of wavelet frames, which we shall not pursue. However, they have had great impact on signal analysis, and it is worth remarking that Daubechies, Grossmann, and Meyer were the founders [D4) . Yet another generalization of an orthonormal basis is a bi-orthonormal basis, and the multi-scale resolution analysis for constructing bi-orthonormal wavelet bases was introduced by Cohen, Daubechies, and Feauveau [CI7) . In particular, they constructed a bi-orthonormal basis of compactly supported class eN wavelets that are more amenable to decomposition and reconstruction algorithms than are the Daubechies wavelets. We shall not pursue this construction either. The aim of this chapter is an introduction to wavelets suitable for Euclidean field theory and to rigorous results on the properties of wavelets. In the first section we discuss the limitations of the discrete windowed Fourier transform, including the BalianLow Theorem. In the following section we introduce the multi-scale resolution anal-

110

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

ysis of Mallat and Meyer, and we apply it to the construction of Meyer wavelets and Lemarie wavelets. We devote the third section to a sketch of the Daubechies construction of compactly supported wavelets. Actually, in mathematical physics there never seems to be any need for compactly supported wavelets; exponentially localized wavelets are always good enough for our purposes. We describe the construction of Daubechies wavelets because it is the crown jewel in wavelet analysis. The succeeding section is devoted to the important issue of vanishing moments. We show how the condition of interscale orthogonality implies that the number of vanishing moments for a wavelet is comparable to its degree of smoothness. This relationship, in turn, implies that interscale-orthogonal wavelets - functions that meet our own definition of "wavelet" - cannot be exponentially localized in both position space and momentum space. In §2.5 and §2.6 we investigate further theoretical restrictions on wavelets in the form of Heisenberg inequalities. In particular, the position-momentum uncertainty of a real-valued wavelet in one dimension is 3/2 instead of the universal 1/2. In §2.7 we describe the constrained minimization method for constructing wavelets. This variational approach can be used to construct a variety of wavelets - including Meyer wavelets and Lemarie wavelets - but it cannot produce Daubechies wavelets. Following this point of view, §2.8 is devoted to a description of the Lemarie-type basis that is not intrascale-orthogonal. For the mother wavelet there is an explicit expression depending on the zeros of Euler-Frobenious polynomials in the unit disk. We devote §2.9 to a description of the Chui-Wang wavelets. Finally, we introduce the extension of the wavelet constructions to the multi-dimensional case in §2.10. At the same time, we construct exponentially localized Sobolev-orthonormal wavelet bases, using the variational method. The multiscale resolution analysis, which is suited for the construction of Daubechies wavelets, does not appear to be suited for the construction of Sobolevorthogonal wavelets with good localization. On the other hand, Lemarie has shown that it can be generalized to construct a Sobolev-bi-orthonormal system of compactly supported wavelets [L26] .

2.1

The Balian-Low Theorem

In the search for orthonormal bases of L2(~) that localize phase space, perhaps the most obvious type to try is a basis of the form

B={ei21TmXf(x_n): m,nEZ},

(2.1.1)

where f is a square-integrable function . The example that comes to mind most readily is f = X, where X is the characteristic function of the unit interval [0,1]. In this case, one is simply applying the orthogonal decomposition 00

L2(~) =

ffi

L2([n,n + 1])

(2.1.2)

n=-oo

to the Hilbert space and then using Fourier series analysis on each interval. Unfortunately, X is a discontinuous function, which means its Fourier transform has peor

2.1. THE BALIAN-LOW THEOREM localization. The latter is regular but not integrable. Indeed,

where p = 0 is a removable singularity. It is natural to wonder if an orthonormal basis of the above form is possible for a function f that has better localization in phase space. Roughly speaking, one would be partitioning phase space into "phase cells" with a rectangular lattice structure given by Fig. 2.1.1. The basis decomposition of an arbitrary function would yield its spread over those phase cells, where the area of each phase cell is 27r. Obviously, sharp localization of a basis function in its associated phase cell is impossible, as no function can have compact support in both position space and momentum space. However, a Gaussian function has excellent decay in both position and momentum. Can such a function generate any kind of basis with these phase space translates? Obviously, a set of the form

cannot be orthonormal, but it does span L2(R); indeed it is over-complete. This set has been called the uon N e u m a n n basis in the physics community and the Gabor basis in the mathematics community. In spite of its drawbacks, this "basis" has actually played a role in the understanding of phase space localization. In particular, the study of this set led to some intuition that resulted in the Balian-Low Theorem. This result states that if a square-integrable function f generates an orthonormal basis of the form (2.1.1), then either fl(x) or x f (x) fails to be square-integrable. The original proof was actually quite ingenious: it was a topological argument involving the winding number of the phase of an expression. There is now a more elementary proof available, which we present here.

Figure 2.1.1:

112

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Suppose that P f and X f both lie in L2 (JR). translation operator defined by

(TmnCP)(x)

Let Tmn denote the phase space

= ei27rmxcp(x -

n).

(2.1.5)

By hypothesis, {Tmnf: m, n E Z} is an orthonormal basis, so by Parseval's identity,

(Pf,Xf)

= "I)Pf, Tmnf) (Tmnf, Xj).

(2.1.6)

m,n On the other hand,

(f, [P, Tmnlf) (f[Tmn, Xlf)

= 0, -n(f, Tmnf) = 0,

27rm(f, Tmnf)

(2.1. 7) (2.1.8)

so the expansion may also be written as

(Pf,Xf) = "I)T- m,-nf,Pf)(Xf,T-m,-nf), m,n

(2.1.9)

which - again by Parseval's identity - implies

(Pf,Xf)

= (Xf,Pf).

(2.1.10)

Since [X, Pl = i, we have the desired contradiction. Actually, one can get around this difficulty if some of the discrete translational symmetry in phase space is sacrificed. Orthonormal bases of the form

B

=

{sin(27rmx)f(x-n): nEZ,mEZ+}U {cos(27rmx)f(x - n): nEZ, mE Z+ U {O}}

(2 .1.11)

have been constructed where f and j both have exponential decay. Such a basis is a Daubechies-Jaffard-Journe basis. Obviously, a given basis element is roughly localized on a phase cell defined by the intersection of a set of the form

{(x,p) : n::;x::;n+l} with a set of the form

{(x,p): 27rm - 71'::; p::; 27rm + 71' or - 27rm - 71' ::; p::; -27l'm + 7r}.

i- 0, the cell consists of two disjoint rectangles, each with area 271', and there are two basis elements associated to this disconnected cell . If m = 0, the cell is a single rectangle, and it has only one basis element. The two separated peaks in momentum space are what make such a tremendous difference in the phase space localization. It has been shown that if one insists on an orthonormal basis of the form

If m

B = {fm(x - n): m,n

E

Z}

(2.1.12)

2.1. THE BALIAN-LOW THEOREM

113

im

localized roughly at 27rm (but not necessarily a momentum-space translate with of by 27rm units), then one can beat the conclusion of the Balian-Low Theorem, but only marginally. X f and P f can both be square-integrable, but either IX l1+e f or IPI1+e f will fail to be. There has long been an industry in phase space localization where one works with a set of the form B = {ei27ramx f(x - /3n) : m,n E Z} (2.1.13)

io

with a/3 < 1. The inner products of an arbitrary square-integrable function with these functions constitute the discrete version of a windowed Fourier transform of that function, and there is a trade-off implicit in the choice a/3 < 1. The expansion functions can have better phase space localization, but they cannot be orthonormal. The latter point can be established with an elementary calculation. Indeed, suppose a set of this form is an orthonormal basis. Then for an arbitrary square-integrable function 'P,

II'PII~

~ a-I

11.: ~ 1.:

2

'P(X)e-i2rrmax f(x - /3n)* dx I 'P(X)'P(X - a-lf)* f(x - /3n)*

f(x - /3n - a-lf)dx

(2.1.14)

by Parseval's identity and Poisson summation. Notice that if the support of'P lies in an interval of length a-I, then only the f = 0 term remains:

II'PII~ =

a-I

L n

1.:

1'P(xWlf(x - /3n)1 2dx,

diam supp

'P:S a-I

(2.1.15)

Since 'P is arbitrary otherwise, it follows that

L Applying the integration

I:

If(x - /3nW

= a.

(2.1.16)

n

dx to this equation, we obtain

1.:

2 If(x)1 dx

= a/3.

(2.1.17)

But IIfll2 = 1, so we have established the claim that the inequality a/3 < 1 rules out an orthonormal basis. On the other hand, the lack of orthogonality does not necessarily pose computational problems in the phase space analysis of an arbitrary function. The windowed Fourier analysis is done with sets that happen to be frames - expansion functions for which the orthogonality property is replaced by a similarity property. We give the definition in the abstract setting. A given set {gd of vectors in a Hilbert space is said to be a frame if Ilgk II = 1 and there are constants ao, al > 0 such that

aoll'P112 :S

L 1('P,gk)1 2 :S alll'P1I 2 k

(2.1.18)

114

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

for all vectors
= ao,

in which case we have (2 .1.19)

k

In the case ao = 1, such a frame becomes an orthonormal basis, but in any case, frames have virtually all the nice properties of an orthonormal basis. If {gd is a frame, then it automatically has a dual frame {gd - not to be confused with a dual basis. In general, (2 .1.20)

but one now has a pair of Parseval-type identities: '2.)
= (
(2.1.21.0)

= (
(2.1.21.1)

k

~)
Existence is an abstract result, and does not determine whether {gd has any of the coherent structure that {gd happens to have. The point here is that one can construct frames for L2(JR) of the form (2.1.13), and that if aj3 < 1, it is possible for f to be class Coo with compact support! Even tight frames with this much phase space localization are possible if aj3 < 1. As far as the value of the constant ao is concerned, the reasoning given by (2.1.14)-(2 .1.17) above implies

L If
I = :j3I1
i2

" " m x f(x - j3n)* dx

(2 .1.22)

m,n

for a tight frame. In particular, we may infer that aj3 = 1 is not only a necessary condition for orthogonality, but also sufficient in the case where the frame is already tight. We have briefly discussed the discrete windowed Fourier transform because it is a familiar approach to phase space localization. It is inappropriate for our purposes, however. The problems of constructive quantum field theory fall into two categories infrared (long-distance) problems and ultraviolet (short-distance) problems. To isolate either the infrared or the ultraviolet behavior of a model, it is more natural to consider different scales of fluctuations . For this reason, we shall use a very different type of phase space decomposition known as a wavelet decomposition. For a fixed positive integer N , a square-integrable function W is an N-adic wavelet if and only if

f

w(x)*W(NTx - n)dx

= 0,

n E Z,r E Z\{O}.

(2.1.23)

We intend to study orthonormal bases of the form (2.1.24)

where 2T / 2 is the obvious L 2 -normalizing factor. Notice, however, that our definition of a wavelet requires neither orthogonality of discrete translates at the r = 0 level

2.1. THE BALIAN-LOW THEOREM

115

nor completeness of the set generated - only orthogonality to N-adic scalings and their discrete scale-commensurate translates. For convenience, we sometimes use the notation (2.1.25) The discrete translates for 'l1 r (x) are therefore 'l1 r (x - 2- r n) with n E Z. According to the convention we have already adopted, 2- r is the length scale at the rth level, so the positive (resp. negative) values of r correspond to the small (resp . large) length scales. In the case of a basis of the form (2 .1.24), 'l1 is referred to as the mother wavelet. If 'l1 is well-localized in phase space, then with (2.1.26) the wavelet basis is represented by the partition of phase space given by Fig. 2.1.2. Many of the wavelets constructed will be real-valued, in which case their phase cells must be symmetric about p = O. On the other hand, the disjoint packing of scaled phase cells separates them from p = 0, and so the cells are disconnected as in the case of the Daubechies- Jaffard-Journe basis. Perhaps the most important observation to be made about this admittedly crude picture is the suggestion that

~(O)

= O.

(2 .1.27)

This is a vanishing moment property, and all of our wavelets will have it. Indeed, the number of vanishing moments a wavelet has will be important.

P

x-I -

I I I '/ ' // /

I/

I p=rt x

Figure 2.1.2: An interesting variety of wavelet bases have been constructed only recently, but the oldest example has been around for some time. It is the Haar basis (i.e. , the

116

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

"square-wave" basis), and it is defined by

o ~ x < ~, ~~x~1.

(2.1.28)

It is easy to convince oneself that

(2.1.29) is indeed an orthonormal basis for L 2 (IR). Also notice that

~(O) = /

w(x)dx

= O.

(2.1.30)

The usefulness of this basis is limited by t he poor localization in momentum space due to the jump discontinuities in position space.

References 1. M. Baastians, "Gabor's Signal Expansion and Degrees of Freedom of a Signal," Proc. IEEE 68 (1980), 538- 539. 2. R. Balian, "Un Principe d' Incertitude Fort en Theorie du Signal on en Mecanique Quantique," C.R. Acad. Sci. Paris 292 , Serie 2 (1981). 3. G. Battle, "Heisenberg Proof of the Balian-Low Theorem," Lett. Math . Phys. 15 (1988),175-177. 4. J . Bourgain, "A Remark on the Uncertainty Principle for Hilbertian Basis," J . Func. Anal. 79 (1988) , 136- 143. 5. I. Daubechies, "The Wavelet Transform, Time-Frequency Localization , and Signal Analysis ," IEEE Trans. Inform. Theory 36-5 (1990) ,961- 1005. 6. I. Daubechies, S. Jaffard, and J. Journe, "A Simple Wilson Orthonormal Basis With Exponential Decay," SIAM J . Math. Anal. 22-2 (1991) , 555-573. 7. I. Daubechies, A. Grossmann , and Y. Meyer , "Painless Nonorthogonal Expansions," J . Math . Phys. 27-5 (1986) , 1271- 1283. 8. R. Duffin and A. Schaeffer , "A Class of Nonharmonic Fourier Series," Trans. Amer. Math . Soc. 72 (1952),341- 366. 9. D. Gabor, "Theory of Communications," J . Inst. Elect. Engng. 93 (1946),429457. 10. A. Haar, "Zur Theorie der Orthogonalen F\mktionensysteme," Math. Ann. 69 (1910), 331- 371. 11. A. J anssen, "Gabor Representation of Generalized FUnctions," J . Math. Appl. 80 (1981) , 377- 394.

2.2. LEMARIE AND MEYER WAVELETS

117

12. A. Janssen, "Bargmann Transform, Zak Transform, and Coherent States," J . Math. Phys. 23 (1982), 720- 731. 13. A. Janssen, "The Zak Transform: A Signal Transform for Sampled Time-Continuous Signals," Philips J . Res. 42 (1987).

14. F. Low, "Complete Sets of Wave-Packets," in A Passion for Physics- Essays in Honor of Geoffrey Chew, World Scientific, Singapore, 1985, pp. 17-22.

2.2

Lemarie and Meyer Wavelets

We now describe a systematic way to construct an orthonormal wavelet basis known as the Mallat-Meyer multi-scale resolution analysis. One starts with a scaling function 1/ defined by the following properties: (a) Let 1fr be the closed span of {1/r(- - 2-rn): n E Z} . Then n1fr is the null T

space and span(U 1fr) is dense in L2 (JR). r

(b) There is a summable sequence 1/(X)

{Cm}mEZ

such that

= L cm 1/(2x -

(2.2.1)

m) .

m

This condition implies 1fr C 1fr+1 and span ( U 1fr) = U 1fr · r

(c) fj is continuous and

2:

r

Ifj(p + 27rf) 12 has no zeros.

fEZ

It is easy to verify that the characteristic function X of the unit interval [0,1] is a scaling function . Indeed, (2.2.2) lX(p + 27rf) 12 = 1,

L

fEZ

and Cm = 80m + 81m in this case. Once the notion of a scaling function has been isolated , the construction of an orthonormal wavelet basis becomes easy. First , define the function

~(p) =

(

L Ifj(p + 27rf) 12

)

fj(p) ,

(2.2.3)

fEZ

and observe that

L I~(p + 27rfW = 1,

(2.2.4)

fEZ

and therefore

J


m,nEZ.

(2.2.5)

118

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Moreover, if we define h(P)

=L

Cme-i~mp,

(2.2.6)

m

then the scale relation (2.2 .1) becomes i)(P)

= ~h(P)i) Gp) .

(2.2.7)

Since h(P) is 47r-periodic, so is the expression

(2.2.8)

The point is that (2.2.9) so if we set h(p) =

L Cme-i~mp,

(2.2.10)

m

we have a scale relation for

<.p:

(2.2.11) m

A scaling function <.p has been constructed that has the additional property (2 .2.5). Finally, the wavelet is defined by (2 .2.12) m

The orthogonality between scales is induced by the identity L(-I)mCmC1 -m

= 0,

(2.2.13)

m

and the completeness of the orthonormal set {w r ( . - 2- r n): n, r E Z} follows from property (a) of <.p together with the observation that 1I. ro

c

span{w r ( ·

-

2-rn) : n,r E Z,r:'S ro -I}.

(2.2.14)

Indeed, if we set Kr = span{w r (-

-

2rn) : n E Z},

(2.2.15)

then Kr is orthogonal to 11., and (2.2.16)

2.2. LEMARIE AND MEYER WAVELETS

119

The iteration of this decomposition implies (2.2 .14) because nHr is the null space. r

This machinery reduces the problem of constructing an orthonormal wavelet basis with given phase space localization properties to one of finding a scaling function with the desired properties. For our first class of examples we take the standard Nth-degree spline with N ranging over all non-negative integers. This means we define T) as the (N + I)-fold convolution of X with itself. Equivalently, (2 .2.17)

and it is easy enough to verify that (2.2 .18)

Hence N+l

~o

T)(x) =

(N m+ 1) T)(2x - m),

(2.2.19)

and T) also has property (c) because

"I '(p 2 f)1 2 ~ T) + 7f

-

1 - ip -

- e

1 ~ (p + 27ff)2N+2 '

112N+2"

l

(2.2.20)

l

.

le- ip - 112N+2 N 2 p-+21rrn (p + 27ff)2 + hm

=

bern .

(2.2 .21)

As far as condition (a) is concerned, it suffices to show that for every orthogonal projection F of L2(~) defined by the characteristic function of some interval , the functions T,n E Z,

span the subspace ran F . Suppose there is an interval for which this is not true. Then there is an element ~ E ran F such that T,n E Z .

Since ~(x) has compact support, €(p) is smooth. Moreover,

so €(p) is orthogonal to e-iTrnpi]r (P),

T,n E Z.

By Fourier analysis on each scale, 2r/2

L €(p + Z-r+l7ff)i]r(P + 2- + 7ff)* = 0 r

e

1

(2 .2.22)

120

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

for all r E Z, where 2r / 2 is used as a normalizing factor in the r also know that lim e(p) = 0

= -00 limit.

ipi-H>O

Now, we (2.2.23)

by the Riemann-Lebesgue Lemma, and

2rN +r le-i2~~~N-:.;IN+l

2r / 2 I17r(P)1 ::;

2rN+N+r+1Ipl -N-1.

(2.2.24)

Since this bound is uniform in r for r ::; 0, we may infer dominated convergence for the f-summation as r -+ -00 . Only the f = 0 term survives this limit, so (2.2.25)

=

but the limit is unity. Thus €(p) 0, and so 'f/ has property (a). The wavelet basis that we can derive from this scaling function is the Lemarie basis. Although 'f/ has compact support, the Lemarie wavelets are only exponentially localized. The point is that the expression -1 /2 ( ~ 117(p + 2rr£) 12 ) is only real-analytic in p - not entire. Therefore, the scaling function c.p intermediate to the derivation has only exponential falloff. As far as smoothness is concerned, the wavelet 'li is in the same class as 'f/ - namely class C N -0. This is most transparent if we write the formula (2.2.12) in momentum space:

~ ·1 'li(p) = - e"Ph(2rr - p)*cp

(12P) .

(2.2.26)

h is a 4rr-periodic function, so ~(p) has the same bound on large-p decay as cp(p) and 17(p) have. How many vanishing moments does a Lemarie wavelet have? The key quantity is h(p)

f

=

(N;l) e-i~mp

m=O

(1

+ e-i~p)N +1,

(2.2.27)

and the point is that

2:= 117(2rr£ + rr _ p) 12) h(2rr - p)* h(2rr - p)*

(

e

21 I17(2rr£ _ p)i2

1/2

h(2rr - p)*,

(2 .2.28)

(1 - e- i !p)N+1

O(lpIN+1),

(2.2.29)

2.2. LEMARIE AND MEYER WAVELETS

121

which implies

0:::; n:::; N.

(2.2.30)

The number of vanishing moments for a Lemarie wavelet is essentially equal to the degree of smoothness. We now turn to the construction of a Meyer basis - a wavelet basis where the Fourier transform of each basis function is a compactly supported Coo function. Again, we start with a scaling function T} , but now we choose T} by specifying f] as a positive Coo function such that ' ( ) _ { 1, Ipl:::; 7r, (2.2.31) T} P 0, Ipi 2: 7r + 8, where we shall adjust 8 > 0 to be as small as we need. The construction of such a function is an elementary exercise; our main task is to show that this is indeed a scaling function. Property (c) is obvious; to find the sequence {Cm}mEZ for property (b), we observe that if we choose 8 < ~7r,

+ 47re) = 8eo f](p)·

f] Gp) f](p

(2 .2.32)

If we define

+ 47r£),

(2 .2.33)

~h(p)f] Gp) ,

(2.2.34)

h(p) = 22:= f](p

then f](p) =

so T}(x) satisfies the scaling relation (2.2.1) with {Cm}mEZ given by (2 .2.6) and our current choice of h(p). There remains the problem of establishing property (a). To this end, we may choose an arbitrary f E L2 (lR) such that j has compact support. Thus supp

j c {-2V(7r - 8) :::; p:::; 2V(7r - 8)}

for some integer v. Let g(p) = 2:=

j(p + 2v + 17r£)

(2.2 .35) (2 .2.36)

e and consider the Fourier series of this 2 v +1 7r-periodic function: g(p) = 2:= a m e i2 -"m p .

(2 .2.37)

m

Since (2 .2.38) it follows that j(p)

g(p)f](TVp)

2:= ame-i2-"mpf](Tvp). m

(2 .2.39)

122

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Hence (2.2.40) m

and so we have the density property. The regularity and decay properties that have been selected for preserved by the derivation because

(

~ 1i)(P + 27rfW)

'T}

in this case are

-1/2

is certainly Coo as well. With (j5 as a Coo function with compact support, ii automatically shares these properties through (2 .2.26). However, h(27r - p) has a more interesting property than smoothness. There is a neighborhood of p = 0 on which this expression vanishes. Since ii(p) shares this property, it follows that all moments vanish for a Meyer wavelet - i.e., (2.2.41) for all integers n

~

O.

References 1. I. Daubechies, "Orthonormal Bases of Compactly Supported Wavelets," Com-

mun. Pure Appl. Math. 41, No.7 (1988), 909-996. 2. P. Lemarie, "Ondelettes (1988), 227- 236.

a Localisation Exponentielle," J . Math.

Pures Appl. 67

3. P. Lemare-Rieusset, "Sur I' Existence des Analyses Multi-Resolutions en Theorie des Ondelettes," Rev. Mat. Iberoamer. 8 (1992),457-474. 4. P. Lemarie-Rieusset and J. Kahane, Fourier Series and Wavelets, Gordon and Breach, Luxembourg, 1995. 5. P. Lemarie and Y. Meyer, "Ondelettes et Bases Hilbertiennes," Rev. Mat. Iberoamer. 2 (1986),1-18. 6. S. Mallat, "Multiresolution Approximations and Wavelet Orthonormal Bases of L 2 (IR)," Trans. Amer. Math. Soc. 315, No.1 (1989),69-87. 7. Y. Meyer, "Principe d' Incertitude, Bases Hilbertiennes et Algebres d' Operateurs, " Seminaire Bourbaki 662 (1986), 209-223. 8. Y. Meyer, Ondelettes at Operateurs I: Ondelettes , Hermann Press, Paris, 1990. 9. Y. Meyer, Ondelettes, Fonctions Splines, et Analyses Graduees, Cahiers des Mathematiques de la Decision No. 8703, Univ. Paris-Dauphine, 1987.

2.3. DAUBECHIES WAVELETS

2.3

123

Daubechies Wavelets

Having constructed exponentially localized wavelets for every desired degree of smoothness, one may ask whether bases of compactly supported wavelets exist. The Haar basis is obviously an orthonormal basis of compactly supported wavelets, but these basis functions are discontinuous. A major development of the subject was the construction of wavelet bases that are smoother analogs of the Haar basis. The mother wavelet of such a basis is a Daubechies wavelet, and the discussion of these wavelets merits a section of its own. As in the construction of Meyer and Lemarie wavelets carried out in the previous section, Daubechies wavelets are constructed within the framework of the Mallat-Meyer multi-scale resolution analysis, but there is an important obstacle to be reckoned with. It is not enough for the initial scaling function 1) to have compact support unless its integer translates are already orthonormal. After all, the initial scaling function in the Lemarie case has compact support, but the orthogonalization -1/2

cp(p) =

(

~ Ir)(p + 21l"f) 12 )

r)(p)

(2.3.1 )

degrades this property to exponential falloff because the inverse square root in the momentum expression is real analytic but not entire. The scaling function 'P must be derived in some other way. On the other hand, once we have a compactly supported scaling function 'P whose integer translates are orthonormal, the multi-scale resolution analysis yields a compactly supported wavelet. Indeed, if 'P satisfies the scale relation 1p(x)

=L

cm 'P(2x - m)

(2.3.2)

m

and has compact support, then only a finite number of the coefficients are non-zero. Since the corresponding wavelet is given by (2.3.3) m

it must have compact support as well. Moreover, q, will possess the same degree of smoothness as 'P. To construct 'P, we find it useful to analyze the properties that 'P must have. In momentum space, the scale relation becomes

cp(p)

~h(P)CP Gp) ,

(2.3.4)

h(p)

LCme-i~mp.

(2.3.5)

m

Iteration of this relation yields (2.3.6)

124

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

and the N = 00 limit yields at least a formalism for the recovery of the scaling function cp from the coefficients C m : (2.3.7) Aside from the property that only a finite number of the Cm are non-zero, what other properties must the sequence {c m } have? The orthonormality property of cp induces a condition on the sequence via the scale relation . We have

J

cp(x)*cp(x - n)dx

OnO

~ c;"c m m,m

J J

cp(2x - m)*cp(2x - 2n - m)dx

~ ~ c;"c m

cp(x' - m)*cp(x' - 2n - m)dx'

m,m

(2.3.8) This may be written as a condition on h(P) . Indeed,

/2" /2"

--2 1 (47r)

dp

-2"

dp' h(p)* h(p')

-2"

2: e-i~mp+i~mp' -inp' m

_1)2 ( /2" dp / 2" dp' h(p)* h(p')e- inp' 47r

-2"

~ /2rr 47r 1

47r

-2rr

Jor

2: 27rO (-21 p' -

1

-2 p + 27re)

l

-2"

dp [h(p)[2e- inp

2rr

e-

inp

([h(p)[ 2 + [h(p

+ 27rW),

(2.3.9)

so it follows from (2 .3.8) that 1

4[h(P)[2

1

+ 4[h(p + 27rW = 1.

(2.3.10)

In summary, we are looking for 47r-periodic functions h(p) which are trigonometric polynomials, which satisfy Eq. (2.3.10), and for which the infinite product in (2.3.7) converges uniformly on compact subsets of the complex plane. The point is that if ip(p) extends to an entire function whose growth in the imaginary direction is bounded exponentially, then by a contour-shifting argument in momentum space, cp(x) has compact support. At this stage, it is instructive to consider the Haar basis from this point of view. In this case, cp = TJ = X and (2 .3.11)

2.3. DAUBECHIES WAVELETS

125

where (2.3 .10) is just the trigonometric identity (2 .3.12) The infinite product identity is (2 .3.13) and the r.h.s. is just the Fourier transform of the characteristic function x. Actually, this formula plays a key role in the construction of Daubechies scaling functions. The strategy is to write (2 .3.14) where No is to be chosen by the desired order of differentiability, and examine the conditions which Q(z) must satisfy. Since (2.3.15) Eq. (2 .3.7) reduces to 00

cp(p)

= cp(O)X(p)NO II (2No-IQ(ei2-k-lp)),

(2.3 .16)

k=O

so the infinite product in Q must be bounded by a certain No-dependent power of p if

II(Z2 + aJi.z + bJi.)

(2.3.18)

Ji.

with aJi. and bJi. real, so for

Izl = 1,

II(1 + a~ + b~ Ji.

2bJi.

+ aJi.(bJi. + l)(z + 2) + bJi.(z + 2)2).

(2 .3.19)

126

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

This makes IQ(ei~pW a polynomial in cos(~p) and therefore in the variable

' 2(1) 4 ="211(1) - "2 "2

(2.3.20)

o :S y :S 1,

(2.3.21)

Y = sm

P

cos

P .

Accordingly, we set and observe that (2.3.22) Equation (2.3.17) becomes (2.3.23) Once a polynomial P(y) is found to satisfy the reduced conditions, one can recover Q(z) by virtue of the Polya-Szego Lemma. One version of this elementary lemma states that if P(y) is any polynomial for which P(y) 2: 0 on the interval 0 :S y :S 1, then there exists a polynomial Q(z) with real coefficients such that (2.3.24) We omit the proof. It is not difficult to find a polynomial solution of Eq. (2 .3.23) . Verification that (2 .3.25) is a solution is based on the combinatoric identities (2.3.26)

(2.3.27) which can be established by induction arguments . The remaining problem is to control the product <Xl

F(p) =

II (2No-lQ( ei2-"-lp)),

(2.3.28)

k=O

and we write the polynomial Q( z ) in the form

Q(z)

= 2- No + 1 L 'Ymzm. m

(2.3.29)

2.3. DAUBECHIES WAVELETS

127

Since Q ( l ) = 2-N0+1 as a consequence of h(0) = 2, we have

and this product converges uniformly on compact subsets in the complex variable w because

Thus F ( w ) is an entire function; moreover, the growth of F ( w ) in Im w is exponentially bounded. Indeed, for Iw 1 2 1,

+

where mo is the largest integer m for which ym 0. This bound on the growth of F ( w ) together with (2.3.32) and a contour-shifting argument imply that cp has compact support, provided F ( w ) has enough decay in Re w for bounded Im w. We focus on

128

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

F(p) for simplicity, as the estimation does not significantly change with a bounded imaginary part. This issue coincides with the remaining issue of regularity of cp, since the latter is determined by how much polynomial decay in p can be extracted from [P(p) . First, we can certainly refine the inequality (2.3.34) on the real axis as follows:

(2.3.35)

for

Ipi 2 1. This implies Ip121,

(2.3.36)

from which we infer (2.3.37)

This initial estimation is clearly not sharp enough. After all , this bound is no better than the bound on X(p), and X is the scaling function for the Haar basis. With only such a bound on [pCP), the whole strategy would be much ado about nothing, so we now describe how to obtain a better bound. The obvious refinement is to take the y-dependence of P(y) into account, but how to do this effectively may not be so obvious. The idea is to pair the factors when we estimate a product as in (2.3.35) above. To this end, we consider 2f+l

II (4NO-lP(sin2(Tk-2p)))

k=O

e

II (4 2NO-2 P(sin2(2-2j-2p))P(sin2(T2j-3p))) j=O

=

e

II (4 2NO - 2P(4cos2(T2j- 3p ) sin 2(T 2

j-3 p ))

j=O

x P(sin2 (2- 2j - 3p))).

(2.3 .38)

2.3. DAUBECHIES WAVELETS Thus, each pair product has the 'form 4 2 N 0 - 2 ~ ( 4 (-1 y)y)P(y) =

(No -: (N o -v,l + ~ ') +

v,v'=O

v,

Since

we have 4 2 N 0 - 2 ~ ( 4 (-1 y)y)P(y)

In each case, the range of y may be exploited to further reduce the bound. Thus,

From this we easily infer that

for all values of y. Therefore,

so if we choose l to be the largest integer such that

2e 5 log2 IPI,

130

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

we obtain

2ff 12: k=O

rmei2-k-lmpI2 ::; (4 NO -2

N~(2 + V2)NO-l) ~ log21pIH

m

=4No-2 N~(2 + V2)NO-1IpINo-2+log 2No+~(No-l) log2(2+..!2).

(2 .3.46)

IT 12:

(2.3.47)

In summary,

2£ + 2> log21pI =>

k=2e+2

rmei2-k-lmp l ::; c

m

as a consequence of the estimation, while

2£ ::; log2 1pI =>

2if 12: k=O

X

rmei2-k-lmpl ::; 2No - 2No

m

(2 + V2) ~No-~ Ipi ~(No-l)(1+~ log2(2+..!2»+~ log2 No-~ .

(2.3.48)

Since £ has been chosen such that both conditions hold, we finally have (2.3.49) from which it follows that Icp(p) I ::; clpl(No-l)( -~+t log2(2+..!2»+~ log2 No-!,

Ipl2: 1.

(2.3.50)

This is a decay law because (2.3.51) The power of the decay is roughly proportional to No, so at last we have constructed an arbitrarily regular <po We have shown how to construct the simplest class of Daubechies wavelets. Variations on this construction can be - and have been - pursued as well. Moreover, the estimation used above for establishing regularity can be significantly refined.

References 1. I. Daubechies, "Orthonormal Bases of Compactly Supported Wavelets," Commun. Pure Appl. Math. 41 , No. 7 (1988), 909-996. 2. A. Haar, "Zur Theorie der Orthogonalen Funktionsysteme," Math. Ann. 69 (1910), 331- 371. 3. G. Polya and G. Szego, Aufgaben und Lehrsiitze aus der Analysis, Vol. II, Springer, Berlin, 1971.

2.4. VANISHING MOMENTS

2.4

Vanishing Moments

In 32.1 we produced some crude pictorial intuition for the expectation that $(o) = 0 for any reasonable wavelet 9.We now wish to make this result precise and to generalize it to moments of arbitrary order. The constructions in the last couple of sections have a number of vanishing moments comparable to the degree of smoothness, and this is no coincidence. Also, this connection will lead to a no-go theorem on the type of phase space localization that a wavelet can have. The latter result is similar in spirit to the Balian-Low Theorem for discrete translational symmetry in phase space. The result on vanishing moments will also lead to more quantitative restrictions on phase space localization of wavelets - in the form of uncertainty relations - but we pursue this in the next section. We first consider a hypothesis for which the intuition is transparent. Let q be an N-adic wavelet such that @(p) is continuous and integrable. In particular, Q(x) is a continuous function that vanishes at infinity. There exist integers no and ro such that

because rational numbers of the form N-'n with n , r E Z are dense in R. $0 = N7'Ono lies on every,finer lattice: define

Now

so that x0 = N-'n(r). The orthogonality property of the N-adic wavelet implies

Combining the integrability and continuity of 5 with the Domin%ted Convergence Theorem, we see that the momentum integral converges to !P(xo)Q(0)' as r -t m. Since @(xo)# 0, it follows that $(o) = 0, so the zeroth-order moment of !P vanishes. This c o n s e q u y e of integrability and continuity in momentum space is easy enough to visualize. If !P(O) were non-zero, the comparison of drastically different scales would have the graphical representation given by Fig. 2.4.1, and the point is that the wavelet with larger momentum scale (smaller length scale) is almost constant over much shorter momentum intervals. The orthogonality of eipxo$(p) to a function that is almost constant over the essential locality of ezpx0$(p) would imply that @(xo) = 0 in the limit as the scale difference increases, which contradicts the choice of xo. Actually, the version of this result that generalizes most naturally to higher-order moments is slightly different. Suppose instead that both x!P(x) and !P1(x)are squareintegrable, which means that $'(p) and pG(p) are both square-integrable as well. This is a stronger hypothesis because it implies on one hand that is locally integrable

132

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Figure 2.4.1 : and therefore ;Pcp) is absolutely continuous, while on the other hand it implies

<

(2.4.5)

00.

Now for order v > 0 we consider the hypothesis that .p(V+l)(X) and xV+1.p(x) are both square-integrable. We propose to show that

J

xl".p(x)dx

= 0,

p. ::; v,

(2.4.6)

and we do so by induction on p.. First, it is easy enough to infer that in particular, .p/(X) and x.p(x) are square-integrable, so this null condition certainly holds for p. = 0, as we have already argued that ;P(O) = 0 in this event. Now suppose the condition holds for p. ::; k - 1, where k is a positive integer::; v, and assume (2.4.7) which means

;P(k) (0)

=1=

O.

(2.4.8)

By the induction hypothesis, (2.4.9) (2.4.10)

2.4. VANISHING MOMENTS

133

and the Taylor remainder makes sense because ~(k+1) (p) is square-integrable and therefore locally integrable. This time we choose integers no and ro such that (2.4.11) but we consider the same orthogonality condition as before: r ~ max{l, ro}.

(2.4.12)

We have the Fourier transform identity (2.4.13) while the boundedness of ~ - which follows from the integrability of IlJ - implies

-

I~(p) ~~(k)(O)pkl

IRk(p)1 :::;

clfJ(1

+ Iplk) .

(2.4.14)

Now the strategy is to combine (2.4.15) with (2.4.12) and (2.4.13) to obtain the equation we intend to exploit: (2.4.16) The idea is to show that the integral decreases faster than N- rk as r -+ 00. This will achieve the contradiction to (2.4.8) that we desire. To this end, we decompose the integral over the regions Ipi :::; cN r and Ipi > cN r . The integral over the latter region is estimated by using (2.4.14) as follows:

r

Ii1pl >eN" :::;

CIfJ

r

eiXOP~(p)RdN-rp)*dPI 1~(p)I(1 + N-rklplk)dp

i1pl >eN" :::;

Cw (

r

Ipk+1~(P)12dP)

1/2 [ (

i1pl <eN"

+ N- rk (

r

i1pl >eN"

r

i1pl>eN"

IPI- 2dP) 1/2]

IPI-2k-2 dP) 1/2

134

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

< _

CI\I

Ilw (k+ 1)11

2

[J

2k

2

+ 1 c -k-!N-rk-!r

+ V2c-!N- rk -!r] O(N-rk- !r).

(2.4.17)

To estimate the integral over the former region, we apply Schwarz estimation to the q-integration in (2.4.10) to obtain for the remainder a bound complementary to (2.4.14) - namely, (2.4.18) Thus

r

Ii1pl S,eW

:S

eiXOP~(p)Rk(N-rp)*dPI

cll~(k+1)1I2N-rk-!r

r

1~(P)llplk+!dp

i1pl S,eW

:S

cll~(k+1) IbN -rk-!r ( r

i1 plS,eW

x (

r

i1pl S,eW

(1

(1

+ IPI)-l dP)

1/2

+ IPI)2k+21~(P)12dP) 1/2

O(N-rk-~r vln(Nr)),

(2.4.19)

so we have the desired type of bound on the large-r behavior of the integral. This completes the induction step in the proof (2.4.6). An interesting consequence of this fundamental theorem is that an N -adic wavelet cannot have exponential localization in both position space and momentum space. Indeed, suppose that w(x) and ~(P) both have exponential decay. Then xl-<1jJ(x) and W(I-<)(x) are both square-integrable for all integers /1-:::: 0, so by our result, ~(I-<)(O)

=0

(2.4.20)

for all /1- as well. On the other hand, if ~(P) actually vanishes to infinite order at P = 0, then it cannot be analytic there. Finally, if ~(p) is not real analytic, then w(x) cannot have exponential decay, and so we have a contradiction.

References 1. G. Battle, "Phase Space Localization Theorem for Ondelettes," J. Math. Phys. 30, No . 10 (1989),2195- 2197.

2. I. Daubechies, "Orthonormal Bases of Compactly Supported Wavelets," Commun. Pure Appl. Math. 41, No.7 (1988) , 909-996.

2.5. UNCERTAINTY RELATIONS FOR (P) 3. P. Lemarie "Ondelettes (1988),227-236 .

= 0 WAVELET STATES

a Localisation Exponentielle,"

135

J. Math . Pures Appl. 67

4. Y . Meyer, "Principe d'Incertitude, Bases Hilbertiennes et Algebres d'Operateurs," Seminaire Bourbaki 662 (1986), 209-223. 5. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis and Self-Adjointness, Academic Press, New York, 1975.

2.5

Uncertainty Relations for (P)

= 0 Wavelet States

The vanishing-moment theorem for wavelets leads to inequalities of Heisenberg type (for wavelets) that are intimately related to scaling. The role of scaling in the proof of such inequalities has never been major for arbitrary states. Even in the proof of the celebrated inequality for an arbitrary state only the self-adjointness of the scaling generator S is used:

\]!

(p2)Ij2(X2)Ij2

2

>

IIP\]!1121IX\]!1I2 I(P\]!,X\]!)I 1((S+iD\]!,\]!)1 1

(2 .5.1)

2

(Although Ii, appeared as a parameter in §1.3, recall that we set Ii, = 1 in §1.8.) For states that are wavelets, the scaling generator will playa more quantitative role than it seems to play in this most basic estimation. First, suppose the state happens to be the derivative of some square-integrable function, with no other assumption made. Thus

\]!

\]! =P(,

(2.5.2)

so we have

IIXP(112 I (s + iD (112 II(S-iD(L Hence

"P\]!"211 (s -iD (II

(p2)Ij2(X2)Ij2

>

I( (s - D()I P\]!,

i

(2 .5.3)

136

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

(2 .5.4) because [P,S]

= -iP.

(2.5.5)

In this case, S plays a useful role before it is discarded. Next, suppose 1J1 is the nth derivative of a square-integrable ( . How is the Heisenberg inequality generalized? With (2.5.6) we propose to obtain a lower bound on (2.5.7) We appeal to the identity (2.5.8) which is easy enough to prove by induction. Indeed, if this identity holds for a given n, then

x

(g (S + (k - D))P i

XPg(S+i(k+D)

(2.5.9)

because (2.5.10) Combining (1.3.37) with (2.5.9) completes the induction step. Now apply this identity as follows: IIxnpn( lI ~

Ilg (S+i (k- D) (II: (g (S2 (k - D2) (, () +

2.5. UNCERTAINTY RELATIONS FOR (P) ~

II(II~

= 0 WAVELET STATES

tl (k - ~) IIS(II~ t II (k _~) tl (k - ~) ) ([ , 2

+

2

I (1nS ± i

( r n

l:II

1n

1=1 ki-I (

Thus,

(p2n)Ij2 (xzn)IjZ

1

k--2)

137

(2.5.11)

2

(2.5.12)

tl (k - D)(liz ±itl (k- D)) (I tl (k - ~) ) tl (k - D)

~

IlpnwllZl1 (1nS ± i

~

I(pn w, (1nS

I(w, (1nS - in1n ± i

pn() I

I(w, (1nS - in1n ± i

w) I'

(2.5 .13)

where we have used

lpn ,S]

=

-inP.

(2.5.14)

Clearly, the better choice in sign is -i, so we may conclude that

(pZn)IjZ (xzn)IjZ ~ n1n +

IT (k - ~) .

(2.5 .15)

k=1

This is the generalization of (2.5.4) that we infer from (2.5.8) . Since (2.5.6) is a translation-invariant condition, it is trivial to extend this result to the observation that (2.5.6) implies a E JR.

(2 .5.16)

Now for an arbitrary observable A, define the nth-order deviation as (2.5.17) Then (2.5.16) amounts to the nth-order uncertainty principle (2.5.18)

138

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

provided that

(P)q,

= o.

(2.5.19)

This additional condition is satisfied by all real-valued W as well as by W that are either symmetric or anti-symmetric about their barycenters (X)q,. For n = 1,2,3 we have: (2.5.20.1) (2.5.20.2) a( 3) (P)a( 3) (X) 'II

'II

> -

3

~v'259 + 15 ~ 77 . 4

8

32

(2.5.20.3)

For large n , the lower bound (2.5.18) is essentially linear in n. What does any of this have to do with wavelets? The point is that if W is a wavelet state and if (p2n)q, and (x2n)q, are finite - otherwise, any lower bound on the product would hold vacuously - then w satisfies the hypothesis of the vanishing moment theorem, with the conclusion that ~(k)(O)

= 0,

k~n-l.

(2.5 .21)

Clearly, the condition (2 .5.6) is a close relative of this vanishing moment condition. Indeed, suppose the wavelet state W satisfies the slightly stronger condition that (2 .5.22)

for some c5 > O. Obviously, this includes all of the wavelets we have constructed thus far: the Meyer wavelets are Schwartz functions , the Lemarie wavelets are exponentially localized, and the Daubechies wavelets are compactly supported. Now (2.5.21) implies the Taylor formula (2.5.23)

and we wish to infer (2.5 .6) - i.e., to show that p-n~(p) is square-integrable. The region Ipi ~ 1 is obviously the issue, so we need only show that

Ipi

~ 1,

(2.5.24)

for some E:: > O. To this end, we apply the Holder inequality - with exponent r > 2 to obtain

(2.5 .25) (2 .5.26)

2.5. UNCERTAINTY RELATIONS FOR ( P ) = 0 WAVELET STATES

139

Since s < 2, we have the desired bound with

provided

I ~ $ ( ~ ) I I<, ca for

some r sufficiently close to 2. Now by the Young inequality,

but with s < 2, we may apply the Holder inequality again to obtain

The first factor is finite, provided r is chosen such that

Thus we have established (2.5.24) and therefore the squareintegrability of p - n $ ~ ) , assuming only the vanishing moment condition and the square-integrability of (1 ( x ( ) ~ + ~ Q (for x ) some 6 > 0. In summary, if an N-adic wavelet state Q satisfies the decay condition (2.5.22), then

+

(P~~)!$'~((x - a)2n)i'2 2 n-y,,

-+

(2.5.31) k=l

and this inequality is an uncertainty inequality if Q is also a (P)= 0 state (real-valued, for example).

References 1. R. Balan, "An Uncertanity Inequality for Wavelet Sets," to appear in Appl. and Comput. Harmonic Anal. 2. G. Battle, "Phase Space Localization Theorem for Ondelettes," J. Math. Phys. 30, NO. 10 (1989), 2195-2197.

3. G. Battle, "Heisenberg Inequalities for Wavelet States,'' Appl. and Comput. Harmonic Anal. 4 (1997), 119-146.

4. A. Messiah, Quantum Mechanics, Vol. I, North-Holland, Amsterdam, 1965. 5. M. Reed and B. Simon, Methods for Modern Mathematical Physics, Vol. 11, Fourier Analysis and Self-Adiointness, Academic Press, New York, 1975.

140

2.6

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Further Constraints of Heisenberg Type

In the previous section, the inequalities we derived essentially depended on vanishingmoment conditions only. We now derive phase space localization inequalities that are more peculiar to the wavelet state condition defined in that section. For an arbitrary N-adic wavelet state IJI, it is interesting to measure the phase space deviations about the points mE Z, (x,p) = (m,O), (2.6.1) in phase space - i.e ., to obtain lower bounds on (2.6.2)

Since every integer translate of an N -adic wavelet state is still an N -adic wavelet state, we may set m = 0 without loss. Consider the n = 1 case and notice the effect of the orthogonality property on the expectation of the scaling group. We have -ie>./2 \ (

S + ~i) e-i>.S )

~

-ie>./2 (X Pe-i>'S)~.

Combining this with ei>'S Pe-i>.s

= e->' P

(2.6.3) (2.6.4)

and the fundamental theorem of calculus, one obtains

VN (e-i(ln N)Sh

- 1

['nN e- i >./2(X e-i>.s Phd>'

= -i J

(2.6.5)

o

for an arbitrary state. For an N-adic wavelet state, (e-ir(ln N)S)~

= 0,

r E Z\{O},

(2.6.6)

so set r = 1. By the Schwarz inequality, 1

~

IIPIJI 11 2II X

['nN

11% J

e->' / 2d>'

o

2(1 - 1jVN)IIP1JI11211X1JI1I2

(2.6.7)

for such a state IJI . Thus we have the lower bound (2.6.8)

In contrast to the last section, we have not assumed the decay condition that IXl 1+01J1 be square-integrable for some c5 > O. The generalization of this estimation to arbitrary n is an application of Rolle's Theorem in elementary calculus together with the property (2.6.9)

2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE

141

for all non-zero integers r. First, observe that

(_i)ne(n-~).\(xn pne-i.\S )w.

(2.6.10)

On the other hand, (2.6.4) obviously generalizes to (2.6.11) so we have (2.6.12) Next, observe that the operator

acts separately on the real and imaginary parts of any function of A. Thus

(_i)ne-A/2(xne-iAS pn)w = Un(A) if we define

+ iVn(A)

(2.6 .13)

e-.\ (e Ad~) k U(A),

(2.6.14.0)

(e d~) k V(A),

(2 .6.14.1)

Uk (A)

=

Vk(A)

= e- A

e.\/2 (e-iAS)w

A

= U(A) + iV(A) .

(2.6.15)

The property (2.6 .6) implies

u(rln N)

= v(rln N) = 0

(2 .6.16)

for all non-zero integers r. In particular, (2.6.17.0)

(2.6.17.1) Now observe that by Rolle ,s Theorem, there are sequences CBP) , ,B~l), ,B~l), ... ) such that

r In N < Ct~ll, ,B~l) < (r + 1) In N,

(1) (1) (1) ) Ct l ,Ct 2 ,Ct 3 , . ..

an d

(2.6.18)

142

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Ul(a~I»)

= 0,

(2.6.19.0)

(,6~1»)

= o.

(2.6.19.1)

VI

Thus (2.6.20.0)

(2 .6.20.1)

Now apply Rolle's Theorem again - this time, to these zeros of e Aul (.>.) and . (2) (2) ) to obtam sequences (a 1(2) ,a2(2) ,a 3(2) ,.. . ) and ((2) ,61 ,,62 ,,63 , . .. where a(1) T

< a(2) < a(l) r r+l'

u2(a~2») ,6(1) r

l l'

cx \2)

(.>.) -

(2.6.21) (2.6.22)

< ,6(2) < ,6(1) r r+l'

V2 (,6~2»)

Clearly,

= 0,

e Avl

= o.

(2.6.23) (2.6.24)

d'>'3U3('>'3)

= - eA2u 2('>'2),

(2.6.25.0)

d'>'3V3('>'3)

= - e A2V2 ('>'2).

(2 .6.25.1)

A2

(3(2)

A2

Obviously, we can iterate this procedure for as long as derivatives exist. In general, we ) (k) (k) (k) . . .. have sequences ( a 1(k) , a 2(k) , a 3( k, » . . . and (,61 ,,62 ,,63 , .. . ) satlsfymg the conditIOns a(k - I) r

< a(k) < a(k-I) r

r+l'

uk(a~k») = 0, f./(k-I)

Pr

<

Vk (,6~k»)

f./(k) }-IT

(2.6.26) (2.6.27)

<

f./(k-I) ,ur+l'

= 0,

(2.6.28) (2.6.29)

and the integration formulas are r\k) d'>'k+JUk+J('>'k+I)

lA,

= -eAkuk('>'k),

(2.6.30.0)

= -eAkvd'>'d·

(2 .6.30.1)

(3~k)

{

lAk

d'>'k+J Vk+d'>'k+d

Pulling this together, one obtains (2.6.31.0)

2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE

143

(2.6.31.1) The second iterated integral equation is useless; the imaginary part of (2 .6.13) contributes nothing to our lower bound. We focus on the first equation and notice that the smallest numbers o~k) in the sequences have the ordering In N

< 0 (2) < . .. < 0 (n-l) . < 0 (I) 1 1 1

(2.6.32)

Moreover,

oik ) < O~k-l) < 0~k-2) < . .. < 0~1) < (k

+ 1) InN,

(2 .6.33)

so we have the estimate

Finally, we apply (2.6.13): (2.6.35) We obtain 1

< (2.6.36)

and so we have the lower bound (2.6.37) We have derived this inequality without the assumption that IXl n +c51J1 be squareintegrable for some /j > O. Notice also that this lower bound can be estimated independently of the basic scale factor N:

::; r

1

10 n

dTI

( ' dT2 ...

10 1

II k_l'

k=1

(n-,

10

dTn

Fn (2.6.38)

2

so we also have the weaker bound (2 .6.39)

144

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

On the other hand, for the dyadic (N = 2) case, the bounds for n = 1,2,3 are:

= ~(2 +../2)

(P2)1/2(X2)1/2> _1_ IJI IJI - 2-../2

-V~

IJI

(p6);f6(X6);f6?:

3

(2.6.40.1)

'

f 6 = V17 /~(5 + 2../2) '

(p4)1/4(X4)1/4 > IJI

2

120 = 54 - 23V2

3

~(54+23../2) . 929

(2.6.40.2)

(2 .6.40.3)

This case is of interest because the wavelets that are routinely constructed are dyadic unless one desires a special property that requires N to have some other value. How can the estimate (2.6.37) enhance the previous lower bound derived from the assumption that IXI~+8 is square-integrable? In the n = 1 case, one certainly has the estimate (2.6.41) from the previous section. To date, no N -dependent enhancement of this lower bound has been found - only the N-dependent enhancement (2.6.8) of the universal lower bound. In this lowest-order case, we can only compare the lower bounds. Clearly, (2.6.40 .1) is stronger than (2.6.41) but for N ?: 3, (2.6.41) is stronger than (2.6.8) . For n > 1 we have more interesting estimates. In these higher-order cases, we can derive an N-dependent enhancement of the corresponding lower bounds derived from this decay in the last section. Consider the n = 2 case first: with IXI 2+°'l1 assumed to be square-integrable, we may infer from the previous section that (2.6.42) for some square-integrable ( . Thus IIX 2P2( 11 ~

II(S+iD (S+iD (II: ( ( S2

+

(S4(, ()

D + D(, () (S2

+ ~(S2(, () +

9 16 ((, ()

1 5 9

?: 4 >,4 ((1 - COS>'S)2(,() + 2(S2(,() + 16(('() 4 :411(1-

cos>'S)(II ~ + II Gv'IOS -

iD (Ii:,

(2.6.43)

where we have used the operator inequality (2.6.44)

2.6. FURTHER CONSTRAINTS OF HEISENBERG T Y P E Therefore,

1

= 4-I(*,

(1- cos X(S - i2))p25)I2

A4

Now set X = In N and apply (2.6.6) to obtain

so we have 4 1/4 x 4 1/4

( P ) (~ )

-

>

(

10 + 16 + (1nN)

Notice that without the N-dependent term, this inequality would be given by t h e n = 2 case of (5.2.15), so we have enhanced the latter inequality with a dependence on the basic scale factor N . How does this new inequality compare to the n = 2 case of (2.6.37)? It is stronger regardless of the value of N . Indeed

so actually, the n = 2 case of (5.2.15) is itself stronger for all N. The advantage of (2.6.37) lies in the absence of the decay assumption. We now derive the N-dependent lower bound for n > 2 with the decay assumption that IX)n+6KPis square-integrable for some 6 > 0. Let be the square-integrable function such that !I = PnC, (2.6.49)

<

and generalize the algebra in (2.6.43) as follows:

146

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Ilg (s+i(k-D)(112 (g

(S2

+

(k - ~r) (, ()

n

L a~n) (Su(, (),

(2.6.50)

l=O

with the coefficients a~n) > 0 defined by (2.6.51) Now for

e=

2m (even

e)

we apply (2.6.44) with A = In N to obtain

(s4m(, () 2: 4m (In ~)4m ((1 - cos (In N)s?m(, () , while for

e= 2m + 1 (odd e)

(2.6.52)

we apply (2.6.44) as follows:

(s4m+2(, () 2: 4m

A~m (S2(1 -

(2.6.53)

cos Ams?m(, (),

m

where the value of Am will be chosen below. Consider the case where n itself is odd, and pair consecutive terms in the following way:

IIxnwll ~ 2:

~(n-l)

[

(n)

+ a~~,!l (S2(1 ~(n-l)

fo

_ cos Am s)2m(, ()

]

mI Ctm Ja~'J.+l (1 - cos Ams)m S

4

- i (In

Hence

(n)

~ 4m (Jna~)4m ((1 - cos(Jn N)s)2m(, ()

~)2m Ja~'J. (1 -

cos (In N)s)m)

(II: .

(2.6.54)

2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE

- i (In

~)2m Va~~ (1 -

~(n-l)

~O

4

m

147

cos (In N)8)m() 12

(it, >.tm va~~+l(l- cos>'m(8 - in))m

1

x (8 - in)pn( - i

(ln~)2m Va~~(l _cos (In N)(8 _ in))m pn() 12

~(n-l)

'" 4 (it ' _1_ g;;J (1 _~ei>·~(S-in) L >.2m V 2 m 1

m=O

m

a2~+1

(2.6.55) Now notice that the expansion

(1 _~eiA(S-in)

_

~CiA(S-in)) m

f (m) (-~)~t

~=O x

J-L

2

ei(~-2q)A(S-in)

q=O

(J-L) q

(2.6.56)

together with (2.6.6) implies

(2 .6.57) Define (2.6.58)

and observe that our inequality may be written in the form

148

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

The significance of wm(A) is the wavelet property w,(lnN)

= wm(21nN) = 0.

Applying Rolle's Theorem to Im wm(X), we may choose A,

such that

For such a choice of A, we obtain an explicit lower bound by discarding the real part of the modulus, which includes the contribution by the real part of wA(Xm). Thus

In the case where n is even, the nth term is not paired with anything, and our estimation yields:

In both cases, we recover the lower bound (5.2.15) in the N = ca limit. In the n = 3 case, (2.6.63) is applicable and the m-summation consists of an m = 0 term and an m = 1 term. We have

2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE

149

as the q-summation consists of only the q = 0 term. It is easily seen that 225 o - 04'

a(3) _

a(3) _ 1

(3) _ -

a2

a 3(3)

259

-10'

-

(2 .6.66)

35

4' 1,

and therefore

(P6)1/6(X6)1/6> [45 v'259 9549 III III 16 + 64

_1_ (3V35

+ (In N)4

+

149)] 1/6 4

(2.6.67)

Since In 2 < V2, the dyadic case enhances the n = 3 case of (5 .2.15) considerably. In the n = 4 case, (2 .6.64) is applicable, but the m-summation still consists of only an m = 0 term and an m = 1 term. We have 2

1 ao + (InN)4 (4j:Jfi + Pf) 16 x (2 M + 2Pf) a + (InN)8 (4)

(4)

(4)

(4)

2

2

1 2

+4

( 1 ( 1) )

,(2.6.68)

where the q-summation in the last contribution consists of both a q = 0 term and a = 1 term. The coefficients a~4) are:

q

(4) _

ao

11025

--zso'

_ 3229 a 1(4) -"10'

_ 987 a 2(4) 8'

(Z

+ 41)

(z

9) +4

(z

25) + "4

(z

49) = t:o ~ at(4) z t , +"4

(2.6.69)

a4(4) -1 , and so we have:

Notice that in the dyadic case, this lower bound is drastically larger than the lower bound provided by (5.2.15).

150

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

References 1. G. Battle, "Heisenberg Inequalities for Wavelet States," Appl. and Comput. Harmonic Anal. 4 (1997), 119-146.

2. A. Messiah, Quantum Mechanics, Vol. I, North-Holland, Amsterdam, 1965.

2.7

Variational Construction

We have already described the Meyer-Mallat construction of various wavelets, and their approach is the most widely accepted one. We now describe a variational method, which is actually more natural to the renormalization group formalism of Chap. 4. It is also the natural way to construct multi-dimensional Sobolev wavelets in $2.10. On the other hand, this method of construction cannot produce Daubechies wavelets. Our starting point is a scaling function q whose integer translates are not necessarily orthogonal. The first and major step in the variational approach is to construct a function $ E L2(JR) such that

in that Hilbert space - i.e., a dyadic wavelet as defined in the introductory discussion of this chapter. If the set generated spans the Hilbert space, then a mother wavelet P for an orthonormal wavelet basis is given by

i.e., the standard Z-translation-covariant orthogonalization of the set of unit-scale basis functions. In position space, this is nothing more than the application of the inverse square root of the overlap matrix to the integer translates of $, and the same operation was applied directly to the integer translates of the scaling function q as a first step in the Meyer-Mallat construction. The function $ to be constructed is first defined as the solution of a constrained minimization problem: $ minimizes the L2-norm llClla with respect to the constraints

where {c,) is the sequence of coefficients in the scaling relation (2.2.1). The driving principle is quite simple. The linear constraints define a closed afine subspace of L2(R) - closed, because the linear functionals are bounded - so $ exists and is unique; it is the point in the affine subspace nearest to the origin. Therefore $ is orthogonal to every vector in the linear subspace associated with the affine subspace - i.e., if

2.7. VARIATIONAL CONSTRUCTION

151

for all m, then $ I C in the Hilbert space. How is this related to the scales? The point is that for r > 0,

..-m, and the transformation ml t+ ml, m2 I-, m2,. . .,m, I+ 1-2rm+2n-2r-1ml of summation indices has the effect of reversing the sign of this multiple sum. Therefore, the sum vanishes, and so -a

This establishes the interscale orthogonality property by the argument above. Now since $ minimizes a quadratic form - namely 1ICII; = (C, C) - with respect to linear constraints of the form (2.7.7) (C, gm) = dm, $ has the formal representation

On the other hand, K can be explicitly realized in momentum space if we set gm(x) = 2q (22 - m) . Indeed, by the Poisson summation formula, we get

(2.7.10)

152

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Thus, an arbitrary power of the operator yields (2 .7.12)

and so it is trivial to verify that (2.7.13)

Since K 9n(P)

Gp)~ g,;(p +

17

e-i~np17 Gp) L

=

e

Gp + r Gp + 12

41ft)17 117

2m:

21ft)

(2 .7.14)

it follows that

e-i~np17 Gp) X

(1 + ~ Gp + e-i~np17 Gp) ~ Gp + 12 + ~ Gp + e-i~np17 Gp) ,

(1

Q

Q

117

Q

117

21ft)

117

21ft) 12) -1

21ft)

n

-1

(2.7.15)

and therefore (2.7.16)

In this case, the linear constraints are (2.7.17)

and so we have the solution explicitly given in momentum space: (2.7.18)

with h given by (2 .2.6) . Property (c) in §2.2 and the summability of {cm } guarantee that this momentum expression is well-defined and continuous, so '1j; is certainly squareintegrable if T/ is. It is worthwhile to remark that (2.7.19)

2.7. VARIATIONAL CONSTRUCTION

153

with llr defined via Property (a) in §2.2. To see this, we set r = 0 without loss of generality (by the dyadic scale covariance of the construction) and note that it suffices to show that 1)(2x) and 1)(2x - 1) can both be expanded in the 'IjJ(x - n) and 1)(x - n). In momentum space this would mean

fJ Gp) = f(p)fJ(P)

+ g(p);P(P) ,

(2.7.20)

e-i~fJ Gp) = j(p)fJ(P) + g(P);p(P)

(2.7.20')

for 27r-periodic functions f, g,j, g with square-summable Fourier coefficients. Inserting (2.2.7) and (2.7.18) and dividing out the resulting common factor fJ(!p), these conditions reduce to 1=

~h(P)f(P) - 2e-i~p ( ~ IfJ Gp + 27r£) r)

1=

~ei~P h(p)](p) - 2 ( ~ IfJ Gp + 27r£)

-1

n

-1

h(p + 27r)* g(p),

h(p + 27r)* g(p) .

(2.7.21)

(2.7.21')

On the other hand, the 27r-periodicity of the unknown functions imposes the additional conditions

n

1=

~h(P + 27r)f(P) + 2e-i~p (~lfJ Gp + 27r£ + 7r)

1=

-~ei~Ph(p + 27r)j(p) - 2 (~Ii) Gp + 27r£ + 7r) r)

-1

h(P)*g(P) ,

(2.7.22)

-1

h(p)*g(p) .

(2.7.22')

(2.7.21) and (2 .7.22) constitute a 2 x 2 linear system for f(p) and g(p), as do (2.7.21') and (2.7.22') for ](p) and g(p). Solving for f(p) and g(p), we obtain

f(P)

g(p)

~

(~I" Gp + ,., + • ) IT'

,,
[h(P),

+ h(p+ ,.)'

(~I" Gp+ "')Ir] ,

= ~D(P)-1(h(P) -

h(p + 27r)),

(2.7.23) (2 .7.24)

where D(P) is the determinant of the coefficient matrix and is given by

D(P)

~

Gp+ ,., Ir '.li' (~I" Gp+ ,.,) IT] ·

,-'I+(P)I' (~I" + Ih(p+

+.)

(2.7.25)

154

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Since D(P + 21l')

= -D(p), f(P)

j(p)

and g(p) are both 21l'-periodic. Similarly,

-215(P)-' [h(P)" -h(P+ z.)"

g(p)

(~I" Gp + 2.[+ • ) I') -,

(~I" Gp+ 2.£) I') -'] ,

(2.7.26)

= ~ei~PD(p)-1(h(P)+h(P+21l')),

,', (2.7.27)

2

where D(p) is the determinant for the latter system. Actually,

D(P)

= -D(P)*,

(2.7.28)

-D(P), and therefore j(p) and g(p) are 21l'-periodic as well. This so D(P + 21l') completes the proof of (2.7.19), which, in turn, implies

spans L2 (R), because

ntir is assumed to be the null space. r

Thus, we obtain an orthonormal basis of wavelets if we orthogonalize this interscaleorthogonal set scale by scale. Clearly, the mother wavelet if for the desired basis is given by (2.7.2) . This basis transformation is the last step in the variational construction and is applied to a wavelet, while the same basis transformation is the first step in the Meyer-Mallat construction and is applied to a scaling function. Given the same initial scaling function 1], however, the resulting mother wavelet if is the same. To show this, note that

L I~(P + 41l'f + 21l') 12+ L

I~(P + 41l'f) 12,

(2.7.29)

l

4lh(p + 21l'W (

~ 11] (~p + 21l'f) 12)

-1

(2 .7.30)

so by (2 .7.25) , we may write (2.7.31) The inverse square root of the bracketed expression in (2.7.25) is involved in this Ztranslation-covariant orthogonalization. On the other hand , the scale relation (2.2.7) implies (2.7.32)

2.7. VARIATIONAL CONSTRUCTION

155

and therefore Ih(P)12

=4L

~ If] (~p + 27rf) 12 + Ih(p + 27r)12 ~ If] (~p + 27rf + 7r) 12 If](p + 27rf) 12

(2.7.33)

l

This reduces (2 .7.31) to

16 (

x

n n~

~ If] Gp + 27rf + 7r)

(~If] Gp + 27rf)

-1

-1

If](p + 27rfW · (2.7.34)

Inserting this formula and the formula (2.7.18) for ;j)(p), we realize (2.7.2) more explicitly as

~(p)

(2.7.35)

Now by (2.2.8) we have

and so we may write (2.7.37)

where we have also applied (2.2.3) . Since the formula is just the momentum-space version of (2.2.12), we have identified the mother wavelet constructed here with the mother wavelet produced by the multiscale resolution analysis of Meyer and Mallat.

References 1. G. Battle, "A Block-Spin Construction of Ondelettes, Part I: Lemarie Functions," Commun. Math. Phys. 110 (1987),601-615 .

2. G. Battle, "Cardinal Spline Interpolation and the Block-Spin Construction of Wavelets," in Wavelets - A Tutorial in Theory and Applications, C. Chui, ed., Academic Press, San Diego, California, 1992, pp. 73-93.

156

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

3. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets and Their Applications, B. Ruskai et al. (eds.), Bartlett and Jones, Boston, 1992, pp. 323-349. 4. P. Lemarie-Rieusset and J. Kahane, Fourier Series and Wavelets, Gordon and

Breach, Luxembourg, 1995. 5. S. Mallat, "Multiresolution Approximations and Wavelet Orthonormal Bases of L2(lR)," Trans. Amer. Math. Soc. 315, No. 1 (1989) , 69-87.

2.8

Without Intrascale Orthogonality

In the variational construction we have just described, the major step is to derive a basis {1/I(2 T x - n) : r, n E Z} of wavelets that are orthogonal for differing length scales but not orthogonal on the same scale. From a computational point of view, this basis is actually more desirable than the orthonormal basis we obtained from it. To be sure, we must use the dual basis to compute expansion coefficients, but the derivation of the dual basis is rather painless. Given the orthogonality between length scales, the dual basis is realized scale by scale. Our focus here is on the case of Lemarie wavelets - i.e ., the case where the initial scaling function TJ is given by (2.8.1) The degree of regularity for 1/1 is parametrized by N, with each value of N corresponding to an Euler-Frobenius polynomial. From such a polynomial one can derive an explicit position-space representation of 1/1 by the calculus of residues. Indeed, 1/1 is a linear combination of discrete translates of TJ with coefficients determined by the complex roots of the associated Euler- Frobenius polynomial. Recall from (2.7.16) and (2 .7.17) that (2.8.2) where the sequence {en} is given by TJ(X)

= L cn TJ(2x -

n) .

(2.8.3)

(1'2 p') ,

(2.8.4)

n

The key observation for what follows is the identity

L: (p' +1271'£) 2 = 41 csc

'"

2

which is a result of the Poisson summation formula. Indeed, 1=

Le n

inp

' Don

=

Le

inp

n

'

J

X(x)X(x - n)dx

2.8. WITHOUT INTRASCALE ORTHOGONALITY

= 2~ L

157

J

Ix(q)1 2e- in (Q-p')dq

n

L IX(p' + 27rfW i

., 2'" (P' + 27rf)2 1

le'P - 11 L

i • 2 4 sm

(12 ') '" (P' +127rf)2 . P

(2 .8.5)

L

i

To apply (2 .8.4), observe that

L IX(P' + 27rf)1

2N 2

+

i

4N+1

sin 2N + 2

(~pl) '" 1 2 L (P' + 27rf)2N+2 i

4++ I)! sm2N+2 (12P') dp'~N2N '" L (P' + 27rf)2 N

1

1

.

(2N

i

(12 ') dp'2N ~N (12 ')

4N . 2N+2 (2N + I)! sm

P

csc

2

P

.

(2.8.6)

On the other hand, even-order derivatives of csc 2 (!p') are polynomials in csc2 (!pl) and cot 2 (!p') , homogeneous of degree equal to two plus the even order, with at least one factor of csc 2 (!pl) in each term. Multiplying this by sin 2N+2 (!p'), we obtain for (2.8.6) a polynomial in cos 2 (!p') only, and with no power higher than COS 2N (!pl). With 1 1 I (2.8.7) cos 2 (12P') =2+2cosP , we may therefore see it as an Nth-degree polynomial in COSp' . Let RN denote this last polynomial, so that (2 .8.8) lij(P' + 27rf) 12 = RN(COSp/).

L i

Combining this with (2.8.2) , setting p' = !p, we obtain

(2.8.9) On the other hand,

L( -1)nCl_ne-i!np n

n

_e-i!p ~(_I)n(N: l)e i !np -e-i!P(1 _ ei!p)N+l

(2 .8.10)

158

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

when the scaling function is given by (2.8.1), so (2.8.11) In position space we have a spline with half-integer knots realized as an expansion in half-integer translates of 1)(2x). Indeed, if we set (2 .8.12) we have (2.8.13) m

(2.8.14) (2.8.15) The calculation of these coefficients is our major concern here. These integrals can easily be written as contour integrals in the complex plane. If we set (2.8.16) o ~ p ~ 47r, we have a parametrization of the unit circle

am

= -1 .

1

27r~ Izl=l

Izl = 1 and the formula

zm- 2(1 - z)N+1

1 1 -1 dz . 1 RN(2z+2z )

(2.8.17)

On the other hand, if we set (2.8.18) then we have the alternative formula (2.8.19) Since (2.8.20) is a 2Nth-degree polynomial with no zero at z = 0, we can avoid computing residues at z = 0 for almost all m. The two formulas become

-1-1

27ri Izl=l

1. --2 7r~

1

zm+N-2( 1 _ z)N+1 _1_dz KN(Z) ,

(2.8.21)

1()dz, z -m-1( z - l )N+1 - K

(2.8.22)

Izl=l

N Z

2.8. WITHOUT INTRASCALE ORTHOGONALITY

159

and we choose to use (2 .8.21) for m ~ 2 - N. If a1, ... , ak are the distinct roots of KN{Z) in the interior of the unit disk, then a

m

= L.. ~ ,=1

) Res ( zm+N-2{ 1_ z)N+1_1_ K (z )' a, ,

m

~

2-N.

(2.8.23)

N

(Note that KN{Z) has no zeros on the boundary Izi = 1 because RN (cos ~p) does not vanish anywhere.) We use (2.8.22) in the complementary case. Thus -

~ ~

a)

1 Res (z-m-1{z _1)N+1__ KN{Z)' ,

,

m :'S min{ -1,1- N},

(2 .8.24)

_,o)

1 -Res (z-m-1{z _1)N+1_ KN{Z)

-t

Res (z-m-1 (z - I)N+1 K l{ z)'

=1

N

a,),

O:'S m :'S 1 - N .{2 .8.25)

Obviously, this last case is vacuus if N > 1 and it is the only case where there is a residue at Z = O. We now derive the position space representation for the dual basis, which can be calculated on each scale because the basis is interscale-orthogonal. As always, we concentrate on the unit-scale functions - i.e., the unit translates of the mother wavelet - without loss of generality. The overlap matrix is Smn

J ~ 1 e-i(m-n)pl~(p)12dp ~ 1" L I~{p +

=

1(J{x - m)1(J{x - n)dx 00

27r

- 00

e-i(m-n)p

27r -"

so the dual basis {u m

}

27r£Wdp,

(2.8.26)

l

is given by

n

~ 27r

L

1(J{x - n)

n

1" -"

e-i(m-n)p

(L I~{p +

27r£)12) -1 dp . (2.8.27)

l

The Poisson summation formula

(2.8.28) n

160

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

implies

~ 27r

2: 17T-7T

ei(x-m)(p+27T j );j(p + 27rj)

~1

00

27r

(2: 1;j(P +

27rfW) -1 dp

l

j

ei(x-m)p;j(p)

(2: 1;j(P +

27rf) 12)

-1

dp.

(2 .8.29)

l

-00

Thus

Um(X)

= uo(x -

m),

(2.8.30)

and in momentum space, (2 .8.31) On the other hand, (2.8.11) implies

(2.8.32) Hence

uo(p)

-~e-i!P(1 -

ei !p)N+1 RN

(-COS Gp) ) fJ Gp) N+1

X[RN (COS Gp) ) 11 + ei !PI2N+2 + RN

(-COS Gp) ) 11 - ei!PI2N+21-1

(2 .8.33)

For the complex integration variable (2.8.16), we obtain Uo(x)

(2.8.34) m

1

1 z - -z 1 -1) -1. z m-2( 1- Z )N+1 RN (-27rt Izl=l 2 2

2.8. WITHOUT INTRASCALE ORTHOGONALITY

161

1 - 2Z-1 1 ) (1- z)N+l(l_ Z-I)N+l + RN ( -2Z

~

1

]-1

dz

zN+m-l(l_ z)N+IKN(-z)(-I)N

27ft 1%1=1

X

[KN(Z)(1

+ z)2N+2 -

(1 - Z)2N+2 KN( -z)tldz,

(2.8 .35)

while the alternate substitution (2.8.18) yields

(2.8.36) As involved as these contour integrals appear to be, the residue calculations are very closely related to (2.8.23)-(2.8.25), as we now show. First note that the variable relation z = eipl obviously yields the relations

4z (z-I)2 '

. dfl

tZ-

dz

(2.8.37) (2.8.38)

,

z=eip'

from which it follows that 1

RN(COSp)

=

4 N . 2N+2 (2N + I)! sm ~N X

2

dpI2N

CSC

(12 PI)

(12 PI) 4N = (2N

(z _1)2N+2

+ I)! (_4z)N+I

d )2N [ -4z J x ( iz dz (z _ 1)2 1

(2N

+ I)!

(z - 1)2N+2 ( d) 2N [ z 1 zN+I z dz (z - 1)2 .

(2.8.39)

Second, we insert (2 .8.20) to get 1

KN(Z)

= (2N + I)!

(z - 1)2N+2 ( d) 2N [ Z ] z z dz (z - 1)2 ,

(2.8.40)

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

162

which immediately implies KN(z)(l

+ Z)2N+2

- KN( -z)(l - Z)2N+2

(z2 _ 1)2N+2 ( d) 2N [ z z ] z(2N + I)! z dz (z - 1) 2 - (z + 1) 2 .

(2.8.41)

Next, we observe that Z

z

(z - 1)2 and the variable relation w

= Z2

(z

+ 1)2

4z2 - (z2 - 1)2'

(2.8.42)

yields

d f( z2) z -d

z

df I = 2wdw w = z2

(2.8.43)

Finally, (2.8.44)

and we may now insert this in (2.8 .35) and (2.8.36) to obtain bm

4-N~1 27l'l

1

4- N - 1 ---.27l'l

1

Izl=1

zm+N-2(1 - z)N+l KN( - z) (-1): dz, KN(Z )

(2.8.45)

(_l)N z -m-l( z - l)N+l KN( - z) - - 2 - dz , KN(Z )

(2.8.46)

Izl=1

respectively. Both 1j; and Uo are exponentially localized, and we are now in a position to determine the rates of exponential decay. Clearly, it suffices to determine the exponential decay rates of the sequences {am} and {b m }, since T] has compact support. Without loss of generality, we consider m ~ 2 - N or m ::; min{ -1,1 - N}; in the former case, am is given by (2.8.23), and therefore am = O(pm) for large positive m, where p =

max , la,1< 1.

(2.8.47)

In the latter case, am is given by (2.8.24), and so am = O(p-m) for large negative m. This means the rate of exponential decay in either direction is the positive number -Inp. Now consider the same two cases for the sequence {b m }; in the former case, we apply (2.8.45) to obtain N 1 bm = _4- - ~ Res (zm+N-2(z - l)N+l i:v~~:1'

Thus bm

= O(pm/2) for large positive m.

±yQ;) .

(2.8.48)

In the latter case, we apply (2.8.46) to obtain

N 1 bm = 4- - ~ Res (z-m-l(l - z)N+l i:v~~:1'

±yQ;) ,

(2 .8.49)

which implies bm = O(p-m/2) for large negative m. Therefore, the rate of exponential decay in either direction is - ~ In p. This incidentally proves that the exponential decay rate of uo(x) is half that of 1j;(x), regardless of the value of N.

2.9. CHUI-WANG WAVELETS

163

References 1. G. Battle, "Cardinal Spline Interpolation and the Block-Spin Construction of Wavelets," in Wavelets - A Tutorial in Theory and Applications, C. Chui, ed., Academic Press, San Diego, California, 1992, pp. 73-93. 2. C. Chui and J. Wang, "A Cardinal Spline Approach to Wavelets," Proc. Amer. Math. Soc. 113 (1991), 785-793. 3. P. Lemarie-Rieusset, "Une Remarque sur les Proprietes Multi-Resolutions des Fonctions Splines," C.R. Acad. Sci. Paris 317 (1993), 1115-1117. 4. P. Lemarie-Rieusset and J . Kahane, Fourier Series and Wavelets, Gordon and Breach, Luxembourg, 1995.

2.9

Chui-Wang Wavelets

There is a class of interscale-orthogonal wavelet bases whose mother wavelets are compactly supported splines. For such a basis, the mother wavelet of the dual basis is not compactly supported, but the interscale orthogonality of the compactly supported splines in the former basis is remarkable. The mother wavelet is a Chui-Wang wavelet, and the point of view of the construction lies somewhere between the Meyer-Mallat approach and the variational approach. We begin with a brief discussion of the wavelets constructed in the previous section to describe this intermediate point of view. Let T)M be the scaling function given by (2 .9.1) and consider the L2-subspace H(M) spanned by the integer translates of T)M . In approximation theory the fundamental M th-order cardinal interpolating spline is the solution to minimizing 11(112 in H(M) with respect to the constraints mEZ .

(2.9.2)

Standard notation for the solution of this problem is LM(X), and we denote the expansion of this fundamental spline by (2 .9.3) n

Obviously, this is not an orthogonal expansion, but the coefficients are uniquely determined. Indeed, they can be explicitly realized in terms of the zeros of the 2Nth-degree Euler-Frobenius polynomial in the unit disk of the complex plane. The crucial observation here is that the derivative function

164

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

is actually the mother wavelet of an interscale-orthogonal basis. Indeed, if we integrate by parts we have

J

Lg)(2x - 1)1)J(2 r+1 x - m)dx

(_I)J2 rJ

J

L2J(2x

2rJ / L2J(2x -1) J

2rJ - r- 1 ~)-I)j

_1)1)~J)(2r+lx -

m)dx

~(-I)j G)8(2 + r

1

X -

m - j)dx

(~)L2J(Tr(m + j) -1).

(2.9.4)

J

j=O

< 0, so

By (2.9.2) these evaluations of L2J all vanish if T

L~~)(2x - 1) ..L span{1)J(2 r+1 x - n): n E Z},

T

< 0.

