Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (Second Edition)

.

(2.17)

The integrability condition of the above equation is obtained by contracting it with either 9M or with elipdp, yielding

2.3 Fermionic Free Fields

29

Therefore ip can be defined by (2.17) only in the massless case. For the two-point function involving (p one obtains

[
(2.18)

and { f (+)f(x),

,

(2.19)

where2

£(+)(,)

l,

n

^_±f

,

(2.20)

£)(+)(a;) = - — In [(i,j,x+ +e)(ifj,x~ + e)]

.

(2.21)

The two-dimensional Lorentz group is Abelian, and takes the simplest form in the light cone variables: f * -> £*' = e±x^ (2.22) with

z± = e±e ,

(2.23)

where x is related to the velocity v by sinh \ = j / 2 - Correspondingly, the Lorentz transformations act on the functions £>(+'(:r) and D^+\x)

as

^ + ) ( r ) = i?(+)(o £ ( + ) ( £ ' ) = £(+)(£) + A .

•

(2.24)

Z7T

As we next show, free Dirac fields may be represented in terms of the exponentials of ip and (p.

2.3

Fermionic Free Fields

The massive free spin-| field obeys the Dirac equation (i @ - m)tp = 0 , which can be derived from the Lagrangian C=^{ip-m)ip .

(2.25)

The Fourier decomposition of the fermionic field is given by

/

dk1 -^={b(k)u(k)e-ik*+J(k)v(k)eik*}

,

with

{bHk),b(p)} = {dt(fc),d(P)} = sik1 -pl) 2

We use In (x + ie) — In \x\ + m9(—x).

.

(2.26)

30

Free Fields

In the Weyl-representation of the 7M-matrices, * = ( ' ; ; )

7i=(_°1

;'j

«(*) = ( ;'k-/ £ )) ', v(k) =\

-Vk

,

(2.27)

the Dirac-spinors take the form v J

fVk+

They have the usual property uaup =fcM7M+ m

,

VaVp = k^j^ - m .

In the zero-mass limit we have U(*)

= V^(^J))

, «(*) = v ^ ( _ J ^ 1 ) )

.

(2.28)

In the massless case, due to the separation of right- and left-movers, we can rewrite (2.26) in terms of the spinor u(k) alone, by defining an operator d{k1) as d(k1) = d(k1) for A;1 > 0 , dik1) =-dik1)

ioi k1 < 0 .

Expression (2.26) then takes the form 4>{x) = [ -^tL=u(k)[b(k)e-ik* J v27r2A;u

+${k)eikx]

.

(2.29)

The canonical anticommutator for the ip(x) fields (2.26) reads, for arbitrary times, {^a(x),i/jp(y)}=iS(x-y)ap , where S{x-y) = (i0 + m)A(x-y)

,

and A(x — y) is given by (2.5). Analogous to (2.4)-(2.6) we also define the two-point functions (0\M^p(y)\0)=(i^

+ m)A^(x-y)=S^(x-y)

,

(O\T-ipa{x)^0(y) | 0) = -i{i $> + m)AF{x -y) = iSF(x - y)

.

These functions are well defined in the zero-mass limit. 3 For m = 0 one has SM(x) = --(_J_ 3

0

In the zero-mass limit, we have D(x) = -\6(x2)e(x°)

l

*--" )

•

and (2.8) for D<+>(x)

(2.30)

2.4 Bosonization of Massless Fermions

31

There exist two conserved local currents: with the usual charge conservation is associated the conserved vector current j^x)

= :^(x)j^(x):

,

(2.31)

where the double points indicate normal ordering; the chiral symmetry of the Lagrangian (2.25) in the zero-mass limit implies in turn the conservation of the axial vector current jl{x)

= :^(Z)7M75(Z):= W

,

(2-32)

where we used the relation (see Appendix A) 7^,75 = eM„7". Conservation of charge and pseudo-charge thus implies V

=0 =e " " ^

,

or d±u=0

.

(2.33)

Majorana representation Another useful representation for the gamma matrices is given by 70 =

0 I i

- A 0) •

71

(0 i "V* 0

In terms of this representation, the Dirac equation leads to d+ip2 - mipi = 0 , d-ipi + mip2 = 0 • The spinors u and v defined in (2.26) are now given by

2.4

Bosonization of Massless Fermions

In two dimensions we may always write a vector field in the form U = d»
,

(2.34)

which corresponds to its decomposition into a curi free and a divergence free part. It follows from (2.33) that (p and 9 are massless fields \j

= —^/w(

j„ = - 4 = M V71"

•

(2-35)

32

Free Fields

One may easily convince oneself by a direct computation, that in (2.31) j M is of the form (2.34), with <j> a zero-mass field. Indeed, using (2.29), one has 27r^_5(A;)7<
+b\k)d>(p)ei(-k+p^

+ d{k)b(p)e-^k+^x

- d}{p)d(k)e-^k~^x}.

(2.36)

Now, from (2.28) we have 1 IkP k k ^ y ^ « ( * h M « ( p ) = ^(^1)
.

(2.37)

We may thus write j1* in the form (2.35) with
cik1) = -=1 ^ W W C P W

+ k1) - J V ^ V

+(?((fc 1 -p 1 )p 1 )d(A 1 -p 1 )6(p 1 )}

i c+(fc) =

+ k1)}

,

-i= y v w ^ ^ v + * w > - <*v+fc1)^1)] +^((fc 1 -p 1 )p 1 )6 t (p 1 )J t (fc 1 -p 1 )}

.

We can now compute the commutator of c and c^, in order to verify that <j>{x) is a canonical field; the g-number terms are found to cancel upon using the support properties of the ^-function, and one finds [c{k),c*(k')] = 6{k-k')

.

Therefore the c's are the usual annihilation operators they commute among themselves, as we readily verify. The vacuum is defined by c(fc1)|0) = 0

.

In turn, the bosonic massless field can be used to build objects having fermionic character. Motivated by the above construction of a canonical free bosonic field out of free fermions, we show now, conversely, how to construct free zero-mass fermions out of bosons. The fact that a fermionic field can be written in terms of bosonic fields has been discovered long ago by Jordan and Wigner [9]. Indeed, define the usual bosonic Fock space by \k) =

ot(fc)|0>

,

with the canonical relations [a{k),<J(k')]=5(k1-k1')

,

2.4 Bosonization of Massless Fermions and [a(k),a(k')] = [at(k),cJ(k')]

33

= 0. One then verifies that the operator

dt(fc) = at(fc)e _ i , r /»~ , l * l o t ( f c l ) a ( f c l )

(2.39)

satisfies the anticommutation relations {d(k),cfl{k')}

= 5(k-k')

.

We have thus arrived at a bosonic representation for the fermionic operator d(k). The above construction shows that statistics in two-dimensional space-time is a matter of convention. Indeed, we may change the statistics by simply replacing 7r in the exponent of (2.39) by an arbitrary parameter A. This is closely related to the fact, that the spin of a field is here also a matter of convention, since there exists no rotation in one space dimension. As we next show, it may be defined in terms of the Lorentz transformation properties of the fields [2], and need not be an integer or half-integer. Thus there exist different solutions of the Thirring model which differ in their "Lorentz-spin"; they correspond to different representations of the conformal group in two dimensions [10, 11]. To be more precise, consider the generalized free field

where Xa,p(x) =a
"" (

7,i

-)

'•"

7 5

'

'

(2.40) (2.41)

(-x))-{a5+0-f)(D(-'>(x)-D(-l{-x))

Using £>(+)(x) -£><+>(- x ) =-l-e{x2)e{x°)

,

D^(x)~D^(-x)=l-e(-x2)e(x1)

,

(2.42)

expression (2.40) reduces to ^°,P) {x)$M) (y) = ^,(7,*) (y)^"*)

(X) X

(2.43)

o o 2 1 1 2 e - l ( a 7 + 0 , 5 ) e ( a ! - i , ) e ( ( a : - i / ) ) + i(a l !+^7) e (x -!/ )(9(-( a :-5 / ) )

From here, we see that for space like distances, the fields ^"'^(x) j i) commute, \ ii) anticommute,

if (3j + a5 = Anir , if @j + a5 = 2(2n + l)7r

and V^'^G/) .

\ • )

34

Free Fields

Otherwise they neither commute nor anticommute. The case of a field that anticommutes with itself, where a/3 = (2n + 1)TT, will be of particular interest for us. An arbitrary correlator involving ip(a'P) (#) may be computed by normal ordering with respect to (+) and using the property <^+) |0) = 0; we have

where F(x1---xn,y1---yn)

=

(2.45)

-a 2 J ]T (DM(Xi - Xj) + Dl+Hvi ~ Vj)) - X><+>fo - Vj) I -/32 i Y, (D(+){xi ~ Xj) + D^\yt - yjj) -YD^Hx, ^ i<3

- Vj) I

i,j

)

+)

-2a/3 J £

(& (xi ~ *>) + D^Hvi - Vj)) ~ £ ^

\ i<3

(+)

( ^ - Vj) \ •

i,3

J

Under Lorentz transformations, D^(x) is invariant, while D^+\x) as in (2.24). Thus we have under Lorentz transformations F{Ax,Ay)=F{x,y)+n—X

transforms

• IT

From here, we conclude that the field ip^a'^(x)

transforms as

U(A)i>{a^)(x)U-1(A)=e^x^{a^{Ax)

.

(2.46)

We define the Lorentz spin s of xp in terms its Lorentz transformation property, U(A)iP(x)U-1(A)=es^i{>(Ax)

.

(2.47)

Thus the Lorentz-spin of the field ip(a'P) is given by a/3 If a/3 = (2n + l)n , s is a half integer and tp^01^ anticommutes with itself; if a/3 = 2WK , s is integer and ip^'PI commutes with itself (see (2.44)). For massless fields, the "spin" is an arbitrary parameter characterizing the realization of the conformal group in terms of the fields. This is the basis of a bosonfermion duality which will be discussed at length later on. Indeed, under dilatations D^ is invariant, while D^ transforms as D^{exx)

= D+\x)-^-

2TT

.

2.5 The RS-Model

35

Therefore F(exx,exy) = F(x,y)-n(a2+p2)±

.

The corresponding transformation law for ip reads

Hence, the scale dimension of ip(a'@> is Aa,0 =

a2+(32 47T

The case a = p = i y ^ corresponds to a canonical spin - | field. For a spinor tp{x) satisfying the massless Dirac equation, we have in the representation defined by (2.27), tpi(x) = 1 /'i( a;_ )> ^(x) = ip2(x+), so that the corresponding Xa/3 only depends on x~ and x+ respectively. Hence, the requirement that the Dirac equation be satisfied implies

a^° + dx*)(ai(^+

^

= (ai

" / ? l ) ( 5 ° + 9l) ^ = ° '

that is, c*i = Pi for the upper component ipi, and

(a2 + /32)(a 0 -a 1 )^ = o , that is, a2 = — @2 for the lower component tp2- Summarizing, we have, for a canonical, spin | fermion satisfying the Dirac equation, x) =

^

JT ( : eiV*{'?>+
yt(':e ^^-'Phl

(2 48)

'

'

with the corresponding current j M (x)

2.5

=:^7MV:=

^<9M<^ V71"

.

(2.49)

The RS-Model

As an application of the ideas developed here, we consider the Lagrangian [12] C = t^7"0„V + \d"^d^

- im2$2 +g ^ l s ^ *

,

with the equations of motion i 0V + S7"75V'^* = °

( 2 - 5 °)

»

(D + m 2 ) $ = (7dM(^757MV)

•

(2-51)

At the classical level, the first equation has the solution iP(x)=ei9^*W^°\x)

,

(2.52)

36

Free Fields

where tp^(x) satisfies the free Dirac equation; the axial vector current is conserved, and $(x) satisfies the free massive Klein-Gordon equation. At the quantum level, Eq. (2.52) is not well defined, and must be normal ordered, ^a(x)=ei9^^~){x)tp{°){x)ei9<^+)^ . (2.53) Since as we show below, $ continues to be a free field, its decomposition into positive and negative frequency parts is valid at all times. With Eq. (2.53) we are led to replace (2.50) by the equation of motion

with 7V[a M $^]=cl M $ ( - ) V' + V ' ^ $ ( + )

-

(2.54)

The above normal product may be rewritten as the symmetric limit 4 Nid^iP)

= lim l{d^(x

+ e)^(i) + 3^{x - e)^{x)}

.

(2.55)

We now compute this limit using equation (2.53). Consider the vector current as defined by the short distance limit 5 J£{x) = lim /(e) h{x + e)1^(x)e-ig

l'+'dre^d"ny)

- v.e.v.]

where v.e.v. stands for "vacuum expectation value" and /(e) is a suitably chosen renormalization constant. Using equation (2.53), expanding the exponential up to first order in e, and using (2.11) in order to move e 875 * ^x+^ to the right of e 275 * (x\ we obtain the following expression J M (z) = lim/(e){ ^ ( 0 ) ( a ; + e )e i 975* ( - ) (^)-is75* ( - ) (x) > e -i 9 75* ( + ) (^>^s75* ( + , (^) e2<0

x (0) (z)e- 92A(+)(c) (l - ige^d^x))

} - v.e.v.

.

(2.56)

Expanding the exponentials and defining the free vector current

jli{x)=:^°\x)1^\x):

,

equation (2.56) takes the form

J,(x)=e-92AW^xL(x) 0)

+ 0)

0 V4 . „ (z + e)v4 (a0 o W [l-ig^efdpQ-ige^ds*]

\

-v.e.v.

> aP 4

Bqs. (2.54) and (2.55) differ by a term of order 0 ( e 2 ) [3„*< + >(z + e), i>(x)} + [d„*(+Hx = *(x)-}-£-K0(my/^») 2ir oe^

5

- e),-4:(x)} - ^ ( . j l ^ - i f o i m v ^ ) = 0(e2) 2.-K Ce^

T h e model exhibits a local invariance at the classical level.

.

37

2.5 The RS-Model Recalling that < A z + e) a <M*) 0 ) = - S < + > ( - e ) =

27TC 2

renormalizing the fermions as xKx) = zi(e)j,R(x)

,

^ ( e ) = e-» 2A<+) W

,

(2.57)

and using tr tT-l-^e ae^d^ 2TTC

as well as (eM„epe" — tpveve^) expression

= e2e£

--e^41 d0$

=

, we obtain, for the renormalized current, the

The corresponding result for the axial vector current reads = jg{x) + ^<9M$

N^^ip]

.

7T

Hence, the divergence of the axial current is anomalous, 7T

and we are led, on account of (2.51), to the field equation „2

(1

- ) • +rn2 $ = 0

We notice that the theory breaks down for strong coupling, i.e. when g > y/ir, due to the appearance of a tachyon. Since there exists a non-trivial fermion wave function renormalization constant, the fermion has an anomalous dimension, which can be computed from (2.57), or else be read off the short distance behavior of the fermion two point function. We have ( 0 1 ^ ( ^ ) ^ ( 0 ) 1 0 ) = ( o | : ( e ^ 5 * < * > ) Q 7 : : {e^*^)sp:

_ -*(i«97*.,- 9 a 7;,7,%A ( 0 >(s) 2TTX2

=

|o) (o|V>f ( s ) $ 0 ) ( 0 ) I o ) ,

J_4e32A^) 2iri x2

For the short distance behavior we find SF(X)

1 27Ti(x2)fc + l

so that the fermion field scales with an anomalous dimension given by 92 Since the coupling constant is not renormalized, the (3 function vanishes identically.

BIBLIOGRAPHY

38

2.6

Conclusions

This study of free fields in two-dimensional space time, reveals unexpected structures, besides providing an important basis for formulating a quantum field theory. Although the main applications of the techniques developed here will be subject of the next chapters, such as the complete solution of the Thirring model, the bosonization technique, and others to come through this book, it is important to emphasize some of the points stated in this chapter. The first concerns the relation between the long range behavior of the bosonic fields and the existence of conserved charges in the fermionic theory. We have demonstrated how to construct conserved charges in a theory containing only real bosonic fields, provided the bosonic theory is massless. We will see that the essential point is actually not the masslessness, but the non-trivial behavior at the spatial infinity. In Chapter 3 we shall demonstrate the equivalence of the massive sine-Gordon field and the massive Thirring model: the essential point will be that the topological properties of the bosonic theory are responsible for the existence of charge sectors of the fermionic model. The free field theory of bosons is not only important in the above context, but it will also provide a non-trivial realization of the conformal algebra, as we shall see in Chapter 16. There we will see how to construct general correlation functions depending on a character defining the realization of the conformal group. The construction depends only on formula (2.15), modified to include charges at infinity. As we mentioned in the introduction, we are able, using the techniques developed in this chapter, to construct the exponential fields, which are the basis of all simple soluble models in two-dimensional space time. This is a characteristic of two-dimensional space-time, where Wick polynomials of a free massive field are well defined [13]; moreover, while an infinite power series of a field does not define, in higher dimensions, a field in the usual sense [14], this is possible in two dimensions, where entire functions of the elementary fields are well defined objects [1]. The bosonization technique makes use of this fact.

Bibliography [1] A.S. Wightman, in Cargese Lectures of Theoretical Physics , ed. Gordon and Breach, New York, 1964. [2] B. Klaiber, in Lectures in Theoretical Physics, Boulder 1967, Gordon and Breach, New York, 1968. [3] B. Schroer, Fortschritte der Physik 11 (1963) 1. [4] J.A. Swieca, Fortschritte der Physik 25 (1977) 303. [5] S. Coleman, Commun. Math. Phys. 31 (1973) 259; N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [6] L. Schwarz, Theorie des Distribuitions, Paris, Hermann, 1951; I.M. Gel'fand and N.YD. Vilenkin, Generalized Functions, N.Y., Academic Press. 1964.

BIBLIOGRAPHY

39

[7] L. Garding and J.L. Lions, Nuovo Cimento (supp.) 14 (1959) 45. [8] A. Jaffe, Ann. Phys. 32 (1965) 127. [9] P. Jordan and E.P. Wigner, Zeitsch. fur Phys. 47 (1928) 631. [10] B. Schroer, J.A. Swieca and A.H. Volkel, Phys. Rev. D l l (1975) 1509; J. Kupsch, W. Riihl and B.C. Yunn, Ann. of Phys. (NY) 89 (1975) 115. [11] Vl.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240 (1984) 312; B251 (1985) 691. [12] K.D. Rothe and I.O. Stamatescu, Ann. of Phys. Banerjee, Z. Phys. C25 (1984) 251.

(NY) 95 (1975) 202; R.

[13] L. Garding and A.S. Wightman, Arkiv Fysik 28 (1964) 129. [14] H. Epstein, Nuovo Cimento 27 (1963) 886.

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Chapter 3

The Thirring Model 3.1

Introduction

The theory of a massless Dirac field with a quartic-self interaction was first introduced by Thirring [1]. Up to Fierz rearrangements, the quartic term is unique, and may be described by a current-current interaction. The theory is exactly soluble, thus allowing an explicit computation of matrix elements, as well as the construction of the eigenstates of the Hamiltonian [1]. Glaser [2] solved the field equations in terms of the free massless Dirac field, but the manipulations were rather formal. Johnson [3] reconsidered the theory, taking due care with the definition of field operator products, and obtained the current in terms of the renormalized operators. By solving the system of equations for the time-ordered functions, he obtained the two- and four-point Green's functions. The operator solution was studied later [4], and the computation of the n-point function has also been performed [5]. The complete solution of the theory, as well as the study of properties such as Lorentz transformations, infrared behaviour, cluster decomposition, and the relation with other solutions in the literature were obtained in the classic paper by Klaiber [5]. In this chapter we construct the complete operator solution, as well as the npoint functions, and discuss some properties of the model. We also discuss the massive theory, introducing the mass perturbatively. The perturbation theory thus obtained is shown to be equivalent to a corresponding expansion in the sine-Gordon theory [6, 7]. This equivalence will first be shown for the zero charge sector and then extended to arbitrary charge sectors by generalizing the bosonization formulae of Chapter 2 to the interacting case. Both, the massive Thirring and sine-Gordon models are shown to possess an infinite number of conservation laws, which survive quantization. At the classical level these conservation laws imply integrability by the inverse scattering method; at the quantum level they allow one to obtain the exact S-matrix of the theory (see Chapter 8) as well as some of the form factors.

42

The Thirring Model

3.2

The Massless Thirring Model

The Lagrangian defining the Thirring model is that of a massless Dirac fermion with a (quartic) current-current interaction [1], £ = ~$i 0V + ^ Y ' H l ^

•

(3.1)

The equation of motion derived from (3.1) reads

nMdMv + gfi^ = o ,

(3.2)

j»(x)=4>(x)^i,(x)

(3.3)

where .

The equations of motion (3.2) implies the conservation laws

V = 0 = e""^,, Hence, at the classical level, the current (3.3) may be written as the gradient of a massless scalar field
In terms of

,

(3.4)

where ip^ (x) is a solution of the free Dirac equation. The solution (3.4) is not unique: given a solution ip of Eq. (3.2), we can construct the one-parameter family of solutions _ 1>{x) = e^^+fo^^ix) , (3.5) with ip the dual of ip (see Eq. (2.17)), where classically we have the restriction a — (3 = —-/=• If we compute the current with the expression (3.5), we verify that classically the current (3.3) is equal to the free current j£ = ip (x)^/ll'4)^(x), and the solution ip(x) of the Thirring equation (3.2) can be written entirely in terms of the massless field ip^(x), and the potential tp. Going over to the quantum case, and choosing tp^ (x) to be canonical, we may use Eq. (2.48), in order to rewrite (3.5) in the form tP(x) = .rK:ei{<*v(x)+PiMx)}.

f]\

)

(3. 6 )

where a = a + ^/n , j3 = /3 + -JTT, and /J, is the IR regulator mass associated with ip. The fermion-number and chirality of ip are contained in the superselection rules implied by the zero mass fields ip and
43

3.2 The Massless Thirring Model

Some remarks are in order at this point. The above construction [8] differs from the original solution of Klaiber [5]: the superselection rules are implicit in (3.6), in such a way as to cancel the IR mass parameter fi. The computations are performed using definition (2.13) and the super selection rule (2.16). Klaiber chose to define infrared finite fields at intermediate steps, by working with non-Lorentzcovariant operators involving an infrared cutoff; this lack of Lorentz invariance could be compensated by adding (non-Lorentz and non-translation invariant) terms involving the charge and chiral charge operators in the exponent, in expressions such as (3.6). Lorentz invariance was only recovered at the end of the computation. The correlators of the fields can be written as (0\rP(x1)---MxnMy1)---Myn)\0)=(±)neF^y>)

,

(3.7)

with (see Chapter 2) F(x,y) = - £ { ( a + b1ljllk)D^\xj

- xk) + A( 7 |,. + llJ^+Hxj

- xk)}

j
- £ { ( a + < 7 2 J 0 ( + ) ( W - Vk) + Hl5Vj + llk)D{+\yj

- Vk)}

j
- £ { ( - a + bj5Xjl5yk)D^(Xj

- xk) + A(- 7 *. + ll)D^{Xj

- xk)}

j,k

and a = a2 , b = f32 , A = a/3. The index in 7 s indicates where it belongs to with respect to the Lorentz indices from the left hand side. Using (2.20) and (2.21) we obtain the more explicit form

= x

11(4 ~xt- ie)Sl(yt - vt ~ i£)Sl 11(4 - vt - ^)~61 j
x

( ^ ( I ^ + ^ V - K ^ - D

j,k x

U^J

ie d2{

- ~k - ) yJ

82 x

- y^ - ^)~&2

- y~k - ^) ll J

j
.

(3-8)

j,k

where 5X = a+\~2X , 62 = a+\+2X • The corresponding correlators of xjj2 are obtained by changing the sign of A and interchanging 5\ and <52, while for mixed correlators we have

(o 1 ni M*J) nr+i M^) ni ^r(%) nr+i r^) i o> n j=l

ft

{x-f-x+-it)s(yf-yt-ie)s{xj-x^-ie)8(yr-y--ie)e

k=l+l

= -t

—-

x

n fibj-Vk-w&t-vt-wfi xfej-vn-w&t-vi-w

3 = 1 k=l+l I

<(0 I YlMxjWiiVj) j=l

1= 1

k=l

n

I 0)(0 I H k=l+l

MxkMiVk)

I 0)

,

(3.9)

44

The Thirring model

where 5 — Sj£-. From the above we see that the Thirring model satisfies the cluster decomposition property for non-vanishing a and 0.1 In particular one has, for the two-point function,

(0 | (a#(0) | 0) = £ e - ^

A

/

rv

/

0

I

a+i-2A

\[M(Z~-«0]

a+b+2A

\p(x ,

-ie)]

a+

[M(X+-«)]

4-

a+b+2A

„

4TT—

0

^—[fi(x+-ie)}

b-2\

4,

The corresponding Fourier transform reads

o+t-2A « T ° — ^-^

( £ ) T (£)' i

a-f-64-2 A

1

Note that the above two-point function does not have any pole, so that ip{x) does not create asymptotic particle states. Using formula (3.6), it is possible to arrive at very simple expressions for bilinears involving the Thirring field. The first of them is the current operator j^{x)

= ii{x)lliij{x)

.

(3.10)

We treat the simplest case of a canonical spin | fermion, for which /?=a

.

(3.11)

Denning J±(x;e) = ^ t (a; + e)(l=F75)V'(a;) = ^ l (x + e)ip2 (x), and using Eqs. (3.6), 1 i

one calculates J±(X;e)

= !±.e-ia('P(x+e)->p(x))±il3((x)){(L_\

2

" (_^2£2y

2-±?-

« ^){-li2e2)-2^1{^)1/\l-iae^d^±i^d^)

.

Recalling (3.11) we define the current ^{x), following Mandelstam [9], as the limit " ^ -

l i m i ^ ) f c + ^){J"(.;e)-(0|J"(^;e)|0)}

,

(3.12)

1

For if = - 1 (space-like) the cluster decomposition property requires

lim (u+pj)) • • • i>(xp+pn)l>(yi.+PV)

• • •?(A(y») |o>

/J—*00

= <0 I V>(xi) • • • V ( z P M i / i ) • • • ? ( » , ) I 0><0 | t/>(z»fi) ' • • i M ^ M y ^ i ) • • • MVr.) I 0).

45

3.3 The Massive Thirring Model where

2

Z(c) is the renormalization constant

Z(e) = 2(-^r-^^

•

One finds jn(x) = --e^d^ipix)

.

(3.13)

7T

Returning to the equation of motion (3.2), substituting for the Dirac field and the current expressions (3.6) and (3.13), and using (3.11), we see that /? is fixed as a function of the coupling constant,

01 7T

1 1 - 4

(3.14)

'

7V

Other bilinears can also be computed. Prom (3.6) we have iPl(x + e)iP2(x) = ^(-H2e2)-^:e-2iMx):

.

(3.15)

We therefore arrive at the identifications N$il>]{x) = J±-:cos 2/3
(3.16)

Z7T

N\frysrl;]{x) = ^:Bm2p
.

(3.17)

Finally, making the Ansatz

W ^ = ^ ( M

2

,

(3.18)

substitution of (3.13) and (3.18) into the Thirring Lagrangian (3.1) reduces it to that of a canonically normalized free zero-mass scalar field, as required by the above considerations. 3

3.3 3.3.1

The Massive Thirring Model Equivalence with sine-Gordon equation

The massive Thirring model is defined by the Lagrangian £ = iip^dfti/j - rrnpip + -g^ilnp-y^ip 2

,

(3.19)

Notice that in general, Lorentz invariance is spoiled in intermediate steps. Nevertheless, it is recovered in the end. The normalization of the current is chosen in such a way that the fermion carries one unit of negative charge. 3 This result can also be derived as follows: Prom the equivalence relation (2.48) between a free, zero mass Dirac and scalar field we conclude that ipi pip = ^(d^tp)2. Evoking Coleman's adiabatic principle, the current-current interaction in the Lagrangian (3.1) amounts to the addition of a term - ^ r ( d M v > ) 2 , where (2.49) has been used. Hence the resulting Thirring-Lagrangian takes the form C = ^(d/iip)2. Normalizing p t o a canonical field, yields (3.13) and (3.18).

46

The Thirring model

which is nothing but the massive version of Lagrangian (3.1). The most straightforward way of studying the above theory, is by means of a perturbation expansion in the mass. This requires one to compute the correlator of an arbitrary number of mass insertions in the Thirring model, that is, (—im)n — — ± r1-{Q\W{x1)---^{xn)

| 0)Th = G(Xl • • • xn)

Til

.

Using equations (3.16) and (3.17), the above correlator can be exactly computed in terms of exponentials of free fields. Defining J3 = 2/3 one has 4 G{Xl---xn)K± where :

( — im)n - i - ( 0 | : c o s Ptp{x1):---:coa n!

pip{xn):\0)

,

(3.20)

: denotes normal ordering with respect to the field
1 I-*

(3.21)

The value p1 — 4TT thus corresponds to a free massive fermion (g = 0). Eq. (3.20) suggests the equivalence of the fermionic Lagrangian (3.19) with the bosonic Lagrangian 1 £ =-(d^)2

+ ^(cos

p

.

