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. <$>. Among these attempts, the operator due to ural" licrmitian D.T. Pegg and S.M. Barnett is best known. The basic idea of their approach is to consider a subspace of Fock space where the photon number is truncated at ,s. Namely, one considers the (s + l)-dimensional space of photon number ip. We then have
and later one lets s become arbitrarily large. In this case one may define the phase operator
with
which shows that e^ indeed defines a unitary operator. In this construction, the annihilation operator in the (s + l)-dimensional Fock space is defined by
and if one defines a\ = (a s ) t , a slightly deformed commutation relation is obtained
From these relations, one can confirm
If one can show that the effects of the state |.s) disappear in the limit of sufficiently large s, one can successfully define a herinitian phase operator for the photon.
44 3.5
QUANTUM THEORY OF PHOTONS AND THE PHASE OPERATOR Index theorem for a harmonic oscillator
We analyze the above problem of the phase operator by using the notion of index, and show that the anomalous behavior of the phase operator is closely related, in mathematical terms, to the quantum anomaly to be analyzed in later chapters. We recall that the ordinary representation of a harmonic oscillator satisfies the index relation where dim ker stands for the dimension of the kernel of the operator a. Namely, dim ker a stands for the number of normalizable states un which satisfy
In the above example
and thus the dimensions of the kernels become 1 and 0, respectively, and the above index relation is satisfied. The index relation is also written as
or by using the trace as
The equivalence of cqn (3.76), which uses a^a, to eqn (3.73) is concluded by noting that au = 0 implies a'au = 0 and conversely, alau = 0 implies (a^au, u) = (au,au) = 0 by considering the inner product, and the positive definite innerproduct implies au = 0. The equivalence to the representation which uses the trace is shown by noting that a'a and aa1 include precisely the same number of non-vanishing eigenvalues. Namely, if An ^ 0 in
the state defined by
satisfies
and thus un and vn satisfy a 1 : 1 correspondence. Consequently, the difference of the number of states with 0 eigenvalues appears on the right-hand side of eqn (3.77). To be explicit, one can confirm
The notion of index has an important property that the index being an integer does not jump to another integer under a superposition of smooth infinitesimal
INDEX THEOREM FOR A HARMONIC OSCILLATOR
45
transformations. Consequently, the index is preserved under the well-known operation of squeezing of the photon in quantum optics (i.e., the compression in a certain direction of the phase space for the photon) . If one assumes, as Dirac suggested,
with a unitary t/(), we have
which shows the unitary equivalence of of a and aa*. We thus have
which contradicts the above index relation. We thus conclude that the unitary U((j)) (or cquivalently, hermitian (!)} docs not exist. On the other hand, if one assumes that as is an arbitrary (s + l)-dimensional square matrix, the relation
is shown. The trace Tr^i) here is defined in (s + l)-dimensional space, and this relation is concluded by rioting that a\as and asa\ share all the non-zero eigenvalues and that, both the finite-dimensional aja s and asa| contain the same number of zero eigenvalues, and thus all the eigenvalues coincide. This shows that in a truncated theory to arbitrary finite-dimensional square matrices, the notion of index is consistent with the presence of a herinitian phase operator
which is obtained by first moving the second term in eqn (3.84) to the right-hand side and then subtracting the same quantity Tr(,,)(e~ asa -<) from both sides. Here Ti(a) stands for the trace over the first s-dimensional subspace in the (s + 1)dimensional space. The right-hand side of this relation is thus the contribution of the state s), and the index 1 appears by noting a s aj|s) = 0, which follows
46
QUANTUM THEORY OF PHOTONS AND THE PHASE OPERATOR
from the definition of as. It is confirmed that the limit of large s of this relation is smoothly defined, and in the limit s —> oo the index relation
is derived. The state s) does not play a special role on the left-hand side in this limit. For example, one may start with Tr( s / +1 -|(e~ 0 '» a *) — Tr(y)(e^ aaa *) for any s' < s, and the limit s' —> oo gives the same result as the left-hand side of eqn (3.86). The existence assumption of a smooth limit of the hermitian phase operator (3.68) for s —>• oe thus contradicts the notion of index. The unitary phase operator is excluded in the limit s —>• oc. The vanishing index in a truncated space, as in the above example, and the recovery of non-zero index in the infinite limit of the truncation parameter (for example, the inverse lattice spacing I/a in lattice gauge theory) is one of the characteristic properties of the quantum anomaly in field theory to be described in this book. In fact, the evaluation of the index in terms of the trace (3.80), if the trace is understood in a general context, is closely related to the evaluation of the Atiyah-Singer index to be explained later, which is a mathematical representation of the chiral anomaly. Prom these considerations, one may expect that the anomalous behavior of the phase operator is regarded as an example of quantum anomalies. From this viewpoint, the anomalous behavior of the phase operator is an unavoidable phenomenon in quantum theory. As physical evidence of the unavoidable presence of the anomalous behavior, one may examine the issue if the minimum uncertainty relation is consistently described by a modified phase operator. In fact, one can show that the hermitian operator
4 REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES We illustrate the regularizatiori of field theory by taking quantum electrodynamics (QED) as an example, and show that the mass of the photon is kept to be 0 even after higher-order quantum corrections. We also illustrate the simplest example of the quantum breaking of chiral symmetry. The one-loop fermionic diagrams become fundamental in these considerations, and the gauge covariant regularization of one-loop Fcynman diagrams for arbitrary theory is explained. It is shown that the chiral anomaly is defined independently of Fcynman diagrams in perturbation theory. The basic idea of the Adler-Bardeen theorem, which asserts that the identity with the chiral anomaly does not receive any higher-order corrections, is briefly explained. 4.1