On the other hand,

n

(2.9.5) Therefore, T

< 0,

and so the interscale orthogonality is established. This wavelet basis is precisely the interscale-orthogonal Lemarie basis constructed in the previous section. To see this, observe first that

JL~~)(2x JL2J(2x-l)1)~J)(2x-m)dx J I) (J) - 1)1)J(2x - m)dx

(_I)J

L2J(2x - 1)

-1)j

)=0

8(2x - m - j)dx

J

"(J)

J ~(-1)1 j L2J(m + j - 1)

_(_l)m

t (~)81-m,j , j=O

J

(2.9.6)

2.9. CHUI-WANG WAVELETS

165

Second, observe that this sum is just -( -1)mci~m' where c(J) m

= ~ (J)8 . . mJ

(2 .9.7)

~

j=O

J

are the coefficients for the scaling relation (2.9.8) n

so the constraints (2.7.3) are satisfied by -L~~~~(2x -1). On the other hand, for any L2-function f satisfying (2.7.3), the orthogonal projection E(N+l) f of f onto 1i(N+l) satisfies (2 .7.3) as well- and it is the only element of 1i(N+l) satisfying (2.7.3). Thus E(N+l) f is the solution to minimizing the L 2-norm with respect to the constraints (2.7.3) . This implies -L~~~;)(2x -1) is that unique element of1i(N+l) and therefore the function 1j;(x) that minimizes the L2-norm with respect to (2 .7.3) . This function is the wavelet given in the previous section, so the claim is established. We could have made just this identification _

(N+l)(

1j; (x ) - -L2N+2 2x - 1

)

(2.9.9)

by simply calculating both quantities in momentum space, but the reasoning we have just given is more to the point. We now consider the Chui-Wang wavelet, which can actually be written as (2.9.10) v

The verification of interscale orthogonality is the integration by parts

J

r 1j; (x) r/J (2 +lx - m)dx

L ( -W'T}2J(1I v

+

1)/ 'T}~~)(2x

x 'T}J(2 r +lx - m)dx (_I)J2 rJ L ( -W'T}2J(1I

+ 1)

- II)

J

'T}2J(2x - II)

v

x 'T}:,J) (2T+lX - m)dx 2rJ L(-lt'T}2J(1I v

+ 1)

J

'T}2J(2x -II)

166

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

This numerical expression is identically zero for r tion exploits the familiar identity

< O! Why? This ingenious construc(2 .9.12)

v

which follows from the observation that the sum is reversed in sign by the index change v t-+ 2n - 1 - v. We have already used it in other wavelet constructions, but this application is more subtle. In this case, clL = 1J2J(/-l) and n = 2- T (m + j), r < O. We may now argue as we did in the case of the spline Lg) (2x - 1) above. Since we have just shown that r

< 0,

(2.9.13)

we conclude that r

< 0,

from the observation that

where the point of this last equation is that '1j;(2TX - m) lies in the span of the 2- r - 1 Z_ translates of 1JJ(2 T + 1 x - 2m). How do we know that '1j; has compact support? Since 1J2J is obtained by convolution of X with itself 2J times, we know that each half-integer translate of 1J~~) (2x) is compactly supported. On the other hand, the linear combination (2 .9.10) of these translates is a finite sum because 1J2J(v + 1)

= 0,

v

< 0 or v

~

2J - 1.

(2.9.15)

Specifically, '1j;(x) is supported in the interval 0 :S x :S 2J - 1. As far as regularity is concerned, '1j; has the same degree of smoothness as 1JJ has - namely, class C J - 1 - , We now examine the dual basis, which is itself interscale-orthogonal. To compute the unit-scale elements, we apply (2.8.30) because general formulas from the previous section apply here. The functions are unit-scale translates of u, where u(p) =

(~I~(p + 211"1!12) ~(p).

(2.9.16)

-1

On the other hand, the Fourier transform of (2 .9.10) is just

~(p)

=

r

(i~P

TJ2J

Gp) 2) Gp) 2)

-W1J2J(V

+ 1)e- i !vp

v

(_1)J (e-i!p - 1)JTJJ

-lt1J2J(v

+ 1)e- i !vp

v

(2.9.17)

2.9. CHUI-WANG WAVELETS

167

where the Fourier sum (2.9.18) v

is 47r-periodic. Hence

le-i~p +

11 2JI>'(p + 27r)12 ~

IfiJ Gp + 27rt) 12

le-i~p + 11 2J I>,(p)12 ~ 117J Gp + 7r + 27rt) 12

le-i~p -11 2J I>'(p+ 27rWR J- 1 (cos Gp)) + and so

2~

u(x)

le-i~p + 11 2J I>'(PW R J- 1 (-cos (~p) )

i:

eipx{i;(p) (

lt

I{i;(p + 27rf) 12 )

11 e,pxilJ (1) 2 >.(p + 00

=

27r

-00

.

P

x I>'(p + 27r)12 R J -

1

(2.9.19)

dp

.

1 1 27r)(e-' 2P - 1) J [ le-'.2P - 11 2J

(cos Gp) )

XRJ-1 (-cos Gp))

-1

,

r

+ le-i~p + 112JI>,(p)12

1dP

-2 I:b m1)J (2x - m),

(2.9.20)

m

where - following the general formulation in the previous section - we have

_~ 47r

tTf ei~pm >.(p + 27r)(e-i~P _ I)J [le-i~P _ 112J

10

x I>'(p + 27rW RJ-1 (cos

(~p ) ) + le-i~p + 112J

r 1

x 1>'(pW RJ-1 (-cos Gp) )

~

1

27r% Izl=1

z-m-1

I: 1)2J(V + 1)( -zt(z -

I)J [( z - I)J (Z-1 _1)J

v

xI: 1)2J(V + 1)1)2J(v' + V1V'

dp

1)( -zt- v ' RJ-1

Gz+ ~ z-1)

168

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

if we set z = e - i a ~ . On the other hand,

[

+

~ - 1 ) expression Same V ~ J ( V l ) ( - ~ ) -(2-l

1-'

LIZ (2.9.22)

if we set z = e e p . The coefficients are computed by residues; to avoid high-order poles at z = 0, it is most convenient to use (2.9.21) for m 5 0 and to use (2.9.22) for m 2 1. By (2.8.20) we may write (2.9.21) as

and (2.9.22) as

x

[polynomial same I-'

dz.

The residues are determined by the zeros of the bracketed polynomial. Since almost none of the coefficients b, prove to be zero, the dual wavelet u(x) cannot have compact support. On the other hand,

so u(x) is exponentially localized.

References 1. G. Battle, "Cardinal Spline Interpolation and the Block Spin Construction of Wavelets," in Wavelets - A Tutorial in Theory and Applications, C.K. Chui, ed., Academic Press, San Diego, 1992, pp. 73-93.

2. C. Chui and J. Wang, "On Compactly Supported Spline Wavelets and a Duality Principle," Trans. Amer. Math. Soc. 330 (1992), 903-915. 3. C. Chui and J. Wang, "A Cardinal Spline Approach to Wavelets," Proc. Amer. Math. Soc. 113 (1991), 785-793.

2.10. MULTI-DIMENSIONAL WAVELETS

169

4. C. Chui and J. Wang, "A General Framework of Compactly Supported Splines

and Wavelets," J. Approx. Theory 71 (1992),263-·304. 5. P. Lemarh~-Rieusset, "Une Remarque sur les Proprietes Multi-Resolutions des Fonctions Splines," C.R. Acad. Sci. Paris 317 (1993), 1115·-1117.

2.10

Multi-Dimensional Wavelets

The extension of the one-dimensional constructions to an arbitrary number of dimensions is quite easy, but it is based on tensor product combinations of the mother wavelet with the unit-scale-orthogonalized scaling function cp given by -1 / 2

ip(p) ==

(

~ Ii)(p + 27reW )

i)(P) .

(2.10.1)

Too often, the remark is made that a one-dimensional wavelet basis can be extended to a multi-dimensional basis by a tensor product. Such a remark is misleading because a straight tensor product of copies of a wavelet basis yields basis functions that can have length scale 2100 in the north-south direction and length scale 2- 100 in the east-west direction! The point is that a multi-dimensional wavelet basis with only one scale parameter is more desirable - actually, a necessity from a practical point of view. For d dimensions, consider the special vectors -; E {O, 1 }d\ {O} and define

WCe-)(T·) ==

II cp(x,) II w(x,). Et=O

(2.10.2)

El.=l

The functions (2 .10.3) with r E Z, r;;E Zd, constitute an orthonormal basis of L 2(JRd) . The functions wCe-) are all mother wavelets in this construction, and there are 2d - 1 of them; the function cp(x,) is not included among the mother wavelets.

n

The moments of a function ~

if the functions j( x)

n x" d

,=1

d

E s, :S M.

,=1

Notice that if

je:;)

are said to vanish up to order M in d dimensions

are integrable and have vanishing integrals for s, 2: 0 and

je:;)

happens to have the product form d

je:;) ==

II j,(x,),

(2.10.4)

,=1

then the moments of je:;) vanish up to order M if and only if the moments of j, vanish up to order M for some £. Since there is at least one factor of W in (2 .10.2) , it follows that every multi-dimensional mother wavelet w(e) inherits the vanishing moments

170

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

property that the one-dimensional mother wavelet I)i happens to have. (Recall that the scaling function 'P - in contrast to I)i - does not even have a vanishing Oth-order moment.) A function f(-;) is said to be class C N in d dimensions if the functions

g d

(

8 ') ~ 8x:' f(x) S

d

are continuous for s, ~ 0 and

L

,=1

_

s, ~ N. Clearly, each mother wavelet I)i(e) inherits

the degree of smoothness possessed by 'P and I)i . Each I)i(-;) obviously has compact support if'P and I)i do, so the existence of multidimensional Daubechies wavelets is no problem. In general, this multi-dimensional extension preserves long-distance decay properties. For example, if 1'P(x)1 ~ eoe-molxl ,

(2 .10.5.0)

II)i(x)1 ~ eoe-molxl,

(2.10.5.1)

then

11)i(-;) (-;)1

~

cgexp(-motlx,l)

~ ci~+~~t.X1)'

(2.10.6)

so multi-dimensional Lemarie wavelets can be built from one-dimensional Lemarie wavelets, preserving even the rate of exponential decay. As another example, consider the decay property 1'P(x)1 ~ eo(l + x 2)-N, (2.10.7.0) 2 II)i(x)1 ~ eo(l + x )-N. (2.10.7.1)

In this case, d

11)i(-;) (-;)1

~

cg II (1 + x~)-N ,=1

~

d )-N cg ( 1 + ~x~ ,

(2.10.8)

so power laws are preserved as well . Finally, consider the smoothness and decay of a Meyer wavelet - i.e., the property rp, ~ E Ccf(lR). Since

---

I)i(-;) (p)

= II rp(p,) c,=O

II ~(p,), e,=1

(2.10.9)

2.10. MULTI-DIMENSIONAL WAVELETS

171

\[Ie<") E CIf (JRd) is an obvious property of the multi-dimensional mother wavelets in this case. Having described the construction of orthonormal bases of wavelets for L 2(JRd), we wish to construct such bases for the Sobolev space defined by the norm

(2.10.10) where s is any positive integer. In particular, (2.10.11) and indeed, this will be the most appropriate Hilbert space norm for the renormalization group analysis of Euclidean field theory. Now observe that it is quite easy to obtain a wavelet basis that is orthonormal with respect to the inner product corresponding to (2 .10.10) . For a given wavelet basis {\[Id that is orthonormal with respect to the inner product for L2 (JRd), the functions \[I k,8 defined by (2.10 .12) certainly form a basis with the desired orthonormality. In this notation k ranges over the triples C~, r , ~) and (2.10.13) The coherent structure with respect to discrete scale-commensurate translation is preserved because the transformation is just multiplication by I 1- 8 in momentum space, while the coherence in scale is preserved by the homogeneity of I I-s . As far as the

P

regularity of

P

\[I~1) is concerned , note that if d

\[1(1) (p)

II

P: '

,=1

is integrable for

2:= s,

~

N , then so is ____

\[1(1 )

d

(p)1 p I-s IIp~' ,=1

for

2:= s, ,

~

N

+ s . This momentum-space description of how the regularity is affected is

-

similar in spirit to the quasi-correct statement that if \[1(1) is class C N then \[I~ e) is class CN+s (The latter statement is false without some technical condition, but the point is that (2.10.12) can only increase smoothness.) The issue raised by this construction lies in the effect of (2.10.12) on long-distance decay properties. For dimension d > 1, for

172

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

example, this transformation obviously destroys real analyticity in momentum space specifically at p= 0 - and therefore destroys exponential decay in position space. On the other hand, (2.10.12) preserves the properties of Meyer wavelets. Recall from the construction in §2.2 that a one-dimensional Meyer wavelet lJt not only has a compactly supported Fourier transform, but also has the property

~(P) = 0,

--

(2.10.14)

Ip,1 ~ 8} :Hp: I p I ~ 8},

(2.10.15)

for some 8 > O. It follows from (2.10 .9) that lJtCE'")(p) vanishes in the region

U {p: g,=l

so each lJt(E'" ) certainly has the property (2.10.16) This means that multiplication by

I PI-s introduces no singularities, so lJt~E'")

is class

Coo as well. Finally, (2 .10.12) preserves the vanishing-moment property for all of the wavelets that have been constructed. If for E s, . ~ M

is continuous and vanishes at

p= 0, then for E, s, ~ M -

s

(g :;;,)(I p I-S~(p)) = (gj;J g(;J ((g :;;,) Ip 1-_) ((g :;;,-:;,) ~(P)) x

is continuous and vanishes at

(2.10.17)

p= 0 because (2.10.18)

(2.10.19) Technically, the momentum-space observation is not quite the same as the assertion '; that if the moments of lJt(E'") vanish up to order M, then the moments of lJt~E'") vanish up to order M - s. Indeed, this latter implication is false without some additional condition. However, we have made the point that (2 .10.12) does not destroy

2.10. MULTI-DIMENSIONAL WAVELETS

173

the vanishing-moment property, but only reduces the order up to which the moments vanish. The strategy given by (2.10.12) fails to preserve the exponential decay of a Lemarii! wavelet. It is worth noting, however, that in the special case d = 1, one can slightly alter this strategy. If we define the Sobolev wavelet Qs by

then any analyticity that 6(p) may have is shared by 6,(p), provided $ ( p ) vanishes at p = 0 up to order 2 s. This means the constfuction produces one-dimensional Sobolev wavelets that are compactly supported as well as wavelets that are exponentially localized; one need only apply (2.10.20) to a Daubechies wavelet. We now turn to the problem of constructing a multi-dimensional Sobolev-orthonormal basis of wavelets that are exponentially localized. The idea is to use the variational approach introduced in 52.7, where q is still given by (2.8.1), but now the Sobolev norm is the Hilbert space norm to be minimized on the affine manifold defined by the linear constraints. Naturally, each of the 2d - 1 constructions $ v S corresponds to a certain set of constraints. Indeed, the constraints are determined by d in a way that reflects (2.10.2)-specifically, $ ,S minimizes 11C112,s with respect to the constraints A

A

The construction guarantees interscale orthogonality of the generated basis functions - G ) with respect to the Sobolev inner product. Indeed, for arbitrary ( in $E*s(2' the Sobolev space such that

C=

$I'S

+ t[ satisfies (2.10.21), so A

But this derivative at t = 0 is just twice the by the variational definition of Since (2.10.23) holds for (i as real part of the Sobolev inner product of ( with well as for 6, the imaginary part of the Sobolev inner product vanishes as well, and therefore $'>s is Sobolev-orthogonal to all ( such that (2.10.22) holds. On the other hand, (2.10.22) applies to all functions $".'(2' - ); for which r > 0. This is a consequence of iterating the scale relation $"gs

-

174

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

and applying the algebraic identity ""' ~ cm,, ""' ~ cm," . .. ""' ~ m~

m~'

Cm,( r)

(_l)m~r) C1_ 2 r m,+2n,_2 r -

1

, " - .. . --m,(r) r-. m,_2 :lm,

=0

m~r)

(2.10.25) for an ~ such that £, = 1. To derive from this a Sobolev-orthonormal basis of wavelets, one simply applies the inverse square root of the overlap matrix to the sequence of integer translates of the 'Ij; -; ,S. In momentum space, the mother wavelets are given by

(2.10.26)

The question of how much smoothness and long-distance decay these wavelets have must now be addressed, as ~sts the value of this construction. It is resolved by analyzing the expression for 'Ij; -; ,8 (p) . The momentum expression is derived in exactly the same way for 'Ij; E ,s as it was derived for the one-dimensional 'Ij; in §2.7, except the factor I p I-s is involved in the Poisson summation. The formula parallel to (2 .7.16) is just

(2.10.27) which can also be written in the form

~(p)

=

llo }I (2e-i~p, h(P,)

x

gIry Op,

h(p, + 27r)*) (

~ 1~ P+27r 71-

2S

e

+ 27rf,)

12) -11 P1-

25

gry Op,) ,

(2.10.28)

where the choice of '" for the construction of exponentially localized wavelets is given by d

II ry(p,) ,=1

=

,=1 (2.10.29)

2.10. MULTI-DIMENSIONAL WAVELETS Now write

so that ( 2 . 1 0 . 2 8 ) assumes the form

Recall that for this choice of q,

and therefore

h(p,

+ 2 ~ ) =* ( 1 - e ' + ~ ~ ) N + l .

(2.10.34)

The replacement of each real variable p L with a complex variable zL analytically continues all of these functions of momentum. We have the analytic continuations

-

rj(pL) ++ 4(pL)*

iN+l(e-izc

- l)N+lz;N-1

( - i ) N + 1 ( ei z . - 1 )N+l h ( p L ) ++ ( 1 + e-'+zL)J"+l h(p, + 2 r ) *

9

-N-l

2,

9

(1-eiiz~)N+l,

~ 6 ) rSG),

which yield the analytic continuation d~y.s(;) c--t ff IS(;)

with

176

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

Obviously,

and since r,(?;) is manifestly positive, it is easy to bound Re J?,(T) away from -1 on the region where: (a) each zL is bounded away from 47rl for all e E Z \{O), (b) Im zL is sufficiently small for each z,. These observations imply that f is analytic everywhere in the region

for some small 6 > 0. The crucial point about (2.10.36) is that analyticity at = ; 0 is manifest. To establish exponential decay of $;ts(;), we need to show that f ;IS(;) is analytic in the Cartesian product region

a ~ has d some mild polynomial decay in Re ; on that region. Pick an arbitrary non-zero m and observe that

so we may rewrite (2.10.36) as

2.10. MULTI-DIMENSIONAL WAVELETS

177

Since

(2 .10.40) (2 .10.41)

the same reasoning as above shows that {e,s(-;) is analytic everywhere in the region

The factor

(z;= (t z,) 2) -s poses no problem, because this region is bounded away from

-; = O. In particular, (2.10.40) and (2 .10.41) establish analyticity at -; = -47r r;;,. Now our choice of non-zero r;;, is arbitrary, so if we take the union of all the regions of analyticity generated, we finally have the desired Cartesian product of strips centered on the real axis. Why does {e,s (-;) have polynomial decay in Re -; for sufficiently small 1m-;? To answer this, consider the analytic continuation of the formula (2 .10.28):

(2 .10.42)

Every factor in this expression is periodic in the real directions except the last tw...2 factors. Moreover, the periodic contribution to this product is bounded for small Imz because

is bounded away from zero in such a region. As for the last two factors,

clearly decays in Re -; for small 1m -;, while

e-i!z, 1

I

2 Z,

11 ~ eiRe z,l- , 1

(2 .10.43)

178

CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION

This decay serves two purposes. On

o~and,

it allows the contour-shifting argument

Ch

for translating the real analyticity of 'IjJ -;,8 (p) into the exponential decay of 'IjJ -;,8 On the other hand , it establishes that the smoothness of 'IjJ -;,8 (-;) is class C N + 2 s- 1 , so the wavelet we have constructed shares the properties of a Lemarie wavelet. We return to (2.10.26) to consider the wavelets III c,. which generate an orthonor~ basis instead of just an interscale-orthogonal basis. The analytic continuation 'IjJ -;,s (p)

I----t

F -;,s (-;) is given by

(2.10.44)

n*

which follows from inspecting the analytic continuation of 'ljJc,s(p +2lT The proof that III -;,8 (-;) shares the regularity and decay properties of 'IjJ -;,s (-;) now follows from arguments very similar to those given above.

References 1. G. Battle, "A Block Spin Construction of Ondelettes, Part I: Lemarie Functions," Commun. Math. Phys. 110 (1987), 601-615. 2. G. Battle, "A Block Spin Construction of Ondelettes, Part II: The QFT Connection," Commun. Math. Phys. 114 (1988), 93-102. 3. J. Kahane and P. Lemarie, Fourier Series and Wavelets, Gordon and Breach, Amsterdam, 1995.

Chapter 3

Equilibrium States of Classical Crystals One of the basic industries of mathematical physics is to exploit the mathematical similarities between phenomena that are physically unrelated. The cross-fertilization between Euclidean field theory and classical equilibrium statistical mechanics is a case in point . Even before the evolution of quantum field theory into Euclidean field theory, there were parallel developments in these two different areas of physics. After all, one of the fundamental problems in both subjects is to control an infinite-volume limit, and the issue of stability is important to each area, both technically and conceptually. Another similarity lies in the basic quantities to be analyzed. Wightman distributions were not unlike the correlation functions of classical spin systems in thermal equilibrium, although the space-time cluster properties of the former were necessarily different from the spatial cluster properties of the latter. Only when the Wightman distributions were analytically continued to Schwinger functions did the similarities become striking. Classical equilibrium statistical mechanics, while far less fundamental than quantum field theory, is also far more ambitious in scope. The aim is to derive the bulk properties of matter from the microscopic interactions of its constituent particles. For a magnetic crystal, the constituent particles are atoms fixed in space, with internal degrees of freedom, and the bulk properties involve specific heat, magnetic susceptibility, regime of spontaneous magnetization, etc. For a gas, the constituent particles are molecules moving in space independently, with potentials defining the interaction, and the bulk properties involve specific heat, compressibility, condensation point, etc. Such diversity means that "the task of rigorous formulation cannot be unified, so a great variety of essentially different mathematical models is called for. The earliest work approaching mathematical rigor was on various expansion methods in the analysis of thermodynamic functions for dilute gases. The grand canonical ensemble is a power series in the activity, and Mayer wrote the density and pressure in powers of the activity with rules for computing the coefficients systematically [M22] . The viTial expansion is the expansion of the pressure in powers of the density, and it is older than the Mayer series. It was first introduced by Ursell [U4], and Mayer him179

180

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

self later simplified it by using graph techniques [M21] . Indeed, it was Mayer's heavy reliance on graphs for both virial and activity expansions that later inspired Feynman to use graphs so extensively in quantum field theory [F36-F39]. Convergence of these expansions (uniform in volume) for small values of the expansion parameters was not immediately established at the time - in part, because it certainly depended on the intermolecular forces assumed for the gas. As an alternate approach, systems of integral equations were formulated to characterize the correlation functions . The Mayer-Montroll system was the first of this type to be introduced [M23], and it is useful for microscopic interactions that are essentially repulsive. The Kirkwood- Salzburg equations were introduced later, but have a greater range of application with regard to the potentials considered [KI3]. Like the Mayer-Montroll equations, they comprise an infinite, linear, inhomogeneous system, and finite truncations have been accurate in low-activity calculations. It was Ruelle who employed the Kirkwood- Salzburg equations, with rigorous estimation, to prove volume-uniform convergence of the Mayer series for small activity and a large class of pair potentials characterized only by regularity and decay conditions [R17- R20]. Indeed, Ruelle and Fisher independently gave the first complete proofs that the infinite-volume limit of the pressure exists for arbitrary activity [F46] . For a special class of interactions, a proof had been given by Lee and Yang years before [LI6], but there had been limited interest in mathematically rigorous statistical mechanics until it was vigorously promoted by Ruelle and others. Subsequently, Lebowitz and Penrose proved uniform convergence (in volume) of the virial expansion for low density [LI3] and this marked the beginning of a renewed interest in investigating a variety of models. Our focus is on classical crystals - especially magnetic crystals, which have a distinctive history of their own. Since the crystalline state of matter is far from understood, the magnetic interactions between atoms at different lattice sites are not derived from first principles. The modeling is empirical in the sense that one investigates various interactions in a search for the model from which one derives the correct bulk properties of the magnetic material, but the derivation itself is mathematical. For a ferromagnet, one may believe the magnetic interactions to be the fundamental long-range coupling for magnetic dipoles fixed at the lattice sites by an unspecified force, but such an assumption only provides yet another input model from the empirical point of view. The earliest model proposed for ferromagnets was the Weiss theory [W2], which is essentially a mean field approximation. The calculations are simple and explicit, and some of the predictions are in qualitative agreement with experiment, but it was immediately clear that a more complicated model was needed. The Ising model - originally proposed and developed by Ising [19] - proved to be such a model, although its basic description is deceptively simple. Peierls gave an ingenious argument (which was not quite a proof) that the two-dimensional Ising model undergoes a phase transition as the temperature is lowered [PI3] . A few years later, Onsager [05] solved the same model explicitly! His solution is difficult, but it yields a wealth of information about the model, including the occurrence of a phase transition at a critical temperature and zero external magnetic field . For the Ising model in arbitrary dimension, it was suspected that a phase transition cannot occur unless the external magnetic field is zero. Lee and Yang extended this wisdom to all ferromagnetic pair interactions with

integrable range, provided the space of spin configurations was still the space of Ising configurations [L17]. Meanwhile, a rigorous study of the thermodynamic limit had been initiated by van Hove [H44]. Both the Onsager solution and the Lee-Yang Theorem were mathematically rigorous as well - a rare virtue at the time. Not until the time of Ruelle7s basic work did rigorous theorems on the classical spin models of crystals began to appear rapidly. Griffiths closed the loop-holes in the Peierls argument [G83] at about that time. In spite of the Onsager solution, this was important because the argument could be adapted to higher dimensions. A couple of years later, Griffiths also proved his celebrated Eorrelation inequalities assuming arbitrary ferromagnetic interactions for Ising configurations [G84-G86]. At the same time, Gallavotti and Miracle-Sole established existence and convexity of the pressure in the thermodynamic limit for a very general class of interactions and spin configurations [G8, G9]. There was also a great deal of interest in the mathematical structures associated with classical spin systems. As long as the space of a single-spin distribution is compact, the entire space of spin configurations for the infinite lattice of spin sites is also compact. The C*-algebra of continuous functions on this space is a complete set of classical observables for the system, while the states of the system are the normalized positive linear functionals on the C*-algebra. By the Riesz-Markov Theorem, this means that a state can also be given by a probability measure on the compact space of configurations. To determine whether an equilibrium state is unique for a given interaction and temperature, one must be able to characterize equilibrium states in the first place. The starting point is to characterize it as the Gibbs state for a finite system (so it is unique in that case), and it is reasonable to define an infinite-volume equilibrium state as the limit of the sequence of Gibbs states associated with some sequence of sets of lattice sites approaching the whole lattice in the sense of set inclusion or perhaps in a stronger sense. However, the basic theory of pure and mixed phases in a regime of multiple phases dictates that the set of equilibrium states be convex, and it is not clear how to obtain the geometry of a set of limit points, except in very simple cases. Dobrushin, Lanford, and Ruelle characterized equilibrium states with equations that link the finite-volume Gibbs states to conditionings of the infinite-volume state, and the set of solutions is automatically convex. Ruelle himself gave another characterization of equilibrium as a solution of a variational problem involving pressure and entropy [R23]. Equilibrium states with lattice-translational symmetry have an elegant formulation in terms of this functional analysis. This convex subset is a simplex whose extreme points are ergodic states, which are the pure phases into which a lattice-translation-invariant mixed phase is uniquely decomposed. Lanford and Ruelle applied the Choquet theory of abstract integral decompositions on convex sets to establish this theoretical result [L2]. This picture also provides a framework for the phenomenon of internal symmetrybreaking. If the single-spin distribution has a group symmetry of its own and if the interaction is invariant with respect to this group as well as the lattice-translation group, then a mixed phase may have both symmetries, but its integral decomposition includes ergodic states which are not invariant with respect to this internal group. The occurrence of spontaneous magnetization in the Ising model is the simplest example of this. The Griffiths inequalities inspired the derivation of other correlation inequalities

182

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

for spin systems in general and the Ising model in particular. Lebowitz proved strong correlation inequalities of Griffiths type for the Ising model, using properties more peculiar to that model [L6). These more powerful inequalities provided bounds on arbitrary correlations in terms of bounds on two-point correlations. On the other hand , Ginibre generalized the Griffiths inequalities considerably - especially to scalar spins where the single-spin distribution is even [G39). For another large class of scalar spin systems, Fortuin, Kastelyn, and Ginibre showed that any two random variables that are monotone increasing in the spin variables are also positively correlated by the Gibbs state [F52). At the same time the rigorous results in equilibrium statistical mechanics began to appear so rapidly, Symanzik was promoting Euclidean field theory [S94-S99). He was also the first to emphasize the formal resemblance between the thermodynamic limit of the expectations associated with the statistical mechanics of a classical field and the infinite-volume limit in space-time of vacuum expectation values associated with the perturbation of the free Euclidean field by an interaction. His idea was to use the methods of classical statistical mechanics to obtain control over the Euclidean field theory, and specifically, he used equations of Kirkwood-Salzburg type [S95) . Unfortunately, no one at that time knew how to analytically continue a Euclidean field theory to a relativistic field theory in the time differences. When Nelson isolated and exploited the Markov property of the free Euclidean field [N4-N7J, Symanzik's program attracted much more serious attention. Guerra, Rosen, and Simon obtained some control over interacting Euclidean scalar fields through correlation inequalities such as the Griffiths inequalities [G103, G104). They proved the Griffiths inequalities in the continuum limit of a lattice approximation in which the free part of the Euclidean field action is actually the Ising interaction and the interacting part defines an even single-spin distribution for a scalar spin system. They also proved the FKG inequalities with this lattice approximation, where the field interaction is not necessarily even. At the same time, Glimm, Jaffe, and Spencer developed a cluster expansion for controlling the infinite-volume limit of two-dimensional Euclidean scalar field theories with weak interaction [G76) . Although it was the first expansion of its kind, this expansion was very much in the spirit of statistical mechanics. Actually, it involved no lattice approximation, since the expansion was designed for the infinitevolume limit of the ultraviolet limit, but the systematic decoupling of random variables in distant regions was still the idea. Glimm, Jaffe, and Spencer were also able to combine a phase boundary expansion with the Peierls argument to show that the quartic self-interacting scalar field has two pure phases for strong coupling in the continuum i.e., with no lattice approximation [G78-G80). The connection between the two subjects attracted even more interest when ideas in quantum field theory found application to problems in statistical mechanics as well. An elegant example of this was the question of phase transitions for the classical Heisenberg ferromagnet, which was more difficult than the question had been for the Ising model. In two dimensions it was believed to have a unique phase for all temperatures because Mermin had proven that it had no spontaneous magnetization in that case [M35) . (The classical Heisenberg model does not satisfy an abundance of correlation inequalities, so one could not infer a unique phase from Mermin's result.) In higher dimensions it was believed to have a continuum of phases at low temperature - one for each orientation

183 of a single spin - but the Peierls argument could not be adapted to such a model. The input from Euclidean field theory was Osterwalder-Schrader positivity, or rather the lattice version of this reflection positivity property. Although the classical Heisenberg ferromagnet does not involve scalar spins, it shares the reflection-positivity property of the Ising ferromagnet . Frohlich, Simon, and Spencer actually derived from this property an infrared bound on the Fourier series whose coefficients are the two-point spin correlations [F77] . This implies existence of a phase transition at sufficiently low temperature in dimension greater than two. A short time later, Bricmont, Fontaine, and Landau proved that in two dimensions the equilibrium state is indeed unique at all temperatures [B54, B55] . An outstanding example of the impact of Euclidean field theory on classical statistical mechanics is the rigorous proof of Debeye screening. It had been shown by Edwards and Lenard that the grand canonical partition function for the lattice approximation of the classical Coulomb gas can be written as the partition function associated with the lattice Euclidean sine-Gordon field [E5] . This identification is called the sine-Gordon transformation. Brydges and Federbush combined this transformation with the phase boundary expansion methods of Glimm, Jaffe, and Spencer to rigorously prove that the high-temperature, dilute, Coulomb gas effectively screens the interaction of test charges [B65, B70] . This phenomenon was an experimental fact for which the only previous explanation had been the self-consistent field intuition of Debeye and Hiickel [D9]. The long-distance analysis of Brydges and Federbush is very difficult, but the intuition of the sine-Gordon representation is that the quadratic contribution to the cosine interaction is an effective mass term. This chapter is devoted to methods of analysis for classical spin models of crystals, with particular emphasis on the analysis of reflection-positive interactions on one hand , and high-temperature expansions for general lattice systems on the other. We confine our attention to three dimensions, not because we have to, but because in this chapter we are not concerned with the dimensional parameter, and dimension d = 3 is certainly the realistic value. In §3.1 we describe a ferromagnetic model where each spin is fixed in both magnitude and position on a cubic lattice, and the interaction is the fundamental one among magnetic dipoles - a pair interaction whose long-distance decay marginally fails to be integrable. We use the most naive high-temperature expansion (powers of the inverse temperature) to illustrate how a long-distance problem develops when one tries to control correlations in the infinite-volume limit. This is an example of an infrared problem, whose solution lies beyond our discussion. Then we shift our focus to lattice models with pair interactions that are integrable over distance and review the basic mathematical structure of equilibrium statistical mechanics in this context. In §3.2 we continue this description from a functional-analytic point of view, where states are realized as positive linear functionals on commutative C'-algebras. The Choquet simplex theory applied to the convex space of states provides an excellent framework for the ergodic decomposition of a lattice-translation-invariant equilibrium state into pure phases. In §3.3 we introduce the property of reflection positivity for lattices and show how it implies the infrared bound of Frohlich, Simon, and Spencer. Actually, there are two slightly different types of reflection positivity for lattices: one involves reflection through a plane of lattice sites, while the other involves reflection through a plane lying between two consecutive planes of lattice sites. The latter property is

184

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

the more important one , but one needs both properties to establish monotonicity in the long-distance approach of the two-point correlation to a (possibly non-zero) limit. This rules out pathologically singular behavior in the infrared estimation. In §3.4 we shift our attention to the Ising model and review its low-temperature phase structure and its critical properties. In §3.5 we introduce the Ginzburg-Landau model and review its similarity to the Ising model. In §3.6 we review a standard version of the abstract theory of polymer expansions in the interests of self-containment. The work of Glimm, Jaffe, and Spencer triggered a whole industry of rigorous cluster expansions for a host of models, each one different in some non-trivial way from every other. It was gradually recognized, however, that all of these expansions - including the MayerMontroll expansion and its contemporaries in early statistical mechanics - have basic combinatorial features in common. A unified framework for automating control over a given expansion by the convenient insertion of estimates peculiar to that expansion was developed and promoted through the efforts of Glimm, Jaffe, Malyshev, Seiler, and others (see [G7, G73, G99, M12 , S35] and references given therein). In §3.7 we review the most basic example of a high temperature expansion in the polymer formalism . The classical Heisenberg model and the Ginzburg-Landau model are the cases considered there. In §3.8 we introduce inductive interpolation as another expansion method cast in the polymer formalism. In §3.9 we consider long-range pair interactions (defined by a decay condition) for fixed-magnitude spins because it calls for a generalization of the polymer estimation given in §3.6. We examine the wholesale expansion of the exponential in all couplings and prove convergence at high temperature. Then in §3.10 we show how an expansion based on inductive interpolation yields a better polymer expansion in the sense that the decay condition on the interaction is weakened as a sufficient condition for high-temperature convergence. We study the combinatorics associated with inductive interpolation as well. For example, an Nth-degree ordered tree graph is a mapping TJ: {2, . .. , N} -t {I, ... ,N I} such that TJ (i) < i . For interpolation parameters h , . .. , t N there is a monomial associated with TJ given by (3.1) The oldest combinatorial inequality involving such monomials is an exponential bound on the sum over ordered tree graphs of the integrals:

L

II N

Nth-degree 71

(

i=1

1

10

dti

)

f 7l (t):S eN-I.

(3.2)

A standard application is to prove convergence of the Mayer series for a sufficiently dilute gas. It was later discovered by Robinson that this inequality can actually be replaced by an identity [B21] which shows how sharp this inequality is. We have

L

Nth-degree 71

NN-l N rl) (II In dti f7l(t) = N t· i= 1

0

.

(3.3)

3.1. CLASSICAL SPIN SYSTEMS This combinatorial estimation has also been replaced by

where d q ( j ) denotes the cardinality of 7 7 - 1 { j } . This refinement has no effect on the convergence of some expansions, such as the Mayer series, but it makes all the difference in the world for other expansions, such as the phase cell cluster expansion in Chap. 5. Shortly after (3.4) was established, it was shown by Speer that this inequality can be replaced by an identity as well [S66]. We have

Again, the identity adds nothing to the application of the corresponding estimate, but it lends a great deal of elegance to the combinatorics. This combinatorial interlude is the content of 53.11. In 53.12 we consider unbounded scalar spins with long-range pair interactions and a large-spin damping condition on the single-spin distribution. We apply exactly the same wholesale expansion used in 53.9 and show how the large-spin estimation burns up some of the long-distance decay of the interaction. In 53.13 we apply the inductive interpolation and show how the combinatoric estimate (3.4) controls the large-spin estimation without burning up any of the long-distance decay. This means convergence of the latter expansion requires no more long-distance decay than it required in the bounded-spin case.

3.1

Classical Spin Systems

From the standpoint of modeling the critical behavior of macroscopic systems in terms of the equilibrium statistical mechanics of their microscopic structure, perhaps no examples have been more heavily studied than the various classical models for the ferromagnet. The atoms are arranged in a cubic lattice of fixed positions, so the only dynamical variables are internal. Each iron atom is elegrically neutral, but has a magnetic moment proportional to its angular momentum s , whose magnitude is assumed to be unity, for example. Such a model is based on the assumption that each atom is a constantly spinning gyros2ope whose inertial axis is affected by the mqnetic field. Accordingly, every element n of a finite set A c Z3has the spin variable s ;t assigned to it, and one possible assumption for this crystal is that the Hamiltonian has the form

186

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

where h is a uniform external magnetic field and J is a constant. This expression for the interaction between atoms at different lattice sites is based on the elementary field of a magnetic dipole. We omit the kinetic terms because they will cancel out in the normalization of a classical equilibrium state, as the momentum variables are independent. For inverse temperature fJ, the equilibrium state of this system is the probability measure on the space of spin configurations defined as the normalization of the measure. e- f3HAC;)

II do-(~;:;),

(3.1.2)

;:; where do- denotes normalized Lebesgue measure on the sphere §2 of possible spin values. In general, do- will denote the single-spin distribution for the systems we study. The exponential factor is the Gibbs factor, and it is the solution of the elementary variational problem that is involved in deriving one probability measure from the constrained statistics of another. In this case, the original measure is just the product of singlespin distributions, the constraint is that the statistical average of the energy is fixed, and the inverse temperature fJ is the Lagrange multiplier. The partition function for a finite set A is the mass of the unnormalized measure (3.1.2) given by the Gibbs factor - Le., (3.1.3)

The probability measure dJ.L~' h for the Gibbs state is just

dJ.L~,h (~) = ZA(fJ, h)-le- f3HA (s) II do-(~ ;:;),

(3.1.4)

;:; M

and so the expectation of an arbitrary product

IT sj=l ni'i

of spin components is given

by

(3.1.5) A standard problem is to control the A = Z3 limit of such expectations for high temperature (small fJ) · From the limiting expectations one can recover a probability measure by applying some functional analysis - the Riesz- Markov Theorem in this case, where the space of spin configurations over Z3 is compact, or Minlos ' Theorem in the case where the space of spin configurations is linear instead of compact. Such a measure is an infinite-volume equilibrium state . To control the A = Z3 limit for small fJ - Le., the thermodynamic limit - one might try to expand a given expectation in powers of fJ. This is the most naive example of

3.1. CLASSICAL SPIN SYSTEMS

187

a high-temperature expansion, but in the case of this long-range interaction, it is only asymptotic - not convergent. Let (')0 denote the expectation functional of the product

n

measure (on the compact space

§2)

of the single-spin distribution, and set h= 0

~EZ3

for simplicity. To first order in the inverse temperature we have the expansion

/IT S~j")

\J=l

=/

IT S~j'j)

\J=l

A

- (3 0

/ (J~_~, --; ~,). --; ~ IT S~j'j)

L

~,~'E{~l""'~M} \ where

J~

j=l

+ 0((32),

(3.1.6)

0

denotes the 3 x 3 matrix defined by

~=O, (3.1.7)

Of the spin variables that are initially differentiated down from the exponent, only those --; ~ for which ~ occurs in {~1"" , ~ M} can appear, because the Oo-expectation is zero if any spin component at any lattice site has an odd power. This also implies (3.1.8) so the (3-derivative of the partition function in the denominator has contributed nothing here . It is significant that in (3.1.6) the first-order contribution does not depend on A at all . The second-order contribution is a little more involved:

d~2 / IT S~j'j) \J=l

1_ = / HX A

~-o

\

IT S~j'j)

J=l

0

- (HX)o / IT S~j'j) \ J=l

0

L /((J~_~" --;~,,) --;~)((J~,_~" --;~,,). --;~,) IT --;~j'j)

L

~,~'E{nl, ... ,nM} ~"EA \ +

J=l

0

/((J~n-rn~ --;~). --;~)((J~ ~ --;~). --;~) rrM --;~nitj ) n n'-m' n' n'

'" ~

\

m

r;,;' ,i1i.,ni.'E{n1! ... ,nM} ni.' :;en;,

j=l

0

(((J~ ~ --;~). --;~)((J~, -m~, --;~,). --;~,))o /rrM S~ m m

'~ "

n'-m

~,n'""m,ni'E{nb" " nM}

n

n

n

\

. .)

nJl- J

j=l

0

~¥~'

L

L

(((J~_r;" --;r;,,)' --;r;)((Jr;,-r;" --;r;,,)' --;r;,))o

n:,Ti"E{nI, ... ,nM} ;IIEA

(3.1.9)

188

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

because, for example,

Now the second and third double sums are clearly independent of A, while the first and fourth are not. On the other hand, the sum over A converges absolutely as A -+ ;[,3 because c

IJ~_~"I

and therefore

~ L,.;

_

(3.1.11)

::; -I~"""--~:--1-'1-3' n-n

IJ-n-n" - IIJ-n'-n" - 1< c -

In(2 + 1 r;; 1~ /I

,_ r;; I) 13 .

n - ~ n

(3.1.12)

n"

Thus, the thermodynamic limit is controlled to second order in /3. Similar estimation controls this limit to all orders in /3, but this control is only perturbative. Does the power series in /3 have a nonzero radius of convergence independent of A? Actually, the number of products of expectations of powers of H" contributing to the Nth-order /3-derivative at /3 = 0 is bounded by cN N!, so the Taylor coefficient has a bound of the form cN , provided each such product of expectations has such a bound. On the other hand, an example of the multiple sums to be estimated is (3.1.13)

Although we have a bound of the form (In(2 + 1 r;;

for every positive exponent

Ct,

'_ r;;

I))N ::; cN,,,, (1

+ 1 r;; '_ r;;

I)'"

(3.1.14)

we also have the dependence CN ,,,,

= O(N!)

(3.1.15)

for any choice of Ct . This is where convergence of the power series fails, and this type of divergence is known as an infrared divergence . The point is that this problem would not arise if J~_~, had just a little more long-distance decay - i.e. , if J~ were summable. The infinite-volume expectations can be controlled nonperturbatively for high temperature by the application of renormalization group methods, but such technology did not exist at the time that rigorous work on ferromagnetic spin systems was initiated. Moreover, magnetic crystals represent a very complicated state of matter from the most fundamental point of view, so it is not at all clear that the most basic interaction between magnetic dipoles is the correct one here. For these reasons, the traditional effort has been a search for shorter-range spin interactions that yield the correct bulk

3.1. CLASSICAL SPIN SYSTEMS

189

properties of ferromagnets. From this point on, we make the convenient assumption that J;. _;., is a scalar and that (3.1.16)

The Hamiltonian now has the form

-;-+h."sn' L n

(3.1.17)

;'EA

for an external magnetic field h . Boundary conditions may be modifications of the coupling coefficient J;. _;., or constraints on the possible spin values at the boundary. As far as high-temperature expansions are concerned, it will be proven later that the power series in (3 does indeed have a nonzero radius of convergence independent of A and its boundary conditions for this class of spin models. The study of the spin product expectations is closely related to the analysis of thermodynamic functions, which are generated by the free energy, defined by 1

~

fA((3 , h) =

~

!AI In ZA((3, h) .

(3.1.18)

First, it is a trivial consequence of (3.1.16) that

ZA((3, h) :S ei3 (C+ 1hI)IAI,

(3.1.19)

so the logarithm has a bound linear in 111.1 . Such a bound is a stability bound, and it plays a central role in the context of Euclidean field theory, as we have already seen in Chap. 1. Its proof was difficult in the field context, but is a triviality in this context because the spin variables are bounded. It obviously provides a A-independent bound on the free energy. In addition, the infinite-volume free energy

f((3, h)

= A-+Z3 lim fA ((3, h)

(3.1.20)

is actually unique for arbitrary values of (3 if the limit is taken in the sense of van Hove. For every positive integer r , define N;!(A) (resp. Nr-(A)) as the number of ~ for which {I, 2, ... ,r P + r ~ intersects with (resp. is contained in) the set A. A sequence {Aj} converging to Z3 in the sense of set inclusion is said to converge in the sense of van Hove if and only if (3.1.21) for every r. The uniqueness of the van Hove limit of fA ((3, h) is a general type of result. The elementary inequality

ZAUN((3, h)

:S ZA((3, h)ZN((3, h)exp (2(3

~ ;= IJ;_;,I) , nEA n'EN

A n A' = 0,

(3.1.22)

190

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

lies at the heart of the proof, which we do not pursue. However, this convergence was one of the early important theorems proven in the original program to develop mathematically rigorous statistical mechanics. Notice also that the free energy has the convexity property on the space of interactions - i.e., with 1

~

/(/3, h, {J.,;}) = lim(van

Hove)

A-+Z3

-IAl ln Z(/3, HA),

(3.1.23)

we have the inequality /(/3, t h +(1 - t) h ',{ tJ.,; + (1 - t)J~)) ~

:::;

t/(/3, h; {J.,;})

~

+ (1- t)/(/3, h

'; {J~}),

0:::; t:::; 1,

(3.1.24)

which follows from the Holder inequality

0:::; t :::;

1.

(3.1.25)

The thermodynamic functions are de~ned to be the partial derivatives of the free energy with respect to the components of h. The A = :£3 limit of either the free energy or anyone of these partial derivatives is the thermodynamic limit of the given function. For example, consider the first-order partial derivatives of the free energy:

(3.1.26) The thermodynamic limit of such a function is an infinite-volume average of single-spin expectations and is therefore a bulk property of the lattice spin system. Its r~gularity properties are important to the study of critical behavior with respect to h at low temperature (large (3). Closely related examples of thermodynamic functions are the second-order partial derivatives:

(3.1.27) The existence of the thermodynamic limit of such a derivative obviously depends on the behavior of the truncated expectation

(3.1.28) If the long-distance decay is summable uniformly in 01'., then this second partial derivative exists in the A = :£3 limit. How does one characterize an infinite-volume equilibrium state? In particular, the limiting state for any convergent sequence of Gibbs states based on a sequence {Aj} of finite subsets converging to :£3 in the sense of van Hove should qualify as an equilibrium state, and the existence of at least one such sequence is guaranteed by a compactness argument. Moreover, there should be only one equilibrium state in the

3.1. CLASSICAL SPIN SYSTEMS

191

case where these sequences of Gibbs states all converge to the same state. This case occurs for high temperature (small P ) , where one may regard the Gibbs factor as a small multiplicative perturbation of the product of single-spin distributions. In the case where uniqueness fails, infinite-volume characterizations are very i m ~ o r t a n t . ~ S2 of the form p( s ) do( s ;), Consider an arbitrary probability measure on

n

2

n

n€Z3

where the density

has the property that p~ l n p ~is integrable with respect to

n da(;;).

The entropy

2

n €A

of the state p on A is defined by

while the internal energy density of this state for a given interaction is defined by

with HA given by (3.1.17). It is yet another basic result that the infinite-volume limits S(p) = lim(van Hove) SA(p),

(3.1.32)

A+23 2

E,(h,{J;})

3

1

= lim(van~ove)~~(h,{~;})

A+23

(3.1.33)

exist. Moreover, the entropy S(p) is actually an affine function on the convex set of states - i.e., S(tp

+ (1 - t)pf) = tS(p) + (1 - t)S(pf),

< t < 1.

0

(3.1.34)

This property is an easy consequence of the concavity and subadditivity of the function E(x) = -xlnx. Indeed,

E(tx + (1 - t)xf) 2 t[(x)

because Eff(x)< 0, while

+ (1 - t)E(xi),

(3.1.35) 0

< t < 1,

(3.1.36)

E(x + xf) 5 E(x) + <(XI)

follows from First, it is clear that

+

+

SA(tp (1 - t)pf) 2 t S ~ ( p ) (1 - ~)sA(P'),

O5

'7

(3.1.39)

192

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

is a result of (3.1.36). On the other hand, (3.1.37) implies WPAC-;)

:S

+ (1 -

t)p;"(s'))

WPA(s'))+W 1 - t )p;"(s')) - tPA (s') In( tPA (s')) - (1 - t)p;" (s') In( (1 - t)p;" (s')) - tpA(s')lnpA(s') - (1- t)p;..(s')lnp;"(s') - tpA(s')lnt - (1 - t)p;" (s') In(l - t),

(3 .1.40)

so we have

J

WPA(s')

+ (1- t)p;"(s')) ]I

da(s'r;):S t

J~(pA(s')) ]I

nEA

+ (1- t)

J~(P;..(s')) ]I

da(s'r;)

nEA

da(s'r;)

nEA - tlnt-(l-t)ln(l-t),

(3.1.41)

from which it follows that SA(tp + (1 - t)p')

:S tSA(p) + (1 - t)SA(p')

-mt t-m(1 1

In

1

t) In(l - t).

(3.1.42)

= Z 3 limit. What is the motivation for introducing these additional thermodynamic quantities? First, it is obvious that

If we combine this with (3.1.39), it is clear that Eq. (3.1.34) holds in the A

(3.1.43) in the special case PA

= Z((3, HA)-le-/3HA.

(3.1.44)

In the A = Z3 limit this means that S(p) - (3Ep(h, {Jr;})

= (31((3 , h, {Jr;})

(3.1.45)

for any state P realized as the limit of a van Hove sequence of Gibbs states. Moreover: (a) S(p) is upper semi-continuous with respect to the weak topology on the convex set of states. (b) For an arbitrary density p, S(p) - (3Ep(h, {Jr;}):S (31((3, h, {Jr;}).

(3.1.46)

The Gibbs variational principle characterizes the equilibrium states as those densities p for which (3.1.45) holds. For our class of spin systems one can show that this equation

3.1. CLASSICAL SPIN SYSTEMS

193

is equivalent to the Dobrushin-Lanford-Ruelle Equations, which are probably the most popular characterization of equilibrium states. We shall not pursue this subject, as it would take us too far afield. is ·ferromagnetic if and only if J~n < An interaction {J~} n - 0 for all r; , and this class of interactions - aside from qualitatively describing magnetic interactions - has advantages that make the models more amenable to analysis. One such advantage at least for our 3-component spin variables ~ ~n - is a Lee- Yang Theorem . Originally proven in the case of the Ising model - which we discuss in §3.4 - such a theorem states that if the components of an external field are replaced by complex variables, then the analytic partition function still has no zeros, except possibly where the real parts are zero. The Lee-Yang Theorem has been established for a small numb:: of ferromagnetic models, including our class of spin models with the components of h ~ the variables for analytic continuation. has a natural

e~ension

Th~signi~ance of this

result is that In Z 11.((3, h)

to a function of h +i k that is analytic everywhere except

possibly where h= O. ]his ~xtension should not be confused with the principal log, as the property Z 11.((3, h +i k) :f. 0 about the origin..:...,. The free energy

doe~not ~clude

fA ((3,

the "winding" of complex values

h +i k) is simply the unique analytic function

for which fA((3, h) is given by (3 .1.18) . Without describing the details, we simply ~ate ~at for any van Hove sequence {Aj} of finite subsets of Z3, the sequence {fA; ((3, ~ +i k)} of analytic functions converg~ uniformly on compact subsets disjoint from {h=

OJ.

Thus, the limiting function f((3, h

~

+i k) has

t~

same domain of analyticity and is the analytic continuation of the free

energy f((3, h) in the infinite volume. Therefore, all of the thermodynamic functions have this property as well. It is generally believed that the thermodynamic ~nctions of a given spin system in the class under study are piecewise real-analytic in h whether the model is ferromagnetic or not, but such a general result has never been established. Anoth~ conjecture is that infinite-volume equilibrium states are unique at every point of real h-analyticity ~

in ((3, h)-space. After all, the partial derivatives have the form (3.1.47)

where each A ~ depends on only a finite number of spin variables ~:;;, . However, it is not clear t£.;,'t the limiting state is uniquely determined by the existence of these ko ). derivatives , even assuming translation invariance, which yields the reduction 2::(A k

In general, one can imagine different states yielding the same sums. On~the other hand both uniqueness of the state and analyticity of the free energy for h:f. O· have been 'shown to follow from convergence and exponential decay of truncated correlations for various ferromagnetic spin systems.

194

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

At any point ({3, h) where real analyticity of the free energy fails, the model is said to have a phase transition . The occurrence of a phase transition can reflect the existence of more than one infinite-volume equilibrium state. In §3.3 we consider a special class of models where this is actually the case.

References 1. R. Dobrushin, "The Problem of Uniqueness of a Gibbsian Random Field and the Problem of Phase Transitions," Funet. Anal. Appl. 2 (1968),302-312 . 2. F . Dunlop and C. Newman, "Multicomponent Field Theories and Classical Rotators," Commun. Math. Phys. 44 (1975), 223-235. 3. G. Ford and G. Uhlenbeck, Lectures in Statistical Mechanics, Amer. Math. Soc., Providence, 1963. 4. G. Gallavotti and S. Miracle-Sole, "Statistical Mechanics of Lattice Systems," Commun. Math. Phys. 5 (1967),317- 323. 5. L. van Hove, "Quelque Proprietes Generales des l'Integrale de Configruation d'un Systeme de Particules avec Interaction," Physica 15 (1949),951-961. 6. K. Huang, Statistical Mechanics, Wiley, New York, 1963. 7. L. Landau and E. Lifshitz , Statistical Physics, Addison-Wesley, Reading, 1969. 8. O. Lanford and D. Ruelle, "Observables at Infinity and States with Short-Range Correlations in Statistical Mechanics," Commun. Math. Phys. 13 (1969), 194215. 9. T. Lee and C. Yang, "Statistical Theory of Equations of State and Phase Transitions I: Theory of Condensation," Phys. Rev. 87 (1952), 404-409. 10. T . Lee and C. Yang, "Statistical Theory of Equations of State and Phase Transitions II: Lattice Gas and Ising Model," Phys. Rev . 87 (1952),410-419. 11. D. Ruelle, "A Variational Formulation of Equilibrium Statistical Mechanics and the Gibbs Phase Rule," Commun. Math. Phys. 5 (1967), 324- 329. 12. D. Ruelle, Statistical Mechanics, Benjamin, New York, 1969.

3.2

Phase Transitions and Ergodic Decomposition

The occurrence of a phase transition is often accompanied by a breakdown in uniqueness of the infinite-volume equilibrium state. In this event, one is interested in classifying these equilibrium states or phases. Sometimes this multiplicity of phases is accompanied by the breakdown of some group symmetry respected by the unique phase one had in the case of high temperature (small (3) . Such a phenomenon is referred

3.2. PHASE TRANSITIONS AND ERGODIC DECOMPOSITION

195

to as symmetry-breaking, and certainly the most obvious candidate would involve the group of discrete translations on the lattice. After all, the class of spin interactions under study are translation-invariant . However, this kind of symmetry-breaking is beyond the scope of our concerns. We shall consider Z3-translation-invariant equilibrium states only . We are interested in the possibility of symmetry-breaking with respect to the internal symmetry group 0(3) acting on the spin variables -; ~ simultaneously. Clearly, the n

~

dot product in the interaction is invariant with respect to this symmetry. For hi- 0 the internal symmetry is obviously broken , but it goes witho~t saying that spontaneous symmetry breaking is the interesting issue, so the case h = 0 is the focus here. Indeed, for ferromagnetic spin systems, such symmetry-breaking is easily described by spontaneous magnetization . For a given unit vector -:;;, consider the translationinvar~nt phase associated with the inverse temperature 13 and the external magnetic

field h = h-:;;, h > o. The uniqueness of this phase in the ferromagnetic case is assumed here, and the spontaneous magnetization M (13) ~ 0 is given by M(f3) = lim(-; 0)(13, h -:;;) . -:;; .

(3.2.1)

h.j.O

By convexity of the free energy, this limit is monotone decreasing, and the idea is that if the model has a phase transition, then M(f3) > 0 for sufficiently large 13. Notice that the lattice site at which the single-spin expectation is evaluated is irrelevant because the equilibrium state is translation-invariant - i.e., (3.2.2)

for all ~ E Z 3. The physical meaning of the case M (13) > 0 is that at sufficiently low temperature the ferromagnet remains magnetized in the direction of the external magnetic field after it is switched off. This also means that ~e model has a continuum of special equilibrium states parametrized by -:;;E S2 when h= O. Let S ~ be the set of translation-invariant equilibrium states for inverse tempera~

f3,h

ture 13 and external magnetic field h. By the characterization (3.1.45), S f3,h~ is a convex set because Ep is linear in p and S(p) is affine in p. S ~ is obviously compact in the f3 , h

topology of pointwise convergence of linear functionals on the commutative C' -algebra of continuous functions on II §2. An extreme point of a given convex set cannot be ~EZ3

written as a non-trivial convex combination of other elements of the set. The extreme points of S ~ are said to be the pure phases or the ergodic phases, and we first consider f3 , h

the problem of decomposing an arbitrary element of S ~ into these pure phases. Let dJ-t denote the probability measure on

II

f3 , h

§2

associated with a given translation-invariant

~EZ 3

equilibrium state p and consider the Hilbert space

1{

= L2(d/J,). Let

{U(~): ~E Z3}

196

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

be the unitary group induced by the group of dj.L-preserving homeomorphisms defined by translation on the lattice - i.e.,

(3.2.3) Let P denote the orthogonal projection onto the subspace of vectors invariant with respect to this unitary group. Moreover, let A denote the commutative CO-algebra §2. of operators on 11. defined as multiplication by the continuous functions on

n

;;EZ3

Then the C' -algebra B generated by PAP is also commutative, so the positive linear functional B EB, (3.2.4) can be represented by a measure dv on the compact space X of characters (multiplicative 'linear functionals) on B. Thus

(Bl, 1h-i

=

L

BEB.

X(B)dv(X),

n

Since df.L is a probability measure on

§2

and PI

(3.2.5)

= 1, it follows that dv is a proba-

;;EZ3

bility measure on X. In the special case B = P A( P, where A( denotes multiplication by some continuous function (on §2, we have

n

;;EZ 3

p(()

/ (df.L (A( 1, 1)1i (P A(Pl, 1)1i

L

X(PA(P)dv(X) ·

(3.2.6)

On the other hand, we may define the state

(3.2.7) so we have a convex decomposition of the arbitrary translation-invariant phase: p(()

=

L

Px(()dv(X)·

(3.2.8)

X is realized as the index set for the states into which P has been decomposed. How do we know this decomposition essentially includes only the states in S - - i.e., how do we show that Px is a translation-invariant phase for dv-almost all X ~'}(? First, the translation-invariance is obvious. Indeed, if ( is a continuous function on §2 and if we introduce the notation

n

;;EZ3

(3.2.9)

3.2. PHASE TRANSITIONS AND ERGODIC DECOMPOSITION

197

then X(PA(-;;P) X(PU(r;;)-l A( U(r;;)p) X(PA(P)

(3.2 .10)

Px(() ·

Next, we verify that Px is an equilibrium state for dv-almost all X, using the variational principle. Since P satisfies Eq. (3.1.45), we have (3.2 .11) because S(p) is an affine function of p and Ep is linear in p. Thus (3 .2.12) while in any case (3.2.13) for all X by the Gibbs variational inequality. Hence, the integrand vanishes for dvalmost all X, but this vanishing condition is again the characterization adopted for equilibrium states. Notice that this last argument does not use the special nature of dv - i.e., it appli~s to any probability measure that decomposes p into other states. This means that if S is the convex set of all states, then the subset of equilibrium states is not "interior" to S, but rather a "face" of S . As far as the measure dv is concerned, we have yet to show that it decomposes p into pure translation-invariant phases - i.e., extreme points of S - - i.e., ergodic phases. (3,h

Consider an arbitrary convex decomposition p = a.Pl + (1 - a.)po of p into two translation-invariant phases Po and Pl. Obviously, the corresponding probability measures dll O and dillon IT §2 are absolutely continuous with respect to dll, so there ~EZ3

exist positive dll-integrable functions

",0

and

",1

such that (3.2.14) (3.2.15)

where we have used the notation (3.2.9). If A, = A1)" property implies

the translation-invariance (3.2.16)

198

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

so both A, commute with P as well . Thus

J

(T/'d/1-

(A(A,I,I) (PA,PA,Pl,lhl

L L

X(P A, P A'I, P)dv(X) X(PA(P)X(PA'I,P)dv(X)

L

Px(()Px(T/,)dv(X)

(3.2.17)

because X ranges over the characters of B. Now there is a partial ordering of probability measures on convex compact sets known as Bishop- de Leeuw ordering, denoted by -<. Given two such probability measures d)" and dA', d)" -< dA'

¢:}

.I

J

(d)" ::;

(d)"',

(3.2.18)

for convex, continuous (. What we have essentially shown above is that the measure dD defined by (3.2.19) D(£) = v{X E X: Px E £} is not only a maximal probability measure on S

~

in the Bishop-de-Leeuw sense, but

(3,h S ~ . This (3 , h

also the maximum probability measure on implies that S ~ is a Choquet (3,h simplex and that Px is an extreme point - thus an ergodic phase - for dv-almost all XEX . We now examine alternate characterizations of extremal translation-invariant phases. For a fixed arbitrary state pES ~ we have the associated construction (d/1-, 1i., U(~ (3,h

),P,B,X,dv) described above. For arbitrary cp E 1i.,

' " U(n)cp ~ Pcp = j~~ 1 ~L.J

IAJ

(3.2.20)

nEAj

for every nested sequence {Aj} of finite rectangular sets of lattice sites converging to Z3 in the sense of set inclusion. This fact is a special case of the Mean Ergodic Theorem, the proof of which we omit. The implication here is that I

lim J--+OO

-IA . I '" ~ J ~

p(e(~)

j~~ I~jl ~2: (A~A(-;: 1, Ihl

n

nEAj

nEAj

1 '" ~ j~~ IAjl ~L.J (A~U(n)A(I, Ihl nEAj

=

(A~PA(I,I)"Il.

(3.2.21)

3.2. PHASE TRANSITIONS AND ERGODIC DECOMPOSITION

199

If p is a non-trivial convex combination of PO,Pl E S{J h as above, then - using the same notation - we have '

(3.2.22) which implies

P

= (1- a)PAo + aPA l .

(3.2.23)

Since both A, commute with P, this decomposition can be nontrivial only if dim P > l. Thus P is extremal {:} ran P = !Cl. (3.2.24) On the other hand, if ran P

= !C1, then (A~l, l)1-/(A
(3.2.25)

p(Op(()·

In summary, P is extremal {:} )-+00 lim

-IA I ~ p(~C-) J _L n 1

.

=

p(~)p(T}),

(3 .2.26)

nEA;

and this equivalence motivates the synonym "ergodic" for the extreme points of S - . {J ,h

This basic theory of ergodic decomposition provides a strategy for proving the existence of multiple phases - and therefore a phase transition - for those values of the parameters at which this phenomenon is expected. To prove that S - is not just a {J,h

singleton, one needs only to construct a state pES - such that (J, h

(3.2.27)

for some pair of continuous ~,C. Suppose we have constructed such a state p with functions Cand ~ where ~ = C* · In this case, the property may be written as the strict inequality (3.2.28)

because p is translation-invariant. In the h= 0 case, this inequality actually implies that the multiplicity of phases is accompanied by a spontaneous breakdown of SO(3) symmetry. Indeed , consider the probability measure dii on S{J,o that decomposes an SO(3)-invariant p into ergodic phases. For every W E S{J ,o,

(3 .2.29)

200

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

by the Schwarz inequality and the translation-invariance of w. It follows from dominated convergence and ergodicity of w E supp ii that

J = J

(IA: I'

dv(w) }!.'! w

(3.2.30)

dii(w)/w((W;

so the strict inequality (3.2.28) implies the existence of ergodic states w for which w(() i- O. What does this have to do with SO(3) symmetry? Let 9 range over SO(3) rotations and dg denote Haar measure in this group variable. Let us suppose that ( has the additional property (3.2.31) where Rg denotes the product action of 9 on

w(()

II

i- 0 cannot be SO(3)-invariant.

We close this section with a discussion of the spin systems. We see from (3.1.27) that

82

1(/3,

8h 2

~

= (32

h)

,

§2.

Then the states w for which

;EZ3

hi- 0 case for

certain ferromagnetic

"" L (p(s-nL s-0, ) - p(s-01. )2)

(3.2.32)

;EZ3

for e:::.ery translation-invariant equilibrium state p arising from the given values of f3 and h. Now the Lee-Yang Theorem implies existence of this derivative, and in the next section we discuss a special class of spin systems for which the p shall be proven to satisfy the condition

p(s-nt s-Ot. ) -< p(ss- ) mL o£ '

(3.2.33)

This can only mean that p(s-nL s-Ot. ) - p(s-0, )2 is summable, but this condition, in turn, implies the ergodicity condition lim

)--+00

1

-/A.. / ]

"" p(sn, s0, )

L-nEAj

= p(s-0, )2

(3.2.34)

The point is that if one could extend this last condition to an arbitrary pair of observables, this would show that every state pES - is ergodic, in which case S {3, h

{3,h

could have only one element for h i- O. The desired result has never been established for §2- value d spin systems in three dimensions to our knowledge - even for the class

hi-

satisfying (3.2.33) . Nevertheless, a unique phase is believed to exist for every o. In any case, the uniqueness of the expectation value in (3.2.1) has now been verified for the spin systems having this special monotonicity.

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY

201

References 1. G. Choquet and P. Meyer, "Existence et Unicite des Representations Integrales dans les Convexes Compacts Quekonques ," Ann . Inst. Fourier 13 (1963), 139154.

2. J. Dixmier, Les C*-Algebres et Leurs Representations, Gauthier- Villars, Paris, 1964. 3. D. Kastler and D. Robinson, "Invariant States in Statistical Mechanics ," Commun. Math. Phys. 3 (1966), 151- 180. 4. O. Lanford and D. Ruelle, "Integral Representations of Invariant States on B* Algebras," J . Math. Phys. 8 (1967), 1460-1463. 5. R. Phelps, Lectures on Choquets ' Theorem, Van Nostrand , Princeton, 1966. 6. D. Robinson and D. Ruelle, "Extremal Invariant States," Ann . Inst. Henri Poincare 6 (1967), 299- 310. 7. D . Robinson and D. Ruelle, "Mean Entropy of States in Classical Statistical Mechanics," Commun. Math . Phys. 5 (1967),288- 300. 8. D. Ruelle, "A Variational Formulation of Equilibrium Statistical Mechanics and the Gibbs Phase Rule," Commun. Math. Phys. 5 (1967), 324-329.

9. D. Ruelle, Statistical Mechanics, Benjamin, New York, 1969.

3.3

Phase Transitions and Reflection Positivity

We have already encountered reflection positivity in the context of Euclidean field theory, where the property was called OS positivity. The most beautiful application of reflection positivity in the context of statistical mechanics is the Fr6hlich-SimonSpencer argument, which produces an SO(3)-invariant state Pf3 E Sf3,Q such that for large j3 (low temperature), . hm

J~OO

1I -IA ·

1

2: r; EAj

pf3(s-nt. s-at. ) = c # 0,

~=1,2 , 3,

(3.3.1)

for a certain class of spin systems, where {Aj} is an arbitrary sequence of finite rectangular sets of lattice sites converging to Z3 in the sense of set inclusion. On the other hand, ~

= 1,2,3,

(3.3.2)

and since Pf3 is SO(3)-invariant, this implies ~

= 1,2,3.

(3.3.3)

202

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

It follows that the strict inequality (3.2.28) holds with

and so we have a phase transition if we can find the desired state. y e begin by defining the class of spin systems. For L = 1 , 2 , 3we define the mappings r ( ~ ) , r ( ~z3 ) :+ z3by = m,, n # L, (3.3.5.0) r(L)(G)K

I'(L) is reflection of the lattice through the xL= 112 plane, while T(L)is reflection through the xL= 0 plane. We stipulate that the interaction has the following properties:

for each value of L and arbitrary complex numbers 7; indexed by Z3. Next, we realize the desired state as an infinite-volume limit of the Gibbs states with periodic boundary conditions - an infinite-volume limit, because uniqueness is not guaranteed for low temperature; existence is guaranteed by a compactness argument. Periodic boundary conditions are chosen because they admit translation in a finite volume. Any sequence of states that are translation-invariant in this sense on their respective finite rectangular sets of lattice sites - with the sequence of sets converging to Z3 in the sense of set inclusion - has only Z3-translation-invariant limit states, even in the absence of uniqueness. The limit states are also isotropic by virtue of (3.3.7) if the cutoff is isotropic. Accordingly, we consider Gibbs states defined by the Hamiltonians

for finite sets of the form

where J: is A-periodic - i.e.,

203

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY For each

L

we now define the A-modified mappings r~), f~) : A --+ A by

r~) (~)

f~\~)

= r(') (~),

= f('l(~),

(3 .3.13)

n.=!= 2N - 1 , n, = 2N-l .

(3.3.14.0) (3.3.14.1)

In the periodic L-direction r~) is the reflection illustrated by Fig. 3.3.1 (where N

= 4),

0

• • •

-1 -2

-3 -4 -5

• • • -6

•

• 2

• •7

•

-7

•

1

•3 •4 •5 •6

8

Figure 3.3.1: while f~) is the reflection illustrated by Fig. 3.3.2. We assume that the A-modified interaction has the following properties:

JA ~

7r(m)

= J~, m

1r

E Perm{l , 2, 3},

J!:()~ =J~,

r;

(m)

(3.3.15) (3.3.16)

m

(3 .3.17)

Finally, we require that

(3.3.18)

so any limit of the Gibbs states

dJ.-L~(s)

=

ZA(f3)-le-{3HAr;)

II da(s;) , ~EA

(3.3.19)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

204

-1 -2 -3

-4 -5

• • • •

-6

,

•

• • -7

0

•

1

•

2

•3 •4 •5

~

8

I

•7

•6

Figure 3.3.2:

ZA ((3)

(]I Jda('";l)

, - PH,(.),

(3.3.20)

mEA

is an isotropic translation-invariant equilibrium state for the interaction {J~}. The inequality (3 .3.17) actually implies the inequality (3 .3.9), and this property of the interaction is reflection positivity adapted to spin systems. Actually, on a lattice we are dealing with two slightly different kinds of reflection positivity - both of which we use. Reflection transformations e~), e~): Il §2 -t

Il

§

2

are defined by

~EA

e~)(s\~

(3.3.21)

e~>Cs\ii

(3.3.22)

First consider an arbitrary continuous function F(-;) that depends on only those spin variables -;;ii for which r;;E A and m, > o. In terms of Gibbs expectations, we have: (3 .3.23) (3.3.24)

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY

205

The reflection-invariance is a direct consequence of the J!:. _ translation-invariance n-m combined with (3.3.16), while the positivity condition follows from applying (3.3.17) to a wholesale power series expansion of the exponential factor

exp

~

[-2f3

L-

J!:.n-m - --; m- . --;-) n

~,~EA ml. > O nL~O

(3.3.25)

that contributes to the Gibbs measure. The residual exponential factor is

(3 .3.26)

11.-;

= {~E A: m, ~ OJ,

(3.3.27- )

11.;-

= {~E A: m,

(3.3.27+)

> OJ.

This type of reflection positivity plays the major role in establishing the existence of a phase transition . Now consider the slightly different reflection transformation 8~) With the (slightly different) condition that F(--;) depend on only those spin variables --; ~ for which ~E A and m, 2: O. We have:

J1~ (8~) (£))

=

J1~ (£),

J(F08~»)F*dJ1~

(3.3.28) (3.3.29)

2: O.

This positivity condition does not follow from (3.3.17) at all; instead, we observe that the symmetry (3.3.16) implies

+2

L

J!:.n-m_

8 -

8-

n

m

~EA? ~EA;-\A?

L

A

J-n-m -

~

8-

n

8m

206

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

(3.3.30)

where

~

A,O = {mE A: m,

= 0 or m, = 2N - 1 }.

(3.3.31)

The point is that (3.3.32) so the

s~ in (3.3.30) may be treated as coefficients with respect to the action of the s~-dependence . Thus m

transformation on the

n

(3.3.33)

s

Note that F(s) may depend on the ~ for which r;;,E A? without violating this positivity. This type of reflection positivity plays a relatively minor role in the analysis, but we need it to rule out pathological long-distance behavior. Let Pf3 denote our choice of limiting state for the sequence - indexed by the volume parameter N - of Gibbs states described above. Our task is to show that it has the

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY

207

property (3.3.1), and the next step in this direction is to establish that the function ) actually has monotonic behavior. In an infinite volume, the reflection

p{3(s~ s~ nit

OK

transformations (}(L), jj(L) are obviously defined by (}(L)(--;)~

(3.3.34)

jj( L) (--;) ~

(3.3.35)

m

m

and P{3 has the positivity properties

2 0, 2 0,

p{3((F 0 (}(L»)F*) p{3((F 0 jj(L»)F*)

(3.3.36) (3.3 .37)

where F(--;) depends on only those spin variables --; ~ for which m L > O. Now we may analyze p{3(s ~ s~) for positive integers q most easily. By translation-invariance and qe" ,K OK by (3.3.37) we have

P{3((S2re",,- - S(2r_1)e",)(Se,,,, - so)) P{3((Sre"" - S(r-lJe,)(S(l-r)e"", - S_re",,)) (3.3.38)

< 0 for all positive integers r, because

S

~

-re",n.

-

S(l -r )~eL l'" is the jj(LLreflection of S re~

L ,/\'

S(r-1)e"" On the other hand, (3.3.36) implies P{3((s(2r+1)e"" - s2re"")(se,,, - so)) P{3((s(r+l)e"" - sre,)(s(l_r)e"" - s_re,)) :S because

S

-T

~e

to ,/';;

-

(3.3.39)

0

S(l -r )~e /. ,,,, is the (}(LLreflection of s( r +1)~e /. ,1'\.

-

sr~e " " Thus

(3.3.40) for all positive integers q, and this inequality may be written as

1 P{3(Sqe""so) :S '2P{3(s(q-1)e""so) Therefore p{3(s ~

q e 1..,'"

1

+ '2 P{3(s(q+l)e""so)

s~) is a convex function of q for q OK.

has the bound

(3.3.41)

> O. Since this function also (3.3.42)

Ip{3(s q ~e ",I'\, s~)I:S 1, 0K

it can only be monotone decreasing for q > O. Since p{3 is isotropic, translationinvariant, and jj(LLinvariant, it follows that p{3(S~ s~ nK

OK

) :S

p{3(s~ s~ mK

OK

)

(3.3.43)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

208

for In,1 ~ Im,l, t = 1,2,3. Moreover, the monotonicity in each coordinate direction together with these symmetries imply (3.3.44) exists, and this long-range condition implies lim

J--+OO

IAI . I ~ L...J

nEAj

P(J(s-n~ s-OK, )

= c(J

(3.3.45)

for every nested sequence {Aj} of finite rectangular sets of lattice sites converging to Z3 in the sense of set inclusion. We have reduced the problem to showing that the more straightforward limit (3.3.44) is non-zero, in which case c(J is known as a long-range order parameter. Aside from the symmetry and reflection positivity already assumed for the interaction, there is yet another property it must have, if we are to show that c(J -I 0 for sufficiently large (3. The strategy is to consider the distribution (3.3.46)

-n

on the torus [-7l',7l'P Since p(J(s-nK. s-OK ) - c(J is a non-negative function on the lattice Z3 that decreases ~ the limit zero in every direction, it follows from elementary Fourier analysis that T(J( k) is actually a measurable function; indeed, one can show that 3

II (e ik ,

,=1

~

-

I)T(J(k) is a continuous function. Thus p(J(k)

= c(J6(k) + T(J(k),

T(J measurable,

(3.3.47)

is the structure of the Fourier series with coefficients p(J(s- s- ). What is the adnK OK ditional property the interaction must have? We shall presently use the reflectionpositivity to derive the bound

J-m

= J- L~ Jm rnO ~ n'

(3.3.49)

~

and this leads to a bound on p(J( k) as follows. If we scale {C;,;} with the replacement C;,; --+ A C;,; and expand the exponential in powers of A, we have 1 - (3A

22:-J-

-

m-n

m,n

~ Q- . ~ Qm n

222:-J-

+2(3 A

-

rn-n

2:-J - -

m'-n'

m ,n

(3.3.50)

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY where

as a consequence of translation-invariance and the normalization

Now subtract 1 from the inequality, divide by X2, and then take the X = 0 limit:

If we make the choice a ; = e ,a;, then 2

2

Since this inequality may be written as

2

As h ( k ) is arbitrary, this implies

and so from (3.3.47) an infrared bound is derived:

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

210

The additional condition on the interaction is that (J( k) - J(0))-1 be integrable. Indeed, since

(3.3.61) we have

c~

>

(g i:

dk,)

(2'llip~(s~

p~(k) - ~/3-1

)-

0"

~(27r)3 - ~/3-1

~r1 2

(g i:

(IT,=1-11" 111"

(gi:

dk,) (J(k) - J«}))-1

dk,) (J(k) - J(0))-1

dk,) (J(k) - J(0))-1

(3.3.62)

because

(3.3.63) and p~ is invariant with respect to the internal symmetry group 50(3) . Thus c~ > 0 for sufficiently large /3 if the integral is finite . At last we have a class of interactions for which a phase transition occurs. It remains to verify the bound (3.3.48), and to this end, we return to the finite box A of lattice sites with periodic boundary conditions. For a technical reason we also approximate da-(; ;;;) with
(3.3.64)

(3.3.65)

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY

211

At this point, we apply a chessboard estimate:

Before deriving this, note that if we now reverse the shift in d ;;-integrations new expectation, we obtain

for each

2

because the sum over n E A now annihilates the exponent by virtue of the normalization (3.3.49). This verifies the bound (3.3.48) in the limit as the regularization of da is removed. How does one derive the chessboard estimate? The input is the reflection-positivity property ( F * ( F 0 B ~ ) ) ) A2, ~0, L = 1,2,3, (3.3.68) where F depends on only those spin variables T; for which A:. It is easy to check that the cp-modification of the single-spin distribution has not destroyed this property. Now, clearly, this kind of expectation is actually an inner product on the space of such random variables, and so the Schwarz inequality applies:

GE

If we apply this estimation to the expectation (3.3.64) we obtain

212

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS G?.

= { G~m'

m

Gr<')(m)'

G~

= {Gr<')(m)'

G

m

m,

~E At, ~E A;,

~E

(3.3.71.0)

At,

(3.3.71.1)

~E A;.

What is different about the new products? Notice that each product still includes all of the spin variables but only half of the functions G~, - e.g., G?. can only be those G~, m m m for which ~ I E At - so this inequality is certainly a step in the direction of (3.3.66). How do we iterate this estimation? First, exploit the discrete translation-invariance of the state to write

(JI G~C;m+2N-2e)) mEA

A#

(JI G~_2N-2-;, (;m)) mEA

(3.3.72) A,'P

where the addition is understood to be periodic - e.g.,

A-; -

2N -

2

~,

A_2N-2~,

{~E A: m,

:s;

_2 N - 2 or

m, > 2 N -

A.

2 },

(3.3.73) (3.3.74)

Next, apply (3.3.69) to this expectation to obtain (3.3.75)

~

mE

A+ "

~EA;, ~

A+

~E

A;.

mE

"

(3.3.76.0)

(3.3.76.1)

Applying this same idea to the G~-product and combining the result with (3.3.75), we m see that (3.3.70) becomes

3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY where G'!!! can only be those G~t for which ~ m m only those G:;:;,.t for which ~

I

E At

n (A;

N

- 2

E

I

-

2

A+ t.

n (A+t. - 2N - 2

213 ~,), the G~ m

~,), the G~ those G:;:;,., such that

~ tEA; n (At - 2 N - 2 ~,), and G~ those for which ~ I E A; n (A; _ 2N - 2 ~,). m The next step in the iteration is applied to each of these four products and involves the translation by 2N -3 ~,. Eventually, we get

(

II

2N-l

G:;:;,.(S":;:;,.) )

A,'":S .,...

(

2

N

-

1

q=JI'+1 r=-!L+1:;:;"~O G:;:;"+qe,(S":;:;"+re)

)1/2N

L

A,CP

(3.3.78) and if we apply this same multiple reflection scheme for all three of the coordinate 3

directions indexed by

£,

we finally have the chessboard estimate because

n A? = {O} . 1.=1

Having described the theory, we conclude this section with an example of an interaction that satisfies all of the above requirements. Let

J_ e, = J e, = -1, J o = 6, Jr:; =0,

£

= 1,2,3,

all other n

(3.3 .79) (3.3.80) (3.3.81)

This is clearly ferromagnetic, but in the present context, it is more to the point to observe that this interaction is isotropic, reflection-invariant, and satisfies the reflectionpositivity condition. Moreover, ~

3

7(k) = 2:L)1- cosk,),

,=1

so 7(0)

(3 .3.82)

= 0 and (3.3.83)

This singularity is integrable in 3 dimensions, and so we have a phase transition. This specific model is the classical Heisenberg ferromagnet. The ferromagnetic property should not be confused with the reflection positivity property; this model happens to have both.