(3.22)

We have thereby identified the fermionic massive Thirring model, with the sineGordon theory defined by the Lagrangian (3.22). Recalling results of the previous chapter, we know that exponentials of massless fields naturally define a superselection rule. The exponential is ab initio IR-cutoff dependent, that is, it will depend on the parameter fj, introduced in the previous chapter. A computation of the two-point function shows that

JV m [e-^]=(-4) S -JV M e-^ , where Nm(N^) particular,

stands for normal ordering with respect to the mass m(/i) [7]. In Nmcos

p
.

Consider the contribution of the above term to the Hamiltonian. For the free Hamiltonian we have ^o = ^ 2 + - ( ^ ) 2 . Using the usual expansion in terms of the creation and annihilation operators, one obtains <#om(M)>=/ 4

dk 2fc 2 +m 2 (/i 2 ) Sir y/ik1)2 +m2(fj,2)

In the usual treatment of the sine-Gordon equation the parameter /3 replaces our f$ = 2/3.

47

3.3 The Massive Thirring Model Hence, NmH0 = N^Ho + (HZ) - (tf0m)

= NtiH0 + ^

2

- m

2

)

•

(3.23)

Adding the mass term in (3.22), Eq. (3.23) is replaced by Ho — a— cos /3

/32

v

mJ/

07T

This expression is unbounded from below as a function of p? , if /?2 > 87r. Thus, the sine-Gordon theory is not well defined in this region. We will come back to the origin of this problem later in this chapter, showing that the theory is superrenormalizable for 0 < P2 < 47T, renormalizable for 47r < /32 < &n, and non-renormalizable for P2 > STT. For the moment, we discuss some general properties of both theories. We shall come back to the equivalence of bosonic and fermionic theories (Eqs. (3.16), (3.17) and (3.18)) when we explore Mandelstam's formula, which is nonperturbative.

3.3.2

Classical conservation laws

The massive Thirring model possesses an infinite number of conservation laws [10]. In order to demonstrate this statement, we consider the field equation in light-cone variables id+ip! =mip2 - 2gip1if}ltp2 ,

(3.24)

id-^2 = rntpi - 2gi/j2ipl'>pi ,

(3.25)

and consider an auxiliary Grassmann-valued field x{x'i e)> depending on a parameter e and obeying the following pair of equations: ed-X - X - 2iepxV'iV'i - teegip^X + i>i = 0 2

d+X + m ex - ligx^i

i

- 4egi/j2x 'x + imfo = 0

, .

(3.26) (3.27)

These equations are compatible for arbitrary e, if one takes account of the fermionic character of x a n d V^ this can be seen using the field equations (3.24) and (3.25) above. If we expand x m powers of e, we see that x(0,z) = ipi{x), which obeys both equations above, for e = 0; the current JM(a;,e) with light cone components given by J-(x;e)-iip\x-ix^i J+{x;e)=em(iplx

, i

+ X '
is conserved, d+J-(x;e)

+ d-J+{x;e)

=0

.

(3.28)

48

The Thirring model

Equation (3.28) implies the existence of an infinite number of conserved currents J£(x) denned in terms of the expansion Jll(x;e) = YienJJi(x)

.

The first few of them are easily calculated considering the expansion

X(e) = 5 > " x ( n )

•

(3.29)

Upon substituting (3.29) into (3.26) and (3.27), one finds for them X(0)=^i (1)

X =cLVi X

(2)

x(

3

,

,

=0^i-2i<^IVi3_ih

) = dl^

- Gig^d-tld-ilt!

, - 4t(7VllM-lh

,

which in turn imply that the first non-trivial currents are given by ji 1 ) =i(^ 1 t d_Vi-d-i t
,

./i 2) = 1 ( ^ 0 - ^ 1 - 0 ^ 1 ^ )

.

2)

ji = m(^a_^i + a _ ^ ) , J!3) = wldifr - ai^I^i) +12^^10-^0-^1 , j | 3 ) = mtyldliln

3.3.3

+ d2_^2)

+ 2igil>lrl>(il>td-tl>1 -

ft-Vllfe)

•

Q u a n t u m conservation laws

In the quantum theory, the conservation laws (3.28) may present anomalous terms. Since the conservation laws above follow from the field equations (3.24) and (3.25), we have to consider the quantum equation of motion. We may use the BPHZ renormalization scheme [11], in which case the equations of motion inside composite operators may present the so called anisotropics [11]. One departs, in e.g. the BPHZ scheme, from the renormalized Lagrangian £ = (1 + b)-t/j%/j - (m - a)ipip + -{g - ^ip^ipip^tp

,

where a, b, c are finite renormalization constants defined at some renormalization point. Making use of the corresponding equations of motion inside the normal ordered product of operators, one has, in the BPHZ scheme (0 | TNd+i[P(i>)i

^\{x)X({Xi})

| 0) =

= (0 I TNd+,[{Pm^rP}(x)X({Xi})

| 0)

49

3.3 The Massive Thirring Model

- ( 0 | TNdH[P(V0^7^V'7'V](z)*({^})

I 0}

+i(0\TN[P(m^7^r-X({xi})\0) ,

(3.30)

where P(ip) is an arbitrary local operator depending on the field ip and its derivatives, while X({xi}) corresponds to an arbitrary product of external fields at separated points Xi. The normal product including curly brackets, N[{P(ip)}], denotes the so-called anisotropic normal product, which following the BPHZ renormalization scheme, generally involves extra subtractions (see Refs. [11] for details). The Zimmermann identity relates the anisotropic normal product to the usual one (subtracted according to the general rules of the BPHZ scheme) N[{PW>)}1>] = N[P(4>)} + Y, OiN[PiM]

,

where the aj's are defined in Nd[P] = Nd+1 [P] + Y, aiNd+1 [Pi] , and where Pi(ip) are operators with one higher dimension as compared to P{ip)One must show that the anisotropies arising in the current conservation equations can all be expressed in terms of total divergences. This allows one to redefine the currents such as to arrive at a new infinite set of consevations laws. An explicit computation has been performed for the lowest non-trivial current [12]. Similar problems are encountered in the quantisation of the sine-Gordon theory [13, 14, 15]. Let us consider the Lagrangian (3.22). In analogy to (3.26) and (3.27), we define at the classical level an auxiliary field ip(x;'y): = f^(x) satisfying the coupled set of equations: d+(ip7 + lp) = 2^sinP^E ^

7

,

- ^ ) = -27^sin/?^_±^

(3.31)

.

(3.32)

These equations are compatible as one readily verifies by comparing the result for d+d-ipj as obtained from (3.31) and (3.32), respectively. They imply the following conservation law: 7

a

+

{cos/3^}

+ 7

-

1

a_{cos/

3

^}=0

,

(3.33)

as can be seen using (3.31) and (3.32). We can now expand {x-1) = Yv(n\x)1n

.

(3.34)

The fields
The Thirring model

50 first three terms
,


= -%dl
,

(3.35)

.

Substituting (3.34) back into (3.33) and equating to zero the coefficient of each power of 7, we obtain an infinite set of conservation laws. The first few of them may be obtained from the explicit expressions (3.35). At the quantum level the conservation laws again suffer from the problem of anomalies. This has been analysed at length using the BPHZ renormalisation scheme; nevertheless an infinite number of conservation laws are again obtained. The problem of anisotropies may, in some cases, be avoided by modifying the renormalization scheme, including subtractions in the mass, in addition to those in the external momenta [16]. In the case of the massive Thirring and sine-Gordon models, one could make soft subtractions in the mass around a given (arbitrary) renormalization point m = /j, (non-zero, in order to avoid spurious infrared divergences). Similarly to the case of QED$, where the anisotropies disappear [16], the problem is also resolved in this case, and conservation laws depending only on field equations are immediately quantized, preserving the classical symmetries. 5 As we shall see, when discussing the problem of computing the exact S-matrix of a certain class of models, it is essential to have an infinite number of higher conservation laws. Higher local conservation laws imply the conservation of the individual momenta in a scattering process. This result may be obtained by direct computation, using the properties of the model [17], or indirectly, by noting that the local conservation laws must be polynomials in the external momenta which are separately conserved. In such cases, the scattering process is simple but nontrivial, and consists in an exchange of quantum numbers, while particle production is prohibited. This property has been checked up to second order in perturbation theory [17]. A general inductive proof of the absence of pair production at the tree level for any number of external lines has also been obtained [18]. One first considers n —J- n particle scattering, and shows that the principal value of the corresponding tree amplitude vanishes. The proof consists in a direct verification for 3 —> 3 scattering, and then proceeds by induction. Using crossing symmetry one can then relate n —> n scattering t o n — p —> n + p scattering. Therefore the intermediate particles are always on shell. These conditions are very strong in two dimensions, and lead to particle number conservation. Let us exemplify this by an explicit calculation [19]. Consider the loop diagram 5 Let P{i>) @4> be an operator with scale dimension d + | . Upon using the equation of motion (3.30), this operator will be replaced by mP(i/>), with scale dimension d+ \ . Since both operators are subtracted in the same way, this implies an oversubtraction as given by the Zimmermann's identity [11].

3.3 The Massive Thirring Model

51

in Figure 3.1, as given by the expression

J (2^/i

(k + Pj)2 — m2 + ie <*,0j

where Sj — 1 if the momentum pj is in the same direction as the charge flow, and Sj — — 1 otherwise. The k0 integration is easily performed by closing the contour

k +p Figure 3.1: One loop contribution to the scattering of 2n particles, in the lower plane, and using the residue theorem, 6

/<»> = -27ri f dk* £ "

s

^

+

" ^

+ m x

n

. (3.36) 2

j-j2 A /[(fci+p))2+m2](pg-p§)-2(A ! i+p})( P J-p}) + ( p f c - p i ) + ' The remaining integral may be performed by making the change of variables Xi

1

y/ik

kx+pj +p\)2 + m2 - m

(3.37)

in each term of the sum above. 7 We have x2 + l 1 1+ 2 + 1 mT

= -\M ^)

dxi = dk] W{ki +Pi)2+m2-

m

m}2y/(ki

+Pi)2+m2

Each term in the integrand may thus be expressed as * = (x2-ir~2

( W , + m ) n

2

Wk«+^

+

^(pl^mpl^xj-xffUxj-xi^ 6

1

1 2

m

+ ie 2

The first term corresponds to the residue at fc? = ^/(A; + p ) + m — pP of the j " 1 propagator, and the remaining terms are obtained from the k ^ j propagators, substituting the above expression. 7 When k1 -> ±oo, we have x —• ±1, while k1 + p1 -> 0± implies x -> ±oo. Thus the new integration variable runs over the ranges [1, oo) and (—oo, — 1].

52

The Thirring model

where pkj = Pk ~Pj, and where a;L- are obtained by requiring the kth and the j t h inverse propagators to vanish simultaneously. We thereby obtain two roots, labeled by (±). _ The integrand of Eq. (3.36) is now a rational function and the integral may be performed by elementary means. The poles at x2 = 1 do not contribute, as one can see from a detailed analysis of the residues, or else from the fact that they correspond to poles at k1 —> ±oo, which according to the power counting theorem [20] do not contribute for n ^ 2 (the case n = 2 is trivial). We now make a partial fraction decomposition

^=E

A"'

<^k

fc=i

Akj = lim (x

xkj)T{a

%kj

Equation (3.36) may then be written as l

X k - X kj

Xk - X kj

kj

We perform the elementary integral above, obtaining the result 1 + xf1-xl 'kj "kj ln1 • * « ! + *£

f(x) = Y,rJ = Y,Y,Akj fc=l j = l

where

4ro2

2mpl ±•Pkj/1

"i? =

2mp°ukj k

^7

(3.38)

Plj fk] and where the coefficient Akj in the partial fraction decomposition is given in terms of the product of all propagators with I different from k, j ; we rewrite the numerator of the k— and j — propagators in terms of the free fermion wave function to compare with the tree approximation; the factor ——f ^_2 arises from the change '

*>f-

of variables (3.37): l

kj

n

co(kj)ui(kj Si(l • h + 7 • p) + m _ 2^ — /-i 4m2" i-J- /,. {ki +, p) — _m,2 ,+ ...t"(fcfcM*fc)

=p27ri

Pkj

^f2ni

1

"h/

(3.39)

-Tkj %

j(k) for 6 = — 1. Substituting for (±) x\-

Here u(k) = u(k) for 5 = 1, and ui{k) expression (3.38), we have

^EE^ k=l

k^j=l

^2iri kj

V

s

kj*kj

In

V~Skj V

s

kj

ftk~j

i

y/^kj

3.4 Bosonization Revisited

53

where Skj = [pk + Pj)2 and tkj = (Pk - Pj)2- Notice that we have a sum of terms given by the product of the tree diagrams obtained by deleting the lines k and j (Tkj), and the loop diagram built from the lines k and j . Our above results concerning the tree diagrams tell us that particle production is again forbidden. We shall use the connection between higher conservation laws and conservation of individual momenta (absence of particle production) as a working hypothesis for calculating the S-matrices of different two-dimensional models in Chapter 8.

3.4

Bosonization Revisited

We have used the representation of the free Dirac field in terms of bosonic fields, as discussed in Chapter 2, in order to demonstrate the equivalence of the (massless) Thirring model to the theory of free zero-mass bosons:

cTh = ~$%w> + ^ s W W ) 2 = £ B o s = \d*
.

Following Coleman [7] we have similarly demonstrated, in the zero-charge sector, the equivalence of a theory of massive fermions with a Thirring like interaction to the sine-Gordon theory, £(F> = ^(* 0 - m)V + i s ( V Y V ) 2 = £ ( S G ) = \{d^)2

+ ^ ( c o s pv-I)

,

provided J3 and g are related by (3.21). The map from the fermion %jj of the massive Thirring model to the boson

I(J was introduced by Mandelstam [9] who was able to write the fundamental fermion of the massive Thirring field, in terms of the bosonic field, thus generalizing the results summarized in Eqs. (3.13)-(3.18). As we discuss next, the fermionic field obtained in this way describes the soliton as well as new particle states of the sine Gordon theory, which cannot be interpreted as conventional bound states; there is no element in the local field algebra which connects the vacuum of the sine-Gordon theory to the non-trivial soliton states. We shall discover in a later chapter, that the particle scattering of the fundamental field of the sine-Gordon theory can, in turn, be described in terms of the scattering of bound states of the soliton states.

3.4.1

Fermions in t e r m s of bosons

In order to find a bosonisation formula for the fermion field, we note that there is a natural candidate for the fermion in the sine-Gordon theory: the soliton. The fundamental property of this soliton operator is encoded in the equal time (ET)

54

The Thirring model

commutator 8 [LP(y),iP(x)}ET = 2irp-1ip(x)8(x1-y1)

.

(3.40)

The commutation relation is in particular realized by the operator

*l

1>(x) = ^ e - t - n e

(^J.

(3.41)

Note that for a zero mass free field
= -•£-: cos/3tp:

.

(3.42)

The definition of N[il>ip] in terms of a Wilson expansion requires now an additive and a multiplicative renormalization, as can be checked already for the case of free massive fermions: ^{x + e)V-(z) = C(e) + [f(e)]N®{x)rl>(x)].

(3.43)

In order to have more explicit formulae, we write a Kallen-Lehmann representation for the two-point function10 of sin /3

(0 I sin M x ) sin /3p(y) | 0) = /

{^fp{^)^+\x

- j / ^ W -

(3-44)

Jo

For the sine-Gordon field theory, we can use the free field singularity of the correlator, Eqs. (2.9) and (2.15), (sin /3
,

to deduce from (3.44) the asymptotic behavior p(M2)~(/i2)£-3

,

(3.45)

for large values of \J? . 8

I n the following we drop the tilde on /3 . T h e phase factor e~ T 7 is required in order to insure Fermi statistics for different components of the field ipa (see Appendix A) . 10 We start from {ip(e)ip(0)) = J p(^2)M+'>(e; fj.2)dfi2 and obtain 9

-4

(sin M e ) sin fiv(0)) = = j {n2)2 P(ii2)^+\e;

^ W

55

3.5 The Soliton as a Disorder Parameter

On the other hand, the equal time commutator [(y)]ET = i5{xl — y1) implies />oo

/ p{ii2W2 = l . Jo Using (3.45), the integral in ( 3.46) is convergent only if j32 <

8TT

(3.46)

.

This is a condition for the sine-Gordon theory to be well defined. In order to define the mass operator (3.42) we compute its vacuum expectation value; using the representation ip(y) = — i& f' ^, we infer that [sin @
;

we thus have im2{0 |: cos p,p(x): | 0><J(a;1 - y1) = (0 | [-n(p{x),
xu=yu

J

=i f

v2p{»2W5{xl-yl).

J

Using again (3.45), we see that the integral is finite if/32 < ATT, and is an infinite constant for An < 01 < 8ir. A logarithmically divergence appears for 01 = 47r, which corresponds to the case of free fermions. Moreover, using (3.43) and requiring (3.40) to have vanishing vacuum expectation value, we obtain for the constant C(e) in (3.43) (e

2p-(^)2 J

We have so far computed the current, the mass term ipip, and the pseudoscalar x/)j5ip. The kinetic term tip glip is again of the form (3.18). As we have witnessed above, the definition of the composite operator N[ipip] deserves special consideration for Ait < p2 < 8rr. It is related with the fact that the exponential field ip\(x) = e*Av(z) develops new (non-leading) singularities when A —>• 1, a phenomenon known as cumulative mass effect [22]. Indeed, Coleman's arguments only hold in the region dim^V < 1 : where the usual Wick product is enough to define the exponential field. For 1 < dimxpip < 2 the sequence of arguments related to equations (3.43) through (3.47) must take account of the cumulative mass effect. The case of dimiptp > 2, corresponding to 01 > 8TT in the sine-Gordon model, is non-renormalizable. In this domain mass corrections of order e2 contribute in the definition of composite operators. We refer the interested reader to the literature [21, 22].

3.5

The Soliton as a Disorder Parameter

We have identified the soliton operator of the sine-Gordon theory with the fermion of the massive Thirring model. In its simplest form it is given by i2f ]dy1doV>(x°,y1) H(x) = e *x .

(3.48)

56

The Thirring model

The representation (3.41) for the fermion field operator suggests defining also the operator
,

ip2(x) =a*{x)p,(x)

.

(3.50)

The operators (3.48) and (3.49) satisfy the equal time algebra u(x)a(v) - I ~a(y^(x)

V1>x1

>

This algebra is analogous to the dual algebra of the order and disorder operators in statistical mechanics, as it was first discussed by Kadanoff and Ceva [23]. The non-local operator fi(x) has the property of creating topological excitations which are orthogonal to the vacuum <0 |/*(*) | 0> = 0

.

This reflects the fact that the operator /j.(x) carries the charge superselection rule of the fermion field (3.50). Similarly one has, in the massive Thirring model (0 | a(x) | 0) jt 0

,

(3.51)

which reflects the absence of a chiral selection rule for the massive fermion. The sine-Gordon Lagrangian is invariant under the translation

— a. Hence (3.51) shows that this symmetry is spontaneously broken. This is however not the most general situation. In fact, given a dual algebra of the form / ei2v/N4>(y)fi(x) IMWW ,

i \M \

, y1 > x1 y <x

.

,2 .

n

it has been demonstrated [24] in the framework of axiomatic QFT, that regardless of the form of the Lagrangian C(, d^), the following properties in this case hold:

i.

W(M) = O ,

2.

(<M>=0

.

3. Whenever

(} = 0 and (/x) = 0

,

the mass gap vanishes.

The above theorem expresses a deep relationship between the spectrum and the vacuum expectation values of order and disorder fields. Let us examine how these ideas may be used to obtain the order and disorder operators of a QFT defined by a Lagrangian £(, d^ifi). It is useful to first review briefly the corresponding situation of the one dimensional Ising model in statistical mechanics. The Hamiltonian reads in this case J

ff=-J^<73(n)a3(n+l)-^a1(n)

,

57

3.5 The Soliton as a Disorder Parameter

where <Ji(n) are the Pauli matrices. The Hamiltonian is invariant under the transformation az{n) -¥ —a3(n), a\(n) ->• ci(n). This system is known to posses two phases [25, 26], characterized by ( 1). Thus J — 1 corresponds to the critical point. Consider a dual lattice, for which each site n* is placed between the sites n and n + 1 of the original lattice, and let ^{n*) be an operator defined via the dual algebra [23, 25] i *\ i \ / -cr 3 (mV 3 (n*) , Vz{n )^zVm) = { \ \ \ „\

n <m .

.

,„ _0s (3.52)

An operator realization of ^ 3 (n*) is [25] ^(n*)

= J J
(3.53)

One finds that for J > 1, (/i3) = 0, while for J < 1, (/J 3 ) 7^ 0. Consider for example one of the ground states for J > 1 (ordered phase), characterized by all spins oriented up:

|o>J>1=|---fr---t--->

•

The action of ^ 3 (n*) on this state turns it into the state I ••• t t U • • • )

•

Hence // 3 (n*) has the character of a soliton operator, which creates Bloch walls from the ordered ground state. In the disordered phase, we have (/J 3 ) ^ 0. Hence the phase transition {fi3) = 0 —> (/z3) ^ 0 at J = 1 can be viewed as a soliton condensation. The above quantum mechanical description of the one-dimensional Ising model is known [25] to have an equivalent description in two-dimensional classical statistical mechanics defined by the partition function

S = 5>-*M

,

where H is the discrete, classical Hamiltonian H[o) = -jY^a{n)°{n+

?{»))

,

(3.54)

• —a(n). The dual lattice is composed of sites n* placed at the center of each plaquette. The disorder variables /i(n*) = ± 1 are introduced on the sites of the dual lattice. The disorder correlation function is given by [23] = E e ~ f f ' W M

'

( 3 - 55 )

58

The Thirring model

where H'[a] differs from (3.54) by the fact that the coupling J is replaced by —J along an arbitrary curve C along the dual lattice sites connecting n\ and n\ (see Figure 3.2). Notice that the curve C cuts the links of the original lattice, each of which is in turn associated with a pair of spin variables.

Figure 3.2: Curves C and C' connecting the points n* and nj of the dual latti We may thus evidently write (3.56) where ff/,w_/2^M -"cL<7J~\0

for

f(/i)GC for f ( / i ) g C

The correlation function (3.56) is actually independent of C. This is easily seen by making the change of variable a(n) —>• <x(—n) for sites n belonging to an arbitrary region R bounded by the closed curve V = C — C (see Figure 3.2), which leaves (3.56) invariant. Hence, {(J.(nl)n(r%))c = (fj,{nl)fj,{n*2))c, . This path independence is important in order to have locality properties of the correlation function (3.56). playing a crucial role in the construction of the disorder operators of QFT. We first apply the above ideas to the construction of the quantum kinks of theories of the type where ^ is a local, complex scalar field [27]. Let us assume the theory to have a Z(N) symmetry,
fv*e-fd2z{c+Cc}

,

(3.57)

where Cc is the analogue of He in (3.56), and is to be determined by the requirement that the correlation function (3.57) be independent of the chosen path C. To this end consider the change of variable <j>(x) -> eta<j>(x), for a; in a region S bounded

59

3.5 The Soliton as a Disorder Parameter

by C — C. Choose a — ^ corresponding to the Z(N) symmetry of the Lagrangian. Since the integration measure is left invariant by this transformation, we have (fi(x)fi*(y))c

= N"1 fvV<j>*e- f

d2x{{C+Cc)+{5C+SCc)}

= N'1 fvV4>*e~I'd2x(c+cc)e-J'd2x(Cc-ccl+sc+6Cc)

_

Hence, independence of the integration path C requires Cc - Cc< +6C + SCC = 0 •

Assuming the potential V ((f), <j>*) to be invariant under the above transformation, we have, 5C = -iaffytftinOiS) + a2cf>*
dlle(S) = (J-Qt2(z-Oellvdzv

.

Making use of this representation, we are thus led to the identification [27] Cc = -ia*(zfy{z)

fV

52(z-0^dtv

Jxfi 2

+a cj>*{z)4>(z) I" d£„ fV dVfl52(z - 052(z - v). Jxfi

(3.58)

Jxfi

With this identification, the correlator {H(x)fj,*(y)) = N-1

fv*e-fd2x{c+Cc)

(3.59)

is independent of the choice of C. The above result can be cast into a more transparent form, by defining the external field Alt(z;C) = [

e^52(z~Od^

•

Jxfi

In terms of this field, the functional integral (3.59) takes the form (H(x)ti*{y)) = N-1

I'vVcf>*e-1'<*2*{(zW(^)+v(^*)}

^

where DM is the covariant derivative D^ = <9M — iaA^. The path independence is now an expression of the invariance of the functional integral under the gauge transformation 4>{x) -> eiae^4>{x) , A^x-C) -> A^{x;C) + dM0(S)

.

60

The Thirring model

From Eqs. (3.58) and (3.59) we extract the following operator representation, in Minkowski space, for the disorder operator a

fi{x)=e

C^*^

It is straightforward to check, that this operator satisfies the dual algebra (3.51), and hence is the quantum field theoretic generalization of the kink operator H3(n*) of the Ising model. We specialize now to the case of a massive 0 4 type theory with potential V(<j>, f) = m V 4 » + A ( > + ^ )

+ -£- {r
.

This potential possess a Z(4) symmetry cj> -> el~*~ <j>. This system can be shown to exist in three phases: 1. symmetric phase: {4>)=Q ,

(/x)^0

,

<M«)A**(l/)>,

-» const.

|i-y|->oo

The phase corresponds to a kink condensate. 2. partially broken phase: (4>)=0 , (M)=0 ,

<ji(x)tS(y))

-f

\x-y\~a

but (<j>2) 7^ 0. Here only the Z{2) symmetry is broken. The \i operator creates massless kinks [28], in agreement with a general theorem [24]. 3. completely broken phase: (0)^0

,

(M)=0

,

< M * K (!/)>,

-?

\x-y\-ae-m\*-v\

.

\x-y\-+oo

In this case fj,(x) creates massive kinks with m 2 ~ (||2). Let us now return to the sine-Gordon theory. Repeating the line of reasoning exposed above, we find this time ((i(x)n*(y)) = Jve-fd2^CsG+Cc)

,

(3.60)

where

+ \(j)

f

^ jV^d^{z-i)^z-n).

(3.61)

61

3.6 Conclusion

From (3.60) we extract the following representation for the soliton operator in Minkowski space: li{t,x)=e

-1

,

(3.62) 1

where we have specialized the curve C to lie parallel to the x axis. The exponential (3.62) is nothing but the soliton operator introduced in (3.53)! The above method of constructing order and disorder operators from a given symmetry of the bosonic Lagrangian in question, can also be used as a tool for constructing bosonic representations of the fermionic operator of other models via the composition (3.50).

3.6

Conclusion

We presented the full quantum solution of the massless Thirring model in terms of free fields. Our building blocks were the exponential fields defined in Chapter 2. The correlators have been computed in closed form, and further properties of the model have been established. The model is conformally invariant at the classical as weel as at the quantum level [29]; as it turns out, it is completely equivalent to a free bosonic theory. We next allowed for a mass of the fundamental fermionic field. Perturbation theory in the mass parameter showed a very important feature: at quantum level, the model is equivalent to the bosonic theory of the sine-Gordon field. The fact that a bosonic theory may describe fermions was suspected long ago [6], before the actual proof of this fact, and has been discussed by several authors. The full quantum equivalence has been proven by Mandelstam (Eq. (3.41), who also provided the required equivalence in the charged sector, as opposed to Coleman's equivalence, Eqs. (3.16), (3.17) and (3.18), valid only for neutral bilinears in the fermions. Nevertheless, the aforementioned features are not the only surprising features of the theory; it also exhibits in both the fermionic and bosonic languages, an infinite number of conservation laws, which are a consequence of the Lax pair (3.26) and (3.27), in the case of the Thirring model, and (3.31), (3.32) in the sine-Gordon case. This is a feature discussed at length in the literature. As a consequence one may exactly calculate the full S-matrix for both of these theories [30, 31, 32], as well as some form factors [33]. These S-matrices will be constructed in Chapter 8. Finally we presented a discussion of the order-disorder operators of the theory, which provides a deeper physical insight, as well as a method for constructing fermions out of order and disorder operators.

Bibliography [1] W. Thirring, Ann. of Phys. 3 (1958) 91. [2] W. Glaser, Nuovo Cimento 9 (1958) 990. [3] K. Johnson, Nuovo Cimento 20 (1961) 773.

62

BIBLIOGRAPHY

[4] F. Scarf and J. Wess, Nuovo Cimento 26 (1962) 150. [5] B. Klaiber, Helv. Phys. Acta 37 (1964) 554; Lectures in Theoretical Physics, Boulder 1967, Gordon and Breach, New York, 1968. [6] D. Finkelstein and J. Rubinstein, J. Math. Phys. Skyrme, Proc. Roy. Soc. A260 (1961) 127.

9 (1968) 1762; T.H.R.

[7] S. Coleman, Phys. Rev. D l l (1975) 2088. [8] J.A. Swieca, Fortschritte der Physik 25 (1977) 303. [9] S. Mandelstam, Phys. Rev. D l l (1975) 3026. [10] P.P. Kulisch and E.R. Nissimov, Pisma Zh. Eksp. Teor. Fiz. 24 (1976) 244; R. Flume, Phys. Lett. 62B (1976) 93; I.Ya. Aref'eva and V.E. Korepin, JETP Lett. 20 (1974) 321; R. Dashen et al Phys. Rev. D l l (1975) 3424; B. Berg, M. Karowski and H.J. Thun, Phys. Lett. 62B (1976) 63, 187; B. Yoon, Phys. Rev. D132 (1976) 3440; R. Flume et al Phys. Lett. 64B (1976) 289. [11] N.N. Bogoliubov and O.S. Parasiuk, Acta Math. 97 (1957) 227; K. Hepp, Commun. Math. Phys. 2 (1966) 301; W. Zimmermann, in Led. on Elementary Particles and Quantum Field Theory, 1970, Brandeis U. Summer Institut; J. Lowenstein, Seminars on Renormalization Theory, Maryland 1972; P. Lam, Phys. Rev. D6 (1972) 2154. [12] M. Karowski and H. J. Thun, Nuovo Cimento 38A (1977) 11; R. Flume and S. Meyer, Lett. Nuovo Cimento 18 (1977) 238. [13] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207. [14] M.D. Kruskal and D. Wiley, American Math. Society, Summer Seminar on Nonlinear Wave motion, ed. by A.C. Newell (Potsdam NY 1972). [15] V.E. Korepin and L.D. Fadeev, Theor. Math. Phys. Zamolodchikov, Pisma v JETP 25 (1977) 499.

25 (1975) 1039; A.B.

[16] J.H. Lowenstein and W. Zimmermann, Nucl. Phys. B86 (1985) 77; E. Abdalla, M. Gomes and R. Koberle, Phys. Rev. D18 (1978) 3634. [17] B. Berg, Nuovo Cimento 41A (1977) 58. [18] B. Berg, M. Karowski and H.J. Thun, Phys. Lett. 64B (1976) 286. [19] G. Kallen and J. Toll, J. Math. Phys. 6 (1965) 299. [20] S. Weinberg, Phys. Rev. 118 (1960) 838. [21] H. Neuberger, Thesis, Tel-Aviv Univ. 1976; M. Bander, Irvine, Prep. (1975); M. Halpern, Phys. Rev. D12 (1975) 1684; H. Lehman and J. Stehr, Desy-Prep. 1976/29. [22] B. Schroer and T. Truong, Phys. Rev. D15 (1977) 1684.

BIBLIOGRAPHY

63

[23] L.P. Kadanoff and H. Ceva, Phys. Rev. B 3 (1971) 3918). [24] R. Koberle and E.C. Marino, Phys. Lett. 126B (1983) 475. [25] E. Fradkin and L. Susskind, Phys. Rev. D17 (1978) 2637. [26] J. Kogut, Rev. Mod. Phys. 51 (1979) 659. [27] E.C. Marino, Nucl. Phys. B217 (1983) 413; B230 (1984) 149; A.A.S. de Macedo and E.C. Marino, Phys. Rev. D40 (1989) 1360. [28] E.C. Marino, B. Schroer and J.A. Swieca, Nucl. Phys. B200 (1982) 473. [29] P. Furlan et al Rivista del Nuovo Cimento 12 (1989) 1; L. Heck, Fortschritte der Physik 27 (1979) 169; M. Gomes, J. Lowenstein, Nucl. Phys. B45 (1972) 252. [30] A.C. Scott, F.Y. Chu and D.Mc. Laughlin, Proc. of the IEEE 61 (1973) 1443. [31] L A . Takhtadzhyan and L.D. Fadeev, Theor. Math. Phys. 21 (1975) 1046; M.S. Alblowitz, O.J. Kaup, A.C. Newell and N. Segur, Phys. Rev. Lett. 31 (1973) 125. [32] L.D. Fadeev and L A . Takhtadjan, Hamiltonian Methods in the Theory of Solitons, Springer, 1987; S.P. Novikov, Theory of Solitons: the inverse scattering method , New York, Contemporary Soviet Mathematics, 1984; R. Rajaraman, An Introduction to Solitons and Instantons in Quantum Field Theory, North Holland, 1982. [33] M. Karowski and P. Weisz, Nucl. Phys. B139 (1978) 455.

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Chapter 4

Determinants and Heat Kernels 4.1

Introduction

Within the path-integral framework, the non-perturbative calculation of n-point functions in gauge theories involves as intermediate step the calculation of Green functions and functional determinants in an external gauge field A^x). By functional determinant we mean the determinant of an operator D[A] defined in (continuum) Euclidean space, whose value depends on the choice of the field configuration A(x). In the cases of interest, such operators will be differential operators depending locally on the external field. The computation of functional determinants requires regularization of divergent expressions. The methods used, although seemingly quite different, are all abstracted from the corresponding definition in the finite dimensional case. Among these, we cite the heat-kernel, ^-function, proper time and Fujikawa method, which play a central role in this chapter. These methods will also allow us to calculate explicitely the functional determinant associated with the Dirac operator in an external Abelian and non-Abelian gauge-field. We shall see that the path-integral framework also provides a suitable framework for dealing with zero-modes, and for establishing their relation to the topology of the manifold. In particular, we shall provide an unconventional derivation of the Atiyah-Singer theorem relating the index of the Dirac operator to the number of zero-modes of positive and negative chirality. In all these considerations the heat-kernel associated with the operator in question will play a fundamental role. We discuss a variety of methods for obtaining its asymptotic expansion and corresponding Seeley coefficients, and illustrate them in terms of specific examples which shall be needed in later chapters. The present mathematical framework also provides a natural basis for discussing Quantum Field Theory at finite temperature. We have thus reserved the final section of this chapter to some selected topics on this matter.

66

Determinants and Heat Kernels

4.2

Functional Determinant, one-loop diagram

In the Feynman path-integral approach to quantum field theory one is naturally led to the consideration of determinants of differential operators. Indeed, functional determinants arise in quantum field theory whenever one is dealing with a functional integration of the Gaussian type; that is, whenever the action in question is a second order polynomial in the fields over which we wish to integrate. This may be an exact characteristic of the Lagrangian, as in the case of the fermionic action of QED/QCD; it may be the result of an approximation (semi-classical or effective potential in a ^ expansion); or it may be the result of rewriting quartic interactions in some fields as a second order polynomial in these fields at the expense of introducing new, auxiliary degrees of freedom ((CPN~X, Thirring, Gross-Neveu models). Conversely, one may represent the determinant of some operator in terms of a Gaussian functional integral over Grassmann-valued fields. This is a very useful technique for dealing with the Faddeev-Popov [1] determinant arising in the process of fixing the gauge. The Mathews-Salam representation [2] of the Euclidean generating functional of fermionic correlation functions in QED illustrates how functional determinants make their appearance in physics. It is given by Z[J, v,rj}=

f dfj,[A][V$\[Vrl>] e-So[A\-sF[i,^,A\

= f dfi[A] e-SaMdet(i

Jw+tn)

0 + iM + e4) e!vG(x,y,A)r,

^

(4J)

where d/z[.<4] is the Faddeev-Popov measure [1] in some suitable gauge, the action SG(SF) is associated with the gauge-field (fermions) and G(x,y; A) is the fermionic Greens-function in an external field1 A^: (i 0 + iM + e 4)G{x,y; A) = 5{d) (x - y)

.

The logarithm of a functional determinant has a simple interpretation in terms of Feynman diagrams: it may be represented as an infinite sum of properly weighted one-loop fermion diagrams whose vertices depend on the operator in question. This is easily seen on a formal level. Thus consider an Hermitian operator D which we decompose into a free (D0) and an interacting part (Dj) as follows: D^Do

+ Dj

.

Borrowing familiar formulae from matrix algebra, we write UJD = hidetD = tx\n{D0 + DI) u D =ln—^—— = tx\n(l

~ (-l)n+1

=E 1

(4.2)

D^1DI)

+

L L2

,

1

— trtt-^/)"},

(4.3)

T h e appearance of the imaginary mass is peculiar to the Euclidean formulation. A^ denotes here a Hermitian - in general matrix valued - vector field.

67

4.2 Functional Determinant, one-loop diagram

where DQ1DI is to be regarded as a (non-local) operator, and where tr denotes the trace with respect to internal indices as well as space-time. The functional U>D possesses the obvious property, U_D. = UJD — W.D0. D

o

The sum (4.3) has a simple graphical interpretation: it is given by the sum of (connected) one-loop graphs as shown in Figure 4.1, with • —Dj for each vertex, • DQ1 for each propagator, a • (—1) overall factor for being a fermion loop, and • (""i1^ = - combinatorial factor for a diagram of order n. nI

n

°

Figure 4.1: Graphical computation of the determinant. As an example consider the Euclidean Dirac operator of massless QED, i p = (i ^ + e 4)

,

(4.4)

with A^{x) some arbitrarily chosen external gauge-field configuration. According to (4.3) we have

-#

L det(i 0 ) 1 ^

n

\i 0 *7

= e f d*xSE(0) £{x)

- y

d4x J diySE{x-y)4(y)SE{y-x)4{x)

(4.5)

+ --- ,

where SE{X) is the inverse of i $ (see Appendix B). As the first term in (4.5) already shows, there arise ambiguities as a result of divergencies, which will have to be treated by an appropriate regularization technique. In this chapter we shall examine various such techniques which do not rely on perturbation theory. The exponential of the series (4.5) has a simple representation in terms of a functional integral. Let {ca(x)} be the components of a complex-valued Grassmannian field, with {ca(x)} their complex-conjugate. The dimensionality of the index a matches that of the matrix-valued operator D; c and c are assumed to obey the

Determinants and Heat Kernels

68

Berezin [3] integration rules given in appendix C. The ratio of determinants of D and D0 is then represented by det£> _ e" D _ fVcVce-fddx'Dc UD detD0 ~ e o ~ jVcV-e-fd*xcD0c

'

where d is the dimensionality of (Euclidean) space-time. The virtue of considering the normalized ratio (4.6) resides in the fact that the leading divergence is thereby eliminated. The formal correctness of (4.6) may be established in different ways. One way consists in expanding the integrand of (4.6) in powers of Dj as follows,2 e-

_ fVcVce-fdd*-cD°cZn=o

^Lf(^/c)]"

(4?)

ddx cDoC

fVcVce-f

'

and then to perform the functional integration. The result is the exponential of the sum of connected one loop diagrams as given by (4.3). Another way of formally verifying the correctness of the identification made in (4.7) is to suppose D and .Do to be operators defined on a compact manifold M, with a discrete eigenvalue spectrum and a complete set of eigenfunctions: Dun = Xn[D]un DQvm = \m[D0]vm

,

(4.8) .

(4.9)

For the time being we shall assume that D and Do have no zero eigenvalues. Expand the fields ca(x), ca(x) in (4.7) in terms of the above eigenfunctions: ca{x) = "^2 anun(x)a

= ^

n

bmvm(x)a

,

(4-10)

,

(4.11)

m

Ca{x)=^anu\l{x)a

= ^2,bmv\n(x)a

n

m

with an, an, bn, bn Grassmann-valued c-number coefficients. Choosing un and vm to be orthonormal with respect to a measure d/j,(x) on M, / djjL{x)u]l(x)un'{x) = 8nn,

,

dnixtyltWvmi(x)

= 5mm>

,

(4.12)

we make the change of variables VcDc -> Yl dandan

,

(4.13)

,

(4.14)

n

VcVc->~[[dbmdbm

2 Dj may be a local or non-local operator. For compactness of notation we shall frequently refrain from specifying the integration variables.

4.2 Functional Determinant, one-loop diagram

69

in the numerator and denominator of (4.7) respectively. Introducing the expansions (4.10) and (4.11) in (4.7) and using (4.8), (4.9) and (4.12), we obtain e" D

SXlndandane-Y,^T^

=

e D

" °

!Y[mdbmdbme-T.^0]lmam

Now, fi-$>[i>]5n-n

=

Y[ e - ^ P P — = J J ( l - \n[D]anan) n

.

n

The last equality only holds, because of the Grassmann properties a^ = a^=^ 0. Furthermore, according to the Berezin [3] integration rules (see appendix C),

f J ] dandan e~ D MDp»«»

=

"* n

J J An[jD ] ,

(4.16)

n

with a corresponding result for the denominator in (4.15). Hence, we conclude that e"D _ II„An[i>]

n„A„[A>]

(4.17)

This confirms once more the identification made in (4.7). If there are zero-modes present, then these should be omitted in (4.17), resulting in det'D e"'° Y['Xn[D] { det'D0 U'K[Do] ' ' e«'Do where the prime indicates omission of zero-modes. Hence det'D is just the subdeterminant corresponding to the non-vanishing eigenvalues of D. The existence of zero-modes is not seen in perturbation theory. The divergencies occurring in the loop-integrations implicit in (4.3) have their reflection in the non-existence of the product (4.17) because of the (in general unbounded) growth of the eigenvalues with r n o o . Thus again, a regularization procedure has to be given for defining the determinant via (4.17). Such procedures will be discussed in detail now.

4.2.1

Determinants and the Generalized Zeta-Function

Elliptic operators Definition (4.17) can be used to calculate wo in terms of an infinite sum involving the eigenvalues A„[Z)]. In order to get an answer for generic field configurations A, we need to know the eigenvalues as functionals of A. This is generally not possible. Hence a direct calculation starting from (4.17) will in general be feasible only for specific field configurations. For a generic field configuration one has to develop special techniques. All of these are based, in one form or another, on the definition (4.17), and correspond to different regularizations of the in general divergent product (4.16), as we shall see.

70

Determinants and Heat Kernels

Let D be a differential operator of order m defined on a d-dimensional, compact manifold M, D

=

h

E

(«){Z)D^

.

(4.19)

\a\<m

Here x = (£1, • • •, x^) £ M. are local coordinates, 3 (a) — (a±, • • •, aj,) is a multiple index, with \a\ = a\ + • • • + a<j, b^(x) = bai. ..ad(xi, • • • ,Xd) are smooth, Ixk matrix valued functions, and f)\a\

£)(") = (_^(a) dx?--dx t ad<

•

We associate with (4.19) the characteristic polynomial

£ ha){x)i^= £ bai-adm?i---ad |a|<m

>

\a[<m

where £ € M. The restriction of the sum to (a): \a\ = m leads to the definition of the principal symbol associated with D,
£

W*K ( a ) >

\ot\=m

The principal symbol a defines a homomorphism from a k- to a /-dimensional vector space. D is said do be elliptic if its principal symbol is invertible for all f ^ 0, i.e., if the homomorphism defined by a turns into an isomorphism. This requires that I = k and that (7 has no zero eigenvalues for ^a' ^ 0. Example 1: Euclidean Dirac operator in Sft2 , D = —ij^d^. In this case m = 1 and bio = 7 i , &oi = 72 a n d cr(a;;£) = £ . The eigenvalues of O~D are determined by the secular equation,

A 2 -(£x 2 +£ 2 2 )=0

.

Hence this operator belongs to the class of elliptic operators. Notice that the corresponding operator in Minkowski-space does not satisfy the criterion of ellipticity! It is for this reason that we shall always work in this chapter in Euclidean space. Seeley expansion For us, the central importance of elliptic operators resides in the following theorem due to Seeley [4] and Gilkey [5]: let D be a positive elliptic operator of order m on a compact Riemannian manifold; then there exists an asymptotic expansion for the diagonal part of the associated heat kernel exp(-tD), as given by (x\exp{-tD)\x) 3

= — l - ^ [ a 0 { x \ D ) + ai(x\D)t (iirt) "•

+ • • •] .

(4.20)

These coordinates may serve to label only patches in M as it happens if a non-trivial topology is involved. We use a hat to distinguish them from the cartesian coordinates x in 9td.

4.2 Functional Determinant, one-loop diagram

71

The expansion coefficients a,k(x | D) are referred to as Seeley coefficients [4]. For elliptic operators of second order, these can all be calculated! Furthermore, we shall see that for a special class of operators D, the calculation of d e t D can be reduced to the calculation of the first few of these coefficients. We may in fact relax the positivity condition on D such as to include semipositive operators. This will be important, in order to be able to discuss the effect of instantons, which turn out to induce zero-modes of the Dirac operator (4.4). The (-function We shall generally suppose D to be a semi-positive, elliptic, (matrix-valued) differential operator of order m defined on a compact manifold M. of dimension d. Let dfi(x) be the measure 4 on M., and let A„, ipn be the corresponding (real) eigenvalues and eigenfunctions of D: Dtpn = Xn
.

We choose n

/

d^(x)fi(x)(pm(x) = 6mn

.

By assumption, all eigenvalues are positive or zero. We may define a generalized (-function [4] by

C(s\D)=^2'X-s[D]

,

(4.21)

where the prime on the sum indicates that zero-eigenvalues are to be omitted. We may write (4.21) in the compact form, C(s\D) = tv(D-°)

,

where the trace includes the internal degrees of freedom and is to be taken over the subspace orthogonal to the zero-modes. £(s | D) can be analytically extended to a meromorphic function of s in the whole complex plane, with s = 0 a regular point [4]. Its value at s = 0 may be calculated in terms of the Seeley coefficients, as we shall see. The quantity of immediate interest to us is obtained by differentiating (4.21) with respect to s: d( (4.22) ds (A„)s

SciJ'i-E

This suggests to define det'D by [In det'D}( = lim s—>0 4

ds

(s\D)

-C'(0 I D)

(4.23)

If we choose to parametrize i b y i = (xx,---,xd) £ tHd, then dn(x) = ddx^/g(x), where g(x) = det(g/Jl/(x)) with 3^t/(x) the induced metric tensor on M.. In that case S(x,y) = g~x/2(x) x

8d{x-y).

72

Determinants and Heat Kernels

The prime in (4.22) indicates the restriction of the sum to the subspace of nonvanishing zero-modes where D is invertible. The calculation of det'D is of course plagued by the usual ambiguities related to the ultra violet divergencies of fermion-loop graphs. The definition (4.23) by analytic extension to s = 0 corresponds to a very specific choice of regularization, the so called (^-function regularization . By normalizing det'D with respect to the free operator D05 l n ^ ^ = -(C'(0|I>)-C'(0|Po))

,

(4.24)

we eliminated the ambiguity associated with the leading divergence. The operators D (D0) of interest in general carry a dimension. This means that the eigenvalues An, and hence the measures (4.13) as well as (4.14) carry dimensions, so that the corresponding Berezin integration rules must involve a dimensional scale parameter M. In different terms: the eigenvalues An appearing in the definition (4.21) of the ^-function should be normalized with respect to that scale. Hence, unless the number of eigenvalues of D and Do match (no zero-modes of D), there will exist an ambiguity associated with scale transformations of the measure in (4.7). It is easily recognized from (4.22) that this ambiguity is of the form £(0 | D)ln M. In order to make this ambiguity explicit, we should thus write [6] w J2 . = - [ C ' ( 0 | Z ? ) - C ' ( 0 | I > o ) ] + [ C ( 0 | / ? ) - C ( 0 | ^ o ) ] l n M

.

(4.25)

This scaling ambiguity still does not exhaust all the possible ambiguities. We return to a discussion of this point later in section 4.8. Meromorphy of ((s\D) The meromorphy of ((s | D) and its regularity at s = 0 are most easily recognized by introducing the (matrix valued) heat kernel associated with D, which satisfies the equation (jt+D\h(t;x,y\D)=0

,

*>0 ,

(4.26)

,

(4.27)

with the initial condition hap(0;x,y\D) where 5^(x,y)

= 6aP6^(x,y)

denotes the <5-function on the compact manifold Ad. We have, [ dfi{x)h(0;x,y

| D) = 1 .

The solution of (4.26) is evidently given by haP(t;x,y\D) 5

= ^2e-tx"ipn(x)aiptl(yh

>

n T h e spectrum of Do will, for the cases of interest, contain no zero-modes.

(4'28)

4.2 Functional Determinant, one-loop diagram where the sum includes the zero eigenvalues. h(t;x,y of the integral trace-class operator e~tD,

\ D) is the continuous kernel

= (x\e-tD\y)

h{t;x,y\D)

73

,

(4.29)

where {|^)} denotes an orthonormal set of states on M:

/•

dn(x)\x)(x\

=l

,

(x\y)=6W(x,y)

.

To prove the meromorphy of ((s\D) we define a generalized (matrix-valued) ^-function £(s; x, y\D) via the Mellin-transform of the heat-kernel (4.24), after subtraction of the contribution from the zero-modes, in order to guarantee the convergence of the (-integration at t = oo [7]: Caf3(s;x,y\D) Here, P^(x,y)

= -— 1 s \ ) Jo

dtt^lK^x^D)

- P^(x,y)}

.

(4.30)

is the (matrix-valued) projector on the zero-modes

i,S)(4.y) = E^0)(*)-^or(»)-8 • n

Note that the integrand in (4.30) decays exponentially with t —• oo. Using (4.28), and performing the t-integration in (4.30) one finds C(';*,y|2>) = £ ' *

2

%

M

,

(4.3D

and in particular f dn{x)tiC(s;x,x\ D) =((s\D) . (4.32) /• For D an invertible operator (no zero-modes) we recognize in ((s; x,y | D) just the meromorphic extension of the kernel of D~s:

as;x,y\D)

= (x\D-s\y)

.

(4.33)

In the mathematical literature [4, 8] also the notation K-s(x,y;D) denote the r.h.s of (4.33). In this notation (4.32) takes the form C(s \D) = J dn(x)tTK_s(x,x;D)

.

is used to

(4.34)

The definition of the kernel K-S(x,x; D) can be extended to include any (Hermitian or non-Hermitian) invertible, elliptic differential operator of order m defined on a compact manifold without boundary [4, 8]. The meromorphy of £(s;x,x | D) is now easily demonstrated. The only singularities in s of (4.30) arise from the lower end-point of the (-integration. Choose e to lie within a small interval around t = 0 for which the Seeley expansion (4.20)

74

Determinants and Heat Kernels

converges. Splitting the integration range into [0,e] and [e, oo), we have for an operator of even order m, (is;x,x I D) = ~ — ~ ^

'

'

f dtt*-1-

r(s) (4TT), (4TT)£ JO

h o r f ) - (47r)™i-P ( 0 ) (x,x)} + regular

x{(a0 + a1t +

,

(4.35)

where r is to be determined below and where regular stands for the remaining holomorphic part. The poles arising from the lower endpoint integration are evidently located at s=

k , k = 0,1,2, ••• . m Now, JTTJY has simple zeroes at s — 0, —1, —2, • • •. Taking d/m to be an integer,6 we see that ((s; x,x\ D) has simple poles at s = 1,2, • • •, — (— integer) , (4.36) m m but not at s = 0. This proves the announced meromorphy of £(s; x,x \ D) in s, and its regularity at s = 0. According to (4.36), r is to be chosen r > ^L. We have [7] for the residue at the pole at s = — — k

In particular, since s = 0 is a regular point, we obtain from (4.35), aO;x,x\D) = ^f^-pW(x,x)

.

(4.37)

(47r)m

Now tr J d»(x)P^

(£, x) = Y,f M&)
,

with the trace taken over the matrix indices, and N is the number of zero-modes. It therefore follows from (4.37) and (4.32) that C(0\D)=ix-^r-N

,

(4.38)

(47r)m

where, in general, ak =

dfj,(x)ak(x)

.

(4.39)

The Seeley coefficients play a central role in the computation of functional determinants. 6

In this book we generally have m = 2, i.e. d/m = 1.

75

4.2 Functional Determinant, one-loop diagram

4.2.2

One Point Compactification

Before we demonstrate how the Seeley coefficients may be calculated, we shall comment on the manifold M. the operator D is to be defined on. Stereographic projection of Dirac operator Theorems about differential operators usually refer to operators defined on a compact manifold, or to boundary value problems. Under suitable asymptotic conditions one may compactify SRd. The most familiar method is that of stereographic mapping of 3?d onto a d-dimensional sphere Sd in d + 1 dimensions, whereby infinity is mapped either on the north or the south pole [9]-[12]. In the case of the Dirac operator (4.4) this presupposes a sufficiently good asymptotic behaviour for the gauge field A^. Thus, a given gauge field tending to pure gauge at infinity will in general live on the stereographic sphere on a patch excluding either the north or the south pole. If the Dirac operator is to exist on the whole sphere, then the winding (Pontryagin) number 7 associated with A^ will have to be integer. Consider, in particular, the stereographic projection of 5t2 onto a two-dimensional sphere of radius R, with infinity being mapped onto its south-pole. Let ? be the vector from the south pole to the point (£1,2:2) G Jt2 (see Figure 4.2). If R is the radius of the stereographic sphere, its length r' is given by 2R

'

(4.40)

cos$

where $ is the angle between f1 and the z-axis. If we denote by

Figure 4.2: Stereographic projection of the real plane onto a two-dimensional sphere. If we denote by r" the segment of the vector f extending from O to the intersection point of f1 with the sphere, we also have r"sintf = i?sin6>

,

r"costf = R(l + cos6)

,

(4.41)

were 9 is the polar angle defined in Figure 4.2 and r" = | r" |. From (4.41) one obtains sin<9 tan v = . l + cos0 7

In the mathematical literature it is referred to as the first Chern number [14].

76

Determinants and Heat Kernels

Combining (4.40) and (4.41) we have

M = 1,2

^ = ir^ ' with f=(R

(4 42)

-

-

sin 9 cos
4R2-x2 ^ i ^ , , ,

,

&=¥>

x

»

, , (4.43)

-

Let (£M) be the angular variables 8 6=0

,

(4-44)

labelling a point on the sphere, and let ds2 be the square of Euclidean element of length ds2 — S^dx^dx,,. In terms of the variables (4.44) on the sphere, the element of lenght is ds2 = g\p(£)d£\d£p, with g\p(£) the induced metric, -

tt\

dx

-

»

dx

»

From (4.42) and (4.44) one finds ds2 = \ *

X 2

) da2 ,

with da2 = R2d82 + R2 sin2 Odtp2 ,

the square of the element of length on S2. Hence the stereographic mapping corresponds to a conformal mapping. In the literature it is common to consider instead the stereographic mapping of yM = 2x^ onto S2, which leads to the slightly modified formulae [12] 2 o D2

D„ Rr„

2R

r

» -

r

2

+ x

2

X

»

r3

>

~

2 / D2 (R -x„ 22 \R2+x2

\

R

,

(4.45)

and correspondingly ds2 = Cl^da2

2R2 ~ nr R T+ x~ x2

, with QR -

2

'

(4.46)

From (4.46) we see that da2 = gliu(x)dxtldxv

,

with gltv{x) = Sl2R{x)5ltv

•

(4.47)

The integration measure on the stereographic sphere is ^fgd2x, with g = det(g^) 8

£i i £2 correspond to £1 , £2 defined earlier.

= QR

.

(4.48)

77

4.2 Functional Determinant, one-loop diagram

The Dirac operator (4.4) will be of central importance in our discussions. We now show how to project it from SR2 onto the stereographic sphere. Proposition 1 i l ^

= ^n2R(x)VHsablab

+ l)V

,

(4.49)

where

and where sab, lab are the spin and orbital angular momentum operators, Sab — 2€abc&c

With pa = \-gr Proof: Using

,

Lb = raPb ~ HPa

,

(4.50)

. M

1 ( dx„\

d

i \drfij

dxv

it is a simple algebraic exercise to show that, • a

i-

iRR2-x2

A ~ M ~ „ ~ „ , -arii

R2

2

J

H

As a consequence, we have for lab, ifxv —

tyx^Oy

XJ/O^J

x

,

- — {R2 -x2)d,j,

h^ = -^ llxvdv

.

Using these results proposition 1 is verified by explicit algebraic calculation. Relation (4.49) converts the Dirac operator «7M9M into the hyperspherical operator SabLb + 1Proposition 2

p = -^n2R(x)vHsabLab + i)v , where Lab = lab + lab, with lab given by e lab = --^(raAb

-rbAa)

,

and A.=

\&

+ *iAp

+

! g A .

,

•

(4.51)

78

Determinants and Heat Kernels

Proof: First of all, we check, that A^ lives in the tangent space of the stereographic sphere: raAa = 0 . The proof proceeds again algebraically. One finds, after some calculation, "

G

"•

G.

+ ^ ( - R 2 - z 2 )(7 • A) + -jj(x • A)(-y • x)

I = sablab = -e^e^x^Av

.

Using the relation l^lulx

= S^-yx - <5MA7„ + ^A7M

,

one finds after further calculation, fi2 2R\

tt-'-rK^-Jr)*" • R J \ R

Combining this result with our previous result for i @ one obtains [12] (i?+4)

= jp2RViDt3V

,

(4.52)

with Ds2 =1 + sabLab being the Dirac operator on the stereographic sphere.

4.2.3

The associated Dirac operator

The eigenvalue spectrum of the (dimensionless) operator Dsi is, evidently, discrete Ds2un(Q) = A n ti n (fi)

,

(4.53)

where fl stands for the variables (4.44) defined on the sphere. The eigenfunctions are chosen to be orthonormal

/ dSlul(n)um(£l) = 6nm , and are complete, u n (ft)„4(fi')/3 = Sa062(Q - n')

^

.

n

In order to establish the link with the usual Dirac operator, we consider instead of the eigenfunctions un, the functions un defined by Un = RVun

.