References 1. J. Frohlich, B. Simon, and T. Spencer, "Infrared Bounds, Phase Transitions, and

Continuous Symmetry Breaking," Commun. Math. Phys. 50 (1976), 79- 85. 2. J. Frohlich, R. Israel, E. Lieb, and B. Simon, "Phase Transitions and Reflection Positivity I: General Theory and Long-Range Lattice Models," Commun. Math. Phys. 62 (1978), 1-34.

214

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

3. J. Frohlich, R. Israel, E. Lieb and B. Simon, "Phase Transitions and Reflection Positivity II: Lattice Systems With Short-Range and Coulomb Interactions," J . Stat. Phys. 22 (1980), 297-347.

4. J. Frohlich and T. Spencer, "Phase Transitions in Statistical Mechanics and Quantum Field Theory," in New Developments in Quantum Field Theory, M. Levy and P. Mitter, eds., Plenum Press, New York, 1977. 5. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987. 6. R. Israel, Convexity and the Theory of Lattice Gases, Princeton University Press, Princeton, 1978.

3.4

The Ising Model

Perhaps no model in statistical mechanics has been analyzed more extensively than the Ising model. It has a deceptively elementary mathematical description, and yet its behavior with respect to parameter changes has provided an interesting laboratory for the rigorous study of critical phenomena. Originally proposed as a crude approximation of the quantum Heisenberg ferromagnet, its experimental predictions are actually less accurate than those of the classical Heisenberg ferromagnet, and yet it has contributed a great deal to an understanding of cooperative behavior at the macroscopic level. This model also has an interesting relationship to a class of Euclidean lattice field theories. At each site on the cubic lattice in three dimensions, we replace §2 with §o = {-I, I} and focus on the interaction between nearest neighbors introduced at the end of the previous section. For finite A C Z3.

'" L...J

s-sn n ' + h '" L....J s-n'

(n,n')eA

sn = ±1,

(3.4.1)

nEA

where (~, ~ ') is notation for two-element sets oflattice sites that are nearest neighbors. Although this model is classical in the sense that the C' -algebra of observables is commutative, the spin values have been quantized into "spin up" and "spin down." The external magnetic field h is effectively a scalar, and for a Z3-translation-invariant equilibrium state at a given inverse temperature (3, the spontaneous magnetization M«(3) ?:': 0 is given by M«(3) = lim(s-)«(3,h). (3.4.2) h.j.O

0

For sufficiently large (3, this model exhibits spontaneous magnetization (Le., M«(3) > 0), which is a breakdown of the global internal symmetry s7i B -s7i of the expectations - a symmetry of the Hamiltonian at h = O. The Lee-Yang Theorem applies to the Ising model- Le., the partition function

ZA«(3,h)=

L SE{ -l,l}A

e-{3H A (s)

(3.4.3)

3.4. THE ISING MODEL

215

has no zeros if analytically continued to h h = 0. This means the free energy

+ ik, except possibly on the imaginary axis

+

has an analytic continuation to all h i k in a neighborhood of the real axis except possibly a set on the imaginary axis. As in the case of S2-valued spins, this leads to convergence of these analytic continuations of { f~~(P, h)) - for every van Hove sequence {Aj) - to an analytic continuation of f (P, h) to the same domain. Since the thermodynamic functions are h-derivatives of the free energy, they have the same domain of analyticity. If (.)(P, h) denotes some translation-invariant, equilibrium state for inverse temperature /3 and scalar magnetic field h, then

at those parameter values. We infer this from the observation that

for every finite set A C Z3, where (.)A@,h) is the corresponding Gibbs state. This average is the A-cutoff on the magnetization, so in the thermodynamic limit it is uniquely defined for h # 0. If there is a discontinuity at h = 0 for a given value of P, this is precisely where spontaneous magnetization occurs. In this case, one can see two thermodynamic phases (.)&(P,0) - distinguished by

Naturally, every convex combination of these two equilibrium states is also an equilibrium state. The second-order h-derivative also sheds light on this kind of critical behavior. The magnetic susceptibility is given by

All of the arguments for the OS-positive class of 8'-valued spin systems apply to the Ising model as well, so the monotonicity condition

- (s;)~ holds. Combining this with the LeeYang Theorem, one can infer that (s;s;) is summable for h # 0. When h = 0, does the sum diverge when a phase transition occurs? Actually, translation-invariant states realized as limits of h = 0 Gibbs states

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

216

with periodic boundary conditions were investigated in the last section. Although not unique in general, such an equilibrium state is invariant with respect to the global internal symmetry (in this case s-; 1-+ -s-;) so the magnetization is zero in such a limit. On the other hand, the OS-positivity of the nearest-neighbor ferromagnetic interaction was also shown to imply (3.4.10) = c{3 > lim (s-s-) m 0

°

m-+oo

for sufficiently large (3. Obviously, the magnetic susceptibility is infinite in this parameter range. Correlation inequalities have played a major role in establishing properties of the Ising model, since the special structure admits a wealth of useful monotonicity relations. We briefly mention some of these inequalities with no attempt at completeness. 1) Griffiths Inequalities. Let A and B be arbitrary subsets of an arbitrary finite A C Z3 Then for h ~ 0,

(SA)A 2: 0, (SASB)A 2: (SA)A(SB)A, where

SA

=

II Sr; .

(3.4.11) (3.4.12)

(3.4.13)

r;EA Moreover, for arbitrary finite A' C Z3 containing A, (3.4.14)

provided (.) A is defined with no boundary conditions. This last inequality implies existence of the A = Z 3 limit for such a sequence of Gibbs states, since the uniform bound (3.4.15) obviously holds. The Griffiths inequalities are also useful in addressing the issue of phase transitions. If a given ferromagnetic interaction yields spontaneous magnetization at a given temperature, so does any interaction that is "more ferromagnetic" . 2) Lebowitz Inequalities. Introduce duplicate Ising spin variables tand set m

Then for h

~

qm

1 -(s..j2 m

+ t-) m'

(3.4.16)

rm

-(s..j2 m - t-). m

(3.4.17)

1

0,

((qAqB))A 2: ((qA)h((qB))A, ((r ArB)h 2: ((r A))A ((rB))A, ((qArB))A 2: ((qA))A((rB))A,

(3.4.18) (3.4.19) (3.4.20)

3.4. THE ISING MODEL

217

where the expectation « .)) A is defined by (3.4.21)

«SAtB))A = (sA)A(tB)A

with 01. the Gibbs expectation in the duplicate spin variables. These inequalities have significant consequences. The last inequality, for example, enables one to bound the expectation of a product of spin variables with a sum of products of expectations of products of fewer spin variables. Such bounds can be used to derive the long-distance decay of general correlations from the decay of the two-point function . 3) Fortuin-Kastelyn-Ginibre Inequality. If F(s) and G(s) are monotone increasing functions with respect to each spin variable s~, then

(F(S)G(S))A

~

(F(S))A(G(S))A.

(3.4.22)

One application of this correlation inequality is that the expectation of such an observable is monotone increasing in the external magnetic field . Since uniqueness of the equilibrium state can be established for large values of the external field by an expansion, this implies uniqueness of the equilibrium state for values of h where the free energy is analytic - i.e., all h 0, by the Lee-Yang Theorem.

t=

h

~c

Figure 3.4.1: The phase diagram in (3 and h for the three-dimensional Ising model is simple. The model has a unique phase for all parameter values other than «(3,0), (3 ~ (3e, where (3e is the critical value of the inverse temperature (3. On this half-line, the model has equilibrium states only of the form 0:':;0:':;1.

For the

0

(3.4.23)

= 1/2 state, the spontaneous magnetization is obviously zero, but (3.4.24)

because

,O) = (s-s-)+«(3,O) (S-S-)-«(3 mom 0

(3.4.25)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

218

by the sr;;

t-7

-sr;; change of variables. Hence

c(3

=

lim (S~S~h/2({3) mo

~ m~oo

=

lim (s~s~)+({3,O). mo

(3.4.26)

~ m~oo

Since O+({3, 0) is a pure phase, lim

~

m-+oo

(s~s~)+({3,O) m

((SO)+({3,0))2

0

(3.4.27)

M({3?, so we have the relation

(3.4.28)

For {3 > {3c, the spontaneous magnetization is non-zero, but it is also continuous in (3, so M({3c) = O. This means that there is a unique phase at {3 = {3c and that the long-range order parameter c(3 vanishes there as well. The properties of the critical point are governed by critical exponents . Consider the expectation (s~s~ ) ({3c, 0) for example. The large-n;, decay implicit in the equation m 0 c(3, = 0 is actually very weak, and indeed, the decay is not exponential in this special case. Instead, it is a power law decay - namely, (3.4.29)

m~oo,

where", is an example of a critical exponent. In the next section we shall see the similarity of the Ising ferromagnet to scalar Euclidean field theory on a lattice. By comparison to (3.4.29), the two-point correlation of the free massless field on a lattice is given by

(rr 1" ) 3

1 (21lV

_" dp,

,=1

.~ ~P e-,m.

3

2

1

L (1 -

,=1

~

~ 1-1, cl m

(3.4.30)

m~oo,

cosp,)

where this lattice function is the Green's function of the lattice Laplacian. For this reason, 3 - ", is called the anomalous dimension of the critical state. The value of ", cannot be computed in closed form, nor is there any known method that determines the value to an arbitrary degree of accuracy, but it is believed to lie between 0.04 and 0.06. In any case, the magnetic susceptibility

x({3, 0)

= {32 "(s~s~)({3,O) , ~ m 0

(3.4.31)

is clearly infinite at {3 = {3c. Another exponent 'Y governs the divergence of the magnetic susceptibility at {3 = {3c: {3 /' {3c. Yet another exponent v controls the divergence of the correlation length by m~oo, (S~SO )({3, 0) ~ cexp( -I n;, 1/~({3)),

(3.4.32) ~({3)

defined (3.4.33)

3.4. THE ISING MODEL for (3

219

< (3c· Its divergence has the form (3.4.34)

and the relation ,,(=(2-1))11

(3.4.35)

is a consequence of the celebrated Widom scaling hypothesis. In contrast to the magnetic susceptibility, the spontaneous magnetization has the property M((3) = 0, and is continuous for all (3 . For (3

(3.4.36)

> (3c, the dependence has the behavior (3.4.37)

where 8 is another critical exponent. The

28 = 311 - "(

(3.4.38)

is another consequence of the scaling hypothesis. The values of these exponents are known to be universal . This means that each exponent has the same value for large families of spin systems that include the Ising model. The point is that each critical exponent is independent of the short-distance details of a model. They reflect long-distance properties shared by many systems. The renormalization group sheds a great deal of light on this universality, as the values of the critical exponents are actually associated with a fixed point of the RG transformation . One of the earliest achievements in the study of the Ising model was the proof that spontaneous magnetization indeed occurs for sufficiently low temperature. (Originally applied to two dimensions, it is adaptable to three dimensions.) This proof did not depend on reflection-positivity methods (which were unknown) and the strategy was to construct each of the two pure phases for h = 0 by taking the thermodynamic limit with boundary conditions that favor the chosen phase. Outside the finite region A of lattice sites, one may specify all spin values to be +1, for example. This was the Peierls argument, and the representation is the earliest example of a low temperature expansion , which we shall not pursue.

See Bibliography for Papers by These Authors 1. D. Abraham

2. 3. 4. 5. 6. 7. 8.

M. Aizenman M. Aizenman and R. Graham M. Aizenman and B. Simon H. Araki and D. Evans T . Asano G. Baker W. Barbosa and M. O'Carroll

9. H. van Beijeren and G. Sylvester 10. J . Bricmont, J . Lebowitz, and G. Pfister 11. R. Dobrushin 12. F . Dyson 13. C. Fortuin , P. Kastelyn, and J. Ginibre 14. G. Gallavotti and S. Miracle-Sole

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

220 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

R. Griffiths R. Griffiths, C. Hurst, and S. Sherman R. Griffiths and B. Simon P. Hohenberg R. Holley W. Holsztynski and J . Slawny C. Hurst and H. Green J . Imbrie D. Kelley and S. Sherman J . Kogut and K. Wilson J. Lebowitz J. Lebowitz and A. Martin-USf J . Lebowitz and J. Monroe T . Lee and C. Yang E. Lieb, D. Mattis, and T . Schultz B. McCoy, C. Tracy, and T . Wu B. McCoy and T . Wu H. McKean A. Messager and S. Miracle-Sole M. Moore, D. Jasnov, and M. Wort is C. Nappi

3.5

36. 37. 38. 39. 40. 41. 42. 43. 44. 45 . 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.

C. Newman M. O'Carroll M. O'Carroll and W . Barbosa M. O'Carroll and F . Sa Barreto L. Onsager R. Peierls J. Percus S. Pirogov and Y. Sinai J. Rosen R. Schor S. Shlosman B. Simon J . Slawny A. Sokal M. Suzuki G. Sylvester D. Thouless F . Wegner B. Widom K. Wilson

The Ginzburg-Landau Model

One of the fundamental connections in mathematical physics is the close resemblance of the Ising model to the lattice approximation of the Euclidean scalar field theory with quartic interaction - more commonly known as the Ginzburg-Landau model. The latter is given by a probability measure on the space of all real-valued configurations on :£;3, but the nearest-neighbor coupling is the same. From the point of view of classical statistical mechanics, there is no longer any constraint on the scalar spin values - i.e., SO = { -1, I} is replaced by ~ at each lattice site. On the other hand, the form of the field interaction assigns a strong probability to "spin-up or spin-down" behavior in a certain parameter range. The lattice approximation of the ~ Euclidean field theory is similar in spirit to the standard ultraviolet cutoff, but the starting point is a lattice field ¢ with a discrete Gaussian covariance: (3.5.1)

As an infinite matrix,

clat

is the following inverse matrix: (3.5.2)

3.5. THE GINZBURG-LANDAU MODEL

221

-6, ~'=~, 1, ~,~' nearest neighbors , { 0, otherwise.

(3 .5.3)

,:llat is the lattice Laplacian, since the quadratic form is given by

L

(_Ll lat ¢, ¢) =

(¢(~ ') _ ¢(~))2.

(3.5.4)

(n,';' )

We define d/-Lbat as the Gaussian measure on IRz S with mean zero and covariance C lat Formally, d/-Lbat is the normalization of the measure exp

(_~((_,:llat + m6)cp,cp))

1J dcp(~) . n

For an arbitrary finite set A C Z3 define XA as the characteristic function of A of Z 3 The interacting probability measure on IRA is given by d/-L~' T (cp)

ZA(A, T)-1 exp( -Ilat(XACP))d/-L&at(XA¢)

(3.5.5)

Ilat(XAcp))d/-L~at(XA¢)

(3.5.6)

J

ZA (A, T)

exp( -

L V(¢(r;;)),

Ilat(¢)

(3.5.7)

~

V(z)

=

Az4

+ TZ2,

A > O.

(3.5.8)

Naturally, the continuum limit involves the lattice spacing parameter, which we set equal to unity here. Also the restriction of the Gaussian measure to XA¢ reflects a choice of boundary conditions. How do we recognize d/-LA as the Gibbs state of a spin Hamiltonian for inverse temperature f3? First we identify the spin variable: (3.5 .9)

Thus

~((_Lllat +m6)XA CP,XA¢) - 13

_~ s,;s~+f3f= (3+~m6)s~ , (n ,m)CA

Af32S~

V(¢(r;;)) Second, we give A and

m

T

(3 .5.10)

mEA

+ f3TS~m .

(3 .5.11)

the f3-dependence that makes the single-spin distribution

dO'( z ) = exp

[-13

(3 + ~m6 + T(f3)) Z2 - A(13) 132Z4 ] dz

(3.5.12)

222

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

independent of f3. Accordingly, we set

T(f3) = )..(f3)

~m6 -

f3-1f -

(3.5.13)

3,

= f3- 2~,

(3.5.14)

with ~ and f the independent parameters, so

ZA(f3)-l exp (f3

~~

sr;sr;;.)]I du(sr;),

(n ,m)CA

(3 .5.15)

nEA

f3-(card A)/2 ZA ()..(f3) , T(f3)),

(3 .5.16)

How do we discern anything resembling the behavior of the Ising model in this Gibbs state? In terms of the independent parameters, we write the single-spin distribution as follows : exp( -fz 2 - ~z4)dz du(z) exp( _~(z2

+ f /2~)2 + f2 /4~)dz .

If f is negative, the completed square has a local maximum at z at

z

(3.5.17)

= 0 and local minima

= ±j-f/2i

(3.5.18)

In this case the graph of the completed square is a double well. The point is that at a given lattice site the single-spin distribution has its maximum probability density at the

±j

spin values -f/2i This is a major reason why the lattice field theory has the same critical behavior as the Ising model for sufficiently negative f. Another, rather different reason is that this theory is approximated by the Griffiths-Simon superposition of Ising models, which we do not describe here. The Lee-Yang Theorem applies to the Ginzburg-Landau model as a consequence. For fixed ~ and f, one expects to have uniqueness of the 11.= Z 3 limit of dp,~,T in the case where f3 is small. This uniqueness breaks down for negative f if f3 is sufficiently large relative to ~/f2, but every possible measure is still a convex combination of two pure phases characterized by the spontaneous magnetization, which is defined exactly as it was for the Ising model. Indeed, if ZA().., T, h) and dp,~'T,h are defined by replacing V(z) with V(z) + hz, then

N

lim

A->Z 3

(rr Jrr ¢J(r:"j)dp,~'T,h(¢J)

¢J(r:"j)) ()..,T,h)

j=l

A

N

lim

A->Z3

(3.5.19)

j=l

exists for h i- 0 - i.e., the infinite-volume state is unique for h i- 0 by the same reasoning that was applied to the Ising model. The spontaneous magnetization is given by

3.5. THE GINZBURG-LANDAU MODEL

223

z

Figure 3.5.1: lim (3-1/2 (¢(O)) ()..((3), 7((3) , h) ,

(3.5.20)

h '-,. O

where this expectation is Z3-translation-invariant. Also, (3-1/2 (¢( 0)) _ ()..((3), 7((3))

=

lim r1/2(¢(O))()..((3), 7((3) , h) h /' O

= -M((3)

(3.5.21)

by the z f-t - z symmetry of V(z). If dJ.L ranges over convex combinations of probability measures realized as limits of the net {dJ.L~'T } , then dJ.L is parametrized by the unit interval as the convex combinations dJ.L

= exdJ.L+ + (1 -

ex)dJ.L_ .

(3.5.22)

If ex = ~, then the ¢C:;;') f-t -¢C:;;, ) symmetry is unbroken for dJ.L and the long-range o~er parameter is given by c((3) = Jim (3-1

/¢(~)¢(O)dJ.L(¢) .

(3.5 .23)

m-+oo

On the other hand, the ¢(:;;,) /

>-t

-¢C-;') change of variables yields

¢(~)¢(O)dJ.L-(¢) =

J

¢(~)¢(O)dJ.L+(¢),

(3.5.24)

so we have the same conclusion as in the case of the Ising model: c((3)

=

Jim (3-1

J ¢(~)¢(O)dJ.L+(¢)

m-+oo

(3-1 ( / ¢(O)dJ.L+(¢)) 2 = M((3)2.

(3.5.25)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

224

There is long-range order for those parameter values that yield spontaneous magnetization. Now consider the phase diagram of the Ginzburg-Landau model in the parameters A and T for fixed inverse temperature (J and variable X, r . In the h = 0 plane, this system has been proven to have the regimes for a unique phase given by Fig. 3.5.2. Actually, there is only one critical curve. We have drawn the boundaries of parameter regions for convergent expansions, where E. > 0 is understood to be sufficiently small. The region {A ::; E.JT} is the regime of the Glimm-Ja£fe-Spencer cluster expansion, while in the region there is a convergent cluster expansion that decouples the Hamiltonian directly - an expansion that will be introduced in §3. 7. Both expansions are often referred to as high-temperature expansions because they establish the properties of a unique phase. In the two-phase regime there is a low-temperature expansion based on the Peierls argument, but we shall not pursue it. The actual phase diagram is given by Fig. 3.5.3, where the boundary is called the critical manifold .

unique phase

---- ---- -- -- .-

umque

/

EArl

I

1=0

I

,,

)

,

--- --- --

A

phase

1=-EJ"l

Figure 3.5.2: The Ginzburg-Landau model has the same critical exponents as the Ising model. The spontaneous magnetization is a continuous function of the parameters, and (3.5.26.0)

3.5. THE GINZBURG-LANDAU MODEL

225

unique phase

)

spontaneous magnetization

Figure 3.5.3: (3.5.26.1) with 8 :::::: 0.31. The magnetic susceptibility has the divergent behavior

X(A,r) ~ e(r - rc(A))-T', X(A,r)

~

(3.5.27.0)

e(A - Ac(r))-T',

(3.5.27.1)

with f :::::: 1.24, where

X(A,r,h) f(A,r,h)

82

(3.5.28)

8h2f(A,r,h), .

1

lim -IAllnZA(A,r,h).

A--+Z 3

(3.5.29)

Clearly, (3.5.30) is well-defined even for h = 0 in the unique phase region arbitrarily close to the critical manifold. The summand is an example of a truncated correlation (already defined in §1.12) and such correlations are known to decay exponentially when this particular one does, and with the same correlation length etA, r, h) . This is a result of the Lebowitz inequalities for the Euclidean lattice 14 field theory. In the case h = 0 and A > Ac(r), the picture is: (3.5.31)

(
I/e(A, r)),

(3.5.32)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

226

~

c(r - rc(A))-II,

r '\, rc(A),

(3.5.33.0)

~(A , r) ~

C(A - AC (r))-II,

A '\t Ac(r),

(3.5.33.1)

~(A,

with v

:=;:j

r)

0.63. Actually, one has the relation 28

=

3v - 'Y by the Widom scaling

hypothesis. At A = Ac(r), the two-point function ((n;,)(O» still decays for large n;" but it obeys an inverse power law. The r = -~m5, A = 0 case is special:

(

21)3

(IT 1" dP') ,=1 -"

7r

--=-3_e,_m._1'2 L (1 - cosp,)

,=1

(3.5.34)

m-t 00,

because in this case the model is just the free scalar lattice field with zero mass - i.e., the Gaussian whose quadratic form is given by the lattice Laplacian (and therefore whose covariance is the lattice Green's function). For every other point on the critical manifold, however, m-t 00 ,

(3.5.35)

where TJ is believed to lie between 0.04 and 0.06. The scaling hypothesis yields the relation I = (2 - TJ)v . The correlation inequalities mentioned in the last section generalize to the lattice ~ field theory, and it is worthwhile to state the generalizations. We suppress the dependence on h, A and r and write dj.£1I. for the measure.

1) Griffiths Inequalities . For finite A C Z 3 and h :::; 0, (3.5.36) m

JfJ. (n;,)"';:;+I<~dj.£II.((n;,)"'~dj.£I1.(
m

(n;,r;;;dj.£I1.(
m

(3.5.37)

m

where", and ",' are arbitrary multi-indices. Such inequalities hold for all scalar-valued spin systems whose single-spin distributions are even and whose Hamiltonians are ferromagnetic. The restriction h :::; 0 is necessary because the external magnetic field term must be included in the Hamiltonian instead of in the single-spin distribution. 2) Lebowitz Inequalities. Introduce duplicate variables 1jJ(n;,) and set 1

~

~

+ 1jJ(m)),

(3.5.38)

1 ~ ~ ..j2((m) -1jJ(m)).

(3.5.39)

..j2((m)

3.5. THE GINZBURG-LANDAU MODEL

227

Then for h 5 0,

for arbitrary multi-indices

K

and

K',

where

3) Fortuin-Kastelyn-Ginibre Inequality. If F ( 4 ) and G ( 4 ) are monotone increasing functions with respect to each variable d ( ; ) , then

Indeed, such an inequality holds for any probability measure of the form

where W(q5) is class C2 and

The Ginzburg-Landau model obviously satisfies this condition.

See Bibliography for Papers by These Authors 1. M. Aizenman 2. M. Aizenman and R. Graham 3. C. Aragao de Carvalho, S. Caracciolo, and J. Frohlich 4. G. Baker 5. G. Baker and J. Kincaid 6. H. van Beijeren and G. Sylvester 7. P. Bleher 8. P. Bleher and Y. Sinai

9. J. Bricmont and J. Frohlich 10. D. Brydges, J. Fkohlich, and A. Sokal 11. G. Caginalp 12. P. Collet and J. Eckmann 13. F. Constantinescu 14. F. Constantinescu and B. Stroter 15. R. Dobrushin 16. F. Dyson

228

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

17. J. Feldman, J. Magnen, V. Rivasseau, and R. Seneor 18. J. Fontaine 19. C. Fortuin, P. Kastelyn, and J. Ginibre 20. J. Frijhlich 21. J. Frohlich, B. Simon, and T. Spencer 22. K. Gawedzki and A. Kupiainen 23. J. Glimm 24. J. Glimm and A. Jaffe 25. J. Glimm, A. Jaffe, and T. Spencer 26. R. Griffiths 27. R. Griffiths, C. Hurst, and S. Sherman 28. R. Griffiths and B. Simon 29. F. Guerra 30. F. Guerra, L. Roson, and B. Simon 31. R. Holley 32. J . Imbrie 33. R. Israel

3.6

34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.

K. Kadanoff D. Kelley and S. Sherman H. Koch and P. Wittwer J . Kogut and K. Wilson J. Lebowitz E. Lieb E. Lieb and A. Sokal V. Malyshev 0. McBryan and J. Rosen C. Newman Y. Park S. Pirogov and Y. Sinai C. Preston J . Rosen L. Rosen R. Schrader B. Simon A. Sokal M. Suzuki and M. Fisher G. Sylvester F. Wegner K. Wilson

Polymer Expansions

There is a fairly unified view of expansion methods in statistical mechanics and q u a tum field theory embodied in the abstract theory of polymer expansions. Historically, the various expansions that were developed over the years to deal with different problems were eventually perceived to have common elements. The abstract formalism that was subsequently developed is quite elegant and neatly separates the technical issues in a specific expansion method from those common elements. Since some of the expansions are very complicated, this axiomatic framework has been very convenient. On the other hand, the formulation that is usually advertised in the literature does not cover all of our needs. Before discussing this, we review the standard estimation strategy for proving convergence. One begins with a set P of objects called polymers. There is a notion of intersection defined by a set of unordered pairs of polymers. Two polymers intersect if there is an unordered pair in which they both appear. Otherwise, they are disjoint. An unordered pair is not to be confused with a two-element set, as a polymer may have a pairing with itself. On the other hand, it may not, in which case the polymer is disjoint from itself. Every polymer ( has a complex number z ( ( ) assigned to it called the activity of

3.6. POLYMER EXPANSIONS

229

( . The partition function for this structure is given by

Z=~~ L.., n!

n

L

n=O

II z ((v),

(. " " ,(n EP

(3.6.1)

v=1

mutually disjoint

so to express the usual partition function of a concrete model in this formalism, one must identify polymers and their activities for that model. An n-polymer is an n-tuple of polymers, and the activity of an n-polymer P = ((1, .. . ,(n) is defined as n

z (P)

= II z((v).

(3.6.2)

v=l

A line of P is any distinct pair {Il, v} of indices such that (I-' intersects (v' The graph of P is the set of all lines of P, while the indices {I, ... ,n} of P are the vertices of the graph. Note here that distinct vertices may index the same polymer, and therefore distinct lines may arise from the same intersecting pair of polymers. Two vertices are connected if there is a path of lines from one to the other. This equivalence relation partitions {I, . . . , n} into equivalence classes called the connected components of the graph. The graph is disconnected if the connected components are the singletons {v}, and the graph is connected if the only connected component is {I, . .. ,n} . An n-polymer P is connected (resp. disconnected) if its graph G p == G((1, . . . ,(n) is. For an arbitrary graph G on {I , ... ,n}, an n-vertex subgraph is just a subset of G - as a set of lines - with {I, . .. , n} as its set of vertices, whether they are connected or not. If G is connected, one defines n(G)

=

(3.6.3) connected n-vertex subgraphs G of G

as the index of G. The index of a connected n-polymer P is defined as n(P) = n(G p

(3.6.4)

).

The fundamental algebraic property of a partition function is an abstraction of what has long been known as the "Linked Cluster Theorem" in different contexts. In short,

InZ

= L, 00

1

n.

L

n=O

PElC n

(3.6.5)

n(P) z (P),

where Kn denotes the set of connected n-polymers . The proof is quite standard by now, and we sketch it right here for the convenience of the reader. Notice that 00

exp

(

?;

1 n!

P~n n(P) z (P)

)

1

£; N! n" '~N=O 00

=

00

N

N

L II n(p II z(p k

pN ElC nN k=1

)

k=1

k

)

(3 .6.6)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

230

must somehow reduce to 1

00

2: n!

n=O

1

n

00

v=l

n=O

II z((v) = 2:, 2: II (1- X((JL'(v)) II z((v),

2: (, , .. . ,(n E"P

n. (, ,... ,(n E"P JL
(3.6.7)

v

mutually disjoint

where X((, (') = {0 ', 1

(and (' disjoint, ( and (' intersect.

(3.6.8)

On one hand, N

L (-1)>=' k=l

connected n,-vertex subgraphs G, of G p'

card G.

connected nN-vertex subgraphs G N of G pN

(3.6.9)

while on the other hand,

II (1- X((JL'(v)) graphs G on {l, ... ,n}

JL
{JL,v}EG

(-1)

card G.

(3.6.10)

n-vertex subgraphs G of G((" ... ,(n)

In this sum, ((1, . . . , (n) is an arbitrary n-polymer, while G is an arbitrary subgraph, not necessarily connected. Now in the expression (3.6.6) define R(n1' . . ·' nN) as the set of all indexed partitions of {I, .. . ,

t

nk} where the subsets have the cardinalities n1,· .. , nN. Given

k=l

r

N

E R(n1, . . . , nN), define pr (PI, . .. , pN) as the

J.t is the 11th integer in the subset indexed by k,

L

nk-polymer defined such that if

k=l (3.6.11)

The key observation is that for any written as

r

E R(n1, .. . ,nN), the identity (3.6.9) may be

(-1) k=l

Ln.-vertex subgraphs

•

r

G of Gr(P' ,... ,p N

is the set of G-components

)

such that

card G

(3.6.12)

3.6. POLYMER EXPANSIONS

231

where Gr(P1,. . . ,P N ) denotes the graph of P r ( P 1 , . . . ,P N ) and G-components are defined by G-connectedness. Two vertices are G-connected if there is a path of lines in G from one to the other. Now since

card R(n1,. . . ,nN) = k=l

k=l

it follows that we have the formula

r

Gr(P1, ...,pN)such that is the set of G-components,

which may, in turn, be written as

(-1)card

C

G

n-vertex subgraphs G of G ~ ( P ' ,...,pN)such that r is the set of G-components

because the restriction on the innermost sum makes the restrictions Pk E En, superfluous. However, this yields the further reduction:

n=O

n=O

PEK,

C

n-vertex subgraphs

r

is the set of

G of G p such that

G-components

n-polymers

(+card

G

232

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

By the summation identification

L

n" ... ,nN;O:O Lnk=n

L

L

all indexed partitions r of {1, ... ,n} using N subsets

rER(n" ... ,nN)

•

(3.6.18)

N! all partitions of {1 ,.. .,n} using N su bsets

the desired identity now follows from the observation that the resulting inner sums, starting with the N-summation, are just a decomposition of the sum over all n-vertex sub graphs of Gp. With the algebraic identity established, we turn to the issue of convergence, which is crucial to the application of polymer expansions. Actually, we need to be a little more concrete at this point. In addition to an activity, a polymer must now have a support and a volume, so the additional structure is a set A that the polymers live in. In applications, this set can be the lattice l,d, the partition of]Rd into unit cubes, or even a wavelet basis in a multiscale analysis of a continuum model. For a given polymer (, a finite subset supp ( c A is assigned to it, and we refer to it as the support of ( . We define the volume 1(1 of the polymer as simply the cardinality of supp ( . The intersection properties of the polymers are assumed to be consistent with this notion of support in the obvious sense: they are disjoint as polymers if and only if their supports are disjoint as sets. The estimates that one often assumes for the abstract theory and verifies for a given model - are:

Iz(()1 :S Coe-col(l, card{( E P:

Ct

E supp

(,1(1 = M} :S C 1 ec ,M,

(3.6.19) (3.6.20)

with Co, C1, Co, C 1 all positive constants, whose relative sizes will be an issue. The first bound is referred to as an energy estimate, while the second is an entropy estimate. Now for every finite SeA, consider 00

Z(S)

1

L n! n=O

In Z(S)

00 1 '~n! "-

n=O

n

L

II z((v),

(3.6.21)

(, ''''' (n mutually v=1 disjoint , supp (veS

'~ "

n(P)z(P),

(3.6.22)

PEK. n supp PeS

where it is understood that for any n-polymer P

= ((1, .. . , (n)

n

supp P

= U SUpp (v '

(3.6.23)

v =1

The problem is to control the behavior of these expansions as S approaches A in the sense of set inclusion.

3.6. POLYMER EXPANSIONS

233

We first consider the expansion of Z(S), as it is easier to control from a combinatorial standpoint. Fix n 2 0 and consider the inner sum: since Cl,. . . ,Cn are disjoint, we may select for each such sequence of polymers a sequence (al,. . . , a n ) of points in S, all distinct, such that a, lies in the support of C,, for each u. The disjointness also implies

xlCvl n

5 card S ~ m ,

v=l so we over-sum as follows:

(3.6.25) ,...,C,

mutually disjoint, supp C,CS C1

v=l

0 1

,...,a,€S

all distinct

...,C,€P u=l a,Esupp C, C1,

C ICvllm The energy estimate implies

while the entropy estimate implies

card{(<^,. . . ,G ) E P n : av E supp Cv and lCvl = mu} 5 CFe

ci

m,

.

(3.6.27)

Hence

C l , ...,C, mutually disjoint, supp C, C S

5

c;c;

v=l cl

):

...,a , E S ali distinct

a1

m,

e

" mu_<m

m,>O

x n

{

5 CgCpeclmcard (ml , . . . ,m,):

mu 5 rn,mu 2 0 v=l xcard{(al, . . . , a n ) E S n : all distinct).

(3.6.28)

The first cardinality counts the number of points with integer coordinates in the positive oct5nt of n-di_mensional Cartesian space enclosed by the simplicia1 face with vertices m e 1, . .. ,m e ,. This number is roughly the volume mn/n! of that region of space. Allowing for boundary effects, one has

The second cardinality is obviously the number

m! -(:In! Perm(m, n) = (m - n)!

234

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

of permutations of m objects taken n at a time. Thus n

II Iz((v)1

L

(l, ... ,(n mutually v=l disjoint , supp (v cS

(3.6.31)

< so if COC1 < 1, then 00

1

Ln!

n=O

n

L

II Iz((v)1

<

(l, ... ,(n mutually v=l disjoint, supp (v cS

X

00

n

n=O

v=O

v

LCr;C~Lm, 1/.

m

e_---,< 2m eel m __ 1- C C O

(3.6.32)

1

This is an abstract stability bound on the partition function Z(S), and it obviously implies 1 (3.6.33) SeA, -dSlnZ(S)::; c, car provided COC 1 < 1. In spite of the algebraic connection between the polymer expansions of Z(S) and (card S)-lln Z(S), this estimation does not control the expansion of the latter quantity uniformly in S - for the very simple reason that In(l + z) has radius of convergence equal to 1. One must control the latter expansion directly. The issue of uniform convergence will now depend on the relative sizes of the constants Co and C1 as well as on the absolute size of Co, which we did not use in the above estimation. The combinatoric estimation is substantially deeper as well. One can no longer sum over n-tuples (0'1' ... ' O'n) of points as we have just done, because the estimation must avoid the occurrence of powers of card S . The connectivity properties of the n-polymers summed over must be exploited, and a tree structure is needed to obtain good control over the n-fold sum. To this end, one normally defines a tree graph on the vertices {I, . . . , n} to be a simply connected graph on {I, ... , n} and applies the standard graph-theoretic result that the index of an arbitrary connected graph on {I, ... , n} is bounded by the number of connected subgraphs that are tree graphs. We omit the proof, as the result is widely known outside of mathematical physics. Thus, for an arbitrary connected graph G on {I, .. . , n},

In(G)1 ::;

1

(3.6.34)

tree graphs T that are connected su bgraphs of G

and so if we insert this estimation in the nth term (with n of In Z(S) and interchange the sums, we get

L PElC n

In(Gp)ll z(P)1

> 1) of the polymer expansion

3.6. POLYMER EXPANSIONS

tree graphs T on {I,...,n}

m l , ...,m,=O

235

P = ( c l r . . . , < n ) ~ K n ,=I T is a subgraph of G p

ICY I=mY

where we have further decomposed the sum with respect to volumes of polymers. Controlling the inner-most sum is the crucial part of the estimation. If one tries to count the connected graphs of which T is a connected subgraph, the resulting oversummation is disastrous. It is important to note that one is summing over all connected n-polymers - not n-vertex graphs - on which T lives. As one uses the structure of T to sum over all such n-polymers, the polymers P for which Gp is not simply connected are automatically included in the summation. To control the sum with T , one finds it convenient to use the standard notion of coordination number. For a tree graph T on (1,. . . ,n ) , the coordination number d,(T) of a vertex v is just the number of lines in T that meet v. Every tree graph can be reduced to a single vertex by the following iterative procedure. Let A be the subset of vertices v for which d,(T) > 1 and let T' be the tree graph on A obtained by removing the single lines meeting the other vertices. Obviously, there will now be vertices v E A such that d,(T1) = 1, so now remove those single lines to obtain the tree graph T" on the subset A' obtained by removing those vertices from A. This procedure can be iterated for as long as there are distinct vertices. The summation for fixed T is carried out as follows. First apply the energy estimate to obtain

and pull this bound outside the inner-most sum, which now becomes (with SF to be defined)

Here we have applied the entropy estimate to each sum over polymers C, such that a, E supp C,,. This replaces the sum over polymers with a supremum over polymers, but one still has to sum over the possible a, determined by the T-connectivity property of a given n-polymer. This vital control is implicit in the definition of SF, which reflects the tree reduction just described. First, for every vertex v such that d,(T) = 1, a, E supp CC, where {v, D) is that single line. Then for every v' such that d,~(TI) = 1, a,, E supp &,, where {v', D') is that single line, and so on. It is easy to see that this definition of S; is the correct one for the bound (3.6.37) and that, since the multiplicity o f p = t# is d,(T) - 1,

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

236

where the last polymer in the tree reduction is not connected to anything. This means that Q" E S is the only restriction for the last remaining vertex v , so card(S) appears as a factor . Combining (3.6.36-3.3.38) with (3.6.35) , one obtains interesting tree graph estimation of the nth term (with n > 1) of the polymer expansion:

I:

00

In(Gp)llz(P)I:S

PEK n

tree graphs T

I:

ffil , .. . ,mn=O

on {I •... . n}

co-cre

(Cl-CO) Em_ card(S)

IT m~p(T)-1 n

(3.6.39)

1'=1

The next step is to decompose the sum over all tree graphs with respect to coordination numbers, and then apply the celebrated Cayley Theorem, which states that the number of tree graphs on {1 , ... , n} with coordination numbers d 1 , . . . , d n is just the ratio n

(n - 2)!/

I1 (dl'

- 1)! . Hence

1'=1

I:

In(Gp)ll z(P)1

PEK. n

(n - 2) !Co-Cr lSI

00

< (n - 2)!Co-Grcard(S)

~

~

e

(Cl-CO) Em_ -

IT e n

mp

~=l

ml, . ·· ,mn=O

(n - 2)!Co-Crcard(S)(1 _ eCl-CO+I) - n,

(3.6.40)

provided CI < Co - 1. This yields the bound 00

1

l:,n . I:

n=O

In(Gp)ll z(P)1

PEK. n 00

1+

1

I: Iz (OI + card(S) I: n(n _ 1) Co-Cr(1- e (EP

Cl

-

co

+1))-n

(3.6.41)

n=2

on the polymer expansion of In Z(S) . Clearly, the n given by

= 1 contribution has the

bound

(3.6.42) where we have summed over volumes and over Q E S , applying the entropy estimate in summing over ( E P with fixed volume and supports containing fixed Q . In other words,

3.7. EXPANSION FOR NEA REST-NEIGHBOR INTERACTIONS the basic estimation is the same as in the n estimation to do. In summary,

>

237

1 case, but there is no combinatoric

and the dominating series converges if CoCl 5 1- ecl-co+l

References 1. G. Gallavotti, A. Martin-Lof, and S. Miracle-SolB, "Some Problems Connected With the Description of Coexisting Phases at Low Temperatures in the Ising Model," in Statistical Mechanics and Mathematical Problems, A. Lenard, ed., Lecture Notes in Physics 20, Springer-Verlag, New York, 1973. 2. J. Glimm and A. Jaffe, Quantum Physics: A hnctional Integral Point of View,

Springer-Verlag, New York, 1987. 3. C. Gruber and H. Kunz, "General Properties of Polymer Systems,'' Commun. Math. Phys. 22 (1971), 133-161.

4. V. Malyshev, "Uniform Cluster Estimates for Lattice Models," Commun. Math. Phys. 64 (1979), 131-157. 5. V. Malyshev, Introduction to Euclidean Quantum Field Theory, Moscow University Press, Moscow, 1985. 6. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969. 7. E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics 159, Springer-Verlag, New York, 1982.

3.7

Expansion for Nearest-Neighbor Interactions

As applications of the theory of polymer expansions, the high-temperature expansions of the classical Heisenberg model and of the lattice $J; model are probably the most elementary. This section is devoted t o describing these two specific examples as part of our progression toward more difficult expansion methods. We consider the hightemperature classical Heisenberg ferromagnet first. S2 and the For a finite set A of lattice sites, the space of configurations is 2

nEA

single-spin distribution is surface measure du on S2 For zero external magnetic field,

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

238

the Hamiltonian is just 3

HA=-J

s;;; (G,~)cA

2

s;,

(3.7.1)

and our goal is to control the A = Z3limit of the Gibbs expectations

for small p (high temperature). These quantities are generated by the generalized partition function

and our task is to develop a disconnected polymer expansion of

where we set

2

2

~ ( he - ) =Jexp(hi

2

2

si)do(si).

(3.7.6)

Once the expansion is developed, we establish the energy estimate and the entropy estimate for A-uniform convergence of the corresponding polymer expansion of

-

1 lnZ(h-: C E A) = card A e card A 2

2

(3.7.7)

e e EA

in connected n-polymers in the high-temperature regime. In this example, the most obvious expansion method works. Simply expand Z(h;: C E A) in powers of p: 2

2

3.7. EXPANSION FOR NEAREST-NEIGHBOR INTERACTIONS

239

However, a power series in {3 is not the most natural point of view, even in this case. We first write this in terms of multi-indices on nearest-neighbor bonds. Let BII. be the set of such bonds contained in A and A~ the set of all multi-indices Ie BII. -t {O, 1,2, ... } such that 111;1 = 11;( (T:t, 1{)) = N . (3.7.9)

L

(;;;',~)Cll. A

.

---lo...-..l...-..l.

---10.

The element of AN associated with a sequence ((ml, nl)' ... ' (mN, nN)) is defined by (3 .7.10) and there are

N!

n

11;( (T:t, 1{))!

(;;;',n)

such sequences with which a given multi-index II; is associated. Since the terms in the expansion depend only on the associated multi-index, we have Z(h~ : l

fE A)

( II Z(h 7 )) fEll. \supp

x

.I (~L f E supp

X

A

~ ~ (m,n)

~

00

A~ and supp

N=O

II;

7)

~

fE supp

K

II (-;;;;.. -;~)K«;;;"~)) II = U

da(-;

II Z(h7)!exp ( L h7·-;7)

fEII.\supp

l E supp

where All.

II f E supp ~

(m,n)

K

L II ((3J):«:,n;) 1I;((m, n)).

KE A

IT (-;; ;. .-; ~)"'«;;;., n))

h7 · -;7)

exp

K

da(-;7)'

K

(3 .7.11)

K

is the set of lattice sites joined by those bonds to

which II; assigns non-zero weights. We define a polymer to be any connected set of nearest-neighbor bonds (T:t,1{). The support of a polymer is just the set of lattice sites joined by the bonds, while the activity of a polymer is to be inferred from a reorganization of this power series expansion. For a given multi-index 11;, there is a uniquely determined set {(I, ... , (n} of polymers with mutually disjoint supports which partitions supp 11;. There are n! permutations of this set of polymers, and each permutation is an element of V n . Thus Z(h~: l

f E A)

240

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

(3.7.12) _

n

lEU supp k=l

<.

and it is apparent that both the multiple integration and the multiple summation factorize over the mutually disjoint polymer supports:

(3.7.13)

because the multi-index a assigns a non-zero weight to precisely those bonds lying in (k . In this way, we have identified the polymer activity :

(3.7.14)

3.7. EXPANSION FOR NEAREST-NEIGHBOR INTERACTIONS

241 ~

z(()

exp(h~ . -;~)

e

x

II

(exp((3J -; r;, . -; r:)

-

1) .

e

(3.7.15)

(r;"r:)E~

This is the reason why a power series in (3 is not the most natural high-temperature expansion in any case . The difference between unity and a nearest-neighbor exponential is a more natural high-temperature quantity in this context. Estimating the activity is simple enough in this case. By Jensen's inequality

Z(h-;) ~ 47r,

(3.7.16)

and so I -; -; I ::; 1 implies

Iz(()1

::;

e E supp

II

Jdoc;,l)

II ",;,1 ( II

(47r)-1~1

e Esupp

~

~

Iexp((3J -; m~ . -; ~) - 11 n

(r;"r:)E~

::;

II ,1',1 ( II

(47r)-1~1

e E supp I(3J -; ~ II (r;"r:)E( m

::;

CII

eE supp (

e Esupp (

~

-; r:

JdoC', l)

exp((3J -; r;, -;r:)1

";'}PNUl'"~ (

(3.7.17)

For (3 sufficiently small, (3J + In((3J) is sufficiently negative for this bound to be the desired energy estimate, since card ( ~ ~I(I . The entropy estimate is simple for the kind of polymer defined here. If M is the given volume (= cardinality of support) for all polymers whose supports contain a fixed lattice site :;;, then such a polymer can have no more than 3M bonds. On the other hand, it follows from the solution of the Konigsberg bridge problem that a connected set of bonds can be covered by a path of bonds such that no bond occurs more than twice. Therefore, the number of polymers is bounded by the number of paths originating at :;; of length no greater than 6M. Since the number of all paths of r . bonds originating at a fixed site is bounded by 6r , we have the desired entropy bound - specifically, card{polymers ( :

:;;E supp (,1(1 = M} ::; 66M •

(3.7.18)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

242

Given these input estimates, the abstract theory of polymers guarantees the A-uniform convergence of the associated expansion of (3.7.7) in connected n-polymers for sufficiently small (3. We now consider the high-temperature expansion of the lattice ¢>~ model discussed in §3.5. For this model, the generalized partition function is

J (!= h-;¢>(i) + _~ ¢>(~)¢>(r;))

Z(h-: eE A)

exp

l

lEA X

]I

(m,n)CA

(ex p (_>.¢>(£)4 -

(T + ~m~

lEA

+

~£A(i)) ¢>(£)2) d¢>(£)),

where £A ( e) is the number of nearest neighbors of

e in

(3.7.19)

A. The inverse temperature

(3 does not appear explicitly unless we make the change of variable (3.5.9), but the expansion corresponding to (3.7.13) should certainly have the required activity estimate

without this scaling. We have

Z(h- : eE A)

(,

l

x

fl (J _II

"~Q. ,. Z(h,))

(~ :! ¢>(~t ¢>(r;t) _ II

x

II e Esupp

+ Z(h- ) l

=

(exp (_>.¢>(£)4 -

exp(h-;¢>( f.))

f Esupp <.

(m,n)E<.

(T + ~m~

<.

~£A(£)) ¢>(£)2) d¢>(£)) ),

i-roooo exp(h-w_>.w4-(T+~m~+~£A(£))w2)dw. 2 2 f

(3.7.20)

(3.7.21)

Since the variables are unbounded, it may appear that the expression (3.7.22)

is a problem, but the single-spin distribution for each variable assigns a small probability to large values. Moreover, large values of the parameter>' provide small values of the hidden expansion parameter. Following the example of (3.7.13)-(3.7.15), we

3.7. EXPANSION FOR NEAREST-NEIGHBOR INTERACTIONS

243

identify our polymer activity:

= _ II

z(()

Z(h )-l

i

J _IJ

l Esupp <

(exp(¢(n;,)¢(n:)) - 1)

(m,n)E<

II

II

l Esupp <

l Esupp <

x

(exp

(_)..¢(i)4 (3.7.23)

Since

we have

II

II

Iexp( ¢(n;, )¢(n:)) -11 :S

(1¢(e)I'A(i)exp

G~A(1)¢(1)2)) .

(3.7.25)

i Esupp <

On the other hand, Jensen's inequality implies

(3.7.26)

i:

so the activity estimate is

II

Iz()1 :S O'o().. , 7)-1<1

Iwl'A(i) exp (hiw - )..w 4 -

l EsupP <

= O'o().., 7)-1<1 _

II

()..- i-i,A(i)

i:

7W

2-

~m6w2) dw

Iw'I'A(i)

l Esupp < X

exp

()..-1 /4h iw

l -

W'4 -

)..-1/2 7W ,2 -

~m6)..-1/2wI2) dw

l

)

.

(3.7 .27)

For arbitrary constant c > 0, consider the regime 7

i:

(3.7.28)

:S c..J)..,

We have the reductions

O'o().., r)

= )..-1/4

exp ( -

W '4 -

)..-1/2

(~m6 + 3 + r)w '2 )dw l ~ c).. -1/4,

(3.7.29)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

244

so the activity estimate becomes

A

for the parameter restrictions. Since L A ( C )

4

> 1 for C E

therefore have

in this regime. It is now clear that we have the desired activity bound for suficiently large A, relative to the arbitrary constant in T 5 c 6 . To see how this regime can be interpreted as a high-temperature regime, we recall that the scaling (3.5.9) of integration variables induces the parameter transformations

with .i and

ifixed independently

of

P. Note that (3.7.28) is equivalent to

while large X is equivalent to small P for fixed choose c = ?/A.

A.

The size of .idoes not affect P if we

References 1. J. Glimm and A. Jaffe, Quantum Physics: A finctional Integral Point of View, Springer-Verlag, New York, 1987. 2. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.

3.8

Inductive Interpolation

Our next example of a polymer expansion is another analysis of the lattice C#J~ model with mass mo and coupling constant X that is similar in spirit to the expansion presented in the last section in the sense that it attempts to decouple the quadratic form

3.8. INDUCTIVE INTERPOLATION (-alat4, 4). Recall that the partition function is given by the integral

for every finite A C Z3. The generalized partition function is given by

The polymer expansion we develop in this section is based on inductive interpolation, which shall be the basis of the phase cell cluster expansion introduced in Chap. 5. In the statistical-mechanical context of this chapter, however, it is just a refinement. Indeed, for this particular model, it is superfluous. Its introduction at this level is useful because of its sophistication relative to the wholesale expansion discussed in the last section. Accordingly, we introduce a linear ordering of A and base an inductive expansion on it - an expansion where eachJerm reflects a history of interpolaticns. The first step is an attempt to decouple 4(n l ) from the other variables, where n is the first element of A. By the fundamental theorem of calculus

'

so this first interpolation yields A

z(hb:

e E A)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

246

This leading term is a decoupled term, while the other terms are remainder terms. The second step in the inductive expansion is applied to each term, but the interpolation now varies with the term. For the decoupled term we apply the same formula to Z (h-:

e

e

E A\ {r; 1}) , except ¢(r; 2) is now the variable we try to decouple from the

other variables, where r; 2 is the successor of r; 1 in A. For a given remainder term labeled by (r; , r; 1), we decompose the d¢>-integral with the interpolation that attempts to decouple both ¢(r;) and ¢(r; 1) from the other variables. In this way, a sum over possible sequences of interpolations is induced by an algorithm that dictates the next step for a given term based on the sequence of past interpolations that developed it. Consider an arbitrary term that has developed at an arbitrary stage in the expansion. Since each previous step was an interpolation followed by a choice of new term, a product ¢(T:t )¢(r;) associated with a nearest-neighbor bond appeared in an integral with each expansion step selecting a remainder term, so sets of variables have also developed . If the term under consideration happens to be a decoupled term, it has the form

where K-y is some complicated

n

d¢( e)-integral developed by the ')'th segment of

e EA, successive expansion steps. The sets A-y of lattice sites are mutually disjoint, and each A-y labels the variables differentiated down by the ~th succession of expansion steps. Given this

d~couPled term, the next step is to decompose Z(he :eE A\ YA-y)

exactly as Z (h-:

eE A) was, except that if r; , is the first lattice site in A\ U A-y with -y

l

respect to the linear ordering on A, then ¢(r; ') is now the random variable one tries to decouple from the others. Now suppose the term under consideration is a remainder term instead. Let {til denote the set of parameters for past interpolations. The term has the form

K.

II K-y(h e: eE A-y) J_II -y

¢(r;)N-;:

(II 11

nEA\UA,'

-y

0

dti) f(t)

3.8. INDUCTIVE INTERPOLATION

247

exp(~ L h-;¢(i)-Ilat(XA\yA~¢)+WA\yA~(¢jt)

x

l EA\UA~ ~

~~ L l

(m6

+

~A(i))¢(i)2) ~ II

EA \ U A~

d¢( £),

(3.8.6)

lEA \ U A~

~

~

where W A\ U A~ (¢j t) denotes the interpolation of ~

L

¢(~)¢C;;)

(3.8.7)

(~ , n:}CA\UA~ ~

that has developed and f(t) is the product of powers of the ti having grown as old t-dependencies have been differentiated down by new interpolations. In this case, the next step is to base the interpolation on the attempt to decouple the variables in A' from the rest of the variables in 11.\ Uk.p where A' includes those ¢(:;, ) that have been "(

differentiated down - i.e.,

(3.8.8) The interpolation formula yields the form

K

= Z(h-;: fEA\(A'UyA"())K'(h-; : fEA')I]K"((h-; : fEA"() +

II

Ky(h-; :

f E A"()

11

J¢(~ ')¢(~)

~ ~L

dt'

"(

(m',m}CA\UA~ ~

~'EA',~I/;A'

x

II

¢(:;')N-;:

n:EA'

xexp

11 (II 11 dt'

0

r~ L l

dti)

f(t)f~,~ (t)

,0

h-;¢(f)_I

1at

(XA\yA~¢) +t'WA\yA~(¢,t)

EA\ UA~ ~

+

(1 - t')(WA' (¢j t)

+ WA\(A'UUA~)(¢j t) - ~

L

(m6

~

l EA\UA~ ~

+ ~A( £))¢( £)2

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

248 x

IT

(3.8.9)

d¢( f).

,

iEA\UA,

Jr;., r;.(t) is just the product of the old interpolation parameters ti associated with

dec~upling those complementary pairs of sets that separated n;, , from

n;,.

This means (3.8.10)

Wi\' (¢; t) denotes the restriction of the sum to those nearest-neighbor pairs both of whose lattice sites lie in 11.', and

K'(h-; : eEli.')

=

J.F!. (TJ.l1 (_2: ¢(:;{)N-;:

h-;¢(i) -

x exp

dti) f(t)

[lat (Xi\' ¢)

+ Wi\'(¢;t)

iEi\'

(3 .8.11)

The next step in the development of a history is to pick either the decoupled term or a remainder term in (3.8.9). This completes our inductive definition of the expansion. Since A is a finite set, it is clear that every possible history of steps eventually terminates with a decoupled term of the form

K=

IT K')'(h-;: eE A,),),

(3.8.12)

')'

Such a decoupled term is a completed term. The A')' are disjoint, and the factorization over this partition of A is due to the history of choices. In the ,th segment of successive expansion steps, every step except the last one chooses a remainder term. The last step chooses the decoupled term, and so one starts all over again with the residual set of variables, initiating the next sequence of expansion steps. We have now expanded the original partition function in these completed terms, and our task is to control the expansion by organizing the terms and writing the K')' more explicitly for estimation. Let Xo be a distinguished element of an N-element set r. An ordered connectivity graph on (r, xo) is a mapping QJ from {2, . .. , N} into r x r such that with QJ(i) = (go (i), gJ( i)),

gl (i) =J: gl (j), i =J: j, go(i) E {xo} U {gl(j): j < i}.

(3.8.13) (3.8.14)

3.8. INDUCTIVE INTERPOLATION

249

The point here is that the ')'th sequence of expansion steps is naturally described by an ordered connectivity graph (1)1' on (AI" r; 1') where go(i) and g1 (i) are nearest-neighbors and r; I' is the lattice site that this succession of steps initially tries to decouple from other lattice sites. Every time a remainder term is chosen for an interpolation step, an additional product ¢,(gJ (i) )¢'(gi (i)) appears in the dcf>-integral, where ¢,(gJ (i)) is a variable that has already been "differentiated down" from the exponent and is among the variables the interpolation is attempting to decouple from the variables that appear only in the exponent thus far - among them, the variable ¢,(gi (i)), which is now "differentiated down" for the first time. This variable cannot occur as a ¢,(gi (i)) for any other i-only as a ¢,(gJ (i)) for later i. This ')'th history of expansion steps has the tree structure described by (3.8.13) and (3.8.14) above. The KI'-factor corresponding to this history in the completed term has the form

KI'(h-;: f

E

xexp

AI')

=

(If 11

dti) fC!37 (t) J

(~L h7¢,(f)-I

1at

If

(¢'(gJ (i))¢'(gi (i)))

(XA,¢,)+WA,(¢,;t)

tEA,

~ ~L

(m6

+ LA(f))¢'(f)2)

tEA,

)1

d¢,Cy)'

(3 .8.15)

tEA,

where the monomial fC!37 (t) is yet to be determined. It is important to note that a factor KI' can be an integral in just one variable. After all, the initial interpolation in a new sequence of expansion steps attempts to decouple the first of the remaining lattice variables from all the other variables. If that first step chooses the decoupled term, then the sequence is terminated already, and the corresponding o.c.g. (ordered connectivity graph) is empty. In this case, N = 1 and

KI'

Z(h~,) n]

=

Jexp(h~,¢'(r; 1') n

V(¢'(r; 1'))

1

-~(m6 + LA(r; I'))¢'(r; 1')2)d¢'(r; 1')

(3.8.16)

occurs with no interpolation parameters at all. The point is that the polymers we define below do not index such factors, and indeed, the single term in the whole expansion consisting of only those factors - i.e., the term developed by the series of decouplings that never "differentiates down" anything - must correspond to unity in the polymer expansion of an abstract partition function . As in the previous section, the abstract partition function in this case is actually the normalization

ZA

= Z(h 7 : f

E

A)

II Z(h7 )-1 tEA

of the generalized partition function .

(3.8.17)

250

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

4

•

6

•

•

10

8

• •

•

11

•

•

13

•

• Figure 3.8.1:

Now from the standpoint ofrealizing f(1j,(t) explicitly, it is clear from the inductive step described above that (if we drop the superscript ')') (3 .8.18) so it is a matter of examining the f m'm (t) introduced there. By way of illustration, suppose the ordered connectivity graph
II

(3.8.19)

i'-1<j
f(1j(t)

=

fdt)f13 (t)f17(t)!I,12 (t)h4 (t)hs(t) x 129 (t)!s6 (t) !s,1O (t)f4S (t)h,u (t) f11 ,13 (t)f11,14 (t) 1 X tl x (tlt2t3t4tS) X (tlt2t3t4tSt6t7tst9tlO) x t2 X(t2t3t4tSt6) X (t2t3t4tSt6t7) x (tst4) x (t3t4tSt6t7tS)

3.B. INDUCTIVE INTERPOLATION

251

xl X (t7tst9) x tu x (tut12) t1 3 t2 5 t3 6 t4 6 t 55 t6 4 t7 4 ts 3 tg 2tlO tu 2 t12 .

(3.8.20)

In general,

g1(i') where

7)(5

= go(i)

=> i'

= 7)(5 (i),

(3.8.21)

is defined by (3.8.22)

with the convention that g11(go(2)) = 1. We combine (3.8.18) with (3.8.19) to obtain the formula (3.8.23) and this completes our description of an interpolated factor in a completed term. The product of lattice field amplitudes that has developed in the dcf>-integral associated with a given ordered connectivity graph (!5 is just

II(¢(go (i) )¢(g1 (i))). Since every (gO(i),g1(i)) must appear as a nearest-neighbor pair for this model, and since a lattice site can occur no more than once as some g1 (i), notice that a given lattice site ~ can occur no more than seven times as some go(i) . With 0 ~ ti ~ 1, we have the trivial estimation (3.8.24) in the formula (3.8.10), so

L

1¢(i7i ')¢(i7i)1

(T;;! ,T;;)cA'

1

~

2 ~L

~

~A' ( e)¢( 1!)2

(3.8.25)

lEA'

Since

~A'

(I!)

~ ~A (I!),

JII

it follows that

(¢(go(i))¢(g1 (i))) exp

,

lEA'

L

1

+WA'(¢;t)--

~ ~II lEA'

(~L h7¢(e) - I1at(xA'¢)

i:

2~

l 'EA'

Iwl(card

~ ~

(m~+~A(e))¢(1! ')2 )

9,1(7))+1

II d¢(I!)

~ lEA'

exp (h7w - V(w) -

~m~w2) dw.

(3.8.26)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

252

The single-spin estimation is now done in the same manner as in the previous section. The combinatorial obstacle in the expansion is to sum over the ordered connectivity graphs. Once again, consider an N -element set r with a distinguished element Xo. An unordered connectivity graph on (r,xo) is simply the range of an ordered connectivity graph on (r, xo). In the combinatorics literature, the synonym for this term is "tree" not to be confused with "tree graph," already defined. An ordered connectivity graph may be regarded as a history of tree growth. Now to apply the abstract theory of polymer expansions, we specify our polymers to be unordered connectivity graphs on sets of lattice sites, where only nearest-neighbor bonds belong to it. The support of a polymer is just the set of lattice sites involved, so the volume is the number of these lattice sites. The entropy estimate is easy in this case. Indeed, if M is the given volume for all polymers whose supports contain a fixed lattice site r; 0, then such a polymer consists of M - 1 bonds. The number of these polymers is therefore bounded by the number of paths (consisting of nearest-neighbor bonds) of length ~ 2(M - 1) originating at r; o. It is obvious that a path can cover a tree of bonds without using any bond more than twice. On the other hand, the number of all paths of r bonds originating at r; 0 is bounded by 8T , so the bound (3.6.20) is established in this case. How do we realize ZII. as an expansion in activities of these polymers? The expansion we have developed is actually a sum over all finite sequences ((!)1, . .. , ~n), n = 1,2,3, . . ., of ordered connectivity graphs on disjoint pointed sets (A-y, r; -Y) such that: (a) (b)

r; -y+1 succeeds r; -y with respect to the linear ordering of A, r; -y

is the first element in A-y with respect to that ordering. n

K-y(h-: eE A-y), where -y=1 e each K-y-factor is entirely determined by the corresponding history of interpolation steps described by ~-y . In particular, there is a mapping

IT

With each such sequence is associated a completed term

~>-+K~(he: eEA~)

from ordered connectivity graphs to the interpolated factors that occur in completed terms. Now for an unorr!:!Ired conn~ctivity graph T consisting of nearest-neighbor bonds on the pointed set (A', n ') with n ' the first site occurring in A', specify the activity of this polymer T to be z(T)

K~(h-I : eE A')

= ordered connectivity graphs ~ on (11.,';') T",=T

where

T~

is the range

of~.

II

Z(h_)-l,

e EA'

e

(3.8.27)

Thus, we have the expansion n

II z(T

i ),

n=O (T, ,... ,Tn)EV~ i=l

(3.8.28)

3.8. INDUCTIVE INTERPOLATION

253

where V~ denotes the set of all n-tuples (T1' .. . ,Tn) of mutually disjoint polymers such that the first element in supp Ti+1 succeeds the first element in supp T i . If Vn denotes the set of disjoint n-tuples of polymers without the order condition, then ~

ZA

00

=L n=O

1

,

n.

n

L

II z(T

(3.8.29)

i ),

(T, •... •Tn}E"Dn i=l

so control of the induced expansion of

IAI- 1 In ZA = IAI- 1 (In Z(h~: t

e A) - '" E

~

In Z(h~)) t

(3.8.30)

tEA

in connected n-tuples of polymers is guaranteed by the general theory in the range of parameters for which we have an activity estimate (3 .8.31) with Co sufficiently large. We infer the desired bound from (3 .7.26), (3.7.29), (3.8.26), and a fundamental combinatorial result . For every unordered connectivity graph T on a pointed set (A', ~ ') , (3.8.32) ordered connectivity graphs I!l on (A'.;')

We prove this identity in §3.11.

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field

Theories," Ann. Phys. 142 (1982),95-139. 2. D. Brydges and P. Federbush, "A New Form of the Mayer Expansion in Classical Statistical Mechanics," J. Math. Phys. 19 (1978),2064- 2067. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360. 4. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987. 5. R. Seneor, "Critical Phenomena in Three Dimensions," in Scaling and SelfSimilarity in Physics , J. Frohlich, ed., Birkhiiuser, Boston, 1983. 6. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.

254

3.9

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

A Generalization of the Polymer Estimation

We now recall the infinite-volume problem brought up in §3.1 (for high temperature). Returning to the lattice system whose single-spin distribution is Lebesgue measure da on §2, we consider the long-range Hamiltonian in a zero external magnetic field given by (3.9.1) 'L" J~n-n' ~ s ~n . s ~n t ) where in this section we assume only the decay condition (3.9.2) which is just summability. This spin model is treated differently from the classical Heisenberg model, as the theory of polymer expansions presented in §3.6 must be generalized in this context, and the Konigsberg bridge-island estimation is too crude as well. The algebraic structure of the expansion is the same, however. One expands Z(h~: iEA)

(3.9.3)

e

IT

Z(h~)

~

e EA

Z(h~ :

e

iEA)

e

J (.?= he· se exp

eEA

x

" -(3 '~

J~n-n' ~ s~n

n,n'EA

II da(se)

(3.9.4)

e EA in all couplings. We have already learned from the nearest-neighbor case that a straight power series in may converge for small but from the point of view of polymers it is more natural to recombine terms into an expansion of the form (3.7.15). Accordingly,

(3

exp

(-(3 '" L.. n,n'EA

(3,

J~ ~, s~ · s~,) = n-n

n

n

'" L...J

II [exp(-(3J~ ~ s~· s~) -1] n-n'

n

n'

rcAxA (n,n')Ef

(3.9.5) is the identity we use in this case. Clearly, our polymers will be subsets of Ax A that connect the lattice sites in their own supports, but we already have a problem with the entropy estimate. If we define the volume of a polymer to be the cardinality of the set of lattice sites supporting it, then - since arbitrarily distant lattice sites in A are directly coupled by a long-range interaction - the bound on the number of polymers with fixed volume and having a fixed lattice site in their supports depends on A and diverges as A /' Z3. An alternative that might seem tempting is to define the volume of a polymer to be the sum of iI-lengths of the line segments associated with the ordered pairs of lattice

255

3.9. A GENERALIZATION OF THE POLYMER ESTIMATION sites. The iI-length of a line segment {n;.,~} is defined to be ~ 1m,

,

- n,l, where the

motivation here is to give integer values to the p0!rmer volumes. With this understanding, we fix an integer L > 0 and a lattice site n E ;;Z3 and try to count all polymers with volume L and ~ in their supports - i.e., all connected sets of line segments having these constraints. By the Konigsberg bridge-island solution, every connected set of line segments has a path consisting of precisely those line segments, originating at any chosen point, and traversing each line segment no more than twice. Therefore, it is enough to bound the number of all possible paths of line segments originating at ~ and having total length L with L :S L :S 2L. Now classify such paths according to the number N of line segments they consist of and the sequence (Ll" ' " L N) of lengths according to the sequence of line segments. With Nand (L 1 , . • . , LN) fixed, it is clear that the number of such paths originating at ~ is bounded by (3.9.6) On the other hand, we know that the number of sequences (Ll" ' " LN) of positive N

A

•

integers with ~ Lk = L is bounded by e L -

1

Summing over all such sequences and

k=1

over all N, we bound the number of paths originating at ~ with length

L00 eL-lc~ ( ~ A

N=1

rN ,L-, (~ ':1' un' Nr

L by the sum

:S

:S :S

eL - 1

(00 ~ (L';;) A

exp(2LJCO + L - 1).

(3.9.7)

In this way, we obtain the desired entropy estimate. Can we balance this with an energy estimate in terms of the polymer volume defined here? Clearly, the activity of a polymer ( is given by

z(()

=

1

IT l Esupp (

x

Z(h 7

(7 II

J II )

[exp(-,lJJr;_r;,

s~

n

8'r;,)-1]

{r;,r;'}E(

~. ~) II

h -;

ell

7E

E supp (

dU

(~87 ) '

(3.9.8)

supp (

so by (3.7.16) we have the bound

Iz(() I:S

II {r;,n' }E(

(,BIJr;_r;, leiJlJn" -n"' I) .

(3.9.9)

256

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

This is inadequate because the number of factors can be much smaller than the volume of ( - i.e., the number of line segments has nothing to do with the sum of their [1_ lengths because the line segments can be arbitrarily long. One can still extract a bound that has exponential decay in the volume of (if the pair interaction J ri _ ri , has

sufficiently fast exponential decay in I ~ - ~ 'I, but such an interaction is essentially "nearest-neighbor" in character - a property much too restrictive. These two failed alternatives hardly represent a crisis, however. After all, hightemperature expansions for long-range interactions were developed long before the abstract polymer formalism was around. Moreover, the latter has a trivial generalization that is ideal for this high-temperature problem. Accordingly, we return to defining the volume of a polymer as the cardinality of its support and re-examine the abstract theory. Instead of insisting on both an energy estimate and an entropy estimate, we note that it is sufficient to require only the estimate (3.9.10) (: <>E supp (

I(I=M

for

C2

> 0 large enough . Recall that the estimation of the inner-most sum n

II Iz((v)1

2:

P=( l •. ·· .(n )EiC n 1'=1 T is a subgraph of G p

I(vl=mv

in (3.6.35) was done by combining the energy estimate with the entropy estimate. The resulting bound was CoC~e

(cl-co)L;m v v

II m~~(T)-l , n

(card S)

1'=1

where Co and Co (resp. C1 and C1) were given in the energy estimate (3 .6.19) (resp. the entropy estimate (3.6.20)) . On the other hand, the estimate (3.6.37) was necessary only because (3 .6.36) estimated the product of activities right out of the summation. If we apply (3.9 .10) to each sum of activities over polymers (v such that (tv E supp (v, the bound on the multiple sum simply becomes

C;e

- C2

Em" v

(card S)

II m~~(T)-l, n

1'=1

which is all that is needed for the convergence theory. In the present context, one has to establish (3.9.10) by using (3.9.9), but the Konigsberg estimation applied in the nearest-neighbor context is no longer sharp enough. Indeed, consider a fixed lattice site ~ 0 and a polymer ( containing ~ 0 with 1(1 = card supp ( = M. Once again, by the Konigsberg bridge-island solution, there is a path o!jine segments consisting of precisely those line segments that are in ( , originating at n 0, and traversing no element of ~ more than twice. Let 7r( denote such a path

3.10. UNIT-VECTOR SPINS WITH LONG-RANGE COUPLING

257

for each (. Then Iz(() 1:S 13 M (: ;oE supp (

(: ;oE supp (

I(I=M

I(I=M

(3.9.11) where D,,< is the number of line segments that have been traversed twice and we have applied 2 (3.9.12) IJ;;_;;, I :S cIJ;;_;;, 11 / to those line segments f~, ~

'}

that have not been traversed twice. Thus M'

L II (IJ;;i_;;i_lI1/2e~IJ;;;i-;;;i-ll)

(3.9.13)

M'=M m ---"I , ... ,m -->OM'i=l

(: ;oE supp (

I( I=M

because the lengths of the sequences 1l'( range from M to 2M. The idea is to estimate the multiple sum from the inside on out, applying a summability condition when summing over for fixed but this iteration depends on the summability condition.

ni

ni-1,

(3.9.14) which is stronger than condition (3.9.2). One can still establish convergence of this polymer expansion without this restriction on the long-range interaction. Indeed, the required combinatorics is standard and predates the polymer formalism. We can label these line-segment polymers with connected graphs and control the multiple summation with those subgraphs which are tree graphs. This control is quite similar to the key estimation for convergence of the abstract polymer expansion of the abstract free energy in §3.6. We shall not pursue this, however. Instead, we devote the next section to the polymer expansion based on applying inductive interpolation to this long-range model. The point of this alternative is that the tree structure is already there in an individual polymer.

References 1. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View,

Springer-Verlag, New York, 1987. 2. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.

3.10

Unit-Vector Spins with Long-Range Coupling

We continue our analysis of the lattice system whose single-spin distribution is Lebesgue measure da- on §2 and whose Hamiltonian in a finite region A C Z3 is given by

HA =

L

~

~

J-n-m - s -n . sm

(3.10.1)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

258

We now describe an expansion to control the A = Z 3 limit of the expectations for high temperature, assuming only that

J o = o.

(3.10.2)

This condition is the weaker of the two conditions mentioned in the previous section. The stronger condition seemed to be called for, but there are various ways to remedy this. Here we generate the products of exponential differences in a more controlled way - using inductive interpolation, as we did in §3.8 - only now we use the more general polymer estimation. Introduce a linear ordering of A and attempt to decouple -; -, from all other spin ---lo.

n

.....:..I.

variables in the integral expression for Z ( h - : eE A) by interpolation, where ri: f the first lattice site in that ordering. We insert the identity exp

(-(3'" J-

n-m

~

-; -n

1

is

-;-) m

n,m

exp

(-(3 '"

J-n-m - -; -n

~

-;-) m

; ,n;:#;; 1

+11o dt1(-(3 x exp

(-t

'"

J-n-m - -;_. -;-) L-, n m ;=;1 or ;;=;1

" J-n-m - -; -n . -; m - -(1 1(3 'L-,

tt}(3

n,m

X

J-n-m - -; -n . -;-) m

' " L...J

(3.10.3)

n,J;#nl

in the integral to obtain ~

Z(h

- (3

~

X

1

or

J-n-m -

T1i=nl

11

---lo.-lo.

= Z(h n,)Z(h

'" ;=n

X

.....),.

7 : eE A)

7 : eE A\{ri: 1})

J-;-.-;n

m

" J-n-m - -; -n . -; m - -(1 - t1)(3 '~ " J-n-m _ -; _ . -;_) o dt1 exp (-t 1(3 '~ n m

II du(-; 7)'

n,m

n,ni#nl

(3.10.4)

f EA

We use the same terminology as before: the factorized term is called a decoupled term, while the other terms are called remainder terms . The second stage in the development

3.10. UNIT-VECTOR SPINS WITH LONG-RANGE COUPLING

259

of the expansion is applied to each term, with the variables we attempt to decouple dictated ~ th:."term. In the decoupled term, the same interpolation is applied to the

eE A\{ri: I}), but -;n is the variable to be decoupled from the other variables, where ri: 2 is the first element in A\ {ri: I}. For a given remainder term labeled by (ri:, n;,) with either ri: =ri: 1 or n;,= ri: 1 - the interpolation is based on the

factor Z(h-;=

2

attempt to decouple both -; ~n and -; m~ from the other variables. The iteration of this procedure develops a sum over histories of interpolations, with each interpolation branching out that development in many possible directions for every partial history previously developed. We still refer to a step as an interpolation followed by a choice of term, so every term that has developed at a given stage is labeled by a history of previous steps. The nature of the next interpolation depends on the term. If a given term is a decoupled term, it has the form

where K'Y is an integral in only those spin variables -; ~ for which eE A'Y' As in §3.8, l K'Y is called a completed factor, and it is developed by the 'Yth succession of steps, where "succession" has the same meaning here as it did there. For such a term, the next interpolation is applied to the

fact~ Z ('h e:

the same manner as it is applied to Z ( h ~ : l

eE A\ YA'Y) only, and in exactly

eE A)

- as an attempt to decouple the

spin -;~, in the integral from the other spins -; ~, where n n

ri: ' is the first

lattice site in

the given set of sites ri: with respect to the linear ordering. On the other hand, if the given term happens to be a remainder term, then it has the form

fft>(t)j~!1

(-f3 J :;;_r;.-;:;;'-;r;.)

(n ,m)EU.,

h~

x

l

s~

l

Let A' denote that set our previous interpolation was trying to decouple from the other lattice sites in A \ UAT This remainder term represents a failure to decouple,

'Y

and it contains a factor J~n-n ~, -; ~n

-;~, in the integrand where n

ri: '

~ A'.
ordered connectivity graph supported by A' and this additional lattice site

ri:

f,

while

260

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

H~\ U A, (t) is the interpolated Hamiltonian that has developed so far.

,

fQ5 (t)

is the

product of interpolation parameters determined by Q) as in §3.8, although Q) represents a partial history of steps at this stage. The K-y are still called completed factors, and the next interpolation is applied to the integral described here. It is based on trying to decouple {ri: '} u A' from the other lattice sites and generates new terms - new remainder terms, and one decoupled term which has the form

Z(h7: eE A\((AIU{ri:I})UYA-y)) K'(h7: fEAIU{ri:'}) x

II K-y (h7:

e

E A-y ).

"I

Every branch in this inductive process eventually terminates. Although the model to be analyzed here is somewhat different from the model considered in §3.8, th:.,.com~inatorial structure of the expansion is exactly the same. We have expanded Z (h -: f E A) in terms of the form

e

II KQ5,(h 7 : eE A-y),

Q) -Y

an o.c.g. on (A-y, ri:

"I),

"I

where ri:

"I

is the first lattice site in A-y, A =

UA-y, the A-y

are disjoint, and

"I

KQ5,(h 7 : eEA-y) x

=

(r:rl1dti)fQ5,(t)

J__II

(n,n'lEU",

(3.10.5)

For a given o.c.g. by

Q)

on a pointed set (A', f '), the interpolated Hamiltonian is given

Sn;, .