From (4.52) and (4.53) it then immediately follows that i flun = X„QRun

,

(4.54)

4.2 Functional Determinant, one-loop diagram where A„ = \n/R.

79

Since

w - ^ - n i ' . * = > ' = $*' it follows from (4.53) that, / d2xftR(x)ul(x)um{y)

= 6nm

,

YJ^n{x)av)n{y)fi=^-R152{x-y)

(4.55) .

(4.56)

Hence, solving the eigenvalue problem (4.53) is equivalent to solving the associated eigenvalue problem (4.54) with (4.55) and (4.56). Since ,/g = fi^j, expressions (4.55) and (4.56) are not yet in the standard form. In order to get them in standard form, define __i

u„ = n R 2 u „

,

(4.57)

i $> = ttx*i # f t |

•

(4.58)

Then the eigenvalue problem reads i $>un = \iiin

,

(4.59)

with

^2un{x)aun(y)0

= —5a05{2)(x-y)

.

One easily verifies that the operator i p is Hermitian with respect to the measure dPxyfg. The usefulness of the stereographic projection resides in the following observation. Consider the functional ew<*>

fVipv^e-f42*^™

fThpThf>e~S'px^*^*

where we have assumed the Jacobian of the transformation tp -»• ip to cancel in the ratio. (This we expect to be true if the dimension of the space spanned by V> and i> is the same, i.e., if there are no zero-modes). Expand now tp in terms of the eigenfunctions of i fi{i 0), r/i = J^r bnun and make the change of variable fVifjVifi -> / Y[n dbndb„. We then obtain (the last manipulation is formal) efrip-ui9)

=

U^njA]

I1M0]

=

detifi

detifi

=

det[Si^i

$0.%) _ detiT/)

det[fi~*i 0fi£]

deti

9

Hence, as a result of the underlying conformal invariance, the dependence on R formally drops out, if there are no zero modes. (See section 4.8 for a more detailed discussion of this point).

80

Determinants and Heat Kernels

Vortex on the stereographic sphere Let us emphasize that the one-point compactification of the Dirac operator discussed above presupposes that the gauge field A^ lives on the corresponding compact manifold M. This will be the case if A^ tends to a pure gauge-field configuration at infinity, i.e. ->• g-1{x)dfig{x)

eA^x)

,

g{x) € SU{N)

.

(4.61)

\x\ —• oo

Afj, then exists on coordinate patches of M., as we now illustrate. Example: Let A^ be given by the U(l) vortex-field

we have A^x)

-»• ~-e^x^

= \d»k

,

(4.63)

with A(x) = narcctn—. Hence the vortex-field (4.62) has the property (4.61) of being pure gauge at infinity. As we now show, it admits, as a consequence, a one-point compactification if n is an integer. Identifying Euclidean infinity with a circle of radius R -> oo , and parametrizing £1,2 on this circle by R and an angle 6, Xi = RsinO, x2 = Rcos9, we have A(R,0)=n6

.

(4.64)

Hence (4.62) corresponds to the Pontryagin number n: 2 fddh x

fdn

According to (4.51) the field corresponding to A^ on the stereographic sphere of radius R is given by > J" =

iM = - ^ e ^ ^

1 2

.

» 43=0

.

Let us write Aa, a = 1,2,3, in terms of the coordinates of that sphere, with the south pole identified with Euclidean infinity. From (4.45) and (4.43) we have R sin 9 cos ip 1 + cos 9

R sin 6 sin

Defining a unit vector tangent to the stereographic sphere, pointing in the azimuthal direction ev = (— sin<£,cos
n

sin<9

„

~

4.3 Calculating Seeley Coefficients

81

We see that A does not exist globally on the stereographic sphere, but only on a patch which excludes the south pole. We could have considered however equally well a stereographic mapping where the north pole is identified with infinity. In that case one obtains -, _ 4

n

sin#

„

-, _

=

2^Rl-cos96'p

'

^3

=

°

'

Thus A and A' live on two different, but overlapping patches defined by (7)

- 7T < 9 < 7T ,

(II)

0 < 6 < 27T

respectively. In the region of overlap, both A and A' are defined and are in fact related by Al

A

U

eRsmv Recalling that „ d Id I d 5 = e'rQr _ — + ~<>rd6 e - — + ~vrsm6d

i->^

f**i

j

'

we therefore see that ^4' is just the gauge transformation of A, A' = A + -VA e

,

with A given by (4.64). A' and A thus differ by a bonafide gauge transformation if n is integer. Correspondingly gauge invariant quantities exist in this case globally on the stereographic sphere.

4.3

Calculating Seeley Coefficients

There exists a variety of methods for computing the Seeley coefficients. Among these the method of Gilkey is quite general and allows for the computation of the Seeley coefficients of a very large class of not necessarily positive semi-definite operators. We shall however discuss more pragmatic approaches such as the perturbative approach, as well as the methods of Schwinger-DeWitt and Fujikawa.

4.3.1

The perturbative approach

The perturbative method involves an iterative procedure for solving the heat-kernel equation in terms of a power series expansion in the potential describing the deviation of the operator in question from the non-interacting case. We illustrate the procedure for the case of the Euclidean Laplace-Beltrami operator A = -^dagaby/g-db v9

,

(4.65)

82

Determinants and Heat Kernels

with g = det gab{x), where gab(x) is the Euclidean metric denned on a two-dimensional manifold M. The operator (4.65) will play a central role in Chapter 18. The functional determinant of (4.65) is invariant under conformal transformations (see Chapter 18), so that we are free to calculate it in the so-called conformal gauge, where the Euclidean metric tensor gab takes the form 9ab(x) = e*W6ab

,

with 4>{x) a conformal Weyl field. Correspondingly we have for Eq. (4.65) in the conformal gauge A = e-^d2 , d2 = Sabdadb • (4.66) To obtain the Seeley expansion of the corresponding heat kernel h = etA we follow the procedure adopted by O. Alvarez [13] by writing Eq. (4.66) in the form A = 82 + V

,

where V=(e-*-l)d2

.

In operator notation, the corresponding heat-kernel equation (4.26) can then be written in the form (dt - d2)h = Vh which, taking account of the initial condition (4.27) is readily shown to be equivalent to the integral equation

h(t;t,V) = h0(t;t-r])+JdtlJd2l;'h0(t-t';t-e)V(l;')h(tl;e-V)

.

(4.67)

where ho(t; £—£') is the matrix element of the heat-kernel ho — eta for the Laplacian —d2 in two-dimensional flat space,

M*;e-?) = ^ e - ^ , also satisfying the initial condition (4.27). By setting £ = r\ and iterating this equation, we obtain from here the Seeley-DeWitt asymptotic power series expansion in the parameter t. It is easy to see that to order 0(y/i) it is sufficient to consider the first iteration of Eq. (4.67) :

h(t; £, 0 ~ Mi; 0) + / df J d2d'h0{t - *'; f - £')Vtf)W\ £' ~ 0 • (4-68) Hence to this order, we can regard the potential as a perturbation. The calculation is simplified by using the conformal invariance in order to make a judicious choice of coordinates. Selecting a particular point f' in our manifold M, one can choose local conformal coordinates such that the Weyl field appearing in the conformal factor in (4.66), as well as its first derivative, vanish at that point:

0(0=0,

0„0(O = O

(4-69)

83

4.3 Calculating Seeley Coefficients

We choose (4.69) to hold at the point £ labeling the left-hand side of equation (4.68). Noting that in the limit of interest, t -» 0, the free heat kernel h0 in Eq. (4.67) is exponentially damped for £' away from the point £, we conclude that we need to know the potential V(£') only in a small neighbourhood of the point £, where properties (4.69) are supposed to hold. Assuming <j>{x) to be a smooth function of x around x = £, it can be regarded as small in this neighbourhood, so that we are allowed to make the approximation

v(o ^ -\(e - zne - tfdadtma2 + o((e - o3) • Equation (4.68) reduces correspondingly to

f dt' f d2C

h{t;Z,0~h0(t;Q)+

x ho(t - t';e - o \-\(e - tne - obdadb(o ^ d2h0(t'-,e-oA straightforward calculation yields h

^^0

= -^-t-^d24>(0+O(Vi) .

(4.70)

This is the desired DeWitt-Seeley expansion for the Laplace-Beltrami operator in the conformal gauge. The corresponding determinant is going to be computed at the end of section 4.5.

4.3.2

The Schwinger-DeWitt method

This method involves a recursive procedure based on the coincidence limit of well known Schwinger's proper-time representation of the heat-kernel. We illustrate it for a class of differential operators of particular interest to us. The result is contained in the following Theorem: Let A be the Hermitian order-two operator defined on 5Rd, A = DlDli+X

,

X* = X

,

(4.71)

with Dfj, = dfj, — iAp, where A^ and X are Hermitian matrix valued fields. Then, (x | e- e A | x) = — — r [ l + ai(*)e + a2(x)e2 • • •] (47re) 2

with ax = -X a

2 = \x2-\{D2X)

+ ^[Dll,Dv]2

.

(4.72)

Determinants and Heat Kernels

84

Proof: We seek solutions to the heat equation 9 {-

+ (DlDtl+X)}h(t;x,y)=0

,

with the initial condition, h(0;x,y)

= 6^{x-y)

.

(4.73)

The singular behaviour (4.73) in x, y suggests making the Ansatz h(t; x, y) = f(t; x, y)h0{t; x, y)

,

(4.74)

with h0(t;x,y)

=

e~» , (Ant) 2 and a = |(a; — y)2. The function ho satisfies the equation (--A)ho(t;x-y) = 0

,

h0(0;x-y)

.

(4.75)

(4.76)

with the initial condition = 6^(x-y)

Now,

£>],£>„ = -d^ + itfM a u

+ 2iA„dll + A^

d a

» u

A simple calculation yields, using (4.76) DlD^fho

= ho(DlD^f)

- h0 ( ^ ) (£>t -

Dll)f

- h0dtf

.

Substitution of the Ansatz (4.74) into the above equation then leads to a differential equation for / itself:

3

Af-?£{Dl-Dlt)f

+ dtf = Q .

(4.77)

Following de Witt [15, 7] we make the Ansatz f(t;x,y)

= b0(x,y) + h(x,y)t

+ b2(x,y)t2

+ •••

,

substitute into Eq. (4.77), and compare powers in t, obtaining r1

: tl :

9

d^aD^bo = 0

,

Abt - ^a{Dl - D^)bl+1 + (I+ l)bl+1=

(4.78) 0,1 = 0,1,-••

(4.79)

We omit specification of the operator A in the argument of the heat-kernel and zeta function.

4.3 Calculating Seeley Coefficients

85

Let us denote the coincidence limit y —>• x by bi(x,x) - [bi](x) = at(x)

.

From the initial condition (4.73) one has a0 = [b0] = 1

•

In general we have for I > 0, ai = --[Abi-i]

,

where the bracket denotes the coincidence limit y —> x. Calculation of a-i From (4.79) we have ai(x)

= [Ab0] = [DlDpboKx) +X(x)

.

Now, from to the parallel-transport equation (4.78) one finds, [£>+DM6o]=0

.

Hence we finally obtain ai = -[Abo] = -X

.

The calculation of the higher Seeley coefficients by the de Witt method becomes increasingly involved. We leave the calculation of a-i as an illustration using the Fujikawa method in the following subsection. Example 1: The Dirac operator of

QED2/QCD2

As we have seen, the Seeley coefficients are defined for compact, semi-positive operators. Hence the above technique is not directly applicable to the Dirac operator (4.4). We can however circumvent this problem by making the following observation: from (4.31) we have

as;x,x\^m

= E'"U*xlyX)

•

Here An and un are the eigenvalues and eigenfunctions of the eigenvalue problem (4.59), and the sum extends over the positive and negative eigenvalues A„. Using (4.57), (4.58) and (4.48) one has

^as;x,x\(i$>)2)

= nR(x)as-,x,x\(im2)

with

tr C ( S ;,,,|(^) 2 ) = E '

! f

^

M

,

(4.80)

86

Determinants and Heat Kernels

Furthermore, defining

A„>0

*•

n

'

A„<0

n

*•

'

and using the fact, that for every eigenvalue A > 0 of i P with eigenfunction un there exists an eigenvalue — A < 0 with eigenfunction 75 un, one has tr((0;x,x\iP)=tr((0;x,x\{ifl)2) tT{'(0;x,x

, \ (i J?)2)

\i $>) =-tiC'(0;x,x

(4.81) .

(4.82)

It thus suffices to calculate the Seeley coefficients a& of D = (i P)2, with i fl = ift+ e 4

,

(4.83)

and A^ a Hermitian Lie algebra valued field. They are related to the Seeley coefficients dfc of (i p)2 by ^fgak = ^Rdk- We have {iip)2 = D\D^X

,

(4.84)

with X=-biinAF^

,

(4-85)

and Fpu = d^A,, - dvA^ - i[A^ Av\

.

(4.86)

Hence it follows from (4.81) and (4.82), ai = - j [ 7 / i . 7 ^ ] ^

-

(4-87)

and in two dimensions, 10 e ai = -zlh^vF^

4.3.3

.

(4.88)

T h e Fujikawa m e t h o d

It is remarkable that the Seeley coefficients do not depend on the dimension of spacetime. This is easily understood in the context of the Fujikawa approach where one departs from h(t;x,y\D)=e-tD*S2(x-y) , (4.89) and represents the ^-function as a Fourier integral. Expression (4.89) then takes the form h(t;x,y\D) = J-^e-^ie-^e^)

.

Let D be again the square of the Dirac operator, D = (i fl)2. We expand the exponential in powers of tDx. To this end it is economical to treat e~tDew'x as 10

In d = 2 our conventions are if^-y5 = e^vlv

and \[l^,lv)

= if-y.v'V'•

(4.90)

4.3 Calculating Seeley Coefficients

87

an operator acting on some function f(x) to be set equal to 1 at the end of the calculation. Noting that e-ipxD»eipx =ip^ + Dfi , and e~ipxDeipx

= p2 - 2ip• D + D

,

we have for the diagonal part of the heat kernel after a rescaling \fip = k, 2

,x\D) = J- d k

-(k2-2iy/tk-D+tD)

or expanding in powers of t, we have h(t;x,x\D)

e~k2 ^

= —

- fey/ik • D - t£>)

.

(4.91)

1=0

We now expand the binomial as i

(2iVik • D - tD) =J2

(-)l~jt'~i/2(2ik-D)jDl-j

J2

,

(4.92)

j=0 dist.perm

where the sum over all distinct permutations replaces the usual binomial factors, since the binomial involves non-commuting operators. Noting that the terms odd in kfj, give a vanishing contribution to the integral, we can set j = 2r. Collecting the terms of a given order in t, we can cast (4.91) into the form of a Seeley expansion h(t;x,x\D)

= Y^tmam{x)

,

(4.93)

ra=0

with the Seeley coefficients given by 11 (4.94) r=0

dist.perm

The Gaussian k integration is elementary. Using in particular d

f ddakk J ird/2 Id 1

/ ^72 ddk

/

e

2 k

k k

* i*k»= <W

.

(4-95)

1

—JJ^ e~ can k^Kk\kp + 5y.\8ofvpthe + S^S^x) Actually these coefficients be shown= to-(S^Sxp be independent dimension, of space-time d. This is the consequence of the independence of the fc-integration of d. 11

88

Determinants and Heat Kernels

performing the permutations in (4.92) and setting equal to zero all terms involving differential operators acting on a constant to the right, one finds for the first two non-trivial Seeley coefficients the result oi = -X

(4.96)

a2 = \x2

4.4

- liD^X)

+ ^[D^,D„][D„

Dv] .

Computing Functional Determinants

Although £(0 | D) may generally be calculated exactly in terms of the Seeley coefficients, a similar computation of UID = In det'D

,

will only be possible for a restricted class of operators. All methods are based on the behaviour of det D under an infinitesimal change D —> D + SD. They however correspond to different regularization procedures. 4.4.1

(-function regularization

Here one departs from the definition [6, 16, 17] wD = -('(0\D)

,

and considers the infinitesimal variation 5UID =

a.c.

-±(((s\D

+

8D)-as\D))

s->0

= a.c.^-\sti'{D-s-15D)} , J s-»o ds L where a.c. stands for analytic continuation and tr' denotes the trace over the subspace orthogonal to the zero-modes of D. In the case where tr'(D~s~15D) admits an analytic extension to s = 0, one evidently has 6wD = [ti'(D-,-16D)]s=o

•

(4-97)

Comparing this result with aetU as obtained by using the familiar property lnde*D = trlnZ?

,

(4.99)

of determinants, we see that (4.97) defines a particular regularization of the generally divergent trace (4.98).

4.4 Computing Functional Determinants

89

The variation (4.97) is explicitely computable in terms of the corresponding Seeley coefficients if D~15D is of the form D-15D

= ^2D~'y"pnD^

,

(4.100)

n

with 7„ natural numbers and {pn} local operators with the property = Pn(x)6d{x-y)

(x\pn\y)

.

Indeed, using the cyclicity of the trace, we find in this case 6wD = {tr[D-sD'18D]}

= {^

= I dn(x)ti[C(0;x,x

| D's

f dfi(x)ti[(x

| D)p(x)}

\ x)pn{x)]}

_

,

(4.101)

where

= ]T>» • Hence for 5D of the form (4.100), cients a,k{x | D).

SUJD

is computable in terms of the Seeley coeffi-

Example : QED2 axial anomaly Consider the Euclidean chiral U(l) transformation i p -s- e 75,5e(l) i jp e^W

,

(4.102)

which implies (actually valid for arbitrary 56), 6(i JJ)) = i #>75<50 + j5S6i

Defining D = (if))2,

TJ) .

we have, correspondingly 6D = lh56D +

2VD-J566\/D

+ Dj556

,

(4.103)

where y/D = i1J). The result (4.103) falls into the category of variations (4.100). Using the cyclicity of the trace we have from (4.101), using (4.37), 5tu{ip) = l-5uj(ip)2 =l-fd2xAtr

fe&

- P<°>(x,x)\

l556(x)

.

In the absence of zero-modes we thus have, using (4.85), 5oj{ip) = J d2x A(x)56{x)

,

(4.104)

where Ax)

= ^r^F^(x)

.

Z7T

is just the anomaly in the divergence of the axial vector current.

(4.105)

90

Determinants and Heat Kernels

4.4.2

Proper-time regularization

Consider representation (4.30) for the ^-function. We have C(s;x,y

T'(s) r°° dt p ^ J jts[h(r;x,y

| D) = -

+

\ D) -

TtSlntlh(t>x'y\D)-p°(x>y^

W)I

P0(x,y)}

( 4 - 106 )

•

For s > 0 the integral converges, and we may replace /0°° by / ° ° . Keeping e > 0, we may now go over to the limit s -t 0, obtaining the alternative (proper-time) representation [13] ytr(e-tD-P0)

.

(4.107)

Note that the projector PQ insures that the integral converges at t = oo. We now derive the analog of Eq. (4.97) in the proper-time regularization scheme. Consider again the case where zero-modes are absent. Then uD+6D - *D = - J" j tr( e - ' < » + " » - e~tD) . Now, e-t(D+6D)

tr( e-^D+SD^

_

e-tD

=

_ f1

dX e-XtD{tSD)

Jo tD - e~ ) = -t tr e~tDSD

e(X-l)tD

^

.

Hence, OO

8OJ

/

dtti e~tD5D = tr{e~eDD^SD)

.

(4.108)

Comparing this result with (4.97) we see that (4.108) and (4.97) represent the proper-time and C-function regularization of the formal expression (4.98), respectively. If D~1SD is a local operator, we need to know only the small e expansion of the diagonal elements of e~eD', as given by the Seeley expansion (4.20), i.e. {x\ e-eD\x)

= -\T[a0+ea1 (47re) m

Example: QED2 axial anomaly From (4.103), (4.108) and (4.109) we have -SU>D

in agreement with (4.104).

= / d2x

A(x)56(x)

+ ---]

.

(4.109)

4.4 Computing Functional Determinants

4.4.3

91

T h e Fujikawa point of view

The functional 5U>D can also be viewed as the logarithm of the Jacobian of the transformation VcDc —> Vc'Vd, in (4.6), as we now show. This is the point of view taken by Fujikawa [18] who was the first to develop a method for studying chiral anomalies in the path-integral framework without resorting to perturbation theory. Fujikawa's observation of the non-invariance of the fermionic measure VtpVip in QED under chiral transformations provided an important break-through in the conceptual understanding of chiral anomalies from the path-integral point of view. In particular, it provides a systematic method for a transparent derivation of anomalous chiral Ward-Takahashi identities within a purely path integral framework, without the need of explicitly adding anomalous terms — calculated from Feynman graphs — to the action. In fact, it is understood by now that all known anomalies can be viewed as being a result of the non-invariance of the functional measure under some symmetry transformation [19, 20]. Nevertheless, as we now demonstrate, Fujikawa's approach is just another variant of what was said above. In the Fujikawa approach the definition of det ^ t " l D D> in terms of the path-integral (4.6) represents the starting point. He considers the situation where the transition from D to D + 6D can be seen as a change of variable of integration. Thus consider det[D + 6D] __ det[D] ~

JVcVce-^D+5D^ JVcVce-SM

(4 110)

'

'

with S[D] = f ddxcDc

.

Now, S[D + 5D] = f{c(l

+ ^8DD~X)D(\

+ 0({6D)2)

+ ^D^SD)^

.

Hence, redefining the fields as c -» c' = (1 + -D-l5D)c

,

c ^ ^ ^ l

+ W

1

)

,

(4.111)

2,

L

we have S[D + SD] = fddxc'Dc' formation (4.111), defined by

+ 0({SD)2).

Hence the Jacobian J of the trans-

VcVc = JVc'Vc1

,

(4.112)

•

< 4 - 113 )

is just the l.h.s. of Eq. (4.110), that is T J

=

det(D + 5D) detD

To evaluate the Jacobian factor we expand c(c) in a complete set of orthonormal functions (fn: c a; = ( ) ~52an
,

a' = a'U

,

where (we assume again D~X5D to be local operators), dfi(x)^prn(x)-D~18D(pn(x)

Umn = Smn+

,

1

+ / dfj,(x)(pm(x)-SDD

(fn(x)

.

Now, using the property (4.99) of determinants, we have Y[ da'nda'n = Y[ dandan e~tr n

ln

< Ft/ >

.

n

Note that due to the Grassmann character of the a„'s, the Jacobi determinant is just the inverse of what one has in the usual (bosonic) case [21]. With the usual replacement

VcVc -» Yl d^n I I d a n n

'

(4.115)

n

one has VcVc=

e=S„/

(i

''(I)^(1)(l!DB~l+D"1,SD)^(I)Dc'Dc'

Since J^ n ipn{x)Tpn{x) — 5(0), the above expression is divergent and in general requires regularization. Formally we have however VcDc=

etrD~l5DVc'Vc'

,

or, with the definition of J given by (4.112), \-aJ = tTD-l5D

= 5ujD

,

(4.116)

in agreement with (4.113) and (4.98). Note that the change in sign in ln J relative to the bosonic case was essential for this agreement. Although the calculation of (4.116) cannot depend on the choice of basis functions, it will depend on the choice of regulator. The choice of regulator depends on the symmetry one wishes to preserve. It should however be chosen such as to make the trace in (4.116) well defined. Fujikawa considered [18] the case of D being the square of the Dirac operator i P, and 8D as corresponding to the chiral transformation (4.103).12 For this case (56 is to be regarded as an operator) 5w{m2 12

= Iim4tr[e-£'D75<5(9]

,

(4.117)

Actually Fujikawa considered the active point of view of calculating directly the Jacobian J of the transformation (4.102).

93

4.4 Computing Functional Determinants

where e~eD is a damping factor introduced in order to regulate the trace. Fujikawa chose a manifestly gauge invariant regularization with D = (i fl)2. His evaluation of the trace (4.116) did not make use of Seeley's expansion, but followed the procedure to be outlined next. Write (4.117) in the form Sutf, = f ddxA(x)56{x)

,

(4.118)

with A{x)=2]imYi
.

(4.119)

n

Note that A corresponds to a regularization of Y^n iPnix)l5'Pn(x) which Fujikawa has referred to as primary definition of anomaly [18]. In terms of Areg the Jacobian (4.116) reads for a finite chiral rotation J(0)

2

=eifd

x8(x)A(x)

_

(4.120)

Although the choice of
as an operator)

(i V>)2 eipx = {-D^D^ + X) eipx , [-DuD* + X] eipx = eipx [(p„ - *£>M)2 + X] . Hence, A(x) = 2 lim t r i 75 / 7 ^ 7d e-^-2ip.D-D»+x] £->o+ |^ J (27r)

\ J

Making the change of variable yfep — k and expanding the resulting integrand in powers of e one sets A(x) = 2£lrm+ ~ ttfafj^

e~k2[l + 2iVik • D+eV2-eX

+
For two dimensions we need not compute higher order terms in e. Using JA-4s e~fc2 = -^, and tr7 5 = 0, we have for the axial anomaly A=-—ti{j5X}=^-e^F^

,

(4.121)

which, when substituted into (4.118), yields (4.105). The fact that no further subtraction is needed is a consequence of working in two dimensions.

94

Determinants and Heat Kernels

The result (4.121) agrees with our earlier calculations, and demonstrates explicitly the independence of the result on the choice of basis. The same result is obtained by working with the eigenfunctions of i qi [13]. On the other hand, working with the regulator 1 = lime_>o e~e(-l^> , one finds A = 0, i.e. a trivial Jacobian [22]. (This is evident from the above calculation leading to (4.121)). , 4 = 0 corresponds, as we shall see, to the absence of an anomaly in the divergence of the axial vector current, although at the expense of gauge invariance. Indeed it is well known, that there exists no regularization respecting both gauge and chiral invariance. Basically one is confronted with defining what has been referred to by Fujikawa [18] as the primary anomaly, i.e. A = 2^
amounts to computing n

where An are the eigenvalues of i JJ). As was shown by Fujikawa, the replacement

where f(z) is any smooth function which rapidly converges to zero as z —> oo, leads to the same result for A, thus indicating that the sum representing the primary anomaly is conditionally convergent. Formally this sum states that A = 2 t r 7 5 . The fact that it does not vanish - contrary to what one would naively expect has its origin [24] in the spectral asymmetry of the operator P, which in fact conforms to the Atiyah-Singer theorem [25] (see section 7). This is exemplified by the aforementioned non-covariant choice of the regulator, which leads to a vanishing anomaly. It is interesting that conventional perturbation theory is in agreement with this chiral asymmetry, if a gauge invariant, Pauli-Villars regularization is employed. In the path-integral framework, the anomalous Ward identity for the divergence of the axial vector current now emerges as follows [26]: the Pauli-Villars regularization of the fermion-loop diagrams is equivalent to the addition of Cpv = (pi ft(p — M is a spinor field obeying Bose statistics. Thus there exists an additional contribution to the functional measure transforming under a chiral transformation with just the inverse of the Jacobian (4.113),

v~4>V4> = j-xv~$v$

,

showing the invariance of VipVipVcpVcp for finite Pauli-Villars mass. The corresponding Ward identity now follows in the normal way [27] from the transformation property of the modified Lagrangian under the simultaneous local chiral transformations ip(x) ->• eia(xh^(x) and • oo. Finally, let us remark that we have supposed in our analysis that D was a positive semi-definite operator. This is true if D = (i ft)2, but is no longer true, if i ft is replaced by the (in Euclidean space non-Hermitian) Dirac operator with axial coupling of the gauge field, ift = i ?)+ A~f5. Various methods have been proposed to circumvent this problem [27, 20, 28, 29].

4.5 A Theorem on a one parameter family of factorizable operators 95 One obvious way consists in considering the operator i ft 4- i ^ 7 5 obtained by analytical continuation from A^ to iA^, and then to continue back at the end of the calculation. It is not surprising that the final result is found to agree with the result obtained by naive application of the ^-function method to the non-Hermitian operator ip. Another approach was considered by Fujikawa [27]. It consists, in considering the eigenfunctions of the Hermitian, positive semi-definite operators ft ft andftft: $P
= \l
;

W V n ( z ) = >?nn{x)

,

An G R

.

For a suitable choice of phase this corresponds to ftn

!

ft

4>n = Kfn

•

(4.122)

The Fujikawa analysis is then repeated by expanding tp(ip) in terms of the complete set of eigenfunctions tpn{4>n)- Since a gauge transformation induces a similarity transformation on ft ft andftft, the eigenvalues A„ are gauge invariant. Hence the outcome of the calculation will be gauge invariant and thus cannot coincide with the result obtained by analytic continuation in A^.

4.5

A Theorem on a one parameter family of factorizable operators

Starting from the proper-time representation (4.107) for the determinant of an operator, we now prove a theorem about a class of operators, which will play an important role in our discussion of two-dimensional gauge-theories. Theorem: [30] Let Kr be an invertible operator parametrized by r having the polar decomposition Kr = nrQfir , (4.123) where fir and Q,r are local operators in d dimensions, (x\nr\y)

= nr(x)6d(x-y) d

(x\nr\y)=Tir(x)6 (x-y)

, ,

satisfying the boundary conditions fi0 = ^o = 1, and where Q does not depend on r. Define (r):= In detK$Kr . Then T"^ ar

= 7 7 - 7 X / d2xti(Wr(x)+Wr(x)) (47re)'« J

(Airs) ™

fd2xtT{Wr(x)a^(x)

+ Wr(x)b^(x)}

(4.124) + 0(e2

96

Determinants and Heat Kernels

where Wr(x) and Wr(x) are the Hermitian matrix valued functions (the dot denotes differentiation with respect to r)

wr(x) = {nr{x)n;1(x)) + ((ir(x)n-1{x))^ Wr{x)

1

= (fi; {x)nr(x))

,

(4.125)

,

(4.126)

1

+ (fi~ {x)Ur(x))i

and where a\- and b\- ' are the coincidence limits of the first non-trivial Seeley coefficients in the small e expansion of the heat kernel associated with KrK\ and K\KT respectively: (x | e-'K*Kl

| x) = - 4 - r [ l + eal»{x) + 0(e2)]

,

(4.127)

.

(4.128)

(Aire) m

(x | e " ^ * - | x) =

+ebW(x)

-±-T[1

+ 0(s2)}

(47T£)-

Proof: To begin with, notice that the theorem makes a statement about a very special variation 5ru>D of a particular class of positive-definite operators which are parametrized by r. We have (for the time being we exclude the possibility of zero modes) U(r)

= - [°°-tie-tKlK*

.

Je t Differentiating with respect to r we have dtti{(klKr

+ KlKT)e-tK*K*}

.

(4.129)

Now, K\Kr

= K\{tlrSl-x)Kr

+ KlKr^tr)

.

Introducing these expressions in (4.129), using the cyclicity of the trace and the property we find ti{(KlKT

+ KtKr) e~tK^}

= ti{WrKrKl

e~tK^

+ Wr K\Kr

e~tK^}

= _ ± tl{Wr e-tKrKi + Wr e-tK±Kr} ) (4130) dr where Wr and Wr have been defined in (4.125) and (4.126). Upon substituting (4.130) into (4.129), the ^-integration may now be trivially performed, to yield ib{r) = tr[W r e-eK^

+ Wr e~eK^\

.

Making use of (4.127) and (4.128), we obtain the announced result (4.124). By integrating this expression in r from r = 0 to r = 1 we can in principle obtain det(KjKi) det S

In-

4.5 A T h e o r e m on a one p a r a m e t e r family of factorizable o p e r a t o r s Example: The QED2

97

determinant

Parametrizing the gauge field A^ in the Lorentz gauge as A^x)

= ^vdv0{x)

,

(4.131)

we have for the corresponding Dirac operator i If) = (ifi + w^duO)

5

b

»i 0e> aPi <> = e„77""t

We define the corresponding one-parameter family of operators Kr = e'^'i

Ve'i*9

,

corresponding to the gauge connection A£' {x) — rA^ (x), and Q = i $, ftr = fir = er^e. We obtain correspondingly from (4.125) and (4.126), Wr = Wr = 2-y59, so that (4.124) reduces for £• = 1 to duj f d2 = r I —^ tr (ai(x)"f56(x)) dr

= re f —^

6(x)elll,Ffiu(x)

Integration from r = 0 to r = 1 yields, upon solving (4.131) for 6(x), w(l)-w(0)=ln

det(t P)2

det(i 0) 2 ,2

= - ^

f d2x f d2yeXpFXp{x)DE{x-y)efu/Ffll/(y)

= -£ J d2x j'd2yAtl(5»v-d-^)Av

, (4.132)

.

In the Lorentz gauge, (4.132) reduces to the familiar Schwinger result [31]. Example: Beltrami-Laplace operator in conformal gauge The general idea underlying the derivation of the above theorem may be applied to a quick calculation of the functional determinant of the operator (4.66). For positive, this operator is negative definite, and we can work directly with it. Define the one-parameter family of operators Mr = -e-^d2

,

and correspondingly w(r) = In detMj. . Using the heat-kernel representation of the functional determinant, we have dt tr (Mre-tM-)

98

Determinants and Heat Kernels

or making use of the Seeley expansion (4.70), 1 iwe

u(r) = - I d2x

r 12?r

(4.133)

Integrating with respect to r from r = 0 to r — 1 we finally obtain In

det(e-^a2) _ 2

det(d )

1 24TT

fd2xd24>.

(4.134)

Since this result is quadratic in <j>, we can now drop the restriction on , and take (4.134) to define the functional determinant also for positive <j>.

4.6

The QCD2 functional determinant

The QCD2 effective action has been computed in a variety of ways. As examples we shall discuss in Chapter 11 the perturbative procedure, as well as the method of Polyakov and Wiegmann based on the integration of the anomaly equation for the axial vector current. Here we shall present a method based on heat-kernel techniques. The first calculations based on this approach have been restricted to the so-called "decoupling" gauge. Shortcomings of these calculations are that the corresponding gauge conditions (Alvarez's integrability conditions [13]) cannot be globally implemented, except for SU{2). The calculation below does not rely on such a choice of gauge [30]. Indeed, we may apply theorem (4.124) to a gauge-independent computation of W[J4]. TO this end we notice that in two dimensions A± can always be written in the form eA.. = Vid-V-1

,

eA+ = U'^id+U

eAll = ^{6llv-iellv)Vdl,V-1

,

(4.135)

+ ±{6llv+iellv)U-1dvU

,

with U and V taking values in SL(N, C), the complexification of We now rewrite the Dirac operator using the identity 7/lJ 4 M

= nP-VidyV-1

+ lvP+U~HdvU

(4.136)

SU(N).

,

with P+(~) the projector on right- (left-) handed fermions, P± = | ( 1 ± 75) . Define

v

0

1 u-M

'

u n-( ~\o

°v

u

Then,

o-ao f ° ^^=[_u-id+u_d+ = ifi + e4

Vd-V^ + d0

.

The Dirac operator thus has the structure of (4.123).

(4.137)

4.6 The QCD2 functional determinant

99

In order to apply the theorem (4.124), we define interpolating fields Ur and Vr depending on a continuous parameter r 6 [0,1], subject to the boundary conditions Ur{x):

U0(x) = l

,

U1(x) = U(x)

,

Vr(x):

V0(x) = l

,

Vl{x) = V{x)

,

with a corresponding definition of Clr, fir and A^J'. Then (4.137) becomes,

Now (compare with (4.84)) (i B[A^}f

= -DMD£)

- ^e^Ftf

,

where, F$ = d„AP-dvAP -ie[AP,AP] = [DP,DP]. Hence the corresponding first non-trivial Seeley coefficients is given by (see (4.88)) a{{\x)

= \l^vF^}

.

(4.138)

Following the notation of (4.125) and (4.126), we furthermore have

^=i'*T c>-Vi+^ Wr=[

U^Ur r r n 0

0

T „>_1 VrV-

)+hc.

Now, from A± = AT we deduce that U* = V

,

V* = U

,

so that Wr = (U-1Ur-VrV-1)l6

.

Furthermore, Wr = Wr. Applying formula (4.124) and using (4.138) we thus finally have w(r) = ^

J d2xtr[(U-lUr

+ W^^Ffi]

.

(4.139)

We next rewrite the r.h.s. in a manifestly gauge-invariant form. In terms of (4.135), the gauge transform of A^, A^ = g~1Alxg + kg~ldlig, reads eA%=i(g-1U-1)(d+Ug)

,

eAB_=i(g-1V)(d-V-1g)

.

Hence, under gauge transformation, U->Ug

,

V-^g-lV

.

100

Determinants and Heat Kernels

Since the functional determinant should be gauge invariant, we expect it to depend only on the gauge invariant quantity G = UV

.

We therefore define (jrr — Ur Vr

We make now a number of observations, which may be explicitely verified [34]: i) d-.{G-ld+Gr)

= ieV^F^Wr

=

ii) d+(G-ld„Gr)

= ieV^F^lVr

-

ili) G~lGr = V-^U^Ur

- VrV'^Vr

ieV^F^lVr [G-ld+Gr,G-ld-Gr] .

Hence, expression (4.139) for w may be cast into the manifestly gauge invariant, expression JcPxtiUVrG^GrV-^iVrd-iG^d+GjV-1)]

w(r) = i = ^-

2TT

= -^

J

[d2xiv{G-lGTd^{G^d+GT)] +iellv)tx[G-1Grdll(G-1dvGr)]

f ^xiS^

=^ f

(fxtl^Grd^G-^Gr)]

~in J'
.

(4.140)

The first term can be put into a more convenient form. Indeed, tilG^Grd^G-^Gr)]

= a M tr[G7 1 G r (G- 1 a M G r )] + \^tv[{d,G-l)(d,Gr)]

.

Hence, dropping in (4.140) the contribution from the surface term at ir-infinity, and doing the r-integration, we finally obtain, for the (Euclidean) effective action W^[A]

= ~(CJ(1)

- w(0)) = -T^[UV]

,

(4.141)

where r(£)[G] = ^ | c i 2 a ; t r P M G - 1 ) ( ^ G ) ] i 4-7T

/

dr f d 2 a;c^ v tr[(G- 1 S r G r )(C?- 1 a M G r )(G- 1 a i ,G r )]

(4.142)

101

4.7 Zero-modes

is the Euclidean version of the Wess-Zumino-Witten action 13 introduced in Chapter 9. The result (4.141) for W^ B ^[J4] is manifestly gauge-invariant and has also been obtained from perturbation theory [35]. It is remarkable that, except for an overall sign, it has the same form as the Euclidean version of the Wess-Zumino-Witten action Swzw, expression (9.8), for the choice n = 1, which we have denoted here 14

b y r( £ ). Using the Euclidean version of the Polyakov-Wiegmann formula (9.24) introduced in Chapter 9, T^[AB]

= T^[A]

+ VW[B] + -!- fd2xtv[(A-1d+A){Bd-B-1)} 4lT J

(4.143)

and recalling (4.135), we recognize that det(i 0 + e4) = e - ^ / d 2 ^ 2 , e r ( E , ^ ] e r < - > ! v - ]

_

As we shall see in Chapter 14, —T^[U] and — r ^ f V ] are the effective actions of the Dirac fermions with pure right- and a left-coupling to the gauge field, for a chirally symmetric regularization. The first factor above thus shows that a gauge invariant regularization is incompatible with a simple factorization of det(i j& + e A) into a right and left piece! This observation plays a fundamental role in the PolyakovWiegmann method.

4.7

Zero-modes

4.7.1

Axial anomaly equation in t h e presence of zero-modes

The Dirac operator i (jl + e A1 anti-commutes with 75. Hence there exists for every eigenvalue A„ a corresponding eigenvalue of opposite sign. Hence, with A„ the " A s s u m i n g that G(x) tends to the identity (.Attends to pure gauge) as | x |-> 00, we may identify the points at infinity. 3?2 may then be viewed_as a large two-sphere S2. The mapping G from S2 into SU(N) can be extended to a mapping G of a solid ball B, with boundary S 2 , into SU(N). This allows us to write (4.142) as (C123 = 1)

r(B)[G] = i - J d2xtT[(dliG'1)(d"G)} "l^r" /

^"^iMiG

l

diG){G

X

djG){G

l

dkG)]

.

This expression is just twice the equivalent bosonic action obtained by Witten for the case of Majorana fermions, where G defines the mapping from B into O(N). 14 The second (topological) term in (4.142) is often referred to as the Wess-Zumino term, and denoted by T[G]. We shall also refer to it as the Wess-Zumino action and denote it by Swz[G]. The renormalization of the theory with action (4.142) can be discussed [36] along the lines of Coleman's work on the sine-Gordon equation.

102

Determinants and Heat Kernels

eigenvalues of the associated Dirac operator (4.58), we have

A„^0

15

(KY

Introduce the short-hand notation

as\(ip)2)o

A

=as\(im2)-c(s\{im

•,

thus, in the ^-function regularization

±C'(0\(ip)2)\*

W[A] = C'(0\i$>)\* = lim

r_l

^1

2

InA^h,^ A (\2jA]y)\o ' A„^0 y

where the summation extends over all (non-vanishing) eigenvalues. Taking the variational derivative with respect to Aa^, one has \

SW[A] 5A?.(x)

A

linJ V

cSA„ «<5Aj(x) s+l

s->o

+ 0(s)

Now, from the eigenvalue equation (4.54) and the normalization (4.55) follows that / d2xuni pun = Xn or

SXn = X"5Aa(x\ / (FxttRulun SA-(x) "5A-(x)

,

+ u\l(x)e'ylj.taun(x)

.

(4.144)

The first term on the r.h.s. vanishes, since the u n 's are normalized. Hence \p, 1 a^ ^ {XIY 5A°(x) = — 7^6 (XI)' SW

(ui(x)e^taun(x)) Xn

(4.145)

Note that the Dirac Green's function satisfying

iPG(x,y\A)=5W{x-y)

,

has the spectral representation

G(x,y | A)ap = E 15

un(x)au*n(y)p Xn

The spectrum of ip consists of all integers different from zero, each eigenvalue being 2n-fold degenerate. This degeneracy (c-conjugation ) is partially split in the presence of a gauge field A^.

103

4.7 Z e r o - m o d e s

in the absence of zero-modes. Hence the s -> 0 limit in (4.145) can be regarded as a regularization of the familiar (divergent) expression J^(x\A) = -ti(ta^G(x,x\A))

,

for the case that zero-modes are present. Noting that

Vf (uttalllUn) = 0 , we obtain from (4.145) our first result, V?Jl(x\A)=0

,

(4.146)

where we have identified the l.h.s. of (4.145) with the external field current, i.e. e oA^yx) For the axial-vector-current, we expect to have, in analogy to (4.145), ^ x | A ) =-ax.E>[U"(8)^r""(!g)] A

n

"

(4-148)

"

To see this, it is convenient to take W[A, A] = - I n det[i ft + e 4 + ie 4J5]

,

as a starting point, with un, A„ the eigenfunctions corresponding to this new (still Hermitian!) Dirac operator, and to compute 18W[A,A) e 6AaJx) From Jd2xQn(x)ull(x)un(x) in analogy to (4.144),

A=0

_ ^"" 2 ? s->0 t-^ (A „) s + l

= 1 and f cPxu^i

( f Ta " J1 . = \5A .(x)< A=O

A=0

ql + e A" + ieA'~f*>)un = An, we find

eulix^^eiinix)

One therefore obtains from here the expected result, Eq. (4.148). Now, Vf{unhnlstbun)

= -2\n(lRuly5taun

•

hence it follows from (4.148) and (4.80) that V°»J5/(x

I ,4) = 2O fi (z)tr* a 75 C(0; x,x\(i

If))2)

.

(4.149)

The r.h.s. of (4.149) represents the axial anomaly. It can be calculated using the Seeley expansion (4.20). Moreover, in two space-time dimensions the combined equations (4.146) and (4.149) may be integrated to give W[A] itself, as we now illustrate for the case of QED2

104

Determinants and Heat Kernels

Example: Effective action of QED2 For QED2 (without zero-modes) we have (see (4.86)) a x

fi«(a:K(0; x, x I (i ipf)\ 2 \ _= -l( )^

_

e

= iz^e^F^ (4TT) ~ 8?r

.

Recalling further 17^75 = eM„7„, we have 75 - e

7

or the equations (4.146) and (4.149) now read dMJM = 0 , dllJll = ~€liVF'lv

(4.150) (4.151)

.

Z7T

This anomaly equation has the solution Jfl{x\eA)

= -^ j ' d2ydlDE{x-y)e^F^+d^

,

(4.152)

where T is an up to now arbitrary function, which is however restricted by Eq. (4.151) to satisfy t\T = 0. If JM is to be normalizable, then T can only be a constant. Thus JM is determined, and it remains only to integrate equation (4.147), with JM(a: | A) given by (4.152). In the present example this is easily done, and the result is just (4.132). This illustrates how in two dimensions the effective action itself may be obtained by the joint integration of the equations for the divergence of the vector and axial vector current. Note that this is a non-trivial property of two dimensions. Although the anomaly is also computable in four dimensions, the same is not true for the effective action. In the preceding example, W[A] was obtained by integrating (4.147). This was easily done, since JM (a; | A) was found to be a linear functional of A^. In the general case this integration is formally also possible, as we next show.

4.7.2

Atiyah-Singer Index Theorem

The Atiyah-Singer theorem [25] is the outgrowth of a long development in algebraic topology. It relates a topological invariant to the index j(D) of a bounded operator on Ti, defined by 7(D) = indexD = dimkerD - dimker/jt

,

(4.153)

where ker£>:= {^ en;D^

= Q} ,

is the null space of D, and D^ is the adjoint of D. D is called a Fredholm operator if the dimensions of the null-spaces of D and D^ are finite: dimkerD < oo,dimkerZ)t < 00. Most operators a physicist becomes confronted with have a vanishing index. This is true for operators (defined on a compact manifold which

105

4.7 Zero-modes

are Hermitian or are of the form 1 + K, with K compact. Whereas the triviality of the index for Hermitian operators follows directly from its definition (4.153), the latter statement can be understood in terms of the equivalence of the vanishing index with the validity of the Fredholm alternative: "Either the inhomogeneous equation (1 + K)ip = x has a unique solution for every x, or the homogeneous equation has a finite number of linearly independent solutions". In the last case the adjoint equation has the same number of solutions. As we now show, the winding number n of a gauge field A^, in two dimensions, is just the index of the operator D = -d1+ id-z + ie{Ai + iA2) Noting that • n

I 0

D

«0=lx>t o we define Indexi^» = k e r i ^ | 7 5 = 1 - k e r i ^ | 7 5 = _ ! = i V + - AT_ where N+(N_)

,

(4.154)

is the number of zero-modes of i Tp of positive (negative) chirality.

Proposition: N+-N-=n

.

(4.155)

The determination of the zero-mode asymmetry from the axial anomaly relation was first suggested by Coleman [37] and further elaborated on by a number of authors [38]-[43]. We shall provide here a physicist's proof [12]. Our starting point is the U{\)- axial vector current in the presence of an (Abelian or non-Abelian) external gauge field A^. Taking account of the possible existence of zero-modes, it is defined in a way analogous to that given by Eq. (4.148):

w^) = -« 6 £,uJ (a;)i7\2s+l7 M (a;) i

M 5

n

Correspondingly we have dpj'lix

| A) = 2fi H (z)tr 7 5 C(0;x,x\(i I?)2)

,

which replaces (4.149). Taking account of possible zero-modes, and recalling (4.37), (4.86), this becomes [12, 38, 39, 43] d»J'l{x\A) = ^Ue^v{x)-2nR{x)YJU(f{x)1bu^{x)

.

(4.156)

Notice that the first term on the right hand side is just twice the Pontryagin density. Since the definition of Jjj excludes the zero-modes, it is natural to suppose that J^ tends to zero at infinity. Hence integrating (4.156) over 3?2, and dropping the surface term at infinity, we obtain 2 '2,. t r e fd x

y^

106

Determinants and Heat Kernels

where tr refers to the trace in group space. For an SU(N) Lie algebra valued field Fpy this trace vanishes, {iri{SU{N)) ~ 0) and N+ - iV_. For QED2, on the other hand, the first term is just equal to the winding number of the gauge-field (7Ti(C/(l)) — Z), and we obtain the result (4.155). Let us explicitly illustrate the above result for the case of the Abelian vortex field (4.62). It is convenient to define the (complex) light-cone variables Z = X\

— %X2

,

Z = X\ + iX2

•

In terms of these variables the associated Dirac eigenvalue problem (4.54) in the external vortex field (4.62) on the unit sphere (R = 1) takes the form 0

2d

-Qd*-"T&;)\(ui\=

n

? + rk;

°

2A

u

/ \ 2/

("i

i + zz\u2

We only consider the solutions to the zero-eigenvalue problem. Making the Ansatz

A0)J~ v(i+^)M0) one finds that ip[ ' and
that is
1>l°J(s) = (£-)iM*i±ix2)]l-1r

,

1

,.tX±

, n^O

.

(4.157)

Here \x is just a mass parameter to give ip\ ' the dimension | . There is no need to orthonormalize these solutions. The condition of normalizability restricts l± to take the values / = 1,2, • • • \n\. Hence there are exactly N± =\ n \ zero-modes, with N+=n

,

N_=0

for

N^=n

,

AT+ = 0 for

n >0 , n< 0

,

(4.158)

so that the Atiyah-Singer index theorem (4.155) is satisfied for the vortex (4.62), with the additional property peculiar to QED2, that the zero-modes associated with a vortex configuration of a given winding are of one chirality, only (vanishing theorem). As we shall explicitly show in Chapter 12, this is also a property of arbitrary gauge-field configurations in two dimensions. 16 For QCDA figurations.

this vanishing theorem has been proven Ref. [59] to hold only for instanton con-

4.8 Ambiguities in Functional Determinants

4.8

107

Ambiguities in Functional Determinants

We have contemplated here various methods for calculating functional determinants. Because of the singular nature of Quantum Field Theory, the results thereby obtained are in general not identical. Thus the proper-time regularization requires in general the introduction of counterterms to cancel the singular terms in Seeley's expansion (4.109).17 However, just as in the case of any renormalizable Quantum Field Theory, the resulting ambiguities can be classified, so that for suitable normalization conditions, unambiguous results are obtained. In the following we examine in general the nature of these ambiguities.

4.8.1

Ambiguities in the regularization

To begin with let us examine the relation between the determinants obtained via the above two regularization schemes. Splitting the integral in (4.106) into the intervals (0,e), (e, oo) and taking the limit s -» 0 one finds [In det'D}i = [In det'D}t + C(0\D)lne

.

(4.159)

Note that the ambiguity is of the same form as that associated with a change in scale of the eigenvalues (see Eq. (4.25)). Now, if we consider ratios of determinants such as , detD U>_D_ = In -—— , D o detD0 then such a scale should not matter, provided that the dimension of the space on which D and Do act is the same. This excludes the case where D, but not Do, has zero-modes. Let us check this idea for the case that D is the Dirac operator (4.83). If the gauge-field A^ has non-zero winding (Pontryagin) number n, then i p will have . In two dimensions the number N of zero-modes will be just n itself, (see Atiyah-Singer Theorem and vanishing Theorem in section 7 of this chapter). From (4.38) and (4.88) we thus have, for two dimensions,

C(P\iP)-C(O\i0)

= -N

,

(4.160)

which indeed vanishes if there are no zero-modes. There exists another method of regularization, which we refer to as the PauliVillars regularization of a determinant [44, 7]. It is given by [lndetD]pv.

£(*..(• IS)

(4.161)

where

w^> = E'{^|><^} 17 It was a peculiarity of QED2 and QCD2 that these, a priori singular, contributions vanished identically for purely kinematical reasons. This however does not mean that there exists no ambiguity in the calculation of the QED2/QCD2 determinant.

108

Determinants and Heat Kernels

Here Mi, • • • Mv are dimensionless Pauli-Villars regulator masses to be sent to oo after setting s = 0. The number v is related to the polynomial bound and spectral density of the asymptotic spectral values. The minimal value of v is that, for which Cp.„.(0 | D) and Cp.«.(0 I D) exist without the need of analytic continuation. This places restrictions on Mk and v which are most conveniently determined by relating the s —> 0 behaviour to the small ^-behaviour in the inverse Mellin transform of (4.161). Thus, writing

i

Cpv{sl D) =

r00

T{s) J0

v

(

dt

''~M

/ dfj,(x)[h(t;x,x

1+

cke~M^

£

| D) -P0(x,x

\ D)]

we see that Cp.«.(0 I D) exists in the ordinary sense if V

V

*=1

k=\

and v = —. m

A simple computation yields [7, 32] [In deW)p.v.

= -Cp.„.(0 | D) = [In detD}^ + ({0 \ D)\n M

A.

T^i

ll

(4?r)m

M

k

k

where ^(0 | A) is the value of £(s | A) obtained by analytic continuation,

In M = Y^ckIn Mfc , k

and ar are the coefficients defined in (4.39). For QED2, — = 1 and a0 = 0, so that the ambiguity is again of the form (4.159), [In det'A\c, = [In det'A]t + C(0 | A)ln e = [In detA]p.v. - C(0 | A)ln M

4.8.2

.

Dependence on the scale parameter

Strictly speaking, our discussion of the QED2/QCD2 determinant has concerned the operator i $> which depends on the radius of the stereographic sphere. The difference (4.160) is seen not to depend on R. This is in fact a general property, which is however not shared by ('(0 \ i P) - ('(0 \ i 0). Hence the normalized determinant will in general depend on R. It is in fact easy to show that it will be independent of R, if and only if the trace of the energy momentum tensor vanishes,

4.8 Ambiguities in Functional Determinants

109

that is, in the absence of a trace anomaly. Such an anomaly will always occur if zero modes are present. This follows from the fact that independently of the dimension d of space [17, 6] /

d W M * ) l o = -K(0 | (* $>?) - C(0 I (i P)2)} •

(4.162)

In four dimensions the trace #MM ^ 0 even in the absence of zero-modes. In d = 2, where trai = 0, the dependence on R will only enter through the zeromodes. However, as we shall see in Chapter 12 this dependence on R cancels in the correlation functions. In d = 4 this is not the case, and the limit R —> oo has to be taken at the end, after proper renormalization. To demonstrate (4.162), we consider as starting point (4.60), with g the determinant of the induced metric g^v defined in (4.47). It is convenient to write the action in (4.60) in the form SF = /

d2xgflv(x)'4)(x)ij^D„ip(x)

This allows us to define the energy-momentum tensor associated with the Dirac field by i — SW 6nv(x) = ^{Hxh^D^^x^A = — . 2 ogfiV(x) In particular, 1 — *=• = -(ip{x)iW^{x))A

9^0^

= g^v{x)-

5W —

.

(4.163)

The left hand side is just the trace of 0 p „. We now want to calculate the r.h.s. We have SW _ IS('{0\ {if?)2) _ = Sg^{x) 2 Sg^x) SXn/Sg^(x)

^

V

u„(x)

- A ^1Q (A2) 7 ^

+^ )

•

(4-164)

is easily calculated; from the Dirac equation (4.59), i.e. / d2xglll/ull(x)i-y^Dvun(x)

= A„

,

(4.165)

one obtains

Hence, substituting this result into (4.164) one finds, upon using once more (4.165), SW 9^{x)--—— = -^gtrC{0;x,x\{i^)2) og^v\x)

,

or ' „ / •

SW 9 v{x)

"

Jg^Ax)

A

o = " [ C ( 0 ' { i P)2) ~ C(0 ' { i ^ )2)] '

(4.166)

110

Determinants and Heat Kernels

which establishes (4.162). Let us now demonstrate the relation between the independence of det(i $)/i 0) and the trace-anomaly. We have from (4.54) and (4.55), / cPxu^i lf)un = Xn

.

The independence of A„ only enters through un. obtains

From the above equation one

From the normalization condition (4.55), we have

or R SXn = - I A„ SR

d2xn2R(x)^ul(x)un(x)

Correspondingly, we obtain J2„n2/„N

a;2

R±W[A\ = -to»\Y. j * * ^ * ) *

u

UX)un(x)

(A,)

= -Jd2x(nR(x)^)^aO;x,x\(i&)2)

;

or, because of (4.163) and (4.166), R ^ ^

= f^x(pR(x)R^)gllveilv(x)

.

This proves that W[A] will not depend on R, if and only if there is no trace-anomaly. Nevertheless, the effective action approaches an R-independent limit as R —> oo. This follows from the fact that the effective action approaches an R-independent limit as R —¥ oo: R

5R

~ 2^(Xl)^HK

-2s J d2x^[nR{x)jp)}_^ = - 2 / d2x^(Ofl^a

5R 2v „

s=0

,it i

—

) [ < ( s ; x>x I (* ^ ) 2 ) ] - ° = ° '

4.9 Mass expansion in proper-time regularization

4.9

111

Mass expansion in proper-time regularization

The proper-time regularization also provides us with a method for obtaining a systematic expansion of the Euclidean effective action In det(d+ im + eJ^) in inverse powers of the fermion mass to all orders of the coupling constant [45], [46]. We shall illustrate the method for the case of QCDi with fermions of bare mass m. As we shall see, the heat-kernel method provides in this case an efficient way of computing the first few terms of such an expansion where, to order 0 ( l / m 2 ) , the triangle and box diagram contribute in addition to the simple second-order self-energy diagram. As we have seen, heat-kernel calculations are performed in Euclidean space. Denote by D the Euclidean Dirac operator

D :=P + m , where Z?M is given by D^ = d^ — ieA^, with A^ a Hermitian, Lie-algebra-valued gauge field in the fundamental representation of SU(N): A„ = raAl

,

tr TaTb = Sab .