'-

~~~,

. {-1(~) -1 ~ = mill g2 m, g2 (m

I

}

) ,

s~"

(3 .10.6)

(3.10.7-) (3.10.7+)

The only difference between the ordered connectivity graphs here and those considered before is that the ordered pairs of lattice sites are no longer nearest-neighbor bonds.

3.10. UNIT-VECTOR SPINS WITH LONG-RANGE COUPLING

261

The summability property of the coupling Jr;_r;, must be exploited in two ways - to control the expansion itself and to bound the interpolated Hamiltonian from below. The latter estimation is obvious: (3.10.8) because the spins are unit vectors. Thus IKG(h·t f E A')I

Z(h e) ~

II

< cf31A'1

.II

(II 11 dti) f(!)(t) .

INr;_r;,1

(n,n')EU"

lEA'

(3 .10.9)

'

The formalism for polymers is actually the expansion of ~~

~

Z(h~ : fE

A) ==

l

Z(h~:fEA) l ~ ,

(3.10.10)

n Z(h~)

~

l

lEA

and - as in §3.8 - a polymer is defined as an unordered

c~nectivity

graph for the

expansion we have developed. For a given u.c.g. U on (A', f '), the activity of U is given by

z(U) ==

(3.10.11) u.c .g .

(!)

on (A',e')

lEA'

U,,=U

and we may group the terms of our expansion into the form (3.8.29). Given (3.10.9) we obtain a bound on the activity:

Iz(U) 1 :S cPlA' l

II

cPlA'l

II

o .c.g.

(I;I 11

L

I,6J-;:;_-;:;, 1

(r;,r;')EU

(!)

dti) f(!)(t)

on (A',e')

U,,=U

I,6Jr;_r;, I,

(3.10.12)

(-;:;,-;:;')EU

where we have applied the u.c.g. identity (3.8.32) . Our point of departure from §3.8 is that we now have to apply generalized polymer estimation , as a standard activity bound is no longer enough for the number of polymers we are dealing with. Accordingly, we observe that

II u.c.g . U: ni,E sUPP U card sUPP U=M

u.c .g . U: ni, Esupp card sUPP U=M

U (r;,r;')EU

Jr;_r;, 1 (3.10.13)

1

CHAPTER 3. EQUILIBRlUM STATES OF CLASSICAL CRYSTALS

262

for fixed n;,l E A. On the other hand, we may write the summation as 1

(3.10.14)

(M - I)!

u.c .g. U: ~lEsuPP U card supp U=M

u .c .g . U

supp U={l , .. .,M}

~2 , ... ,r;;.MEA

rr'i 1 ,rr'i 2 J... ,ni M distinct

and since J~n-n' ~ = J~ ~ the terms are the same for those U having the same tree n ' -n' graph. The tree graph associated with U is given by (3 .10.15)

Tu={{i,j}: (i,j) EU} ,

and for a given tree graph T on {I, . . . , M} there are exactly M unordered connectivity graphs U such that Tu = T , because the directedness of the line segments in a u.c.g. are determined by the point of origin only. Thus c/3 M

L

Iz(U)1 ~ (M _ I)! (2f3)M-l

u .c. g. U: ~lEsupp U card supp U=M

L

x

L

tree graphs T

on {l , ... ,M}

II

IJ~i _~; I,

(3.10.16)

~2 ' .. . '~MEA {i,nET

where we have also removed the summation restriction that n;,1,n;,2 , . . . ,n;,M be distinct. We now have the familiar situation where the tree graph controls the multiple sum. The rule at each stage of the summation is to sum out those n;,i for which i is connected to only one j, and this enables us to use (3.10.17)

to obtain

L

Iz(U)1

u.c.g . U: ~lE supp U card supp U=M

<

C/3 M

(M _ I)! (2f3)M-l cM-l card{tree graphs T on {I, . . . , M}}. (3 .10.18)

By Cayley's Theorem, the number of such tree graphs is M bound of the generalized type (3 .9.10) if f3 is small enough.

M -2,

so now we have a

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann . Phys. 142 (1982) , 95-139.

2. D. Brydges and P. Federbush, "A New Form of the Mayer Expansion in Classical Statistical Mechanics," J. Math . Phys. 19 (1978), 2064-2067.

3.11. COMBINATORIAL PROPERTIES OF INTERPOLATION WEIGHTS

263

3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360. 4. J . Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987. 5. R. Seneor, "Critical Phenomena in Three Dimensions," in Scaling and SelfSimilarity in Physics, J. Frohlich, ed., Birkhiiuser, Boston, 1983. 6. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.

3.11

Combinatorial Properties of Interpolation Weights

We have already defined a number of graph-theoretic objects, and it is appropriate at this point to collect the purely combinatoric results that are relevant to inductive interpolation. We begin by proving the u.c.g. identity (3.8.32), which we have applied twice already. Let U be a fixed unordered connectivity graph on a finite set with distinguished point Xo and associate to each (p, q) E U a complex variable Wpq. The idea is to apply :nductive interpolation to the function

L

exp

Wpq

(p,q)EU

based on attempts to eliminate those variables Wpq for which p is "old" and q is "new". In that sense, we first try to "decouple" Xo from the other points and obtain

L

exp

Wpq

= exp

(p,q)EU

+

L

L

Wpq

(p,q)EU popxo

W xOq'

(xo ,q')EU

11 dt1

exp

(t1 L

WXOq

+

(xo,q)EU

0

L

Wpq) ,

(3.11.1)

(p,q)EU popXo

where the first step is to pick one of these terms. The first term is the "decoupled" term, and a term labeled by (xo, q') is a "remainder" term. If the first step chooses the decoupled term, then we are "done" . If the remainder term for a given q' is chosen, then the second interpolation is based on trying to decouple {xo, q'} from the other points. Thus

exp

(h

L (xo,q)EU

WXOq

+

L (p ,q)EU popxo

wpq)

= exp

(t1

w xOq'

+

L (p,q)EU popq' ,XO

wpq)

264

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

VXOq,(W;tl,t2)=tlWXOq'+

L

Wpq

(p,q)EU popq'

+

t2 (1 L

WxOq

L

+

(xo,q)EU qopq' ,Xo

(3.11.3)

wq,j,

(q' ,q)EU

where the first term is - once again - the "decoupled" term, and there are now two types of remainder terms. The second step is to choose a term in (3.11.2), and the decoupled term is regarded as a completed history. If either type of remainder term associated with a given q' is chosen, then the third interpolation attempts to decouple {xo, q', q"} from the other points. More generally, at any given stage in a history of steps, the set that the next interpolation tries to decouple from its complement is the set of all points met by the variables Wpq that have already been "differentiated down" from the exponent. Clearly, the next step either completes a history by choosing a decoupled term or introduces a new point to that set by choosing a remainder term. Therefore, every branch of the expansion eventually terminates because supp U is a finite set. Once the expansion is developed, the only remainder terms are exactly those terms where all of the variables Wpq appear. A decoupled term will always have at least one variable missing from the exponent that has not been previously "differentiated down" from the exponent. Hence exp

L

IT

Wpq

(p',q')EU

(p,q)EU

IT

(p' ,q')EU

fJ -fJ-- exp Wp'q'

fJ:" P q

L

Wpq

(p ,q)EU

sum of all ) remainder , ( terms

(3.11.4)

and these remainder terms are labeled by those ordered connectivity graphs (!; for which UG = U. For a given o.c.g. (!; representing a history of attempts to decouple points, the differentiation with respect to ti-l associated with the (i - l)st interpolation brings down the product i-2 ZG(i)

IT

tk,

(3.11.5)

k=1/.,(i)

from the exponent. Therefore, we may write the sum of all remainder terms as

3.11. COMBINATORIAL PROPERTIES OF INTERPOLATION WEIGHTS

265

where VQ; (w; t) is the final interpolated form of the exponent for the history represented by <5 , and (3.11.6) On the other hand,

II

exp VQ;(w; t) =

(1

+ O(wpq)),

(3 .11.7)

(P,q)EU

so the only contribution having only one power of each Wpq is

and therefore the constant contribution to the r.h.s. of (3 .11.4) is just

Since the constant contribution to the l.h.s. is unity, we have the desired identity (3.8.32). There are other combinatorial identities , which are based on the notion of ordered tree graph. An Nth-degree ordered tree graph is defined to be any mapping 'f/: {2, .. . , N} -+ {I, . .. , N - I} such that 'f/(i) < i. Clearly, for an o.c .g. <5, 'f/Q; is an ordered tree graph, and the point is that the quantity fQ; (t) depends only on 'f/Q; - i.e., f1/(t) is well-defined:

f1/(t)

= II ( i

IT

(3.11.8)

tk)'

k=1/(i)

There is an ordered tree graph identity known as the Robinson identity:

L

Nth·degree 1/

N (1)

(

II in i=l

dti

NN-1

f1/(t) =

~'

(3.11.9)

0

and the u.c.g. identity may be used to prove it . To this end, we first consider the effect of permutations of underlying sets on ordered and unordered connectivity graphs. Let (f, xo) be a pointed set with cardinality N and consider an arbitrary o.c.g. <5 on (f, xo) . Let h be a permutation of f with h(xo) = Xo . Clearly, (h x h) 0 <5 is an o.c.g. on (f, xo) and U(h Xh)oQ; 'f/(hXh)oQ;(i)

(h x h)(UQ;), (h

0

(3.11.10)

gl)-l(h(go(i)))

g11(go(i)) 'f/Q;(i)

(3.11.11)

266

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

Indeed, if (5 and (5' are any two ordered connectivity graphs on the equivalence:

T]e;' = T]e;

¢}

(5' = (h x h)

0

(r, xo),

(5 for some permutation h of (r, xo).

then we have (3.11.12)

An automorphism of an unordered connectivity graph U on (r,xo) is a permutation h of r such that h(xo) = Xo and (3.11.13) (h x h)(U) = U.

In the case Ue; = Ue;, (3.11.12) becomes: T]e;'

= T]e;

¢}

(5'

= (h

x h)

0

(5 for some automorphism h.

(3.11.14)

Finally, if U and U' are unordered connectivity graphs on (r, xo), we say that U if and only if there is a permutation h of r with h(xo) = Xo and

U'

= (h

x h)(U) .

~

U'

(3.11.15)

Let v(U) denote the number of U' ~ U and O'.(U) the number of automorphisms of U (including the identity). Since there is a one-to-one correspondence between the U' ~ U and the left cosets of the automorphism group of U regarded as a subgroup of the group of permutations h of r with h(xo) = Xo, we have

O'.(U)v(U)

= (N -

1)1

(3.11.16)

We are now ready to prove the Robinson identity. For a given unordered connectivity graph U on (r, xo), we first decompose the sum in the u.c.g. identity (3.8.32) as follows:

ryE~U) e;:~=ry (IJ 11 dti) fe;(t) = 1,

(3.11.17)

U,,=U

N(U)

= {T]e;:

Ue;

= U}.

(3.11.18)

Since fe;(t) = fry" (t) , the identity reduces to:

L

card{(5: T]e;

= T],Ue; = U}

ryEN(U)

(II 11 i

dti) fry(t)

= 1,

(3.11.19)

0

but (3.11.14) implies that the set cardinality is precisely O'.(U). If we combine this observation with (3.11.16), we obtain:

v(U) L (II• 10r1) dti fry(t) = (N - 1)1'

ryEN(U)

(3.11.20)

0

Now (3.11.12) implies U' ~ U ¢} N(U') = N(U), so choose a representative u.c.g. from each equivalence class defined by ~ and index this selection: U1 , • •• , Um. Since every

3.11. COMBINATORIAL PROPERTIES O F INTERPOLATION WEIGHTS

267

Nth-degree ordered tree graph q can be realized as qa for some 0.c.g. 6 on (r,xo), it follows that the disjoint N(Ul), . . . ,N(Um) comprise the set of all Nth-degree ordered tree graphs. Thus

-

-

On the other hand, the disjoint equivalence classes {U Ul), . . . , {U Urn) comprise the set of all unordered connectivity graphs on (r,xo). These objects are actually trees whose line segments are directed, and the directedness is determined by regarding xo as the point of origin. It follows from Cayley's Theorem that

and so (3.11.9) is finally verified. It is interesting that in statistical mechanics this sum of interpolation weights arose in expansions long before the Robinson identity was known. Instead, the inequality

was verified and used. This bound now follows from the identity. This is not the end of the story. If q is an Nth-degree ordered tree graph, define the weight function dq such that dq(j) 1 is the coordination number at j for the tree graph associated with q - i.e., define

+

dq(j) = card{i: q(i) = j).

(3.11.24)

Then we have the Speer identity:

and an immediate consequence is the bound

The introduction of the factorial growths associated with coordination numbers of a tree graph do n o t change the exponential character of the bound! The key to the Speer identity is the interpolation weight identification

268

CHAPTER 3. EQUILillRIUM STATES OF CLASSICAL CRYSTALS

where 71"(7]) is the number of permutations p of {2, ... , N} such that 7] 0 p is still an ordered tree graph. To prove this identity, we first calculate the interpolation weight directly. If we set card{i: 7](i) :S k:S i - 2}

= w1/(k),

(3.11.28)

then (3.11.8) may be written as N-l

f1/(t)

II t~"(k),

=

(3.11.29)

k=l

from which we obtain (3 .11.30) Therefore, the problem is to show that N-l

71"(7])

=

II (w1/(k) + 1),

(3.11.31)

k=l

and the best approach is to count the possibilities at each step as we build a permutation p such that 7] 0 p is an ordered tree graph. We may determine the values p(2), . .. , p(N) in order, where the only constraint is that 7](P(i)) < i. The candidates for p(2) clearly comprise the set of i' such that 7](i') = 1. More generally, the candidates for p(i) are those i' f= p(2), . . . ,p(i - 1) such that 7](i') < i . Note that for j < i , the requirement 7](p(j)) < j < i governing the previous selections implies {p(2), . . . ,p(i - I)} C {i': 7](i') < i},

(3.11.32)

so there are precisely card{i': 7](i') < i} - (i - 2) = card{i': i'

> i-I and 7](i') < i}

(3.11.33)

candidates for p(i) . The latter set has the same cardinality as the set of candidates because {2, . .. ,i - I} C {i' : 7](i') < i} . (3.11.34) On the other hand , {i': 7](i') < i < i' + I} {i' : 7](i') :S i -1 :S i' -I},

{i' : i' > i-I and 7](i') < i}

(3 .11.35)

so it follows from (3.11.28) that the number of candidates for p(i) is w1/(i - 1) Therefore, the number of possible p is just N

II (w1/(i i=2

1)

+ 1),

+ 1.

3.11. COMBINATORIAL PROPERTIES OF INTERPOLATION WEIGHTS

269

and this completes the proof of (3.11.27). The Speer identity now develops as follows. For any two ordered tree graphs TJ and TJ', we can find a permutation p of {2, . . . , N} with TJ' = TJOp if and only if d7)' (j) = d7)(j) for all j. Pick a representative TJ from each of these equivalence classes and index these ordered tree graphs: TJI, . .. , TJM. Then

N,"~'~' OI d'(il') (fJ l' dt.) I,(ti ~ OI .,,(il) "E,.. (fJ l' "t.) I,(ti ,

(3.11.36)

On the other hand, the permutations p of {2, .. . , N} for which TJ 0 p = TJ are precisely those p that permute the sets TJ- I ({j}) independently, so the number of such N-I

permutations is

IT

d7)(j)! Hence

j=l

.D

N_I

1l'(TJ)

(

)

card {TJ': TJ'

d7) (j)!

= TJ 0 p for some permutation p} (3.11.37)

Combining this with the interpolation weight identity (3.11 .27), we reduce (3.11.36) to:

1

M

L L

k=I7): d,=d'k

card{TJ' : d7)'

= d7)k}

M

L1 k=l

(3.11.38)

M,

so the dependence of M on N is the answer. M enumerates the equivalence classes defined by the equations d7) (j) = d7)' (j) , so M is the cardinality of the set WN

= {(nl,. ' " nN-I):

nj

= d7)(j)

for some ordered tree graph TJ} ·

(3.11.39)

On the other hand, integer sequences (nl, . . ' ,nN-I) arising from ordered tree graphs are characterized by the constraints nj ~ 0 and k

k ~ Lnj ~ N-1. j=l

(3.11.40)

270

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

Given this alternative definition of WN, it is a standard combinatoric fact that M

1(2NN _-12) '

(3 .11.41)

= cardWN = N

and so (3.11.25) is proven. We conclude our discussion of pure combinatorics by stating the Anghel-Tataru identity without proof. For complex variables WI, ... , W N,

p

permU~{I,

... 'N}

Nth-~ree ( 1/

N

}] W

p

1) !! 10

)(N (1/(i))

dti

f1/(t)

(N )N-2 = ~ Wi

p(I)=I

(3.11.42) This result is a master identity for interpolation weights. It reduces to the Robinson identity if we set Wi = 1 for all i, while we can obtain the Speer identity by applying the integration

to this equation.

References 1. V. Anghel and L. Tataru, "New Identities for Cluster Expansions," Lett. Math. Phys., to appear. 2. G. Battle, "A New Combinatoric Identity for Cluster Expansions," Commun. Math. Phys. 94 (1984), 133-139. 3. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 4. G. Battle and P. Federbush, "A Note on Cluster Expansions, Tree Graph Identities, Extra liN! Factors!!!" Lett. Math. Phys. 8 (1984), 55-57. 5. D. Brydges and P. Federbush, "A New Form of the Mayer Expansion in Classical Statistical Mechanics," J. Math. Phys. 19 (1978), 2064-2067. 6. F. Harary and E. Palmer, Graphical Enumeration, Academic Press, New York, 1973. 7. E. Speer, "Combinatoric Identities for Cluster Expansions," Lett. Math. Phys., to appear.

3.12. SCALAR SPINS WITH LONG RANGE COUPLING

3.12

271

Scalar Spins with Long Range Coupling

We return to the case of scalar spins with assumptions defining a rather large class of models. Recall that the generalized Ising model associated with the Euclidean lattice r/lj theory (i.e., the Ginzburg-Landau model) had only nearest-neighbor couplings. We now complicate this unbounded spin case with the introduction of long-range couplings as we have already done in the §2- valued spin case - and consider once again the wholesale expansion in couplings. Let

HA

=

~ ~

J~ s~s~ m-n m n'

(3 .12.1 )

r;;.r;EA

where Jr;, -n satisfies the decay condition (3 .12.2)

This condition is stronger than condition (3.9.14), let alone condition (3.9.2), but we reserve refinements for the next section. The non-inductive expansion together with the Konigsberg estimation is the most convenient context in which to describe how the extra long-distance decay can be used to deal with the large spin values whose uncoupled probabilities are quenched by a single-spin distribution da(z)

= e-U( z)dz

(3.12.3)

with U(z) a continuous function such that

U(z) 2: f z 2

-

c,

f

> O.

(3.12.4)

The Ginzburg-Landau model meets this single-spin requirement for arbitrary,..,. The transcription of the non-inductive expansion formula (3.9.5) to the present model is obvious. The polymers themselves are exactly the same - subsets of Ax A whose supports are connected by their elements. We have

exp (-fJ

~~

Jr;;_r;sr;;sr;)

n ,mEA

L II

{exp(-fJJr;;_r;sr;;sr;) -I}

rcAxA (r; .r;;)H 00

1

Ln!

n=O

l

L

<, .....
E'P mutually disjoint

IT ~ II ,=1

(n .m)EC

{exp(-fJJr;;_r;sr;;sr;) -

l}l '

(3.12.5)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

272

and the

n J da(s~ )-integral of the product over i

~

is the product over i of the

l

lEA

n e Esupp

J da(s~e )-integrals. This yields the polymer expansion of (i

fE A) II Z {l} ~ (h~)-1, e

Z(h~: e

Z(h~ :

e

(3.12.6)

~

J (~ h~s~ 13 ~ J~ -s-s~)

fE A)

exp

~ -;EA

e e-

~ ; ,;nEA

m-n m

n

II da(s-;) ,

x

(3.12.7)

lEA

where the activity of a polymer is given by

z(()

= ~

n

1

Z

l supp (

x exp

(h-)

~

e

{l}

(~L

J II

h-;S-;)

e Esupp

{exp(-f3J- -s-s-) - I} m-n

(; ,;n)E(

(

~ II

n m

(3.12.8)

da(s-;).

l Esupp (

Clearly, the preliminary bound on the activity follows from the basic estimate lexp(-f3J;n_;s;s;n) -11

S f31J-m-n~s-s~1 dt exp(-tf3J-s-s-) m n J m -n n m

to

S

f3ls-s~JI exp {~f3IJI(s~n + S~)}(3.12.9) n m m-n 2 m-n m

and the summability (3 .12.10) which is an obvious consequence of (3.12.2). We also apply the Schwarz inequality as follows :

x exp

(~L l Esupp (

S (/

h-;S-;)

~e II

~I] (s~s~) ~ II (n,m)E(

da(s-;)

E BUPP (

lE sUPP (

da(S-;))

1/2

3.12. SCALAR SPINS WITH LONG RANGE COUPLING

x ( / exp

{(3 _~

273

IJ;;_;I(s~ + S~)}

(n ,m)E(

(2 _L

x exp

II

heSe) _

eE

l Esupp (

dU(Se)) 1/2

(3.12.11)

supp (

By (3.12.4) we have

(S~S~)

/ II (; ,;;)E(

e

II

du(se)

s e cl (Lf-

2

card (

E supp (

(3.12.12)

eE

supp (

where N( ( f.) denotes the number of line segments in ( that meet f.. J du(s-e )-integral is bounded by _

n

eEsupp

The other

(

for sufficiently small

(3, where (3.12.13)

If we combine all these estimates with the Jensen inequality

1

00

Z - (h-) > Z - exp (-h-Z-':; {e} e - {e} e {e}

ze-{3Joz

2

dU(Z))

-00

(3.12.14) '

we obtain the following bound on the activity:

Iz(()1

S cl(l{3card (r card (f({3)-!I(1

II

(NJe)!)1/2

II

IJ;;_;I

eEsupp (

x exp _

eE

L (C({3)h e + ~f({3)-1 h7) .

(3.12.15)

supp (

Before we try to sum over polymers, we must find something in this product bound against which to cancel the factorial growths. Since powers of small numbers cannot accomplish this, the product of varying coupling strengths is the only candidate. Accordingly, one writes

(3.12.16)

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

274

and observes that for aI, . .. , aN > 0, (3 .12.17)

by log concavity. This means

(3.12.18)

and in our context we get

II

(3.12.19)

The point is that (3 .12.2) bounds the sum with a universal constant. Thus

II

IJ-m-n_12/3

(;:; ,;;)E(

x exp _

L

(C(fJ)h 7 + ~f(fJ)-lh7) ,

(3 .12.20)

f Esupp (

and we are ready to sum over polymers. The estimation is now done exactly as in §3.9, because the bound no longer differs in any essential way from that which occurred in the §2- valued spin case. To be sure, a fractional power of each IJ;:;,_;:; 1 has been burned up, and the Konigsberg bridge-island estimation will cut the remaining power by half. This is the reason why the assumption (3.12 .2) is convenient. Once again, there are ways to refine the expansion technique. Although estimation such as (3.12.19) is important to a phase cell cluster expansion, we show in the next section that this consumption of small factors is unnecessary for the class of unbounded scalar spin systems considered here.

References 1. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981) , 327- 360.

2. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer- Verlag, New York, 1987.

3.13. EXTRA liN! FACTORS IN THE INDUCTIVE EXPANSION

3.13

Extra

1/N!

275

Factors in the Inductive Expansion

We now replace the wholesale expansion of the previous section with the inductive expansion already applied to the §2- valued spin case in §3.1O. Compared with the inductive expansion in that case, the expansion rules have not changed, but the estimation on the polymer activity has changed, since there are factorial growth bounds on single-spin expectations in this case. The point to be made in this section is that the combinatorial properties of the interpolation weights enables us to cancel the factorial growths without using any of the long-distance decay of the interaction! Thus we can replace the long-range requirement (3.13.1) of the less refined expansion and estimation with the weaker requirement (3.13.2) The challenge in this section is to find that alternative cancellation of the factorial growths. The inductive interpolation described in §3.1O generates the desired polymer expansion of a generalized partition function: ~

1

00

Z(h-: fE A) l

= L, n. n=O

n

II Z(Ui)

L

(3.13.3)

(U" ... ,Un)E'Vn i=l

where the polymers are unordered connectivity graphs. The activity is given by

z(U)

II

= eE

Z _ (h- )-1 {f}

(II 11 dti) fQ)(t) ,

X

l

o.c .g. Q) : U"'= U 90(2) = first site in supp U

supp U

exp (_

L

_II

(-(3J-;;,_-;)

J _II

(n,m)EU

his

i

-

(Sn S-;;')

(n,m)E U

(3H~pp U(t)) _ II eE

l E supp U

du(s-;),

(3.13.4)

supp U

where "first site" refers to the fixed linear ordering chosen for A, and H~pp u(t) is the interpolated Hamiltonian for the history of attempted decouplings associated with ~. It is easy to establish by induction that it has the convex form:

H~pp u(t)

L L

I:. partitions supp U

I:. partitions supp U

w~(t)

L

(3.13.5)

HA"

NEI:.

w~(t) = 1,

w~(t) ~

o.

(3.13 .6)

276

CHAPTER 3. EQUILillRlUM STATES OF CLASSICAL CRYSTALS

On the other hand, (3.13.2) implies

L

HI\' -> -J _

(3.13.7)

s:'t

tEl\'

for an arbitrary restriction of the Hamiltonian. Thus

S:.,

(3.13.8)

l

t Esupp

U

and so we have the activity bound: Iz(U)1

:s

II

Z - (h_)-I(3card {l}

1:

l E supp U

II

X _

II

U

l

IJ;,_~I

(~.;')EU

IzINu(7)l7Z+2~JZ2 du(z) ,

(3.13.9)

l E supp U

where we have also used the u.c.g. identity (3.8.22). Nu (f) denotes the number of (~ , iii) E U meeting f. - as in the previous section - and (3.12.4) implies

1:

IzINu(l\h7z+~cz2 du(z)

:s

(1:

:s

(f - 2(3J)-1/4cr~Nu(7) exp(~U'-2~J)-lh7) (Nu( f)!)1/2

z2NU(t)dU(Z)) 1/2

(1: e2h7Z+2~Jz2

dU(Z)) 1/2 (3 .13.10)

for small (3. Combining this with (3.12.14), we reduce (3.13.9) to: Iz(U)1

:s

c IUI(3 card Ur card

II

U

t E supp

x exp

L

(C((3)h t

(Nu (f)!)1 /2

II

IJ;,_~I

U

+ ~(f - 2(3J)-lh7) ,

(3.13.11)

l E supp U

which is certainly more refined than (3.12.15) . Following the general polymer estimation scheme , we seek to reduce the obvious bound (with card U ~ lUI - 1)

L

Iz(U)1

u.c.g . U : ~oE supp U

IUI=M u .c .g . U : ~oE supp

IUI=M

u

(t:;,ni.)e U

3.13. EXTRA liN! FACTORS IN THE INDUCTIVE EXPANSION

II

x

277

((Nu(i )!)1/2 X((3, h- )),

(3.13.12)

+ ~(f -

(3.13.13)

i

i E supp U

X((3,h)

exp (C((3)h

2(3J)-1 h2 )

,

to a bound of the form (3.9.10). We cannot throwaway the combinatorial advantage of summing over unordered connectivity graphs if we wish to control these factorial growths without burning up fractional powers of the factors IJ~_,:;I. The point is that we were indeed throwing away that advantage in §3.10. To see this, we write the sum in terms of tree graphs on {I, . .. , M} with line segments directed by choosing 1 as the root. First, we have the summation identity

=M u.c.g. U: ':;°E supp U !U!=M

because the summand is symmetric in pairs. Second,

(3.13.14) u.c.g. U: !U!=M ':;0= root of U

en:, r;;,) and the root only orders the individual 1

L

u.c .g. U: !U!=M ':;0= root of U

tree graphs T on {l, oO .,M} rooted at 1

(M -I)!

(3.13.15) Q: {l, oO .,M}-+A injective, Q(1)=-;:-O

where the summand is altered as follows:

II

IJ~_,:;I

II

IJQ(j)-Q( i) I,

(i,j)ET M

X((3, hi)

f----t

iEsupp U

eE

II

f----t

(,:; ,~)EU

II X((3, hQ(i)), i=l

II

i=l

supp U

The (M -I)! in the denominator compensates for the over-counting of subsets of A due to the permutation of {2, . .. , M} corresponding to different injections with the same range. Hence (with Ihil ::; c)

L

Iz(U) 1

u.c.g. U: ':;°E supp U !U!=M

cM (11:- 1 (3)M-1 M (M - I)! x

L

tree graphs T on {l,oO.,M} i=l rooted at 1

II

Q: {l,oO .,M}-+A (i,j)ET Q(l)=':;o

IJQ(j)-Q(i) I,

(3.13.16)

278

CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS

where we have also over-estimated by dropping the requirement that Q be injective. For fixed T, the sum over Q is just a multi-index summation, with the iterative application of (3.13.17) IJQW-Q(i) I ~ J

L

Q(;)EA

indicated by the structure of T . At each stage, only an index that occurs in only one of the remaining factors is summed over. We described this in the context of abstract polymer expansions in §3.6. Thus

M

II (d;(T)!)I/2,

x

(3 .13.18)

tree graphs T on {I •. .• M} ;=1 rooted at 1

so we may now apply Cayley's Theorem, which states that the number of tree graphs on {I, ... , M} with specified root and coordination numbers d I , . ·., dM is just M

(M - 2)!/

II (d; -

I)!

;=1

The sum over T is bounded by M

II(d !)-I/2d; ~ 2M/2(M -

(M - 2)!

i

d, •. ..• dM>O

2)!

i=I

L:i di=2M-I

x card

{(d

I , . ..

,dM):

d

I , . . . ,

dM

> 0 and

~ d; = 2M -

I} ,(3.13.19)

but the cardinality of the set is known to be bounded by 4M, so we finally have the desired bound for convergence of the polymer expansion at high temperature (small (J) . The reciprocal factorials were essentially hidden in the summation over unordered connectivity graphs, which, in turn, arose from the u.c.g. identity (3.8.32) . That combinatorial identity exploits the smallness of the interpolation weights to a degree comparable to the estimate (3.11.26) rather than (3.11.23) . One can derive the required polymer estimate by using ordered tree graphs instead of the polymers (u .c.g.) themselves. To this end, we write u. c .g. U: IUI=M ;:;0= root of U

o.c.g. (1; : deg (1;=M 90(2)=;:;°

3.13. EXTRA liN! FACTORS IN THE INDUCTIVE EXPANSION

279

M-I X

II ((d1)'" (j) + 1)!)1/2 ,

(3.13.20)

j=1 where the previously defined coordination numbers are related by (3.13.21) Thus u. c .g .

~UI=M Iz(U)1 ::; (7-1 eM

f3)M-I

Mth-~ree 1) (;g 11 dti) f1)(t)

~o= root of U M-I

M

II ((d1)(j) + 1)!)1/2

X

j=1

II 119, (i)-90(i) I,

o.c.g . 15 :

(3.13 .22)

1)", =1) i=1

90(2)=~0

and we estimate the inner most sum by the iterative applicatio~ of (3.13 .2) dictated by ~e ordered tree graph TI, which specifies which site variable f. j a given site variable

ei is "connected back to." u .c. g .

This reduces the bound to:

~UI=M Iz(U)I::; e

M

I

(r- f3)M-I

Mth-~ree 1) (;fill dti) f1)(t)

~o= root of U

M-I

X

II ((d1)(j) + 1)!)1/2,

(3.13.23)

j=1 and this is the point where (3.11.26) may be directly applied. The more traditional but weaker estimate (3.11.23) would require the use of some of the long-distance decay of the interaction to cancel the factorial growths in the manner described in the previous section. This, in turn, would require the condition (3 .9.2) on the interaction , in spite of our use of inductive interpolation.

References 1. G. Battle, "A New Combinatoric Estimate for Cluster Expansions," Commun. Math. Phys. 94 (1984), 133-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 3. G. Battle and P. Federbush, "A Note on Cluster Expansions, Tree Graph Identities, Extra liN! Factors!!!" Lett. Math. Phys. 8 (1984),55-57. 4. F. Harary and E. Palmer, Graphical Enumeration, Academic Press , New York, 1973.

Chapter 4

A Wavelet Introduction to the Renormalization Group In classical statistical mechanics and quantum field theory, no tool has had greater impact - either analytically or computationally - than the renormalization group. In the analysis of systems with infinitely many degrees of freedom, the iteration of a renormalization group transformation systematically eliminates irrelevant behavior and reduces the set of degrees of freedom. This reduction is most efficient because the idea is to realize the entire set of degrees of freedom as a scale hierarchy, with the functional integration over the smallest-scale variables implementing the transformation at a given stage. In the lattice formalism of the renormalization group, these variables are actually wavelet amplitudes. The natural connection between wavelets and the renormalization group is the central theme of this chapter. In an attempt to derive the non-analytic behavior of thermodynamic functions for scalar spin systems from microscopic interactions, Kadanoff and his co-workers introduced the idea of considering a block of strongly correlated spin variables on the lattice as a single spin variable [K3]. For parameter values close to critical values - i.e., close to singularities in thermodynamic functions - the exponential decay of correlations is very slow, so a block of spins can be treated in this way. The difficulty lies in writing down the effective interactions between blocks and then regarding super-blocks as single spin sites. The iteration of this procedure necessarily weakens correlations, so the block-spin description loses validity for blocks of blocks of .. . blocks of spins. Nevertheless, Kadanoff considered the effective interaction between super-blocks as a function of the effective interaction between super-blocks corresponding to the next level down the hierarchy. With no information about this function, he and his team assumed that it was analytic and derived non-analytic behavior for the thermodynamic functions! Naturally, the result was not quantitative at all, but it inspired the work of Wilson [W16, W17]. From the standpoint of developing Kadanoff's idea rigorously, Wilson recognized that there was no help for it but to successively integrate out fluctuations in successively larger blocks of spins, rescaling the coarser lattice to the unit-scale lattice with 281

282

CHAPTER 4. THE RENORMALIZATION GROUP

each step. The partial functional integration and rescaling for a given partition of the lattice into blocks came to be known as the renormalization group transformation, and the primary aim was simply to calculate Kadanoff's effective action at each stage by this functional integral transform. The iteration was complicated at the very outset, but the success of this approach was unparalleled in the study of critical behavior. In the case of three dimensions, for example, a non-Gaussian state was discovered which is actually fixed with respect to the renormalization group transformation. Moreover, every critical state is driven toward this fixed point by iterations of the transformation. There is a Gaussian fixed point as well, but the flow of the iteration on the manifold of critical states is away from that state. Since the RG (renormalization group) transformation changes the correlation length by a constant factor and a critical state has infinite correlation length, the critical manifold is invariant with respect to the RG iteration. On the other hand, the manifold has unstable directions in the space of possible interactions. Indeed, if a state is not critical, it has finite correlation length and is therefore driven away from the critical manifold by the RG iteration because the transformation scales down the correlation length. This implies the non-Gaussian fixed point is semi-stable, and such a property enables one to calculate critical exponents by analyzing the linearization of the RG flow in a neighborhood of that fixed point. Moreover, this picture of critical behavior explains the important principle of universality i.e., the principle that very large classes of models share the same values of critical exponents. The point is that any scalar spin interaction in a neighborhood of the critical manifold is drawn by the RG flow into a neighborhood of the non-Gaussian fixed point before it is driven away from the manifold. Since the correlations after many iterations are supposed to reflect the long-distance behavior of the original correlations, the fixed point governs that long-distance behavior. Early and very qualitative calculations were done in second-order perturbation theory with reasonable results. In an attempt to analyze the RG iteration beyond perturbation theory, Wilson introduced an approximation in the functional integration that reduced the RG transformation formula to an integral transform in one integration variable [W17] . There is no more control over the error in this approximation than over the error in the crudely modified second-order perturbative calculations, but this simple integral transform promotes RG calculations to the non-perturbative level. Wilson gave a qualitative justification of his reduction by describing the fluctuations at all scales in terms of an orthonormal basis of expansion functions for the Euclidean scalar field. The basis had a multi-scale coherence and its existence was not proven. Wilson simply assumed that the functions were well-localized in phase space [W17], but he anticipated wavelet analysis fifteen years before the subject really began to develop . We shall refer to his integral transform as the Wilson wavelet recursion formula . Shortly after Wilson's derivation, Baker pointed out that this integral transform gives the exact RG transformation for a toy model where the interaction between nearestneighbor blocks is indirectly realized through their interaction with a larger-scale block containing both of them [B4] . This model is a variation on a scheme originally introduced by Dyson [D45-D47] and is known as a hierarchical model. The study of such models proved to be very fruitful [B36- B38, C22, K22-K24] , so naturally the attention of the mathematical physics community was drawn away from the wavelets postulated by Wilson.

The program to obtain rigorous results for realistic models through renormalization group techniques was enthusiastically promoted by Gawedzki and Kupiainen [G17G28]. For example, the approximate RG calculations referred to above predict that in four dimensions the Gaussian fixed point is the only fixed point for scalar selfinteractions and that it is attractive on the critical manifold. Gawedzki and Kupiainen proved that this fixed point is attractive for the iteration of the full RG transformation with no approximations [G26]. An immediate application was to control the infinitevolume limit of the massless @: model, whose long-distance behavior poses problems for any straightforward expansion method. A convergent phase space expansion in the infrared regime was developed by Feldman, Magnen, Seneor, and Rivasseau [F30]. Their ideas were similar to the ideas in the RG analysis of Gawedzki and Kupiainen. While the Ising model was the original focus in the study of critical exponents for scalar spin systems, Kadanoff and Wilson worked with the Ginzburg-Landau model, which can be regarded as a generalization of the Ising model. Let 4 be the free lattice field in d dimensions with mean zero and covariance matrix H;', where Ho is minus the lattice Laplacian, already given in 53.5 for d = 3. For arbitrary d, 2

2d, -1, 0,

2

nl=n, n and n ' nearest neighbors, otherwise. 2

2

(4.1)

Equivalently, 2

(H04, 4) =

(4(;

'1 - 4(n))27

(4.2)

where (., .) denotes the inner product on e2(Zd). Let dPHo denote the Gaussian measure associated with the covariance of this free lattice field - i.e.,

Consider the Euclidean lattice action

The non-Gaussian perturbation corresponding to this quartic interaction is given by the A = Zd limit of

CHAPTER 4. THE RENORMALIZATION GROUP

284

where X A denotes the characteristic function of the finite set A c Z d We discussed this model in $3.5, and the advantage in working with this variation on the Ising model is that the spin configuration 4 can be averaged over blocks of lattice sites with no restrictions on spin values. This block-spin averaging is realized as the KadanoffWilson transformation, given by

where the reason for the averaging weight 2-4-I is to preserve the order of magnitude of the quadratic form (4.2). The renormalization group transformation 'RTassociated with the averaging transformation T acts on probability measures d p on the space of spin configurations 4, and it is given by

for arbitrary random variables X. This innocent-looking definition is computationally convenient in cases where the expectations for dp are explicitly known - if d p is a Gaussian measure, for example. Otherwise, one must deal with what this transformation means in terms of functional integration. The fluctuations to be integrated out must be defined, and the residual functional integral must be expressed in terms of the blockspin variables. This complicated transformation drastically affects the form of dpH. Not only does the interaction I($) lose its locality, but the quadratic form (Ho4,d) loses its nearest-neighbor structure. The iteration of the transformation escalates these complications. Recognizing the essential irrelevance of some of this complexity, Wilson replaced the Euclidean lattice field with its continuum analog, where the short-distance regularization is realized as the standard ultraviolet cutoff. In this setting, the averaging transformation is replaced by a truncation in momentum space. Let be thz gee continuum field in d dimensions with mean zero and covariance kernel -A-'(x, y). Consider the regularization

of that kernel. The action (4.4) is now given by

but the non-Gaussian perturbation of the Gaussian measure dpTegassociated with is given by the 0 = IRd limit of ~1(;,

G) p

=

(z,H)-~ exp (-u2

a,g(32d

;

z,H

=

/

exp ( - u 2

Jn area

(;)'d

; - ~

4

where R is an arbitrary bounded region in Rd. Let dfiregbe the Gaussian measure with mean zero and covariance kernel

The RG transformation is defined in this formalism by

Since -~

4 , =~/ () v~* : ~ ~ ) ~ ,

(4.20)

the free part of the action is already a fixed point - clearly an advantage in calculating the effect of R on a perturbation of this free action. The early calculations done by Wilson were based on this modification of the generalized Ising model [K25, W16], except that the mass was included in the covariance. Nevertheless, this scheme poses a serious technical problem as it stands. Since the ultraviolet cutoffs are sharp, the correlations among fluctuations have slow longdistance falloff. On the other hand, the entire renormalization group program owes its success to the fact that fluctuations are weakly correlated - specifically, with exponential falloff in the correlations. This conflict can be resolved by the following scheme. For r E Z+ define the covariance kernel

CHAPTER 4. THE RENORMALIZATION GROUP

286 Clearly,

~ c(r)(~, y) = (2~)d Jd P

eip,(;-y)

~

1 12

Cp I~ + 1) ,

(4.22)

which is a Pauli-Villars regularization [G28, P12] of the covariance kernel (4.23) For each r E Z+, the covariance kernel c(r)(~, y) has exponential decay with correlation length 2- r + 1 , while on the other hand, the decomposition (4 .22) can be realized by a direct sum 00

cp(~)

= EB cp(r) (~)

(4.24)

r=l

of Gaussian random fields . Let covariance c(r)(~, y):

djl(r)

be the Gaussian measure with mean zero and (4.25)

If dfJP) is the Gaussian measure with mean zero and covariance

( ~) (~))

(cp x cP Y

_ dl'(l ) -

1 (27l')d

Jd~

ip.(;_y)_l_ (

pel

P12

1)

1P 12 + 1 '

(426) .

then 00

dJ.L(l)

= EB djl(r) ,

(4.27)

r=l

(4.28) where

cp(r)

(~) is not to be confused with

This rigorous improvement over the sharp ultraviolet cutoff is used in much of the work of Feldman, Magnen, Seneor, and Rivasseau as well as in some of the work of Gawedzki and Kupiainen. We do not use this "momentum-slicing" formalism, but return to the block-spin transformation of lattice variables. Gawedzki and Kupiainen [G28] introduced a formalism based on the lattice Gaussian fixed point. Perturbing the lattice field action associated with that Gaussian, they carried out an RG analysis of interacting lattice field theories based on the Kadanoff- Wilson transformation which also had the advantage offered by the standard ultraviolet cutoff on the continuum field . We shall display

287 the wavelets that are hidden in this lattice formalism. Indeed, there is a continuum picture canonically associated with the lattice picture. If Hoo denotes the matrix for the free lattice field action one has in this formalism, then the covariance matrix is given by

= _1_ ( H-1)~~ n n' (211")d 00

(rr 1" d

d )

,=1 -"

p,

e

ip.(-,:;--,:;')

~ I£(p +211" ~

~e EZ d 1

l:')12

P+211" l:' 12

(4.29)

where X denotes the characteristic function of the standard cube [o,l]d. On the other hand, (4.30)

where X-,:; (;I;) X( x n) and 00 denotes the expectation functional of the free (massless) continuum field
and if we choose the s

= 1 wavelets from

§2.1O with scaling function X, then (4.32)

if we insert the expansion (4.33)

of the continuum field. The continuum version of block averaging induces a cutoff in the wavelet expansion that is rather similar to the standard ultraviolet cutoff. Moreover, 00

(Hoo¢,¢)

=L

L

r=1 iiiEZ

L d

(a~~;;y

(4.34)

E' E{0.1}d\{ O}

if we make the natural identification of ¢(r;) with
288

CHAPTER 4. THE RENORMALIZATION GROUP

diagonal form (4.34) is not only valid, but exact. On the other hand, Wilson required an additional condition in the derivation of his recursion formula, which has the flavor of square-integrable orthogonality, and on the surface, it may appear that we still have a problem. Specifically, it was assumed that each wavelet is approximately constant on each dyadic sub-cube of the cube supporting most of the wavelet. This means the wavelet resembles a Haar function, which is the wavelet associated with the scaling function X if one requires orthogonality in the L2-sense. Far from implying a lack of consistency, however, Wilson's piecewise constant expansion functions reflect yet another RG formalism. It is related to the lattice-continuum formalism in much the same way that the original lattice formalism is related - through RG iterations that throwaway the fluctuations - except that it is an inverse limit instead of a starting point. When the renormalization group transformation Rr acts on a lattice Gaussian, the operation of integrating out the fluctuations is equivalent to throwing them away. When a perturbation is introduced, it is no longer equivalent; the latter operation is a zeroth-order contribution to the former. Let 00

I(u; ¢)

= LUI' L ¢C:;{)"

(4.35)

r;EZ d

1'=1

be the perturbation of the nearest-neighbor lattice action 1

= 2(Ho¢, ¢) .

Ho(¢)

(4.36)

Now it is standard to regard Rr as transforming the action rather than the associated probability measure, and as we have already remarked, (4.37) Indeed, it shall be proven in §4.4 that lim R!f (Ho) = Hoo. N->oo

(4.38)

We shall introduce the zero-fluctuation transformation R'f', which throws away the fluctuations instead of integrating them out. If we define the interaction

f J~l Uv

(;;)" d

;;,

(4.39)

v=l 00

LQ~)~W~~)(;; -2~), r=1

<,m

(4.40)

then the effect of R'f' is just

R'f'(Hoo

+ loo(u))(¢')

= Hoo(¢')

+

f v=l

J (;;)" d;; .

2v-v~+duv ~~

(4.41)

289 Moreover, if we define

UO,N v

= 2-(v-v~+d)Nu v,

(4.42)

then (4.43)

as we shall see in §4.5 by a Riemann sum argument. Thus the original lattice action is related to the lattice-continuum action by a convergent sequence of transformations. At the same time, both actions are expected to be governed by the same fixed points with respect to the full RG transformation RT . The same universality principle has always been assumed when the lattice formalism is simply replaced by an ultraviolet cutoff in the continuum. In §4.7 we shall introduce another lattice configuration sq" which has a linear dependence on ¢ and corresponds to a piecewise-constant configuration in the continuum. Needless to say, the action of Hoo on these configurations is not second-order differentiation, but an action qualitatively similar to that of the Laplacian, with a complicated scale-dependent structure. It will be shown that

Tsq,

= STq"

(4.44)

and if we define

Loo(u; ¢) = I(u; sq,),

(4.45)

we have 00

R'f'(Hoo

+ Loo(u))(¢')

= H oo (¢')

+

L 2v-v~+duv L

sq,.(~t ·

v=l

In addition, we shall construct a sequence (IN: that

-00

R'f'(Hoo + IN(UO ,l)) = Hoo lim IN(u) = loo(u), N->oo lim IN(u) = Loo(u). N->-oo

< N < (0) of interactions such

+ IN+1(U),

(4.46) (4.47) (4.48)

Our point is that the action Hoo + Loo(u) is really implicit in Wilson's assumptions. If we extend the universality principle to the very similar connection provided by the sequence (IN), we expect this action to be governed by the same fixed points as H 00 + 100 (u) . The non-perturbative approximation proposed by Wilson actually involves this inverse limit manifold of actions. In §4.8 we discuss the well-known connection - from the wavelet point of view between Wick ordering and the linearization of the RG transformation in the neighborhood of the Gaussian fixed point. In §4.9 we derive the wavelet version of the second-order correction formula. In §4.1O we show how Feynman diagrams are replaced by wavelet diagrams and in §4.1l describe the one-loop-per-scale approximation in this

CHAPTER 4. THE RENORMALIZATION GROUP

290

context. After deriving the Wilson wavelet recursion formula in §4.12, we introduce the Baker-Dyson- Wilson model in §4.13 independently. In the context of that model, we revisit the second-order correction formula in §4.14 and the one-loop-per-scale formula in §4.15. In particular, we explain why the one-loop-per-scale approximation is not expected to shed much light on critical behavior. Beyond this, we describe how the wavelet formalism modifies the Baker-Dyson-Wilson model. There is a notational shift in this chapter, since the renormalization group formalism focuses on arbitrarily large length scales. We make the wavelet notational change iIt(<'") >--+ iIt(<'")

r'

-r

since positive r indexes large-scale wavelets in this discussion.

4.1

Averaging Transformations

Any renormalization group transformation of a probability measure on a space of lattice configurations will be based on an averaging transformation of some kind, so it is appropriate to discuss the latter operation in its own right. It is based on a selection of weights for the rriE Zd which are often - but not necessarily - positive. Indeed, they can have complex values, but we restrict our attention to the real-valued case. Accordingly, we consider a fixed real-valued function on Zd - with the index notation {em) - such that (4.1.1)

For a given scalar configuration
{

1, 0,

~

d

mE {O,I} , otherwise,

(4.1.3)

with s = 1. In this case, the mapping T is the block-spin transformation:

(T
= 2-~-1

L


+ "€).

(4.1.4)

<'" E{O,l}d

The significance of the parameter s can be seen eventually in a Sobolev space of continuum configurations with the unitary group Us (>') of scalings. Specifically, the Sobolev norm is (4.1.5)

4.1. AVERAGING TRANSFORMATIONS

291

and Us (>') is given by (4.1.6)

In the case s

= 1, the square of the norm is J 1V' (1 2 .

What does the Sobolev space of continuum configurations have to do with the lattice averaging? The simple answer is that scaling relates the averaging transformation to a mapping from the Sobolev space to the phase of lattice configurations. Suppose that for the signed weights c~ one can find a measurable function 'f/ on JRd, with reasonable decay properties, such that (4.1.7)

In the case (4.1.3) associated with the block-spin transformation, 'f/ is just the characteristic function of the unit cube. As in Chap. 2, we refer to any function satisfying (4.1.7) for some real-valued {c~} on Zd as a scaling function. For an arbitrary function ( E Gel" (JRd ), define the lattice configuration ¢>( by (4.1.8) The averaging transformation of ¢>( reduces to

T~-s

L c~¢>d2 ~ + ni) ~

.I ~ 2-~-s .I G -; -~) 2~-s J ~)((2 T

~-s 2; c~

'f/(-; -2

- ni)((-;)d -;

m

((-;)d-;

'f/

'f/(-; -

¢>u,d~),

-;)d-;

Us = Us (1n2),

(4.1.9)

so if T denotes the mapping ( t-+ ¢>( from the Sobolev space to the space of lattice configurations, we have the answer to our question:

ToT=ToUs.

(4.1.10)

How does the averaging procedure extend to vector-valued configurations in a geometrically natural way? First of all, a vector-valued function on the lattice may be regarded as a scalar-valued function on the set of neare~-~igh~or bonds. ~n oriented nearest-neighbor bond is any ordered pair of the form (n, n ± e It), ~here e It denotes the j.tth-coordinate unit vector. An arbitrary vector-valued function A on Zd assigns a number A(b) to each oriented bond b such that (4.1.11.+)

292

CHAPTER 4. THE RENORMALIZATION GROUP

(4.1.11.-) Secondly, one would like to define the vector averaging such that it is induced by the scalar averaging in the case where the vector configuration is given by the lattice gradient of a scalar configuration. If A(b) is given by

(\71at
+ -; /L)

-


(4.1.12)

for some scalar configuration
+A(2 ~ + m,2 ~ + T~-sLcr; m

L

m+ -;/L») (4.1.14)

A(b)

bEr~(n)+m

with r /L(~) the shortest path of bonds from 2 ~ to 2 ~ +2 -; /L

r /L(~) = {(2 ~, 2 ~ + -; /L)' (2 ~

-

i.e.,

+ -; /L' 2 ~ +2 -; /L)} '

(4.1.15)

We simply extend this equation to define T' A for an arbitrary configuration on the set of bonds. ~ Now consider an a~bitrary vector function:;; on IRd . We wish to define the mapping v >--+ A -;; such that if v = \7 <" for some scalar function (, then (4.1.16) This implies

(4.1.17) We now extend this formula to define:;; >--+ A-;; for an arbitrary vector function on IRd : (4.1.18)

4.1 . AVERAGING TRANSFORMATIONS

293

The fundamental relationship (with T' : -;; >--+ A~) is just v

T'

n \7

= \71at T .

(4.1.19)

0

On the other hand, the averaging transformation of A~v reduces to

(T' A-;;)(n:,

n: + -;1')

T~-S L>~ r;

L

A-;;(b)

bEr ~ C;)+n;

(4.1.20)

so we also have:

T'oT'=T'oUS _

(4.1.21)

1'

We now turn logically to a description of averaging for anti-symmetric tensor quantities. Our first observation about such configurations on lattice sites is analogous to our first observation about vector-valued lattice configurations: an anti-symmetric tensor-valued function on tl d may be regarded as a scalar function defined on the set of oriented nearest-neighbor squares called plaquettes. A plaquette is any square whose four sides are nearest-neighbor bonds and whose orientation is -;1' 1\ -;1' or -;" 1\ -;1' if !!:e square lies in the j.£v-coordinate plane. An arbitrary anti-symmetric tensor-valued

F on tl d assigns a number F(P) to each oriented plaquette P such that (4.1.22)

where PI''' denotes the plaquette with orientation -;1' 1\ -;" and minimum-coordinate vertex at the origin. We consider the continuum first. Naturally, if --u; is an antisymmetric tensor function on m,d, we wish to define the associated plaquette configuration F~ under some mapping Til such that if --u; = \71\ -;;; for some vector function ~

w

v , then ((\7 1\)latA-;; ) (PI''' +

n:)

+ -;1" n: + -;1' + -;.,,) - A-;;(n:, n: + -;.,,)) -(A~(n: + -;.", n: + -;." + -;'1') - A~(n:, n: + -;',,)) . (4 .1.23) v v

(A-;;(n:

~

CHAPTER 4. THE RENORMALIZATION GROUP

294 Equivalently,

T" 0 (\71\) = (\7I\)lat 0 T'

(4.1.24)

is the condition we want . Clearly, (4.1.23) reduces to

F'";;;(p/tv+

J -J

TJ(:; -

n:)

n:) (/-; ~,-; ~+ -; ")+x :; . d -; -

(1 ~)+x ~ . J n:) (1 ~ ~ . TJ(:; -

n:)

d -; -

(-; ",-;"+ -;

TJ(:; -

/0,-; ")+x

r

~ . d -;) d:;

J(O ,-; ~)+x

~ .d -;) d :;

d -;) d:;,

(4.1.25)

ap~"+x

where oP/tv denotes the boundary of the oriented plaquette in Fig. 4.1.1. It follows from Stoke's Theorem that (4.1.26) where da denotes element of surface area. We now extend this equation to define F.,; for arbitrary

"Vi.

Figure 4.1.1 : As far as the averaging transformation T" on the lattice is concerned, our definition is guided by the same geometrical criterion that guided our definition of T" . If the plaquette configuration F is given by

F(P/tv+

n:)

n: + -;/t + -;v) - A(n:, n: + -;v)) -(A(n: + -;v, n: + -;v + -;/t) - A(n:, n: + -;/t))

=

(A(n: + -;/t,

(4 .1.27)

for some bond configuration A, then we require

(T"F)(P/tv+

n:) =

((T'A)(n: + -;/t , -(T' A)(n:, -(T'A)(n:,

n: + -;/t + -;v)

n: + -; v)) n: + -;'/t)) .

((T' A)(n: + -; v,

n: + -; v + -;. It) (4.1.28)

This condition reduces to

(T" F) (P/tv+

n:) = 2-~-s ~ cni ( m

f=~ ~ A(b) bEr"(n+e~)+m

4.1. AVERAGING TRANSFORMATIONS

L

295

A(b)-

(4.1.29)

'" '" '"

2n + 2e J1+e v

'" '"

'"

2n

2n + 2 e ~

Figure 4.1.2: by the definition of T' . If we collect these bonds, adjusting the orientations to absorb the minus signs, we obtain the TTi.-translate of the loop given by Fig. 4.1.2. Clearly, we can replace this sum by the sum over the four plaquettes inside this loop, because the terms corresponding to a bond common to adjacent squares will cancel as shown in Fig. 4.1.3.

00 00

•

. Figure 4.1.3:

Hence

(T"F)(PjJ.v+ ~ n)

= 2-2d

8 ""

L.Jcr;,

F(p),

(4.1.30)

where r I'vC~) denotes the set of plaquettes inside the square whose vertices are 2 ~, 2 ~ +2 ~ 1" 2 ~ +2 ~ v, and 2 ~ +2 ~ I' +2 ~ V · Thus

rl'v(~) = {Pl'v+2 ~,pl'v+2 ~ + ~jJ.,pl'v+2 ~ + ~v,pl'v+2 ~ + ~I' + ~v}. (4.1.31) As usual, we extend the equation to define Til F for any plaquette configuration F. Finally, we compute the effect of Til on F.,; for an arbitrary anti-symmetric tensor field "u;:

m

CHAPTER 4. THE RENORMALIZATION GROUP

296

m

(4.1.32) Thus

T"oT=ToUS _

2'

(4.1.33)

References 1. T . Balaban, "Averaging Operations for Lattice Gauge Theories," Commun. Math. Phys. 98 (1985), 17-52. 2. G. Battle, "A Block Spin Construction of Ondelettes, Part I: Lemarie Functions," Commun. Math. Phys. 10 (1987),601-615. 3. G. Battle, "A Block Spin Construction of Ondelettes, Part II: The QFT Connection," Commun. Math. Phys. 14 (1988), 98-102. 4. G. Battle and P. Federbush, "Divergence-Free Vector Wavelets," Michigan Math. J. 40 (1993), 181-195. 5. P. Federbush, "A Phase Cell Approach to Yang-Mills Theory I: Modes, LatticeContinuum Duality," Commun. Math. Phys. 107 (1987), 319-329. 6. K. Gawedzki and A. Kupiainen, "A Rigorous Block Spin Approach to Massless Lattice Theories," Commun. Math . Phys. 77 (1980), 31- 64. 7. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer- Verlag, New York, 1987.

4.2. THE RENORMALIZATION GROUP TRANSFORMATION

297

8. J. Imbrie, "Renormalization Group Methods in Gauge Field Theories," in Critical Phenomena, Random Systems, Gauge Theories. K . Osterwalder and R. Stora, eds., North-Holland, Amsterdam, 1986.

9. K. Wilson, "Renormalization Group and Critical Phenomena I," Phys. Rev. B4 (1971), 3174-3183.

4.2

The Renormalization Group Transformation

Once we choose the averaging transformation T for the space of scalar configurations on Zd, we introduce an important transformation - based on T - for probability measures on the configuration space. The most elegant definition of the renormalization group transformation R T ( d p ) of a probability measure dp is given by

for every random variable X. However, the calculation of such an innocent-looking transformation can be very complicated for most probability measures. Our interest is in the A = Zd limit of some net { d p A )of measures of the form

for finite A c Zd, where H(4) is a functional that is bounded below. The limit is taken for only finite in the sense of expectations of random variables that depend on 4(;) sets of lattice sites .; H(4) is the Hamiltonian in the context of classical statistical mechanics, while the Euclidean field-theoretic interpretation - if there is one - identifies H(4) as the Euclidean Lagrangian. In any case, we shall refer to H(4) as the action. It is usually more natural to regard RT as a transformation on some space of actions. Accordingly,

where A' = 1lr-12~~This limit is controlled by expansion methods based on decoupling the integration variables remaining after the delta functions have been eliminated. The preliminary task is usually non-trivial because - although it is clear that we are integrating out fluctuations at a given scale - the fluctuation variables are not specified by the formula. On the other hand, this problem can be avoided in the special case of a Gaussian measure, which we study first.

CHAPTER 4. THE RENORMALIZATION GROUP

298

We consider the class of translation-invariant Gaussian measures on the space of configurations. This corresponds to the cone of translation-invariant actions which are positive quadratic forms. If H ( 4 ) is such an action, we introduce the following abuses of notation:

Let d p H denote the probability measure on IRZd realized as the A = Z d limit described above. Then d p H is the mean-zero Gaussian measure with covariance matrix H-'. The expectation functional has the structure:

where D is the set of those partitions of ( 1 , . . . ,N ) consisting of disjoint pairs. These formulas are generated by

(exp i

- a - - 4 ( ~ ) ) d p H= exp (-12 - -( ~ - l ) ; , ; a ; , a ; m

(4.2.10)

m' ,m

The advantage here is that we may use the functorial definition (4.2.1) of the renormalization group transformation to compute R T ( H ) . We have

so the Gaussian structure of the measure is preserved by the transformation, and the new covariance matrix is T H P I T t . This is all that we need to calculate:

4.2. THE RENORMALIZATION GROUP TRANSFORMATION

299

(4.2.12)

where we have defined (4.2.13)

exactly as we did in Chap. 2. Indeed, it is more natural to work in periodic momentum space. For T itself, the Fourier series is rewritten in ~ to obtain

p

T
= T3~-s _

L

h(p +211'

1)*¢

G

p +11' 1)

(4.2.14)

e E{O,1}d

for an arbitrary lattice configuration
p

P

(4.2.15)

so we may write (4.2.16)

with

(T H-1Tt)(p)

=T

2d - 2s

L

Jh(p +211' 1W H

Gp

+ 11'

1)

-1

(4.2.17)

eE{O ,1}d

Since this momentum expression is just the diagonalization of the covariance of Rr( dJ-LH) , the new action Rr(H) is given by (4.2 .18)

E E{O ,1}d

CHAPTER 4. THE RENORMALIZATION GROUP

300

We have also established that the cone of translation-invariance positive quadratic forms is invariant with respect to the transformation. It is now easy to identify the action Hoo(¢» in this cone which is a fixed point under RT . If there exists a scaling function 'T/ for the set of coefficients {cr;i} defining T, then the relation (4.1. 7) can be written as

~ d ~ (1~) fJ(p)=T h(p)fJ 2P .

(4.2.19)

This implies

2: IfJ~ +27r~£W 7

1

2s

p +27r £

1

~

~

~

2-2d-2s2:lh(p+27r£W

7 T2d-28

~

2

~

IfJ(lp+7rp.)1

l~p+7rP.12S

Ih(p +27r E')1 2

L

~

2

~ IfJ(~ P+2~£' + 7r E' 12 11~ ~128 zp+27r£'+7r€

L

7,

eE{O,l}d

(4.2.20)

(4.2.21)

because h(p) is 47r-periodic. This proves that for

2: IfJ~ +27r~£W)-1 ( 7 1P +27r £ 128

(4.2.22) (4.2.23)

= 1, we have

In the case where T is block-spin averaging with s

Hoo (p) =

!w)

(2: Ix~ +27r 7

1

-1 ,

(4.2.24)

P +27r P. 12

where X is the characteristic function of the unit cube. For what initial actions can Hoo be realized as the limit of iterations of the renormalization group transformation? Let Ho(¢» be an initial action and consider

HN = R!'j. (Ho)

(4.2.25)

for an arbitrary number N of iterations. By the formula (4.2 .18) it is easy enough to verify that

l E{O ,1, ... ,2 N _l}d

x Ho(TN

P+2- N+ 7r £)-1, 1

(4.2.26)

N

IT h(Tr+1 p). r=l

(4.2.27)

4.2. THE RENORMALIZATION GROUP TRANSFORMATION

301

Notice that hN(P) is a periodic function with period 2N +171". Clearly, the convergence of the sequence {HN} of actions depends on the convergence of the sequence {hN(p)} as well as on the small-p behavior of HoCp) . The latter requirement is the s-related condition: lim Hr:.,(p)

P-->0

1

p

= c # O.

(4.2.28)

2s

1

In the case where H o((m

+ ~e ,) -

~ (m))2

(4.2.29)

m

discussed in the context of Euclidean lattice field theory, s = 1 is the necessary choice for T because (4.2 .30) In the general case, (4.2.28) implies

2- 2Ns

lim

N-->oo Ho(2-N

P+2- N+171"

c- 1

~

€)

1

P+271" e

(4.2.31) 2s

1

The former requirement - i.e., convergence of the infinite product involving h - is an important issue in the study of compactly supported wavelets, as we have seen in Chap. 2. Since the normalization condition (4.1.1) implies h(O)

= 2d ,

(4.2.32)

the infinite product is 00

hoo(p)

= II (2- d h(2- k + 1 p))

(4.2 .33)

k=l

if it exists. Indeed,

(4.2.34)

if ij(p) is continuous and if the infinite product converges uniformly on compact sets. Under these conditions, we may conclude from (4.2.26) that ~

. H (~)-1 _ -1" Ihoo(p +271" eW 11m N P -c ~ ~ N-->oo 1 +271" e 12s

c

P

,

(4.2.35)

which is precisely Hoo(p)-l up to a proportionality constant involving c and ij(O). The action H oo (
CHAPTER 4. THE RENORMALIZATION GROUP

302

for arbitrary test functions

f and g. Then

(<J>(1]ni,)<J>(1]ni))dl'.

= (1 V7 1- 2s 1]ni" 1]ni) = (2!)d

P1- 2S lii(p) 12,

1]-(-;) m

=

I:

Cfl dP')

eiP .(ni' -ni) 1

(4.2.37)

1](--; -~).

(4.2.38)

Thus

while (4.1.8) may be abbreviated to (4.2.40)

so we have the identification (4.2.41)

References 1. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets and Their Applications, B. Ruskai et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349.

2. K. Gawedzki and A. Kupiainen, "A Rigorous Block Spin Approach to Massless Lattice Theories," Commun. Math. Phys. 77 (1980), 31-64. 3. J. Kogut and K. Wilson, "The Renormalization Group and the c:-Expansion," Phys. Rep . 12 (1974), 75-199. 4. S. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, 1976.

5. K. Wilson , "Renormalization Group and Critical Phenomena I," Phys. Rev. B4 (1971),3174-3183.

4.3. MINIMIZERS AND ORTHOGONALITY

4.3

303

Minimizers and Orthogonality

As we have already pointed out, the more elegant definition of the renormalization group transformation is useful only in the case where the structure of the expectation functional is quite explicit - such as for the translation-invariant Gaussian measures. For models that are not exactly solvable - such as a model whose action is a quartic perturbation of a positive quadratic form - we have to integrate out fluctuations explicitly to eliminate the delta functions in (4.2.4) . To this end, we introduce the minimizer associated with the averaging transformation and the quadratic form. For a given translation-invariant positive quadratic form (Ho¢, ¢), we define the minimizer to be the right inverse mapping M of T such that for every configuration ¢' on Zd, the configuration M ¢' minimizes (Ho¢, ¢) with respect to the constraint T¢

= ¢'

(4.3.1)

In this context, the fluctuation configuration is given by

¢ = ¢ - M¢/. Clearly, T¢ sition

(4.3 .2)

= 0, while the variational property of M (Ho¢,¢) = (HoM¢/,M¢/)

implies the orthogonal decompo-

+ (Ho¢,¢).

(4.3.3)

Hence (4.3.4) for an arbitrary random variable X. Now suppose the action has the form

H(¢)

1

= "2 (Ho¢, ¢) + I(¢),

(4.3.5)

where I(¢) is bounded below and involves terms that are of higher order in the amplitudes ¢(~). First, we may replace the formula (4.2.4) with

e-RT(H)(¢'l

=

lim [( A..... Z·

r

e-I(¢XAldP.HO(¢)) -1

JT¢=O

r

e-I(¢XAldP.HO(¢)] , (4.3.6)

JT¢=¢'

where the A = Zd limit of the Gaussian part has already been taken. Second, if we now apply (4.3.4) to this formula, we obtain

e-~(HoM¢',M¢'l

lim [( A..... Z·

x

r JT¢=O

r_ e-I(hAldp.HO(¢))-1 JT¢=O

e-I(M¢'+;PXAldP.HO(¢)] .

(4.3 .7)

CHAPTER 4. THE RENORMALIZATION GROUP

304

Actually, we have only replaced the delta functions in the integration by constraints on the integration. To write the transformation in terms of an unconstrained integral, we must expand J in a basis of fluctuation configurations. Let

(4.3.8)

and note that the condition

TJ = 0 reduces to (4.3.9)

which may be scaled and decomposed into

"" (lId_ L" r dp, ) etn·Ph(p .- - ~ +211" ~c)*w (12 ~p +11" ~)c = 0,

_ L

"E{O,l}d

(4.3.10)

,-1

and so the fluctuation property becomes

""

_ L

~ +211" ~c)*w h(p

(12 ~

P +11" ~) E:

= O.

(4.3.11)

" E{O,l}d

p,

For fixed but arbitrary this is a homogeneous linear equation in 2d unknowns +11" "1), so one can fine 2d - 1 linearly independent column matrices

wa p

solving this equation. Once such a basis for the solution space has been chosen at each point p in the cube {-11" :S p, :S 11"}, an arbitrary solution has the form

(4.3.12) Obviously, this implies

4.3. MINIMIZERS AND ORTHOGONALITY

305

(4.3.13) I<

~

m

where

L a -e -in;,.p I<m

(4 .3.14)

'

(4.3.15)

Thus we have expanded ¢ in the special lattice configurations '1/;(1<)( . - 2 71i.), and the amplitudes al<~ are unconstrained. Now in our renormalization group context it is most desirable to choose this basis such that (Ho'l/;(I<) (. - 2 71i.), '1/;(1<') (- - 2 71i. ')) = 81<1<, 8~~, . (4.3.16) This is equivalent to the condition

_L

Ho

Gp

+71'

'1)

w(I<')

Gp

+71'

'1)

* w(l<)

Gp

+71'

'1) = 81<1<',

(4.3.17)

, E{O ,l}d

p

which can easily be arranged. For each point we simply pick our basis in the solution space of the homogeneous equation (4.3 .11) to be orthonormal with Ho(! +71' as the weight defining the norm. Our motivation is the reduction

p

'1 )

(4.3.18) I<,m

The point is that if we slightly change the limiting procedure again by making the replacement (4.3.19) ¢XA r--+ ¢A = al<~'I/;(I
L L I<

2~EA

in the formula (4.3 .7) we get e-nT(H)(¢,')

exp

= e-~(HoM"",M"")

(-~ L ~ a!~ - I I<

2mEA

lim A-+Zd

[Z_l

fM¢1 +

~

A

(II II Jroo da -) -00

I< 2~EA

I<m

L ~ al<~'I/;(I
2mEA

)

~

CHAPTER 4. THE RENORMALIZATION GROUP

306

(4.3.21)

The formula is now written in terms of unconstrained integrals. In particular, this formula yields an alternate expression for the renormalization group transformation in the absence of an interaction - namely,

(RT(Ho)¢/ , ¢>')

= (HoM¢/ , M¢/),

(4.3.22)

which means that in this special case the fluctuations are simply thrown away. We now extend this description to the iteration of the renormalization group transformation . Since the minimizer M is a right inverse of the averaging transformation T, we have

(MT)2

= =

M(TM)T MT.

(4.3.23)

and so MT is idempotent. On the other hand, T(I - MT)¢>

=0

(4.3.24)

for the same reason, and the variational property of M implies that M ¢/ is (Ho¢>, ¢»orthogonal to any configuration that T annihilates. Thus

(HoMT¢>l, (I - MT)¢>2) (Ho(1 - MT)¢>l, MT¢>2)

0,

=

(4.3.25) (4.3.26)

0,

which together imply

(4.3.27) and so MT is self-adjoint with respect to this inner product. Therefore MT is an (Ho¢>, ¢»-orthogonal projection. Set M = M(l) and define the rth minimizer M(r) in the following inductive way: M(r)¢>, minimizes the quadratic form (HoM{l) ... M(r-l)¢>, M{l) ... M(r-l)¢» with respect to the constraint T¢> = ¢>' . It is obvious that all of these operators are right inverses of T. If we set Mr

= M{l) .. . M(r) ,

(4.3 .28)

then it is also obvious that M r¢>' minimizes (Ho¢>, ¢» with respect to the constraint T r ¢> = ¢>' . With Mr also a right inverse of Tr, we can infer that MrTr is an (Ho¢>, ¢»orthogonal projection. Moreover, for r ~ q, MrM(r+l) . .. M(q)Tq

(MqTq) (Mrrr)

=

Mq(TqMr)rr Mq(Tq-r)T r = MqTq,

= MqTq,

(4.3.29) (4.3.30)

4.3. MINIMIZERS AND ORTHOGONALITY

307

so the sequence of projections MrTr is decreasing in the operator-theoretic sense. Now for an arbitrary configuration ¢', we can easily derive the formula for Mr¢' in momentum space:

p).

(4 .3.31)

where hr is defined by (4.2 .27) . This follows from a lattice version of the derivation given for the variational problem in §2.1D. This formula will be useful, but the decreasing sequence of projections enables us to introduce a scale hierarchy of fluctuation configurations for an arbitrary configuration ¢ - namely, (4.3.32) Clearly, N


= MNTN¢+ L~(r),

(4.3.33)

r=l

(4.3.34) N

(Ho¢, ¢)

= (HoMNTN ¢, MNTN ¢) + L(Ho~(r), ~(r)).

(4.3.35)

r=l

It is also clear that

(4.3 .36) . so we have the relation (4.3.37) as well. If we define the (Ho, ¢, ¢)-orthogonal projection (4.3.38) then Pr¢

= ~(r)

(4.3.39)

and we may write (4.3.35) as the operator decomposition N

Ho

= TtNn!j.(Ho)TN + LHoPr . r=l

Obviously, these projections commute with Ho ·

(4.3.40)

CHAPTER 4. THE RENORMALIZATION GROUP

308

We now introduce expansion functions for any scale of fluctuation configurations. First, Mr_1Tr-1rjJ - MrTrrjJ

Mr _ l (l - M(r)T)Tr-1rjJ.

(4.3.41)

Second, (1 - M(r)T)Tr-IrjJ has the property that T annihilates it, so by (4.3.13), we have (4.3.42) K,m

for some set {a(rJ.} of rjJ-dependent coefficients. Thus Km

(4.3.43) K,m

{Mr _ I (1/J(K)(. - 2 n:i,))} K,m - is a basis for the rth subspace of fluctuation configurations, but in general, this configuration basis is not orthogonal for r > 1. However, M r _ I (1/J(Kl(. - 2 n:i,)) = (Mr _ I 1/J(K))(. - 2r n:i,), (4.3.44) and M r _ I 1/J(K) is given in momentum space by (4.3.31). In this case we have _ ~ M r _ I 1/J(K)(p)

Ihr _ I (2 r -

=

2(3~+s)(r-l)

~ Ho(p)

(

l

1~p +21T ~£)1 2

L E{O,I, ... ,2

~

1

r -

1

-I}d

~

)-1

Ho(p +2- r + 2 1T e) xh r _ I (2 r - 1

p) L1/J(I<)(~)exp(-i2T-I ~ p).

(4.3.45)

n We now have a scale hierarchy of expansion functions from the unit scale on up to the length scale 2N - 1 - namely,

with 1 ::; r ::; N, n:i,E Zd, and K, ranging over its 2d - 1 values. Notice that the inverted sum in the momentum expression is periodic with period 2- T + 2 1T - as is the sum over ~, which involves 1/J(I<) . These shorter periods in momentum space correspond to the larger length scale of the expansion functions associated with larger r. By construction, Mr is a right inverse of TT, but it is a useful exercise to verify this particular property in momentum space. Given (4.2.14), we can easily write the momentum expression for the iteration of T:

eE{O,I, .. .

,2 r

x~(Tr

-l}d

p +TT+I1T),

(4.3.46)

4.3. MINIMIZERS AND ORTHOGONALITY

309

where we recall the definition (4.2.27) of hrCih If we now insert ¢ = Mr¢' in this expression, (4 .3.31) implies (4.3.47) Again, Mr is not a power of M , and the larger-scale expansion configurations are not scalings of the expansion configurations 'Ij;(I<} (. - 2 r;:;.) that we originally defined on the unit scale. The construction is tied to the quadratic form (Ho¢, ¢) on the original lattice. Although these expansion functions are not intra-level-orthogonal with respect to (Ho¢, ¢), they are (Ho¢, ¢)-orthogonal for different r . Translation-covariant orthogonalization among the basis functions at the same r-level is a standard procedure. Consider the overlap matrix

((HoMr-1'lj;(I<'})( · - 2T r;:;. '), (Mr-1'lj;(I<})(- - 2 r r;:;.))

s(r) __ KK';m'm

(2:)d

(g i: dP') ei2r(~'-~}.p

Ho(p)

x M~I<}(p)* M~K'}(p)

2(27r)

rd d

(IT r

2r "

,=)_2r"

dP') ei(~'-~}'P Ho(Tr p)

x ~I<}(2-r p)* M~I<'}(p) r

2- : (27r)

(IT,=)_"f"

d P')

ei(~'-~)'P

L

_eE{O.1 •...• 2

r

-l}d

P +Tr+ln f. )M~I<'} (Tr p +Tr+l7r f.) X~K.}(Tr p +Tr+l7r f.)*

Ho(2-r

(4 .3.48)

Then the configurations 'Ij;}K.}( . - 2r r;:;.) defined by (4.3.49)

Ls ---" m

(r}

__ e - i~ ' p ,

(4.3.50)

K,'Kim'm

are orthonormal and span the same r-level subspace. Summarizing the analysis, we see that an arbitrary configuration ¢ has a decomposition of the form N

¢

= MNTN¢+ LLa~k'lj;~Kl(. -

r

2 r;:;.) ,

(4.3.51)

r=l K. . ~ and from the standpoint of integrating out the fluctuations in the iteration of the renormalization group transformation the random variables a~k are independent with

CHAPTER 4. THE RENORMALIZATION GROUP

310

respect to the probability measure d/-LHo! The formula for nrj. (H) is now quite easy to write: e- R !j.(H)(4>')

X

= e-~(HoMN4>',MN4>')

A

(I]t,JU~: da~k) (-~ s:tE(a;k)' exp

-/ (MN4>1 + L

t

"

zf')

lim [(Z(N))-l A-+Zd

~ a~k1/J~")(' - 2T ~))) 1,

(II IT II i: da~k) (-~ L t (s:tE a;k',w'( -" ;;i)) ) exp

"

T=12"mEA

(4.3.52)

T- 1 2"mEA

"

T-

~ (a~k)2

1 2"mEA

-J

(4.3.53)

The integration prescription for the iteration is finally obtained.

References 1. T. Balaban, "Ultraviolet Stability in Field Theory: The 4>~ Model," in Scaling and Self-Similarity in Physics, J. Frohlich, ed., Birkhauser, Boston, 1983.

2. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets and Their Applications, B. Ruskai et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981),327-360. 4. K. Gawedzki and A. Kupiainen, "A Rigorous Block Spin Approach to Massless Lattice Theories," Commun. Math. Phys. 77 (1980) , 31-64. 5. J . Imbrie, "Renormalization Group Methods in Gauge Field Theories," in Critical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds., North-Holland Publishing, Amsterdam, 1986.

4.4

Connection to the Continuum Wavelet Cutoff

We have already seen in §4.2 that the Gaussian fixed point has a continuum description, but we have not described it in terms of fluctuations . First, we define the continuum

4.4. CONNECTION TO THE CONTINUUM WAVELET CUTOFF

311

minimizer to be the right inverse M of the mapping T such that M¢' minimizers (1V'1 2.(, () with respect to the constraint

T(=¢' .

(4.4.1)

It is routine to derive the formula for this mapping:

(4.4.2) and it is easy enough to infer from (4.3.31) that (4.4.3) Second, despite the functorial relation (4.4.4) established in §4.1, one has to be wary about the dual relation. Notice that (4.4.5) because (4.4.6)

{Mr } has only the groupoid structure (4.3.28). On the other hand, if we define the operator MOO such that M oo ¢, minimizes (Hoo ¢, ¢) with respect to the constraint T¢= ¢',

(4.4.7)

then this right inverse of T satisfies

M

0

(MOOr

= U.- r 0

M.

(4.4.8)

Third, the formulas (4.2.22) and (4.4.2) imply (4.4.9) so the continuum minimizer is directly related to the Gaussian fixed point Hoo(¢). The formula for Moo is

CHAPTER 4. THE RENORMALIZATION GROUP

312

so the momentum expression for the iteration (MOOr is just the formula for Mr with Ho replaced by Hoo · We now examine the continuum expansion functions (M1/J(K»)(;;- -2 m) induced by the level-one lattice configurations 1/J(K) (. - 2 n;) of the previous section. One can easily verify the orthogonality

(M1/J(K'»)(2- r '+1 ;;- -2 n; ')

.l(1V"12'(,()

(M1/J(K»)(T r +l

as follows . On one hand, M1/J(K') minimizes

(1\71 28 (, 0

-;

-2 n;),

r'

# r,

(4.4.11)

with respect to the constraint (4.4.12)

while on the other hand, U8- r + 1M1/J(K)(. - 2 n;) minimizes the constraint Tr-1T(

= =

(1\71 28 ('0

with respect to

TU;-l(

1/J(Kl(. - 2 n;) .

(4.4.13)

However, we know that

0,

r

> 1,

(4.4.14)

so u;r+l M1/J(K) (- - 2 n;) minimizes (1\71 25 (,0 with respect to a constraint whose linear transformation annihilates M1/J(K'). Hence r > 1,

(4.4.15)

and this conclusion suffices. Thus the basis generated by this scaling is a wavelet basis with respect to a Sobolev norm. We may also write (4.4.11) as ((M oo )-r'+l1/J(K'»)(. - 2r '

n; ') .l(H~
((M oo )-r+l1/J(K»)(. - 2r n;),

# r. (4.4.16) This lattice version of interscale orthogonality is not to be confused with the different orthogonality property r'

(4.4.17) of the different basis of lattice configurations generated in the preceding section. Those configurations are not wavelets because the action Ho(¢) on the unit lattice Zd was built into the construction on each r-Ievel. Now if we apply the translation-covariant orthogonalization among the wavelets on the same scale, the functions we obtain are given by iJ!~K) (-;

_2r

n;)

~)(p)

2(~-8)(r-l)iJ!(K)(2-r+l -; -2

n;),

~)S(2 p)-1/2)"K'~)(p),

(4.4.18)

4.4. CONNECTION TO THE CONTINUUM WAVELET CUTOFF

313

(4.4.19) On the other hand, (4.4.20)

where {'IjJ~I<) (- - 2r rr;,n K,m ~ ,T is the set of basis functions employed in our expansion of the fluctuation configurations on the lattice. This is not to be confused with the lattice picture of the continuum wavelets, which is easily described as follows. Introduce 'Ij;';'(I<)(. - 2 rr;,), where (4.4.21) and the configurations 1j;oo(l<) are defined in the same way as we defined the 1j;(I<) , except that the condition (4.3.16) is replaced by: (4.4.22) Since

Hoo

is a fixed point, this orthogonality implies (4.4.23)

for r > 1 as well. The canonical relationship is between the continuum and the lattice is given by (4.4.24) The functions 'Ij;~I<) (. - 2r rr;,) of the previous sections can be regarded as wavelets only in some approximate sense. We can certainly decompose arbitrary lattice configurations by applying (MOOr instead of Mr. We define the r-Ievel fluctuation configuration for a given configuration rPas (4.4.25) Obviously, N


= (MOO)NTN
J;oo(r) ,

(4.4.26)

r=l

T r J;oo(r)

= 0,

(4.4.27)

but in this case the orthogonality property is N

(Hoo
= (Hoo(MOO)NTN
(4.4.28)

CHAPTER 4. THE RENORMALIZATION GROUP

314

Since Hoo is a fixed point, we have (4.4.29) so the remainder term reduces to (4.4.30) If we introduce the (Hoorp , rp)-orthogonal projections

PToo

= (Moo y-1TT-l

_ (MooyTT,

(4.4.31)

then we may write N

Hoo

= TtN HooTN + L HooP;:"

(4.4.32)

T=l

Fluctuations can be expanded in the special configurations 1f;';"(I<) (- - 2T 11i.) corresponding to the continuum wavelets. By reasoning identical to that employed in expanding ;P
Now consider the Gaussian random field (q" dp,s) introduced at the end of §4.2, and expand the random configuration Tq, on the lattice in these new basis configurations: (4.4.34) where we have obviously taken the N = 00 limit in our application of (4.4.26) . Again, this is not to be confused with the expansion 00

Tq,

= LLa~k1f;~I<)(· - 2T 11i.) ,

(4.4.35)

T=l I< ,m ~

which is related to the continuum only by a continuum limit. If we now apply M to (4.4 .34) we obtain 00

MTq,

=L T=l

L a~k 1lt~1<\ - 2T 11i.)

(4.4.36)

~ I<,m

from the formula (4.4.24). This is the wavelet cutoff of the continuum random field q,. In the absence of this regularization, we have the decomposition 00

q,

=

L L a~k 1lt~1<\ - 2T 11i.), r=-oo I<,m -'"

(4.4.37)

4.4. CONNECTION TO THE CONTINUUM WAVELET CUTOFF

315

so MT is the (1'V1 2 8(, ()-orthogonal projection onto the subspace spanned by wavelets whose length scales are the unit scale or larger. This cutoff is very similar to the standard ultraviolet cutoff on in a sense that we discuss in the next section. The point we wish to make here is that we also have the unit-scale expansion MT

= 2)T
of this regularization, where 9 minimizes

(1'V1 2 8(, ()

~)

(4.4.38)

with respect to the constraints (4.4.39)

This function is given by .....lr.

C

~-lo.

g(p) = 1P128 Hoo(p )ry(p)

(4.4.40)

in momentum space, and each unit-scale expansion function in (4.4.38) can certainly be expanded in the wavelets with length scale 2: 1: (4.4.41)

The unit-scale functions g(.- ~) are not (1'V1 2 8(, ()-orthogonal, but it is straightforward to verify that the dual basis in the generated subspace is just the set {1V'1- 2 87](-- ~ ) }. As for the wavelet amplitudes themselves, we have (4.4.42)

i.e., these random variables are independent with respect to the Gaussian measure dJ.L8. It is important to remember that the a-variables can be expressed purely in terms of the lattice quantities as well. Just as

a(rl

I<m

= (1'V1 28 , w~I
2 r;;)),

(4.4.43)

it is also true that (4.4.44)

so (r') (r) ) (0 ",'m' ---" a Itm ---" dllHoo = ,...

;:;:;: uk'ku---", ....... ur'r· mm

(4.4.45)

Compare this to the property ;:;:;: (a (r' . .). a (r») ---" duHo = U""K,U---",---"Ur'r Kim' K,m ,... m m

(4.4.46)

of the 1/J~I<) ( . - 2r r r;;)-amplitudes derived in the previous section. The lattice averaging transformation T has an elegant description in terms of the wavelet amplitudes a(rl. I<m

CHAPTER 4. THE RENORMALIZATION GROUP

316 Notice that

(4.4.47) where we find it convenient to set (4.4.48) (4.4.49) Observe that if r

> 1, then (4.4.47) reduces to (4.4.50)

while on the other hand, (4.4.51) This means that with <jJ expanded in wavelet amplitudes as in (4.4.34) , we have (4.4.52)

The averaging transformation has the effect of throwing away the r = 1 wavelet ampitudes and shifting the assignment of the remaining amplitudes down in length scale to fill the r = 1 vacancies. As far as the lattice-continuum connection is concerned , the most important formulas to remember are

L g(-- ~)<jJ(n:) = M<jJ, Lg(·- ~)1/J;:O(I<)(~)

= q,~k) ,

(4.4.53) (4.4.54)

r; because 9 plays an important role in the wavelet description of the renormalization group transformation of a perturbed action. As for the coefficients in the expansion (4.4.41) of each unit translate of g, they have the property (4.4.55)

4.4. CONNECTION TO THE CONTINUUM WAVELET CUTOFF

317

which follows from evaluating (4.4.54) at -; _2T r;;., making the shift ~f--t~ _2T the summation index, and then inserting (4.4.41) . This identity, in turn, implies

~ . y<':l (~)4>(~) L ~m

= O(T.!., Km

r;;. in

(4.4.56)

n

which is a key relationship in the wavelet formalism . What are the basic quantities for s = 1 and the case of block-spin averaging? We have already observed that 1/ is just the characteristic function X of the standard unit cube and (4.4.57)

In the construction of the configurations with '" and then considering ~ ( )(p) wI<

1j;oo(l<)

we begin by first indexing {O, 1 }d\ {O}

d

. «) = II (1- e'P,)"

(4.4.58)

1.=1

as our candidate functions w(I<) (p) . These certainly generate a linearly independent set of solutions to the homogeneous equation (4.3 .11) because d

h(p) = II(1 +ei!Pi) ,=1

(4.4.59)

in this case. The point here is that

d d d II(1+eirr"ei!p' )*II(l-eirr"ei! p,)"~<) = II((1+ei!p')* (l-ei!p,)"~<) .. E{O,l}d ,=1 ,=1 ,=1 -I -IP (Ie:) ( +(1 - e'2 P,)* (1 + e'2 , )" ) , 4.4.60)

L

d") =

1 for at least one

L,

and

(1 + ei~P')* (1 - ei~P') + (1 - ei~P' )* (1 + ei~p,) = O.

(4.4.61)

On the other hand, the column matrices

[w(l<)

Gp

+7r1) : 1E {0,1}d1

P

are not orthogonal with respect to the weights Hoo (~ +7r 1 ) . This can be remedie~ however, by Gram-Schmidt orthogonalization, and this procedure is continuous in p to the extent that Hoo(p) is. Having constructed orthogonal

CHAPTER 4. THE RENORMALIZATION GROUP

318

for -7r ::; p, ::; 7r, we have constructed the functions WOO(K)(p) on the same p-domain and the desired configurations are (4.4.62)

References 1. G. Battle "Wavelets: A Renormalization Group Point of View," in Wavelets and Their Applications, B. Ruskai, et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349.

2. G. Battle, "Wavelet Refinement of the Wilson Recursion Formula," in Recent Advances in Wavelet Analysis, L. Schumaker and G. Webb, eds., Academic Press, Orlando, 1994, pp. 87-118. 3. K. Gawedzki and A. Kupiainen, "Asymptotic Freedom Beyond Perturbation Theory," in Critical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds., North-Holland, Amsterdam, 1986.

4.5

Canonical Wavelet Manifold and ZF Transformation

We return to the original lattice quadratic form H o(' minimizes Ho(=,. (4.5.2) This is not really the iteration of the initial mapping (4.5.3) Instead, the modification associated with n(O)

T,N

(H) =

nTf is inductively given by

n(O)

T,N-l

(H)

0

M(N)

,

(4.5.4)

and we refer to each step as the zero-fluctuation transformation. We have already discussed the minimizers MN and M(N) in §4.3, and we recall the momentum space

4.5. CANONICAL WAVELET MANIFOLD AND ZF TRANSFORMATION

319

formula (4.3.31) here:

__ ~ MNq;'(p)

=

2(3~+s)N

~ Ho(p)

(

L

~

~

IhN(2N

P +27l' eW

l E{O ,I ,oo. ,2 N _l}d

X

~ Ho(p

1

+2- N +1 7l'

~

)-lhN (2 N p) Lq;'(n;.)e- i2N ;;;.P. f) ;;;

(4.5.5)

In position space this becomes:

= 2(s-~)N L

q;'(n;.)I{JN(2- N T: - n;.) ,

(4.5.6)

;;; where we have set

(4.5.7)

The negative powers of 2 are distributed in the momentum expression for the implementation of the N = 00 limit of this function. We consider local perturbations

/(q;)

= L V(¢(T:))

(4.5.8)

of the quadratic form H o(¢), where 00

V(z)

= LUvz v

(4.5.9)

v=l

is a real-analytic function bounded below on the real axis. (This would obviously include, for example, the case U VO > 0 and U v = 0, v > vo, where vo is even.) Such

CHAPTER 4. THE RENORMALIZATION GROUP

320

an interaction is a generalized Ginzburg-Landau interaction, and we first discuss the simple effect that of R~~ has on it. To this end, we exa~ine the pointwise limit of the sequence {CPN} . It is obvious that (4.5.10) where g is the function introduced by (4.4.38) and given by (4.4.40) . The point here is that CPN stabilizes pointwise, while

2,·-j}·N

~ ( ~ "'(;;;)~N(TN;; -;;;)) •

2''"-' jH}N

We have written the sum over

~ 2- N' ( ~ .' (;;;)~N (2- N ;; - ;;;)

r

(4.5 .11)

nas a Riemann sum approximation of the integral

to see that for a given configuration 4/,

:L)MNr/>')(nt

= O(2(SII-V~+d)N)

(4.5 .12)

;:; for large N. Clearly, the lattice action loses its local property with the application of R~~ but on the other hand, locality in the continuum sense is approximated for large N . From this point of view, each term in the perturbation is changing in a roughly simple way. If we set AO,N(U) = 2(sv-v~+d)N Uv, (4.5.13) v then the iteration of the zero-fluctuation transformation is given by

R~~(H)(r/>') = Ho(MNr/>')

+ ~ A~,N (u) ~ 2- N'

(

~ .'(;;;)~N(2-N ;; - ;;;)) •

(4.5.14)

Naturally, A~, N(u) is the crudest approximation of the true output parameter which develops after N iterations of the full renormalization group transformation.

'

4.5. CANONICAL WAVELET MANIFOLD AND ZF TRANSFORMATION

321

The manifold of initial actions H(4) parametrized by s and u, with u = 1,2,3, . . . is called a canonical manifold and we shall call this particular one the canonical LandauGinzburg manifold. Any modification of the RG (renormalization group) transformation RT - such as the ZF (zero-fluctuation) transformation - can be used to create a new canonical manifold that may be technically more convenient for initiating the RG analysis. We now describe how the sequence of the ZF transformations lead to the canonical wavelet manifold, which is actually similar in nature to the canonical manifold corresponding to the standard ultraviolet cutoff so often used in the literature. First consider the continuum limit of the lattice model from the renormalization group point of view. Clearly, the rescaling of the averaged configuration to a unit lattice configuration has the effect of reducing the spacing of the original lattice, so the iteration of RT - or the sequence of ZF transformations - has the desired qualitative effect of driving the original lattice spacing to zero. Quantitatively, one must hold the output parameters fixed with respect to the iteration, as they now reflect the unit-scale behavior of the physical system after each transformation, with the configurations realized as unit-scale averages of increasingly irregular ones. Thus, with each application of the transformation the original value of each parameter must be readjusted accordingly. Even in principle, this is not as simple as it sounds, since the form (4.5.8) of the perturbation is destroyed by the full transformation. However, for the zero-fluctuation transformation, the output parameters are essentially X$N(u), as the form of the perturbation is stabilizing for large N. In this approximation, the readjustment of u , is just an obvious rescaling. O n the other hand, the N = oo limit of this scheme simply replaces a lattice regularization of the continuum model with a wavelet regularization. No short-distance behavior is really introduced until we consider an approximation of RT that includes fluctuations. In this discussion it is useful to parametrize the original action with the lattice spacing 2-N that it has after N iterations. For arbitrary lattice spacing b > 0 the configurations are related to the original unit lattice configurations by the scaling 2

d6(;)

= 6-4+"4(6-'

x),

and the action is given by ~'~'(46 =)H(4). Thus

For convenience, we set v -

~v$-sY-~ V

because it is the form of the uth coefficient when the sum over lattice sites is written as a Riemann sum. The problem is to find a &dependence XV(6)that controls a nontrivial 6 = 0 limit of a physically meaningful class of expectations for the probability

CHAPTER 4. THE RENORMALIZATION GROUP

322 measure

Zi,~ exp (- 2: All (0) _2: ¢>/i(-;)"Od) dp,g(¢>/i), Z/i,O

J

exp (-

(4.5.20)

XE 0 6

II

L:>v(o) _L

¢>/i(-;)"Od) dp,g(¢>/i),

(4.5.21)

XE 0 6

II

(4.5.22) where n is a fixed, bounded region in IRd and p,g is the Gaussian measure on the space of ¢>/i-configurations with mean zero and covariance (4.5.23) The dependence of All on the lattice spacing is not at all the dependence suggested by (4.5.19) - Le., with the U II held fixed. Indeed, in the context of constructing the continuum limit associated with the sequence of the zero-fluctuation transformations, it is the All that are independent of 0 = 2- N . The o-dependence of All appears only in the continuum limits associated with approximations of RT that include fluctuations. Now, in this crude approximation, fixing the All dictates the iterative readjustment ue,N (A) of the original

U II •

= 2("~-811-d)N All

(4.5.24)

This means that for N iterations of R~) we take H(¢» == H(u; ¢»

= H(uO,N; ¢»

(4.5.25)

as the initial action. Thus

,)i!'oo n!;' N(H (."N)) (.') = H

00 ( . ' )

+

~ '. Jd'; (~? (;;)g('; - ;;)

f

(4.5.26) where we have taken the N = 00 limit of (4.5.14) with UII = u~,N . On the other hand, if we consider the Gaussian random field <J> with covariance 1V'1- 25, we recall from the previous section the alternative formulas

(MT<J»(-;)

(4.5.27) r; 00

(MT<J»(-;)

2: 2: Q~k 1lt~1<)

(-;

_2r

ni)

(4.5.28)

r = lm,1< -

for the (1V'1 28(, ()-orthogonal projection MT. Also recall that

H oo (¢>')

= ~(1V'128 M¢>',M¢>'),

(4.5.29)

4.5. CANONICAL WAVELET MANIFOLD AND ZF TRANSFORMATION

323

which implies (4.5.30) Hence K('\; q>')

= N-+oo lim n(O) (H(uO,N))(q/) r,N

(4.5.31)

is the effective action on the unit-scale lattice for the formal continuum limit with a fixed wavelet cutoff at the unit scale. Alternate expressions are:

K('\; T4!)

~J(IVISMT4!)(-;)2d-; + 'f'\v J(MT4!)(-;td-;, (4.5.32) v=l

K(,\;T4!)

(4.5.33) Note that the limiting behavior of UZ,N is given by (4.5.24) . We have lim uO,N = 0

N-too

lim ue,N

N -+00

v

2d v

,

< d _ 28' 2d

= oo(sgn ,\),

v> d _ 28'

(4.5.34-) (4.5.34+ )

and the latter case would be of serious concern if a genuine continuum limit of the lattice N measure dJ-Lr (¢2-N) were involved. To control the effective action as N -+ 00, one has to reckon with the effect of the smaller-scale fluctuations that have been integrated out, and the infinite uZ,N -limits pose a problem for the 2- N -dependence of the '\V needed for this purpose. This issue does not come up in the sequence of ZF transformations, which throwaway those fluctuations. The manifold of these lattice actions K (¢) parametrized by s, and '\V with v = 1,2,3, ... is the canonical wavelet manifold, and it resembles the ultraviolet cutoff on a Euclidean field theory that is yet to be renormalized. The property shared by the wavelet cutoff and the ultraviolet cutoff is that the free part of the action is a fixed point with respect to integrating out the fluctuations and rescaling. However, in the case of the wavelet cutoff, this functional integration is implemented by the averaging transformation, while in the case of the standard ultraviolet cutoff, the connection is only qualitative. If we now apply the iteration of the exact renormalization group transformation Rr to K(u) = K(u; ¢), we have e-'R.¥(K(u))(')

= e-~(H=',')

lim [(Z(N))-l A-+Zd

A

CHAPTER 4. THE RENORMALIZATION GROUP

324

x

(OXp ( - ~ u.

+

~

ZY')

f

tE ·;k \

d? {

"f"(? -2';;;)

;;)((Moo)N ¢')(;;)

ntJ

(4.5.35)

~uv Jdx { ~~2r~A a~k

exp ( -

XW~K)(X _2r ni) where the partial expectation

~ g(? -

Or~N

f) )r~N'

(4.5.36)

is defined by (4.5.37)

with the relation (4.5.38) We have used

L9(X -

~)'l/!::"(K)(~)

= W~")(x),

(4.5.39)

n and the leading exponential is unaffected because (4.5.40) Notice that the integration variables are no longer the a-variables that appeared in (4.3.52), but the a-variables introduced in (4.4.33) - i.e., the hierarchy of fluctuation amplitudes on the lattice have been replaced by wavelet amplitudes in the continuum. This formula is an immediate consequence of the connection between Hoc> and the continuum wavelet cutoff discussed in §4.4. It is natural to base the zero-fluctuation approximation on the wavelet amplitudes in this context, because the form of the interaction does not change under the transformation, and the sequence is now an iteration of (4.5.41) Thus

r

n~oo' (K(u))(¢') = ~(Hoo¢" ¢') + ~ u. f d? ( ~ g(? - ;;)(MOO¢')(;;)

(4.5.42)

4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS

325

Recall that (4.5.43)

-n It follows from (4.4.8) that

(4.5.44)

(U;l M1/)(:;)

2'-~(M1/) (~ :;) ~ ~) ¢'(n). ~ 2-'L: g (1"2x-n •

d '"

(4.5.45)

n

Thus

~)(K(u))(¢')

or more concisely, the iteration has the effect n~OO)N (K(u))

= K(>..O,N (u)).

(4.5.47)

with >.~,N given by (4.5.13).

References 1. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets

and Their Applications, B. Ruskai et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349.

2. G. Battle, "Wavelet Refinement of the Wilson Recursion Formula," in Recent Advances in Wavelet Analysis, L. Schumaker and G. Webb, eds., Academic Press, Orlando, 1994, pp. 87-118. 3. K. Gawedzki and A. Kupiainen, "Rigorous Renormalization Group and Asymptotic Freedom," in Scaling and Self-Similarity in Physics, J. Frohlich, ed., Birkhauser, Boston, 1983, pp. 227-261.

4.6

Relevant, Marginal, and Irrelevant Parameters

As non-interactive as the zero-fluctuation transformation is - at best, it is a zeroth-order approximation of the renormalization group transformation - it maps the canonical

CHAPTER 4. THE RENORMALIZATION GROUP

326

wavelet manifold into itself. Its only effect on any action belonging to this manifold is a scaling of each coefficient of V(z), so it is actually a linear transformation on this linear manifold of the V(z) with the functions z" as eigenvectors. This is the transformation used to describe the notions of relevant parameter, marginal parameter, and irrelevant parameter. Henceforth, we set s = 1, as this is the Sobolev index of physical interest. Thus

(4.6.1) with <}i = Mr/>. The associated wavelets are well-localized in this case. They have exponential falloff because their momentum expressions are real-analytic in the case s = 1 as we have already shown in §2.1O. This exponential localization improves with the localization of the averaging transformation upon which one chooses to base the wavelet construction. After N iterations of the ZF transformation R'f', the output parameter corresponding to the input parameter u" is given by

(4.6.2) as 2v tion

vd

/2+d

is the eigenvalue for the eigenvector z". For those v satisfying the condiv - vd/2 + d

< 0,

(4.6.3)

the output parameter is driven to zero - i.e.,

(4.6.4) because the corresponding eigenvalue is less than unity. In this case, the parameter is said to be irrelevant. If there is an integer solution Vd of the equation v - vd/2 + d

= 0,

(4.6.5)

then the corresponding output parameter remains fixed:

(4.6.6) because the eigenvalue is unity. There is no V2, but V3 = 6 and V4 = 4. This parameter is dimensionless from the standpoint of dimensional analysis, and it is said to be a marginal parameter. Corrections to the ZF transformation would introduce logarithmic behavior of the iteration with respect to length scale for such a parameter (Le., algebraic behavior in N). Finally, for those 1/ such that 1/ -

vd/2

+ d > 0,

(4.6.7)

= (sgn uv)oo

(4.6.8)

the output parameter is driven to infinity: lim >.~,N (u)

N-t oo

4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS

327

because the eigenvalue is greater than unity. In this case, the parameter is said to be relevant. Note that the v = 2 parameter is always relevant and that its dimension is the square of a mass. Corrections to the ZF transformation may drive relevant parameters to finite limiting values. Now if all of the parameters of an initial interaction are irrelevant, then iteration of the ZF transformation damps it out: lim R~OO)N (K(u)) = Hoo .

(4.6 .9)

N-+oo

Indeed, this property extends - for such an interaction - to iterations of the full RG transformation: lim R!j. (K(u)) = Hoo. (4.6.10) N-+oo

Moreover, one can prove existence and uniqueness of the infinite-volume limit of the expectations arising from the initial action K - namely, existence and uniqueness of

}~~}Zr)-l

zr

JX(
Je-I~(¢xA)df.LH~(rP),

(rP),

r

~".J d;: ( ~;.(;: - ;;).(;;)

(4.6.11) (4 .6.12) (4.6.13)

There is no general theorem of this nature, but the methods developed to establish such a fixed-point result as (4.6.10) for a specific model invariably provide an expansion method for controlling the A = Zd limit of an expectation. The intuition here is that if the large-scale behavior of the model is asymptotically Gaussian to a degree that is exponential in N, then the long-distance effects of the perturbation are weak enough for the expansion to converge. More generally, if lim

N-+oo

R!j. (K(u)) = Koo

(4.6.14)

exists, then the limiting action Koo is an infrared fixed point. It is in the special case where (4.6.10) holds that the infrared fixed point Hoo is said to be trivial. Now, just as long-distance behavior is analyzed by fixing the input parameters, short-distance behavior is analyzed by fixing the physical quantities derived from the action obtained after N iterations. In the latter analysis, the fixed quantities are renormalized parameters, while the corresponding variable input parameters are ultravioleteffective parameters. The ultraviolet-effective parameters are not to be confused with the bare parameters, which were regarded as variable input parameters when renormalized parameters were discussed in Chap. 1. Actually, the (variable) bare parameters are the output parameters with respect to iterations of the zero-fluctuation transformation applied to the (variable) ultraviolet-effective parameters. Thus (4 .6.15)

328

CHAPTER 4. THE RENORMALIZATION GROUP

where the u~N) are the ultraviolet-effective parameters. The iterative readjustment of either these parameters or the bare parameters to hold the renormalized parameters fixed can be a difficult inverse mapping problem, but the point is that (4.6.16) Now suppose (4.6.3) holds - i.e., that U v is irrelevant - and that the renormalization does not drive the bare parameter to zero. Then obviously lim u~)

N-too

= (sgn uv)oo,

(4.6 .17)

and this means the renormalization cannot yield a continuum limit of the theory. If the bare parameter is driven to zero, then this limit is possibly finite . In the marginal case, the bare parameter and the ultraviolet-effective parameter are obviously identical. Finally, the case (4.6.7) implies lim

N~oo

u(N)

=0

(4.6.18)

1/

if )..~are,N is not driven to infinity, but the bare-parameter assumption is actually unnecessary here. The relevant ultraviolet-effective parameters are indeed driven to zero by the renormalization of the field theory. In general, if lim K(u(N») N-too

= K oo

(4.6.19)

exists, then the limiting action Koo is said to be an ultraviolet fixed point . In particular, if all of the parameters of K are relevant, then (4.6.20) As in the case of an infrared fixed point, any ultraviolet fixed point that is a quadratic form is trivial . In the previous section we described the short-distance problem in terms of the lattice regularization. Since the wavelet formalism has a lattice picture, the description of the problem in terms of a wavelet cutoff looks rather similar. However, for minimum length scale 8 = 2- N , the lattice random field rP~N on the lattice 2- N 71 d is related to the continuum field q, by (4.6.21) where the unitary operator U1 = U1 (In 2) is defined by (4.1.6). This is not the same choice as (4.5.15) unless the unit-lattice configuration is identified as TU1Nq,. Our basic motivation is that (4.6.22)

4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS

329

by (4.1.10). After N iterations of the RG transformation, the initial lattice field - now living on the lattice 2- NZ d - is chosen such that the terminal unit-lattice field is T . Thus the free part of the fine lattice action is the quadratic form associated with the covariance w ~ w ~ W Nd-2N -N N~ (¢2-N(x)¢2-N(Y))2-N = 2 ((TU1 <1»(2 x) N <1» (2N Y))dlJ.1> X (TU :i, TNZ d. (4.6.23)

YE

1

Now by (4.4 .37), we have the continuum wavelet expansion 00

(U1N
=

TN~+N L

L a~kw~I<)(2-N:i _2T~)

r=-oo m,1< ...... 00

r=- oo -"

m,1<

(4 .6.24) and therefore

(TU1 N)(n)

00

=L

L

a~~N)1/1;"'(I<)(n _2r~)

(4.6.25)

T=l m,I< -

because T is a left inverse of M. Since the wavelet amplitudes are independent with respect to the probability measure dJ.tl, the covariance washes away the scale-index shift and we obtain 00

2Nd - 2N L

L 1/1;"'(1<) (2N :i _2T ~)

T=l ~,I<

x1/1;"'(I<)(2 N Y _2T~) for:i,

(4.6.26)

YE 2- N Z d . On the other hand, 00

L

L 1/1;"'(1<) (n _2T ~)1/1;"'(I<)(n ' - 2T ~)

= (¢(n)¢(n '))dIJ.H~,

(4.6.27)

T= l m,I< -

so it follows that

(4.6.28)

CHAPTER 4. THE RENORMALIZATION GROUP

330

(4.6.29) In the end, our wavelet definition of the fine lattice covariance coincides with the simple lattice scaling given by (4.5.17). However, the wavelet definition of the fine lattice field alters the natural choice for the iteration on 2- N 7!..d We do not define the total action by a formula like (4.5.16) but rather by the formula (2- N

Hoo

f ,,=1

)

+ 12w-

u" j d

N

(4.6.30)

(>'),

Y(MTU1- N~ )(y)"

(4.6.31)

In the context of fixing the output (renormalized) parameters while iterating the RG transformation, this formula becomes

I~N(>.bare,N;I/>':i!-N) = fuS N) j

dy

(MTUIN~)(y)"

(4.6.32)

,,=1

for the interaction needed on the lattice 2- N 7!..d to yield the renormalized interaction on the unit lattice. We may also write this in terms of the bare parameters themselves:

I~N (>.bare,N ; I/>':i!-N) =

f f f

2J1Nd/2-vN-Nd>.~are,N j

d

Y (MTU1 N ~)(y)"

1'=1

>.~are,N j d Y [2N ~ -N (MTu1- N~) (2N yW

1'=1

>.~are,N

1'=1

I

.

d Y (U(l MTU1 N~ )(y)"

(4.6.33)

The composition MT is actually the (-~ . " )-orthogonal projection onto the continuum configuration subspace spanned by the wavelets with length-scale 2: 1. By the nature of a wavelet basis, this means the scaling Ui'MTU1 N is the (-~ " ·)-orthogonal projection onto the subspace spanned by the wavelets with length-scale 2: 2- N . This is the type of short-distance cutoff that this fine lattice field really imposes on the bare interaction in the continuum. In this wavelet context, the short-distance problem is to find an N -dependence of the bare parameters that controls a non-trivial N = 00 limit of the expectations M

=

jrrl/>':i!-N(-;j)dP,hN\I/>':i!-N),

(4 .6.34)

j=l

(Zt))-l exp( -I~N (>.bare,N j I/>':i!-N Xn(N»))

dp,!/-N (I/>':i!-N),

(4.6.35)

4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS Z(N)

n

331

J

exp( -I~N (Abare,N j ¢>~NXrHN»))

dp,~~ (¢>~N),

(4.6.36)

!1 n 2- N Z d ,

(4.6.37)

where the region !1 C IRd is fixed and dp,r~ denotes the Gaussian measure on <5Z dconfigurations with mean zero and covariance given in momentum space by HJ6) (p)-1 If all of the parameters of K are relevant, the short-distance limit is super-renormalizable. We have already mentioned that the ultraviolet fixed point is trivial in this case - i.e., that (4.6.20) holds. The point is that the continuum limit of (4.6.35) exists as well, and that the methods for proving (4.6.20) also prove existence. If K has a marginal parameter but no irrelevant parameters, then the short-distance behavior is strictly renormalizable, regardless of the existence of the continuum limit. Finally, if K has irrelevant parameters, then the model is said to be non-renormalizable . An action is said to be critical if the correlation length of the expectations is infinite - i.e., if the expectations do not decay exponentially with respect to the separation between local random variables. Obviously, a manifold of critical actions is closed under iterations of the RG transformation, as the effect of that complicated transformation on correlation length is to simply scale it down. It is equally clear that every fixed point must either lie on a critical manifold or have zero correlation length. In the latter case, the action is called a high-temperature fixed point, and since it is reasonable to expect the actions close to it to have finite correlation length, such an action is expected to be an infrared fixed point.

References 1. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets and Their Applications , B. Ruskai et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349. 2. G. Battle, "Wavelet Refinement of the Wilson Recursion Formula," in Recent Advances in Wavelet Analysis, L. Schumaker and G. Webb, eds., Academic Press, Orlando, 1994, pp. 87- 118. 3. K. Gawedzki and A. Kupiainen, "Rigorous Renormalization Group and Asymptotic Freedom," in Scaling and Self-Similarity in Physics, J. Frohlich, ed., Birkhauser, Boston, 1983, pp. 227-261. 4. J . Kogut and K. Wilson, "The Renormalization Group and the c-Expansion," Phys. Rep . 12 (1974),75-199. 5. K. Wilson, "Renormalization Group and Critical Phenomena I," Phys. Rev. B4 (1971),3174-3184. 6. K. Wilson, "Renormalization Group and Critical Phenomena II," Phys. Rev. B4 (1971),3184-3205 .

332

4.7

CHAPTER 4. THE RENORMALIZATION GROUP

Canonical Inverse-Limit Manifold

We have already used a sequence of ZF transformations to replace the original canonical manifold of generalized Ginzburg-Landau actions

Ho(¢)

H(u;¢)

+ I(u; ¢),

(4.7.1) (4.7.2)

I(u; ¢) with the canonical wavelet manifold of actions

r

+ loo(u; ¢),

Hoo(¢)

K(u;
(4.7.3)

~ u" Jd; ( ~>(; - ;;).(;;) ~

Uv

J (~~ a~k 1J!~"l(-; iii)) d -;

_2T

1/

(4.7.4)

",m

In the case I == 0, the ZF transformation coincides with the RG transformation, and iterations drive Ho to the Gaussian fixed point. In the interacting case, the sequence of ZF transformations - with the parameters 'xv

= 2(v-vd/2+d)N Uv

(4.7.5)

held fixed - drives H('x; ¢) to K('x; ¢). The motivation to change the canonical manifold in this way is to replace the free part of the action with a Gaussian fixed point, and the effect on the interaction - while it still has a lattice picture - is to replace a sum over lattice sites with a d -;-integral. From this point on, we choose the averaging transformation to be the block-spin transformation. Thus

J

(T
L

= 2-~-1

¢(2

r; + 'E:\

(4.7.6)

e" E{O,l}d and what we propose to do here is to introduce yet another canonical manifold, which is as natural to this choice of averaging transformation as the associated wavelet manifold, and yet involves piecewise constant functions in the continuum picture. This manifold consists of the lattice actions

H(u; ¢)

Hoo(¢)

+ Loo(u; ¢),

(4.7.7)

00

Loo(u; ¢)

=

L L sq,(r;t Uv

1: I(u;sq,),

v=l

(4.7.8)

4.7. CANONICAL INVERSE-LIMIT MANIFOLD

333

where the new lattice field 8", has a non-local, linear dependence on the original lattice field rjJ. In spite of this non-local dependence, however, the averaging transformation induced on the new lattice field is still the block-spin transformation - i.e.,

L

T~-l

s",(2 ~

+ €)

.. E{O,l}d (Ts",)(~).

(4.7.9)

This relation will be a straightforward consequence of the definition of so As a result of this property together with the identification of the new interaction in rjJ as the old interaction in s, this canonical manifold resembles the original generalized GinzburgLandau manifold, but of course it is not the same. One can certainly write Hoo (rjJ) in terms of the new variable configuration 8, but the result is assuredly not Ho(8",) . The local interaction in the lattice field s is also given by a continuum expression like (4.7.4), except piecewise constant configurations replace the wavelet configurations . The new canonical manifold has an inverse-limit relation to the canonical wavelet manifold that we shall derive in this section. Consider actions of the form 00

Hoo(rjJ)

+ L uvJv(rjJ), v=l

where Jv(rjJ) is not necessarily local in rjJ, but is v-fold multi-linear in rjJ and Zd_ translation-invariant. This class is preserved under the ZF transformation and includes the wavelet manifold and the manifold of actions (4.7.7), but not the canonical Ginzburg-Landau manifold. Recall that

RT(K(uO,l(>.))) = K(>.) ue,N (>.) = 2(v~-sv-d)N >'v.

(4.7.10) (4.7.11)

This transformation property of K extends to iterations:

RTN (K(uO,N (>.))) = K(>.).

(4.7.12)

Once we have defined the lattice field 8"" we shall see that H(>.; rjJ) also has this transformation property. Moreover, there is a sequence {h(>.;rjJ): -00 < k < oo} of interactions such that

RT(Hoo + h(UO,l(>.))) lim I k (>.) = 100 (>') , k ..... oo lim h (>.) = Loo(>') · k ..... -oo

= Hoo + h+l(>'),

(4.7.13) (4.7.14) (4.7.15)

This is a sequence of canonical manifolds driven toward the canonical wavelet manifold and away from the new canonical manifold by iterations of the transformation

Hoo

+ ~ >'vJv t - t RT (Hoo + ~ Ue,l(>')Jv)

CHAPTER 4. THE RENORMALIZATION CROUP

334

The latter manifold is the limit of extrapolating back, so we call it the canonical inverse-limit manifold. Let X denote the characteristic function of the standard unit cube. The idea of defining the lattice field s¢ is to write 00

Loo(u;4»

= LUv Ls¢(it v=1

(4.7.16)

and realize

l: xC:Z;- - -; )s¢ (-;)

as a step-function approximation of l: g(x - n)4>(n).

~

~

n

Now (4.7.17) since 9 is determined by our choice of averaging transformation - namely the block-spin transformation (4.7.6). It may seem obvious to set stf> (n) = 4>(n) and take X as the stepfunction approximation of g, but then Loo would be precisely J, and jj(u; 4» would not have the transformation property (4.7.11) in that case. The piecewise constant approximation of 9 must have a multi-scale structure. We introduce the step-function replacement

~(It)(x) -+ Lw(It)(f)x(x - e),

(4.7.18)

l

w(I<)(e)

=

J

d y x(y -

e)~(It)(y),

(4.7.19)

and we implement this at all levels through the relation (4.7.20) For arbitrary r, the replacement is

~~It)(x) -+ 2(1-~)(r-1) L w(lt) (e)x(2- r +1 x - e),

(4.7.21)

l

so we need only the expression of 9 in terms of the wavelets to obtain the multi-scale step-function approximation of g. This approximation shares a property of 9 that is important to the iteration of the RG transformation when applied to Loo. Recall that (4.7.22)

4.7. CANONICAL INVERSE-LIMIT MANIFOLD

335

so (4.7.21) induces the replacement 00

g(-; -

r;)

-+

b(-;, r;) ==

L L 'Y~1 (r;) L r=1

-

w(lt)

(e)

-l

Kim

x2(1-~)(r-1)X(2-r+1

-; -

e

rri).

-2

(4.7.23)

To obtain the desired relation 00

L ¢>(r;) L L 'Y~1 (r;) L n

j

r=1

l

1t .;;

x2( 1-~)(r-1)X(z-r+1 we need only define the lattice field

w(lt) (e)

e

-2 rri),

-; -

(4.7.24)

s'" by

d

x2(1-~)(r-1)

II O[2- +'j,J,l,+2m"

(4.7.25)

r

<=1

where [.J denotes the greatest-integer function. Now recall that an important property of the function 9 is its scaling relation to the lattice minimizer Moo - namely, (4.7.26) n

n

Inserting (4.7.22), we obtain

~~ (~'Y~1(~)(MOO¢»(~)) iT~It)(-; -2 Kim

~

,-jH

r

rri)

n

~ ~ (p;~(;;)4>(;;)) "r" G"-";;;)

~ ~ (~'Y~1 (r;)¢>(r;)) iT~~1 (-; Kim

_2r+l

(4.7.27)

rri).

n

Since each side of the equation is an orthonormal expansion, we may equate the (r, rri, k)-coefficients with the index shift r t--+ r + 1 on the left-hand side. Thus (4.7.28)

0, r

2: 1,

(4.7.29)

CHAPTER 4. THE RENORMALIZATION GROUP

336 and therefore

%; E(p~1 (;;)¢(;;)) ~;W(.' (') x2(1-~)rX(Tr -; - -; -2

T~+I~)

G-;, ~)

r;;,)

¢(r;).

(4.7.30)

n

This is the important property that b shares with g . As far as the lattice field s¢! is concerned, we have yet to establish the property (4.7.9). The iteration of the RG transformation has already been realized in terms of integrating out the variables a(r]., with Hoo(¢) realized as half the sum of the squares I<m of these variables. We can write s¢! in terms of them as well. We have (4.7.31) if we combine (4.7.25) with (4.7.32) More generally, we define the variable configuration

s~q)(h

~

00

=

L)Tq¢)(~)LL,~1(~)Lw(I<)(e) ;:;

r=q

x2(1-~)(r-q)

II
7

I<,~

d

r

+q j,J,l,+2m, ,

(4.7.33)

,=1

for which the alternative formula is

d

x2(1-~)(r-q-l)

II


(4.7.34)

,=1

Note that this large-scale tail of s¢! has been scaled back to the unit lattice. Moreover, the block-spin averaging of the original lattice field ¢ induces the same averaging of the new lattice field s¢! . Indeed (with s~O) = s¢!)

(Ts~q-l))(])

=

T~-1

L E"E{O,l}d

s~q-l)(2 -; + ~)

4.7. CANONICAL INVERSE-LIMIT MANIFOLD

337

00

2-~-1 L)(1-~)(T-q) LQ~k L w(lt) (C) T=q It,m d

II (b[2,=1

r

+o+ 1 j,J,e,+2m,

+ b[2-

r

+O(2j,+1)),e,+2mJ .

(4.7.35)

A crucial observation here is that r

> g,

(4 .7.36)

from which it follows that the r > g contribution collapses to

00

L 2(1-~)(T-q-1)LQ~kLw(It)(C) T=q+1 It,m d

X

II b[2-

r

= s~q)(h.

+ O+ 1 j,J,e,+2m,

,=1

What about the r

= g contribution?

T~-1 L Q~~

(4.7.37)

We have

L

w(lt) (2 -; -2

r;;, + 1) =

0,

(4.7.38)

;;" E{O,1}d

It,m

and so we have verified the desired compatibility:

T s¢(q-1) -- s¢(q)

(4.7.39)

s~q) = Tqs¢ .

(4.7.40)

This implies It is important to notice that in (4.7.38) we have used the wavelet property

L

r;; + 1) = 0,

w(lt) (2

(4.7.41)

;;" E{O,1}d

which we shall verify later.

This is not to be confused with the vanishing moment

property

L

e

w(Itl(C) =

J

w(lt) (X")d

X" = 0

(4.7.42)

CHAPTER 4. THE RENORMALIZATION GROUP

338

that all wavelets have. (4.7.41) is a stronger property that reflects the block-spin construction of our wavelets, which are certainly not supported on cubes. Given that H(u;
e-R~(K(u))(4>') = e-!(H~4>',4>') }~~d [(.ZlN))-l ( exp ( +

- 100 (u; (Moo)N 1//

L t L a~k1/J;:"(Itl(. - 2 ~)))) J, r

It

(4.7.43)

r5,N

r=12rniEA

where we have inserted (4.5.39) in the fluctuations. Obviously, the effect on obtained by replacing 100 with Loo :

e-R~ (H(u))(4>') = e-!(H~4>' ,4>') A~~d [(Zt'))-l ( exp (

+Lt L a~k1/J;:"(It)(--2r~)))) It

Zt')

r=12rniEA

= ~XpCLoo(Lt \

\

It

H is

- Loo (u; (Moo)N 1//

],

(4.7.44)

r5,N

~ a~k1/J;:"(Itl(.-2r~0\) J) r5,N,

(4.7.45)

r- 1 2 r mEA

with the partial expectation Or5,N defined by (4.5.37). On the other hand, (4.7.46)

so if we iterate (4.7.30), we also have 1_ 00 (u; (Moo)N '

+L It

t

L a~k 1/J;:"(Itl(. - 2r ~))

r=12 r niEA

~ ,i"-"'i'H)N"" Jd; { ~ b(;, n)ql' (n) + 2(~-1)N L It

f

L a~k L b(2N ~, ~)1/J;:"(It)(~ _2r ~)r, (4.7.47)

r=l 2rniEA

-;;

J

where we have made the change of variable ~ I-t 2N ~ . Hence

e-R~(H(u))(4>') = e-!(H~4>',4>') Al~~d [(ZlN))-l( exp ( _ ~2(V-Vd/2+d)Nuv

4.7. CANONICAL INVERSE-LIMIT MANIFOLD

XL

L

2- Nd

-; 7 E{O,1, ... ,2 N

339

{sq,.c'}) + 2C~-1)N L f)(1-~)(r-1)

_l}d

K

r=l

~ Q~kwCK)(2N-r+l -; + €,Cr) -2 n;)}"))

J,

(4.7.48)

r5,N

2"mEA

(4.7.49) where we have used the identity (4.7.50) This completes our description of the action Loo (u; 0, we still make the replacement (4.7.21) when r S q, but when r > q, we make the replacement

'l1~I<)(~) --+ 2Cl-~)(r-l) Lw~~/i)x(Tq+1 -; - e),

w~~)(e) =

(4.7.51)

e

J

x(y - e)'l1 CI<)(T r '

y)d y .

(4.7.52)

Thus b(-;, T:) has been modified:

r=l

i

K. ,m

+ L2Cl-~)(r-1)L,~1(T:) r>q

K,m

(4.7.53)

-e We define the interaction for k

= -q by (4.7.54)

It is straightforward to verify the relation

CHAPTER 4. THE RENORMALIZATION GROUP

340

R'T(Hoo

+L

q_

1(uO,l(A)))

= Hoo + Lq(A),

(4.7.55)

and it is obvious that lim Lq(uj ¢) q--+oo

= Loo(uj ¢).

(4.7.56)

For k 2': 0 we define the interaction by (4.7.57) so the group property (4 .7.55) is extended to all integers. Finally, (4.7.58) because h(uj ¢) reduces to a Riemann sum approximation of the loo(uj ¢), and the refinement of that Riemann sum has scale 2- k

4.8

J d-;;-integration for

Wick Ordering and Linearized RG Analysis

In this section we discuss the linearized renormalization group transformation in the neighborhood of the trivial fixed point Hoo - more precisely, the Frechet derivative of the RG transformation at Hoo on a manifold of actions. We first consider actions of the form

where

(4.8.1) and the kernels Jv (-;; 1, ... , -;; v) are symmetric in their arguments. This manifold contains the canonical wavelet manifold and is closed under iterations of the linearized RG transformation. The linear transformation is defined by (4.8.2) for an arbitrary interaction J, where we suppress the T-dependence because T has been specified as the block-spin transformation. Clearly,

LH_ (J)(.')

~

\ J (

M~.' + ~ "om":;;'O») )'n••

-\' (~"om":;;"») )'0.• )

1

(4.8.3)

4.8. WICK ORDERING AND LINEARIZED RG ANALYSIS

341 (4.8.4)

Thus 00

(4.8.5) v=l

II) II! ( d-;,

[~(1I-1)l J II (-;l"",-;II)

,=1 11-2/J

X

L (;)

/J=O

/-L

L

II (U1 M¢')(-;,) 1

pairing parti tions P of {1I-2/J+1, ... ,1I}

l.= 1

(4.8.6)

where we have applied the symmetry of the kernel together with M'Ij;'::(I<) m

MM

OO

= iT~),

(4.8.7)

m

1

(4 .8.8)

= dKKIOmml1

(4.8.9)

= U1- M ,

(aK~QK'm/)O

and the standard formula for Gaussian expectations. Let

JII /J(-;l ,"" -;1I-2/J)

=

(=It+l! d -;,) J II (-;l ,"" -;11)

pairing~rtitions P of

{1I-2/J+1, ... ,1I} (4.8.10)

so that

342

CHAPTER 4. THE RENORMALIZATION GROUP

(4.8.11)

,=1 If we make the change of variable -; : = ~ -;, for 1 ~

L

~

V -

21-£, we obtain the formula

v-2J1-

J V J1- (2 -; ~, . .. ,2 -; ~-2J1-)

IT (M¢') (-; ~).

(4.8.12)

,=1 Therefore, if we set

(4.8.13) we have (4.8.14)

(4.8.15) This completes our description of the linearized RG transformation in the neighborhood of the Gaussian fixed point on this non-local extension of the canonical wavelet manifold. In the renormalization group context, the most natural parameters of an interaction are determined by the expansion of the interaction in those special interactions which are eigenvectors of the linearized RG transformation. The eigenvalues determine the change in the parameter values to first order. On the other hand, recall the definition of Wick ordering in §1.12, and note that since the Gaussian covariance (4.2.36) in the continuum for s = 1 is actually diagonal in the random variables (4.8.16) we have : F(a)G(a):

=

: F(a) : : G(a) :

(4.8.17)

if F(a) and G(a) depend on disjoint sets of a-variables. If we consider the interaction

4.8. WICK ORDERING AND LINEARIZED RG ANALYSIS

343

(4.8 .18)

we use the properties of Wick ordering rather than the formula. Thus LH_ (J)(¢')

~ (J (Moo¢, + ~ a.i i ¢::;'·)) ) I<,m

("'.;;;)

-(1 (~a'iii¢::;")) ) I<,m

=u

("'.;;;)

~ (~) J \= (MMoo¢/)(-;)f (~al<;;;w~)(-;)) "-') d -;

I< ,m

=u

~ (~) Jd -;

: (MMoo<jJ,)(-;)t : \

("' .;;; )

(~a'iii"~)(i) "-'; I<,m

)

(4.8.19) 0

because (MMoo<jJ')(-;) does not depend on the variables a I<m -. Since the expectation of any Wick power is zero, this sum consists of only the £ = 1/ term:

u u

J J

d -; : (MMoo<jJ')(-;t : d -; : (Ul 1 M<jJ') (-;)" :

u 2(1-d/ 2),,+d

J

2"-,,d/2+ dJ(<jJ').

d -;' : (M<jJ')(-;')" . (4.8.20)

This means that the interaction is an eigenvector of £H~ with eigenvalue 2"-,,d/2+d, and so the linearized RG transformation has the same effect on the class of interactions of the form (4.8.21)

as the zero-fluctuation transformation has on the class of interactions of the form (4.8.22)

CHAPTER 4. THE RENORMALIZATION GROUP

344 For

(4.8.23) v=1

the transformation is simply U v >--+ 2v-vd/2+duv in each case. The space of interactions on which we defined .cH~ may be regarded as an extension of the canonical wavelet manifold to a manifold of non-local interactions. This whole construction applies to the inverse limit manifold as well. Accordingly, we now consider the space of actions of the form 00

where (4.8.24) and the kernels have the piecewise constant structure (4.8.25) Thus ~

V

-.lo

J

Bv(j 1" ", j

~

II sq,(j J.

(4.8.26)

<=1

The linearized RG transformation in the neighborhood of Hoo on this manifold of actions is still defined as the Frechet <;lerivative (4.8.2) and given by the formula (4.8.3), but in this case it reduces to

g

v-2J1. ( X

~ b (~ x" ~ )
)

~rtitions

pairing

P of {v-2J1.+1, ... ,v}

4.8. WICK ORDERING AND LINEARIZED RG ANALYSIS

""' ~ ~ ~ ~ L.J L.J b(x" n)b(x", n II (""' K,m

{t.,,,/}E'P

I

00(1<)

)'I/J;.

~

00(1<)

(n)'I/J;.

345

~

I

(n)

)

,

(4.8.27)

;',;'1

where we have applied (4.7.30). Introduce

(4.8.28)

(1<) _

(4.8.29)

s_ - s.,.=«), m

'+'~

which induces

=

) IIV (,=v-2/L+1 J d X, ~

""' ~ ~ ~ ~ ;;,~, b(x" n)b(x", n II (""'

{",'}EP

""' L.J

~ J v (X1, .. . , ~ xv)

pairing partitions "P of {v-2/L+1 , ... ,v}

I

00(1<)

)'I/J;.

~

00(1<)

(n)'I/J;.

~

I

)

(n)

(4.8.30)

Thus

v-2/L

B V /L(}1 , .. . '-; v-2/L)

II xC~, - 7')

,=1

(4.8.31)

CHAPTER 4. THE RENORMALIZATION GROUP

346

and, as before, we make the change of variable

x~ = ~ x, for 1 ::s

£

::s /J -

2p, to obtain

11-211BIIJ; l' ... ,

j 11-211-)

II X(2 x; - j,) ,=1

(4.8.32)

Once again, we use the identity

(4.8.33)

and now observe that

J

X(2

x; - j ,)x(X; - j;) d x; = 2- d _

L

(4.8.34)

e E{O,1}d

If we set ~

B~, (j ~ , . .. , j~,)

(4.8.35) then

(4.8.36)

---l.

.....10.

' ( J. ,1 " ' " J. ,II' ) B II'

B~,(¢') = "

J

"

1' .. ·• J

v'

II St/>' (.J,') , --10.

(4.8.37)

,=1

vi

and so £H~ is now derived for the non-local extension of the inverse limit manifold as well. The special Wick-ordered interactions realized as eigenvectors of £H~ in this new manifold are analogous to those found above. If we apply £H~ to an interaction of the

4.8. WICK ORDERING AND LINEARIZED RG ANALYSIS form

J(1))

u

U

JdO:. (~b(O:, n).(ii)

347

r"'

L :s",Cit :

(4.8.38)

j

the calculation is exactly the same as in (4.8.19) . Hence

LH_ (J)(.') =

r r r

JdO:. (~b(O:, n)(M 1>')(n) 2('-''')"u JdO:. (p G 0:, n) .'(0) oo

u

2" -"'f a,u 2v -

vd

JdO:'. (~b(O:, n)
/2+ d J(¢/),

(4.8.39)

where we have applied (4.7.30) again. This means the linearized RG transformation has the same effect on

J P(O:, n).(n)) • dO: V (

=~

V(,.U ))

(4.8.40)

J

as it has on

References 1. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets

and Their Applications, B. Ruskai et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349. 2. K. Gawedzki and A. Kupiainen, "Asymptotic Freedom Beyond Perturbation Theory," in Critical Phenomena, Random Systems, Gauge Theories , K. Oster walder and R. Stora, eds., North-Holland, Amsterdam, 1986, pp. 185-292. 3. J . Kogut and K. Wilson, "The Renormalization Group and the E-Expansion," Phys. Rep. 12 (1974), 75-200.

CHAPTER 4. THE RENORMALIZATION GROUP

348

4. S. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, 1976. 5. K. Wilson, "Renormalization Group and Critical Phenomena I," Phys. Rev. B4 (1971),3174-3183.

4.9

The Second-Order RG Transformation

As far as the linearized renormalization group transformation is concerned, one can easily verify closure of the sub-manifold of interactions given by kernels of the form v

JV(~l' . .. ' X"v)

= uvCa;d II 8(X"1 - X",)

(4.9.1)

,=1

in the case of the extended wavelet manifold, or by kernels of the form (4.9.2) in the case of the extended inverse limit manifold. It is reasonable to wonder why nonlocal interactions were considered. Actually, such interactions become unavoidable when one applies the second-order approximation of the RG transformation in the neighborhood of Hoo. This approximation is just the second-order truncation of the cumulant expansion. We consider the extended wavelet manifold first. Thus

n (Hoc + ~ JV) +

}!.~,

(rP')

H

exp (-

-1++ ~J"

= Hoo(rP')

~J" (~£ ".~v~))) )'a.•' (U11MrP' +

~ ~ QI<;;;IJ.i~)))) l' 2mEA

(4.9.3)

(a.~)

while the cumulant expansion is formally given by

00) n (HDO + ~ Jv

1dn

=

DO)I ~ n! dtn n ( HDO + t ~ J v t=o · D0

Therefore, the second-order RG transformation is given by

(4.9.4)

4.9. THE SECOND-ORDER RG TRANSFORMATION

and we already have the formula for

LH~ (~Jv)

349

from the previous section.

To calculate the second-order correction, we find it convenient to write the interaction in terms of Wick-ordered products. Since 1!(v-1)]

v

II(U1-1Mcf»(~,)=

,=1

2.:

L

11-=0

rC{1, ... ,v}

'Ilr

card r=211-

it follows from the symmetry of J v (~ 1, .. . , ~ v) that 00

00

11=1

11'=1

2.: Jv(cf» = 2.: Xv' (cf»,

(4.9.7)

v'

XX211-+V"I1-(~1""'~v') : II(UI1Mcf»(~J : , ,=1

(4.9.8)

CHAPTER 4. THE RENORMALIZATION GROUP

350

L

J V '+2/l(:;1, ... ,:;v'+2/l)

pairing partitions

l' of

{v'+1, ... ,v'+2/l} (4.9.9)

X/l V is not to be confused with J/l V , as the definition (4.8.10) of the latter quantity involves the r = 1 level of wavelets only. However, (4.9.10)

and therefore,

-~ \(~1 Xv' (U M¢/ + ~al<mW~)))') 1

1-

m,1<

+

(OK;;;)

~ (~Xv,(UI1MIjII)r

(4.9.11)

Applying (4.9.8) and interchanging the double sums with the a-expectation, we obtain

g(~al<mili~)(:;')) :g(~al<mili~)(Yi)) =) -\ g ~al<mili~)(:;')) :

x [ (:

K,m

(U1-1 MIjI I)(:;,)

K }m

+

I< ,m

(QI(~)

4.9. THE SECOND-ORDER RG TRANSFORMATION

X:

g

351

~aKmW~)CYJ) :)

((UI1M¢I)(YJ+

~,m

(OK;;;)

g g (: g(~al<mW~)(-;')) fi (~al<mW~)(YJ) (UI 1M¢/)(yJ : ].

(U1-1 M¢/)(-;,) : :

+:

(4.9.12)

By the properties of Wick-ordering,

::

K,m

I'i.,m

L,

O,N

6(~w~)(-;')W~)(YP(')))

PEPerm{1,oo.,v} ,-1

(: g g((U,-'

,

:) (01(;;;;-)

(4.9.13)

I<,m

((U1- 1M¢/)(-;,) + ~al<mW~)(-;')) :

X

I<,m

M¢')(Yd +

~ "~'CY'))) a.iii

(nK~)

K.,m

L

L: II (U

1-

rC{1,oo.,v'} X :

rC{1,oo.,ii'}

1

M¢/)(-; ,) :

,~r

II (UI 1M¢/)(y i) :

0eard

r,

card

r

i~r

X

p~

f

!! (~?~' (;;,) ,,~, (y

P(,') )

(4.9.14)

Note that when f)' = v', the r = {I, . .. , v'} term in the latter expression is just the former expression, so there is a cancellation in the f)' = v contributions to (4.9.12). In fact, the last contribution in the brackets is also a term in (4.9.14), specifically, the r = 0 term. These observations together with the symmetry of X 2 1'+v',1' (-; 1, .. . , -; v' ) in its arguments reduce the second-order correction to

CHAPTER 4. THE RENORMALIZATION GROUP

352

X21'+v' ,1'(:;; 1, ... , :;; v' )X2/HIi' ,ily l' .. X

.,

Y Ii')

g(~ifJ~)(:;;')ifJ~)(YJ) ,Q.1(U1.Ii (IT J (n J 1

:

Mr//)(:;;,):

It,m

/ 1 (U1- M¢/)(Y;) : - Jv,£o,v !

X :

,=q+1

d :;;,)

d Yi)

,=1

,=1

X 21'+V',I'(:;;l,"" :;;v' )X2/Hv' ,{t(Yl" ' " Yv') X

i1 (~?~)(;;')W~)(Y.)) }

(4.9.15)

The summand is clearly symmetric with respect to the interchange v' ++ iY, so we separate out the iY = v' terms, replace the double sum over ii' '" v' by twice the double sum over iY > v', and then introduce the shift i/ = v' + ij. Thus

X

tq!(VI) (VI +ij)3S~~v'+
q=l

(4.9.16)

q

where 3S~~';I'{t (¢/) denotes the qth multiple integral in (4.9.15) and the q = v' contributions to the v' = v' part have been cancelled out. Now in the multiple integral, we wish to express each of the two Wick-ordered products in terms of ordinary products. We have the standard formula [~(v' -q-l)]

v'

: II

L

(UllM¢/)(:;;.):=

L

(_l)l

rc {q+l , ... ,v'}

l=O

card r=2l

v'

II ql r

II

L

(Ull M¢/)(:;;,)

t. =q+l

{",'}E'P

pairing partitions 'P of r

'" '" ~ ,) ifJ~ ( ~~~~ifJr~(x (It)

(It)

~

(X,,)

)

,

(4.9.17)

4.9. THE SECOND-ORDER RG TRANSFORMATION

353

and therefore, by symmetry of the kernels in their arguments (again),

3(~), ' . . (q/)= v ,v +q ,I-' ,1-'

x ,."0,

[~(v'-q-1)J [~(v'H-q-1)J

~;,;"O,

P of {q+1, ... ,q+2f}

X 2I-' +v' ,I-'(? 1, .. x

'~ "

f=O

f=O

,""0,

(UI 1M
i=q+2l+1

x

E,,,"o,' (g Jd;?.)

? v' ) X 2!H V'H,I-'(Y\"

K,m

VIr

,

'A

-q)(y +~-q) 2e

(_I)f+l(Y 2e

of {q+1, ... ,q+2i}

tr (~I]i~)(?,)I]i~)(YJ) ,-1

x

.,

'LJ "

(tv dy, )

. . , YV'H)

IT

(U1- 1 Mq/)(?,)

,=q+2l+1

II (fLLI]i~~(?,)I]i~~(?L'))

{L,L'}EP

r=1 '"

un. (%; ~ p;~ (Y')";~

~

rY,,) )

(4.9.18:

Inserting this in (4.9.16), we see that we have finally expressed the second-order correction in the desired form: (4.9.19)

(4.9.20) where ~;

=~

~ L and ~, ranges over the variables x q+2f+1, . . . , x v', Y q+2f+1 "

..,

Y v'+
the admittedly complicated formula for J~" (~~, . .. , ~~,,) is based on the collection of terms for each = + q2£ - 2l. (4.9 .21)

y" 2y'

2q -

We shall not write out that formula. One may easily obtain it by changes of indices in the multiple summation. The second-order calculation is combinatorially identical for the extended inverse limit manifold . In that case we have

354

CHAPTER 4. THE RENORMALIZATION GROUP

~:2R(Hoo+t~Bv)(~I{=O = (Formula (4.9.16) with BS~~';/LJi. replaced by nS~~';/LJi.)'

[~(v'-q-l)][!(v'+
X

"(_l)l+i

~ f=O

~

!1 ~ g F V'

~tions ~

v'

(S¢'

(j ~))

L=q+2l+1

[I _ L ( L-l

~

v' +q

II

)

) (v,+q

J

L

( v'

II L

(S¢' (kD)

i=q+2i+l

v,+q

( ' ) ( '+ ' ) 1/ - q 1/ ~- q 2f 2f.

i=o

pairing P pairing Eitions fj ( of {q+l, ... ,q+U} of {q+l, ... ,q+2f}

II

(4.9.23)

)

i

)

L=17,E{O,I}d

~

W 2/L+v',/L(2 j~+

~ '€\, ... ,2 j~,+ Ev')

6,E{O,I}d

(4.9.24)

x

L

II

pairing partitions P {L,L'}EP of {v'+1, ... ,v'+2/L}

LLLS~~(iJ<~(i'L')' 00

(

r=l

)

I<

-

m

(4.9.25)

4.9. THE SECOND-ORDER RG TRANSFORMATION

355

(1<) (-:-) S - J

rm

x2(1-~)(r-1)

d

II t5[2-r+lj~J,l~+2m~ .

(4.9.26)

1'=1

Thus (4.9 .27)

(4.9.28)

where f. : ranges over the discrete variables .,

.,

J q+2i+1' · .. ,lv"

and the

index~relatio~ is

k'

q+2f+1'···'

k'

v'+
still (4.9.21) . The combinatorics for collecting the contribu-

tions to B~" ( f. ~ , ... , f.~,,) is exactly the same as before. The formula for the second-order approximation n[ [ of n becomes computationally useful for iteration only when further approximations are made. It is much too complicated as it stands, whether one works with the extended wavelet manifold or with the extended inverse limit manifold. Wilson's celebrated second-order calculation first defines the renormalization group transformation n in terms of the standard ultraviolet cutoff in the continuum and then derives nIl in much the same way as above, except the mass term is absorbed into the definition of the free covariance and the multiple integration is in the momentum variables. The complicated formula obtained in that context is then simplified by very crude approximations - specifically by replacing many of the propagators in the momentum integrands with constants! This is qualitatively justified by the presence of the mass in the denominators of these momentum expressions. Finally, the irrelevant terms are discarded - i.e., the terms made negligible by the scaling effect under RG iterations. In both of our formalisms it is interesting to search for the most appropriate qualitative approximations for the simplification desired for iteration. We shall not pursue this issue in the case of the extended wavelet manifold. In the case of the extended inverse limit manifold, we choose Wilson's hierarchical approximation, which can be introduced at the non-perturbative level. We discuss this in subsequent sections.

References 1.

J . Kogut and K. Wilson, "The Renormalization Group and the c-Expansion," Phys. Rep. 12 (1974) , 75-200.

CHAPTER 4. THE RENORMALIZATION GROUP

356

2. S. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, 1976.

3. K. Wilson "Renormalization Group and Critical Phenomena I," Phys. Rev. B4 (1971), 3174-3183.

4.10

Wavelet Diagrams

At this point an introduction to the wavelet analog of Feynman diagrams is appropriate. Consider a quantity arising in the perturbation theory when the (more familiar) sharp ultraviolet cutoff on the Euclidean field is used in defining and iterating the renormalization group transformation - say, the quantity

Jd-; J

dy

~1(-;)~1(y)(C,,(-;,y)-C1(-;,y))3,

where ~,,(-;) is given by (1.11.52) and the field ~ is the continuum argument of the effective action after N iterations. Clearly,

((~,,(-;) - ~l(-;))(~"(Y) - ~l(Y)))O 1 ~~~ _1_ d ~ e-i(x-y).p (27T)d P i p 12 l:5lpl:5"

J

(4.10.1)

with (j = 2N after N iterations of the RG transformation defined by (4.19) instead of by block spin averaging on a lattice. The Feynman diagram of the integrand is given by Fig. 4.10.1, where the external lines correspond to the field amplitudes ~1(-;) and ~1 (Y). This diagram does not have the global meaning that it had in Chap. 1, since the internal lines correspond to the smaller-scale fluctuations that have been integrated out by the RG iteration, while the external lines correspond to the larger-scale fluctuations that are still present as field amplitudes .

Figure 4.10.1: What is the analog of this quantity in the wavelet formalism? The key observation is that the wavelet cutoff 00

~w,,,(-;)

L L Q~k w~"l(x - 2T i7i),

=

r=-N+1

(j

= 2N ,

(4.10.2)

",r;;

replaces the standard ultraviolet cutoff as a regularization. Following the discussion in §4.6, we observe that with (j = 2N , W

~


(j

d/2

1

N

~

-(TUl~w ,,,)((jX),

= uf' M(r;t-t (j-d/2+1
~

1

d

XE(j-Z,

r;)) .

(4.10 .3) (4.10.4)

4.10. WAVELET DIAGRAMS

357

Clearly, the wavelet analog of CO' (;;, y) - C1 (;;, y) is given by

Cw,O' (;; ' y) - Cw,t{;;, y)

((~w,O'(;;) - ~w,t{;;))(~w,O'(y) - ~W,l(Y)))O

L L W~I<)(;; _2T ~)W~I<)(y _2r~)

(4.10.5)

-N
because (r) (r') ( a ~a -)0 Km K'm'

= 81<1<,8--,8 ,. mm rr

(4.10.6)

Therefore, the wavelet quantity is just

so if we insert (4.10.7) we obtain the reduction

Each term is represented by a diagram of the type given by Fig. 4.10.2. The long dashes represent modes, while a solid vertical line represents an integral of the product of those wavelets whose dashes lie on that vertical line. Each dashed horizontal line is an identification of the modes it connects, and that mode amplitude has been integrated out; the mode appears only numerically. The height of a long dash on a vertical line illustrates the scale of the wavelet, where the smaller length scales are "lower" The mode identification lines correspond to the internal lines of the Feynman diagram given by Fig. 4:10.1. Just as one integrates over the position variables in the Feynman diagram to calculate its contribution, one sums over the modes satisfying the case structure of the wavelet diagram; the position integrals are just numerical factors in the wavelet context. If the bare parameters are held fixed during the RG-iteration, Figure 4.10.1 is an ultraviolet-divergent diagram for dimension d > 2. In the wavelet setting, this divergence manifests itself in the sum over (r"~,, K;,), ~ = 1,2,3. Indeed, for the sake of simplicity, consider the sub-sum where (r / , ~ I, K;I) = (r,~ , K;) and the modes (r"~,, K;,) are identical. Dimensional analysis of the integral reveals (4.10.8)

CHAPTER 4. THE RENORMALIZATION GROUP

358

for the modes (r1, n;,1 , 1\:1) in the vicinity of (r, n;" 1\:), while the set of such (r1, n;,1, I\:d has cardinality O(2(r-r d d) . Since each term is the square of the integral, the sum over all (r1, n;,1, 1\:1) on one level s S 0 obviously has magnitude O(2(6-2d)82 2r ). For d > 3, this single-level sum diverges as s becomes arbitrarily negative. For d = 3, each one,2- N +1 is clearly level sum is O(22r), but then the sum over the levels s = I, O(N22r), in which case the sum over small-scale modes still diverges as N --+ 00.

!, i, .. .

Figure 4.10.2: The second-order (two-vertex) perturbative contribution we have just discussed is obviously generated by the 1/ = 4 term in the interaction

00

100 (uO,N (>.); -:;:;>---+ 17-d/2+1
-:;:;))

= L 17vd/2-v-d >'v v=1

~ L.J >'v

J ~,

~, d x
t,

(4.10.9)

v=1 Other perturbative contributions generated by the 1/ = 4 interaction term are given by the Feynman diagrams in Fig. 4.10.3. The integrations in the vertex variables yield the quantities

Figure 4.10.3:

4.10. WAVELET DIAGRAMS

359

J d -; J d -; , J d -; "C 1(-;, -; '?C 1(-;,-; ")2C 1(-; "-; ")1 (-; ')1 (-; "), Jd-; Jd-;IJd-;1I u•

u•

U•

J d -; J d -; J d -; ", C 1(-;, -; ')C 1(-;, -; ")C 1(-; u•

I

u•

U•

I, - ;

")1 (-;)2,

respectively, where (4.10.10) The corresponding wavelet diagrams are given by Fig. 4.10.4, in which we have

Figure 4.10.4: introduced the abuse of notation (4.1O.11)

with L# treated as a variable index ranging over the modes. The sums over the L# yield the quantities

L

L

C¥1C¥2

1,2

C¥l,C¥2'

T l , T2>O

Tt' ,T2'

L 3.4

I' ,2'

>0

T3,T4::;O

2,3 ,4 T2,T3,T4$O

L 1,2 rl,T2> O

L

C¥1C¥2

I' 2 ' Tt'

,r~, > 0

C¥l,C¥2'

JII w, J 4

W1,W2,W3 W4,

,=1

CHAPTER 4. THE RENORMALIZATION GROUP

360 4

L jII'li, ,=1 3,4

2',3' ,4'

T31r4~O

T2"T3"T 4 , ~O

respectively. From the standpoint of analyzing ultraviolet divergences, wavelet diagrams create a mind-set that is somewhat different from the more familiar Feynman diagram point of view. As we have seen above in the example given by Fig. 4.10.3, the analysis reduces to counting sub-cubes in a given cube!

References 1. G. Battle, "Ondelettes: The QED3 Connection," Ann. Phys. 201, No.1 (1990), 117- 151. 2. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360. 3. J. Kogut and K. Wilson, "The Renormalization Group and the c-Expansion," Phys. Rep. 12 (1974), 75-200.

4.11

The One-Loop-per-Scale Transformation

Having extensively discussed the canonical structures associated with the zero-fluctuation transformation and the linearized RG transformation, we now examine a more interesting modification of the RG transformation that is quadratic in a certain sense, but quite different from the second-order approximation. There can be problems with the stability of this approximation, but it is actually a scale-by-scale version of what is known in quantum field theory as the one-loop approximation. The traditional version is defined at the perturbative level for all scales, with all Feynman diagrams discarded except the one-loop diagrams. What is introduced here will not be that simple, but each application of the transformation is just Gaussian integration with the (highly nonlinear) contribution of the background field changing with each step. On the other hand, it is widely believed that the only fixed point for this modified transformation is the trivial (Gaussian) fixed point. We shall discuss this issue in a later section. First consider an action K(u) from the canonical wavelet manifold together with its RG transformation R(K(u)) given by e-'R(K(u »)(¢')

e-~(H~¢' ,¢') AEZ. lim [z-11 ex p { A \

-JdX

-

V

(~g(; ~)(Moo.,)(;;)

+

L ~ a,,~'li(")(~ -2~))}) "

2mEA

(Q.~)

],

(4.11.1)

4.11 . THE ONE-LOOP-PER-SCALE TRANSFORMATION

361

where we suppress the T-dependence because the choice ofT has been fixed by (4.1.3). The modification discards all terms in the exponent that are of order three or higher in the fluctuation amplitudes O:"m' If we denote this transformation by R 2 , we have e-R.2(K(u))(¢')

L ~ "

= e-R.~(K(u))(¢')

lim A ..... Zd

O:"m'l1(")(--; -2 n;)V

[Z-1 / exp { - Jd--; 2,A \

(~g(--; - r;)(Moo¢,)(~))

2mEA

n

(4.11.2)

Z2,A

\ exp { -

~V(O) Jd--;

L L

O:"m'l1(")(--; -2 n;)O:"'m' i[J("')(--; -2 n;

",,,, 2m ,2m'EA

,)l) f

.(4.11.3)

(ar.;;::)

The first-order term in the expression for Z2 ,A is zero because the wavelets have vanishing moments, while the zeroth-order term has been absorbed into the zero-fluctuation transformation. The calculation of R2(K(u)) is just an application of the Gaussian formula

(II 1 dti 00

i

)

e

-! Et~-! L;A.;t.t;+L:.B. t •

-.J

-

-

-00

det(l

+ A)-1/2 exp

(-~ L:[(1 + A)-lli /J;/3 j

j )

(4.11.4)

',J Thus Roo(K(u))

+ ~ ~ ~ [(1 + A)-lt""mm,/J"m/J"'m' K.,K

m,mi

+ ~ In det(l + A) - ~ In det(l + A(¢' = 0)), 2

A KK' ,mm' --

2

J

(4.11.5)

d --; 'l1(")(--; -2 n;)'l1(1<')(--; -2 n; ')

xV

(~g("

-

';)(M OO , , ) ( ' ; ) )

,

(4.11.6)

CHAPTER 4. THE RENORMALIZATION GROUP

362

where the lower bound on V(z) insures positive-definiteness of the matrix I+A. Clearly, the new action is highly non-local, but extending the application of this one-loopper-scale transformation R2 to non-local interactions - a generalization necessary for iteration - is not really a problem. Accordingly, we consider an action H(¢) = Hoo(¢) + J(¢) of the more general type - i.e., where 00

J(¢)

=L

(4.11.8)

u"JA¢)

,,=1

with J,,(¢) non-local in ¢ but Zd-translation-invariant and v-fold multi-linear in ¢. In this case,

2

e-'R. (ii)(4>')

= e-'R.= (ii )(4>') }~~d

-w(I<)(x -2:r;,.) OJ('CO 8~(x)

Ct

I<m

-~ J J dx

Z2.A

;,~~:~~ (

') (,

exp { -

2 •A )-1 /

d

'"

I<

2mEA

n

CtI<;;CtI<';;' w(l<) (x

2ni.,2ni.' eA

~ ~ g(- ;;) (MOO .') (;;)) }

~ Jd x

J x L r.

exp { -

\

(~= ~ g( .- ~)(Moo¢,)(~))

L L

d x'

#t , K '

x

[(2

J

dx '

~ _~

-2 :r;,.)w(I<')(x ' - 2:r;,.)

t.J

« .119)

CtI<;;CtI<';;' w(I<)(x

-2:r;,.)

"" , It 2m,2m' EA

8:J(T~ (~== O)})

xw(I<')(x' - 2:r;,.,)

8~(x)8~(x ')

,

(4.11.10)

(0.;;;)

where the zero-fluctuation transformation is given by

(4.11.11) Thus (4 .11 .5) holds with

AI
=

J J dx

d x' w(I<)(x -2 :r;,.)w(I<')(x ' - 2:r;,.,)

2

x

8 J(H)

8~(x)8~(x')

-7

(_ '" g

~

~

( ·- n)(M

00'

~)

¢ )(n)

,

(4.11.12)

4.11. THE ONE-LOOP-PER-SCALE TRANSFORMATION

363

~ f d X "("(x -2 ;;./~\;i) ~ ~ P(- n)(MOO,p')(n~

p.m

(4.11.13)

It is important to note that N iterations of this transformation are not equivalent to the same approximate transformation associated with N iterations of the full renormalization group transformation - i.e. if K is the initial Ii, then clearly, e-'R.f(K(u))(')

i- e-'R.~N(K)(')

lim

[(Z(N))-l / exp { - Jd-;

A->Zd

2,A

\

~t."E Q~~"~·'(x -2' ;;')1; (~g(X - n)((MOO)N,p')(n)) N

- ~2 Jd -; ""' ""' ~ ~

""' ~

a(r2a(r'2 1lJ(I<)(-; _2r Km

Kim'

r

r;;,)

/\',;:;.' r,r'=1 2 r ni. ,2r' n;,/EA

Z~~)

p \ ex { -

~V(O) Jd-; ~r'~12r;;;' ~;;;'EA a~ka~~,

xllJ~I<)(-; _2r r;;,)IlJ~~'}(-; _2r' r;;, ')})

.

(4 .11.15)

r~N

Throwing away all fluctuations of order ~ 3 for all iterations at once is a much simpler truncation that is similar to the global one-loop approximation mentioned above. The global one-loop approximation is the basis of semi-classical field theory. If one simply uses the standard ultraviolet cutoff and considers the sum over connected Feynman graphs representing the log of the generating functional in the sense of formal perturbation theory, this traditional approximation discards all graphs having more than one loop. Only graphs such as those in Fig. 4.11.1 are allowed to be generated by the /J = 4 term of the interaction, for example. In the context of a wavelet cutoff, the corresponding diagrams are given by Fig. 4.11.2. In neither context does this restriction have anything to do with the RG iteration but is introduced at the global, perturbative level. To be sure, one could make the further restriction that the Feynman diagrams have the RG meaning described in the previous section - e.g., that the diagrams in Fig. 4.11.1 correspond to the quantities

364

CHAPTER 4. THE RENORMALIZATION GROUP

Figure 4.11.1:

1--Figure 4.11.2:

Jd-; Jd-;IJd-;IIJd-;1II C",l (-;, -; ')C",l (-;

I, - ;

")Co-,l (-; ", -; III)Co-,l (-;

III, -;)

X
1--Figure 4.11.3: The left-hand side of the non-equality is most appropriately called the one-loop-perscale approximation, since it is given by N iterations of the one-loop approximation of a single RG transformation. We can describe this phase-localized approximation scheme

4.11. THE ONE-LOOP-PER-SCALE TRANSFORMATION

365

in terms of wavelet diagrams as follows . Each RG step creates new wavelet diagrams and develops old ones, generating identification lines for all of the modes integrated out by that step, so we describe the scheme inductively. Consider a connected wavelet diagram at a given stage in the iteration. The RG-components of that diagram are defined to be: (a) new vertical lines created by the given RG iteration step, (b) connected wavelet diagrams developed by previous steps. The omission of diagrams is governed by the following restrictions: (1) No new identification line has connected an RG-component to itself unless it is the only RG-component. (2) If there is only one RG-component, then there is only one new identification line. (3) The new identification lines connecting the RG-components can form no more than one loop.

.-----------

, \

.~

:If

equal scale

Figure 4.11.4: These rules allow roughly as many "loops" as there are scales appearing in a connected wavelet diagram. For example, the wavelet diagrams in Fig. 4.11.4 are admissible, while the wavelet diagrams in Fig. 4.11.5 are dropped. The scheme automatically rules out the diagrams in Fig. 4.11.6 as well, because an application of the RG transformation integrates out all wavelet amplitudes at the given scale. Every such mode is connected somewhere by an identification line . Form a qualitative standpoint, all ultraviolet divergences are preserved by this approximation . For example, consider the wavelet diagram given by Fig. 4.11.7 once again. As we have already discussed in the previous section, the sum over all modes (r" rr;", 11;,), £ = 2,3,4 , is ultraviolet-divergent in dimension d > 2. Actually, we discussed the smaller sum represented by the case of identical (r" rr;,,, K;,) and obviously such a case is excluded by the approximation, but this restriction was only a convenience. The crude estimationyresented at t~e end of the previous section wo~d also apply to the sum where (r4,m4,1I;4) = (r3,m3,1I;3) and r3 = r2 - 1 with (r3,m3,1I;3) confined to the 2r2-scale vicinity of (r2 ,rr;,2,K;2) ' The approximation retains diagrams

366

CHAPTER 4. THE RENORMALIZATION GROUP

L~

'-equal

scale

\

I

/

equal scale

/

'------------", Figure 4.11.5:

equal scale

Figure 4.11.6:

Figure 4.11 .7:

r'\ 77 equal scale

4.11. THE ONE-LOOP-PER-SCALE TRANSFORMATION

367

of this type, while the sum over (r2,n;2,1\;2) and (r3,n;3,1\;3) produces the same divergence. Having described the application of the one-loop-per-scale transformation to actions of the form K = H 00 + 100 , we now consider its application to the corresponding actions H= Hoo + Loo. Recall that if I has the form

I(
L V(
(4.11.16)

then the full RG transformation is given by

e-~(Hooq,',q,') }~~d [ZA1 I exp { - ~ ~ L \

+L K

~

0Kmw(K) (2 -;

j

g

V(2

E{O,l}d

+ --: -2 n;))})

2mEA

1 d/2 sq,,(h

],

(4.11.17)

(ar.;;;)

lexp{-L L \

j

.. E{O,l}d

(4.11.18)

Clearly, the one-loop-per-scale truncation is given by

K.,K,'

2n:i,2n;., eA

(4.11.19)

Z2,A

I exp {- ~V(O) ~ ~ L \

xw(K)(2 -;

j

+ --:

g

E{O,l}d

~ ~~

K,K

-2n;)w(K')(2 -;

°Km°k'm'

2m,2m'EA

+ --:

-2n;)})

, (4.11.20) (ar.;;;)

CHAPTER 4. THE RENORMALIZATION GROUP

368

but for each first-order term in the exponent of (4.11.17) we see only the w(I<)-factor depends on -;. Since (4.11.21) w(l<) (2 j + -; -2 n;,) = 0,

L

-; E{O ,l}d

it follows that the first-order terms cancel out, and we obtain ~ 1 1 , ROO(H) + 2lndet(1 + B) - 2lndet(1 + B(
B ttx,',mm' ~~

=

L L

w(I<)(2

= 0)),

(4.11 .22)

j + -; -2 n;,)w(I<') (2 j + -; -2 n;,,)

-; -; E{O ,l}d

(4.11.23) which is a simpler formula than (4.11.5). Even in the more general (non-local) case where the interaction is given by (4.11.8) with

J(
= J(84) ),

(4.11.24)

we have

Q

~Q ~ w(l<) (2 ~J' K'm'

.Km

x w(I<')(2 j'+

-;, -

2n;,')

~J(r)~

(r

+ -; -2 n;,)

= 21- d/ 284»\

Z'A

( oxp { Q

-

~ E,"'~" }' ~'m~'
~ w(l<) (2 ~J'

",,'m'

+ -;

-2n;,)

j'+ -;' - 2 n;, ')

x w(I<')(2

J (4.11.25)

'fl(o •.:.) ,

8r(j )8r(j ')

~J(r)~

(r

= O)})

8r(j )8r(j ')

,(4. 11.26) (0 • .:.)

because each first-order term QI<~w

() I<

~ (2 j

+ ~c:

~ 8J(r) / -2 m)--~-(r = 2 1 - d 2 8 4>')

8r(j) still has all -; -dependence in the w(l<) -factor. Thus (4.11.22) holds with the matrix B now given by

j ,j'

-;,-;'E{O,l}d

4.12. THE WILSON WAVELET RECURSION FORMULA

369 (4.11 .27)

We investigate the hierarchical version of this transformation in §4.15.

References 1. D. Amit, Field Theory, Renormalization Group, and Critical Phenomena, McGraw-

Hill, New York, 1978. 2. J. Kogut and K. Wilson, "The Renormalization Group and the c-Expansion," Phys. Rep. 12 (1974), 75-200.

4.12

The Wilson Wavelet Recursion Formula

Although the zero-fluctuation transformation preserves the form of the interaction, it is not very interesting, as its only effect on either K(u, ¢) or H(u, ¢) is to scale the parameters according to dimensional analysis. The one-loop-per-scale transformation is a step up the ladder of approximations that is still computable in some sense, but only the trivial action Hoo is expected to be a fixed point with respect to that transformation . The second-order RG transformation is a better qualitative approximation, but it is only perturbative. We now introduce a rather different amputation of the renormalization group transformation that captures a substantial amount of physics and still preserves the locality of the interaction. It is a nonlinear mapping of the space of parameters into itself, but it is given by an explicit integral known as the Wilson recursion formula. The transformation can only be a qualitative approximation to the RG transformation, but instead of throwing away either the fluctuations themselves or certain orders of the fluctuations, one throws away contributions of the fluctuations to all but a few of the sub-cubes. As we shall see below, it is best to use very sharply localized wavelets, but in §4.7 we chose the block-spin transformation , which has the sharpest localization possible. The resulting wavelets are not quite class C 2 still enough regularity for any short-distance expansion that one might base upon the wavelets, should one wish to control a continuum limit . The canonical Haar manifold is the canonical manifold on which this transformation is to be defined, and indeed we introduced the manifold primarily for this purpose. The formula for a single application of the full renormalization group transformation is the N = 1 case of the formula (4 .7.48)- namely,