We seek an expansion of the Euclidean effective action

-W[A] 1 J =

lndet{f-'e4\m)

det($ + m)

in powers of 1/m. Heat kernel methods require a semi-positive operator. Because of the nonhermiticity of the Euclidean operator D = If + m we have In det D ^ | l n det DDK We shall therefore follow a procedure differing somewhat from the ones adopted previously in this chapter. Formally we have for an arbitrary variation D -» D+SD, Sin det D = tr(5DD~1) (see (4.97)). As we have seen, this expression is in general divergent. The propertime regularization amounts to the replacement of (4.97) by (compare with (4.108)) «51n det D = tr(5Z>D _1 e- ei
,

with K some suitable positive operator, and e a parameter regulating the expression. For an arbitrary variation 5D, the operator 5DD~l is non-local. This means that off-diagonal elements of the heat kernel e~eK are involved in the computation of the trace, rendering the trace computation in general untractable. 18 As we now show, we can nevertheless obtain a power series expansion in 1/m, by making the gauge-covariant choice 19 K = £>£>* = - TJ)2 + m 2 . Making use of the representation D~x=

dtD^e-tDD' ./o

18

For gauge transformations &DD~X is local, which is the reason for being able to compute gauge-anomalies exactly. 19 For m j£ 0 D and Z)t have no zero-modes.

112

Determinants and Heat Kernels

we observe t h a t in this case

20

we can write [45] oo

/

dtti[5DDh~tDDl]

.

(4.167)

Since 5DD^ is a local operator, only knowledge of t h e diagonal elements of the heat kernel h(t;x,y\Drf):=(x\e-tDDi\y) are now required. T h e price we have t o pay is t h a t these elements are now required for general values t & [e, oo]. We now consider variations of the one-parameter family of connections A^=rA„.

,

D ( r ) = $ - ie4{r)

+m

.

(4.168)

This parametrization has proven very useful in connection with t h e Zumino descent equations [47]. W i t h

SAW =dr A^

,

8DW = —iedrA'

we have, using (4.167) dW[A(r)] _ dr ie /

(4.169)

d3xtr{A(x)(-E>ir)+m)h(t;x,-:v\ - (V>(-

dt

This integral is not computable in general. In t h e limit m —>• oo it is, however, dominated by the lower endpoint, so t h a t we can replace h(t;x,y\— P)2) by the asymptotic series in t (we omit from now on the superscript "r", unless explicitly required): oo

h(t;x,y\-(p)2)

= h0(t;x-y)J2>>i(x,y\A)tl

,

(4.170)

1-0

where bi(x,y\A) are the off-diagonal Seeley-DeWitt coefficients h0(t;x - y) is the heat kernel associated with t h e two-dimensional Laplacian as given by (4.75). 20

An alternative way of arriving at this result is to note that det£>2 = T T ( - > 2 + m 2 - 2im\)(-X2 A>o

+ m2 + 2im\)

oo

=

[J

(\2+m2)=det(tfD)

,

A=— oo

where A are the eigenvalues of i p, and where we have used the fact that the non-negative eigenvalues A come in positive and negative pairs ±A. A quick way of also arriving at the above result is to note that [48] det(f) + m)2 = det(j{D + m)det[-y5(fZ> + m)^5] = det DD*, where we have made use of the cyclic property of the trace, one the one hand, and the anticommutativity of 7 5 with 7^, on the other.

4.9 Mass expansion in proper-time regularization

113

Substituting (4.170) into (4.169), performing the i-integration, and further integrating both sides of (4.169) with respect to the parameter r, we obtain the desired expansion in terms of powers of 1/m: det(Q - ie4 + m) ^ -ie y ^ T(l)

(A 171)

2

dettf + m)

V

(4TT) £ j j (m )< fd2x

f

drtr{4(a:)(#(r) +

'

'

m)bt{x,y\A^)} x=y

The computation of the Seeley-DeWitt coefficients bi(x,y\A) and their derivatives in the coincidence limit y = x proceeds now as in subsection 4.3.2. Substitution of (4.170) into the heat-kernel equation

§i-m2)h{t;x,y\-m2) = o , leads to the recursion relations (I = 1, 2,...) = ((DD2-X)bl-1

lbl + (x-y)tiD*bl

,

(4.172)

and the parallel transport equation ( i - y)MZ?»6o = 0

,

(4.173)

where X is given by Eq. (4.85), that is X = — §75eM„i<)l,/, and where riiv — o^A.v

O^A.^

ie[A.fi, Av\

As an illustration, let us calculate (4.171) up to order 0 ( l / m 2 ) . To this order only the terms with / = 1,2 need to be considered in the expansion. We therefore need to calculate the coincidence limits [DM60], [&i] and [D^b{\. By construction, b0 is normalized such that [&o] = 1The coefficient bo(x,y) is obtained by solving the parallel transport equation (4.173) subject to the normalization condition bo(x,x\A) = 1: b0(x,y\A)=Peiefcdz»A^

,

(4.174)

where " P " stands for "path ordering" along a path C connecting the points x and y. Choosing for this path a straight line parametrized as follows, z(r;x,y)

= XT + (l-r)y

,

r = [0,1]

we may write (4.174) in the form i. I

I A\

IT, ie I

bo(x,y\A) =Te

Jo

dTB(T;x,y\A)

' ,

where "T" now denotes the usual time-ordering operation with respect to T, and where B(r;x,y\A) = {x - y ) ^ ^ ; x,y)) .

114

Determinants and Heat Kernels

Defining the auxiliary functional 7

/

i A\

b0(T-,x,y\A) we see that bo(r;x,y\A)

m ie I

=Te

dr'B(T':x,y\A)

'y|

Jo

;

,

satisfies the integral equation

6 0 (r;x,y\A) = 1 + ie [ CIT'B(T';X,y\A)b0{T';x,y\A) Jo

.

(4.175)

Prom the definition of B(T; X, y\A) one immediately computes the coincidence limits

[d*B(T;x,y\A)]=All(x) [d;dxvB(T; x, y\A)] = r (OMx)

+ dvA»{x))

2

[&Z&l&lBiT; x,y\A)] =T (d^A^x)

+ d„dvA^x)

+dfldvAa{x))

.

By straightforward differentiation of (4.175) we then find after some calculation, upon setting T = 1, [—b0(x,y\A)]=ieAli(x) ox n

(4.176)

' dx^dx ^ -balxMW^^M^+dvMxV-liM^M*)} ' 2 v

idx

gx

dx

(4-177)

b0(x,y\A)} = ^(dadMx)

+dadvAll(x)

~ ^[(^A^x)

+ d„All(x))Aa

- ^\Aa{x)(d»Au{x) + iAa(x){Ati(x),Av(x)}

+

dlldvAa{x))

+ (
+ {a -B- LI)}

+ duA^x))

(4.178)

+ (cr o v) + (a o /j,)} .

Using these results, one finds after some further algebra [Dlbo(x,y\A)]=0, [D;D:b0(x,y\A)}

(4.179)

= - y ^ ( i ) ,

[Dl(D*v)2bo(x, y\A)} =

e

-DvFv»{x)

(4.180) (4.181)

where Dv acts on a Lie-algebra valued field F in the usual way: DliF =

dF-ie[A^,F}.

From (4.179) and the tracelessness of the gamma-matrices it follows that the I = 0 term in the asymptotic expansion (4.171) vanishes, showing the expected absence

115

4.10 The F i n i t e T e m p e r a t u r e H e a t K e r n e l of a linear growth in the mass. Hence we have

Now, setting I = 1 in the recursion relations (4.172) and using (4.180), we obtain [bx] = {((Drf

- X)bo] = -X

(4.182)

with X given by (4.85), that is X = -§7 5 e,,„i' , /1 „. Setting again Z = 1 in (4.172), applying D^ to both sides of this equation, and using (4.179), we further obtain in the coincidence limit 2 [D,M] = [D^Dv)2b0] - [D^Xbo] ie e r = -DUFVIM + -^D^expFxp

.

Using (4.182) we see that the Z = 1 contribution to the O(-) term also vanishes since tr7 5 7 M = 0. Hence the leading term is of order O(-^z), and is given by

det(9-ie4 + m) = det($ + m)


487rm J

f1 J0

dr

() w

"

(,)

(

}

vX

v

'

Finally, performing the r-integration one finds the expected gauge invariant result , det(d — ieA + m) e2 f ,, ,„,..•> v M w ; det($ + m) 1927rm2 J This completes our illustration. The method has also been applied for the calculation of the fermion determinant in 2+1 dimensions, where one is led to order 0(l/m) to a Maxwell-Chern-Simons theory [46].

4.10

The Finite Temperature Heat Kernel

The calculations in the preceding sections all refer to zero temperature. The usual experiments conducted in the laboratory can be described in terms of QFT at zero temperature. Standard cosmology however predicts matter at extremely high temperatures in the early universe. The usual point of view is that in the very early universe higher symmetries were present which disappeared in the course of the cooling process. The phase transitions involved in the breakdown of such symmetries are of great interest, and must be described in the framework of QFT at finite temperature. Dolan and Jackiw [49] have been the first ones to investigate the symmetry behaviour of QFT at finite temperature. They showed, for instance, that in massless electrodynamics in two dimensions the gauge boson acquired a mass which was

Determinants and Heat Kernels

116

independent of the temperature. Their results were rederived by Reuter and Dittrich [50], using the C-function prescription. There exists an extensive literature discussing the finite temperature formalism in QFT [51]. There are essentially two approaches refered to as the real and imaginary time formalism. For the study of the thermodynamical properties of a system at equilibrium only the partition function Z = t r e _ / 3 i f is required, and the latter formalism is the appropriate one. The same applies to the heat kernel treatment of functional determinants. The real-time (or closed-path) formalism, on the other hand, is most appropriate if one is interested in Minkowski Green functions at finite temperature, as defined by <*(*,)*(„) • • • « ( « . ) ) , =

£

" ' -

W < n |

* ' ^

2-in

1

--•*

M

M

•

(4.184)

e

where /? = 1/T, the sum is taken over a complete set of states, and $ stands for a generic field. In what follows we shall only be concerned with the calculation of the partition function Zp at finite temperature. As it is discussed in a number of textbooks [51], for a Hamiltonian quadratic in the canonical momenta, one is led (after Gaussian integration over the momenta) to the simple result Z0=

f V[fields] e-fo

dTL fields

^

^

,

(4.185)

where LE is the Euclidean Lagrangian obtained by analytically continuing the Minkowski Lagrangian in time to the negative imaginary time axes: x° —• —IT, the Euclidean time-integration is restricted to the range [0,/? = 1/T], and where the boson (fermion) fields are required to satisfy periodic (antiperiodic) boundary conditions in time with period A T = /?. In Fourier space this means that only integer (half-integer) Matsubara frequencies [52] are allowed, so that the usual Fourier Integral over frequencies turns into a discrete sum. In the case where the Lagrangian is quadratic in some matter fields, the computation of the partition function (4.185) involves at an intermediate stage the computation of a functional determinant in the presence of an (Euclidean) "external potential". This is where the heat kernel methods to be discussed below become relevant. A finite-temperature (T > 0) or thermal quantum field can be visualized as a sea of virtual particles occupying space together with a thermal gas of real particle excitations. The virtual particle sea is independent of the temperature T. It can however be deformed by coupling it to a static background potential. This is generally known as the (static) Casimir effect (see e.g. the general references cited in [55]). Gauge theories present a Casimir problem with particular features related to the underlying gauge invariance. These concern in particular the spacial energy distribution of a charged thermal matter field when coupled to a static background electromagnetic field. The resulting distortion of the virtual sea and thermal plasma (as we now refer to the thermal gas consisting of both positively and negatively charged

4.10 The Finite Temperature Heat Kernel

117

particles) is revealed by a local analysis in terms of the ground state expectation value of the thermal stress tensor. The problem of charged quantum fields coupled to a uniform electromagnetic background field is an old one, going back to famous papers of Euler, Heisenberg [53] and Schwinger [54]. There has been done a great deal of subsequent research, reviewed in books [51] and elsewhere (see e.g. Refs. in [50]). In recent times modern quantum field theoretical methods have been applied to this traditional problem (see e.g. Ref. [55] and references therein). In the following we illustrate how heat-kernel methods can be used in order to gain some insight into the local response of scalar thermal matter fields to a background electromagnetic field. Although the discussion concerning scalar fields can easily be carried out in arbitrary spacial dimensions d, we shall keep in line with the general spirit of this book, and choose d = l. 2 1 Hence we shall work on a cylinder of circumference /? = 1/T along the (Euclidean) "time" direction, corresponding to space being infinite. In order to better appreciate the special effects resulting from the minimal coupling to background gauge potential we begin by considering a thermal scalar field in presence of a static background potential V and then extend this discussion to the case where bosonic and fermionic matter fields are minimally coupled to a static gauge potential A^. Special attention is drawn to the periodicity features and their relation to the topology R x S1 of our space-time. These features become explicit for the choice A^ = (Exi + const., 0) of the gauge potential, and are also found to be characteristic of the finite temperature Schwinger model (see Chapter 13.)

4.10.1

Scalar field in a static background potential

Let us begin with the simple case of a scalar quantum field interacting with a static background potential V(xi) in one-dimensional space. We wish to study the thermodynamical properties and vacuum Casimir energy of this system using heat kernel methods. As already mentioned we shall work in Euclidean space-time using the imaginary time or Matsubara formalism. Correspondingly we impose periodic boundary conditions in Euclidean time on the scalar field <^>(a;i,a;2), 22 {xi,x2) = <j>{x1,x2+P)

•

(4.186)

Euclidean space-time is then a cylinder RxS1. The spectral operator for the theory in question is [—A + F(a;i)], where A is the two-dimensional Laplacian. The vacuum and thermodynamical properties of the system can be computed from the bilocal heat kernel

h^(t;x,y) = J2^tXlMx)4>l(y) ,

(4.187)

k

where A| and k (x) are the eigenvalues and respective eigenfunctions of the spectral operator, [-^ + V(x1)]4>k{x) = \Uk{x) . 21 22

For a general discussion in d dimensions, see Ref. [56]. Throughout this section we use the Euclidean notation {x^} =

(x\,xi).

118

Determinants and Heat Kernels

With k(x) subject to the boundary condition (4.186) we have k(x) ->• mn{x) - - ^ e * T l l V n ( a ; i ) , A 2 fc -^A^„ = ( ^ )

2

+ ^

,

(4.188)

where the spatial modes ifin(x\) a n d associated spectum {w 2 } are determined by solving the auxiliary eigenvalue problem [-dl + V(x1)]tpn(xi)

= w^„(a;i) .

subject to the condition that the eigenfunctions be normalizable in 1Z. The Euclidean thermal correlator 23 has the spectral representation (

^

)

V

=

E

/*oo Jo

l

^

#

(4-189)

-i

P

m

n

We may perform the Matsubara sum by using the Jacobi identity ^2

e-b(m-a)*

e-^e~i2™1

= J^Y1

m= — oo

(4.190)

l= — oo

with the result °°

/»00

({x)
dth(t;x,y)T=0

^O

"£

2 2

e-^e'^^-^,

(4.191)

J=-oo

where hit- cr. Tii\ h(t;x,y) =0

\ I i^2)1J2e~tUlllP^x)'Pn(yi) ==zJ^-e4irt

(4-192)

is the T = 0 bilocal heat kernel of the operator [-A + V ^ ) ] . Hence, for a static background potential the T > 0 correlator decomposes into two distinct and well defined parts: the virtual sea part (1 = 0 contribution) which is independent of T and coincides with the T = 0 correlator, and the thermal gas part (1^0 contribution) which exhibits the full temperature dependence and vanishes exponentially as T->0: (4>(x)4>(y))p

= ((x)
+ {{x)
Now, from ((j>(x)
We do not distinguish between operator and c-number valued functions.

4.10 The Finite Temperature Heat Kernel

119

then shows that the finite part of the stress-tensor can, for T ^ 0, be written in the form n

i_e-/3w„

— 1 sea "T" /-~i n rpfiU

n

'

epun

—I

'

. /TT/if

-* sea "•"

gas

At finite temperature the mode sum is thus modified by the familiar Bose-Einstein distribution, in agreement with one's expectations. This completes our brief review of a "usual" Casimir problem at T > 0.

4.10.2

Scalar field in a static background gauge potential

We next consider scalar fields minimally coupled to an Abelian background gauge potential A^(x). On a space-time cylinder R x S1, the periodic boundary condition in Euclidean time for the field (x1,x2) = 4>(x1,x2+/3)

, .

(4.193)

On the cylinder R x S1 we cannot gauge A2 to zero, if one is to respect the periodicity properties (4.193) of A^ and 4>. All configurations eA2 can however be mapped into the interval [0, ^S-] by a bonafide gauge transformation. (See Chapter 13 for a more detailed discussion of this point.) Only constant configurations of the form eA2 = ^ p can be gauged to zero by making the (allowed) gauge transformation 24 eA2 ->• eA2 + d2A with A = ^ x 2 . Consider now, in particular, the static background gauge potential eA2(Xl)

= Ex! + 2ira/0,

Ax = 0

,

(4.194)

corresponding to a constant background electric field along the xi direction. 25 This potential will be of particular interest in Chapter 13, where we discuss the finite temperature Schwinger model in the presence of instantons. The relevant spectral operator in this case is the gauged Laplacian — .D2, where D^ = d^ — ieA^ couples the quantum scalar field to a static background gauge potential Ali(x). 24 Another way of stating this is to observe that in this exceptional case the gauge transformation can be absorbed into the Matsubara index m via the transformation m —> m + N. For this reason eA2(xi) is always gauge equivalent to a configuration taking values in the range [0, 2K]. In the zerotemperature limit, on the other hand, we may always gauge ^2(11) to zero. Indeed, the discrete Matsubara frequencies ki = ^?p. t u r n m t 0 a continuous variable ki in the range - c o < ki < 00, and the corresponding gauge change in A2 may be absorbed into a shift in &2 under the integral

Jdk2.

25 Note that the configuration (4.194) is gauge equivalent to the configuration eA2(xi) 27ra//3, Ax = A(n).

= Ex\ +

120

Determinants and Heat Kernels

For later comparison it will convenient to first consider the (at first sight, trivial) case E = 0. The thermodynamical and vacuum properties of the system can be computed from the bilocal heat kernel (4.187), where A;2 and i are the eigenvalues and respective eigenfunctions of —D2, idi + —a\

+{id1f

ct>l{x) = \2l4>l{x)

,

(4.195)

with 4>i(x) subject to the boundary conditions on the cylinder. This eigenvalue problem can be solved by separation of variables with the Ansatz = -j=ei2!fLx0(pk(Xl)

Mx)->4>mk{x)

,

(4.196)

where the periodicity in X2 has been implemented, and where ipn{x\) are normalized eigenfunctions of (idi)2:

The eigenvalues are correspondingly given by + *J

t = » i

>

(4-197)

where am = -m— + —a

,

m = 0, ± 1 , ± 2 , • • •

.

(4.198)

We thus have for the diagonal elements of the heat kernel oo

h^\t;x,x\D) = Y,

,

e a

"' ™ / dke-tk'ipkixjtptix!) ,

(4.199)

m = — oo

Using the definition of the theta function [57] Q(z,p)=

oo ]jT

2

ei«m

pei2xmz

(4.200)

m=—oo

we may write (4.199) in the form h^x,x\D)

= I e - « ( ¥ ) V ( ¥ - ) ' c - ^ e (i%t x

+

f ^e-tk*

ip

,i%t) (4201)

J 27T

Making use of the identity [57] e f e r i ^ / i e - ^ e f - . - l , -ip \p p

(4.202)

4.10 The Finite Temperature Heat Kernel

121

find

h^\t;x,x)

= h(t;x,x) 1 + 2 Y^ ( - 1 ) " 1 cos(m27ra)e-

(4.203)

m—l

where h(t; x, x) is the zero-temperature diagonal heat kernel h{t;x,x)

1 f dki dkx = —== / — : V47Ti J

-tkf

(4.204)

27T

Indeed, for /3 —> oo,

yl-

e - ' < ->

J

2TT

2

e

=

(4.205)

/4?rt

so that we are led from (4.199) to (4.204) in the limit /? —>• oo. Notice that in this simple case of a constant gauge field the finite temperature heat kernel factorizes into the zero-temperature part, and a temperature dependent part depending periodically on a. This periodicity is a consequence of the periodicity of A2 and the allowed gauge transformations on this cylinder. Expression (4.203) also provides a clean separation between the temperature-independent ("sea") and temperaturedependent (thermal gas or "plasma") part. We now turn on the electric field in (4.194) by choosing E ^ 0. The eigenvalue equation (4.195) is now replaced by 2-7T

(f>i(x) = \fi(x)

(id2 + EXl + — a ) 2 + {idtf

(4.206)

Making again the Ansatz -2^m

1

4>i{x) - > mn(x) = -=e%

0

X2

yn{x-i_)

,

(4.207)

and defining 2na

2TT

Xm = Xl m

- EP

+

(4.208)

W

we arrive at the eigenvalue equation d2
dxl

-

\2„Wn{xm)

(4.209)

This is just the harmonic oscillator eigenvalue problem in Schrodinger theory with orthonormal eigenfunctions n 2

lfin(xm)^2-

/ ~(-Ye-i^2mHn{^Xm)

A2=2E(n+i),

n = 0,1,2,...

.

(4.210)

122

Determinants and Heat Kernels

Here the Hn (z) are Hermite polynomials satisfying Hn — 2zH'n + 2nHn = 0. In Eq. (4.209) the m-dependent backgrounds Vm(x\) = E2x2n are identical harmonic oscillator potentials centered at equidistant positions x\ = (m — a)2Tr/(3E. This periodic arrangement of identical potentials leads to periodic structure along xi (with period Azi = 2ir/PE) in the Euclidean stress tensor T^u{x) and other local quantum functions [56]. The m-dependence of
1

^ ) («;^y) = ^E e < m ^ ( * a ~ w ) E e " < 2 B ( n + i Vn( a : m )^(y f n ) p n=0 E 2n sinh(2.E£)

4E«'

imZg-(x2-y2)

xe The mode sum ^ Ref. [57])

ie(x1-»/i)2coth(2St) 2

tanh(St)

(4.211)

here has been performed with the help of the identity (see e.g.

Y,^X^yHn(x)Hn(y) n=0

-Exmym

e

(l-z2)-1/2exp\-^[2xy(l-z)-z(x-yM.

=

'

*>

'

Alternatively one can use the Feynman-Kac formula [58] for the propagator in the harmonic oscillator problem (see Chapter 13) . The diagonal part of the heat kernel (4.211) may be written in the form hW(t;x,x)

=

E l+2 2^cos[nf3(Exi 4-7T sinh Et n=l

2-nas + —— )]e

(4.212)

o™"'

where we have made use of the identity (see (4.190))

- y

„ — Ext, t a n h S t

E

•in(2iva+xi0E)

4-7T t a n h Et

(4.213)

n= — oo

Note that already in this simple case the factorized form of the heat kernel deviates substantially from expression (4.203) obtained in the E = 0 case. In the limit E —> 0 the heat kernel (4.211) smoothly becomes the free-space heat kernel as it should h,W{t;x,y)

-)• - ^ e - f ^ - w )

2

/^!

-*(-™^+f)2e™^(x2-y2)

E

^

0

.

4TT£

For the diagonal heat kernel this reduces t o expression (4.199), and correspondingly t o (4.203). Remarkably, the dependence on m so conspicuously absent from the

123

4.11 Conclusion

E > 0 spectrum (4.210) reenters in the E = 0 spectrum "through the back door" via xm labelling the sites of the harmonic oscillator potentials [56]. Let us finally observe that the heat kernel (4.212) has the form of a Fourier expansion, hW(t;x,x) = ^hl?)(t;x1)cos(n(3eA${x1)), (4.214) n

with periodicity ^ in the variable CAQ(XI), and "Fourier coefficients" /ijj?'(t;x\) only depending on the electric field. As is argued in Ref. [61] this is expected to be a general result for static gauge fields, also applicable to 3+1 dimensions. In that case, /i„ (t; xi) will be a function of the electric and magnetic fields. The periodicity of h^(t;x,x) under the gauge transformation eA0 -+ eA0 + ^ is, however, not a property of Minkowski space.

4.11

Conclusion

In a path integral framework, the non-perturbative calculation of n-point functions in gauge theories typically involves as an intermediate step the calculation of Green functions and functional determinants in an external gauge field. As we have seen, in 1+1 dimensions such quantities may be exactly calculated in the case of fermions minimally coupled to an Abelian and non-Abelian gauge field. We have emphasized in this chapter heat-kernel techniques, which have been shown to provide powerful tools for summing one-loop diagrams, and are particularly suited for dealing with zero-mode problems as well as Quantum Field Theory at finite temperature. There exists a very extensive mathematical literature on this subject, which evidently is outside the scope of this book, and which the mathematically inclined reader is asked to consult. An extensive list of references to this literature can be found in the article on topological methods for gauge theories [38] by B. Schroer, as well as in Refs. [7, 60]. Gilkey's method for calculating the Seeley coefficients, including the case of non-invertible operators, has been extensively discussed in a "digestible" way for physicists by Gamboa-Saravi et al in Refs. [8]. The Fujikawa approach has similarly been discussed and applied to a variety of problems by Fujikawa himself. As we have tried to exhibit, all of these approaches are equivalent if due attention is paid to the ambiguities, and we shall have ample opportunity to apply them in the chapters to follow.

Bibliography [1] L.D. Fadeev and V.N. Popov, Phys. Lett. 25B (1967) 29.. [2] P.T. Mathews and A. Salam, Phys. Rev. 94 (1954) 185. [3] F. Berezin, The Method of Second Quantization, Academic Press, New York, London (1966). [4] R.T. Seeley, Am. Math. Soc. Proc. Symp. Pure Math. 10 (1967) 288.

124 [5] P.B. Gilkey, Proc. Symp. Pure Math. 601.

Bibliography 27 (1973); J. Diff. Geom. 10 (1975)

[6] S.W. Hawking, Commun. Math. Phys. 55 (1977) 133. [7] K.D. Rothe and B. Schroer, Field Theoretical Methods in Particle Physics, ed. W. Riihl, Plenum, N.Y., 1980. [8] R.E. Gamboa-Saravi, M.A. Muschietti and J.E. Solomin, Commun. Math. Phys. 89 (1983) 363; ibid 93 (1984) 407; R.E. Gamboa-Saravi, M.A. Muschietti and J.E. Solomin, Commun Math. Phys 93 (1984) 407; R.E. GamboaSaravi, G.L. Rossini and F.A. Schaposnik, Int. J. Mod. Phys. A 11 (1996) 2643. [9] P.A.M. Dirac, Ann. Math. 36 (1935) 657. [10] S.L. Adler, Phys. Rev. D6 (1972) 3445; R. Jackiw and C. Rebbi, Phys. Rev. D14 (1976) 517. [11] J. Kiskies, Phys. Rev. D16 (1977) 2535. [12] N.K. Nielsen and B. Schroer, Nucl. Phys. B127 (1977) 493. [13] O. Alvarez, Nucl. Phys. B216 (1983) 125; ibid B238 (1984) 61. [14] J.W. Milnor and J.D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press 1974. [15] B. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach 1965; N.K. Nielsen, Nordita preprint 78/24. [16] R.E. Gamboa-Saravi, M.A. Muschietti, F.A. Schaposnik and J.E. Solomin, Ann. of Phys. 157 (1984) 360. [17] J. Singe, in Relativity: The General Theory, North-Holland, Amsterdam, 1960; J.S. Dowker and R. Critchley, Phys. Rev. D 1 3 (1976) 3224; J.S. Dowker, J. Phys. A l l (1978) 347; N.K. Nielsen, H. Romer and B. Schroer, Nucl. Phys. B136 (1978) 445. [18] K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; Phys. Rev. D21 (1980) 2848; Phys. Rev. D22 (1980) 1499(E). [19] K. Fujikawa, Phys. Rev. Lett. 44 (1980) 1733; Phys. Rev. D23 (1981) 2262. [20] K. Fujikawa, Proceedings of Nato Advanced Research Workshop Super Field theories, Simon Fraser University, Burnaby, B.C. Canada, 1986; Proceedings of the Kyoto Summer Institute, 1985, ed. T. Inami (World Scientific, Singapore, 1986). [21] C. Itzykson and J.B. Zuber, Quantum Field Theory , Mc Graw Hill, 1980. [22] G.A. Christos, I. C. Trieste preprint 82/97, 1982.

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Chapter 5

Self-interacting Fermionic Models 5.1

Introduction

In this Chapter we consider self-interactions of a multiplet of fermionic fields. The symmetry group will be taken as being either O(N) or SU(N) ® U(l) ® 17(1). These models have a long history, having been introduced by Nambu and Jona Lasinio in four dimensions, in order to describe the phenomenology of elementary particles [1]. They are referred to as the Gross-Neveu and chiral Gross-Neveu model, respectively [2]. At the classical level they describe the interaction of massless fermions, being conformally invariant. At the quantum level however, a mass gap is generated by the S-matrices of quantum fluctuations, leading to asymptotically free theories. These are solvable in the N —S- co limit, and an expansion in powers of 1/N may be defined. Both models possess an infinite number of conservation laws [3, 4] which survive quantization [4], and as a consequence, the S-matrices of the models may be computed exactly. In section 5.2 we discuss the (^(AQ-invariant model, and in section 5.3 the chiral model. The interaction of these fermionic theories with a models will be discussed in Chapters 6, 7 and 8. An important issue concerns chiral symmetry breaking in the chiral Gross-Neveu model, suggested by the dynamical generation of mass, since the appearance of Goldstone bosons is forbidden in two-dimensional space-time [5]. We shall see, in terms of the operator solution of the model, that this paradox is resolved by the fact that the chirality carrying field decouples from the physical fermion fields, which are massive, but carry no chirality.