e-!(H~q,',q,') 2v-vd/2+duv

lim

A~Zd

[(2A)-1 / exp (-

~_L 2jEA EE{O,1}d

\

2- d {

f=

=1

sq,,(j)

+ 2~-1 ~

~ 2mEA

CHAPTER 4. THE RENORMALIZATION GROUP

370

(4.12.1)

(4.12.2)

In this transformation formula, the kinetic part ~ (Hoo
~ _L

e-!(H~¢' ,¢') AI~~d [(Z~)-l ( exp ( - ~ 2v-vd/2+duv Td{s¢,(j)

+ 2~-1 LQI<;W(I<)("£)Y)) I<

2jEA EE {O ,l}d

Z~ =

(exp (-

~uv ~

_

L

2jEA EE{O,l}

d

J,

(4.12.3)

(",~;;;)

{~QI<;w(I<)("£) r))

(4.12.4) (",~;;;)

Obviously the integration now factors 2.nto a j -indexed product of integrals , each with respect to the variables Q ""7 for each j. With 1<]

00

L uv z~ = V(Uj z)

( 4.12.5)

11=1

we recall that

Loo(Uj ¢» = J(Uj s¢) =

L V(Uj s¢(~)),

(4.12.6)

which yields the reduction ~

R' (H(u))(¢>')

1

~

= 2(Hoo¢>' ,¢>') + ~ R(V(u))(s¢, (j )), j

(4.12.7)

4.12. THE WILSON WAVELET RECURSION FORMULA

371

e-R(V)(z)

(4.12.9)

This is one version of the celebrated Wilson recursion formula. Note that the application of this transformation still requires an integration in 2d-l variables, e.g., 15 variables in four dimensions. There is a way to reduce the formula further with the same kind of brutal approximation, however. Recall that our wavelets were never completely specified; the mother wavelets were derived from an arbitrary basis of a (2d - I)-dimensional space. By (4.4.2), (4.4.24), and (4.4.62), we have

(4.12.10)

where the set {woo(l<) (p)} of 271"-periodic functions satisfies certain linear and quadratic constraints, which by no means uniquely determine the set. The fluctuation condition is a consequence of the linear constraint

L

h(2

P+271"""[ ')*WOO(I<)(p +71"""[ ') == 0

-;'E{O,l}d

because

d

XX(p

+271" e)*e-i2~,pII(I+e-iP')

,=1

(4.12.11)

CHAPTER 4. THE RENORMALIZATION GROUP

372

( 2~)d (IT i~rr dP') e-i2~.p X

t.=l

L

27r

woo(,,)(p +7r

€ ')h(2 P +27r € ')*

(4.12.12)

e'E{O,I}d

Moreover, the intra-scale orthonormality follows from the quadratic constraint

L

Hoo(p +7r

€ ')woo(,,)(p

+7r

€ ')*woo("')(p

+7r

€ ') == 8"",

(4.12.13)

e'E{O,I}d

because

J

dp

I P 12~)(p)*~(p)ei2~P =

(fli:

dP')

ei2~'P~ l

I P+27r -; 12~(p c

c

(g i:

dP')

+27r

-;)*~(p +27rf)

Hoo(p)ei2~·pwoo(,,)(p)*woo(,,')(p)

(~it!: dP') ei2~'p _ L ,-1

2

E

Hoo(p +7r €)wOO(")(p +7r

€)*

E{O,1}d

(4.12.14) Beyond the constraints (4.12.11) and (4.12.13), there is a freedom in choosing the functions woo (,,) (p). In contrast to the choice we made in §4.4, let K- range over functions on intervals {i + 1, ... , d} with values 0, 1. Let ,.-1

(J,,(p) =

IT (1 + e

d

iP

')(1 - eiP ,.)

IT

(4.12.15)

t=1

where

~"

denotes the i defining the domain of

K- .

We now make the choice

(4.12.16)

where D(p) is the matrix (4.12.17) The condition (4.12.13) is automatically met; note that d

L IT(1 +e-i(p,+rr<:))(J",(p +7r € ') e'E{O, l}d ,=1

4.12. THE WILSON WAVELET RECURSION FORMULA

'TIl C~l

i (1 + e- (p,+1T/'<))(1 + ei (p,+1T/.<)))

x (1 - ei (p,.+1T/.<)))

iI (L ,=,. +1

(1

373

(1'~1 (1 + e- i(p ,. +1T/.< ))

+ e- i(p,+1T/.<))e il< ,(p,+1T/.<))

/.<=0,1

0,

(4.12.18)

where we have applied (4.4.61) to the L = LI< factor. This verifies (4.12.11) for our choice of woo(l<) (p) . Having co~leted the specification of our wavelets, we consider the expression we have for w(I<)(c) . By (4.7.19), (4.12.10) , and (4.12.15) ,

J Y XCy d

-1-fd (271')d.

t:')Il1(I<)(y)

P x(p)*e- ie . p~)(p)

_c_ (iIl1T dP ) e-ie,pwoo(I<)(p) (271')d ,=1 -1T ' (2:)d

~ (tJ

[( _ L

I:

P d ')

DCp +71' t:' f))

-!]

(}I<,(p) e- ie ' .p . (4.12.19)

1<1<'

"E{O,l}d

Observe that if the bracketed expression were the identity matrix, (4.12 .15) would reduce this integral to ,.-1

II (60., + 61 •.)(60.,.

d

- 61.,')

II

61<,.,.

,=1

The guiding principle in this further approximation is to ignore those w(l<) (t:') that vanish under such a change. This is really the same kind of choice that was made above in the derivation of (4.12.8) . By way of illustration, consider the case d = 2, where K is either the null function or the two singleton functions. The smearings of the corresponding mother wavelets against the characteristic functions of squares in a dyadic block have the crude representations given by the respective sub-divided squares in Fig. 4.11. For arbitrary dimension d, we have made the replacements c, ""

K"

for any L> LI<'

(4.12 .20)

CHAPTER 4. THE RENORMALIZATION GROUP

374

tIm

K,

= null function (~I< = 2),

[QJ]J

Figure 4.12 .1:

crr..=:J crr..=:J

[QJ]J It is straightforward to verify that

w(l<) (cI' .. . ,1- c,<, .. . ,cd)

= -W(I<)(c:I , ... ,c,<, ... ,cd)

(4.12.21)

in any case. The invariance of D(p) with respect to the sign-reversal of any single momentum coordinate is the reason why this anti-symmetry holds. This invariance obviously extends to sign-reversal for arbitrary subsets {PI, ... ,Pd}, which, in turn, implies

w(I<') (-; ')

~I<'

= w(l<) C£\

= ~I<

and

<- c; =

K" -

C" ~ > ~I< .

(4.12.22)

The upshot of (4.12.20)-(4.12 .22) is that there are really only d numbers involved namely, W(l
e-Ri(V)(z)

= Zj(V)-1

i:

dTe-~T2 exp (- ~V(2~-~ Z ±TWj~,

(4.12.23)

i:

dTe-~T2 exp(-2V(TWj)),

(4.12.24)

2( ~- ~ )(j-I)W(Ki )(0).

(4.12.25)

= Rd 0 R d- l

(4.12 .26)

Then (4.12.20) reduces RT to R~

0···0

RI.

Each of the d transformations involves integration with respect to one variable. (4.12.23) is the original and most basic version of the Wilson recursion formula. Given our description above, it is easy enough to verify (4.12.26). Consider the twodimensional case once again, where Wj variables in (4.12.8), and

e-R(V)(z) =

Zo11°O -00

x exp (- _ €

L E{O.l}d

V(z

= W(l
1

00

dTnuII

-00

Then there are three integration

1

00

dTo

dTI

e-~T;u"e-~Tge-~T;

-00

+ wnull(1)Tnull + WOC-;)TO + WI (1)T) ')

,

(4.12.27)

4.12. THE WILSON WAVELET RECURSION FORMULA

375

where our notation for the three possible", is suggested by Fig. 4.11 . Now the effect of (4.12.20) in this case is to make the replacements

Wo (0, 1) t---t 0, wO(l, 1) t---t 0,

WI WI

(0, 0) t---t 0, (1, 0) t---t 0,

(4.12.28)

while (4.12.21) and (4.12.22) make the identifications

wO(O,O) = WI (1,1) = -wO(l, 0) = _WI (0,1), wnull(O,O) = w null (l, 0) = _wnull(O, 1) = _w null (l , 1) . These two numbers are in this case. Thus e-R'(V)( z)

=

W(I
and w(I<') (0\ respectively, as

d J OO Z I_ljOO - 00 Tnull - 00 dTo

°

(4.12.29) (4.12 .30)

",2

is the null function

J OO dTI e- "Tnu ll e - "TO e- "T 1

2

1

2

1

-00

2 1

x exp[- V( z + Wnull(O, O)Tnull + WO(O, O)TO) - V(z + w null (1, O)Tnull + w°(1, O)TO) -V(z + wnull(o, l)Tnull + wl(O , l)TI) - V(z + w null (l, l)Tnull + WI (1, l)TI)]

Zlolj OO

dTnulle-~T;ull { j OO dToe-~T~ exp (- L

- 00

V( z

-00

+ w(I<') (0) Tnull ± w(I
x exp (-

~ V(z -

ZIOI ZdV)21.: and the z

w(I<') (0) Tnull ±

1.:

±

dTle-~T~

w(I
(4.12.31)

dTnulle-~T~ull exp ( - ~(RI V)( z ± w(,,') (O)Tnull))

,

= 0 integration shows (4.12 .32)

This establishes (4.12.26) for d = 2, since Wj = w(I<j) (0) in this case. The idea is the same for arbitrary d, except that (4 .12.25) becomes relevant to the d iterations.

References 1. G. Battle, "Wavelet Refinement of the Wilson Recursion Formula," in Recent Advances in Wavelet Analysis, L. Schumaker and G. Webb, eds., Academic Press, Orlando, 1994, pp. 87-118.

2. K. Wilson, "Renormalization Group and Critical Phenomena II," Phys. Rev. B4 (1971),3184-3205.

CHAPTER 4. THE RENORMALIZATION GROUP

376

4.13

The Baker-Dyson-Wilson Hierarchical Model

We have just introduced the Wilson recursion formula for a severe truncation of the RG transformation as applied to the class of models comprising the canonical inverse limit manifold. While this formalism is new, it justifies only some of Wilson's assumptions. The historic value of the formula itself lies in the alternative point of view that it has to offer. One may introduce a toy model for which the Wilson recursion formula is the exact RG formula! While a qualitative approximation of the RG transformation is always suspect with regard to the critical properties its iterative behavior predicts for a given model, the exact RG transformation is reliable in the analysis of the toy model itself, regardless of how much the critical properties of that model may differ from those of the realistic model. In this section we introduce a toy model known as the Baker-Dyson-Wilson hierarchical model, which is not quite the desired toy, as it sets Wj = 1 for all j . The values of these numbers were unknown at the time the model was proposed. Accordingly, we introduce the standard Haar (square-wave) basis {~rn:J for e2 (Zd), where - as in the preceding section - the index K, ranges over functions on intervals {i + 1, . . . , d} with values 0,1. We set (4.13.1) 2- ,·/2

'L

X(~ - ~)(8oe,. - 8le ,.)

d

II

80 , 1<"

(4.13.2)

-; E{O ,l }d

where £1< denotes the i defining the domain of nearest-neighbor quadratic form

K,.

The usual idea is just to replace the

(4.13.3)

of the Ginzburg-Landau model with (4.13.4)

where the variables Trn:1< are defined by expanding ¢l in this basis: (4.13.5)

Equivalently,

Trn:1< = T r -,./d+l/d+l/2

'L ~rn:I«n;,)¢l(n;,),

(4.13.6)

;:; and the reason for our choice of exponent will become apparent below. Clearly, we may also regard H 0 as the operator whose eigenfunctions are these basis functions and

4.13. THE BAKER-DYSON-WILSON HIERARCHICAL MODEL

377

whose eigenvalue for ~rr;1< is simply the square of the power of 2 in the expression for Trr;1< This spectrum has the same infrared behavior as the spectrum of Ho , and this observation is the reason why the long-distance behavior of the new model is expected to be similar to that of the old model. The technical motivation for introducing the new model is that the couplings between lattice variables on a given scale lie entirely inside each block of lattice variables associated with one application of the block-spin averaging transformation. There are no nearest-neighbor couplings between lattice sites lying in adjacent blocks, but on the larger scale there are couplings between the lattice variables corresponding to adjacent blocks within the larger block. To see this, we write (4.13.7) in terms of the block-spin averaging - namely, (T¢)(~) = 2- d/2-1

2:=

¢(2 ~

+ 7).

(4.13 .8)

-; E{O ,1}d

We have

d

DO

= ~ 2:= 2:= 2:= 8 r=1

x

~n

c.~?,~,

i=1

2:=

T 2(1/d+1 / 2)(i-1)

,<=i

1<:

((1"-'.)(2;;

+ , + ;;;j -

(T'-'.)(2;;

+

'll) ,

(4.13.9)

Now define the following directional averaging: (Tp.¢)(r;)

=

T(1/d+1/2)(¢(n1, .. . , 2np., . . . , nd)

+ ¢(n1, . . . ,2np. + 1, . .. ,nd)) and note that T

= TdTd-1 . . . T 1. More to the point, (4.13.9) becomes 1

Ho(¢)

d

DO

= 82:= 2:= ~ r=l

;

L.

£.=1 It: itt=t.

(4.13.10)

CHAPTER 4. THE RENORMALIZATION GROUP

378

(4.13.11) and a model with such an expression for its kinetic term is a hierarchical model. Let Clearly,

CCr;/" iii ') denote the Gaussian covariance associated with this quadratic form . 00

C(iii, iii ') = L r=l

L L 22r+2LK/d-2/d-l~rnk(iii)~rnK(iii ')

n

(4.13.12)

It

and if OSier denotes the Gaussian expectation functional, we have ~ T ~ )hier ( T rn", r'n',..,' 0

= 6rr ,6~~ 6 nn'

Kit ·

,

(4.13.13)

= Ho(
(4.13.14)

Consider the class of actions

H(u;
and V(z) given by (4.12.5). The corresponding models are Baker-Dyson-Wilson hierarchical models, and they have the very desirable property that the exact RG transformation preserves this class - i.e., not only is the kinetic action a fixed point, but the locality of the interaction is preserved. Specifically, (4.13.15)

Z~l (If [: dTK) exp [ - ~ ~ T~ e" E{O,l}d

V

(Td/2+l

(If [: ~L 'E{O,l}d

Z

+ ~ 2LK/d-l/d+l/2TK~K(~)) ],

dTK) exp ( V (L

(4.13.16)

~ ~ T~

2LK/d-l/d+1/2TK~K(~)) ).

(4.13.17)

K

Since

L 2-LK /2+LK /2(60"K - 6l "K) ( K

IT

6e'K') TK

t.=l.K+l

K.

d

i """ W 2- /2+i/d(£UOe , i=l

-

£) T(e"li)

Ul"

(4.13.18)

4.13. THE BAKER-DYSON-WILSON HIERARCHICAL MODEL

379

with (-: It:) defined as the,.. on {i: + 1, ... , d} such that ,.., = c" we have very much the same situation we had in the preceding section . Indeed,

Z~l

i:

1

dTexp

(RO)d(V),

00

-00

(4 .13.19)

dTexp

(_~T2 - ~V(Tl/2+1/dZ±T») ' (4 .13.20)

(_~T2 - L V(±T») , 2 ±

(4.13.21)

so the renormalization group analysis of the Baker-Dyson-Wilson class of models can be carried out by iterating a formula that involves integration in one variable! For a range of non-integer values of the dimension, much has been rigorously determined about the critical behavior of these hierarchical models. The fundamental drawback of such models is that the critical exponent 7) is necessarily zero . On the other hand, this approximation simplifies the ultraviolet renormalization of asymptotically free field theories in the continuum limit to a degree that clarifies the issue of asymptotic freedom. Moreover, hierarchical models appear to confirm the conventional wisdom about the existence of a non-trivial fixed point. In the case 3 < d < 4 with U v = 0 for odd II, the flow of the renormalization group has the graphical representation given by Fig. 4.13.1 with the interaction parameters U v irrelevant for even II > 4. The critical manifold is indeed an unstable manifold, while the non-trivial fixed point is attractive on that manifold. The trivial fixed point is repulsive in both parameters. There is also a high-temperature fixed point, which is attractive in the high-temperature region bounded by the critical manifold. As d /' 4, the non-trivial fixed point approaches the trivial one, and this observation is the basis of the c-expansion. The idea is to regard the non-trivial fixed point as a small perturbation of the trivial fixed point in d = 4 - c dimensions and to expand thermodynamic quantities - and therefore the critical exponents - in powers of c. For hierarchical models, the c-expansion has been shown to be asymptotic. In the case d = 4, the only known fixed point on the critical manifold - even in the space of realistic models - is the trivial one, and in that dimension it is attractive on that manifold. In this case, the RG flow has the graphical representation given by Fig. 4.13.2, and this behavior of the RG iterations has been used to control the infinitevolume limit of a realistic critical theory in four dimensions. It has also been used to prove that any successful renormalization of the continuum limit of the hierarchical ¢1 model is only Gaussian. In the case d > 4, the continuum limit of the hierarchical ¢~ model is nonrenormalizable because the parameter U4 is now irrelevant. The quantitative difference is that U4 is now driven to zero exponentially fast by the flow of the RG iteration. In contrast to the d = 4 case, there is a non-trivial fixed point with U4 < 0, but it is repulsive. Long-distance behavior is governed by the trivial fixed point. Perhaps the most remarkable achievement in the study of the Baker-Dyson- Wilson models has been the proof of existence and analysis of the non-trivial fixed point in the case d = 3. The single-spin distribution of this fixed-point hierarchical model analytically continues to an entire function! It is also bounded by exp( -cz 6 ) on the

380

CHAPTER 4. THE RENORMALIZATION GROUP

U

2

~! /1\ u4

trivial ,,~

fixed point

critical manifold

\

non-trivial fixed point

Figure 4.13 .1:

U2

trivial fixed point critical manifold Figure 4.13.2:

4.13. THE BAKER- DYSON-WILSON HIERARCHICAL MODEL

381

real axis - i.e., the potential V(z) has z6-positivity. This is consistent with the fact that the parameter U6 is no longer irrelevant, as it becomes marginal in the d = 3 case. In order to replace the Baker-Dyson-Wilson model with a hierarchical model that reflects the wavelet refinement, we need only make the replacement ~I< f-t 2(~-~)«<-I)W(I<)(0)~1<

in the expansion (4.13.5) for ¢(~) . Equivalently, Trnl<

= rr-«

/2+ 1W(I<) (0)-1

L ~rnl«~)¢(~)

(4.13.22)

;:;; with Ho(¢) still given by (4.13.4) . Following the derivation (4.13 .7)-(4.13.11), we now obtain

(4.13.23) for the hierarchical kinetic action. The associated Gaussian covariance becomes

C(~, ~ ')

00

=L r=1

d

L L 22r+2j/d-2/d-lw; L n

j=1

It:

~rnl«~)~ rnl«~ f),

(4.13.24)

LI(=j

but (4.13.13) still holds. As far as the Wilson recursion formula is concerned, it is important to remember that a fixed point actually corresponds to a periodic sequence of potentials with period d. Since (4.13.25) a fixed point is given by (VI (z), V2(z) , . . . , Vd(Z)) with the property Rj(Vj) = Vj+l,

Rd(Vd)

= VI ·

j

< d,

(4.13.26) (4.13.27)

References 1. G. Baker, "Ising Model with a Scaling Interaction," Phys. Rev. B5 (1972),2622-

2633. 2. P. Bleher and Y. Sinai, "Investigation of the Critical Point in Models of Type of Dyson's Hierarchical Model," Commun. Math. Phys. 33 (1973), 23-42. 3. P. Bleher and Y. Sinai, "Critical Indices for Dyson's Asymptotically Hierarchical Models," Commun. Math. Phys. 45 (1975), 247-278.

CHAPTER 4. THE RENORMALIZATION GROUP

382

4. P. Collet and J. Eckmann, A Renormalization Group Analysis of the Hierarchical Model in Statistical Physics, Lecture Notes in Physics 74, Springer-Verlag, New York,1978. 5. F. Dyson, "Existence of a Phase Transition in a One-Dimensional Ising Ferromagnet," Commun. Math. Phys. 12 (1969),91-107. 6. F. Dyson, "An Ising Ferromagnet with Discontinuous Long-Range Order," Commun. Math. Phys. 21 (1971), 269- 283. 7. K. Gawedzki and Z. Kupiainen, "Triviality of ¢i! and all that in a Hierarchical Model Approximation," J . Stat. Phys. 29 (1982),683-698. 8. K. Gawedzki and A. Kupiainen, "Non-Gaussian Fixed Points of the Block Spin Transformation in the Hierarchical Model Approximation," Commun. Math. Phys. 89 (1983), 191-220. 9. H. Koch and P. Wittwer, "A Non-Gaussian Renormalization Group Fixed Point for Hierarchical Scalar Lattice Field Theories," Commun. Math. Phys. 106 (1986),495-532. 10. H. Koch and P. Wittwer, "On the Renormalization Group Transformation for Scalar Hierarchical Models, Commun. Math. Phys. 138 (1991),537-568.

11. H. Koch and P. Wittwer, "A Non-Trivial Renormalization Group Fixed Point for the Dyson-Baker Hierarchical Model," Commun. Math. Phys. 164 (1994), 627-647. 12. J. Kogut and K. Wilson, "The Renormalization Group and the c:-Expansion," Phys. Rep. 12 (1974), 75-200.

4.14

The Hierarchical Model to Second Order

In §4.9 we described the celebrated second-order perturbation theory of the RG transformation on our extended inverse limit manifold. Originally exploited for the standard ultraviolet cutoff on the scalar field <1>(;;), this perturbatively truncated RG transformation qualitatively predicts some of the correct critical behavior. We revisit this calculation in the simplifying context of the hierarchical approximation, considering the Baker-Dyson-Wilson model first. In powers of the interaction , the approximation is the next-higher-order correction to the linearized RG transformation. Accordingly, we expand

R(V)(z )

=

In (ex p ( -

~ V(±r)) )

-In (ex p (-

(T)

~V(2~-~Z±r)) )(T)

(4.14.1)

4.14. THE HIERARCHICAL MODEL TO SECOND ORDER

383

1

where

00

1 1 2 rrc e- 2T F(T)dT. v 27r -00 The expansion is the one-variable version of the cumulant expansion. Thus

(F(T)}(T)

R(V)(,)

~

In

=

(1 -~ (V(±r))", + ~ ( (~v(±r))

(4.14.2)

't

-~( (p(±T)n" +)

-In (1- ~(V(2~-!Z±T)}(T) +~( (~V(2HdT))\,

-~ ((~V(2I-ldT)) ,)", +}

(4.14.3)

and second-order perturbation theory discards all contributions of order 2 3 in the potential V This modified transformation is given by

RII(V)(Z)

=

LH~(V)(Z) - ~ (( ~V(2~-!Z±T) _

/LV(2~-!Z ±TI)) )2) \ ±

(T)

+~ / (L V(±T) - / L \

±

\

(T')

1

V(±T ) )

±

)

2) ,

(T')

(4.14.4)

(T)

where LH~ denotes the linearized hierarchical RG transformation:

LH~ (V)(z)

= L(V(2~-! Z ± T)}(T) ±

- L(V(±T)}(T)'

(4.14.5)

±

Inserting (4.12.5) in this linearized expression, we obtain 2

~

LH~(V)(z)=2L(Tq}(T)~ 00

q=O

Similarly,

v=l

(2q v+ v) 2";;-2U +2qZ. v

v

V

V

(4.14.6)

CHAPTER 4. THE RENORMALIZATION GROUP

384 00

00

q=O

11=0 1I+2q>0

2L L

(4.14.7)

and therefore

/(L V(2~-~z

\

± T) - /

±

\

LV(2~-~z ±

(T')

00

4

00

L 2q

+

v)

(2ij

V

+

_

iI)

V

(T)

00

L

2 2q q ((T(2 +2 iJ )(T) - (T )(T) ( T iJ)(T»)

q,q=0

(

7/)) ) 2)

±

L

II+~~O v.t~~o 2 !dE _!dE d 2 U II +2q

U v +2ijZ II+V .

(4 .14.8)

This means the second-order contribution to (4.14.4) is given by the subtraction of 00

00

2

L

((T

2q 2ij + )(T) 11=0 v=o 1I+2q>0 v+2ij>0 V+II>O

q,ij=O

2~-~UII+2q

x

U V+ 2ijZ II+V .

Thus 00

RII(V)(Z)

=

L~£(u)z£,

(4.14.9)

£=1

00

L

- 2£-~+1

{(T

2q 2ij + )(T) -

(T

2q

)(T) (T

2ij

)(T»)

q,ij=O l

X

L

V (

+v 2 q )

(f -

v

f _

+V 2ij) Ul - II +2ij U II +2q .

(4.14.10)

11=0

1I+2q>0 II
This transformation is certainly defined if V{ z ) is a polynomial, as all of the infinite series become finite sums in that case. As discouraging as this second-order correction to the linearized transformation may appear, it has an interesting closure property. It is well known that the space of potentials of the form (4.14.11)

is actually closed under the transformation. Whether the potentials given by higherdegree polynomials are necessarily drawn to this manifold by second-order RG iterations depends on the dimension. Before we discuss this issue, it is worthwhile to

4.14. THE HIERARCHICAL MODEL TO SECOND ORDER

385

verify that the polynomial form given by (4.14.11) is indeed preserved. For such a potential, the only terms in the linear contribution to (4.14.10) that do not vanish are the q ~ 1 terms. In the quadratic contribution, the terms for which either q or ij is greater than 2 must vanish. Indeed, v and e - v cannot both be zero, so the terms for which q and ij are both equal to 2 must vanish as well. Moreover, the (q, ij) = (0,0), (0, 1), (1,0), (0,2), (2,0) terms vanish because the truncated expectation is zero in those cases. Thus for ~ 1,

f.

~Au)

_

, '1 " +1 2 = 2<1-"-+ Uf + 2r..(T )(T) (f. + f.

2)

(T2)(T)(T4)(T))

~

2£-~+1((T6)(T) _

(T2)(T)(T4)(T))

~

4) (T) 2d1_ {+1(( 2 T

(T 2) 2(T) )

-

C: 4) (f. ~::

e

2£-~+1((T6)(T) -

Ul+2

e

(v: 2) (f. ~::

2)UV+4 Uf-v+2

4)UV+2 Uf-v+4

~ (v + 2) (f. - v+ 2) ~ V

v=o

V

(4.14.12) Clearly, ~f(U) =

because either v

> 2 or

f.> 4,

0,

(4 .14.13)

f. - v > 2 in this case. Also, f. odd, ~e(U) = 0,

(4.14.14)

because v and e- v cannot both be even in that case. This proves closure of the space of even quartic potentials given by (4.14.11) . We calculate: 2 2 a ~ 2 (4.14.15) 2<1U2 + 2<1 X 6U4 - 2 d X 24u2U4 - 2 d X 144u 4, 2~-1U4 - 2~-1

72u~,

(4.14.16)

= 1,

(4.14.17.1)

(T4)(T) = 3,

(4.14.17.2)

= 15.

(4.14.17.3)

X

where we have inserted (T2)(T)

(T6)(T)

Therefore, a fixed point satisfies the equations U2 U4

= =

2

2<1U2

+ 2<1 2

X

II

6U4 - 2 d

2~-1U4 - 2~-1

X

X

II

4

2

24u2U4 - 2 d x 14 U4,

72u~ .

There are two solutions for d < 4: the trivial solution U2

(4 .14.18) (4.14.19)

= U4 = 0 and (4.14.20) (4.14.21)

CHAPTER 4. THE RENORMALIZATION GROUP

386

For d 2: 4, the trivial solution is the only solution. For any d and for any polynomial of degree > 4 - as well as for any quartic polynomial with a cubic term - the second-order transformation increases the degree. We return to the issue of whether the potentials given by such polynomials are actually driven toward some manifold of low-degree polynomials by iterations of HI I. Recall from §4.6 that a parameter U v is irrelevant if (d- 2)v > 2d (as we have set s = 1

since §4.7) . Indeed, the calculation of llt(u) involves scaling by the factor 2(~-!)l+1, which is less than unity for (d - 2)£ > 2d. This means that for d > Jj the scale factor is less than unity for £ 2: 5. One expects that iteration drives every polynomial with sufficiently small coefficients toward the manifold of quartic polynomials for these values of d, but we shall not give the proof. Instead, we further our discussion by introducing yet another approximation. Let

~~(V)(z)

=

L

lll(U)

(4.14.22)

e>1 (d-2)t:::;2d

be the principal part of the second-order transformation. In addition to irrelevant terms, the contributions of previous values of irrelevant parameters to current values of relevant or marginal parameters are thrown away by the iteration of this approximation. Clearly, for d > Jj the image of this modified transformation is the space of quartic polynomials, but the case d = 3 is of greater interest. In three dimensions, the constraint reduces to £ :S 6, so the range of ~~ is the space of 6th-degree polynomials. Indeed, ~~ preserves the space of even 6th-degree polynomials: (4.14.23) For such potentials,

~~(V)

+ ll4(U)Z4 + ll2(U)z2 ,

lle(u)ze

360U4Ue - 5400u~ ,

(4 .14.24)

lle(u)

Ue -

ll4 (u)

+ 15u6 - 60U2U6 - 2520U4U6 - 72u~ - 24300u~),(4.14.26) 2 ~ (U 2 + 6U4 + 45u6 - 24u2U4 - 360U2U6

ll 2(U)

(4.14.25)

2t(U4

-144u~ - 3960U4U6 - 1800u~).

(4.14.27)

In this case the fixed-point equations are (4.14.28)

(2t - 1)u4

+ 2t

-8100u~)

X

3(5u6 - 20U2U6 - 840U4U6 - 24u~

= 0,

(2i - 1)u2 + 2i X 3(2u4 + 15ue - 8U2U4 - 120u2u6 -48u~ - 1320u4u6 - 600u~) = 0,

(4.14.29)

(4.14.30)

4.14. T H E HIERARCHICAL MODEL T O SECOND ORDER

387

and us = 0 is obviously a solution of the first equation. That choice reduces the other two equations to (4.14.18) and (4.14.19) with d = 3, and one can see that the nontrivial solution given by (4.14.20) and (4.14.21) is still the only physically meaningful non-trivial fixed point. While there are two non-trivial solutions for the choice

both of them are spurious in the sense that the potential is not bounded below us < 0. To verify this, we insert (4.14.31) in (4.14.29) and (4.14.30) to obtain

-

i.e.,

respectively. The first equation reduces to the choices u4 = 0 and

If we now insert the non-trivial choice in the second equation, we finally obtain

The discriminant of this quadratic equation is

and its square root is approximately 1.532. The linear coefficient is approximately 1.679, so the solutions are

+

It is obvious that 4u2 2 - i < 0 in both cases. It follows from (4.14.34) that u4 > 0 in both cases, and so by (4.14.13), us < 0 in both cases. On the other hand, we have already mentioned in the previous section that the exact non-trivial fixed point for the hierarchical model in three dimensions is given by a potential whose growth is at least that of a 6th-degree polynomial. The rigorous, non-perturbative analysis underlying this celebrated result highlights the very qualitative nature of the above approximation, which realizes the non-trivial fixed point as a quartic polynomial. This crude perturbative approximation throws away the stability associated with the 6th-degree term in three dimensions, and stability is a non-perturbative property. We now return to an analysis of the fixed points on the 7711-invariant manifold of potentials of the form (4.14.11). The trivial fixed point is repelling for d < 4. To see this, we need only revisit LH, and note that the eigenvalues in this two-dimensional space are greater than 1. Indeed,

CHAPTER 4. THE RENORMALIZATION GROUP

388

so the eigenvectors are

v~~ (z)

(4.14.38)

V~~ (z)

(4.14.39)

For d = 4 the eigenvalue of V~~ (z) is equal to 1, so we return to the nonlinear transformation formula (4.14.19) to investigate the quadratic behavior. Observe that if U4 has a sufficiently small positive value in this case, then the iteration of Q drives U4 to zero. The eigenvalue of V~~ (z) is still greater than 1, so the trivial fixed point U2 = U4 = 0 is still repelling in the u2-direction. In addition to showing that U2 = U4 = ois attractive in the u4-direction on the U4 > 0 side, (4.14.19) shows that U2 = U4 = 0 is repelling in the u4-direction on the U4 < 0 side. Finally, for d > 4, the eigenvalue of V~~ (z) is less than 1 while the eigenvalue of V~~ (z) is still greater than 1. This means U2 = U4 = 0 is attractive in the u4-direction (on both sides) and repelling (still) in the u2-direction. Now consider the nontrivial fixed point that exists for d < 4 and linearize the second-order transformation there. We have a/::.2 (U2, - U4 -) -;,-UU2

= 2
a/::.2 _ _

22d

X

24U4, 1.

2

-;,--(U4,U4) = 2<1 x 6 - 2d UU4

a/::.4 (U2, - U4 - ) = -;,-UU2 a/::.4 (U2, - U4 - ) -;,-UU4

(4.14.40) _

1.

24U2 - 2d

X

X

0,

288u4,

(4.14.41) (4.14.42)

2~-1 =.

21.-1 d

X

144U4. -

(4.14.43)

Clearly, this matrix has eigenvectors V(O)(z)

= cz 2,

(4.14.44)

V(1)(z) = c(6(1 - 4U2 - 48u4) z 2

+(2~-1 - 1 + 24(1 - 2~ x 3)U4)z4)

(4.14.45)

with eigenvalues 1.

1.

2d - 2d

_

X

24U4

2~-1 - 2~-1

X

2

2

2

= 3(2<1 + 2- d ), 144u4 = _2~-1 +

(4.14.46) 2,

(4.14.47)

respectively. In the direction of the former eigenvector the nontrivial fixed point is repelling because (4.14.48) In the direction of the latter eigenvector the point is attractive for dimension d < 4 because 2 - 2~-1 < 1, d < 4. (4.14.49) This completes the fixed-point analysis for the second-order approximation of the RG transformation applied to the Baker-Dyson- Wilson model.

4.15. HIERARCHICAL ONE-LOOP-PER-SCALE TRANSFORMATION

389

References 1. G. Baker, "Ising Model with a Scaling Interaction," Phys. Rev. B5 (1972),26222633.

2. P. Bleher and Y. Sinai, "Investigation of the Critical Point in Models of Type of Dyson's Hierarchical Model," Commun. Math. Phys. 33 (1973), 23-42. 3. P. Collet and J. Eckmann, A Renormalization Group Analysis of the Hierarchical Model in Statistical Physics, Lecture Notes in Physics 74, Springer-Verlag, New York, 1978. 4. F. Dyson, "Existence of a Phase Transition in a One-Dimensional Ising Ferromagnet," Commun. Math. Phys . 12 (1969),91-107. 5. F . Dyson, "An Ising Ferromagnet with Discontinuous Long-Range Order," Commun. Math. Phys. 21 (1971), 269-283. 6. H. Koch and P. Wittwer, "A Non-Trivial Renormalization Group Fixed Point for the Dyson-Baker Hierarchical Model," Commun. Math. Phys. 164 (1994), 627-647. 7. J. Kogut and K. Wilson, "The Renormalization Group and the c-Expansion," Phys. Rep. 12 (1974), 75-199.

4.15

Hierarchical One-Loop-per-Scale Transformation

We have already described the one-loop-per-scale transformation for two initial manifolds - the canonical wavelet manifold and the canonical inverse limit manifold . We now describe it in the hierarchical setting, where a slight modification of the wavelet diagram restriction is introduced. As in the previous section, we focus on the BakerDyson-Wilson model in the interests of simplicity. The wavelet diagram restriction is redefined by regarding each scale as possessing d sub-scales, where the i-th sub-scale of a given scale includes those wavelets for which the index K, has domain {i + 1, .. . , d}. Rules (1)-(3) in §4.11 hold exactly as before, except that it is a "one-loop-per-sub-scale" restriction. A single RG transformation in the statement of these rules is now only R - the transformation based on a single directional averaging transformation TJ1. for some f.L . On the other hand - as we discussed in §4.11 - the wavelet diagram restriction is an alternative description of the approximation. One may also regard the approximation as a modification of the RG transformation, which is given by discarding all terms in the exponent that are of order ~ 3 in the fluctuation amplitudes - for that single RG step. Thus we replace exp( -R(V)(z))

z~11OO dr exp (_!r2 - LV(2~-! z ± r)) , (4.15.1) -00

--1

Zo

1 dr 00

-00

exp

2

±

(_!r2 - L v(±r)) , 2 ±

(4.15 .2)

CHAPTER 4. THE RENORMALIZATION GROUP

390 with

(4.15 .3) (4.15.4)

No where we have imposed the condition

V(O)

=0

(4.15.5)

without loss of generality, and the first-order terms obviously cancel in the change

defining the approximation. The advantage we have gained is that the formula (4.15.3) is just a one-variable Gaussian integral. We obtain

exp(-2V(2~-~z)) ( ~ + V(2~-~z) )

exp(-R2(V)(Z)) =

_1. ( 2

)1/2 ,

~ + V(O)

(4.15.6)

where the condition (4.15 .5) is clearly preserved. We may write -

R2(V)(Z)

1 (1Z + V(2d'-2 .. = 2V(2 'r2 z) + Zln z) 1

1

1

1

)

1 (1Z + V(O) .. ) , - Zln

(4.15.7)

and the leading term is just the effect of the zero-fluctuation transformation on the interaction. The transformation R2 is entirely different from RI I , as the latter is second-order in V . The corresponding fixed-point equation is just

V(z)

1 (1- + V(2r2Z) .. = 2V(2 r2 z) + -In 2 2 1

1

1

1

)

1 (1- -In 2

2

.. ) + V(O)

'

(4.15 .8)

but for d > 2 this equation is difficult. First we concentrate on the two-dimensional case .. ) - Zln 1 (1Z + V(O) .. ) , 1 (1Z + V(z) (4.15.9) 0= V(z) + Zln which can be analyzed by elementary ODE methods. This ordinary differential equation is nonlinear, but separable in the second-order sense. We may write it as

wexp(-2V(z)) = w=

1

..

1

..

Z+ V(z),

Z+ V(O) > 0,

(4.15 .10) (4.15.11)

4.15. HIERARCHICAL ONE-LOOP-PER-SCALE TRANSFORMATION

391

and multiply it by V(z) for the first integration of the equation. This yields the firstorder equation -wexp(-2V(z)) = V(z) + V(Z)2 - Co, (4.15.12) where CO is an arbitrary constant. With the normalization YeO) = 0 still specified, this constant is determined by (4.15.13) if the value of V(O) is specified. Although we have not yet solved for V(z), the basic ODE theory applied t~ (4.15.10) assures existence and uniqueness in a neighborhood of z = Zo if V(zo) and V(zo) are given . To assure global existence and uniqueness for a specified value of V(O), we need a priori bounds on both V(z) and V(z) . Actually, we shall see that both functions are bounded by constants. Notice that (4 .15.12) implies

V(z)

+ w exp( -2V( z )) S co,

(4 .15.14)

which, in turn, immediately implies

V(z) S Co ·

(4.15.15)

On the other hand, we have the standard inequality (4.15.16) so if we set

t for arbitrary ,B

= -2V(z) -In,B

(4.15 .17)

> 0, we have exp(-2V( z ))

~,B

- 2,BV(z) - ,Bln,B

(4.15.18)

as well. Combining this with (4 .15.14), we obtain the inequality (1 - 2w,B)V(z)

+ ,B(l-ln,B) S Co ·

(4.15.19)

Thus we obtain a constant lower bound on V(z) by choosing ,B > ~W-l. The boundedness of V(z) together with (4.15 .12) immediately implies the boundedness of V(z) as well. This verifies global existence and uniqueness. For a given value of V(O), it is worthwhile to examine the behavior of V( z ). If we assume V(O) > 0 for the sake of definiteness, then (4 .15.12) may be written as

V(z)

= Jco -

wexp(-2V(z)) - V(z)

(4.15.20)

for z in some neighborhood of zero. Since this equation is separable, we can solve the equation on this interval with an integral formula for the inverse function z(V) of the solution. We have (4.15.21)

CHAPTER 4. THE RENORMALIZATION GROUP

392

assuming V (0) = 0 as before. Clearly, the domain of this inverse function is the interval V_ < V < V+ , where (4.15.22) We claim the z-interval involved here is definitely a finite interval, but this is not quite obvious. The problem is to show z+

= =

lim z (V) v /, v+ v +(Co-u-wex P(-2U))-!dU<00,

l

where the problem of showing

L

(4.15.23)

> -00 is similar. We have

d

dU(c O - U -wexp(-2U))lu=v+

= 2we- 2V+ -1,

(4.15.24)

and the improper integral converges if this derivative is negative. (The derivative at U = V+ cannot be positive because this function of U vanishes at U = V+ and has non-negative values for U :::; V+ .) The improper integral diverges when this derivative is zero, and divergence is the case we intend to rule out. Accordingly, suppose (4.15 .24) vanishes, or rather V+

1

= 2In(2w) .

This means z+ = 00 and V(z) increases monotonically as z -+

(4.15.25) 00.

Thus

2V(z) < In(2w),

(4.15.26)

but if we combine this with (4.15.10), we conclude that

V(z) > 0,

(4.15.27)

which contradicts the boundedness of V(z) . Having established that z+ < 00, we consider the problem ofrealizing V(z) beyond this critical point. After all, (4.15.28) and we propose to determine whether this is a turning point. We can answer this question immediately because convergence of the improper integral implies the derivative in (4.15.24) has a negative value. But this means the right-hand side has that negative value, so we infer that (4.15.29) V(z+) < 0 from (4 .15.10). Therefore, the critical point is a local maximum, and so V(z) =

-Jco - V(z) -

wexp( -2V(z))

(4.15.30)

is the first-order equation that holds over the next interval of z-values before the next turning point.

4.15. HIERARCHICAL ONE-LOOP-PER-SCALE TRANSFORMATION

393

The reasoning we have given can easily be extended to show that V(z) is an oscillatory function as well as a bounded one. We now have a two-dimensional manifold of fixed points parametrized by V(O) and w. This is contrary to physical expectations based on the second-order calculation done in the previous section. Besides the trivial fixed point, only one isolated, (nontrivial) fixed point was found. This is the first indication that the one-loop-per-scale approximation cannot reflect true physical behavior, but the d = 2 case has always been a marginal one from the standpoint of renormalization group analysis. For dimension d > 2 the fixed point equation (4.15.8) has never yielded any solution except V(z) == O. No non-trivial solution is believed to exist. The assumption of analyticity at z = 0 sheds some interesting light on the issue. We have

V(z)

= 2aV(az) + ~av(az)

G+

V(az))

-I ,

a=2~-~,

(4.15.31)

if we differentiate (4.15.8). Consider the equivalent equation (4.15.32) where we have made the substitution z >-+ ~ + W(z), the equation becomes

a-I z.

If we set

V=

Wand multiply by

12"W(z) . = (a- W(a- z) - 2W(z)) (1.) 2" + W(z) I

I

(4.15.33)

Now differentiate this equation N times to obtain

(a-N-IW(N)(a-Iz) - 2W(N)(z))

G+

W(z))

+ ~ (:)W(/.<+I)(z)(a-N+/.<-IW(N-/.<)(a-Iz) _ 2W(N-/.<)(z)),

(4.15.34)

where w(v) (z) denotes the vth derivative of W(z). Setting z the recursion relation

W(N+2)(0) = (a- N- I - 2)W(N)(0)(1 N

+ 2L /.<=1

= 0, we have now derived

+ 2W(0))

(N) W(/.<+I) (0) (a- N+/.<-1 - 2)W(N-/.<)(0).

(4.15.35)

J.L

On the other hand, the radius p of convergence for the Taylor series

=L 00

W(z)

v=l

vuvz v-

1

=L 00

N=O

W(N) (0) N! zN

(4.15.36)

CHAPTER 4. THE RENORMALIZATION GROUP

394 is given by

(4.15.37) The recursion formula does not suggest an easy way to control this limit, but note that if we let UN(Z) denote the leading term in (4.15.34), we have W(N)(O) UN(O)

Since a < 1 for d

=

a N +1 1 1 + 2W(0) 1 - 2a N+1 ·

(4.15.38)

> 2, (4.15.39)

While this does not prove that p = 0, the cancellations by the neglected terms would have to be rather delicate to prevent this conclusion. The point is that if p = 0, then a non-trivial solution cannot be analytic at z = O. According to conventional wisdom, a non-trivial solution does not exist. This is the reason why there has never been much interest in the one-loop-per-scale approximation, at least for long-distance problems.

4.16

Wavelet Correction of the Hierarchical Model

The only geometry in the Baker-Dyson-Wilson model is the value of the dimensional parameter d. With the renormalization group transformation broken down into d steps - one step for each coordinate direction - there is no distinction between any two steps. On the other hand, for the wavelet-modified hierarchical model, the renormalization group transformation is decomposed into d directional steps that differ from one another in a way that reflects the ellipticity of the Laplacian. Recall that

R' e-Rj (V)(z)

Rd

Zj(V)-l

exp

i:

R d- 1

0

0 ··· 0

R1 ,

dT

(_~T2 - ~V(2~-~ Z ±TWj»),

i:

dTexp

(_~T2 - ~V(±TWj») ,

2(~ - ~ )(j-l)W("i) (0).

By (4 .12.19) we have

(4.16.1)

(4.16.2)

(4.16.3) (4.16.4)

4.16. WAVELET CORRECTION OF THE HIERARCHICAL MODEL

[( ~ L

D(p +71'

~ I)) -~l I
e'E{O,l}d

X (

II (1 + eiP')) (1 - eip'K') II ei
(4.16.5)

t>tk/

HooCp)(l - eip'K )(1 - e-ip'K')

(II (1 + eiP,) II eil<'p,) t.
x

I,>LK.

(II (l+e- iP,) II e-i
(4.16.6)

t.>i tt ,

L
where K,j is defined to be zero on {j contained in

395

+ 1, . . . ,d}

and the ellipticity of the Laplacian is

(4.16.7)

with X as the characteristic function of the unit cube [O,ljd As for a given mother wavelet q, 1<, the frequency of variation is 0 ( ~ in [I< - 1 directions and 0 (1) in d - [I< + 1

1

p

directions, so when the quantity 1 +271' f 12 is dropped from (4.16.5), the usual hierarchical approximation replaces (4.16.7) with

in (4.16.6) . Partial differentiation is replaced with multiplication by a geometric average of crude numerical values of the frequencies of a given mother wavelet over the d directions . Thus we have the replacement

D(p)I
t-+

T('K-lJ/ d2-('K,-1)/d(1 - eip ," )(1- e-ip'K')

(II

(1 + eiP ,)

t
X

II eil<'p,) (II (1 + e- ip ,) II e-i1.,,.

t
t>tl(l

where we have also inserted the standard identity (4.16.8)

CHAPTER 4. THE RENORMALIZATION GROUP

396

In this case, we have the replacement

L

(~Il ± e

D(p +71' -; ')1<1<' >--+ r(,·-l)/d8,.,.,

iP

"1

2

)

<" 'E{O,l}d

x

11 (L 11 ± e

ip

1,<1.1(

,

12)

±

11 (L(±I)"'-<e (I<'-<)P') i

L>il(

because

±

L(1 ± eiP , )(1 T e- ip ,)

= O.

(4.16.9)

±

Moreover,

L 11 ± e

ip

,

12= 4,

(4.16.10)

±

L(±I),,'-<e i (I<,-<)P,

= { ~'

±

because

K, - K:

(4.16.11)

'

can only have the values -1,0, 1. Thus we have the further reduction

L

D(p +71' -; ') >--+ 2d4-(,·-l)/d+,./28/t/t"

<"'E{O,l}d

which induces the replacement

because Klj) = 0 for ~ > j . Since the multiple integral has the value (271')d, the numbers Wj have been equalized by the elimination of ellipticity. The price we pay for keeping the ellipticity lies in the mathematical nature of a fixed point. As we have already pointed out in §4.13, such a potential is the initial element of a sequence (Vl(Z), V2 (z), . .. , Vd(Z)) of potentials where Rj(Yj) = Yj+l, Rd(Vd )

j

< d,

= Vl·

(4.16.12) (4.16.13)

Note that for j > 1, Yj is not a fixed point for R' but rather for R j - l R j - 2 ... RlRdRd-l ... R j . Now the linearization of R j at the trivial fixed point V(z) == 0 is given by

Lj(V)(z)

= L(V(2~ -tz ± WjT))(r) ±

L(V(±WjT))(r), ±

while the approximation that is second-order in V is given by

(4.16.14)

4.16. WAVELET CORRECTION OF THE HIERARCHICAL MODEL

397

Equation (4.14.7) trivially generalizes to

00

2LwJq(r q=O

00

2q

- (r,2 q)(T'))

L v=O v+2q>O

V (

+V 2q) Uv+2q 2V( d _1) z, v 1

2

(

4.16.16)

and therefore

RII,j(V)(Z) 00

00

2(~-!)lzl L

- 2L l=l X

WJq+2ij

q,ij=O

((r 2q+2-q)(T) - (r 2q)(T)(r 2-q)(T)) l

X

L v=O

v+2q>O v
(4.16.17)

The space of potentials of the form (4.16.18) is still closed under the transformations, and (4.14.15) and (4.14.16) are now replaced by (4.16.19) (4.16.20) respectively, where tlj,l(U) denotes the total coefficient of zl in (4.16.17). The problem is to find the fixed points, not for each transformation, but for the composition of the transformations.

CHAPTER 4. THE RENORMALIZATION GROUP

398

As in the Baker-Dyson-Wilson case, the transformation of the parameter U4 is decoupled from the parameter U2 in this second-order approximation, so the fixedpoint values of U4 are the fixed values of a one-variable transformation - namely,

f = fd 0 fd-l 0 ' " 0 Jr, /j(v) = 2~-IV(1 - 72w1v) .

(4.16.21) (4.16.22)

Clearly, f(v) is a 2d-degree polynomial with vasa factor, so the fixed-value equation f (v) = v has v = 0 as a solution. Any other fixed value is just a root of the polynomial p( v) defined by p(v)v == f(v) - v. (4.16.23) We claim this (2d-l)-degree polynomial is a decreasing function provided the Wj satisfy a relative magnitude condition . Such a polynomial has precisely one real root. Since

p(O)

= 2~-1 -

1,

that root is positive for d < 4, zero for d = 4, and negative for d our claim that p(v) = f(v) - 1 v is decreasing, we show that

(4.16.24)

> 4. Now to verify (4.16.25)

p'(v) = vf'(v) ; f(v) ::; 0 v or rather

v!'(v) ::; f(v)

(4.16.26)

= lov !,(u)du

(4.16.27)

because f(O) = O. Clearly, this inequality holds if f' is a decreasing function as well, so the problem now is to verify that

!" ::; O.

(4.16.28)

On the other hand, the chain rule implies (4.16.29) If we differentiate again, the chain rule and product rule yield

n

!" = (f~

0 fd-l 0 •. . 0 h 0 fr)(f~-l 0 . . . 0 h 0 Jr)2 ... (f~ 0 fr)2 + (f~ u fd-l 0 ..• 0 h 0 fr)(f~-l 0 •.• 0 h 0 fr) ... (f~ 0 fr)2!'i + ... + (f~ 0 fd-l 0 . . . 0 h 0 fr)(f~-l 0 ••. 0 h 0 fr) . . . (f~' 0 fr)!'i + (f~ u fd-l 0 ' " 0 h 0 fr)(f~-l 0 '" 0 h 0 Jr) ... (f~ 0 fr)f{'. (4.16.30)

Since

1'/ == -2~72w1,

(4.16.31)

4.16. WAVELET CORRECTION OF THE HIERARCHICAL MODEL

399

we need only show that those first-derivative factors

fj 0 Ii-I 0 ... 0 h not appearing as squares are manifestly positive - although

fj(v)

= 2~-1

- 2~

X

72WjV4

(4.16.32)

certainly is not. The key observation here is that j > 1 for this kind of first-derivative factor and that

fj

0

fj-I ~

°

(4.16.33)

is the final reduction of the claim. On the other hand,

fj(v) ~ 0,

1 - 144wJ'

v<--

(4.16 .34)

while the maximum value of the quadratic function Ii-I (u) occurs at 1

U

= 144w4 ' _ j

(4.16.35)

1

imposing the inequality 2~

fj-I(U) ~ 576w 4

I

(4.16.36)

]-

This is enough to establish (4.16.33), provided that

2~-!lwjl ~ IWj-ll·

(4.16.37)

By (4.16.4) this requirement reduces to (4.16.38) which has never been proven, as these numbers have not been calculated numerically. If this inequality is not true, then our argument no longer applies, but it is no longer necessary if the numbers are known. The polynomial f(v) is explicitly realized in that case. One expects uniqueness of the nontrivial fixed point to hold for these numbers, but it is important to remember that we have only been discussing the second-order approximation here. The issue would still have to be settled for the full RG transformation.

Chapter 5

Wavelet Analysis of § The greatest early achievement in constructive quantum field theory was the construction of the ~ Euclidean field theory. This was done concurrently with the construction of scalar field theory in two (space-time) dimensions, but - as we have seen in Chap. 1 - three (space-time) dimensions is very different with regard to one's approach to the ultraviolet problem. Indeed, that problem cannot be so neatly separated from rigorous control of the infinite-volume limit. The phase space analysis developed by Glimm and Jaffe [G45, G57] was necessarily long and arduous at the time, but ideas that involve scaling can be found in their work. Using their methods, Feldman established existence of the ultraviolet limit in a finite volume [F26] . Existence of the infinite-volume limit for weak coupling was subsequently proven by Feldman and Osterwalder [F31, F32]. Their proof included verification of the Osterwalder-Schrader axioms. What discouraged so many mathematical physicists at the time was the daunting observation that - as difficult as a rigorous analysis of the ~ model seemed to be - it is far less difficult than any rigorous approach to four-dimensional field theories. Even at the level of formal perturbation theory, the ultraviolet renormalization in three dimensions is dwarfed in difficulty by such cancellations in four dimensions. As we have already seen in §1.13, the cancellation of divergences with counter-terms is perturbatively easy in the ~ case because there is more large-momentum decay in those cancellations than needed to estimate them. Magnen and Seneor substantially refined the work of Glimm and Jaffe with technical ideas of their own [M6], but the analysis of the ~ model was still far from elementary. This model became the testing ground for the development of rigorous expansion methods in quantum field theory. Recognizing the importance of scaling, Benfatto, Cassandro, Gallavotti, and their co-workers introduced a multiscale decomposition of the free covariance - at the same time preserving the Markov property, which plays such an elegant role in proving ultraviolet stability of the scalar field interactions in two dimensions. The formalism partially simplified the phase space analysis in three dimensions in this ingenious way [B29], but it still involved heavy-duty functional analysis. It was Federbush who first advocated the phase space analysis of fields in terms of expansion functions based on phase cells. Originally applied to an infrared prob401

402

CHAPTER 5. WAVELET ANALYSIS OF ~

lem [FlO], his orthonormal basis had a multi-scale coherence that was most appealing. In short, it was a polynomial generalization of the Haar basis aimed at extracting a large number of vanishing moments from the phase cell functions. Unfortunately, such functions are necessarily discontinuous, and this limitation led to problems in the ultraviolet analysis of the cf>j model. Actually, it was natural to expand cf> in the Bessel potential (-~ + m5)-! of these basis functions, so there was enough regularity to carry out the perturbative analysis in terms of these phase cells. The point is that this formalism created a technical problem for the non-perturbative analysis of the cluster expansion developed in [B2l, B22]. Nevertheless, important combinatorial ideas were introduced in the wavelet (phase cell) formalism [Bll, B2l-B23] which made it clear that phase cell cluster expansions should apply to a broad range of Euclidean field theories. The authors simply had the wrong kind of wavelet at the time, and even the resulting technical problem could be avoided by complicating the new expansion in substantial but familiar ways. In the meantime, Brydges, Frohlich, and Sokal found an entirely different and elegant approach to the construction of the cf>j model. Instead of using expansion methods, they controlled the continuum limit with a class of correlation inequalities based on a random walk representation [B72, B73]. Not long afterward, a phase space analysis based on a multiscale momentum-slicing of the free covariance ((;;)(y))o was developed by Feldman, Magnen, Seneor, and Rivasseau [F27- F30]. It was shown to be applicable to asymptotically free models that are strictly renormalizable - not just j, which is super-renormalizable. Even in the case of the cf>! model (which is not asymptotically free) these authors controlled the long-distance behavior for a fixed ultraviolet cutoff and zero mass. They proved existence of the infinite-volume limit for weak coupling - a difficult infrared problem. Gawedzki and Kupiainen had proven the same result by showing the Gaussian fixed point to be attractive on the critical manifold of models with respect to the flow of the renormalization group [G26]. A couple of years later, the subject of wavelets began to develop rapidly as a result of widespread interest in signal analysis. The type of wavelet needed for short-distance analysis in a phase cell cluster expansion was found at this time. The Lemarie class of wavelets has already been discussed in Chap. 2. It has an integer-valued constructibnparameter that determines both the class of smoothness and the number of vanishing moments. A Lemarie basis of wavelets is ideal for a phase cell expansion of the ~ model, or any other scalar model that is asymptotically free . Such expansion functions are not sharply localized like the piecewise-polynomial expansion functions originally tried, but they have exponential decay, which is the long-distance decay that is actually required of a function associated with a cube of a given scale [B12 , B13]. The wavelet cluster expansion developed in [B2l, B22] was finally vindicated [B24]. In this chapter we describe and control the wavelet cluster expansion of the cf>~ Euclidean field theory with the advantage of hindsight . The starting point is the decomposition cf>{-;;) = L:>:tkUk{-;;) (5.l) k

of the free Euclidean field with mass mo, where

Uk = {-~+m6)-h}lk

(5.2)

403 and {\II kl is an orthonormal basis for L2 (I~.3). This basis is actually a modification of a Lemarie wavelet basis inspired by the presence of the mass rna in the theory. There is no infrared problem for weak coupling because the correlations of the free Euclidean field have exponential decay. With no multi-scale decomposition required for the longdistance analysis, the idea is to include all elements of the Lemarie basis with length scales :::; 1 and then complete the orthonormal set with special unit-scale functions . The most natural set of such auxiliary functions is the set of all unit-scale translates of a function which is not quite a wavelet but plays a role in the construction of the wavelet basis. In the end, one must obtain an interacting continuum field theory satisfying the Euclidean Axioms . As we discussed in Chap. 1, the most important properties to be verified are Euclidean invariance and reflection positivity - both of which are destroyed by the wavelet cutoff. However , once the wavelet cutoff is removed by a phase cell cluster expansion, the proof that the limiting correlations have these properties is not as difficult as one might believe at first glance. To obtain rotational invariance, for example, one superimposes a spherically symmetric ultraviolet cutoff on the wavelet cutoff and applies a double-limit argument, although one has to be careful with the type of ultraviolet cutoff. While the spherical symmetry insures rotational invariance of the easily-controlled infinite-volume limit of a fixed ultraviolet cutoff with the wavelet cutoff already removed, the double-limit argument cannot work unless the convergence of the phase cell cluster expansion is uniform with respect to the removal of the ultraviolet cutoff. This can be guaranteed by uniformity of the input estimates , so the point is that the dependence of these bounds on length scales must be uniform in the ultraviolet cutoff. Such a condition cannot hold for the standard (sharp) ultraviolet cutoff, which we used in Chap. 1. However, the Pauli-Villars regularization [P12] adapted to Euclidean fields works very well . If we apply the cutoff

with M = 00 as the ultraviolet limit, it is easy to show that the essential estimates are uniform in M ~ 1 for wavelets regularized in this way, provided that arbitrarily large length scales are ruled out . On the other hand, our choice of orthonormal basis has done just that. Another advantage offered by the choice of expansion functions is that Z3-translational invariance of the correlations is guaranteed by the Z 3-translational invariance of the basis, provided the infinite-volume limit is unique. On the other hand, it is a nice grouptheoretic exercise that rotational invariance together with Z3-translational invariance imply full Euclidean invariance, as continuous translations are easily generated by the combination. Symmetry arguments of the type we have given above are the reason why the expert is so often satisfied with the convergence of a phase cell cluster expansion, provided that the superposition of ultraviolet cutoff and phase cell cutoff poses no problem in itself. As long as the interaction Lagrangian is formally invariant, there is no cause for concern beyond model-specific details of a routine argument. Nor does the verification of Osterwalder-Schrader positivity pose any problem again, provided that the Euclidean probability measure is formally OS-positive. Sup-

CHAPTER 5. WAVELET ANALYSIS OF §

404

pose one wishes to verify this property in the x3-coordinate direction. If the interaction Lagrangian is formally local, then the ultraviolet cutoff , ~

(p)o--+

2

M2 2

, ~

M2<1>(P)

PI +P2 + preserves locality in the x3-coordinate and that locality property can be rigorously exploited to establish the refiection positivity because - although this regularization is not smooth for sample fields - it is still continuous. The double limit proof of the x3-refiection positivity in the absence of both this cutoff and the wavelet cutoff applies here because the wavelet estimates are uniform in M ~ 1 for this ultraviolet cutoff as well. The wavelet cluster expansion we develop is based on the polymer formalism described in Chap. 3, so the expansion rules themselves determine the definition of an arbitrary polymer, which we shall refer to as a phase cell polymer. In [B21, B22], where the polymer formalism was not fully exploited, the same type of polymer was developed, but it was called a Representation 2 graph . Naturally, such polymers have a complicated structure that contain renormalization cancellations, but they also involve sums over "histories" of the type explained in Chap. 3. The expansion will be inductively defined, but the algorithm is nontrivial because the inductive interpolation with which we are already familiar must be interrupted by integration by parts whenever a short-distance divergence begins to develop. One great advantage of a phase cell cluster expansion is that no energy counter-terms are really needed for the renormalization because infinite-energy diagrams never develop if our inductive expansion is clever enough. Now in §1.13 we saw that the counter-terms required in formal perturbation theory for the § model are a second-order mass term, a second-order energy term (which compensates for the Wick-ordering of the mass counter-term), and a third-order energy term. This means that all we need for our expansion is the mass counter-term, and we do not even Wick-order it. It is important to remember that there is no heavy-duty functional integration involved in this expansion. At any given time, only an elementary integral for a finite number of the variables Ctk is ever considered. They are accompanied by products of numerical factors given by f d ?-integrals of products of the expansion functions Uk(?) . We have seen in Chap. 4 how such quantities are associated with wavelet diagrams, which we shall use here as well. In this chapter, however, we are not using the renormalization group formalism at all, but rather a grand cluster expansion that encompasses arbitrarily small length scales. The identification lines in our wavelet diagrams here are generated by integration by parts, which are expansion steps developing the diagrams. It is also important to remember that the free part of the Euclidean action is diagonal in these phase cell variables, and so the expansion does not decouple covariances. The convergence proof consists of two intricately interwoven but very different parts - the perturbative part and the large-amplitude part. The former contains the cancellation of short-distance divergences and is the part of the phase space problem that guides the development of the expansion. Experts have a more routine attitude toward this problem than toward the large-amplitude problem. The latter is very critical because it can only be solved by both stability and phase cell positivity of the interaction.

405

To be sure, the Gaussian (free) part of the measure quenches the probability of large amplitudes too, but such estimation only yields the bound (5 .3)

for high powers of the amplitude Ok of each mode k . This is too much factorial growth to admit convergence of the expansion, and we shall refer to this phase-celllocalized growth as the number divergence . A major task is to show that each factorial growth is really much weaker and then cancel that weaker growth with a product of numerical factors that decrease in size at a certain geometric rate. The extraction of these factors from the small factors differentiated down by the expansion steps will be referred to as the assignment of numerical factors . The phase cell positivity of the interaction is applied to weakening the factorial growths, as that positivity is quartic in the amplitudes Ok in contrast to the quadratic positivity of the kinetic (free) part of the Lagrangian. The extra liN! factors arising from the combinatorics in Chap. 3 plays an indispensable role in the assignment of numerical factors. This chapter is the most technical and, at the same time, the most elementary. Only the most basic concepts are invented to deal with the convergence problem. The expansion rules never involve any operations more sophisticated than interpolation and integration by parts, so the generated terms are represented by Representation 1 graphs [B2l, B22J, which consist of simple wavelet diagrams and chains thereof. Our terminology differs in a couple of respects from the terminology used in the original papers. For example, we refer to the simplest wavelet diagrams as links, and any link u associated with a space integral of the form (5.4)

is a 4-link. We still use the term composite [B22) for a link (u, u ' ) associated with w{u)w{u ' ), where at least one mode in u is a mode in u ' and the case requires that u and u ' be considered together. On the other hand, we use the term combination link for those special cases where a composite link has to be combined with an additionaI4-link. Moreover, since Representation 2 graphs are the polymers in this phase cell expansion, we speak of a "Representation 2 graph" and "phase cell polymer" interchangeably. The notion of Representation :1 graphs [B2l, B22) is avoided by applying combinatoric estimation that is more widely used (see §5.1O) , but of course one can argue that the concept is still there. In the first section we describe the <J>~ Euclidean field theory in the wavelet formalism, where the regularization is just the restriction of the field configurations to a finite but arbitrary set of wavelet amplitudes. We refer to the wavelet indices as modes, but actually, the discussion in that section is independent of the wavelet analysis . The Federbush stability bound derived there applies to any orthonormal basis. Related results have been proven by Lieb [L37). In the second section the analysis becomes wavelet-specific. We exhibit the most basic estimates on Wk(a\ uda;), ~k(p), and partial derivatives of these functions.

CHAPTER 5. WAVELET ANALYSIS OF ~

406

The primary objective here is to establish the basic quartic positivity bound (5.5) where Lk and

= 2-

rk

is the length scale of

wd:;;), A is a finite but arbitrary set of modes, (5.6) kEA

The scale parameter r ranges over the non-negative integers, but - in contrast to the notation and circumstances of Chap. 4 - the length scales are arbitrarily small instead of arbitrarily large . Since we are solving a short-distance problem with a phase cell cluster expansion, it is most convenient to parametrize shrinking length scales with increasing r - a convention we followed in Chap. 2. In the third section we introduce the two inductive operations for the expansion interpolation and integration by parts. The former is based on attempts to decouple variables, but it is a little more elaborate than the interpolation used in Chap. 3. As the coupling agent for the phase cell variables, the quartic part of the interaction is interpolated in almost the same straightforward way as the nearest-neighbor interaction for a spin system - actually, its Wick-ordering is interpolated in exactly the same way but the mass counter-term is interpolated in a different way. Indeed, the latter interpolation is more convoluted than that chosen for the mass counter-term in [B22]. The aim is to avoid the development of any infinite-energy diagrams - an option made available by the nature of phase cell cluster expansions. The price we pay is that the proof of stability in the number of modes for the total interpolated interaction is a little more involved. The proof is a major part of §5.8 and depends on a combination of Federbush stability with estimates on numerical factors for bounding the negative contributions to the interpolated mass counter-term. In §5.4 we establish those estimates - indeed, all estimates on numerical factors that will be needed for proving convergence of the expansion. This includes the estimate on the type of numerical factor containing the mass renormalization cancellation. In §5.5 we introduce the expansion rules. With the interpolation of the interaction already defined, the rules specify exactly when to interrupt interpolation with integration by parts along a given branch of the inductively defined expansion. Naturally, the algorithm chooses the a-variable with respect to which the integration by parts is done, and wavelet diagrams are very useful visual aids in understanding the numerical factors that are generated by a sequence of integration by parts steps. The algorithm we choose is by no means unique - and it is not the same one chosen in [B22] - but there is a guiding principle for all of these choices. The set of expansion rules is motivated by the need to avoid ultraviolet divergences in the perturbation theory, and nothing more. Cancellation of the number divergence - the nonperturbative part of the problem - is done by clever estimation, and none of the difficulti es therein influence the expansion rules . As far as the expansion rules in this chapter are concerned, we choose to integrate by parts as little as possible . In §5.6 the completed expansion is realized as a sum over histories, which we proceed to organize with the notions of links and chains. The renormalization cancellations are

5.1. THE CUTOFF - A FINITE SET OF MODES

407

automated here by the appropriate combination of histories. This means that every such cancellation is realized as a special numerical factor, on which the necessary estimate has already been established in §5.4. Some important notation is fixed in this section as well. The actual terms in the completed expansion are labeled by Representation 1 graphs. In §5.7 we relate this expansion to the polymer expansion formalism and identify the polymer as a Representation 2 graph, which is basically a Representation I graph without its history of development. This means that the activity of the polymer is the sum of all expansion terms whose Representation I graphs all correspond to the given polymer. Therefore, the preliminary estimation of the expansion terms in §5.8 is not an activity estimate, but this is the stage where the stability of the interpolated interaction is proven and the quartic positivity is extracted. The preliminary estimation of the polymer activity itself is the content of §5.9, and this is very important, as it depends on a combinatorial identity summing the weight factors associated with the Representation I graphs and realized as integrals over products of interpolation parameters. It is a generalization of the Federbush identity, which has already been discussed in Chap. 3. In §5.10 we consider the problem of summing activities within the framework of the polymer formalism, and this is where the important notion of attachment is introduced. From the combinatorics of the summation we extract the extra liN! factors associated with the attachments, and in §5.11 these factors are used to reduce the problem of cancelling the number divergence to a problem concerning the assignment of numerical factors. §5.12 is devoted to reducing the whole convergence problem to the consideration of just a finite number of cases. Chains are a main source of concern here, as the necessary smallness their development extracts must be distributed over the links before the chains are broken up. The notion of attachment is extended to the notion of pinning for each link in a chain, which depends, in part, on whether a link precedes or succeeds the attachment link of the chain. The case structure of a chain is further complicated by the possibility of a chain being internally connected in a way that affected the expansion steps. As in [B22J, this case is dealt with by introducing a hook and a hook link - important topological notions that affect the distribution of small factors over the links of the chain. In the end, we leave the case-by-case assignment of numerical factors to the reader as one great exercise. However, in §5.13 we bring back the wavelet diagrams already used to describe the original expansion rules. The diagrams provide the best way to think about the assignment of numerical factors. It is similar in spirit to the power counting for Feynman diagrams, and in §5.14 we illustrate this game by successfully assigning numerical factors in a number of cases.

5.1

The Cutoff - A Finite Set of Modes

Let {wd denote an orthonormal basis in L2(1l~.3) which we specify in the next section as a certain wavelet basis. Our initial remarks are independent of the multi-scale structure so vital to a phase cell analysis. As we have already indicated above, we expand the

CHAPTER 5. WAVELET ANALYSIS OF ~~

408

Euclidean field ~(X") in random amplitudes ak as follows :

L akuk(X"),

(5.1.1)

k

(-~

+ m6)-1/2 wk .

(5.1.2)

Equivalently, we define the random variables (5.1.3)

with the motivation that since (5.1.4)

these variables have the property (5.1.5)

which makes them independent with respect to the Gaussian measure dJ1.o . The rules for Wick ordering are simple:

II .

. II U.k' ~Nk .

.

~Nk

.

. Uk

(5.1.6)

"'

k

k

~N e -!"'~ .. U.k

dN

1

2

(-1) N --e- ,"'k

..

daf:"

(5.1.7)

Let A be a finite subset of the set of indices and define

~A(X")

=L

(5.1.8)

akuk(X").

kEA

Let the expectation functional (-) A be given by ZA'I (F(a)e-I(A))o,

(5.1.9)

(e-I( A))O,

J:~A(~f + J ~A(X")2 J

(5.1.10)

: d 'X"

A

48A

2

d'X"

dy

(~A('X")~A(Y))~.

(5.1.11)

In this a priori renormalization of the interaction, the mass counter-term is not Wickordered, nor are the second-order and third-order energy counter-terms included. This omission is possible because at each stage in the removal of the cutoff, only a finite number of variables are involved. Naturally, the phase cell expansion must still be inductively defined such that the infinities cancelled by the omitted counter-terms can never develop.

5.1. THE CUTOFF - A FINITE SET OF MODES

409

No functional analysis is needed to make sense of this regularized functional integral (5.1.9) because it is an ordinary integral in a finite number of variables. More explicitly,

=

(F(Q))A

(II 1

00

(27r)-!

cardA ZA"1

kEA

e

-!

a~-I(A)

L kEA

,\

dQk)

-00

F(Q),

(5.1.12) 4

L

w(kl,k2,k3,k4):IIQk,:

k"k2,k3,k.EA

,=1

k, , ... ,k5EA

(5 .1.14) Our goal is to control the limit of the expectations

\Jl ~(fj))

A

k'~kN \Jl

Qk; ) A

Jl J

/j(-;)ud-;)d-;

(5 .1.15)

as A approaches the set of all modes in the sense of set inclusion. The interpretation of the A-expectation when not all k1 , . . . , kN lie in A is understood to be:

II

N Qk; ) \ j=1 A

=

\

II

Qk; )

j: k;EA

II

\

A

j: k;rlcA

(5.1.16)

Qk; ) 0

It suffices to control the limit of such an expectation for a fixed product, provided the bounds obtained imply the condition

(5.1.17) The point is that if

L 1(( -~ + m6)-no

Uk )(-;)1

~e

(5.1.18)

k

for some positive integer no, then (5.1.17) implies

N

~

(N!)1/2e N

II II( -~ + m6tofjlll j=1

~

N

(N!)! eN

II sup 1(1 + 1-; 12)no (-~X" + m6)no Ij (-;) 1 j=1

x

(5.1.19)

CHAPTER 5. WAVELET ANALYSIS OF <.t>~

410

for the test functions h, ... , f N if we also choose no > 1. This scheme already imposes a significant condition on the orthonormal basis {'l1 d, but (5.1.18) is a property that is typical of a wavelet basis. The generating functional to study for the expectations (5.1.16) is obviously given by (5.1.20)

and the partition function for the polymer formalism will be (5.1.21)

The polymers for the phase cell expansion will be rather complicated, but the lower bounds on I(<.t>A) required for estimates on polymer activities will be relatively simple. Indeed, the most basic lower bound holds for any orthonormal basis - not just a wavelet basis. First, one completes the square to obtain: (5.1.22) Second, one expands this last integral: (5.1.23) Observe that the lower bound (5.1.24)

follows from the Federbush stability bound :

L k,lEA

JUk(~)2Ul(~)2

d ~~ c card A,

(5.1.25)

which holds for an arbitrary orthonormal basis {'l1 k}. Although this quantity is a double sum in the modes, we have a bound that is linear in the number of modes. To see this, set


= L 'l1k(~d'l1kC~2)

(5.1.26)

J~

(5.1.27)

kEA

M(~I' ~2; ~3' ~4) =

d

4

IIC-6. + m~)-1/2C~, ~.).

,=1

5.1. THE CUTOFF - A FINITE SET OF MODES Thus

411

1

L

Uk (;;)2Ue(;;)2 d;;= (Mt.p,t.p),

(5 .1.28)

k,iEA

and since 1I t.p1 1 ~ ::; card A,

(5 .1.29)

the problem is reduced to showing that M is a bounded operator. Now in momentum space we have the kernel

M(Pl' P2; P3' P4)

(IT 1

d;; ,) e ip I'? 1+iP2'? 2- i P3 '? 3- i P. ·? M(;; 1, ;;2 ; ;;3, ;;4)

,=1

1d;;

4

e i (PI +P2-P3-P.),?

IIu p, 12 -I- m6)-1/2 ,=1

II (I p, 1 -I- mo)- 1/2 , 4

~

3

~

~

~

(27r)- 8(Pl -I- P2 - P3 - P4)

~

2

2

(5 .1.30)

t=l

so for an arbitrary 9 E L2 (JR3) ® L2 (JR3),

(Mg)(Pl' P2) x

1

d P3

~

= (27r)-3(1 PI (I P3

~

12 -I- m5)-1/2(1 P21 2 -I- m6)-1/2

12 -I- m5)-1/2 ~

x (I PI -I- P2 - P31

2

2 1/2

-I-m o )-

~

~

~

~

g(P 3, Pl -I- P2 - P3)'

(5.1.31)

By the Schwarz inequality and a standard convolution estimate,

I(Mg)(Pl ' P2W

::;

(27r)-6(1 PI 12 -I- m5)-1(1 P2 12 -I- m5)-1

[j dP3 (I P31 2 -I-m5)-1(1 PI -I- P2 - P3 12 -I-m5)-1] x (1 d P3Ig(P3' PI -I- P2 - P3W) x

::;

c(1 PI 12 -I-m5)-1(1 P21 2 -I-m5)-1(1 PI -I- P21 2 -I- m5)-1/2 x

(1 d P3 Ig(P3' PI -I- P2 - P3W) .

Now write

1 1 d PI

d P2 I(Mg)(Pl' P2W =

1 J d PI

d P I(Mg)(Pl ' P - Pl)1

and integrate with respect to PI first . Then (5.1.32) implies

1

d PI I(Mg)(Pl' P - PIW

(5 .1.32)

2

(5 .1.33)

CHAPTER 5. WAVELET ANALYSIS OF 4lj

412

s

P 12 + m~)-1/2

c(1 X

/

( / d P3

Ig(P3' P - P3)1 2)

~ ~ 2 2 1 ~ ~ 2 2) 1 d PI (I PI 1 +m O)- (I P - PI 1 +mO -

S c(1 P 12 + m~)-3/2 / d P3 Ig(P3' P - P3W,

(5.1.34)

where we have applied the convolution estimate again. Hence

/ dp / dpll(Mg)(pl>p - PIW <

c/ c/

dp / dP3Ig(P3'P - P3)1 2 dP3 / dp Ig(P3,P)1 2

cllgll~,

(5 .1.35)

and so M is a bounded operator.

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982), 95-139. 2. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis and Self-Adjointness, Academic Press, New York, 1975. 3. E. Lieb, "An LP Bound for the Riesz and Bessel Potentials of Orthonormal Functions," J. Funct. Anal. 51 (1983) , 159- 165.

5.2

Wavelet Estimates

In the previous section we discussed the mode decomposition of the Euclidean field as thoroughly as possible without specifying the orthonormal basis. This can no longer be avoided if we want estimates that cut to the heart of phase cell analysis. We shall use a basis consisting of wavelets having arbitrarily small scale but completed by different functions at the unit-scale level. We need no large-scale wavelets in the basis because this model has no long-distance problem in the weak-coupling regime. The small-scale wavelets we use are three-dimensional Lemarie wavelets with large degree parameter N. The mother wavelets are:

= w(Xd
(5.2.1.100)

WlOl (:;) = W(Xl)
(5.2.1.101)

WlOOCX")

(5 .2.1.110)

3

W111(:;) =

II w(x,), ,=1

(5 .2.1.111)

5.2. WAVELET ESTIMATES

413

I]!Ol1C;;;)

= cp(Xdl]!(X2)I]!(X3),

(5.2.1.011)

I]!OlOC;)

= cp(Xdl]!(X2)CP(X3),

(5.2.1.010)

I]!OOl (X') = CP(X1 )CP(X2)I]!(X3),

(5.2 .1.001 )

where I]! is the mother wavelet for the one-dimensional Lemarie basis described in §2.2 and cP is a scaling function given by (2.2 .3). For length scales strictly less than unity, the mode k is labeled by three indices: 1) ~ E {O, 1P \ { which indexes the mother wavelet 'lII': ' 2) ~E Z3, which indexes the discrete translate of I]!_ I': ' 3) r E Z+, which indexes the dyadic scaling of 'lI-.

0},

I':

For k

= (r,~,~)

we have (5.2.2)

while at the unit-scale level, the mode k is indexed only by (~ , ~) but with ~ = 0 included: (5 .2.3) The functions l]!ooo(X' - ~) are the "different functions" that complete the basis. I]!ooo is not a wavelet, but a three-dimensional scaling function - i.e., 3

l]!ooo(X')

= II cp(x,).

(5 .2.4)

,=1

N is the degree of the spline from which cp and I]! are derived, so the functions 'lI k are all class C N - 1 in each coordinate. Moreover, the functions

o :s v, :s N

- 1,

have exponential falloff, and obviously the "correlation length" for each function is proportional to the length scale of that function. Finally, the vanishing-moments condition (5.2.5) holds for small values of some momentum coordinate p" provided ~ E {O, 1P\{ O} . Bear in mind that I]!ooo does not have any vanishing moments. Now once the operator (-~ + m6)-1/2 is applied to I]!k, the scale-commensurate character of the exponential decay is destroyed. The exponential decay of the resulting function Uk is tied to a fixed scale regardless of how small the length scale of k may be. We introduce the notation k = (rk' ~k' ~k) and let (5 .2.6)

CHAPTER 5. WAVELET ANALYSIS OF ~

414

which is the length scale of k. These small-scale Wk may be written as (5 .2.7)

On one hand, (

g3) a~:

1

-3/2- L' e", -

Wk(;;) ::; cLk

cL

;;

1

~ ~

Ix-x"l,

(5.2.8)

1

where -; k denotes, say, the minimum-coordinate vertex of the Lk-scale cube in which ilk is approximately localized - i.e., (5.2.9)

On the other hand, for the Bessel potentials Uk the exponential decay bound is drasticall y different: 1

(g a~:)

Uk(-;)I::;

cL~1/2-~"'e-cmoI7-7"1 .

(5 .2.10)

Such an estimate cannot control sums over all modes at a given small scale, as the dense packing of modes must be balanced by a scale-commensurate localization. The requirement (5.1.18) already imposed on the basis in the previous section is a case in point. It is the vanishing-moments property of the wavelets which provides all of the scalecommensurate falloff of the Bessel potentials that we will ever need. Pick any mode k whose length-scale is strictly less than unity. We claim that for v, ::; N and N + L: v, even,

-1/2-L", ::;cLk

~

~

'(I+L;;11 x - Xk I)

-N-L,,- 2 ,' ,

(5 .2.11)

and by (5.2 .2) , it suffices to show that (5.2.12)

Since Lk is arbitrarily small, the estimation needed to prove this necessarily throws away the massive contribution to the denominator in momentum space. We have

(g a~:) ((-~ + L%m~)-1/2W (-;)1 : ; J p I~¥ ((g p~,) (I P 12 + L%m~)-1/2\ji (P)) 1' 1-; 12M 1 d

.. .)

.. "

(5.2.13)

5.2. WAVELET ESTIMATES

415

where the integrand is bounded by

"~,

~IJ.,=M

X

For 1

1

1

(u %,) rrM~,! U('n I(u a;: ) ~,. (pl I ,=1

(g 8;;'-U') ((gP~') (I P

12 + L%m5)-1/2) I·

P ::s 1 we use the estimate 1

(g 8;;'-U') ({I P

12 + L%m6)-1/2

gP~')

1

::s cl P 1-2M+~u'+~V'-1

(5.2.14)

and observe that (5 .2.5) implies (5.2.15) by the real-analyticity of ~-: .(p). Thus the integrand is integrable over provided

1p 1-

2

1

p ::s 1

1,

is the worst small-p divergence - i.e., (5.2.16)

For

1

pi:::::

1, one should estimate more carefully than (5.2 .14) to exploit (5 .2.17) 3

but since

1 P 1- 1/ 2 IT {I + Ip,I)-1 ,=1

is integrable, it is obvious that our integrand is

integrable in this region as well, provided //, allows the choice 2M

::s

N. Since N

+ L //,

is even, (5 .2.16)

= N + L, //, + 2, so (5.2.12) is established.

This useful estimate will often be applied in conjunction with an estimate on a pile-up of Bessel potentials of Uk . The argument just given can be adjusted slightly to verify that for even Nand 2// ::s N + 2,

I{{ -~

+ m5)-V Uk )(a;)1 ::s CL;1/2+2V {I + L;;11 a; - a; k I)-N+2V-2.

(5 .2.18)

This is also the key to the bound (5 .1.18) with // = no = 1. Indeed,

::s

c2-~r,

(5 .2.19)

CHAPTER 5. WAVELET ANALYSIS OF
416 and therefore

2: 1(( -~ + m~)-luk)(X")1

00

:s: C 2: 2-!r = c.

k

(5.2.20)

r=O

Technically, the r = 0 summation cannot use (5 .2.18), since some of the unit-scale functions are built on ifl ooo , but the non-scale-commensurate exponential decay I(( -~ + m6)-V uk )(X")1

:s: CL~!+2V e-cmolx - x.1

(5.2.21)

is more than enough at the unit-scale level. Another fundamental inequality for the phase cell expansion is a-positivity. For any finite set A of modes, we have

~4 dx2:cL...,(1+lnL;;)~ '"' 1 2
J

Let FA be the space of functions k T : L 2(JR3) -T FA by

I-t Zk

on A, and define the linear transformation

T(fh = L~I/2(f, (-~ + m~)1/2if1k)' Now for 1 :s: q <

00

(5.2.22)

kEA

(5.2.23)

we define the norm I · Iq on FA by

Izig = 2:(1 + InL;;I)-2L~lzklq

(5 .2.24)

kEA

and note that 1(( -~ + m6)1/2if1 k)(X")1

:s: cL~5/2(1 + L;;II X" -

X" k

I)-N-4

(5 .2.25)

This estimate implies (5.2.26) 2:

1(( -~ + m~)1/2if1k)(X")1

:s: c25r / 2

(5.2.27)

kEA £.=2- r

By the first inequality, sup IT(fhl kEA

:s: :s:

SUp(L~I/211( -~ - m~)1/2if1klh 1I!1I00 ) kEA

cll!lIoo,

(5.2.28)

and by the second inequality, 2:(1 + In L;;I)-2L~IT(fhl

:s:

kEA

J

d X" 1!(X")I2: (1 + In L;;l )-2 L~/2 kEA

X 1(( -~

:s:

c

f

+ m6)1/2if1 k )(X")1

dX" 1!(X")II:(1+rln2)-2

.

cll!lll.

r=O

(5.2 .29)

5.3. INDUCTIVE OPERATIONS

417

This means T has both an A-independent (11 · 1100, I 'Ioo)-bound and on A-independent (II, IiI, l'ld-bound, so by the Riesz-Thorin Theorem, T is (II · 114, I '14)-bounded independently of A. Thus

2:(1 + InLk'1)-2 L~IT(f)kI4 :s c JIfCX")1 4dx,

(5 .2.30)

kEA and since

(5.2.31) (5 .2.22) follows immediately. This quartic lower bound will be vital to the nonperturbative part of the convergence proof for the phase cell cluster expansion. We remark in passing that the (II . 112, I . 12)-bound provides the inequality

J

A (X")2d

X"~ c 2:(1 + InLk'1)-2 L~a%,

(5.2.32)

kEA

but there will be no occasion to use this quadratic lower bound. After all, the free part of the action is precisely ~ 2:= a~, which obviously has much stronger positivity. kEA

References 1. G. Battle, "A Block Spin Construction of Ondelettes, Part I: Lemarie Functions," Commun. Math. Phys . 110 (1987),601-615 . 2. G. Battle, "A Block Spin Construction of Ondelettes, Part II: The QFT Connection," Commun. Math. Phys. 114 (1988), 93-102. 3. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 4. G. Battle and P. Federbush, "Ondelettes and Phase Cell Cluster Expansions, a Vindication," Commun. Math. Phys. 109 (1987), 415-417. 5. P. Lemarie, "Ondelettes (1988) , 227-236.

a Localisation Exponentielle,"

J. Math. Pures Appl. 67

6. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis and Self-Adjointness, Academic Press, New York, 1975.

5.3

Inductive Operations

Our starting point is the generating functional (5.1.20) with a linear ordering imposed on A. As in the context of statistical mechanics, we try to decouple variables through interpolation, but instead of knocking out two-variable interactions, we knock out fourvariable interactions. In our wavelet formalism, the kinetic part of the Euclidean Lagrangian (which in the lattice approximation would contribute nearest-neighbor couplings between site variables) is actually a diagonal quadratic form with respect to the

CHAPTER 5. WAVELET ANALYSIS OF 4>~

418

wavelet amplitudes. This means that the local quartic field interaction contains all couplings in those amplitudes . Another difference is that our inductive interpolation may be interrupted by some sequence of integration by parts aimed at cancelling ultraviolet divergences with the mass counter-term. Suppose at some stage in some history of inductive operations a free expectation of the form / exp

(L

TeCtl - I(t ; rPA')) F(Ct) : Ctko )

lEA'

\

0

has developed, where the t-dependence represents the modification of the interaction by previous interpolations and F(Ct) does not depend on Ctko' This condition must be met by any situation where integration by parts is called for in our inductive expansion. The integration by parts is based on the variation (5.3.1)

on the formula (5.1.7) . We have / exp

\

(L

lEA'

/ :

\ X

TlCtl - I(t; rPA')) F(Ct) : Ctk o :)

Ct~.;-I : F(Ct) exp

0

(L

TeCtl - I(t ; rPA'))

lEA' (5.3.2)

(TkO - a!k/(t; 4>A')) \

and the next expansion step in this event is to select either the term contributed by

Tko or a term contributed by an Ct-product in I(t;4>A') ' This means that after such a step, one still has a free expectation of the basic form (ex p

C~, TeCte - I(t; 4> A')) G(Ct))

0 '

where the details of G(Ct) determine the next expansion step. The decoupling procedure is qualitatively the same as in §3.10, where we applied inductive interpolation to long-range couplings on a lattice. Let Ctl = Ctk' be the first variable with respect to our fixed linear ordering. The first expansion step is the interpolation based on trying to decouple Ctl from the other variables in A . Accordingly,

Z(Tk : k E A) = Z{kl}(Tt}Z(Tk : k E A\{kl}) - [

dti (ex p

(~TeCte -

J(tI; 4> A))

d~/(tI ; 4> A))

, o

(5.3.3)

5.3. INDUCTIVE OPERATIONS

419

where the interpolation of the interaction (5.1.11) is given by I(t1j<J>A) = t1'\ /

: <J>A(;;)4: d;;

+(I-td [,\/ : <J>A\{kl}(;;)4 : d;; +,\:af:.I U1(;;)4d;;1

L

+ 48,\2

W(k1' k 2, k3) / d;; Uk , (;;)Uk2 (;;)Uk3 (;;)

k, .k2 .k3EA x {[t 1<J>A(;;)

+ (1- t1)(8{t~}k3<J>{kl}(;;) +8~\{2k~3}<J>A\{kl}(;;)W

- [t <J>k 1k2k3(;;) + (1 _ t )(8klk2k3 <J>klk2 k3(;;) + 8klk2k3 <J>klk2 k3 (;;))]2 A

1

1

{k'}

A\{k'} A\{k ' }

{k'}

+ t1 <J>~' k2 k3(;;)2 + (1 _ td (<J>{k~}kS (;;)2 + <J>~\{2kk,\ (;;)2)),

(5 .3.4)

(5 .3.5) kEB

Lk~max{Lkl.···.Lkn

}

I, k 1 , ... , k n E B, { 0, otherwise.

(5.3.6)

The quartic part of the interaction is interpolated in the most straightforward manner, while the counter-term is interpolated in a more subtle manner that anticipates the cancellation of ultraviolet divergences. The latter interpolation still has the desired decoupling property because rk,k2k3 rk,k2k3 - 0 (5.3.7) U{kl} uA\{kl} - . Now we may write (5.3.4) in the spirit of (5.1.13) to obtain

I(t1j<J>A)='\ x [h

4

w(k1,k2,k3,k4):IIak,: ,=1

+ (1- td(8{k~}k3k4 + 8~Wk~3}k4)]

+48,\2 X

L

L

W(k1,k2,k3)W(k1, . . . ,k5)ak4ak5 k, •... .k5EA + 8klk2k3k4)][t + 8klk2k3k5)] [t 1 + (1 - t 1)(8klk2ksk4 {k'} A\{k'} 1 + (1- t 1)(8klk2k3k5 {k'} A\{k ' } (same summand) 48,\2

L

kl , .. ·,k5EA Lk4 .Lk5 ~max{Lkl .Lk2 .Lk3}

kl ..... k5EA Lk4 .Lk5 ~max{Lkl .Lk2 .Lk3} X

(8k4k18k5kl

+ 8~\{k' }8~5\{kl })].

(5.3.8)

CHAPTER 5. WAVELET ANALYSIS OF 4'~

420

Hence

k3k4 _ Oklk2ksk4] [1 _ Oklk2 {k'} A\ {k'}

+

L W(kl,k2,k3)W(kl, .. . k5)ak4aks k ... ,k5EA " (1 - t )(Oklk2 k3k4 + Oklk2 k3k4)][1 _ Oklk2 k3kS _ Oklk2 kSkS] [t +I x I {k'} A\{k ' ) {P} A\{k'} 96>.2 (same summand) 96>.2

L

k11 .. ·, ks

EA

L'4 ,L'5 ~ max{L" ,L'2 ,L'3}

+

L

96>.2

w(k 1, k2) k3)W(kl' k 2) k 3, k 4 ) k 1)ak4 al ·

(5.3.9)

k" .. . , k 4 EA

k4#k'

The first step in developing a branch of the expansion is to select either the contribution of one of these terms to (5.3.3) or the decoupled term. If the decoupled term is selected, then one interpolates Z (Tk: k E A \ {kl }) by trying to decouple the second variable a2 = ak2 from the other variables, as the first variable al = ap is already decoupled from them. If any other term - a remainder term - is selected, then an integration by parts may be called for. Consider now a sequence of steps - possibly null- involving integration by parts. At the conclusion of this sequence we have a term of the form

where G(tl; a) is a polynomial in some of the a-variables. The set B of modes k such that G(tl; a) depends on ak is the interior set at this stage. Now we have yet to specify the expansion algorithm - and we shall do so in §5 .5 - but if we assume that no further integration by parts is needed at this stage, then the next expansion step is based on interpolation. This interpolation attempts to decouple the interior set B from A\B. Accordingly,

11 dtl \ exp

(~Tlal -

I(tl; 4'A)) G(tl ; a)) 0

11

(~Tlal -

= Z(Tk : k E A\B)

dt1 \ exp

J(tl; 4'B)) G(tl; a)) 0

-11dtiexP(LTlal-I(tl,t2;4'JG(tl;a)aa o '\ lEA ') t2

J(tl,t2;4'A~

!

, (5 .3.10) 0

where the interpolation of I(tl;4'A) is defined as follows. Let I'(tl;4'A) denote the

5.3. INDUCTIVE OPERATIONS

421

Wick-ordered quartic part of I(tii iI>A) and set

J

L

QA(tl i iI>B) = 48>,2

w(k l , k2, k3) d:; Uk, (:;)Uk2 (:;)Uk3 (:;) k"k2,k3EA x [t i iI>~'k2k3(:;)2 + (1 - td(iI>k'k2 k3 (:;)2 + iI>k,k2k3 (:;)2)] (5 .3.11) B\{k'} , {k' }nB

(5.3.12)

Then

I(tIiiI>A) x

L

= I'(tIiiI>A) +48>.2

.I d:;

w(k l ,k2,k3)

k, ,k2,k3EA Uk,

(:;)Uk2(:;)Uk3(:;){E~,k2k3(tli iI>(:;)) 2

- E~'k2k3(tli iI>k,k2 k3(:;))2}

+ QA(tli iI>A),

(5.3 .13)

and the new interpolation is given by

I(tii t2i iI>A)

= t2I'(tl i iI>A) + (1- t2)[I'(t l i iI>B) + I'(iI>A\B)]

+ t2QA(tl i iI>B) + w(k l ,k2,k3)

.I

(1 - t2)QA(t l i iI>A\B) + 48>.2

L

k, ,k2,k3EA ~ ~ ~ ~ kkk ~ d x Uk,(X)Uk2(X)Uk3(X){EA' 23(tl , t2iiI>(X))2 (5 .3.14)

E~,kok3(tl,t2;iI>(:;))

=

t2E~,k2k3(tIiiI>(:;))

+ (1-t2) x[E~'/;lk3(tli iI>(:;)) + E~\kt3(tli iI>(:;))] . (5 .3.15)

The remainder terms now are contributed by the a-products generated by the t2derivative of the modified interaction I(tl, t2i iI> A)' If the decoupled term is chosen by this expansion step, then the next step is to interpolate Z(Tk: k E A\B) by trying to decouple the first variable in A\B from the other variables in A\B. The iteration of this procedure generates a sum over histories of expansion steps, with each step a choice among many possibilities provided by the partial history already developed. For a given partial history, the term is either a decoupled term or a remainder term. In the former case, it has the form

IT dak-integral developed by the £th sequence of previous expansion kEA, steps , which is composed of successive interpolations occasionally interrupted by a where K, is the

CHAPTER 5. WAVELET ANALYSIS OF ~

422

succession of integration by parts operations and finally terminated by choosing the decoupled term. The sets A, are mutually disjoint sets of modes, and the next expansion step is to interpolate Z A\

(Tk:

k

E

A\

YA,)

by trying to decouple the first mode in

UA, with respect to our fixed linear ordering of A from the other modes in A \ UA, . , '

Suppose the term associated with the partial history is a remainder term instead. It has the form

(II 11 ,

L

dti) J(t) / exp (

0

Teae - J(t;

A\YA,)) G(t; a))

eEA\ UA,

\

,

0

where {til is the set of parameters for past interpolations and J(t) is some product of powers of the ti having developed as old t-dependences have been differentiated down by new interpolations. The quantity G(t; a) depends on only those ak for which k E A\ UA" and the modes which indeed occur in G(t; a) comprise the interior set for

,

the next interpolation of the

J IT

dak-integral, provided the next step is indeed

kEA\UA.

interpolation. An integration by parts may be in order - with AI

= A \ UA, in

(5 .3.2)

- but only if

G(t; a)

= F(t ;a)

: a~a :

(5.3.16)

for the variable aka with respect to which the integration by parts is to be done, where F(t; a) does not depend on aka . Since A is a finite set, every possible history of expansion steps eventually terminates with a decoupled term of the form

and such a decoupled term is a completed term (as it was §3.1O, where we were decoupiing lattice variables). The generating functional is now expanded in these completed terms, but of course the estimates for controlling this expansion must be independent of A. Bear in mind that we have only described the inductive operations themselves and not the rules governing their application, so the expansion we propose to generate in this way is given only schematically at this point. As we shall see later, we have brutally swept some messy but important issues about G(t; a) under the rug as wellissues concerning what type of expression constitutes a term when a formula is applied. However, our next step is to derive the input estimates that will actually motivate the rules for the integration by parts.

5.4. ESTIMATES ON NUMERICAL FACTORS

423

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 2. G . Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical j Model," Commun. Math . Phys. 88 (1983),263- 293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math . Phys. 81 (1981), 327-360.

5.4

Estimates on Numerical Factors

The basic estimation of terms in the expansion obviously falls into two categories - estimation of a-integrals and estimation of space integrals involving products of functions corresponding to modes. We refer to these space integrals as numerical factors, and the quantity w(k 1, ... , k n ) defined by (5.1.14) is the simplest type of numerical factor. There is more than one way that we will have occasion to estimate this quantity, but the most straightforward way is to insert the bound (5.4.1) which is a special case of (5.2.11) . Since w(k1, ... , k n ) is symmetric in the modes, we may assume without loss that (5.4.2) Clearly, n

Iw(k 1, ... , kn)1 :::; c

IT L~,1/2 sup((1 + L;21-; - -;k n I)-N+1+' x

t=l

x IT(1+L/;,11-; -

-;k,

I)-N-2)

,=1 cL~~2

Jd-;

(1 +L/;n11 -; -

-;k n 1)-3-,

n-1

IT L;;,1/2s~~'p((1+L/;:I-; - -;k n I)-N+1+' t.=1

x

n-1 X

IT (1 + L/;,11-; - -;k, I)-N-2),

(5.4.3)

,=1 and the desired estimate involves the use of one mode k, as a fixed point of reference. By the triangle inequality,

(1+L/;,11-; -

2: 2:

1

---lo.

-;k, 1)(1+L/;,11-; - -;k, I) -..lo.

---lo.---lo.

1+L/;, (I x - Xk, 1+1 x - Xk, I) --" --" L
(5.4.4)

CHAPTER 5. WAVELET ANALYSIS OF j

424

For [ ::::: £ < n (smaller length scale for k,) we use

(1+L;,

1

---l.

1x

---.J.

- Xk,

1

---l.

1

.....).

x - Xk,+1 I) ~ 1+L;'

1)(1+L;'+11

-.l..

1Xk,

---l.

- Xk,+1

I·

(5.4.5)

Thus Iw(k 1, ... , kn)1

c L!~2

:::::

n-l

i-I

,=1

,=1

II L;,I/2 II (1 + L;,11 -; k, -

-; k, I)-N'

n-l

X

-ll ~ Xk, - ~ X k,+1 1)-lN-l 2 II(1 + L k,

(5.4.6)

i=i

because the application of (5.4.5) uses each £-factor twice for £ < £ < n, while for £ = [ we reserve an exponent -~N - 1 for the application of (5.4.4). We must choose

1 (1-N+1)

N f < -- [-I

(5.4.7)

2

to provide enough [-factors for that estimation. There is a refinement of (5.4.6) that will be important to the case structure of the expansion. We may apply a spatial integration by parts as follows :

= m~fUkl ... Ukn_l(-6+m~)-IUkn

w(k 1 , ..• ,kn)

-

~ i~1 f

(TI 8~'i+c5,j

Uk,) (-6 +

m~)-IUkn ·

(5.4.8)

Consider a single term with u, i, and j fixed. By (5.2 .11) and (5.2.18),

(TI 8~"+c5
(-6

-;k n I)-N+3+ e

< c

t.=1

x

n-l

(1

)

C

X

d -;

(1

n

kn I)-N+3e

k, I)-N-2).

(5.4.9)

where we have used (5.4.2) to estimate the Kronecker deltas in the exponents. Applying (5.4.4) and (5.4.5) as above, we obtain

5.4. ESTIMATES ON NUMERICAL FACTORS

425

t=1 (5.4.10) if N' again satisfies (5.4.7) . For the special term in (5.4.8) the estimate is actually better:

< CL~~2

n-l

i -I

t=1

t=1

II LZ,I/2 II (1 + L-';,11 7l

k, -

7l k,

I)-N'

n-l

X

II (1 + L-';,I I7lk, -

7l k ,+, I)-!N-l

(5.4.11)

L=i

Thus (5.4.10) is a bound on the whole numerical factor w(k 1, .. . , k n ). In summary, the spatial integration by parts has altered (5.4.6) by the replacement L~~2 >-+ L~~2 L-';:_1 Now these estimates on w(k 1, ... , k n ) are applied in the context of establishing finiteness of sums like

and

k, ,k2,k.: L' 4 $L' 3$L' 2SoL., where k2 is fixed for the double sum and k3 is fixed for the single sum. In the former case, one applies (5.4.6), as the finer estimate is both useless and unnecessary. In the latter case, the finer estimate is necessary, and we consider this single sum first. Since k3 is fixed, we set i = 3: 2 12 Iw(kl' k 2, k 3, k4 )1 < CL~~2 L;35 / LZ2 / L;,1/2(1 + L-';3117lk3 - 7l k• I)-!N-l (5.4.12) Clearly for Nil > 3,

L L k: L.=2- r

(1

+ Li l l7l k - 7l l l)-N" :::; c(2rLl)3,

(1

+ L-,;II7lk - tll I)-N" :::; c,

T

r

:::;

Ll ,

(5.4.13)

(5.4.14)

CHAPTER 5. WAVELET ANALYSIS OF ~

426 Thus

::;

(5.4.15)

c.

Notice that (5.4.6) would not have been enough to provide convergent geometric series over the length scales. Now consider the double sum. Since k2 is fixed , we set i = 2 for the outer summation; for the inner summation the natural choice is i = 4, but we must extract something from w{k~, k~, k~, k 4 ) that is related to k2 or we will not have a convergent double sum. By (5.4.6), 3

Iw{k~, k~, k; , k 4 ) 1::; c L!~2 II[L;:1/2{1 + L;;:11 -;; k:

-

I)-N'J,

-;; k.

(5.4.16)

t=l

and we combine this with (5.4.14) - setting Nil k~ first:

1W{kI' ' k'2' k'3' k 4 )1

5

::; CL k 4/

= N'

and

e= k4 -

2L- 1 / 2{ 11~ k' 1 + Lk' X k' 2

2

2

to sum out

-

k~

and

~ I)-N' X k.

k;: L k. 5,L kS k~:

Lk2 =::;L k1

(5.4.17) Actually, we may sum out k~ with (5.4.14) as well, except k~ satisfies the stronger condition Lk; 2: L k2 · Therefore, the inner summation is bounded by

<

C

L2k. L-k21 ·

(5.4.18)

I)-~N-IJ

(5.4.19)

On the other hand, (5.4.6) applied to the other factor yields 3

IW{kl,k2 ' k3, k4)1 ::;

cL!~2 II[L~!{l + Lk.11-;;k, ,=1

-;;k,+1

5.4. ESTIMATES ON NUMERICAL FACTORS

427

because j; = 2 is the choice for that factor. Applying (5.4.14) with ~N + 1, we bound the double sum with

e=

k2 and Nil

=

c

k3. k,: L.,$L· 3$L' 2

(1

X

+ Lk31

1

~k3 - ~k,I)-!N-l,

where Lk2 has lost another ~-power because the geometric series estimation (5.4.20) uses up that degree of smallness. Now apply (5.4.13) with obtain the bound

e= k3 and Nil = ~N + 1 to

c where Lk3 has gained an extra ~-power from (5.4.21) Applying (5.4.13) again with

e=

k2' we bound this k3 -summation with

CL2k2 '~ " 2r.2- r3 =cL 2k2 -
(5.4.22)

r3~rk2

and so convergence of the double sum is established. Besides (5.4.6) and (5.4.12), there is the input estimate for the cancellation of ultraviolet divergences :

IW(kl' k2 ' k3, e)W(kl' k2 ' k3, £') - W(kl' k2' k3)W(kl' k2' k3, e, £')1 ::;

c L~36 Lk21 L k,l L"I/2 L; ~ - (1

+ L£.11 -ll x (1 + L k2 x(l

~k,

+ L(11 ~ e - ~ k, 1)-!N-l(l + L k,ll ~k,

I)-N

~i'

-

- ~k2 1)-N-2

~ X k2

L > - ~ X k3 1)-N-2 , e _ L t' > _ L k, > _ L k2 > _ L k3 . (5.4.23)

The point is that

e' .k,.k2.k3 : L. 3$L. 2$L., $L" $Lt - W(kl' k2' k3)W(kl' k2' k3, e, £')1 ::; c,

(5.4.24)

while the same summation of each product diverges. Given (5.4.23) , one only has to iterate (5.4.13) to obtain the successive bounds c

CHAPTER 5. WAVELET ANALYSIS OF ~~

428

+ L l-11 ~Xl - ~Xk. 1)-N(1 + L-l' 11~Xl' - ~X k. 1)_lN-1 2 ~ 1)-N-2 < "L..J L7/2+L-l/2L-l+ -ll ~ x (1 + Lk. Xk. - Xk2 _ C k. l' e X

(1

X

(1

+ Ll-ll ~Xe - ~x k.

1 Xl' - ~Xk. 1)-N(1 + Ll' 1~

1)_lN-l 2

l': L,,$L,

(5.4.25)

where the third bound is obtained by applying the triangle inequality (5.4.26)

to just a (3 + c)-power of the decay factors before summing out kl with (5.4.13). To derive the basic estimate (5.4.23), we write the difference between products of integrals as a double integral: W(kl' k2, k3, e)W(kl' k2, k3, f.') - W(kl' k2' k3)W(kl' k2, k3, e, f.')

J~J

d Y Uk. (~)Uk. (Y)Uk2 (~)Uk2 (Y)Uk3 (~)Uk3 (Y)Ul' (~)

d

X(Ul(Y) - ue(~)).

(5.4.27)

The key observations are that (5.4.28)

1~ -

Y

1 :::: 1~ - ~ k3 1+ 1Y ::::

1~

- ~ k3

1

~

Lk3(1+L/;31 x - xk31) (1

1~

+ L/;3

1

~

Y - X k3 I).

(5.4.29)

By (5.4.1), we also have

lud~)udY)1

::::

C

X

lue(Y) - ul(X')1

::::

C

+

+ L/;,II~ - ~k, 1)-N-2

L/;,I(1

(I+L/;,1IY_~k,I)-N-2,

Le (1

1

/

2

[(1

+ Le

+ Le 1 1X' - X'e 1 1

Y - X'e n-

We split the estimation between (5.4.28-5.4.29) and (5.2.31):

IUk3 (X')Uk3 (Y)llue(Y) - ul(~)1

N

-

(5.4.30)

n2

).

N

-

2 (5.4.31)

5.4. ESTIMATES ON NUMERICAL FACTORS c LI:3°(1 + LI:311~

:s

x(l + LI:311

-

~ k3

429

Il- N - 1 - O

y - ~k3 Il- N- 1- OL~ ~ +-¥[(l + Lell ~ Y - ~e 1)-(N+2)oJ,

~e 1l-(N+2)0

+ (1 + Le 1 1

(5.4.32)

where the exponent <5 > 0 parametrizing this split is chosen to be as small as we wish. Inserting this in (5.4.27), we now estimate the double integral in the spirit of (5.4.3) to obtain

/2 sup {( 1 + L-k 1 1 ~ -o L-1L-1L-l < - cL k3 k2 kl k' X 3 l

.......

-

---0.

~ X

k3 1)-N+2

X,Y

X( 1+LI:31Iy

2

-

~k3 1)-N+2 II(1+Lk , l~ - xk,Il- N- 2

,=1

2

x II(1+L;2 1 y - ~k, 1)-N-2(1+Le; 11 y - ~f' I)-N-2

x L~~+-¥ [(l + Le 1 1~ - ~e 1)-(N+2)0 + (1 + Le 1 1Y - ~e 1l-(N+2)0]} x

J~JY d

d

(1

+ LI:311 ~

-

~k31)-3-c5(1 + LI:31 y - ~k31l-3-c5. 1

(5.4.33)

The residual double integral is O(L~3)' Applying the triangle inequalities (5.4.4) and (5.4.5) in the same way as before, we obtain the desired bound (5.4 .23) if we choose ~

1

N:S 2(N + 2)<5.

(5.4.34)

Obviously, the condition N > 3 needed for the summation requires us to choose the wavelet parameter N to be larger for smaller <5. Finally, there is a rather special numerical factor whose estimation plays a key role in establishing stability. For a quintuple (k 1 , k 2 , k 3, k, e) of modes with k 1 , k 2 , k3 in scale-lexicographic order and Le, Lk < L k1 , we introduce (5.4 .35) and estimates shall depend on Lk and Le relative to the length scales L k,. For the cases

Lh

Le > Lk, > Lk 2 Le > L k3 ,

we estimate no more carefully than (5.4.3) for w(k 1 , ... , kn ). For the remaining case

CHAPTER 5. WAVELET ANALYSIS OF q>~

430

we shall need an estimate in the spirit of (5.4.10), but we need to be more careful with this numerical quantity. There are essentially two successive integrations by parts in the formula (5.4.8), and the point is that derivatives must fall on the factors IUk, (;;) 1 if we integrate (5.4 .35) by parts. A first derivative of such a function is already discontinuous, although still bounded, so to avoid technical problems, we choose to integrate by parts only once. Therefore, we abandon the elliptic scheme (5.4.8) in favor of a slightly more subtle trick. Recall that for every basis function that is a wavelet - and that includes all basis functions except the special functions on the unit scale - there is at least one coordinate variable in which the function has some vanishing moments for fixed but arbitrary values of the other coordinates. This is a stronger property than having some vanishing moments in the multi-variable sense, and it is equivalent to the vanishing of the Fourier transform on a whole coordinate plane up to a certain order. Thus (5.4.36) for some coordinate direction ;;, while (5.4.37)

(5.4.38) Combining these estimates with the integration by parts

JII 3

'IiJ(k1 ' k2' k3; k, f) = -

IUkl (;;)I(oi u k)(;;)(o;-1 ut )(;;)d ;;

t.=l

-~ J(.g lu., I) (a;lu.,. Il (>')

(>')u. C;oJ(

a,-' u,) (>')d >',

(5.4.39)

we sup out every factor in each integrand except the function O;-1 Ut . Integrating IO;-1Utl, we obtain

x (l + L;;11 ;;k -1

~

;;t I)-NO (1 + L;;311;;k3 - ;;e I)-No ~

N

X(l+Lkll Xkl - Xk 1)- 0 x (l + L;;21 ;;k 2 - ;;k3 I)-No 1

(5.4.40)

for a positive integer No which can be made large by adjusting the wavelet construction parameter N.

5.5. EXPANSION RULES AND WAVELET DIAGRAMS

431

References 1. G . Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No. 1 (1982),95-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical 4>~ Model," Commun. Math. Phys. 88 (1983), 263-293.

5.5

Expansion Rules and Wavelet Diagrams

As we have already described in §5.3, there are two types of expansion steps - the type arising from interpolation and the type arising from integration by parts. The latter type of step is aimed at avoiding ultraviolet divergences, so naturally the cases in which it is called for involve certain combinations of small length scales. In this section we describe those cases, and this description will complete the expansion algorithm. Recall the terminology we have already introduced. A completed term has the form

where each factor has been developed by a history of expansion steps finally terminating with an interpolation step that chooses the decoupled term. A completed term is a decoupled term where the complementary set of variables is empty. No further expansion steps are called for in the case of a completed term . More generally, a decoupled term has the form

Such a term calls for an expansion step, but we have previously specified that in this case, the next expansion step is a choice of term arising from the interpolation of

that attempts to decouple the first mode in A \

U , A, from the other modes in that residual

set. By "first mode" we mean first with respect to a fixed linear ordering of A . A remainder term has the form

x

II K , (Te : eE A,),

and the expansion step that such a term calls for is the issue here. The modes occurring in the expression for G(t; 0:) constitute the interior set.

CHAPTER 5. WAVELET ANALYSIS OF ~

432

We require a little more terminology. The interpolated interaction may be written in the form I (tjA\YA,) =>.

L

fk, ...k,(t)w(k 1 , ... ,k4 ):ak, ... ak, :+48>.2

k, , .. . ,k, X

L

ak,akJk, .. ks(t)w(kl,k2,k3)W(kl, .. . ,k5),

(5.5.1)

k" .. ,ksEA\UA, , where fk, ...k,(t) and lk, ... ks(t) are products of powers of all previous interpolation parameters tl, .. . , tn determined by the history regarding which subsets of A \ U A, we

,

have attempted to decouple from their complements relative to A \

U A,. ,

Note that in

contrast to fk, ... k,(t), the case structure of lk, ... k5(t) will be a little more complicated because - as we have already described in §5.3 - the interpolation of the mass counterterm is a little involved. Each of these products of variable expressions with parametric monomials and numerical factors in the exponent will be called a unit. The units Teae are form 0 units, while the Wick-ordered a-product units in the O(>.)-contribution are form 1 units. The units that are second-order in >. are form 2 units . Now every remainder term is the result of an expansion step that has either differentiated some unit down from the exponent or simply a-differentiated an already-differentiated unit very previously brought down from the exponent. In the former case, that previous expansion step could arise from either interpolation or integration by parts. In the latter sub-case, that step could have been called for by a situation where the preceding step a-differentiated an already-differentiated unit already down from the exponent. This means the induction hypothesis for the expansion rules must involve several cases, some of which reach back two or three expansion steps into the history of what has developed thus far. On the other hand, there are a couple of general rules, which we state immediately in the interests of case reduction. Form 0 Rule. Whenever a form 0 unit has been differentiated down from the exponent (which can happen only as a result of integration by parts), the next step is generated by interpolation based on the current interior set (which is also the previous interior set). Form 2 Rule. Whenever a form 2 unit has been differentiated down from the exponent (whether as a result of interpolation or of integration by parts) , the next step is generated by interpolation based on the current interior set.

These two rules imply that a chain of integration by parts steps is automatically terminated if a form 0 or form 2 is differentiated down from the exponent. Therefore, the only cases left to be covered by our expansion rules involve form 1 units only. We first consider the form 1 case where the previous step was induced by interpolation - designated henceforth as Case I. Let G(t j a)

= F(tj a)U(tj a),

(5.5.2)

5.5. EXPANSION RULES AND WAVELET DIAGRAMS

433

where U(t; Q) is the differentiated unit - differentiated with respect to the most recent interpolation parameter tn. Thus (5.5.3) The modes occurring in the expression for F(t; Q) obviously form the interior set on which the interpolation is based, and F(t;Q) does not depend on tn. Let B c A\UA,

,

denote that interior set. The occurrences of modes in U(t; Q) that do not lie in Bare new occurrences, while those that lie in B are old occurrences. Recall that we fixed a linear ordering of A for the expansion. We now stipulate that the linear ordering must be chosen such that a latter mode has equal or smaller length scale, and we shall often refer to it as the scale lexicon or the scale-lexicographic ordering. One may now state the Case I rules concerning how to determine the next step. Consider (5.5.3) and assume without loss of generality that (k 1 , ... , k 4 ) is the linear ordering of the occurrences. Thus (5.5.4) by our scale-lexicographic convention. The case structure is based on which of these occurrences are new. Case la. If either k3 or k4 is old, then the next step is induced by interpolation, where B U {new occurrences} is the interior set. Case lb. If both k3 and k4 are new, then the next step is induced by integration by parts with respect to Qk,. Now consider the case where the previous step was integration by parts but the step before that was interpolation. We classify this scenario as Case II, and in this event,

F(t;Q)

= F'(t;Q)U'(t;Q),

(5.5.5)

where the circumstances of U'(t; Q) are Case Ib, because Case la did not call for integration by parts. Let B' be the interior set for the interpolation that differentiated down U'(t; Q) - i.e., the set of modes that occur in the expression for F'(t; Q) . Then B=B'U{k~,

. .. ,k~},

(5.5.6)

where k~ , .. . ,k~ are the modes occurring in this tn-differentiated form 1 unit. We again assume without loss of generality that (k~, . .. , k~) is the linear ordering of those modes, so by Case Ib, the integration by parts is done in the variable Qk~, and therefore

U'(t;Q)

= -)..8~/k~ ... k~(t)W(k~, . .. ,k~): Qk~Qk;Qk; :,

(5.5.7)

while the Case II unit is differentiated with respect to Qk~' The possibilities for the latter unit are

U(t; Q)

(5.5.8)

CHAPTER 5. WAVELET ANALYSIS OF j

434 U(t; a) U(t; a)

=

but we already know that (5.5.8) and (5 .5.10) call for interpolation, so (5.5.9) is the expression whose expansion directive is not yet given. As far as Case II is concerned, we shall say that an occurrence is new if it occurs in either unit but does not lie in B'; the modes in B' are old occurrences . Thus every new occurrence in the preceding Case Ib scenario is still regarded as new. In particular, ka and k~ are new. Case IIa. If either k3 or k4 is old, then the next step is induced by interpolation with interior set B U {k 1 , •.• , k4 }. The same step is called for if k2 is old and either k3 = k~ or k4 = k~ - also if the latter occurrence in {k 1 , k;} is old and either k3 = k~ or k4 = k~. Case lIb. If both k3 and k4 are new and k~ # k 3, k4, then the next step is to integrate by parts with respect to ak•. Case IIc. If none of the above cases hold - i.e., if both k3 and k4 are new, if either k~ = k3 or k~ = k 4 , and if the latter occurrence in {k 1 , k;} is new - then the next step is to integrate by parts with respect to ak;'

This third possibility is the case where a divergence may develop that has to be cancelled by the mass renormalization. Next we consider Case III, in which the last two steps were induced by integration by parts and the third step in the past was induced by interpolation, but the integration by parts that has just taken place only a-differentiated one of the two units differentiated down by those two preceding steps. We have the same basic form G(t; a) = F'(t; a)U'(t; a)U(t; a)

(5.5.11)

that we had in Case II, only this time U and U' have been altered by this most recent integration by parts step, which could have been called for only in Case lIe with ka = k3, with ka = k 2, or with ka = k 1 • The rules for Case lIa and Case lIb can only differentiate down a new unit . Therefore, the differentiated units are given by

U(t; a)

{)2

->"fk, ... k.(t)w(k 1 ,oo.,k4){)

{) Qka

U'(t;a)

>..

Qk~

{)~n fk; . k~ (t)w(k;, . .. , k~) : ak; ak;

:ak,oo.ak.:, :.

(5.5 .12) (5.5.13)

In this case, the next step is to integrate by parts in the variable ai, where:

(i) If ka = k 2, then e is the latter occurrence in {k 1 , k 2} with respect to the scale lexicon. (ii) If ka = k 1 , then e is the latter occurrence in {k 2, k 3, k 4 } \ {kU. Case IV is the designation we give to the case where the last three steps were generated by integration by parts, the fourth step in the past was due to interpolation,

5.5. EXPANSION RULES AND WAVELET DIAGRAMS

435

and neither of the two most recent steps have differentiated down a new unit, but have only a-differentiated the units that are already down from the exponent. This means that previous to the integration by parts that has just taken place, the situation was clearly Case III, and since this integration by parts did not differentiate down a new unit, we have f = k~ . The differentiated units are now given by

U(t; a)

-)..jk'oo.k4 (t)w(k 1 , • . . , k 4) ()3

x ()

U'(t;a)

ak~

()

"

ak;uak~

: ak, ... ak4 :,

-A {)~n fk~ oo.k~ (t)w(k~, . . . k~)ak~ .

(5.5.14) (5.5.15)

In this case, the next step is interpolation based on the interior set B U {k 1 , ... , k4 }. We still have to consider the same hierarchy of cases where the first unit was differentiated down by an integration by parts instead of by interpolation. We designate the corresponding cases by Case I, Case fi, Case ill, and Case iV, but we need to allow for overlap with the cases already considered. With this constraint in mind, we state the expansion rule for each case. Case Ia. If either k3 or k4 is old, then the next step is induced by interpolation with B U {k 1 , ... , k4 } as the interior set, where B is understood to be the interior set prior to the integration by parts that has just taken place. Case Ih. If both k3 and k4 are new, then we have a situation that may be Case lIb with the given unit as the second unit. We extend the Case lIb rule to this generalization - i.e., we integrate by parts with respect to ak.. This is also the Case Ib rule. Case fia. If either k3 or k4 is old, then the next step is induced by interpolation with interior set B U {k 1 , ... , k 4 } where B is now given by (5.5.6) and B' is understood to be the interior set prior to the initial integration by parts. The same step is called for if k2 is old and either k3 = k~ or k4 = k~ - also if the latter occurrence in {k 1 , k~} is old and either k3 = k~ or k4 = k~. Case fib. If both k3 and k4 are new and k~ i- k 3, k 4, then the second unit is now Case lb. Therefore we integrate by parts with respect to ak • . Case fie. For the remaining Case fi situations - i.e., if both k3 and k4 are new, if either k~ = k3 or k~ = k4' and if the latter occurrence in {k 1 , k~} is new - the next step is to integrate by parts with respect to ak;. This is also the Case IIc rule.

m.

Case This case does not overlap with any case previously considered. We apply the Case III rule. Case IV. This case does not overlap with any previous case. Using the same interior set as in Case fia, we interpolate in this situation.

This completes the description of our expansion algorithm. Wavelet diagrams are useful visual aids in understanding these cases, but here they are a little different from those introduced in the renormalization group context . The mode identifications correspond to selective integration by parts for a term rather than to integrating out all variables for a given length scale and then looking at a term.

CHAPTER 5. WAVELET ANALYSIS OF ~

436

However, the correspondence between markings and quantities is the same. Following the terminology already introduced, we refer to indices k, that appear in a unit as occurrences of modes, since the k, need not be distinct. We use vertical line segments to represent spatial integration variables, and any occurrence that labels a function in a given integrand will be on the corresponding line segment, either as a dash or as an xmark. We also use dotted lines to connect occurrences of the same mode. The position of an occurrence as a vertical coordinate will represent the length scale of the mode, where the smaller-scale modes occur below the larger-scale modes. A real occurrence is any index k, where the variable Ci!k, actually appears in the unit ; otherwise, k, is a ghost occurrence. We use the dash for a real occurrence and the x-mark for a ghost occurrence. Thus, the three basic types of units have the graphical representations exemplified by Fig. 5.5.1. For the actual cases , we enhance the graphs as shown in Fig. 5.5.2, where Fig. 5.5.2.1a gives examples of Case la, Fig. 5.5.2.1b gives examples of Case Ib, and so on. -f.

Form 0

k J k2

f k4

k3

Form 1

k4

~~! --

ks

kJ k2 k3 Fonn 2

~ ~]

--

kJ

k4

k2 k3

ks

Figure 5.5.1:

k I old

k I new

I I k 2 new

k 2 old

k 3 old

k 3 new

k 4 new

k 4 old

Figure 5.5.2.1a:

5.5. EXPANSION RULES AND WAVELET DIAGRAMS

f

I

k I new

k 2 new

k 2 old

k 3 new k 4 new

!

k , I old k 2 new

k4 new

k ' I Old

j

k I new

k ' 2 old

k 3 old - - -

f

k 3 new

Figure 5.5.2.1h:

k I old k 2 new

k 3 new

k' - n~w-

k I old

437

k 2 new

k ' 3 new

-------

k '4 new

k 4 = k '4

4

k 4 new

k I old k I old

k 2 old k 2 new

k 3 new

k 4 new

Figure 5.5.2.I1a: k I old

:: : : f~ -t: ~~e:~ ;_n:w_____

k 4 new

j

k 2 old

k 3 new

- ------ -- -

k 1= k,j

k 4 new

k3 0ew

k 4 new

k 2 new k3

new

k 4 new

Figure 5.5.2.I1b:

CHAPTER 5. WAVELET ANALYSIS OF ~~

438

: :: !

k' l old

k 'J new

k 3 new

----- --k'4 new

l

, k 4 = k4

k'l old

:~

old

.-L.

new

kj

new

k 2 new

- -- -- - - - -

k 4 new

k 4 new

Figure 5.5.2.lIc: k ' l old

k I new

k'2 new

k 2 new

k I old k I new

k 2 new k '3 new

- - ----- - -

k 3= k 3

k ')

new

- -- -- - - -

---- - ----

k 2= k 3

k'4 new

k'4 new

k 4= k 4

--- -- - -

k 3= k4 k 4 new

k I old

:~~;~j

k I old k 1 new

k 2= k 3

k'2 old

-~;-";: --]

k 1= k 3 k 2 new

k 3 new

k 3 new

k '4 new k 4 = k 4

-- -

--- -

-

k 4= k4

k 4 new

Figure 5.5.2.111:

The Case IV graphs are cancelled by the respective graphs in Fig. 5.3 to implement the ultraviolet renormalization, as we shall see later. For future convenience we define a composite as the pair of differentiated form 1 units that occur in a Case lIa, Case IIc, Case III, Case fia, Case fic, or Case ill

5.5. EXPANSION RULES AND WAVELET DIAGRAMS

!-T~:-:-:-]

k

439

3 new

k 4 ne w

k I o ld

F~~----] _____

k I new

k '2 ne w

k 2= k 2

---------

k '3 new

=k 2

K

1

K

2= k 3

_k ~ ~e:

k ) = k 3

k 3= k 4

k 4 new

k 4=

k~

,

Ie.

4 new

k 4 ne w

Figure 5.5.2.1V:

k I old

I

---f:::~:

k lold k I new

--jk neW --Tk )neW 2

- - -X

--

"2 ncw

--

k3new k 4 new

k 4 new ,

Figure 5.5.3:

k I new

k I new

k 2 old

k 2 new

k 3 new

k 4 new - --- ---

k I new

~

-

- --

k I new

k 2 old

k 2 new

k 3 new

k 3 new

k 3 new

k 4 new

k 4= k 4

Figure 5.5.4.IIa:

----- -Figure 5.5.4.lla:

k 4= k4

CHAPTER 5. WAVELET ANALYSIS OF
440

scenario, where the next step differentiates down a new unit - but that unit is not included. Examples of graphs for composites are given in Fig. 5.5.4 with Fig. 5.5.4.I1a giving an example of a Case I1a graph, etc. We recall the hat symbol signifies that integration by parts initiated the development of the case instead of interpolation. The following rule for composites is implicit in the algorithm, but we choose to state it here explicitly.

!

I k

K

I o ld

k

~ new

k 2 new

- - --x :~:c: k

~

1 new

__ _

k

3 new

x- - --

k 1 old

k I new

k;

new

K

2 new

k 3 new

K

3 new

- - --X k 4 new

new

Figure 5.5.4.llc:

Figure 5.5.4.IIc: k I old

k I new

k lold

k I new

k 2 new

k 2 new

----X

~------

k 2 new

k 3 new

k 4 new

Figure 5.5.4.III:

------X

k 2 new

k 3 new

k 4 new

Figure 5.5.4.III:

Composite Rule. Whenever a composite has developed (whether initiated by interpolation or by integration by parts) the next expansion step - by definition differentiates down a new unit, which is not included. The step after that is determined by the rule for Case la, Case Ib, Case la, or Case Ib, as the case may be, applied to the new unit that has been differentiated down but not included . Examples of graphs for such a unit are given by Fig. 5.5.5. Notice that if the graph in either Fig. 5.5.5.Ia or Fig. 5.5.5.Ib is indeed following a composite, then that composite is Case IIa or Case lla. The following rule for form 1 units is also redundant, but we consider it worth stating explicitly. Form 1 Rule. Whenever a form 1 unit has been differentiated down but is not the second unit in a composite, the next expansion step is determined by the rule for Case la, Case Ib, Case la, or Case Ib, as the case may be. Depending on the next expansion step, such a form 1 unit can still prove to be the first unit in a composite.

5.5. EXPANSION RULES AND WAVELET DIAGRAMS

J

kl

new

kl old

k2

new

k2

new

k3

new

k3

new

k4

new

k4 old

Figure 5.5.5.1a:

~-

441

--I

k I old k2 new k3 new k4 new

Figure 5.5.5.Ia:

J

Figure 5.5.5.1b:

~---t

k]

new

k2 old k3 new k

4

new

Figure 5.5.5Th:

We conclude this section by illustrating the consequences of the expansion rules for - shall we say - the case where the two previous steps were integration by parts steps, the third step in the past was induced by interpolation, and the step just made actually differentiated a new unit down from the exponent. We further assume the form G(tjO:)

U(tj 0:) U'(tjO:) U" (tj 0:)

F" (tj o:)U" (tj o:)U' (tj o:)U (tj 0:), a -Afk1 ... k4(t)w(k1, . .. ,k4)-a-: Qk~'

-Afk; ... k~ (t)w(k~, ... , k~)r!-- : a.k~'

(5.5.16) 4

II O:k.:,

(5 .5.17)

£,=1

4

II O:k: :,

(5 .5.18)

t.=1

-A a~/k;' ... k~' (t)w(k~', .. . , k~) : O:k;' O:k~ :,

(5.5.19)

where U' and U" describe the Case II situation that was previous to the last step , and f. is determined by that previous situation. Possible graphs are given by Figs. 5.5.6-5.§,.9 below. In Fig. 5.5.6 the first two units form a composite, while the third unit is Case Ia. In Fig. 5.5.7 the 2nd unit is Case Ib and the 2nd and 3rd units form a composite. Figure 5.5.8 describes a situation where the first two units form a composite and the third unit is Case lb. In Fig. 5.5.9 the 2nd unit is Case Ib and so is the 3rd unit, so there is no composite.

CHAPTER 5. WAVELET ANALYSIS OF 4>~

442

k I old

k I old

k I new

k 2 new

k 3 new

.....

k 3 new

I

----

I

I

I

I

k 4 new k 4 new

------------(1 st and 2nd steps yield Case lIe)

Figure 5.5.6:

k.~

!

old

k I old

k I new

k 2 new

K

2 new

k 3 ne w

k 3 new

-----------k 4 new

k 4 new

(1st and 2nd steps yield Case lIb)

Figure 5.5.7:

k 1 old k I new k I new

I

k 3 new

k 2 new

I I

k 3 new

I

k 3 new

;,. 4 new J... 4 new

( I st and 2nd steps yield Case lIe)

Figure 5.5.8:

5.6. ORGANIZING THE COMPLETED EXPANSION

443

k ; new k 1 old

k

~

new

----------------k

~

k 2'" k

new

~

Ie

3 new

k

4 new

(I st and 2nd steps yields Case lIb)

Figure 5.5.9:

References 1. G. Battle, "Ondelettes: The Spinor QED3 Connection," Ann. Phys. 201, No.1 (1990),117-151. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical ¢>~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981),327-360.

5.6

Organizing the Completed Expansion

For the arbitrary finite set A of modes, every sequence of steps allowed by the expansion rules eventually terminates with a completed term because the interior set grows monotonically for every possible history. It is possible that an integration by parts step does not introduce new modes, but the rules allow at most three successive steps of this type. Even when the growth of the interior set is arrested by choosing the decoupled term in an interpolation, the next step gives birth to a new interior set in the residual set of modes. Thus, the completed expansion of Z(T/ : P. E A) consists of terms of the form A, = A, A, n A" = 0,

U

and our next task is to describe an arbitrary factor K,(T/: P. E A,) in a manner suitable for estimation. Unfortunately, these completed terms are not indexed by the possible partitions {A,} of A, but rather by the histories (of expansion steps), many of which induce the same partition. Our notation for a factor in a term has been somewhat tentative.

CHAPTER 5. WAVELET ANALYSIS OF j

444

We define an n-link as an n-tuple of occurrences, ordered by the fixed scalelexicographic order of modes, together with an indication of which occurrences are real. Thus to a form 1 unit that has been differentiated down - and possibly altered further by integration by parts - we associate a 4-link. We associate a pair of links to a form 2 unit that has been differentiated down - a 3-link consisting of ghost occurrences only and a 2-link having at least one real occurrence. We call such a pair of links a mass link . Obviously, in the case of a form 0 unit we have a I-link due to integration by parts - i.e., a ghost occurrence of a single mode. We associate to a composite the pair of 4-links corresponding to the pair of modified form 1 units that constitute the composite. We call this pair a composite link. Moreover, it will be important to the case structure of the convergence proof to consider combination links . If a sequence of integration by parts steps terminates with a 4-link preceded by a composite link, then we regard the pair as a single link - a combination link. For any history of expansion steps, consider a sequence of integration by parts steps such that the step preceding the first step and the step succeeding the last step are both induced by interpolation - i.e., a complete, uninterrupted sequence of steps induced by integration by parts. These steps generate links because they differentiate down units, possibly differentiating them again, and possibly forming composites. In this sequence of links the expansion rules imply that the last link can be anyone of the four types, but that the others are either 4-links or composite links. We call such a sequence of links a chain, but this does not include the case where the sequence of integration by parts steps is so short that it develops a single composite link or a single combination link and then calls for interpolation. Such isolated links are not regarded as chains, even though several integration by parts steps may be involved. After all, our convention does not regard a mass link due to interpolation as a chain, and in the renormalization scheme it may be paired with Case IV composite link. Now a link is only recording the unit or composite that has been differentiated down and/or developed by the expansion steps, and our notation for those quantities has been tentative up to now. Notice that in the case of, say, a 4-link a, the quantity U(a; t) depends on the past history of expansion steps - not just on a. It consists of the factors:

U(a;t) = -Av(a)w(a)q" (t)F,,(a),

(5.6.1)

where q,,(t) is a product of previous interpolation parameters, v(a) is the number of permutations of distinct modes in a, w(a) is the numerical factor w(k 1 , .. . , k4 ) associated with the occurrences k 1 , •. . , k4 in a, and F" (a) is the modification of : 4

IT ak, : appropriate to the situation.

,=1

In contrast to q,,(t), w(a) is independent of the

previous history, with F,,(a) almost independent because a also designates the ghost occurrences among kl' . .. , k 4 . It is worthwhile to inspect this more closely. If a is not in a chain, then 4

F,,(a) = :

II ak, :, ,=1

(5.6.2)

5.6. ORGANIZING THE COMPLETED EXPANSION

445

while if a is the first link in a chain, then 3

Fu(O:)

= : II O:k,

:.

(5 .6.3)

L=l

If a is the last link in a chain , then (5.6.4)

for some i > 2, while if a is any 4-link in a chain that is neither first nor last, then (5 .6.5)

for some i < 3. Except for the multiplicity factor, we may think in terms of ghost occurrences without regard to the place of a in a history. If a has no ghost occurrences, then a is not in a chain and Fu(O:) is given by (5.6.2). If only k4 is a ghost occurrence, then a is either the first link or the last link in a chain, but notice that the integer factor is the only dependence of Fu(O:) on these cases, because ~o = 4 in the latter case. If only k3 is a ghost occurrence, then a is the last link in a chain and (5.6.6)

If k4 and k i are the only ghost occurrences with i < 3, then Fu(O:) is given by (5 .6.5). There are no other possibilities for a with regard to ghost occurrences. In the case of a composite link (a',a), the general observation is the same: the only dependence on previous expansion steps (that is not already implied by ghost occurrences) lies, for the most part, in the product of previous interpolation parameters. The associated quantity is given by

U(o: ; t)U'(o:; t)

= >.?v(a)v(a')w(a)w(a')qu(t)q", (t)F" (o:)Fu' (0:)

(5.6.7)

with w (a#) and Fu# (0:) virtually determined by a# alone. We mention only a couple of cases here, where (kr, .. . , kt) denotes the quadruple associated with a#. For example, suppose k4' k~ , and k~ are the only ghost occurrences. Then k~ = k4 and the composite is Case IIc with the next step (integration by parts in O:k;) differentiating down a unit from the exponent. The composite link is the first link in either a chain or a combination link, but these remarks are superfluous to the formula 3

II O:k, : . L=l

(5.6.8)

CHAPTER 5. WAVELET ANALYSIS OF ~~

446

As another example, suppose k1 , k 2, and k~ are the only real occurrences. Then k~ = k3 and k~ = k 4 , where the composite is Case III with the next step (integration by parts in ak ) differentiating down a unit from the exponent. Moreover, kl precedes k~ in the 2 scale lexicon - a constraint on (a', a) itself. (5.6.9) for this composite. As yet another example, suppose k~, k~, and k2 are the only real occurrences. Then, again, the composite is Case III with k~ = k 3 , k~ = k 4 , and the next step differentiates down a unit from the exponent, but in this case, the integration by parts is in ak,. Thus kl succeeds k~ in the lexicon, and (a' , O") itself obviously determines the sub-case. (5.6.10) for this composite. There is a very special type of composite link that can reflect two different developments with the same composite expression. Suppose k, = k;, £ = 2,3,4, and that kl and k~ are the only real occurrences. The composite must be Case IV, and the a-expression is: (5.6.11) but suppose kl as well as k~ is old with respect to the previous expansion steps. In this case, the development that differentiates down the primed unit after the unprimed unit is also a possibility, and the a-expression is still given by (5.6.12). Thus (a, a') and (a' , a) have identical composite expressions as well as identical previous histories in this situation. This case is combinatorially relevant to the renormalization cancellation. Case IV and Case iV composites in a given history are important because they contribute to the ultraviolet divergence. In Case IV, the composite link (0"', a) has the property that (k~, k~, k~) is a subsequence of (k 1 , ... , k 4 ) and the only real occurrences are k~ and the occurrence in (k 1 , ... , k 4 ) not in the subsequence. In Case iV, the composite link has the same properties, except that k~ is a ghost occurrence - i.e. , the only real occurrence is the occurrence in (kl' ... ' k 4 ) not in the subsequence. An exact link is either type of composite link. The possible composite expressions are: (5.6.12) for Case IV with k;; not in

(k~, k~ , k~),

and (5.6.12)

for Case iV because qui (t) = 1 in that case. The mass link we need for the renormalization cancellation against the exact link is given by the pair ((k~, k;;) , (k~, k~, k~)),

5.6. ORGANIZING THE COMPLETED EXPANSION

447

and such a mass link we call a counter-link. There is a minor abuse of notation here: (k~, k,) is in scale-lexicographic order except possibly when l = 1. Clearly, the possible form 2 expressions are:

U (a ; t)

= -48), 2v(k~, k,)3!w(k;;, k~, ... , k~)w(k~, k~, k~)ak; ak,qk; k"k~k;k~ (t)

(5.6 .13)

in the case where the form 2 unit has been differentiated down by interpolation, and (5.6 .13) in the case where the form 2 unit has been differentiated down by integration by parts in the variable ak; . Now suppose l = 1 and kl' k~ are both real occurrences with kl preceding k~ in the scale lexicography. Further assume that kl is a new occurrence. In the cancellation of (5 .6.13) against (5.6 .14), the graphs to be compared are given by Fig. 5.6.1. Clearly, the cancellation cannot be implemented unless k I new

k I old

k I new

k I old

k 2= k 2 k 3= k 3 k 4= k 4

newk 4

----1 ----l

k 2 new k 3 new k 4 new

Figure 5.6.1:

U(a ;t)U'(a;t)

+ U(a;t)

DC

w(a-)w(a-') - W(kl,k~, ... ,k~)w(k~ , k~ , k~),

(5.6.14)

so we need the equation (5.6.15) Since kl i- k~, we have V(kl' kD = 2, and so the integer products match. Also, qu (t) because kl is a new occurrence. The desired equation reduces to

=1

(5.6.16) Now by inspection of the interpolation defined for form 2 units described in §5.3, qk,k' ,k' k' k' (t) is the product of all interpolation parameters associated with those previods i~te;ior sets containing k~. But that is exactly what qu' (t) is. Having dealt with this particular case, we need to organize this matching. Consider an arbitrary sequence of chains and isolated links corresponding to a term in the expansion and pick any exact link. Can we replace that exact link with the corresponding counter-link such that a history of allowed expansion steps is still reflected

CHAPTER 5. WAVELET ANALYSIS OF ~

448

by the sequence? If so, do the products of integers and interpolation parameters match for that replacement so that a proportionality like (5.6.15) holds? The first question is easily answered in the affirmative because: (i) The interpolation that generates the step initiating the development of a Case IV composite also generates the step differentiating down the corresponding form 2 unit. (ii) The integration by parts that generates the step initiating the development of a Case IV composite - i.e., the step differentiating down the first form 1 unit in the composite - also generates the step differentiating down the corresponding form 2 unit. There is an extra wrinkle here. If the exact link reflects Case IV with k; = k" t = 2,3,4, then kl' k~ are both real occurrences, and in the case where kl is an old occurrence, ((7, (7') and ((7', (7) are both given by (5.6.12), as we have already mentioned above. The point to be emphasized here is that replacing one with the other does not violate any rules. We state this as a third observation: (iii) Consider the interpolation that generates the step initiating the development of the Case IV composite whose exact link is ((7', (7) with kl an old occurrence as well as k~ . The same interpolation generates the step initiating the development of the Case IV composite whose exact link is ((7, (7') . To answer the second question, we begin by observing that in any case, k~ is old and k~, kg, k~ are new. Therefore, qu' (t) is the product of all interpolation parameters associated with those previous interior sets containing k~. Beyond that observation, we must inspect each case. In the example given by Fig. 5.6.1, we have already established the match. Consider the case where the occurrence k~ is real and t: = 4. Then k4 is real and new, and the corresponding counter-link is ((k~, k4)(k~, kg, k~)) with k~ the only old occurrence. The graphs to be compared are given by Fig. 5.6.2, while k I old

k I old

----l

k 1= k2 k 3 new

k 4 new

- - - - - - - - - - -

----l

k 2= k 3

k I new

k 2 new k 3 new

k 3= k 4

k 4 new

k 4 new

Figure 5.6.2:

U(aj t)U' (aj t)

+ U(aj t)

ex: W((7)W((7') -

w(k~, ... ,k~ , k4)W(k~, k~, k~)

(5.6.17)

is the desired proportionality in this case. We need the identity 48v(k~, k4)3!qk;k4,k2k3k~ (t)

= (4!)2qu(t)qu' (t),

(5.6.18)

5.6. ORGANIZING THE COMPLETED EXPANSION

449

but this holds for the same reason that (5.6.16) held : kr is a new occurrence, so qu(t) = 1, v(k~, kr) = 2, and (5.6.19) Now consider the Case IV situation, where k~ is a ghost occurrence. For an example, pick "[ = 3, in which case k3 is real and new. The corresponding counter-link is ((k~,k3),(k2,k~,k~)), and the graphs to be compared are given by Fig. 5.6.3. The required matching in this case is the identity k I old

Figure 5.6.3:

96(3!)qk'1 ka 'k'2 k'3 k'4 (t)

= (4!?qu(t)qu' (t),

(5.6.20)

which is even more trivial than (5.6.19) because there are no products of previous interpolation parameters - i.e., (5.6.21) in this pure integration by parts scenario. The initial integration by parts in the variable o.k'1 means that k~ has never been in an interior set. k I old

k I old

k 2 new

-------

k 3 new k'4 new

-------

k I old

k I ?Id

k 3= k 3

---] ., . ----

k3 new

k 4= k 4

----

k new 4

k 2= k 2

Figure 5.6.4:

"

CHAPTER 5. WAVELET ANALYSIS OF ~

450

The scenario relevant to the observation (iii) above is the most interesting one. Suppose "[ = 1 but kl is an old occurrence strictly preceding k~, and k~ is a real occurrence. The graphs to be compared are given by Fig. 5.6.4. If we inspect the interpolation defined in §5.3 for this case, we find that qklk~,k;k;k~ (t) is twice the interpolation parameter just introduced times the product of all interpolation parameters associated with those previous interior sets containing either k~ or kl' where the interior sets containing both modes have the squares of their interpolation parameters appearing in the product. The interior set for the interpolation parameter just introduced contains both modes, but this parameter has just been designated as special - it is doubled instead of squared, by differentiation. Now clearly, (5.6.22)

from which we obviously get

48v(kl' k~)3!qklk;,k~k;k~ (t) where V(kl' kD

= 2 because kl i- k~.

2U(0:; t)U' (0:; t)

+ U(o:; t)

= 2(4!)2 q".(t)q"., (t),

(5.6.23)

This match yields the proportionality

ex: w(O")w(O"') - W(kl' k~, . .. ,k~)w(k;, k~, k~),

(5.6.24)

so the renormalization cancellation requires two copies of the composite expression U(o:; t)U' (0:; t). But that is exactly what the two exact links (0"',0") and (0",0"') provide. To automate the renormalization cancellation, we now collect the equivalence classes of completed terms into single terms, where the equivalence relation is that two terms are equivalent if one can be obtained from the other by some set of link replacements allowed by the observations (i-iii) above. By way of illustration, suppose a completed term is given by the sequence [0"1,0"2, (0"3,0"4, (0"~,0"5)), 0"6, (K;7,K7),0"8,0"9], where (0"3, 0"4, (O"~, 0"5)) is a chain terminated by an exact link (O"~, 0"5) and (K;7, K7) is a counter-link whose two real occurrences are both old. Denote the corresponding expression by

There are five other terms equivalent to this one in the sense we have just defined, because the replacements (O"~, 0"5) (K;7, K7)

f---t

(K;5, K5),

f---t (O"~, 0"7)

or (O"~, 0"7 )

are allowed. The idea is to collect these six terms into the product

U1 ... U4(U~U5

+ U5)U6(U7 + 2U;U7)U8U9

and denote the corresponding sequence by

[0"1,0"2, (0"3,0"4, {(O"~, 0"5), (K;5, K5)}), 0"6, {(O"~, 0"7), (0"7, O"~), (K;7, K7)}, 0"8, 0"9] . Each grouping represents a cancellation of graphs. We call each grouping a renormalized link.

5.7. THE PHASE CELL POLYMER

451

References 1. G. Battle, "Ondelettes: The Spinor QED3 Connection" Ann. Phys . 201, No. 1 (1990),117- 151. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical 4>~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981) , 327-360.

5.7

The Phase Cell Polymer

As involved as our inductively defined phase cell cluster expansion appears to be - and a certain degree of complexity is inescapable for the 4>~ Euclidean field theory - we still realize it as a polymer expansion to which the abstract theory applies. It is the polymer that is complicated and the input estimates that are hard to establish. The terms of the completed expansion are labeled by histories for which we now have a concrete description, and we have just combined those terms in such a way that renormalization cancellations are all contained in the consolidated terms. The quantity expanded is the generating functional Z(re : f. E A), and the sequence of interior sets in a given history consist of segments where the sets are increasing (in the sense of inclusion) in each segment and the maximum interior sets in these monotone segments actually partition A. Any sequence of links and chains giving rise to such a segment of the sequence shall be called a Representation 1 graph. Many can have the same monotone sequence of interior sets, while on the other hand, there is one kind of monotone segment with no Rl (Representation 1) graph - namely, a segment consisting of only one interior set. Such an interior set can only be a singleton, so a history is reflected by a sequence of isolated modes and Rl graphs, where the growth of each Rl graph is terminated by the selection of the decoupled term in the interpolation, whose interior set is therefore the maximum interior set in the monotone segment. Now, actually, the partition function for the polymer formalism is given by the normalization Zh : f. E A) = Z(rl: f. E A) (5 .7.1)

n Z{l}(re)

eEA

and the denominator is the leading term in the expansion of Z(re: f. E A) . It is the result of choosing the decoupled term in every interpolation; no situation calling for integration by parts ever arises and the interpolations never differentiate down anything, so there are no Rl graphs in the leading term. More to the point, those factors in an arbitrary term that correspond to isolated modes are cancelled out in the normalization, so the terms in the expansion of Z(re: f. E A) are labeled by sequences of Rl graphs only. Let (g1, .. . , gn) be a sequence of Rl graphs consistent with the development of a term and define the support of an Rl graph as the set of all modes occurring in the links and chains of links - as both real and ghost occurrences. This is actually the maximum interior set associated with the development of the Rl graph.

CHAPTER 5. WAVELET ANALYSIS OF j

452

Then the sets supp gi are mutually disjoint, but the union is not necessarily A, as the isolated modes are not included. However, there is another restriction: if k i is the leading mode in the development of gi - i.e., the singleton for the initial interior set then (k 1 , . . . , kn) must be in lexicographic order. Let En be the set of all such n-tuples of Rl graphs . Schematically,

~( ) ~ Z Te: £ E A = 1 + L

'" L

lIn Kgi(Te : £ E supp g')

n=1 (g', ... ,gn)Ecn i=1

n

eE supp gi

Z

( ) ,

(5.7.2)

{e} Te

and our task here is to write the factors Kgi more explicitly. To this end, we choose the notation 9 for the sequence of links obtained from an Rl graph 9 by ignoring the chain structure. Enhancing previous notation, we denote by UJ(o.; t) the differentiated unit or units associated with the jth link in g. On the basis of our organization of the expansion, we observe that UJ (0.; t) can be a composite, so we adopt the notation (5 .7.3) in that case, where UJo (0.; t) is the development of the form 1 unit initially differentiated down from the exponent and UJI (0.; t) is the development of the subsequent form 1 unit. It is important to remember that UJ(o., t) can also be the expression associated with a combination link, in which case we write

(5.7.4) where UJ+ (0.; t) is the form 1 unit following the development of the composite. If UJ(o.; t) is a renormalized expression, we let UJx (0.; t) be the counter-term, in which case we write

UJ(o.; t)

= UJo(o.; t)UJl (0.; t) + UJx (0.; t)

(5.7.5)

unless gj has two real occurrences that occurred previously in 9 (which also means cannot belong to a chain in g). Under that condition ,

gj

(5.7.6) as we have already discussed in the previous section. As far as the links themselves are concerned, it seems appropriate in the case of a composite link to use the notation

gj = (aJo,aJl)'

(5.7.7)

and in the case of a mass link we choose the notation

(5.7.8) For a combination link we write

(5 .7.9)

5.7. THE PHASE CELL POLYMER

453

In the case of a renormalized link,

gj

= {(aJo,aJl)' (II:J, ~J)}

(5 .7.10)

unless the above condition holds, and · (5.7.11) under that exceptional condition. The other types of links are the I-links, with the single occurrence automatically a ghost occurrence, and the 4-link, which has at least two real occurrences. If gj is a I-link with its single mode occurrence denoted by f gj , then the corresponding differentiated unit is just

UJ(cv.; t)

= Te,

f = f gj ,

(5 .7.12)

because a I-link can appear only at the end of a chain - i.e., a form 0 unit can be differentiated down from the exponent only by integration by parts, and the subsequent expansion step is induced by interpolation. If gj is a 4-link with ~th occurrence denoted by k" then we write (5 .7.13) This notation is appropriate because the q-factor and F-factor depend on the previous development of g, in contrast to the numerical factor, which depends only on the link. Now in the previous section we saw how every UJ(cv. ; t) can be written as this type of product - in particular, those expressions arising from renormalized links - and we now adopt some master notation accordingly. Let w(gj) be defined by (5 .7.14) if gj is a 4-link, (5 .7.15) if gj is a I-link,

w(gj)

= w(aJO)w(aJl)

(5.7.16)

if gj is a composite link given by (5.7.7) , (5 .7.17) if gj is a mass link given by (5 .7.8) with k4' ks the occurrences in II:J and kl' k2 ,k3 the occurrences in ~J, (5.7.18) if gj is a combination link given by (5.7.9), and

w(gj)

= w(a'JO)w(aJl) -

w(k 1 , ... , ks)w(kl' k2' k3)

(5.7.19)

CHAPTER 5. WAVELET ANALYSIS OF j

454

if 9j is given by either (5.7.10) or (5.7.11) with k1,k2,k3,k4 the occurrences in oJo and k1,k2,k3,k5 the occurrences in OJ1 ' Let qJ(t) denote the product of interpolation parameters in UJ(a; t) for all cases. In the case of a composite, for example, (5.7.20) where qJo(t) (resp. qJ1 (t)) is the product of interpolation parameters appearing in the differentiated form 1 unit UJo(a; t) (resp. UJ1 (a; t)). In the case of a differentiated form 2 unit, we introduced the notation (5 .7.21)

"'1

in the previous section, where, again, k4' ks and k1' k2, k3 are the occurrences in and ;;.,1, respectively. In the case of a renormalization cancellation, qJ(t) is still given by this formula. Let the notation Fl(a) be extended in the same way. In the composite case, (5 .7.22) 4

where Flo(a) (resp. Fl1 (a)) is the modification of:

IT ak:

,=1

4

: (resp. :

IT ak.

,=1

:) that

appears in the differentiated form 1 unit UJo(a; t) (resp. UJ1 (a; t)), where k~, . .. , k~ (resp . k1, .. · , k4) are the occurrences of I7Jo (resp. I7J1)' In the case of a differentiated form 2 unit, then - with k 1 , . .. , ks defined as in (5.7.17) -

Fl (a) = ak4 aks if the mass link

9j

(5.7.23)

is not a link of any chain in g, and (5.7.24)

for ~ = 4 or ~ = 5 if 9j is the last link of a chain (which is the only other possibility for a mass link). Fl(a) is given by these formulas in the case of a renormalization cancellation as well. Finally, we extend the notation for the combinatorial factor in UJ(a; t). Let (5.7.25) if 9j is a 4-link, (5.7.26) if 9j is a I-link, V(9j)

= v(k 1, . .. ,k4)V(k~, ... ,k~)

(5.7.27)

if 9j is a composite link, and V(9j) = v(k 1, ... , k4)V(k~, ... , k~)v(k~/, ... ,k~)

if

9j

(5.7.28)

is a combination link. (5.7.29)

~,

5.7. THE PHASE CELL POLYMER

455

is the combinatorial factor - again, with k 1 , .. . , k5 defined as in (5 .7.17) - for both mass links and renormalized links. We can write the Kg; factors more explicitly with this master notation. Let [. denote the set of all I-links in a representation one graph g. Consider the cardinalitie~ ng n~

= card{j: 9 is a 4-link},

= card{j: gj

(5 .7.30.0)

is a mass link},

(5.7.30.1)

= card {j: gj is a composite link}, n~1 = card{j: gj is a renormalized link},

(5 .7.30.3)

ngIV = card{'J: gj

(5.7.30.4)

n~

A

IS •

(5.7.30.2)

a comb"matlOn I'mk} .

With Ig(t;
and this will be our starting point in the next section. We define a Representation 2 graph on a pointed set (B, ko) of modes as the set of all links and chains of links comprising a representation one graph with support B and initial mode ko. There can be many Rl graphs associated with the same R2 (Representation 2) graph. Clearly, we may write (5.7.2) in the form

Z(Te: £ E A)

=

II . n

00

1+ "

" o 0. n=1 (Ul ,... ,un )EV

X g:

E,u;

n

,=1

1

(

IT Z{£} (Te) eEsupp U;

Kg(Te: £ E SUpp U

i )),

(5 .7.32)

where Ug denotes the R2 graph that g belongs to. If 'Ye denote the distinguished mode associated with an R2 graph U by ku, we define Vn as the set of all n-tuples (U 1 , .. . , un) of R2 graphs such that supp U i n supp U i ' = ¢, and (kUl, . .. , kun) is in scale-lexicographic order.

i:l i' ,

(5 .7.33)

CHAPTER 5. WAVELET ANALYSIS OF 4i~

456

We now define our phase cell polymers to be arbitrary R2 graphs on pointe~ sets of modes . The set Vn of disconnected n-polymers is clearly the extension of Vn to n-tuples (U 1 , ... , un) of R2 graphs still satisfying the disjointness condition but no longer satisfying the order condition on (kUl, ... , kun). Taking this into account, we get

(5.7.34)

so we have identified our phase cell polymer activity as well :

z(U) =

II lesupp

1

u

()

Zit} Tl

L-u Kgb : R. E supp U) .

(5.7.35)

.U

g.

g-

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field

Theories," Ann. Phys. 142, No.1 (1982) , 95-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical 4>~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981),327- 360. 4. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987.

5.8

Estimating the Activity I: Stability and Quartic Positivity

The first stage in estimating the activity z(U) of an R2 graph U is to estimate Kg(Tl: R. E supp g) for an arbitrary R1 graph g. The focus of this preliminary task is to obtain a useful lower bound on the interpolated interaction, and the foundation has already been laid in previous sections, where we obtained inequalities involving the quartic part of the interaction and without considering interpolation. The interpolation of the Wick-ordered quartic may be written in the form

')'~(t) P partitions supp 9

L BeP

I'(4i B ),

(5 .8.1)

5.S. STABILITY AND QUARTIC POSITIVITY

457

where 'Y~(t) 2: 0 and 'Y~(t)

= 1.

(5.8.2)

P partitions supp 9

This coefficient is zero for those partitions that are never involved in the history. For a partition P that has developed, 'Y~ (t) is some product where a given interpolation parameter tm can appear only in the form tm or 1 - t m . It is the convexity of the interpolation that preserves the desired inequalities. Indeed,

J

J

~,\ iP~ + ,\ GiP~ - 6(iP~)oiP~ + 3(iP~)~) 2:

L (1 + In Lk"l )-2 LkQi - c'\ card(B),

c'\

(5 .8.3)

kEB where we have applied (5.2 .22) to the first integral and (5.1.24) to the second integral. Inserting this inequality in the sum over B E P, we obtain a lower bound (on that sum) independent of P : (5.8.4)

BEP

kEsupp 9

It follows from (5 .8.1) and (5.8.2) that I~ (t;

iPsupp g)

L

2: c'\

+ In L;;l )-2 LkQk

(1

- c'\ card(supp g).

(5.8.5)

kEsupp 9

Our main goal here is to extend this lower bound to

Ig(t; iPsupp

I~(t; iPsupp g)

g) =

+ I~/(t; iPsupp g),

(5 .8.6)

and the first step is to write the interpolation of the counter-term in its own convex form:

L

48,\2

I~' (t; iPsupp g)

w(k 1, k2, k3)

k, ,k2 ,k3 Esupp

J

d:Z; uk, (:Z;)Uk2 (:Z;)Uk3 (:z;)

9

x{ (L'Y~(t) L
BEP

- (L'Y~(t) L
BEP

+ L'Y~(t) P

L iP~k2k3(:Z;)2}.

(5.8.7)

BEP

For convenience, define

Wk, k2k3 (:z;)

= 48w(k 1, k2, k3)Uk, (:Z;)Uk2 (:Z;)Uk3 (:z;)

(5 .8.8)

CHAPTER 5. WAVELET ANALYSIS OF 4>~

458 and consider the decomposition