5.2

The 0(N) Invariant Gross-Neveu Model

This model, first introduced by Nambu and Jona-Lasinio [1], describes a quartic Fermi-type self-interaction of an Applet of Majorana fermions transforming as a fundamental representation of the orthogonal group O(N). The Lagrangian is given

128

Self-interacting Fermionic Models

by

C = fo Hi + \g(^i)2

,

(5.1)

where the summation over repeated indices is implied. This is the most general O(N) invariant, renormalizable and conformally invariant theory for self-interacting Majorana fermions in two dimensions. 1 It presents the discrete symmetry tp -> 75^-

5.2.1

Classical conservation laws

The classical model is integrable, having an infinite number of conservation laws.2 The semiclassical study of the model reveals a very rich structure, and the time independent solutions of the equation of motion can be computed systematically [6]. This is a consequence of the fact that the Noether current associated with O(N) rotations, given by

jjf = hfil^

,

satisfies &*$ = 0

(5.2)

as a consequence of the O(N) symmetry. Using the equation of motion of the Lagrangian (5.1) as well as a Fierz transformation, it also satisfies3 drfi-dvjj* + 2g[jfl,jl/]i^0

.

(5.3)

This is a crucial identity. It implies integrability by assuring the compatibility of the differential equations [7, 3] 8llU = 2gUj^

.

(5.4)

Indeed, taking the derivative with respect to xv of (5.4) and subtracting the resulting equation from the one with / i H i / w e obtain the integrability condition 1 We have, for two-dimensional Majorana (anticommuting) spinors, the identities ipfstp = 0 = i/rfpil), which follow from the anticommutation of the components, and from the definition i/> = ^ i t y = ^,-yO for a Majorana field, with a charge conjugation matrix being unity; in the Weyl representation, the charge conjugation matrix is C = 75; we use the Majorana representation, (A.8)-(A.10), in this Chapter. We also have the identity <5a/3<57^ = |[5a<5<57^ + {ly,)as{lt')-1p +

(7 5 )a<s(7 5 )7/3], for Majorana fields the Fierz identity ip^jipjipi = — §(V'iV'i)3> where a sum over repeated indices is implicit. In the particular case of 0 ( 2 ) symmetry the Lagrangian (5.1) is just that_of the Thirring model as one sees by defining a Dirac field V D = i>i + tyz , and using 2 We have discussed integrability of the sine-Gordon equation, and will in Chapter 6 discuss integrable non-linear sigma models. For the moment, integrable means that the model has higher conservation laws, which are not directly associated with a Lie group symmetry of the Lagrangian. It is rather related with the existence of a set of differential equations for a matrix valued field. These differential equations depend on a potential, which is in turn, related to the integrable model in question [3]. _ 3 We use t h e equation of motion iftipi = —gV'fcV'AV'i. which implies idli{i>jl,1lbi>i) = +2g^i/nj)jf5i(>i, as well as the Fierz identity ^jl^l^k^h^^i = ^jfsi'f'l'k^kTherefore, using (A.10), we have e^d^J,, = -2gt'"'ipiiuil>k,>l>k/y^j< il0ia which (5.3) is obtained.

5.2 The 0(N)

Invariant G r o s s - N e v e u Model

129

(5.3). Using current conservation and (5.3) it is also possible [8] to establish the following linear system of equations for any value of the parameter A: d^U^

= -gUW

[(1 - cosh 2A) j M + sinh 2AeM1, j v ]

.

(5.5)

As a result of equations (5.2) to (5.5), there exists a one parameter family of conserved currents given by the expression jW = uW {(CoSh 2A - 2) j M - sinh 2AC/1„ j " } t/ ( A ) _ 1

.

(5.6)

The current (5.6) can be shown to be conserved. Expanding j / , in powers of A, and proceeding as in Chapter 3 one obtains an infinite number of conserved currents and charges (for a more detailed discussion see Chapter 6), thefirstnontrivial (matrix-valued) charge being given by Q(2) = / dy1dy2e{y1 - y2)jo(t,yi)jo(t,y2)

+-

dyji (t, y)

. (5.7)

One may construct an infinite number of conserved currents in terms of any conserved current J** satisfying the zero-curvature condition [Dli,Dv]=dlJv-dvjlt

+ 2g\jll,jv]=0

,

(5.8)

where £>M denotes the covariant derivative as given by D^ = d^ + 2gj^. The algorithm is very simple [9]. We start out with fy = j M . Consider a conserved current jjT , and the corresponding potential, that is

Construct yj1

' as j< n+1 >=Z?„j< n >

.

(5.9)

This new element is also conserved. Indeed, using the conservation of j M , one has

or recalling definition (5.9) of j/P , one finds ^•(n+l)

= eitvD*j(n)v

=

e^D^D-jin-l)

=

Q

;

(510)

where use has been made of Eq. (5.8). Therefore we have an infinite number of conserved charges in the classical theory. They obey a non-linear algebra (see section 7.5).

5.2.2

Effective potential a n d /3-function i n a | expansion

We study now the quantization of the Gross-Neveu model. The Lagrangian (5.1) may be rewritten in terms of an auxiliary field a{x) as C='$W>-j-
+ \<>W> •

(5.H)

Self-Interacting fermionic models

130

Since the dependence on the auxiliary field a is quadratic, and does not involve derivatives, the equation of motion of a is given by the constraint a = giptl)

,

which, when substituted into (5.11), reproduces (5.1).4 The idea of introducing the auxiliary field is to enable one to first integrate over the fermions, what is equivalent to computing the functional determinant det(z $ + a). The effective potential V^(a) is defined as (— \ times) the logarithm of the determinant for constant a: V^'(a) = —-In det(i $ + cr) ,

a = constant

.

Its computation corresponds to summing the one loop fermion diagrams at zero momentum [2] (see Figure 5.1); we have the result

The sum (5.12) corresponds to the expansion of the logarithm rW(„\'.//<")

iV ff JLI cPk " 2 J (27T (2^ ln

N

I1"*

87

In—r- - 1 A2

where a cut-off has been introduced, after Wick rotation, at k2 = A2 in order that the integral be well defined. Taking into account the tree contribution as well, we finally obtain TV 2 1 2 . + + — Veff(
This effective potential, whose form is given in Figure 5.1 has a maximum at the origin, and a minimum at the point a1 -mz

= Az2e

—N -*2-

(5.13)

The maximum at the origin has several consequences. If we try to quantize the theory around a = 0 in the large N limit, we obtain a tachyonic contribution to the effective four fermion interaction [2]. On the other hand, quantization around one of the minima at a — ±m (Eq. (5.13)) means that the discrete symmetry (5.14)

i> - » 7 5 ^ 4

Alternatively, we can write - — a2 +-.aipip =4g 2

— (a-gi>ip)2 Ag

+-g{tpip)2 4

•

The first term may be completely eliminated integrating over cr, leaving no trace of that field.

5.2 The 0{N) Invariant Gross-Neveu Model

131

I F i g u r e 5 . 1 : Calculation and form of the effective potential.

of the original Lagrangian (5.1) is spontaneously broken by a mass term ^mtjjip. Hence, for this choice of the vacuum, a mass is spontaneously generated for the fermionic field. The interpretation of m as the mass of the fermionic field requires that it be renormalization group invariant, i.e.

A2

^+«4H

0

(5.15)

This permits the computation of the /3 function of the model,given by 2

m = -Ng

(5.16)

2TT

The sign of the /? function shows that the theory is asymptotically free. Moreover,

•- + --0-

+ --O-O- + - - 0 0 - 0 -

F i g u r e 5.2: Summation of bubble diagrams contributing t o fermion interaction in the large N limit.

the cut-off A and the bare coupling constant g(A) can both be eliminated from the theory, in favor of the generated mass parameter alone. This "mass transmutation" simplifies considerably the discussion of the theory. The "mass transmutation" phenomenon might provide a dynamical mechanism for generating masses in theories which are massless ab initio, worthwhile to understand [10]. In the limit N -> oo with gN fixed, the bubble diagrams of Figure 5.2 have to be summed (this is a geometric series) and the theory may be solved. We are going to look for corrections in powers of 1/N. To this end, we rewrite the theory in a way as to allow for the systematic calculation of the corrections. The generating functional is given by Z[rj\ =

IV^VaeH'*'*{W9+<')1'-4;<'3+*N>}

(5.17)

expanding a around the vacuum expectation value (a) = — m, a = —m +

a

3

(a) = 0

(5.18)

132

Self-Interacting fermionic models

and defining $L[cr] = i ft — m + -?=, we can write the functional integral in the form Z[7?] = / 2 ? ^ a e * ^ 2 ^ ( ? + ^ [ " r l ) A F [ " ] ( ^ + A [ S ] " S ) - ^ " 1 ^ ^ " 2 - 7 ^ m S } .

(5.19)

We integrate out the fermions, writing the determinant thus obtained as det #[cr] = tr In $.[0]

.

Substituting this expression in (5.19) we have the result Z[rj\=

fvaei-l'd2x^~^MarlTI'^a2+^^ma}+^tT^~m+^)

.

(5.20)

The fact that a has vanishing vacuum expectation value, implies that all tadpoles cancel against the linear term in a, that is m / d2xa + — / (fxatr

J

/ -—rr^

= 0

J (2TT 2 k2 - m2

2J

.

(5.21) V

'

Using a cut-off A for the momentum integral, 5 we obtain Eq. (5.13) for the mass. The P function has been computed in (5.15), (5.16). A comment concerning this procedure is in order at this point, if one aims at computing higher order corrections. The higher order /3 function may depend on the renormalization procedure used to calculate the fermion four point function. The two loop 0 function computed by normalizing the fermion four point function as in the symmetric theory (m = 0) gives again (5.16).6 If instead, the renormalization constants are computed using the non-symmetric theory, a different j3 function is obtained [12, 13, 14],

«9) = - ^ + Wa /*i 2 l + 4 o 2'/ / u 2 2TT with a2//J,2 related to g by, y ' l + 4a 2 //i 2 - 1 _

4TT

where the second expression is a modification of the gap equation (5.21) [14]. The 1/iV correction to the above result has been also discussed in the literature [12]. The two apparently conflicting results are seen to be correct, with different interpretations for the respective /3 functions [14]. Indeed we can rewrite the generated mass (5.13) in terms of renormalized parameters. The renormalized coupling constant g([i) is defined, at one loop level, by 1

1

JV

A

- —7r I n -n 5(A)

5

,

g(fj-) We perform a Wick rotation in the momentum integral, obtaining 1

mN

, A/ 2m/ "

r, /

J0

2

dx — - s2

x +l

(. =zm I

\

1 iv,- , A- ^-\ i „ 1 In —^r 2g 4TT m2J

This procedure is correct. See, in this respect, Ref. [11].

(5.23)

5.2 The 0{N) Invariant G r o s s - N e v e u Model

133

and we obtain from (5.13), m2 = i?e"^)

.

(5.24)

In general we can write this result in an explicitly renormalization group invariant way in terms of the /3 function as o - f9

,

mz

5.2.3

-is

= /x 2 e J ^

.

The ^ Expansion: Feynman rules

From (5.20) we are able to derive systematically new Feynman rules corresponding to the 1/N expansion of the model. The leading order contribution in 1/N to the IP I two point function of the field a is given by the second term in the expansion of the the logarithm in (5.20), plus the tree graph contribution (see Figure 5.3). k+p

—h-*-h-

-v~<^i>--f

-TTTT

k

~P~

2gN

Figure 5.3: Tree-and one-loop diagrams contributing to the a two-point function.

The corresponding momentum space result reads,

f(p)-

'

yP

'

2gN

\rfd2k

(* + ")[*+*+"1

(525)

2

J (2TT) {k - m2)[(k + p)2 - m2} 2

2

{

'

'

We perform a Wick rotation, and use the fact that the logarithmically divergent part of (5.25) cancels the cut-off dependent constant —— upon use of (5.13), (5.23), and (5.24), leaving a finite result: the renormalization of the model within the 1/N expansion is very simple due to the mass transmutation. Performing the elementary integral in (5.25) one obtains, after cancellation of the divergent term, 7

f^'J^^i^zS . 471-y 7

W n havp W e haVe

P d2k

t

-p2

(¥+m)W+rf+m]

J j2^tT^-m2Wk+P^-'m'2]

v

v - p * + 4m2 + v c ^ _

-

2

f1 .

(2^T Jo

dx

f j2i.

J

d

k2-p2x(l-x)~m2

fe

(5. 26 ) >

,. ,

[fc2_m2+'p2,(1_,f

WhlCh Can

be trivially integrated in the angle variables; the radial integration can be performed and results

•h Si dx { i l n A 2 - ila {~m2 +P2x^ ~ *)) + -ffiffi^i*-,)}-

See 15 for this in

I l

tegral.

Self-Interacting fermionic models

134

The
(5.27)

The Feynman rules can now be read from (5.20) and are summarized in Table 5.1 where we have defined p2 = 2m 2 (1 — cosh#) = —4m2 sinh 2 | . Table 5.1 a- propagator

(0 | a(p)a(-p)

| 0)

ip propagator

<0|Va(p)^(-p)|0>

iprpa vertex

0'(p)'/ , a(9)'/ , /3(-P-9)

(l»)Af(p)''=

-j^

In the 1/N expansion, the renormalization of the model is very simple. By the power counting theorem [16], the only divergent diagrams are those for which the superficial degree of divergence, given by 5{Y) = 2 - ±N* - Na

,

is non-negative, where T denotes the diagram, N$ the number of external fermion lines, and Na the number of external a lines. Thus, we have divergencies in the two- and four-fermion correlation functions, in the two-fermion one-cr correlation function, and in the inverse a propagator. The latter, as well as the divergence of the fermion two-point function, can be absorbed in the fermion and a wave function renormalization, and in the mass renormalization (or equivalently, due to mass transmutation, in the coupling constant renormalization). The superficial degrees of divergence of the remaining correlation functions are actually negative in the 1/iV-expansion, if we consider an appropriate sum of diagrams of the same order. The cancellation mechanism occurs between the diagrams displayed either in Figure 5.4 or in Figure 5.5: consider a diagram with the external lines tpi and ipi', then form a new diagram, joining those lines into a a line, which in turn reproduces the pair I/J* and ip again (see the diagram of Figure 5.4), i.e. the above mechanism was used in the first two, and in the last two diagrams of Figure 5.5). It is easy to verify that the diagrams are of the same order in I/TV, due to a factor of N arising from the fermion loop. The cancellation is due to the fact that the blob is logarithmically divergent, which is thus a (cut-off dependent) constant when shrunk to a point. In this case,

/ Vcj(j{x)(j{y)ez

82 SJ(x)SJ{y)

J

<TI

SJ(x)5J{y)

!*>«*/«-

•fvaeif1'

Ta-2Ja)

-jr-1)r(
5.2 The 0(N) Invariant Gross-Neveu Model

135

\ /

F i g u r e 5.4: Cancellation mechanism of lowest order divergencies.

A F i g u r e 5.5: Cancellation mechanism of higher order divergencies.

due to the definition (5.27), the divergent pieces cancel as shown in figures 5.6 and 5.7 (the a propagator is minus the inverse of the renormalized fermion loop).

F i g u r e 5.6: Cancellation mechanism concerning diagrams of figure 5.4: the same divergent constant appears in both diagrams, differing by a minus sign.

5.2.4

Leading order S-matrix elements

The l/N expansion of the model is very useful to check several results concerning the S-matrix, which will be computed exactly for the Gross-Neveu model in Chapter 8. The essential property allowing for the computation of the exact S-matrix is its factorization in terms of 2 body S matrices. In turn, this property is equivalent to the statement that the probability of pair production is zero 9 . 9 In a factorizable amplitude, the intermediate states (after 2 body collisions) are on shell. Consider the amplitude of pair production, and the 3 body amplitude in Figures 5.8 and 5.9, related by crossing. In Figure 5.8 the intermediate particle momentum r = —pi + P2 + P3 is on shell if either pi = p 2 or p i = p 3 , which are correct configurations. However, in Figure 5.9,

Self-Interacting fermionic m o d e l s

136

F i g u r e 5.7: Cancellation mechanism concerning diagrams of figure 5.5.

"Pl + P 2 + P3

Pi

P2

P3

F i g u r e 5.8: Three body scattering amplitude.

This has been proven in general [17]. Indeed, the absence of pair production in this model is verified by a very simple computation, using the 1/N expansion [4, 18]. The lowest order amplitude is given by diagrams of Figure 5.10, which are of the order -^. We will use the fact that in two-dimensional space-time, one loop diagrams are exactly computable in terms of tree diagrams times a two-propagator loop (see [19], [20], Chapter 3 and Figure 5.11). This is due to the fact that the result of performing the k° integration using the theorem of residues is a rational function, after a convenient change of variables. The rational function thus obtained may be decomposed into partial fractions, with coefficients calculated at the respective residua. This leaves us with a coefficient given in terms of a sum, where each term of the sum is given by the product of all but two propagators and vertices. The momentum integration only receives a contribution from points where the poles of the two selected propagators coincide. The remaining integral is trivial, and it is given in terms of the function AI ^ f A(p)=

1

d2k

7 (2TT) w (k + 2

Pi)

2

1 -m (k

27ryV(p 2 - 4m 2 ) 47rm2sinh#

2

In

2

+ Pj)

-

m2

y/—p2 + 4m 2 — \f— i \f-p2

+ 4m 2 + (5.28)

r = p\ +/>2 +P3, consequently r 2 > 3m 2 , by the two-dimensional kinematics. Therefore, the total amplitude for pair production (or annihilation) must be zero, in a factorizable amplitude, when all diagrams are added.

5.2 The 0(N)

Invariant Gross-Neveu Model

137

Pl+P2+P3

Pi

P2

P3

F i g u r e 5.9: Amplitude for pair production related to figure 5.8 by crossing.

with p = Pi~Pj,p2 have used

= —4m2 sinh 2 6/2, pi = m(cosh#;, sinh#;), 8 = 9i —9j. Here we

J (2n ) 2 [k2 + a ( l - a)p2 — m2 + is]2

An m 2 - a ( l - a)p2

'

(5.29)

as well as a ( l — a) / da 2 m -a(l-a:)p2 Jo

v/p2(p2_4m2)

In

^/p2(p2-4m2)-p2 I

x

/p2(p2_4m2)+p: (5.30)

-rU.IH l.X.

V

F i g u r e 5.10: Diagrams of order ^

V -ct>-

contributing to the amplitude of pair products

V

—

-o-

V

V

+ ^ -

+ -<&-

F i g u r e 5 . 1 1 : Explicit computation of the first diagram of figure 5.10. The dashes in the intermediate figure mean use of the cutting rule (Chapter 3), where the momenta are such that the dashed lines are on shell; each diagram here cancels two corresponding diagrams in figure 5.10.

138

Self-Interacting fermionic models

The above procedure permits one to compute the first diagram of Figure 5.10 exactly in terms of the other six tree diagrams, and shows that they add up to zero. Using the 1/N expansion we are able to obtain the S-matrix up to second order. The two particle scattering amplitude for fermions with isospin i,j and rapidities 61,62, has the general form 10

(e'1k,0,2i\e1i,e2j) = = [ax{6)5ikPl a ik

- [ai(e)5 6

+ a2(e)SilSkj

+ a 3 (0) W ]

ik

+ <j2(9)5 6i' + (73(0) W ]

<J(0i - 0i)6(02 - &2) S(BX - 6'2)5{e2 - 9[) ,

where 0 = 61 — 62. To lowest order perturbation in jj, we have the contribution due to exchange of the auxiliary a particle (Figure 5.12). Using the Feynman rules given in Table 1 one finds [4] 2TT

<7l(0) = l +

(5.31)

JVsinhtf

*<»> = £

(5.32)

2-Ki

(5.33)

N(iix - 9)

The computation can also be extended to one loop order. A perturbative expansion of the S-matrix, may also be obtained in this case (see also [21] for further details). p

q

q I

p

q

p

(a)

(b)

(c)

Figure 5.12: Lowest order contribution to (5.31), (5.32) and (5.33).

5.2.5

Quantization of t h e non-local charge

Our next aim is to quantize the non-local charge (5.7) in order to obtain nonperturbative results from the model. Divergencies in this non-local charge arise from the short distance singularities of the product of two currents in the first term. We therefore, consider the Wilson expansion

\j»{x + e)M*)\

= Y,C${e)0{ay(x)

10

,

(5.34)

Notice that instead of the usual momentum delta function, we use deltas in the rapidity. To compare with the usual perturbative calculations, we need to include a factor m2 *inh g > since

S(P0 -P'OMPI - P i ) =

mi^(e_ei)S(0i

- «i W * - «i)-

5.2 The 0(N)

Invariant G r o s s - N e v e u Model

139

where 0{a}(x) are local field operators and CJi" (e) are c-number functions. The dimension of the operators 0{ay (x) can be classified by making use of the asymptotic freedom of the theory [2, 24]. Since the left hand side of (5.34) has dimension 2, there are only contributions to the right hand side coming from algebra valued operators of dimension 1 and 2. Therefore one finds [25, 26] \3v{x,e),jv(x)\

= C%{e)jP{x) + D%{e)dajp(x)

.

(5.35)

There are constraints from P T and CP invariance, as well as current conservation, restricting the coefficients G^v{e) and D^(e). The first two imply ^ „ ( e ) = - C ^ ( - e ) = C^(e)

,

Using Lorentz invariance, we may thus obtain the general structure of the coefficients, which read

^nA£) - Ci~^r~+ ° 2 — p ^(e)-^i

-T2

+ G3_

+ D2

pyr

~2

K £v6 TO+W» \ h°+\C2Z£a»EVEPE ° 2 ' e2 (e 2 ) 2

+iMw Asymptotic freedom implies that 1 1

Ci = 0 ( | £ | - ° )

,

A = 0(|£|-°)

•

Current conservation implies several constraints. One of them is d G1 = - 4 ^ - 2^ {D1(-x2)

- D3(-x2))

d = -2x— (Dx(-x2) da;

- D3(-x2))

dl

(5.36)

_ cc, or, for e° — 0, -is1 =

Cx{-x2)

.

(5.37)

We define the cut-off charge <9i =

/

dyidy 2 e(yi - y2)jo(t,yi)j0(t,y2)

- Zs

dyjx(t,y)

\yi—j/2\>S

In Ref. [25], one supposes that the fundamental fields have only logarithmic singularities; therefore, a smearing in the space coordinate would be enough to define correlators. In this case it is possible to compute the C;'s exactly. We do not need them, however [28].

Self-Interacting fermionic models

140

which will be finite in the limit 6 —> 0, if we define the cut-off as 12 Z5 = 2(D1(-62)

- D3(-52))

.

(5.38)

From the fact that the expansion (5.35) contains no further term, we also have conservation of the non-local quantum charge, which follows from ^

L

= fdy{[jo(t,y)Jx(t,y

+

6)]-(5^-6)}

-2(D1(-S2)-D3(-52))Jdy^(t,y)^0

as

S -»0

.

We may thus define the quantum non-local charge as the limit Q« = lim Q]j We examine now its action on asymptotic states. We could follow reference [25] and compute Qm/°ut in terms of asymptotic states. Although the results turn out to be correct, the procedure is not rigorous. We shall thus follow [28] and [29], computing first the action on one particle state and then on two particle states, which is our aim. Details not shown in the text are presented in Appendix G. The Lorentz transformation acts on the non-local charge as N —2 J*'

[T,QV] =

,

(5.39)

IT

where J y is the isospin operator acting on a 0{N) vector | 9, k) as

jij\e,k)

= isik\9,j)-isjk\e,i)

= (iij)kl\e,i)

.

(5.40)

Since the non-local charge commutes with translations, it must annihilate the vacuum: Qij\0)=Q

.

Moreover, it is a second order antisymmetric tensor and, on a one particle state, must satisfy Qij\0,k)=g(e){5'k\6,i)-Sik\9,j)}

.

On the other hand, the generator of Lorentz transformations acts on a one particle state as T\e,i)

= -^\e,i) d9'

.

12

The divergent piece arises from the first, linearly divergent term in the Wilson expansion (5.35). We have Qs =

finite

+ /

dx\ -

C l

^

' \ / dyji(t,y)

Therefore we find (5.38) after using (5.36) and (5.37).

- Zs /

dyji(t,y)

141

5.3 Chiral Gross-Neveu Model Applying (5.39) on a one-particle state, we find .N-

dg

2

-i

de

7T

Since Qij is odd under parity, we have ,N - 2 9(0) = ~iTherefore QH\e,k)

= ]^e{Vk\6,i)-6ik\9,j}}

.

(5.41)

ITT

The action of the charge on two-particle states is described in Appendix G. For a two-particle state with different momenta we have (fa ® ^41Q"^'|i ® h) = limt_,.±oo / dxdy e(x - y ) x X {2) + {l\Q\ P2, in the limit t —> — oo, we have only a contribution from x < y (in the far past, there was no interaction, and the faster particle 1 is to the left of 2). There is a change in sign if either pi < y>2, or t —>• +oo. Therefore, performing the integrals, we may summarize the action of the non-local charge on asymptotic states by

Qmn\0i,ii02j) =

\91,k-,e2,i){-[{rnr)ik{rnyl-(imryl(rn)ik'\ ITT

Z7T

J

and (0i,*;02,j I Qmn = {01,k;e2,l

| {[(J m r )' f c (I™)'' - (J™")"(I™H

TV — 2 + —: 9AImn)ikVl

+

ITT

/V — 9 -92{Imnyl5ik\ ITT

-i . i

The constraints on S-matrix elements [4] implied by the action of the non-local charge on asymptotic states are discussed in Chapter 8 and Appendix G, where we also show that as a result of these constraints, there is no pair production.

5.3

Chiral Gross-Neveu Model

This model is a generalization of the previous one, promoting the discrete symmetry (5.14) to a global continuous chiral (7(1) symmetry [2] V> ->•

eie~*5ip

142

Self-Interacting fermionic models

and the trivial symmetry tjj -> —ip to a global U(l) symmetry i> -> eiaip

.

In fact, ip now is a complex Dirac field (see footnote 1), so that the symmetry is now SU(N) <8> U(l) ® U(l). The Lagrangian is given by C = i & fal>i + ^g[(^i)2

- bPils^i)2}

,

(5.42)

where the summation over the SU(N) index i is understood. The Lagrangian (5.42) again defines an integrable model [3, 8]. The Noether current associated with the U(N) symmetry is given by Use of the equation of motion and Fierz transformation, shows that it satisfies Eq. (5.2), as well as equations (5.8) to (5.10). This demonstrates the integrability of the chiral Gross-Neveu model, and implies the existence of a non-local conserved charge of the form (5.7).

5.3.1

Cancellation of infrared singularities

In order to obtain the 1/iV expansion of this model, we have to reformulate it. As in the previous case, the theory can be reduced to a quadratic form in ip at the expense of two auxiliary fields, C =ii> flip- —(a2 +7T2) +:ip(a +iTT-f5)ip .

(5.43)

However, the 1/JV expansion of the model using the above Lagrangian cannot be performed, due to serious infrared (IR) problems [30]. Indeed, if we try to do it, we find a massless pole in the IT propagator, which plays the role of a massless ("would be") Goldstone boson, whose appearance is forbidden in two dimensions by the Coleman-Mermin-Wagner theorem [5]. In order to deal with this problem, it is common practice to write the fields in terms of a + iir = pe1^, such that the Lagrangian reads C = hp jhl>~ —P2 + pipe^'ip

.

(5.44)

We are going to discuss the quantum theory associated with the above classical Lagrangian in three different ways, obtaining similar results. The more pedestrian approach consists in the extensive use of the bosonization formulae of Chapter 2. The fermionic fields are bosonized in terms of an iV-plet y>»- The situation is analogous to the massive Thirring model, and one obtains the equivalent bosonic Lagrangian N

C=-^2(dtl
N

+ ^PJ2™s( +
(5.45)

143

5.3 Chiral Gross-Neveu Model

It is now convenient to use "fermionization" formulae in order to rewrite (5.45) in terms of new fermion fields xpi [31, 32], by making the identifications irpi Wi = - \d„ ((pi + Z L

-=)

\

V47T'

ipiipi = 7rcos(

+ l

Piv4'ir)

,

Z7T

The Lagrangian (5.45) then takes the form [32] —

-

1

-=- -

\

-=-

N

C = 4>ii @i)ilg- —p2 + pipiipi + -d^HnYl^i »7T + ^-{d^Y z

(5.46)

The important point is that the massless field

±i
the Lagrangian (5.44) takes the form

C = *[iV + P

+

1

! ^ +

PJ-^1>--p+P-

(5.47)

It is convenient to consider an auxiliary Lagrangian £ ' obtained from (5.47) by the substitution <9M -» £>M = dM — iA^, with A^ an "external" £/(l)-field:

£ = i> (ip>+P+lA^+p-1—^-)

v- - \P+P-

(5.48)

The corresponding effective action is given by iW[A,p+,p-}=

In det (ip

+ p+^-^+p. 2

1 _ 7 5

(5.49)

2

The effective Lagrangian (5.48) is invariant under the local transformation

Afi -»• Ap + e^vdv(p P±^e^p± ,

,

the generator of the corresponding infinitesimal transformation being T[
5 = J d2xL^dv "uS**

s

p-1 5p-

s p+1— 5p+

}•

(5.50)

144

Self-Interacting fermionic models

The effective action (5.49) however breaks this local invariance. Indeed, the external field axial vector current SW oAv

= tpVJv(x,A)=etlv—-

Jti(x;A)

,

(5.51)

exhibits the well known Adler-Bell-Jackiw anomaly [33, 34, 35], which in 1+1 dimensions takes the form (see Chapter 4) &kh

= Tr?vFi™

•

(5-52)

Using (5.51) and (5.52) one finds [23, 22]

Hence W[A] is not gauge invariant as a result of the anomaly (5.52). From here it immediately follows, using (5.50), that T2y\W

f d2x{d»
=~

Tn[
,

n>2

,

(5.53)

.

Since finite transformations are obtained by exponentiating T, the corresponding series truncates when eT is applied to W: eT^W

= W+^

2TT

[d2x vt^Fuu

J

- ^

/ d 2 x (d^f

2-7T J

.

Therefore

W[Alt-elll/d,'
P-} + ^ J

= d2x e^F^v

- ^ J d2x (d^)2

.

In particular, we obtain for the actual case of interest, A^ = 0, TV r

W[0, p+, p-] = W[e^d"p,

e - * p + , e * V ] + ^ J
.

(5.54)

The left hand side is just the effective action associated with (5.47), while the first term on the right hand side is the effective action associated with (5.46), if we make the identification ip — f. This proves the equivalence of Lagrangians (5.46) and (5.47).

5.3.2

T h e ^ expansion

The large TV expansion of the model defined by the Lagrangian (5.46) can be explicitly performed [32], following the method described in Eqs. (5.17) to (5.27). The

145

5.3 Chiral Gross-Neveu Model

propagator of the p field is exactly the same as that obtained for the a field in the 0{N) case. 13 The zero'th order contribution to the p-propagator is thus given by expression (5.26), with the diagrams depicted in Figure 5.3. From (5.25) and (5.26) we obtain for f (p) 27T tanh § where 8 is defined by p2 = —Am2 sinh

"S£" 9,lv F i g u r e 5 . 1 3 : Diagrams contributing to the two-point function of the field Ali—y/Nclt„a'/(t:.

Since only d^ occurs in (5.46), we just need the two point function of A^ = vNttxvdv4>. We thus have to compute the diagrams of Figure 5.13, as given by 14

""W ~

Sn9^

4try

(# + m)7„(#+ ji + m) m2)[(k +p)2 — m]

{2n)2^{k2_

Performing the integral one finds / — p2 + Am21 \/—p2 + 4m 2 — y/—p2 2TT V -P2 y/-p2 + Am2 + v / 3p* 1 f) — 0 t a n h -(# M „p 2 - p^pv) 1

^w=^\H^F-^ y ,

f

2

\ ~ v , a>,^ - p**)

where p2 = -Am2 sinh 2 | . The vertices are shown in Figure 5.14. The amplitudes for particle scattering are thus all free from IR divergencies, and may be computed without difficulty. We shall compute the two particle scattering amplitude in lowest order. The S-matrix elements have the general structure (0iM£Z I <M;02j> = {ux(9)8ik5jl + u2(e)Sil6kj)5(e1 {Ul(e)5a5ki + u2{9)Sik5jl)5(e1

- 0'1)5{e2 - 0'2) + - d'2)8(62 - e[) .

13

We first redefine p — — m + -y^p; the condition that the field p has zero vacuum expectation value, implies that the mass parameter m be given by the expression (5.13). 14 This integral is convergent if a reasonable regularization is used. Indeed, the only divergent piece comes from / d 2 fctr7 f ,7 P 7„ 7
/ d2kk2f{k2)

,

which vanishes due to the identity 7^7^ j p = 0. Therefore, the integral may be performed using the residue theorem. The anomaly cancels between the two terms, as it should, since the second term corresponds to the effective action.

146

Self-Interacting fermionic models

;YuY 5

F i g u r e 5.14: Interaction vertices corresponding to the Lagrange density (5.46).

The lowest order contributions to ui(8) are given in Figure 5.14. The p exchange diagram contributes a term N " h e (the p-propagator at zero momentum is a constant), while the A^ exchange contributes /

47ri\

(—-k)2

i

V AT/4msinh<9 The resulting amplitude is found to be U l (0)

,

ffi

M

,

.

N

= l + ^coth(^

.

(5.56)

We shall verify in Chapter 8 that this is indeed the lowest order approximation of the exact solution. Particle-antiparticle backward scattering vanishes. This may also be verified by a perturbative calculation, using an IR cut-off [27].

5.3.3

Operator formulation

This model may also be studied in the operator formalism, which leads to the 1/7V expansion, and a correct understanding of the relation between the "would be" Goldstone boson and chiral symmetry. Since the fields tpi lie in the fundamental representation of U(N), we have the following bosonic representation [36]

with i = 1 , . . . , N. Since the x'is satisfy15

are

SU(N) valued; they are not independent, but

N

J2xi(x)=0

.

(5.58)

»=i

The field x i s the potential of the conserved U{1) current. Its zero -mass character will ensure that the U{1) symmetry is not spontaneously broken. 16 15

The independent fields are given by Xf = X ) T / / ^ l r > w h e r e T%D a r e t h e generators of the "torus" of SU(N), and
147

5.3 Chiral Gross-Neveu Model

In the above, K;is a Klein factor, necessary to enforce the correct anticommutation relations among different \p[s. Due to the U{\) x U{1) symmetry, the divergence and the curl of the 1/(1) current vanish, so that the field xix) is massless. Therefore the fermion fields contain the so-called infraparticles [37], and we need to extract them in order to arrive at the physical fields of the theory. They are given by j,i(X)=K

/Ze^{75Xi(x)+/JVxi(x°,y1)} V 27T

_

(559)

The ip fields (5.59) will be found to correspond to the field ^ in (5.46). These fields no longer carry E/(l) x C/(l) charge, and transform as a representation of SU(N). The constraint (5.58) implies V'i ~ 7 TTie»i•••<"-!^i ' " ^ i v - i ' ( 5 - 60 ) (n-1)! where on the right hand side a suitable redefinition of the Klein factor and the normal product prescription is required. Eq. (5.60) states that the antifermions of the chiral Gross-Neveu model can be viewed as a bound state of N — 1 fermions. We shall use this fact to determine the S-matrix and its pole structure. Asymptotically one expects ip to describe massive particles, so that one should have [36] rp(vt,t) -> ^ { e - i m 7 _ 1 ' a ( m 7 t ; ) + e ' ^ ^ ' S ^ n v y t ; ) }

,

(5.61)

vl*l The fields ipi carry spin s = | ( 1 — 1/N), as can be seen from Eq. (2.47). They have the property (see (2.40)), 4>(x, t)j>{y, t) = e2'"<x-y^(y,

t)j,(x, t)

,

implying an unusual statistics for the creation and annihilation operators defined in (5.61) tf{p)ai(p') = e ^ ^ - P V f c V t e ) • Since no scattering theory is known for particles with the above statistics, it is necessary to replace the field xf> by another field tp' with a well defined statistics. This is achieved by introducing in (5.57) free massless scalar and pseudoscalar fields B and A, quantized with metric opposite to that of x{x)i i n such a way that the divergent infrared behavior of ip induced by x(^) is compensated, without affecting the statistics. We define [36] xP[(x) = eVfrfr

A

^+B^^i(x)

.

Correspondingly, the operators a* ,a,tf ,b are related to a\ a, ¥, b by a

JF in\P) = ain(P)e -t I \ t ( \ 2iri(s-|) f N0„t(p')dp'

(5.62)

148

Self-Interacting fermionic models

where N m are the corresponding particle number operators. Since we expect ip\(x) to be a local field describing massive degrees of freedom, we should have in the far past and future [38] ip'(vt,t)

->

-i=[e-imT_1'aou«(m7u)+eimT"lt6L(m7u)]

.

(5.63)

Substitution of ip in terms of ip' in (5.42) leads formally to the Lagrangian

£ = & Hi + \9[$rt'i)2

- bP'nsri)2} - \{d,Af

+ -?=lp'jS1»iP>dfiA--{Lj1»ip'dtiB

-

\{d,Bf

,

(5.64)

where we allowed for general couplings a and /?, which after renormalization should reduce to y/n as the renormalized value. We will came back to this point after obtaining the ^ expansion, which we consider next. The effective action obtained from the Lagrangian (5.64) after introduction of the auxiliary fields a and 7r (compare with (5.43)) is given by Seff = -iN ti In i i ft + a + in-y5 + - ^ = 7 5 dA--^=

I

VN

- i J d2x{a2 +*2)-\j

VTT

dB\

J

d2x[(d,A)2 + (d.B)2} .

The field a is found to have a non-vanishing vacuum expectation value (a) = — m, so that it is convenient to write a 7=

u = —m H

,

and

IT =

w —=

The second order contribution to the effective action is then given by s 2

i ff=l2tTfd2xd2y(iP-m)~1 x

i7NHy)+7N*{yh5+7N15 1

x (i

fi-m)-1

—a(x)

my)

my)

~7w

%

)

Ot

+ y=Tr(x)^s + -T=l5

@A{x)--

- ^ fd2x(a2+n2) + ^ jd2x{AuA + BUB) . The proper functions are given in terms of the one loop diagrams tabulated in Figure 5.15. There are mixed functions as well. We summarize the results. For the a two-point function

d2k

fi + m 2

2

2

ji+tf + 'i

) k - m (k + p)2 - m2

gN

5.3 C h i r a l G r o s s - N e v e u M o d e l

iy5

149

iy5

i

-#»<^>»//-

~i'-<^>-#"

(a)

«/W<^>W>.

(b)

p

P

(d)

-iap v y 5 y,

lap"ir s Y„

x

(c)

(e)

F i g u r e 5 . 1 5 : Diagrams contributing to the jr(o),
i / — p2 + 4m 2 \/—p2 + 4m 2 — \/—p2 n ~ 2n Y -p2 ^ / - p 2 + 4m 2 + y/^P _-j

e

(5.65)

~ 2TT tanh §

'

which is the same as (5.55). For the •n two-point function we have f d2k

-

(2TT)

i

ji + m

2 /0

2

fc - m

— p2 Y -p2 + 4m 2 /

2TT

n

2

fi+ p1 + m

i

'° (fc + p)2 - m2

gN

y/—p2 + Am2 — \/—p2 ^/-p2 + 4 m 2 + y /Z^2

t 61 2TT coth f

~

(5.66)

For the 1PI two-point function involving A and B one finds f

(n\

2 n

,r

f

= -i—p2

d2k

7 M 75(^ + m ) 7 ' / 7 5 ( ^ + ^ + m )

.2

- ip2 + (2ma)2fW7r

7T

= Wsinh^{l4(l--^)} r B B ( P ) = /3 t r y {2ir)2{ka_ma)[{k

+

(5.67)

p)i_m2]p^-v

Q

= -ip2 = 4 i m 2 sinh2 -

,

(5.68)

where we used, in (5.67), the contribution of the Adler anomaly, and in (5.68), current conservation. We also have the mixed term k

d* 5 f^) = -atr|A^ 72
jt + m 2

' k -m

2

ji+ p'+m lo

{k+p)2

- , . ma / c o t h | \ = -2mar™(p) = - — I — j ± \

-m2^*

.

(5.69)

150

Self-Interacting fermionic models

All other 1PI two-point functions involving B are found to vanish. In order to obtain the a and B propagators we just invert (5.65) and (5.68) respectively. The computation of the A and it propagators requires on the other hand, the inversion of the 2 x 2 matrix p

/

A 7T7T

1

\l\rA

TA

TAA

whose determinant is given by (using Eqs. (5.66), (5.67), (5.69)) det T = - i p 2 ( l + a2/ir)Tww

= 4im2 sinh2 i(i

+ —) 2ni — coth §

We find D{p)-

iT

~ ^

^(p2)

( p 2 )

with DA(P2)=

1

p2 1 + a2/n

2,_0_cothf

Dn(p2) =2TT

1+ ^ ( 1 - - ^ )

1 2ma p2 1 + a 2 /7r

£>,rA(p2)

We now fix the parameters a and /? in (5.64) by requiring that the IR divergencies cancel. We expect to obtain for the non-renormalized values, a = oo (corresponding to aren = y/n and j3 = j3ren = ^/TT, since B couples to a conserved current). As a matter of fact, if we consider the fermion-fermion interaction to lowest order in the JJ expansion (see Figure 5.16) we verify that for a —> oo, the mixed propagator D^A gives the contribution (Figure 5.16 b,c). The subscripts refer to 0+0

i 2ma2 p2 i + «i It

{PnU(Plh5u(pi)u{p2)'y5ltlu(p2)

+PMM(pi)757/iu(Pl)"(P2)75M(P2)}

(5.70) We use the equation of motion for the free spinor « ( p ' ) 7 s ( / - ?0«(p) = -2mu(p')7 5 u(p)

,

in order to reduce Eq. (5.70) to the form Ai66+i6c = 2T5--5-U75 puu'7 5 P"' 1 + v_p* 7T

2 Q:

= 2

rl 1+ —

2 -P\lPv

~ 9nvP

—

o P

wy^uwy^u

u

—1 a

1

/c 7, \

,

(5-71)

151

5.3 Chiral Gross-Neveu Model

O n (a)

U~<^>.

(b)

[wuwuwJ

(c)

tOOMWOwd

(d)

(e)

(e)

F i g u r e 5.16: Contribution to fermion fermion scattering at order i-.

where we used the identity e^ppC^p^ = pPp" - g^p2. The exchange of a 7r gives a vanishing contribution for zero momentum transfer. Indeed, w(p)75u(<7)u(
tr

0 — -v 0— v 7 5 ( cosh — h 75 sinh — h

• sinh • i *—+ zjo

X 1. •171 sinh • i * —-— + X ^J

vanishes in the limit q —>• p because of the trace, so that Ai6a = 0

.

Finally, the B exchange gives the contribution A 1 6 e = -^Pz2uru(p)u'l"u^ p

,

p*

which partially cancels (5.71) , leaving us for a —• 00 and /3 = ir, with Aie = A i 6 / - 2m7 M uu'7 M u'

,

which leads again to the result (5.56) . We have thus verified in the N —> 00 limit that both Lagrangians (5.46) and (5.64) lead to the same result for U(l), as they should. In the limit a —> 00, the renormalized coupling aren indeed turns out to be y/n, as one reads off from the four-point function, and the pole in the 7r-propagator vanishes. To summarize, we conclude that part of the field (5.57) which carries chirality decouples [31, 32, 36, 39] from the physical spectrum and the remaining part describes an SU(N) multiplet with a well defined factorizable S-matrix (see Chapter 8). As we shall show in Chapter 17 there exists however a critical point in the coupling constant, where these conclusions are not realized, and the theory is conformally invariant.

5.3.4

Quantization of non-local charge

The discussion of the existence and conservation of a non-local charge in the quantum chiral Gross-Neveu model follows exactly the same pattern as in the O(N) invariant model. No anomaly exists in this case, since (5.35) gives all possible candidates to the Wilson expansion. As for the asymptotic result, there exist however

152

BIBLIOGRAPHY

differences due to the different symmetry group in question; thus, in (5.39) we have to replace N - 2 by• N, and in (5.41) only the first term is present. It is not difficult to see that the action of the charges on asymptotic states is given in this case by

Qab\0ii;02j) = \6ik;e2i)\-(iac)ik(rby' "• ac ki cb lj (Oii;02j\Qab = (e1k;02l\\-{I ) (I ) L

Qab\eii;e2j) = \0ik-,02i)\-(iacyk(i<:byl L

ac ki cb l (Oii;hj\Qab = (e1k;e2l\\-(I ) {I ) i L

+ —e1{iab)iks^>1 + —e2(iab)jlsik] in

in

J

+ —01(Iab)ki5'i + —62(Iab)ljSik] J in in + — ^ j 0 6 ) ' * ^ ' - —0i(iab)1^ sik] J in in ab ki l ab l ik + —0l(I ) 6 > - —0~2{I y 6 ]J in in

, , , ,

where 7 a b are the SU(N) generators defined analogously to (5.40). These formulae will be used in Chapter 8 to fix the exact S-matrix of this model as well.

5.4

Conclusions and Physical Interpretation

The Gross-Neveu models are simple but physically rich models. The semi-classical analysis, both in the O(N) and in the SU(N) x U{\) x [/(l)-symmetric cases reveals that the models have a rich bound-state structure [4]. As we shall see in Chapter 8, in the O(N) symmetric case the complete S-matrix can be computed, thus providing a reasonable understanding of the Gross-Neveu model. The case of the chiral Gross-Neveu model is particularly interesting, due to the chiral symmetry breaking issue. Since, as we saw in the previous section, a mass term is dynamically generated for the fermion, one could be led to conclude that the chiral symmetry is broken, which is prohibited in two-dimensional space-time17.[5] This problem has been discussed at length by several authors [31, 32, 39]. The interesting outcome is that the chirality carrying field decouples from the theory (Eq. (5.62)). In the operator language, this is realized by the factorization of the auxiliary fields A and B. The physical fermions, as given by either (5.59) or (5.62), though exhibiting a non-vanishing mass gap, are chiral singlets. This physical picture is carried over to the supersymmetric C P W _ 1 model, and reflects the fact that antiparticles are bound states of particles, in both, the Gross-Neveu model, and in the supersymmetric ( C P ^ - 1 model. This permits the computation of the S-matrix for these two cases [40] The chiral Gross-Neveu model will also appear in Chapter 12, when we study the so-called U(l) problem in the context of QED2, where an interesting physical interpretation is obtained.

Bibliography [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [2] D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [3] A. Neveu and N. Papanicolau, Commun. Math. Phys. 58 (1978) 31. 17

Gauge theories play a special role in this respect; see Chapter 10.

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[4] A.B. Zamolodchikov and Al.B. Zamolodchikov, Phys. Lett. B72 (1978) 481. [5] S. Coleman, Commun. Math. Phys. 31 (1973) 259; N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [6] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D12 (1975) 2443. [7] A.C. Scott, F.Y. Chu and D.Mc. Laughlin, Proc. of the IEEE 61 (1973) 1443. [8] T.L. Curthright and C. Zachos, Phys. Rev. D24 (1981) 2661. [9] C. Brezin, C. Itzykson, J. Zinn-Justin and J.B. Zuber, Phys. Lett. 82B (1979) 442. [10] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888; D l l (1975) 3040. [11] K. Symanzik, Renormalization 1970.

of theories with broken symmetry,

Cargese,

[12] H. Fleming and K. Furuya, Nuovo Cimento 49A (1979) 101. [13] H.D.I. Abarbanel, Nucl. Phys. 130 (1977) 29. [14] H. Fleming, K. Furuya and J.L. de Lyra, Nuovo Cimento 53A (1979) 405. [15] I.S. Gradshteyn and M. Ryzhik, Table of integrals, series and products, A. Press, 1980. [16] S. Weinberg, Phys. Rev. 118 (1960) 838. [17] D. Iagolnitzer, Phys. Rev. D18 (1978) 1275. [18] B. Berg, M. Karowski, V. Kurak and P. Weisz, Phys. Lett. 76B (1978) 502. [19] B. Berg, Nuovo Cimento 41A (1977) 58. [20] G. Kallen and J. Toll, J. Math. Phys. 6 (1965) 299. [21] A.B. Zamolodchikov and Al.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. [22] A. DAdda, A. Davis, P. di Vecchia and P. Salomonson, Nucl. Phys. (1983) 45.

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[23] A. Polyakov and P. Wiegmann, Phys. Lett. 131B (1983) 121; 141B (1984) 223. [24] K. Wilson, Phys. Rev. 179 (1969) 1499. [25] M. Liischer, Nucl. Phys. B135 (1978) 1. [26] E. Abdalla, A. Lima-Santos, Rev. Bras. Fis. 12 (1982) 293. [27] E. Abdalla and M.C.B. Abdalla, Nuovo Cimento 57A (1980) 334. [28] M. Liischer. Erratum of Nucl. Phys. B135 (1978) 1, unpublished.

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154

[29] D. Buchholtz and J.T. Lopuszanski, Lett. Math. Phys. 3 (1979) 175. [30] B. Berg and P. Weisz, Nucl. Phys. B145 (1978) 205. [31] E. Witten, Nucl. Phys. B145 (1978) 110. [32] E. Abdalla, B. Berg and P. Weisz, Nucl. Phys. B157 (1979) 387. [33] B. Zumino, Les Houches, 08/1983, (Notes by K. Sibold); Nucl. Phys.B253 (1985) 477; B. Zumino, Wu Young Shi and A. Zee, Nucl. Phys. B239(1984) 477; L. Baulieu, Nucl. Phys. B241 (1984) 577; R. Stora, in New Developments in Quantum Field Theory and Statistical Mechanics, eds. M. Levy and P. Mitter, Plenum, N. Y., 1977. [34] W. A. Bardeen, Phys. Rev. Cimento 60A (1969) 47.

184 (1969) 1848; J. Bell and R. Jackiw, Nuovo

[35] S. Adler, Phys. Rev. 177 (1969) 2426; in Lectures on Elementary Particle Physics and Quantum Field Theory, ed. S. Deser et al., MIT Press (1970). [36] R. Koberle, V. Kurak and J.A. Swieca, Phys. Rev. D20 (1979) 897; E 2638. [37] B. Schroer, Fortschritte der Physik 11 (1963) 1. [38] H. Araki and R. Haag, Commun. Math. Phys. 4 (1967) 77. [39] A.J. da Silva, M. Gomes and R. Koberle, Phys. Rev. 20 (1979) 895. [40] E. Abdalla and A. Lima-Santos, Phys.. Rev. D29 (1984) 1851.

Chapter 6

Non-linear a Models: Classical Aspects 6.1

Historical development

Non-linear sigma models have extensive applications in statistical mechanics as well as in quantum field theory. Historically they have been introduced by Schwinger [1] and used in field theory by Gell-Mann and Levy in the sixties to describe the phenomenology of pions [2], realizing the expected current algebra [3, 4] in terms of a quantum theoretic model. The main drawback was the lack of renormalizability [5]. Moreover, the great success of non-Abelian gauge theories, which have become a cornerstone in the field theoretic description of elementary particles, has left the sigma models at a secondary level [6]. Their revival is due to various developments which we will describe in this and the following chapters. First of all, there have been activities on the statistical mechanics of sigma models, with applications to spin dynamics [7]. Also, it became clear in the late seventies that sigma models are, in several ways, closely related to Yang-Mills theories [8, 9, 10, 11]; they were therefore used as a field theoretic laboratory for understanding the theoretical basis of more realistic theories. More recently, sigma-models have appeared in string theory, whose Lagrangian is of the sigma-model type [12, 13]. These models also appear naturally in the low-energy description of supergravity and of string (or superstring) theory, mainly in the framework of Kaluza-Klein schemes [14]. In addition, they provide realizations of certain infinite-dimensional algebras, such as the Kac-Moody algebras [15], which are very important in the context of string theory and conformal field theory. Finally, sigma models appear quite naturally in several other field-theoretic problems that we shall discuss: integrable models [11, 16], confinement [10], -^-expansion [10, 17], Wess-Zumino effective actions [15], non-Abelian generalization of the bosonization technique [15, 18], and others.

156

6.2

Non-linear a Models: Classical Aspects

Sigma models and current algebra

The sigma model was originally introduced in four dimensions and used to describe the phenomenology of P.C.A.C. (partially conserved axial current). It was not a purely geometrical model, but rather given by the Lagrangian [2, 3] C = Co + &

,

(6.1)

which is the sum of two terms, a SO(4) symmetric Lagrangian

, , 2 1 ^ 2 _i_ ^?2\

^ /

2 , =;2\2

and the perturbation C\ given by Ci = ca . Co is symmetric under the group 50(4), or equivalently, under the chiral group, SU{2) x SUX2), which is realized as %

i -*

•>/>->• ip + -a • rip - -/3 • f'ysip 7T —)• 7T — S A 7T + /3<7

a —> a — (3 • -it

,

,

.

The Noether currents associated with these symmetry transformations are the vector current

and the axial vector current A

l

= ~ ^ M T S T V + •*%
,

which are associated with the transformations labelled by the parameters a1 and /?*, respectively. L\ breaks this symmetry; more precisely, it respects the vector part but breaks the axial part. Therefore, the vector current remains conserved (CVC hypothesis) but the axial current develops a non-vanishing divergence which is, in fact, proportional to the pion field (PCAC hypothesis) [2, 3]: d»A\

= C7T*

.

This is in accordance with the fact that the operators 7rs interpolate between the vacuum and the one-pion states [3]: (01^(0)1^(9)) =zhij

•

157

6.2 Sigma models and current algebra We therefore have the following relation for the pion decay constant:

f„ml = cZ% . A more detailed look at the quantum theory associated with the Lagrangian (6.1) reveals that C\ induces a non-vanishing expectation value v for a. We thus define a new field s with vanishing vacuum expectation value by a = s+ v

,

(s) = 0

.

In terms of s (rather than a), the Lagrangian is of the form C = Ca + £b, where Cb contains the piece linear in s: g{s + in • TIS)]4> + ^(O 9 ^) 2 ~

Ca=i>[^$-m-

A*2)

+ i((aS)2-M^2)-A^(52+7r2)-^(52+7r2)2 Cb = {c-vnl)s

,

(6.2)

,

(6.3)

and where we have introduced the new mass parameters m = gv

;

/z2 = fi2 + Xv2

;

^ 2 = /j? + 3\v2

.

For v ^ 0, the pion fields if and the sigma field a are no longer degenerate in mass, as they were in the original Lagrangian CQ. The parameter v is determined from the requirement that (s) = 0. In lowest order, equating the coefficient in Eq. (6.3) to zero gives v = cjp?v. Obviously, when £ i ^ 0, the SU(2) symmetry is broken explicitly. But even when £i = 0, we may have two phases: for p? > 0 , we are in the symmetric phase: the minimum of the potential occurs at a2 + 7?2 = 0 , i.e., a = irl = 0,v = 0, and the fermions remain massless. For /j,2 <_0j on the other hand, we are in the socalled Goldstone phase [19, 6], where the SU{2) symmetry is spontaneously broken: the minimum of the potential occurs at cr2+7r2 = — /U2/A ^ 0 , i.e., V——/J,2/X^0, and the fermions acquire a non-zero mass, while the pion fields TT describe massless (Goldstone) bosons. The problem with this model is the fact that the particle corresponding to the field a, has not been observed experimentally. One possibility to get rid of this particle is to realize the chiral SU(2) symmetry non-linearly, by imposing the constraint cr2+ff2 = / 2

.

(6.4)

In this case f^ is the only parameter of the theory (formally this corresponds to taking the infinite coupling limit A -> oo). The model is now described entirely by the geometry of the target manifold on which it is defined, that is, the sphere 5 3 . For example, we can use Eq. (6.4) to eliminate the a field from the Lagrangian, obtaining

A) =^t+

9{\/W^i

+ in • 775)]^ + \(dn)2 + \ ( J

2

^

,

(6.5)

158

Non-linear a Models: Classical Aspects

C^c^Jl^p .

(6.6)

In four dimensions, the non-linear sigma model as denned by Eqs. (6.5) and (6.6) is unfortunately not renormalizable. A study of its renormalization properties has been performed long ago [5]. In lowest order new counterterms such as {dH • di?)2 are required. Therefore, this theory is at best a phenomenological low-energy approximation of some more realistic model. In two dimensions, the non-linear sigma model as defined by Eqs. (6.5) and (6.6) (without the matter interaction) is a special case of the general class of nonlinear sigma models that we are going to study from now on. These models have a geometric origin - just like non-Abelian gauge theories. In the present chapter we shall, at the purely classical level, define non-linear sigma models, and their interaction with matter fields, from the geometric point of view, and investigate their integrability. In the next chapter we shall discuss quantization of these models, and consider further formulations.

6.3

Two-dimensional a models: preliminaries

Our aim is to provide a general framework for defining classical non-linear sigma models. We are particularly interested in theories that are integrable, i.e., that exhibit an infinite number of conservation laws, as well as in models relevant to more realistic quantum field theory. We begin with the simplest series of non-linear a models, namely the spherical models, more commonly known as the 0(./V)-invariant non-linear sigma models or, somewhat abusively, the 0(N) (sigma) models; these have played an interesting and peculiar role in the recent development of quantum field theory. Classically, the field configurations in this kind of model are maps (f from some base space (usually flat Minkowski space or Euclidean space) to the (N - l)-dimensional sphere S ^ - 1 , so any such map can be viewed as a multiplet of N real scalar fields
,

(6.7)

with dynamics governed by the Lagrangian

C=±(d
•

(6.8)

(Here and in the following, summation over repeated indices is understood, as usual.) This is, of course, just the Lagrangian for the free field, but the presence of the constraint (6.7) implies that the model is not a free field theory. In fact, variation of the Lagrangian (6.8) with respect to