(5.8.9) into positive and negative parts. Now by convexity of the square, we certainly have

s

S

L 'Y~(t) (L O~,k2k34>~k2k3 (X")) 2 P BEP L 'Y~(t) L O~'k2k34>~k2k3(X")2 P BEP L'Y~(t) L 4>~k2k3(X")2 (5.8 .10) P

BEP

because BnB'

= 0.

(5.8.11)

This means that the total quantity in braces in (5 .8.7) is positive, and so from the standpoint of obtaining a lower bound we may throwaway the Wt,k2 k3-contribution. As far as the w k'k2 k3-contribution is concerned, we may obviously discard the second combination in the braces because it is manifestly negative. Thus

I~(t;4>suppg)

2

L

_,).2

jdX"W k'k 2k 3(X")

k"k2,k3Esupp 9

x{ (L'Y~(t) L o~'k2k34>B(X"))2 P

+ L'~ (t) P

2

_,).2

BEP

L 4>~ k2 k3(X")2 }

BEP

L 'Y~(t) L P

L

BEP k, ,k2,k3Esupp

x{O~,k2k34>B(X")2

j d X" Wk,k2 k3(X") 9

+ 4>~k2k3(X")2},

(5 .8.12)

where we have now applied convexity and (5.8.11) to the first combination in the braces. We must use the Wick-ordered quartic part I~ (t; 4>supp g) to dominate this negative quadratic quantity. Combining (5.8.12) with (5 .8.1) and (5.8.2) eliminates the interpolation from the problem once again. We need only show that

1'(4)B)

L j d X" Wklk,k3(X"){O~'k2k34>B(X")2 +
kl,k2l k S

2

kEB

5.B. STABILITY AND QUARTIC POSITIVITY

459

for an arbitrary set B of modes. Accordingly, we write 4 3"2A / <J>B+A

1'(<J» B

2:

/(14

22 22) 3"<J>B-6(<J>B)O<J>B+3(<J>B)O

~A/ <J>~+CA2)1+lnL;;1)-2LkQ:k -

cAcard(B),

(5.8.14)

kEB where we are estimating as in (5.8.3) above, except that we have set aside part of AJ<J>~. With the negative part of WkJk2k3(~) dominated by its absolute value, the desired bound (5.8.13) will follow from:

f

~A <J>B(~)4d ~ _A 2 .

L / d ~ IWklk,k3(~)I<J>B(~?

kJ,k2,k3EB

2: - CA card(B),

(5.8.15)

L

cAL(1+lnL;;1)- 2L kQ:k- A2 kEB ~,h , ~ 2: -C(c)A card(B),

!d~lwkJk,k3(~)I<J>~k2k3(~)2 (5.8.16)

for c > O. We prove the inequality (5.8.15) first . The estimation of W(k1' k2, k3) is our initial concern. We have

Iw(k 1,k2,k3)1 :scLtL~2~L~J~(1+L;;J11 ~kJ - ~k31)-No 1 x (I+L;;J 1~kJ - ~k21)-N°(I+L;;211 ~k2 - ~k31)-No by virtue of (5.4.3). It follows from (5.4.14) that

(5.8.17)

(5.8.18) On the other hand, if we complete the square, the integrand in (5.8 .15) is bounded below as follows:

(5.8.19) with

L kJ ,k2,k3EB

=

288

IWkJk2k3 (~)I

:s 6 kJ,k2,k3EB: LkJ ?:Lk2?:L.3

CHAPTER 5. WAVELET ANALYSIS OF ~

460

By the Schwarz inequality together with (5 .8.18) , we also have

~ C.

, " "B,

E,

h,

,L., Iw(k" k" k, )I') ,

(5.8.21)

At the same time, (5.4.1) implies (5 .8.22) from which we may infer that

(5.8 .23)

Finally, combining this with (5.8.19) and integrating, we obtain (5 .8.15) from the Federbush stability bound (5 .1.25). We now turn to the verification of (5.8 .16). Ordering the scales of kl ' k2' k3 with the estimate (5 .8.20), we have

~k,k3CX')

= ~CX' ) =

L

Uk(a;)Ok ,

(5.8.24)

k EB Lk < L . ,

and therefore

L ~

288

J a; IWk,k,k3 (a;)I~k2k. (a;)2 d

k, ,k2,k3

L kl ,k2 ,k.

Iw(kl' k2' k3) I

J a; d

IUk, (a;)Uk2 (a;)Uk3 (a;) I~' (a;) 2

L kl ;;: Lk2 ;;: L • •

288

(5 .8.25)

5.8. STABILITY AND QUARTIC POSITIVITY

461

(5.8.26) we see that c>.

L (1 + In L;1)-2 Lkak -

>.2

kEB

L k"k2,k3

Jd;

IWk1k2 k3(;)I~1kok3 (;)2

L

2: c>'L(1+lnL;1)-2Lk(at-288>.c-1L;1(1+lnL;1)2 kEB

eEB: k k2,k3 L. , ?L'"2 ?L' 3 L.,L,
(5 .8.27) so if we complete the square in each mode k (in contrast to completing the square pointwise in an J d ;-integrand, as we did before), the problem is reduced to proving:

Iw('"'" ',IUl('" k" :S c(c).

k,; k,

l)1) , (5.8 .28)

Now by (5.8.17) we have 1

1

sup Iw(k 1, k 2, k3 )1 :S CL%3 min{ L~2, L~32}, k"k2: L' 1>L.

(5.8 .29)

L' 1 ?L' 2 ?L' 3

so the desired inequality will follow from :

Divide the sum as follows:

B~

Br

Bk == {(k 1,k2,k3 ,P) : Lk1 2: Lk2 2: Lk3 and Lk1 > Le > L k},

(5.8.31.0)

== {(k 1,k2,k3 ,P): Lk1 2: Lk2 2: Lk3 and Lk1 > Lk 2: Le > Lk3}' == {(k 1,k2 ,k3 , P): Lk1 2: Lk2 2: Lk3 2: Le and Lk1 > Lk 2: Le}.

(5.8.31.1)

For the set Bk we do not have to be particularly subtle. We estimate

(5.8.31.2)

CHAPTER 5. WAVELET ANALYSIS OF j

462

~

Iw(k 1,k2,k3;k,£)1

Jd~IIIUd~)IIUk(~)llueCX")1 3

(5.8.32)

,=1 and we apply (5.8.22) to the indices k1 and k2 in the following way:

IUk, (~)I L k,: Lk, >Lk

~

IUk2(~)1 L k2 : Lk;?Lk3

~

Thus

L~~-6LLt.Hluk,(~)1 ~ cL~~-o,

(5.8.33)

k,

L~}-6LLt6Iuk2(~)1 ~ CL;3~-6.

(5.8.34)

k2

3 Lf3Iw(k1' k2, k3; k, £)1 L (k"k2 ,k3,e)EBk

J

~ L~~-6

(5 .8.35) L L~~6 d ~ IUk3(~)llud~)llue(~)I· k3,e L,>Lk We have just taken the crucial step, as the restriction Lk, > Lk was essential. If we now apply (5.8.22) to the indices k3 and £, this inequality further reduces to: 3

Lf3Iw(k1' k2, k3; k, £)1

~ cL~~-6 l:

~

CL;1-20

L L,>L k

J ~ IUk(~)llue(~)1 d

Jd~ IUk(~)ILLtH lul(~)1 l

~

CL;1 - 20 /

dxlud~)1

CL~-20

(5 .8.36)

k

If we choose 8 < ~, we have the bound (5.8.30) on the Bk-sum. For the set B~ we apply the same crude estimate (5.8.32), and the crucial step is still to sum over k 1 :

(5 .8.37) where we have also summed over both k2 and £ by using (5.8.34). Since

J~ d

I

Uk 3 (~)lluk(~)1

L i': L t l=2- r

(1

+ L;11

~ cLtL~~ (1 + L;11 ~ k3

~f' - ~k I)-No ~ C (2~kr ) 3,

-

~k 2- r

I)-No,

(5.8.38)

~ Lk ,

(5.8.39)

5.S. STABILITY AND QUARTIC POSITIVITY

463

= k3)

it follows from (5.8.37) that (with f'

(5.8.40)

!,

Again, if we choose 8 < the bound (5.8.30) holds for the B~-sum with room to spare. For the set B~ we apply the more delicate estimate (5.4.40), but the crucial step is still to sum over k1 with respect to k and over k2 with respect to k3. We obtain:

L

L%3 min{L;!, L;3! }lw(k 1, k2, k3; k, f)1

(kl ,k2,k3,l)EB~

~ ~

l,k3: L,
L~L1-0min{L-! L-!}(L- 1 +L- 1) l k3 k 'k3 k k3

L.3~L,

x( l + L';-311~k3 - ~l I)- N°(l + L,;-11 ~l - ~k I)-No (1

+ min{L,;-1, L;;.1} 1 ~k - ~k3

I)-No

(5 .8.41)

where we have applied

L k~:

(1

+ L';-,1 1~k,

- ~o I)-No::; C

(5 .8.42)

Lk, =2- r

to ~ = 1 with ~O=~k and to ~ = 2 with ~0=~k3' Now apply (5.8.39) with e' = f in this context, but still with reference mode k as long as Lk < L k3 . If Lk3 ::; L k , then we replace k with k3 as the reference mode. Thus , our upper bound has the two contributions

S 0= CL 1k- O k3: S1

=

CL,;-2-0 k3:

L

LtO(1+L';-311~k3 - ~k I)-No

L

Lt°(1

L. 3>L.

L. 3<:;L.

+ L,;-11 ~k3 - ~k

L

T~.

r:

2- r <:;L.

r:

2- r <:;L. 3

I)-No

L

2-~ .

(5 .8.43.0)

(5 .8.43.1)

Clearly, the geometric series in So is bounded by L! -0, while the geometric series in S1 is bounded by Lto Finally, we apply (5.8.42) (resp. 5.8.39) to the k3 -summation in So (resp. Sd. We have (5.8.44.0) (5.8.44.1)

where in (5.8.44.0) we cancelled (5.8.45)

with the geometric series in (5.8.43.0). As far as the exponents in (5.8.44.1) are concerned, note that we bounded the geometric series in (5.8.43.1) with CL%3' which is

CHAPTER 5. WAVELET ANALYSIS OF 4>~

464

more than we need. We have now established {5.8.30} for the B~ -sum, and therefore {5.8.16} is proven. This completes the proof of the inequality for the total interpolated interaction namely, the lower bound

Ig (t;

4>supp)

L

2 c>.

(1

+ In LJ:1? Lkaj. -

c>.(supp g) .

(5.8.46)

kEsupp 9

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139.

2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical ¢>j Model," Commun. Math. Phys. 88 (1983), 263-293.

5.9

Estimating the Activity II: Internal Combinatorics

For an arbitrary representation one graph g, we have obtained a lower bound on the interpolated interaction Ig(t; 4>supp g) that depends only on supp g. The bound we now have on Kg(Tl: f. E supp g) is given by

x

(II 11 m

dtm)

0

(Teal - c>'(1

II qJ(t) / II IFJ(a)1 exp L J

\

lEs upp 9

J

+ In LeI )- 2 Le(1) ) 0 e >' C

card(supp g),

(5.9.1)

where we have applied (5.8.46) to (5.7.31). The only dependence on the interpolation parameters that remains in this bound lies in the qg-factors. The preliminary step in controlling the activity (5.7.35) of the whole phase cell polymer (representation two graph) is to bound the other factors in (5.9.1) by quantities depending only on Ug . The goal in this section is to use the J dt-integrals of the qg-products to control the combinatorics of the sum over Rl-graphs in (5.7.35) . The combinatorial factors v(gj) are irrelevant to the combinatorial problems of the expansion, and we have worried about them up to this point simply because they are there. The value for a composite link can be as large as (4!)2, while the value for a mass link can be as large as 96(3!), but since there are only finitely many possible values, we can bound every combinatorial factor associated with some link by the same universal constant: v(gj) ::; c. (5 .9.2)

5.9. ESTIMATING THE ACTIVITY II: INTERNAL COMBINATORICS Note that the set of links in

465

9 of a given type depends only on Ug , so we may also write n# 9

supp 9

#

nu. '

(5.9.3)

supp Ug .

(5.9.4)

Moreover, the numerical factor w(gj) associated with a given link on how gj appears in g, but only on gj. Thus

II Iw(gj)1 = II

g. does not depend J

Iw(J)I,

(5.9.5)

JEU.

where t1 is the set of all links in U and of the chains in U . Finally, in estimating the Fg-product, we find it convenient to define the number NJ(f.) as follows: (i) If J is a 1-link, then NJ(f.) == O. (ii) If J is a 4-link, then NJ(f.) is the number of times that the mode f. occurs in that 4-tuple. (iii) If J is a composite link (a',a), then (5.9.6) where k~ is the last occurrence in a' and Nu#(f.) is defined as in (ii). (iv) If J is a mass link (11:, ii:), then NJ(f.) is the number of times that the mode f. occurs in the pair II: of occurrences. (v) If J is a renormalized link whose mass link is (11:, ii:), then NJ(f.) is defined as in (iv). (vi) If J is a combination link ((a" , a'),a), then (5.9.7) where

k~, k~

are the last two occurrences in a"

Actually, it would be more natural to define NJ(f.) simply as the number of real occurrences of f. in the link J, but this choice would also make it slightly dependent on how .J arises . Our definition of NJ(f.) over-counts the real occurrences in a manner dependent on J alone. The occurrence in a 1-link is always a ghost occurrence. For a mass link (11:, ii:), the occurrences in ii: are always ghost occurrences. A composite link (a', a) cannot even develop without integration by parts in the variable ak;'. This means k~ is a ghost occurrence in a' and that the same mode occurs at least once in a as a ghost occurrence, so one may safely subtract 2ok ;,e in (5 .9.6) for all cases without throwing away any real occurrences. Similarly, a combination link ((a", a'), a) can develop only through integration by parts in ak~ and ak~ . There may also be integration by parts in ae, where f. is the latter occurrence in {k~, k~ , .. . , kD \ {k~, kn, but (5.9 .7) does not subtract possible ghost occurrences - only those that necessarily occur. What does this have to do with estimating Fg-factors? If we introduce the obvious estimate

N':S:N:S:4 ,

(5 .9.8)

CHAPTER 5. WAVELET ANALYSIS OF iI>~

466 we may write

(5.9.9)

rlj.

The step from (5 .9.8) to (5.9.9) may treat a ghost occurrence as a for every link real occurrence, but the resulting bound depends on the link Qj itself and not on how it arises in g. Pulling all of these preliminary observations together, we may now bound the activity of a representation two graph U as follows :

Iz(U)1 ~ oX

+2

n U

t

<=1

n «) +3n IV U

U eC>'

card(supp U)

1 II (clw(J)I)-=----:--:II Z{l} (rl)

JEU

x /

II(1 + lall)Jfa NJ(l) exp L

\

X

+ In L/1)-2 Lla:))

lEsupp U

l

g:

(real - coX(l

lEsupp U

f=u (I] 11 dt I} qJ(t). m)

0

(5 .9.10)

Since 00 is a product expectation, the a-integration now splits into a product of one-variable integrals. For convenience, we introduce (5.9.11) so that we may write

Iz(U)1

(5 .9.12) It is the remaining sum of interpolation weights over R1 graphs that we now need to control, and this is the internal combinatorial problem for the phase cell polymer (R2 graph). To this end, we define an Mth-degree generalized ordered connectivity graph on a set A of modes to be a mapping A : {2, . .. , M} ~ P(A) with the following properties:

(i) A(2) contains the first mode in A with respect to the scale lexicon, (ii) For 2 < m ~ M, m-1

A(m) n

U A(i) i= 0,

i=2

(5.9.13)

5.9. ESTIMATING THE ACTNITY II: INTERNAL COMBINATORICS

467

m-1

A(m)\

U A(i) # 0,

(5.9.14)

;=2 M

(iii) A

= U A(i). ;=2

Obviously, an Rl graph induces a generalized ordered connectivity graph on its own support with the sets in the sequence identified as the supports of the links and chains of links in the sequence defining the R1 graph, deleting the I-links. We return to this point below. For an Mth-degree generalized o.c.g. A on A, we define the mapping TJA : {2, ... ,M}-+{I, .. . ,M-l}by

TJA

(

M)

={

1, if A(m) contains the first mode in A, min{i: A(m) n A(i) # 4>} otherwise.

(5 .9.15)

By the property (i) we have TJA(2) = 1, while (5.9.13) insures TJA(m) < m for A(m) not containing the first mode in A, so TJA is an ordered tree graph. Let (5 .9.16) with f'1(t) defined by (3.11.8), and define on Mth-degree generalized unordered connectivity graph on A as the range TA of some Mth-degree generalized ordered connectivity graph A on A. We claim (5.9.17)

for an arbitrary generalized u.c.g. T on A. The argument is a little different from the proof of the o.c.g. identity given in §3.11, but the basic strategy is the same. To each set B in T we assign a complex variable ZB, consider the function exp 2:= ZB, and our first step is the interpolation: BET

exp

L

sup

ZB

L

ZB

+

BET iortB

BET

W( Z; t)

h

L

L

ZB'

B'ET ioEB'

ZB+

BET ioEB

L

11

dt 1e W (z ;t),

(5 .9.18)

0

(5 .9.19)

ZB ,

BET iortB

where fo denotes the first mode in A with respect to the scale lexicon. The next step in the expansion of our multi-variable function is to interpolate the exponential appearing in each B'-term as follows :

exp (

L

BET iortB

ZB

+ t1

L

BET ioEB

ZB)

= exp

L

BET BnB'=0

ZB

CHAPTER 5. WAVELET ANALYSIS OF 4i~

468

+ B"ET loEB" ,B'nB"#0

(5.9.20)

+ B"ET lo rtB" ,B' nB" #0

L

WB'( Z ; t)

ZB

+

L

t2tl

ZB

BET loEB,BnB'#0

BET BnB'=0

L

+ t2

ZB·

(5.9.21)

BET lortB ,BnB'#0

The iteration of this expansion is inductively defined by pursuing an arbitrary branch. Suppose we have chosen remainder terms m - 2 times, thereby introducing the sets B', B", . .. , B(m-2): our (m -1)st step is to interpolate between the current remainder term and the expression obtained from it by keeping only those variables ZB for which m-l

U B(i-l) = 0,

Bn

(5.9.22)

i=2

and then choose either that terminal expression or one of the remainder terms generated by that interpolation. Since T has only M - 1 elements and a new element is chosen every time a remainder term is chosen in the inductive expansion, every branch must terminate in less than M steps. Thus, every term is completed and is labeled by a history - specifically, a sequence (B', B", . .. , B(m-l)) of elements of T. The point is that the term depends only on the complex variables ZB',ZB", .. . ,zB(~-l) together with those ZB for which m

Bn

UB(i-l) = 0.

(5.9 .23) i=2 If there are such B, then the connectivity property of T implies that the term cannot depend on all of the complex variables. If there is no such B, then the term depends only on Z8', ZB", .. . , ZB(~-l), in which case the term cannot depend on all the complex variables unless m = M. Hence

exp

L BET

aM - 1

ZB

IT

a ZB

exp

BET

aM - 1

IT

L

zB

BET

aZB

(

sum of all terms whose ) histories have M - 1 steps .

(5.9 .24)

BET

On the other hand, it is clear from our expansion procedure that the sequences (B', B", . . . ,B(M-I)) we have built are exactly the generalized o.c.g. A on A such that TA =

5.9. ESTIMATING THE ACTIVITY II: INTERNAL COMBINATORICS

469

T Moreover, the differentiation with respect to t m - 1 associated with the (m - l)st interpolation step in the development of A brings down the product m-2

z A(m)

IT

ti

i=1JA(m)

from the exponent for the choice of remainder term labeled by A(m) . Therefore, we may write sum of all terms whose ) ( histories have M - 1 steps =

IT ZB BET

generalized o.c.g. A TA=T

(5.9.25) where W A (z j t) denotes the final form of the exponent for the sequence of interpolations determined by A. Since this function is linear in the complex variables, we have eWA(z;t)

IT (1 + O(ZB)),

=

(5 .9.26)

BET

and so if we combine (5.9.24) with (5.9.25), we obtain 1 =exp

L ZB/

BET

z =O

(5.9.27)

We have proven the identity (5.9.16) . What application does this have to (5.9.1l)? For a given representation one graph g, we let Ag denote the induced generalized o.c.g. Clearly, TA. depends only on Ug , as it is only the set of supports of the links and chains of links in g. Also note that while Ag does not determine 9 uniquely, Ag together with Ug does. Therefore, summing over all 9 such that Ug = U corresponds to summing over all A such that TA = T For convenience we set

= fA. (t), ng(m) = nA.(m) . fg(t)

(5.9.28) (5.9.29)

Now for an Rl graph 9 consisting of M - 1 links and chains of links - again, deleting the I-links - there are M - 1 interpolation parameters tl, .. . , tM-l and M - 1 factors

CHAPTER 5. WAVELET ANALYSIS OF ~~

470

b~(t), . .. , bL_l (t), each a product of old interpolation parameters brought down from the exponent by differentiation with respect to a given interpolation parameter. Thus M

II qJ(t) = II b~_l (t)

(5.9.30)

m=2

j

because b~_l(t) is a single q9-factor in the case of a single link and a product of q9-factors in the case of a chain. On the other hand, M

f9(t)

#

II b~_l(t)

(5.9.31)

m=2

because the factor b~_l (t) is certainly not as simple as the product m-2

II

ti

i=1)g(m)

we had at the more abstract level. Recall that the interpolation of the mass counterterm was not straightforward. If the single link is a mass link or a renormalized link, then b~_l = qJ can include the square of an interpolation parameter or twice that parameter. Nor is the additional complexity confined to this issue. The square of an interpolation parameter can appear in the case of a composite link as well. In the case of a chain, b~_l can definitely be a complicated monomial, because a large number of q9-factors may comprise b~_l. An arbitrarily high power of a given parameter can appear in such a monomial, which we have never explicitly described; each q9-factor was implicitly given by the expansion procedure. On the other hand, the factor 2 can appear at most once in b~_l (t) - even in the case of a chain - and (5 .9.32)

Therefore, j-2

II

bJ-l (t) :::; 2

ti,

(5.9.33)

f9(t).

(5.9.34)

i=1)g(j)

which, in turn, implies M

II bJ-l (t) :::; 2

M

-

1

j=2

Combining this estimate with the generalized o.c.g. identity (5.9.17), we obviously obtain the bound (5.9.35)

5.10. COMBINATORICS FOR SUMMING OVER POLYMERS

471

and so we have reduced the activity estimate (5 .9.12) to Iz(U)1

(5.9.36)

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical ¢~ Model," Commun. Math. Phys. 88 (1983),263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360.

5.10

Combinatorics for Summing Over Polymers

Having eliminated representation one graphs from the bound on the activity z(U) of a given representation two graph U, we now have the freedom to pick a representation one graph gU such that (5.10.1) To fix a choice for each U is essential because we still need the notion of "old" and "new" occurrences of modes in a given link. We also need the notion of attachment, which we now define. As important as it is, this notion is simply described. Whether a given element g~, m > 1, is an individual link or a chain of links, we say the attachment of g~ is the last mode in the set m

supp g~ n

U supp g~-l i=2

with respect to the scale lexicon. In the special case m = 1, the attachment is just the first mode in supp U with respect to that order. We shall denote the attachment of g~ by e~ . Our foremost concern at this point is the combinatorial scheme for summing over the R2 graphs (= phase cell polymers). For an arbitrary R2 graph U, we can certainly pick a tree structure for this connected set of links and chains of links based on selecting some of the support intersections and ignoring others. Since the chosen tree structure still connects all elements of U, distinct representation two graphs lead to distinct tree structures because the sets of elements are not equal. (The structures may still be isomorphic.) Therefore, we can replace the sum over U by the sum over these representatives. However, we have to be careful in our selection of a representative

CHAPTER 5. WAVELET ANALYSIS OF ~

472

for each U because, at some point, the gU-dependent estimation implied above must be inserted in the combinatorial scheme. Accordingly, we define the representative in terms of gU as follows . For a given link or chain r E U we have r = g~ for some m. Let i(U ,m) = min{i : f~ E supp g~}, (5.10.2) If indeed f~ -I f~ , we regard r = g~ as connected to g~u .m) and to all g~, such that i(U, m') = m . If f~ = f~, we say r = g~ is connected to f~ and to all g~, such that i(U,m') = m . Thus we have defined a tree on the set U U {fn. Ultimately, we must establish the bound demanded by the polymer expansion formalism - namely,

Iz(U)1 ~ c

(5.10.3)

R2 graphs U fEsupp U

for an arbitrary mode

e. We focus on the more restrictive sum f SML Iz(U)1

(5.10.4)

R2 graphs U card U=M-l fEsupp U

with the intent of proving (5.10.5)

Since f. must be the fixed mode for the tree summation, we cannot choose f~ as the root of the representative tree in general, and this is a problem. Our first concern is to estimate with a sum over R2 graphs U such that f~ is the fixed mode. To this end, consider the sequence (m2, " " mv) of integers such that

st

i(U,m,+d = m" i(U,m2) = 1, mv

(5.10.6) (5.10.7)

= min{i : f. E supp g~},

(5.10.8)

and define (5 .10.9)

with pr(k) defined for r E U and k E supp U according to a few cases. If individual link other than a combination link, then

r

is an

(5 .10.10) If r is an individual combination link ((a", a'), a), then

+ (max{Le,Ld)-ll ~k E supp(a", a'),

P(u" .u,)(f)(1

f¥

Pc(k)

={ Pu(f)(1

f¥

+ (max{Le,Ld)-ll

E supp a,

-

~e 1)-3-E,

~e - ~k 1)-3-",

(5.10.11)

5.lD. COMBINATORICS FOR SUMMING OVER POLYMERS

473

where e is the last occurrence in supp an supp(a" , a'). If r is a chain, there is a finite sequence (k 1 , • • • , kN) of modes such that kl = k and kN = and

e¥

N-l

prCk)

=

II (1 + (max{Lkj,Lkj+,))-l\-;kj -

-;kHl 1)-3-, .

(5.10.12)

j=l

We postpone the description of this sequence to §5.12; all we need here is the estimate prCk)

~

(1 +

(max Lk 15,j5,N

)

)-l\-;k _ -;eur

\)-3-"

(5 .1O.13)

which follows from the triangle inequality. Extending this simple estimation for the case of a chain to the entire product in (5.10.9), we obtain (5.1O .14) The point is that Le~ ~ every length scale that arises in the journey from f back to e~ , because e~ is the first mode in supp U with respect to the scale lexicon. This enables us to estimate:

L

R2 graphs U card U=M-l iEsuPP U

(5.10.15)

x R2 gra phs U card U=M-l =e' ,iESUPP U

e':.

On the other hand,

(I

+ L i'-1 \ Xi

~

--10.

-

X i'

I)

-3-E

(5.1O.16)

~ c,

so by geometric series estimation, (5 .10.17)

Hence

L

R2 graphs U card U=M-l iESUPP U

\z {U)\ ~

C

L

SUp

e' E A

L,,?L ,

R2 graphs U card U=M-l =e' ,iESUPP U

e':.

Pu{f)-l\ Z{U)\,

(5.10.18)

CHAPTER 5. WAVELET ANALYSIS OF ~~

474

and therefore we have the desired reduction - to prove (5.10.19) R2 graphs U card U=M-l e~ =e' ,iESUPP U

for Ll' 2: Le' Now the hardest part of the estimation - done in the next section - is to control the number divergence. That control will yield an activity estimate which can be written in the form

Iz(U)1

:s du(it)! II du(r)! II rEu

w(g~, g«u,m))

II

w(g~),

(5.10.20)

m : e';!,i-e~

where the du-integers are the coordination numbers. For the combinatorics, we use this as our input estimate. Let [. denote the set of all pointed links and chains of links supported by the original set A of modes. The case-by-case determination of w(g~,g«U,m)) when e~ -j. e~ and of w(g~) when e~ = e~ is postponed to the next section, but in the meantime we extend this function w to the domain (£ x £) U £ in the following trivial way. If there is no R2 graph U such that (r, r') = (g~, g«u ,m)) and e~ (resp . e«u,m)) is the distinguished point of the link or chain g~ (resp. g«u ,m)) for some m, then w(r, r') = O. If there is no R2 graph U such that r = g~ and e~ is the distinguished point of the link or chain g~ for some m with eZ;:. = e~ , then w(r) = O. The idea is to replace the sum over R2 graphs with the sum over all rooted trees connecting M - 1 elements of £ and one mode £' E A, where that fixed £' is the root and precedes every mode in the given tree with respect to the scale lexicon. Now label the trees - i.e., introduce one-to-one mappings !1 from the set {2, .. . , M} into £ - so that we are now summing over both rooted tree graphs T and maps !1 while dividing by the number of permutations. We still need to bound Pu(i)-l with a quantity depending on!1 and T only - not on U. We can make the obvious over-estimation (5.10.21) where m = mv in the sequence defined by i and we have simply included extrapr(k)-lfactors. The factor Pg~ (e~2) in (5.10.9) has not been dropped, since i(U , m2) = 1. The loose cannon here is m defined by i, which can lie anywhere in supp U. Therefore, we must sum over m: M

PU(i)-ll z(U)I:S R2 graphs U card U=M-l e~=e' eEsupp U

I:

(5 .10.22)

m=2 R2 graphs U card U=M-l

ft.;=t' lEsupp g';:,

5.10. COMBINATORICS FOR SUMMING OVER POLYMERS

475

The bound (5.10.19) is certainly implied by the same type of bound on the inner sum, so we fix m with this constraint on i. If we let fr denote the distinguished point of a pointed link or chain r, then by (5.10.20) and (5.10.21),

L

Pu(i)-llz(U)1

R2 graphs U card U=M-l e~=e' tEsupp g~

~

(M

~ I)! (~e:

e

r

tree graphs T on

{! •...• M} rooted at 1

M

II dT(m)! m=l

one-to-one mappings 11 : {2 •...• M}-+C (l.m)ET~en(~)=f'

iEsuPP l1{m)

X

II

(

PI1{m') (fl1{m»)_l)

(m'.m)ETm

II

X

w(O(m),O(m'))

II

(5.10.23)

w(O(m)),

{l.m)ET

{m'.m)ET m'#l

where Tm is the unique path along T from 1 to m. Here we are permuting M -1 labels and the coordination numbers of the tree graph are the coordination numbers of the tree. By Cayley's Theorem, there are exactly (M - 2)! I1(d m - I)! m

tree graphs T on {I, ... , M} rooted at 1 such that dT(m) Therefore,

= dm

gd (Le')<

for 1

~

m < M.

m

M - 1

R2 graphs U card U=M-l e~=e' tEsuPP g~

Li

SUp tree graphs T on

{l ....•M} rooted at 1

II

PI1{m)(i)-l ( one-to-one mappings

PI1{m')(fl1 {m»)_l)

(m' ,m)ETm.

11: {2 •...• M}-+C (l.m)ET~en(~)=e'

iESUPP l1{m) X

II {m'.m)ET m'#l

w(O(m),O(m'))

II {l .m)ET

w(O(m)),

(5.10.24)

CHAPTER 5. WAVELET ANALYSIS OF j

476

while the geometric mean inequality implies (5.10.25) m

m

On the other hand, it is a standard combinatoric fact that car d{(d1,"" d) M : d 1,· ··, d M -> 1,

"'d _4M L m =2M-2} <

1

,

(5.10.26)

m

so we finally obtain 8M-1

Pu(i)-llz(U)1 ~ M _ 1

(LL )C i

: i

R2 graphs U card U=M-l l~=e'

sup

tree graphs T on {1 •...• M} rooted at 1

ieeupp g~

one-to-one mappings

(1.m)ET

0: {2 •...• M}-+L (1.m)ET=?en ( ~)=e'

eEsupp O{m)

X (

II

II

Po{m')(l!o{m))-I)

{m'.m)ETm

w(!1(m), !1(m')) . (5 .10.27)

{m' .m)ET m';o!1

This concludes the description of the combinatorial strategy. Controlling the sums for a fixed tree graph T is the counting problem, and it is one of the two major tasks remaining. The other task is to cancel the number divergence i.e., the factorial growths due to the f-factors in the activity bound (5.9.36). The two problems are related, since the w-factors needed for solving the counting problem are determined by the activity estimate (5 .10.20) that reduces the number divergence to the d-factorials we have just cancelled combinatorially. We tackle the number divergence first, but we must do so with an eye on counting. An important scheme for controlling the number divergence lies in exactly how each f-factor in (5 .9.36) is estimated - i.e., in how the quadratic positivity of the free part of the Euclidean action is complemented by the quartic positivity of the interaction to weaken the number divergence. The parameters Te play no essential role in this. Since

i: J i:

d~exp ( - ~e -),e Jue(x)4d -; -48),2e

x

Ui(-;)3d -;

J

ueCy)5d

d~ exp ( - ~e -

Y +Te~)

),CLie - ),2cLie

+ Ti~)'

(5.10.28)

5.10. COMBINATORICS FOR SUMMING OVER POLYMERS

477

it follows from Jensen's inequality that Z {e} (Te) has a lower bound independent of both Te and Le - namely,

== C(A) 2:

c,

(5.10.29)

because Le :S 1. On the other hand, we may estimate:

[eT'~ (1 + IWJ~U NJ(e)]>.,e x

s~p (exp (-~e

ce Tl sup(e-te(1

-

:S ( [ : de e-te+T'~)

cALe(1

+ InL(1)-2e)

(1

+ IWJ~U NJ(l))

+ Iwnewu(l))

~

x sup(exp( -cALe(1

L:

+ In L( 1)-2e4)(1 + IWJEU

old~(e)

),

~

(5.10.30)

where the integer newu (f) counts the first 3 occurrences of f in gU if the total number of real occurrences is > 3, and all occurrences of f otherwise, while old~ (f) is just NJ(f) minus the number of these special occurrences that happen to belong to the link J as well. Since newu(f) :S 3, the first supremum is bounded by a universal constant. Combining this estimation with (5 .10.29) we reduce the activity bound (5 .9.36) to nu+2

Iz(U)1 :S A

t n~)+3nLv II (clw(J)I)ccard(supp

,=1

L:

TF

U)e'E'uPp u

JEU

L:

x

II sup(exp(-cALe(1 + InL(1)-2e)(1 + IWJEU e

old~ (l)

).

(5.10.31)

~

It is the remaining supremum that contains the number divergence. This number divergence is much weaker with the quartic exponent than it would have been with the quadratic exponent. Nevertheless, the latter exponent has just played an important role by separating out certain occurrences that will not have to be reckoned with when we extract small factors from the w(J) to implement cancellation of the number divergence. A more basic necessity met by this estimation is to save powers of the small parameter A. The point is that we pay a certain price when we use the quartic exponent - namely in negative powers of A and Le. We have

L:

sup (exp ( -cALe(1 ~

+ In Lei )-2e4)(1 + IWJEU

old~ (e)

)

CHAPTER 5. WAVELET ANALYSIS OF
478

(5.10.32)

so the power of >. in the activity bound becomes (5.10.33) as we shall see in the next section. By contrast,

(5.10.34)

5.11

Strategy for Number Divergence Cancellation

In order to reduce the sum over polymers to the combinatoric estimates (5.10.27), we used an estimate of the form (5.10.20). We have yet to realize the w-factors, and our next concern is to apply (5.10.31) and (5.10.32) to this end. Logically, the proof of (5.10.20) precedes the estimation carried out in the previous section, since we are now concerned with a delicate estimation of the polymer activity. The basic issue here is the number divergence

If ((~ Old~(e)) ,) ,/. For a given R2 graph U, define the sequence (J';;.) of attachment links as follows : (i) If g~ is an individual link, then

J';;.

= g~ .

(ii) If g~ is a chain, then J';;. is the latest link in g~ such that the attachment £~ of g~ lies in the support of the link. Let [1' be the set of all links in f] that are not attachment links , and decompose the number divergence accordingly:

(

~ old~ (e)) ! (2: old~~ (e))! JEU'

(5.11.1)

m

Small factors will have to be burned up to cancel these factorial growths . For each J E f] and e E supp U we assign a factor 'Y~ (e, L) to the Lth old occurrence of in the link J. This is the assignment of numerical factors, for which the rules must be determined. Obviously, these factors cannot be the same small size, as an exponential decay in the number of old occurrences cannot beat a factorial growth in that number. For each cancellation, the relative sizes of the factors are governed by

e

5.11. STRATEGY FOR NUMBER DIVERGENCE CANCELLATION

479

a summability condition, which is exploited by some variation of the geometric mean inequality. For the fl' -assignments the summability conditions are old~ (t)

L L

,~(e,~)4 S c

(5.11.2)

suggested by the inequality old~ (e)

II II JEU'

,~(e, ~)4

<=1

<

(5.11.3)

The attachment-link-assignments involve a different type of summability condition. For at least one old occurrence ~ of a mode e in J~, the factor ~u (e,~) must depend on the attachment and the occurrence only. This requirement will become clear later, but the consequence is that a summability condition like (5.11.2) is impossible, because an attachment £' can be e~ for many m . in the previous section. Recall du (g~, ) Consider the tree defined on U U denotes the coordination number for g~, and - by the way in which we defined the tree - note that (5.11.4) du(g~,) - 1 = card{ m : i(U, m) = m'}

{en

with i(U, m) defined by (5 .10.2). On the other hand, e~

= elj, => i(U,m)

= i(U,m),

(5 .11 .5)

so it obviously follows that

du(g~,) - 1 =

L

card{m: e~ = £'},

(5.11.6)

f.'EA~,

where A~, is the set of all attachments in supp Thus

g~, \ ({en u m'~m' supp g~" )

du(g~,)! 2:

II f.'EA~,

(card{m: e~ = £'})! ,

(5.11.7)

CHAPTER 5. WAVELET ANALYSIS OF j

480

while in the special case of the attachment e~, du(~)

= card{m:

= en.

e~

(5 .11.8)

Since {e~} and the A~, partition the set of attachments, we therefore have the factorial inequality (5.11.9) (card{m: e~ = e'})!:S du(~)! du(r)!,

IT

IT

U-attachment l'

which is the essential connection between (5 .10.20) and the geometric mean strategy for controlling the number divergence. As far as the summability conditions on the numerical factors are concerned, fourth powers are still involved, but summation over distinct attachments is necessary: old;:.
L

card{mlf: leu"

U-attachment i'

= f.'}

m

m:

L L I~(e, L)4 :s c, i~ =£'

(5.11.10)

£=1

(5 .11.11)

=

old~ (e, L)

old~~ (e, L) .

(5.11 .12)

We apply geometric mean estimation as follows: old;:'(l)

IT IT IT e

,z:,(e, L)4

,=1

m

IT

(card{mlf : e~"

= e'})4 card{m":

e;:',,=l'}

U-attachment l'

x

IT U-attachment l'

IT

(card{mlf: ~"

= e'}!)4

U-attachment l'

I] (L: Ol~~ (e) u-atta~ent e' L L

old;:'(l)

x

m: e~=l'

m

,z:,(e, L)4 card{mlf : e~"

E ~

£.=1

old;:'(l)

(5.11.13)

= f.'} )

where the preliminary step uses

L

old~(e)

:s 4.

(5.11.14)

5.11. STRATEGY FOR NUMBER DIVERGENCE CANCELLATION

481

By (5.11.9) we have old~(l)

IIII L l

:s

Ccard

'Y~(f,£)4

,=1

m

U

(du(e~)! II du(r)!) 4 rEU

x

It (

1 ) ~ old;;,(e) ! old~ (l)

L

x

IIe (

L

m: l~ =l'

L

Card{mll.1eu _ f'}

U-attachment l'

)

m"-

.

~ old~ (l)

'Y~(e, £)4

(5 .11 .15)

£=1

in consequence. We are now ready to pull this scheme together. Combining (5.11.2) and (5 .11.3) with (5 .11.10) and (5.11.15), we get the master inequality

(5.11.16)

Hence

II~ lw(J)III ((~ Old~(e))!)1/4 II ((L Old;;,(e))!)1/4 lEU

:S

e

lEU'

l

m

L: L: old~ (l) C JEU l

xdu(e~)!

X

II du(r)! II~

rEU

lEU

(

Iw(J)1

IIe

old~(e)

1

~ i1 (e, £)

,_1

)

,

(5.11.17)

1

and if we combine this result with (5.11.1), (5.10.31), and (5 .10.32), we obtain

CHAPTER 5. WAVELET ANALYSIS OF ~~

482

x

11 (c1W{J)1 IT olIt Y1 (~, £))

JEU

l

xdu{i~)!

J

£-1

IT du{r)!

(5.11.18)

rEU How do we derive from this bound an inequality of the form (5.10.20)? The key is to estimate with a product of factors having a dependence only on the corresponding pairs of elements in U. We certainly have ccard(supp

U)

IT ccard(supp r) ,

~

(5.11.19)

rEU

IT tEsupp

2

eT ,

IT IT e rEU tEsupp r

~

U

2

T,

,

(5.11.20)

but our main concern is with the assignment of numerical factors and the old-occurrence function (5.11.21) old¥{i) = Lold~{i) , JEr where we have adopted the abuse "J E r" to mean J E r if r is a chain and J = r if r is an individual link. The nuisance is that both -If (i, £) and old~ (i) can depend on the distant past in the history gU. Therefore, if r, r' E [. and (r, r') can be identified as (g~, g~u ,m)) with i~ = ir and i~u,m) = ir' for some R2 graph U, then we define Old{r,r') simply as the set of finite sequences (old~IJ E r) arising from some U in this way. Thus we have the desired bound (5.10.20) if we define w{r,r')

=

max (oldIJEr)Eold(r ,f') { x

x

t n(')(r)+3n'V(r)-t L:, JErL: .x n(r)+2 .=1 2

IT (eT'L lEsupp r

old(t)

( r) ccard supp

-(t+,) L: OldJ(l)) JEr oldJ(l)

1

£=1

r, oldJ (f) "IJ ,£

IT clw(J)1 IT IT JEr ( l

) }

,

(5.11.22)

where n{r) (resp. n'{r), n / {r),n'/{r),n IV (r)) is the number of 4-links (resp. mass links, composite links, renormalized links, combination links) in f, and we have already put a constraint on the "I-factors with the notation . Old{r,e') and w{f,e') are defined in exactly the same way for (r, i') as for (r, f') . The strategy for cancellation of the number divergence has proven the inequality (5.1O.20), but the assignment "15' oldJ (i, £) of numerical factors has not yet been made. In the previous section we promised to verify (5.10.33) - i.e., to show that there are still enough powers of .x for convergence after this scheme for cancelling the number

5.12. FROM INFINITELY MANY CASES TO FINITELY MANY

483

divergence has introduced the large factor

This claim follows from the simple bounds

L L L L

J is a 4-link,

(5.11.23)

oldJ(e) ~ 5,

J is a comopsite link,

(5.11.24)

oldJ(e) ~ 2,

J is a mass link,

(5 .11.25)

oldJ(e) ~ 2,

J is a renormalized link,

(5.11.26)

oldJ(e) ~ 5,

J is a combination link.

(5.11.27)

These integer bounds follow, in turn, from inspection of the circumstances under which these types of links can arise.

References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann . Phys . 142, No.1 (1982),95-139. 2. G. Battle and P. Federbush , "A Phase Cell Cluster Expansion for a Hierarchical 4>j Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981),327-360.

5.12

From Infinitely Many Cases to Finitely Many

Recalling the central estimate (5.10.27), we now turn our attention to controlling the residual sum for a fixed tree graph T - i.e., we focus on proving

( ~ee:) c

L

P(O(m) (i)-l

one-to-one mappings 0: {2,oO.,M}-+C

II

PO(m')

(fO(m))-l

(m',m)ET

(l,m)ET=?en(~)=e'

iE supp O(m) X

II (m' ,m)ET

m';el

w(!1(m), !1(m'))

II (l,m)ET

w(!1(m), e') ~ eM )"cM

(5.12.1)

CHAPTER 5. WAVELET ANALYSIS OF ~

484

for fixed modes i and £', fixed integer m, and tree graph T on {I, ... , M} rooted at 1. This is a major task, but the combinatoric estimation carried out in §5.1O shows that it is all we need for the polymer formalism because the inequality (5.11.18) that we have just established in the previous section yields (5.10.20) with w(f, ff) given by (5.11.22). w(f, e') is given by the same expression with the change Old(f, ff) t-+ Old(f, ff), where Old(f, £') denotes the set of all finite sequences (oldJ: J E f) of weight functions of the modes arising from some R2 graph U, as oldJ = old~ with f = g~ for some m and f~ the distinguished point £' of f . The numerical factors 'Y~ldJ (f, L) have yet to be assigned. We find it convenient to estimate the multiple sum with the p-factors absorbed into the w-factors, but not in the obvious way. We can no longer postpone the complete definition of Pr' (k) . We have to deal with the fact that an element of L can be a chain with arbitrarily many links. If ff = (J1 , .. . , I n ) is some chain, remember that it has a distinguished point fr' that is identified as its attachment in some history that ff is a part of. Let Jr' be the last link J i in the sequence such that fp E supp J i and call it the attachment link. With k E supp rf, consider the last link Jk in whose support k lies. Let (Jio '· · . , J il ) be the sequence of J i between J k and Jr' inclusive. If there is no combination link, we set i 1-l

pr,(k)

II (1 + (max{Ll" Ll i+J)-ll Xli - Xl i+1 1)-3-0

=

i=io

+ (max{Lk, L lil _I })-ll Xl il _1 - Xk 1)3-0 x(I + (max{L lio ,Llr ,})-ll Xl r, - Xl io 1)-3-0

x(I

(5.12.2)

with fi the last (smallest scale) occurrence in supp J i n supp Ji+1, where we have assumed J io = Jr' and J il = J k for the sake of definiteness. Obviously, the reverse case interchanges the fio and f il - 1 in this formula. As for combination links, only the last link in a chain can possibly be such a link, so a combination link (( u" , u f ), u) occurs in (Jio , . . . , J il ) only as J io or J il - depending on the direction in which the sequence is ordered - if at all. In the former case we replace the fr,-factor in (5.12.2) by

(1 x

+ (max{Ll io ,Le})-ll Xl io - Xl 1)-3-0 (1 + (max{Ll,Llr ,})-ll Xl - Xl r, 1)-3-0,

where f is the last occurrence in supp un supp(u",u f ). In the latter case we replace the fk-factor by (1 x

+ (max{Ll;I_" Ll})- 1 I Xe - Xl;I_1 1)- 3 -0 (1 + (max{Le,Ld)-ll Xl - Xk 1)-3-0 . ~

---lo.

In all cases, it is clear that Pr' (k) has the required form (5.10 .12) for the estimation (5.10.13- 18) done in §5.IO. In the present context, we over-estimate pr,(fr}-l in a mushy way to erase the fr-dependence. Specifically,

5.12. FROM INFINITELY MANY CASES TO FINITELY MANY

~ ,U[,.,.,Q" '; (1 + (m~{L .. L,. W'I ~, - ~,. D'+< 1

Pc'('cl-' S

485

II [k,k' II

JEr'

(1

+ (max{Lk,L k, })-11-;k

-;k' 1)3+ 0 1

-

(5.12.3)

E supp J

and it is important that all of these factors can be cancelled against the decay in the numerical factors w(J). We apply this bound to each of the factors Prl(m') (£rl(m))-1 in (5.12.1) . Similarly,

II [(

Ll"Prl(rh)(f)-1 S

JErl(rh)

II

X

(1

min

Lk)-O

kEsupp J

+ (max{Lk,L k, })-11-;k

-

-;k' 1)3+ 0 ] .

(5 .12.4)

k' ,kEsupp J

These are the bounds we now absorb into the w-factors. Actually, there is a twist in the association, as we have already hinted. We choose the pairings

, II [ II

w(D(m),D(m))

JErl(m)

(m' , m) E T, m'

i-

~

~

1 x k- x k,1) 3+ 0 , (1+(max{L k ,L k,})-l

k' ,kEsupp J

1,

II

w(D(m))

JErl(m)

(I,m) E T, of factors, where we over-estimate unity with such a J-product for every w-factor that is without one. On the other hand, the path T rh played an important role before this elimination of the Trh-dependence . The legacy is that each D(m) has only one J-product except D(m), which has only two. This is vital, since the degree of decay in each numerical factor w(J) is high, but bounded. Accordingly, if we define

w(r, r') = w(r, r') ( min Lk)-O kEsupp r

x

II [ II JEr

k' ,kEsupp J

(1

+ (max{ Lk, L k, })-11

-; k

- -; k' 1)3+0 ]2, (5.12.5)

CHAPTER 5. WAVELET ANALYSIS OF j

486

w(f, f') == w(f, €') (

x

min

kEsupp r

II [ II JEr

Lk)

-3

(l+(max{L k ,L k,})-ll-;k

-

-;k'

l)3+ e ]

2,

(5 .12.6)

k' .kEsupp J

then the multiple sum in (5.12.1) is bounded by the multiple sum L

II

t,

II

w(O(m), €')

one-to-one mappings (l,m)ET

w(O(m),O(m')),

(m',m)ET

n: {2, ... ,M}->,C

(l,m)ET""ln(=>=l'

where we have applied the m-bound to every O(m)-factor - not just to the O(m)-factor. This choice eliminates the i-dependence as well as the Tm-dependence. We now reduce the problem to an estimation of single sums. The standard tree graph summation is essential here - first summing out the factors with coordination numbers equal to 1, then summing out those factors whose subsequent coordination numbers are equal to 1, etc. However, there is an extra wrinkle in the issue of "connecting back" because we have to sum w(O(m), O(m')) over all possible fn(m) is supp O(m') for fixed O(m') as well as over all possible O(m) for fixed fn(m) ' This is easily dealt with by counting the possibilities in the obvious way:

L

w(O(m),O(m')):<:-:: card(supp O(m'))

n(m)E'c

L

sup kEsupp n(m')

w(O(m),O(m'))

(5.12.7)

n(m)E'c In(=>=k

because w(f, f') == 0 unless fr E supp f'. The iteration leads to no serious counting divergence because card(supp r) Aen(r)n(f)

< cn(f),

(5.12.8)

c.

(5.12.9)

:<:-::

The establishment of (5 .12.6) is now reduced to proving that Aen(r')n(f')

L

A-en(r)w(f,f'):<:-:: CAe,

(5.12.10)

rE'c lr=k

L

A-en(r)w(f,f')

:<:-::

CAe,

(5.12.11)

rE'c ll' =l'

where the telescoping powers of A in the iteration are motivated by the elementary cancellation (5.12.9).

5.12. FROM INFINITELY MANY CASES TO FINITELY MANY

487

The problem would already be reduced to a finite number of cases if there were only individual links to reckon with, but we have to pursue a further reduction in the case of chains, which can consist of arbitrarily many links. Now by (5.11.22), we may certainly write

II

max

w(r, r')

(oldJ iJEr) EOld(r .r')

[>.J.'J-t

1

oldJ(e)

x ( clw(J)III II f

y

oldJ(f)

JEr

<=1

)

r.oldJ(e)

'YJ

,~

I

L~(t+e)

II

(e T £

OldJ(f))

fEsupp J

(5.12.12)

,

where fJoJ is the order of the link - e.g., fJoJ = 1 if J as a 4-link, fJoJ = 2 if J is a mass link, and fJoJ = 3 if J is a combination link. It is important to remember that each J -factor except the last one is too big to sum out . This is the reason why the expansion rules develop the chain. Accordingly, we must distribute the extra smallness in the last link over the links in the chain, and we need some additional notions to implement this scheme. We say that a chain is hooked if the following conditions hold. (i) The last link is not the attachment link (which implies the only old occurrences in the last link occurred previously in the same chain) . (ii) The last old occurrence in the last link has a strictly smaller scale than does the variable for the integration by parts differentiating down the first unit in the link. The last link in a hooked chain is therefore a little special. The hook link J~ is the latest previous link whose support contains the mode whose occurrence in the last link is the last old one. We call that mode the hook. Let r be an arbitrary chain - hooked or not. For an arbitrary link J E r we define the pinning of J as follows.

eS

es

(1) If J = Jr, then = er - i.e. , the pinning of the attachment link is the attachment of the chain. (2) If J precedes Jr , then is the mode for which integration by parts with respect to the a-variable differentiates down the next unit - initiating the development of the link succeeding J . (3) If J succeeds Jr and if the chain is not hooked, then is the mode for which integration by parts with respect to the a-variable differentiates down the unit that initiates the development of J itself. (4) Suppose r is hooked, J succeeds Jr, and either J = J~ or.] precedes J~. Then is defined as in (3). (5) Suppose r is hooked, J succeeds J~ , Jr precedes J~, and J is not the last link in r . Then e5 is defined as in (2) . (6) Suppose r is hooked , J succeeds Jr, and Jr succeeds Jf. Then e5 is defined as in (3) . (7) Suppose r is hooked, Jr precedes J~, and J is the last link in r . Then e5 is the hook.

es

es

es

By careful inspection of all the cases for integration by parts in the expansion rules, one can verify the following principle.

CHAPTER 5. WAVELET ANALYSIS OF ~

488 Ordering Principle. IT f scale lexicon as well.

= (J1 , ... , I n )

is a chain, then lSi precedes lS'+l in the

This simplifying observation will be valuable to us in our distribution of extra smallness over the chain. First consider the case where f is not hooked. IT (JiD , .•• , I n ) is the segment of the chain where JiD = Jr and I n is the last link, then we can write w(f,f')

<

II wJ(f,f') JEr ( x

with Li

WJ(f,f')) Lt:lWJr(f,f')

II JEf\{J·D.···,Jn}

(

nII-l (Li+l)!+ - (f f,)) L-!+ - (f , f') L. WJ., n WJ n i=io+l

(5 .12.13)

"

= L,rJ. , where we have set

and inserted a telescoping product of ratios of length scales, where OldJ(f,f') is the projection of Old(f, f') onto the J-entry. By the Ordering Principle, the ratios are ~ 1. (5.12 .13) is an inequality instead of an identity only because we have made the gross over-estimation

( min Lk )-e < II ( min Lk )-e kEsupp

r

-

JEf

kEsupp J

(5 .12.15)

Next consider the case where f is hooked and either J~ = Jr or J~ precedes Jr in f . Let (JiD , .. • , I n ) be the same segment as before. In this case we choose almost the same propagation of length scales, except the hook link supplies the last link with a small factor. We have w(f, f')

~(

II JEf \ {J'D •.. .,In ,J~}

WJ(f,f')) Lt:lWJr(f,f')(L'J/./Ln)!+

5.12. FROM INFINITELY MANY CASES TO FINITELY MANY

489

(5.12.16) with Li

= Ll~,

as before. Finally, consider the case where r is hooked and J~ succeeds

Jr in r. Let (Jio," " J i,) now be the segment from Jr to J~ inclusive. Then we estimate

II

w(r,r') S (

WJ(r,r')) L!:lWJr(r,r')

JEr\{J'o, ·· ·,J" ,In

}

x)f (( Lt1) ~+ !WJ,(r,r'~ <=<0+1 \

,+

xL;;'i WJ n

(Lr

L~~+ )WJ~(r,r')

')

(r, r'),

(5.12.17)

where I!}n is the hook in this case. In our progression of careful over-estimation, we now complete the elimination of the r'-dependence from these major factors . Let (5 .12.18) and write

II eJ(r).

e(r) =

(5.12.19)

JEr

This represents the relaxation of (5.12.14) to the maximum over (oldJIJ E r) E Old(r),

(5.12.20)

where Old(r) simply denotes the set of all sequences of the form (old~IJ E r) for some

R2 graph U including r with I!r as the attachment. We return to our target bounds (5 .12.10) and (5 .12.11) to take stock of the current situation. Both have been reduced to proving:

3+

xL;;'i A.-eeJ" (r) S CA. e ,

r

( JEf\{J,oII,... ,J"

hooked

lr=e'

J~ succeeds

Jr

(5 .12.21)

p. -e eJ (r))) L!:l A. -e eJr (r) ,In

}

CHAPTER 5. WAVELET ANALYSIS OF ~

490

1+

X

LJ

3+

1+

)..-'L-:'2 ~J:(r)L;;2 )..-'~Jn(r):S c)..',

r

(,"lUg

hooked

ir=i' J:=Jr or J: precedes

.J••

(5.12.22)

mp,-'" (r)) )

Jr

(5.12.23)

As far as the successive summation is concerned, the notion of "connecting back" is specified for a chain. For summation over chains that are not hooked, we first sum over all possible last links, then over all possible next-to-last links, and so on, until one reaches the attachment link. Now sum over all possible first links, then over all possible second links, and so on, until one reaches the attachment link. Finally, sum over all attachment links. For summation over chains that are hooked and where the hook link either coincides with the attachment link or precedes it, we first sum over all possible links succeeding the attachment link with the successor of the successors held fixed. Then sum over all possible successors of successors, and so on, until one reaches the last link. Now sum over all possible last links with the hook link held fixed. Next we sum over all possible first links, then over all possible second links, and so on, until one reaches the hook link from that direction. Finally, sum over all hook links, then over successors of hook links, and so on, until one at last arrives back at the attachment link. As always, the last step is to sum over all attachment links. For summation over chains that are hooked and where the hook link succeeds the attachment link, we first sum over all successors of the hook link with the successor of the successors held fixed . Then sum over all possible successors of successors, and so on, until one reaches the last link. Next we sum over all possible last links with the hook link held fixed . The rest of the iterative summation is based on regarding the remaining part of the chain as an unhooked chain with the hook link regarded as the last link. There is a subtlety in this iteration that needs to be addressed. Naturally, each step over-sums in a way that disregards the global structure of the chain, but if we throwaway too much information, we sum over links that need to be ruled out to avoid divergent sums - that were indeed ruled out by the chain development of the expansion for precisely this reason. With this constraint in mind, what information about a link J must be retained? We propose:

(1) J must have a distinguished mode k, and one sums over J with the distinguished mode held fixed. Moreover, there must exist some chain in which J is a link with k as the pinning, but such a chain is not included with (J, k). Existence alone is a

5.12. FROM INFINITELY MANY CASES TO FINITELY MANY

491

restriction. We denote this distinguished mode of J by k J .

(2) J must also have associated with it an integer-valued weight function on supp J which is realized as X E Old J (f) for some chain f of which J is an element and kJ is the pinning of J . We denote this weight function by XJ . (3) One of the following alternatives must be specified. (a) J is the attachment link - but not the last link - of some chain satisfying the above requirements for the old occurrences and the pinning. (a') J is both the attachment link and the last link of a chain satisfying the above requirements. (b) J precedes the attachment link in a chain that is not hooked. (b') J precedes the attachment link in a hooked chain but is not the hook link. (b t ) J is a hook link and precedes the attachment link. (c) J succeeds the hook link-but is not the last link-in a hooked chain where the hook link succeeds the attachment link. (c') J succeeds the attachment link of some chain but precedes the hook link if the chain is hooked and the last link if the chain is not hooked. (c t ) J succeeds the attachment link and is a hook link. (c) .J succeeds the attachment link - but is not the last link - in a chain that is hooked with the hook link either equal to or preceding the attachment link. (d) J succeeds the attachment link and is the last link in a chain that is not hooked. (d') J succeeds the attachment link and is the last link in a hooked chain where the hook link is either equal to or precedes the attachment link. (d) J is the last link in a hooked chain where the hook link succeeds the attachment link. (4) If Alternatives (a),(b t ), (c'), (c t ), (c), (d'), or (d) are specified, J has ~nother special mode associated with it. We call it the exit mode and denote it by kJ. For Alternatives (a) , (c'), or (c), it is the occurrence for which integration by parts in the corresponding a-variable differentiates down the unit initiating the development of the next link. For Alternatives (b t ), (c t ), or (d'), the exit mode is the hook. For Alternative (d) , it is the occurrence for which integration by parts in the a-variable differentiates down the unit initiating the development of J itself. This reduces the proof of (5.12 .21)-(5.12.23) to the estimation of single sums over links J with this additional structure - enhanced links. We now have enough information associated with J to drop the f -dependence from the assignment of numerical factors. If J is an enhanced link, we set (5.12.24) as X is also specified.

CHAPTER 5. WAVELET ANALYSIS OF i)~

492

5.13

Wavelet Diagrams for Cases

The entire proof of convergence of the phase cell polymer expansion has now been reduced to establishing the following bounds: ~J:::;

cN',

f.'

E A,

(5 .13.1)

J an individual link in some polymer

lJ=l'

k' E A,

(5.13.2)

J an Alternative (a) link in some chain in some polymer

kJ=k'

(5.13.3) J an Alternative (a'), (b), (b'), or (c) link in some chain

kJ=k' 3+

L

~JL~}i:::; CAo,

(5.13.4)

J an Alternative (b f) link in some chain

kJ=k'

(5.13.5) J an Alternative (e'), (e) , or (d') link

kJ=k'

(5.13.6) J an Alternative (e f ) link

kJ=k'

(5 .13.7) J an alternative (d) link

kJ=k'

(5.13.8) J an Alternative (d) link

kJ=k'

~J

II

AI'J-q::v(i)

(eT1 L~(t+o)xJ(i))clw(J)1

iEsupp J

1

XJ(i)

X

II II --( 'YJ(f., £) i

X [

,=1

II kEsupp J

(1

min kEsupp

J

Lk)-o

+ (max{Lk,LkJ})-II:;k

- :;kJ

1)3+ 0 ]2

(5.13.9)

5.13. WAVELET DIAGRAMS FOR CASES

493

Recall that the integer-valued weight function we associate with a given link need not be unique, so when we sum over enhanced links, we are summing over these possible weight functions as well as the mode occurrences themselves. As far as old occurrences are concerned, it is important to remember that XJ never counts ghost occurrences, even when they are old. We adopt the convention that enhanced links include individual links J of a polymer with kJ = f J . Each of the eight sums over enhanced links include finitely many species of terms, where each species is defined by a case. For each case we must choose the assignment of numerical factors such that the sum of terms of that species is bounded by some universal constant. The constraints on this scheme are the requirements (5.11.2) and (5.11.10) - i.e. , XJ~(e)

L L m

IJ~(f,t)4::; c,

(5.13.10)

L=l

for every collection {Jm } of enhanced links that can be realized as the collection [J' for some polymer U with the kJ~ realized as the pinnings, and

(5.13.11)

for every collection {Jm } of enhanced links that can be realized as the collection of attachment links of some polymer with the kJ~ realized as the attachments. These are the same conditions as in §5.11 with the terminology, notation, and choices introduced here. Since these sums obviously mix the species of enhanced links, these conditions may appear formidable at first glance, but now notice that both bounds are implied by

L

XJ(l)

sup

L

IJ(f,L)4::; c.

(5.13.12)

an enhan~ed - 1 l- E A J link, kJ =l ,-

(5.13.11) obviously follows from this; (5.13.12) follows as well, since the elements of[J' have distinct pinnings. The advantage of this requirement is that it no longer depends globally on a polymer - an advantage anticipated by the dependence already relaxed for the assignment of numerical factors. Moreover, it can be reduced to the same demand for the sum over one species of terms, as there are finitely many species. To visualize the assignment of numerical factors, we employ the wavelet diagrams used to illustrate the expansion rules. Examples of individual enhanced links are given by Figs. 5.13.1 through 5.13.5 with their species given by the associated descriptions. Examples of Alternative (a) enhanced links are given by Figs. 5.13.6 and 5.13.7. Combination links, mass links, and renormalized links cannot be Alternative (a). Examples of Alternative (a') enhanced links are given by Figs. 5.13.8 through 5.13.11. Examples of Alternative (b) links are: Figure 5.13.6 with the change kJ = k4 and no kJ .

494

CHAPTER 5. WAVELET ANALYSIS OF ~~

4-link (kl' .. . ,k4 ) kJ = k3 2 old occurrences counted by XJ k1 old

k3 old

Figure 5.13.1:

mass link ((kl' k2' k3), (k', k)) kJ = k k succeeds kl 1 old ghost occurrence 1 old occurrence counted by XJ k'

new

k I old k old

k 2 new k 3 new

Figure 5.13,2:

5.13. WAVELET DIAGRAMS FOR CASES

495

composite link ((k~, .. . , k~), (k 1 , ... , k4)) kJ = k2 k4 = k~ 3 old occurrences counted XJ

k' new 2

Figure 5.13.3:

renormalized link {( (k~ , k2, k3 , k4), (k 1 , ... , k4)), ((k 2, k3 , k4), (k~, k1 ))} 1 old occurrence counted by XJ k I old

k I old

k I new

k I new

r--

- - ---

k 2 new

new k 2

k 2= k 2

k 3 new

new k 3

k 3= k3

k 4 new.

new k 4

k 4= k4

Figure 5.13.4:

CHAPTER 5. WAVELET ANALYSIS OF j

496

combination link ((k~/, k3, k~, k~), (k~, . .. , k~), (kl' ... ,k4 )) kJ = k~' 3 old occurrences counted by XJ k I old k ; old

r"

k 2 new k 2 new

//" , _k~:e:_ I I

..----1

k;= k; '

"

,

k 3 new

I \ \

,

-,:---

k)= k;'

k 4 new

k 4= k4

k 4 new

--Figure 5.13.5:

4-link (k1 , .. . , k 4 ) kJ = k2 kJ = k4 not part of a composite link 1 old ghost occurrence 1 old occurrence counted by XJ

(j0--occurs in preceding link

kl old k2 old k3 new

----~ k4 new

"'V

occurs in next link

Figure 5,13.6:

497

5.13. WAVELET DIAGRAMS FOR CASES

composite link ((k~, k~ , k3 , k4 ), (k 1 , ... , k4 )) kJ = kl kJ = k~ k~ succeeds k2 1 old ghost occurrence 1 old occurrence counted by XJ

19---

occurs In preceding

,

"

link I

\

--------Figure 5.13.7:

composite link ((k~, . . . , k~) , (k 1 , .. . , k4 )) kJ = k~ k2 succeeds k~ k~ = k4 1 old ghost occurrence 2 old occurrences counted by XJ

cG---occurs in preceding k 2 new

link k 3 new

k 4 new

Figure 5.13.8:

k 3 new

CHAPTER 5. WAVELET ANALYSIS OF ~~

498

combination link ((k~' ) k~) k3) k~) kJ = k2 1 old ghost occurrence 3 old occurrences counted by XJ

0--

occurs

In

(k~)

k 1 old

k 1 old

k 2 new

k 2 new

preceding k 3 new

link

, ,,

,

k 3 new

-- - ----

k 4= k 4

k 4 new

. . . ) k~) (k 1 ) . .. ) k4 ))

(I

Figure 5.13.9:

mass link ((k 1 ) k2) k3) (k') k)) kJ = k3 k succeeds kl 1 old ghost occurrence 1 old occurrence counted by XJ

G----

k old

occurs in preceding link

k old

k 2 new

Figure 5.13.10:

k 1 old

k 2 o ld

k )= k 3

k 4 new

5.13. WAVELET DIAGRAMS FOR CASES

renormalized link {((k;, kz, kg, k4), (k1, .. . ,k4)), ((kz, k3, k4), (k; ,ki))) kJ = kl kl succeeds ki 1 old ghost occurrence 1 old occurrence counted by x J

occurs in

I

I

occurs in preceding link

link

1

---

K

----

k 3 new

new

Figure 5.13.11: Figure 5.13.7 with the change kJ = k; and no kJ. Examples of Alternative (b') or (c) links are given by these Alternative (b) examples as well. Mass links, combination links, and renormalized links cannot be Alternative (b), (b'), or (c). As for Alternative (bt) links, only composite graphs qualify, as the Ordering Principle implies that every hook link must be composite. An example of this case is Fig. 5.13.7 with kJ = kh and k~ = k4 as the only changes. Examples of Alternative (c') links are: Figure 5.13.6 with kJ = kl and

LJ

Figure 5.13.7 with kJ = k; and

iJ= k;,

= k4,

and it is clear that Alternative (c') links cannot be mass links, combination links, or renormalized links. As for Alternate (ct) links, an example is Fig. 5.13.7 with kJ = k; and LJ = k4. As in the case of Alternative (bt) links, only composite links can be Alternative (ct). Examples of Alternative (?) links are given by the Alternative (c') examples as well, and no Alternative (c') link can be a combination link, a mass link, or a renormalized link. Examples of Alternative (d) links are given by Figs. 5.13.12 through 5.13.16. Note that Alternative (d) is the only instance where a 1-link appears. Also, since these are links succeeding the attachment link in a chain that is not hooked, all old occurrences must have a scale larger than the scale of kJ. Examples of Alternative (dl) links are: Figure 5.13.8 with the changes

kJ

= k; and kJ = k;,

Figure 5.13.9 with the changes

LJ

= kz and kJ = k;',

Figure 5.13.10 with the changes

kJ

= k3 and kJ = k',

Figure 5.13.11 with the changes

kJ

= k and kJ = k'

CHAPTER 5. WAVELET ANALYSIS OF ~

500

composite link ((k1, ... , k4), (k1 , .. . , k4 )) kJ = k~ k~ succeeds kl k2 succeeds k~ k~ = k4 1 old ghost occurrence 2 old occurrences counted by XJ

C;Z~~~~ preceding link

I :::: j klold

k 2 new

--.----k 4 new

Figure 5.13.12:

mass link ((k 1 , k2 , k3), (k', k)) kJ = k k succeeds kl 1 old ghost occurrence 1 old occurrence counted by XJ k old

G----

k old

occurs in preceding link

Figure 5.13.13:

k 1 new

5.13. WAVELET DIAGRAMS FOR CASES

501

renormalized link {( (k~ , k2' k3, k4 ) , (kl' . . . , k4 )) , ((k2' k3, k4 ), (k~, k1 ))} k~ succeeds kl kJ = k~ 1 old ghost occurrence 1 old occurrence counted by XJ

8----

@---occurs in preceding link

-

k I old

k I old

k 2 new

k 2= k 2

occurs preceding

k 3 new

k 3= k 3

link

k 4 new

k 4= k 4

k I old

k I old

k I new

--1 --

k 2 new

k 3 new

Figure 5.13.14:

e;,

combination link ((k~/ , k3 ' k.i), (k~, k3, k3' k.i), (kl' ... , k4 )) kJ = k~' k~ succeeds k~ 1 old ghost occurrence 2 old occurrences counted by XJ

8-----k-, o~ occurs in

k

2

new

preceding

link

k ) new

k4 new

Figure 5.13.15:

CHAPTER 5. WAVELET ANALYSIS OF ~

502

I-link k k an old ghost occurrence nothing to sum over

Q---x

k old

occurs in preceding link

Figure 5.13.16: Finally, examples of Alternative (d) links are given by the examples of Alternative (a') links , where kJ is the hook instead of the attachment . Having illustrated some of the species of terms that are involved, we need to examine the sizes of these terms. In the formula (5.13.9) for wJ, note that all the necessary smallness must be extracted from the factor w(J). Temporarily disregarding the scaled distance-decay factors of which w(J) has an abundance, we focus on orders of magnitude in length scales and ratios of the length scales. For our examples of individual links we have (5.13.13.1) (5.13.13.2) (5.13.13.3) 6-

1

1

_1

-;! +

w(J) = O(Lk3 Lk2 Lkl Lk 2 L k/ w(J) = O(L!

4

),

Lk3L~!L~!L~,Lk-,lLk-'!Lk-'!Lk-!) ' 32 14321 J

(5.13 .13.4) (5 .13.13.5)

respectively. Our Alternative (a) examples yield (5.13.13.6) w(J) =

O(LtLk31L~2! Lr;} L~} L~;!) ,

(5.13.13.7)

respectively, while our Alternative (a') examples yield (5.13.13.8 )

(5.13.13.9) (5.13.13.10)

5.13. WAVELET DIAGRAMS FOR CASES

503

O(L%~ Lk21 Lkl1 L~} L~F) ,

w(J) =

(5.13.13.11 )

respectively. The Alternative (b) examples respectively give (5.13.14.1) w(J) = 0(L%.Lk3 L~2! L~} L~,! L~;!),

1

(5.13.14 .2)

while the Alternative (b') examples (resp. Alternative (c) examples) above are the same. For our Alternative (b t ) example, we have w(J) = 0(LtLk31L~} L~} L~,! L~;!) .

(5 .13.15)

For the Alternative (c') examples we have the orders of magnitude (5 .13.16.1) (5.13.16.2)

respectively, while the Alternative (c) examples above are the same. For the Alternative (c t ) example, we have

w( J) = O(Lt Lk31 L~2! L~~! L~,! L~}) .

(5.13.17)

The respective orders of magnitude for the Alternative (d) examples are (5.13.18.1) (5.13.18.2) (5.13 .18.3) ~

-3

-~

-~

5

-1

-~

-~

-4

w(J) = O(Lk 4 Lk 3 Lk2 Lk1L4 L k, Lk J ), k, L3 k, L2 k, 1

(5 .13.18.4) (5 .13.18.5)

w(J) = Oh) ,

while the orders of magnitude for the Alternative (d') examples are (5.13.19.1) 9

1

1

1

1

1

1

1

w(J) = O(Lt Lk33 L"i} L~,2 L~,: L~: L~;2 L~l L~}) , w(J) =

0(LtLk21L~!Lkl1L~}), 6-

-1

-1

-!

- ~+

w(J) = 0(Lk3 Lk2 Lkl L"'J LkJ

),

(5.13.19.2) (5.13.19.3) (5.13.19.4)

respectively. Finally, recall that our Alternative (d) examples are the same as our Alternative (a') examples, where kJ is the hook instead of the attachment in this instance.

CHAPTER 5. WAVELET ANALYSIS OF
504

References 1. G. Battle, "Ondelettes: The Spinor QED3 Connection," Ann. Phys . 201 , No. 1 (1990), 117- 151. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical ¢>~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush , "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360.

5.14

How to Assign Numerical Factors

For every enhanced link J and every mode f. E supp J, the assignment of the numerical factor to the ~th old occurrence of f. in J is yet to be made. This is highly casedependent, but it is clear from (5.13.12) that we can make the universal assignment (5.14.1) without loss of generality. The challenge to the reader is to find a minimal set of rules that will cover all of the many cases to be reckoned with. In this section we illustrate the nature of the game by assigning numerical factors in a few of the cases. Consider the species given by Fig . 5.13.1. There are only two old occurrences counted by XJ, and one of them is k J . Thus (5.14.2)

(5.14.3) Combining this with (5.13 .9), we must establish (5.14.4) Fig. 5.13.1 links J kJ=k'

with the choice of , (k 1 ) constrained by the requirement (5 .14.5) lEA Fig . 5 13.1 links J kJ=l.kl=f

In this case we choose (5.14.6)

5.14. HOW TO ASSIGN NUMERICAL FACTORS

505

because on the set of enhanced links we sup over, we want

-'{j(kd

= (1 + (max{Le,Ll})-II-;e

-

-;ll)-~+ (~!) ~ +

(5.14.7)

(5.14.5) now follows from the basic estimates

L

(1

+ (max{Ll,Ll})-ll-;e

- -;[1)-3+

l: L i =2- r

(5.14.8)

L

(2-rLil)"~C

(5.14.9)

r: 2- r $L ,

already discussed in §5.4. Does the extraction (5.14 .6) from w(J) leave enough smallness for (5.14.4) to hold? To answer this question, we insert (5.14.3) and (5.14.6) in (5.13.9) to obtain f"J

x

[g(l +

(m=(L,,,L',,W'1 C;" - C;",

-~ +

1+

x lw(J)ILkJ4 Lkl (1

+ (max{Lk"LkJ})-

1

1)'H1'

-I.

I

Xk , -

-~

XkJ

In the summation, we sum over all of the occurrences except kJ = k' over kl and k2' exploiting the estimates

3+

1)4 .(5. 14.10) First we sum

k, : L k ,=2- r

2- r 2': L kJ ,

L

(2r LkJ)" ~ c.

(5.14 .11) (5.14.12)

r: 2 - r~LkJ

Then we sum over k4 ' exploiting the estimates (5.14 .8) and (5.14.9) with the replacements f. H kJ, I H k 4 . Now, in this game, there are always enough long-distance decay factors in w(J) for every purpose; we need only to adjust the construction parameter of the wavelet to obtain enough vanishing moments for the Bessel potentials of the wavelets to have the degree of polynomial decay required. For this reason, we can focus on length scales only , with the understanding that when we sum over smaller-scale modes k relative to a fixed mode k', we "use up" the small factor

CHAPTER 5. WAVELET ANALYSIS OF
506

when we sum over larger-scale modes k relative to k', we "use up"

only. Recall that for this species of enhanced links we have the order of magnitude (5.13.13.1) for w(J). Hence (5.14.13) in this case. Summing over kl and k2 gives us the order of magnitude

O(L ~ -t: L - 4-t:) k4

kJ

(5.14.14)

'

which is an ample degree of smallness for summing over k 4 • The situation is a little tighter if k2 is an old occurrence as well - i.e., if the wavelet diagram is given by Fig. 5.14.1. In this case there are two assignments of numerical factors: 4-link (k}, ... , k 4 ) kJ = k3 3 old occurrences counted by XJ k I old

k 2 old

k) old

k 4 new

Figure 5.14.1:

(5.14.15)

(5.14.16) so we choose rJ(k.) = (1

+ (max{Lk" LkJ})-11-;k

1

-

-;k J

I)-~+ (~:~

r

3+

(5.14.17)

5.14. HOW TO ASSIGN NUMERICAL FACTORS

for

~

507

= 1,2. Thus

~J

=

x lw(J)I(1

1

-\

+ (max{Lk2' LkJ})- I Xk2

-\

3+

- XkJ I).

(5 .14.18)

so (5.13.13.1) now implies

~

O(L~- e L -¥ - e L- e L-'). k4 kJ k2 k.

=

J

(5.14.19)

Summing over k1 and k2 yields the order of magnitude O(L ~ -'L-¥-e) k4

kJ

'

which is still enough smallness for summation over k 4 . Consider the species of enhanced lines given by Fig . 5.13.2. This case deals with a mass link ((k1 , k2 , k3),(k',kJ)) with

Lk' ~ Lk. 2: LkJ 2: Lk2

(5.14.20)

and kJ as the only old occurrence counted by XJ . Thus

XJ(t)

1

IIt II

'"U (£,

t=1

~)

= 1,

(5.14.21)

(5.14.22) and so we have

~J

=

c>.~

(

II

Tl ) L;} - ' Lk3'

e

tEsupp J

x [_

~ II

(1

+ (max{LJ.:L;;;})-11 :l" - :l"k

1)3+,]2 IW(J)I .(5.14.23)

k . kEsupp J

By (5.13 .13.2) it follows that

c = 0(L 5- , L -1 L -~-, L -1 L -}) k3 k2 kJ k. k '

<,J

(5.14.24)

CHAPTER 5. WAVELET ANALYSIS OF.pj

508

and summing over kl and k' relative to kJ yields the order of magnitude

Now sum over k2 relative to a fixed k3 to obtain the order of magnitude

0(L4- " L-~- " ) k3

kJ

'

which is sufficient smallness for summing over k 3 . There are other species of mass links to consider, but recall that for Lk l ;::: Lk the expansion rules imply that k and k' cannot both be old occurrences. Now consider the case given by Fig. 5.13.3, which is a composite link ((k~ , .. . , k~) , (k 1 , ... , k 4 )) with k~ = k 4 , kJ = k2 preceding k 2, and k~ , k 1 , and k2 as old occurrences. Thus (5.14.25)

(5.14.26) so we choose (5.14.27) for

#

= "primed, unprimed ." In this case,

~J

=

c).~

~ II

2

x [_ (1 k,kEsupp J

+ (max{L;.:,Lk})-II;;;.:

x lw(J)I(l + (max{LkJ,Lk1})-11

- X

?+"]

kl

- Xk1I) t+

;;kJ

(5.14.28) so (5.13.13.3) implies 1

]

1

9

~J = O(Lt" L~3'i L~;'i L~22 L~} - " L"k1" L"k;") . Summing over kl and k~ rela tive to kJ yields the order of magnitude

0(L 5 -" L -! L -! L - ! L - ~ - " ) k4

ka

k;

k;

kJ

'

(5 .14.29)

5.14. HOW TO ASSIGN NUMERICAL FACTORS and subsequently summing over

k~, k~

509

and k3 relative to a fixed k4 yields

O(L~-"L-~-") k4 kJ '

which is more than enough smallness for summing over k 4 . Suppose we tighten the situation described by Fig. 5.13.3 with a species of enhanced links having the same wavelet diagram but involving more old occurrences. Consider Fig. 5.14.2 and note that kJ is no longer k 2 . Thus composite link ((k~, ... , k~) , (k 1 , . . . , k 4)) kJ = k3 k~ = k4 5 old occurrences counted by XJ k

old

k I old

k 2 old

k 2 old

I

k 3 new

k 3 old

k 4 new

-- -----

k 4= k 4

Figure 5.14.2:

(5 .14.30)

(5 .14.31) so we choose 3+

'"U(kt") = (1

In this case, ~J

+ (max{LkJ,Lk;,})-ll-;kJ

,;). (II ~ II

_ -;k;'

I)-~+ (~:;);;

(5 .14.32)

Te2) L~-"L~-"L~-"L!-"L-¥-"L-t : kJ

e

c/\ 4

k,

k;

k2

k"

k4

fEsupp J

X [ _

k , kEsupp J

(1 + (max{LiC , Lk})-ll -;iC -

-;k 1)3+t:] 2IW(J)1

CHAPTER 5. WAVELET ANALYSIS OF ~

510

x

ITo + (max{LkJ'

Lkt'} )-11

-; kJ

-

-; kt'

I)~ +,

(5.14.33)

#,L

so (5.13.13.3) now implies c = O(L 5-£ L - ~ L -

..,J

Summing over k1 ' k~, k2 , and nitude

k.

k~

k;

¥- -£ L k2 -£ L -,£ L -£ L -'£) . k2 k, k,

kJ

(5.14.34)

relative to kJ, we obviously obtain the order of mag-

and subsequent summation over k3 relative to a fixed k4 yields

O(L ~-£ L k.t

¥- - e)

kJ

'

which is more than enough for summing over k 4 . We now turn to the species of enhanced links given by Fig. 5.13.4. This is a rcnormalized link involving the mode k', k, kl. k2, k3 in scale-lexicographic order. k' is the only old occurrence and kJ = k'. Thus (5.14.21) and (5.14.22) hold, and so ~J is given by (5 .14.23) . Combining this with (5 .13.13.4) , we obtain (5.14.35) so if we sum over k, k1, and k2 relative to a fixed k3, we obtain the order of magnitude O(L~-£L-t-£) k3

kJ

'

which is enough for summing over k 3 . Consider the same wavelet diagram, where both k1 and indicated by Fig. 5.14.3. Thus

k~

are old occurrences, as

(5.14.36)

(5.14.37) so we choose (5 .14.38) In this case,

5.14. HOW TO ASSIGN NUMERICAL FACTORS

511

renormalized link {( (ki , k2, k3, k4 ) , (kl' . .. , k4 )), ((k2' k3 , k4 ), (ki, k 1 ))} kJ = kl k1 succeeds ki 2 old occurrences counted by XJ k I old

k I old k 2 new

r--r---

k 2 new

k 2= k 2 k J new k J= k J k4 new

k 4 new.

k 4= k 4

Figure 5.14.3:

x [_

2

~ II

k,kEsupp

x lw(J)I(1

(1

+ (max{L k,L k })-II?k

- ?k I)3+E]

J 1

~

+ (max{LkJ' Ld)- I XkJ

~

-

Xk'

3+

1)4 ,

(5.14.39)

so (5.13 .13.4) now implies ~ = O(L 6- EL -1L -1 L -~-E L -I-E). J

k3

k2

k.

kJ

k'

(5 .14.40)

Summing over k' relative to kJ , we obtain the order of magnitude

and subsequent summation over kl and k2 relative to a fixed k3 yields O(L4-EL-~-E) k3

kJ

'

which is a small enough magnitude for summing over k 3 . Now we consider the case of a combination link - specifically, the species given by Fig. 5.13.5. We have kJ = k~, k~ = k~, k!{ = k3 ' and k~ = k3 with kl' ki , and k~ as old occurrences. Thus (5.14.41)

(5.14.42)

CHAPTER 5. WAVELET ANALYSIS OF ~~

512 so we choose (5.14.27). In this case,

eJ

c>.~ ( II

Lt-' Lt;-' L~}-' L-;;: L-;;~'

eT ; )

lEsupp J

X [_=

II

(1

+ (max{L k ,L;))-II-;k - -;;;; 1)3+,]2

k , kEsupp J2

X [_

=

II

(1

+ (max{LkJ,Lk,})-II-;kJ - -;k,

1)3+,]2 IW (J)1

k,kEsupp J 1-'-

-.l.

3+

+ (max{Lk" L kJ })- Xk, - XkJ 1)4 1-'--" 3+ x(l + (max{Lk;,LkJ})- Xk; - XkJ 1)4 ,

x(l

1

(5 .14.43)

1

where J 1 and J2 are the 4-link and composite link comprising 1. By (5.13.13.5), it follows that (5.14.44)

Now, in general , we handle a combination link by summing first over the possibilities for the constituent link other than the one whose support contains k J . In this example it means summing over the possible J 1 = (k 1 , . .. , k 4 ) with k3 fixed. If we sum over kl and k2 relative to k 3 , we obtain

and subsequent summation over k4 yields

Since kJ succeeds k~ and precedes k~ in this case, our next step is to sum over k~ relative to kJ to get 0(L 1 - ' L 5 ; , L -,I L -} L -~- ' ). k3

k4

k3

Now since k3 = k 2 , we now sum over k~ , k3 obtain the order of magnitude

This is all we need in this case.

k2

= k3'

kJ

and k2 relative to k~

= k~

to finally

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