Current conservation and Ward—Takahashi identities
We start with the analysis of the one-loop photon self-energy, which was mentioned in Chapter 1, and show that there exists a calenlational scheme which preserves gauge irivariance and thus the vanishing photon mass. The basic Lagrangian density of QED is given by (see Appendix A)
where t\lv = dfj,Av—dvA^. The terms with a derivative of F^v in this Lagrangian density are introduced to implement a rcgularization, which is called higherderivative regularization, namely a method to reduce the degree of divergence in the theory. The parameter M with the dimension of mass provides a truncation of the momentum integral. If one lets M —> oo before any calculation, one recovers the unrcgularized Lagrangian density of QED. The last two terms in £eff stand for the gauge fixing term and the Faddeev Popov ghost term. To simplify the notation, we use the natural units
In these units, the dimensions of all the quantities are represented by the mass dimension [M]
47
48
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
For example, length has the same dimensionality as the Compton wavelength h/(mc) — 1/m, and the charge e2/(hc) ~ 1/137 becomes dimensionless. The requirement that the factor \' d4x C,ef[, appearing in the exponential in the path integral formula, should be dimensionless determines the mass dimensionality of various field variables as given above. The mass for the photon field
does not appear in the Lagrangian density (4.1) of QED. This is because such a mass term changes its form (i.e., is not invariant) under the gauge transformation of the electromagnetic field
Experimentally, it is known that the mass of the photon is 0, and thus such a mass term not only does not appear in the starting Lagrangiari but also should not be induced by higher-order corrections. In gauge theories such as quantum electrodynamics, it is of fundamental importance to perform calculations by preserving gauge invariancc in any finite order of perturbation theory. The important relations which express the gauge invariance of the starting Lagrangian in terms of Green's functions are called Ward-Takahashi (WT) identities. We write the path integral with source terms added as
where the source terms are defined by
The WT identity resulting from the gauge invariance of the starting action is derived by considering the following change of integration variables (gauge transformation)
and the following identity in the path integral formulation holds
SELF-ENERGY OF THE PHOTON
49
This derivation is based on the fact that the value of a definite integral does not depend on the naming of integration variables. This relation when combined with the invariance of the path integral measure under the above change of variables (which is established in Chapter 5)
and the change of the action in the order linear in a(x)
finally gives the identity
by keeping the terms linear in a(x). We also performed a partial integration by using the fact that a(x) is local, namely, it vanishes at space-time infinity. We also defined
for a general operator O(x). This notation is often used in this book. If one chooses a(x) in the identity (4.12) to be a function with a d-functional peak at x, one obtains
which is the basic relation known as a WT identity. If one functionally differentiates this identity with respect to J" (y) once and then sets all the sources to be 0. one obtains
The Fourier transformation of this last relation means that the probability amplitude for electron-positron pair creation from the current j^ = w-j^tl; and then pair annihilation into a photon vanishes when multiplied by the momentum. This process in the lowest order in perturbation is represented by the diagram in Fig. 4.1. In common language, this corresponds to current conservation 4.2 Self-energy of the photon The above WT identity gives a constraint on the photon self-energy if one recalls that the current is the source for the photon. We are going to explain this fact below.
50
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
FIG. 4.1. Feynmaii diagram corresponding to current conservation (4.15) The actual calculation is performed in Euclidean theory, so we start with the definition of Euclidean theory. The basic definition of Euclidean theory is to write the time component of coordinates as
and to regard x4 as a real number. At the, the same time, the 7 matrix of Dirac is defined as The hermitian 7° is thus converted to an anti-hermitian 74, arid in our convention all the 7 matrices become anti-hcrmitian (7 M ) T = —7**. The general contravariant vector V*M is also replaced by V° —> — iF4 in Euclidean theory. The covariant vector is then replaced by AQ —>• iAj with a real quantity A±. Consequently, the inner product of two vectors V^A^, remains unchanged in Euclidean theory, but the metric of space-time is replaced by g^ = ( — ! , — ! , — 1 , — 1 ) . To make the path integral convergent in the Euclidean metric, one may replace B —> iB and (: —>• ic in eqn (4.1). The Dirac operator in Euclidean theory is given by
and its form does not change from the Minkowski one. The conjugate ib of the Dirae spinor w is treated as an independent variable in the path integral, and in Euclidean theory it is understood as transforming like i^ under a Lorentz transformation (i.e.. SO(4) transformation) of Euclidean theory. Consequently, the operator ]/) becomes a hcrmitian operator defined by the inner product
in Euclidean theory.
SELF-ENERGY OF THE PHOTON
51
The path integral in Euclidean theory is given by
and the propagator of the free photon is derived by considering all the terms quadratic in _4M in eqn (4.1). By recalling eqii (A. 20) in Appendix A. with £ = 0, the momentum representation of the propagator when converted to the present Euclidean metric is given by
The second propagator stands for the propagator for the auxiliary field, and it is not used in the present discussion. The counting of the degree of divergence on the basis of the propagators given here and the ordinary fcrmionic propagator (A. 20) in Appendix A shows that all the diagrams except for the fermionic one-loop diagram are either convergent or logarithmically divergent, which are made finite when combined with gauge in variance. In fact, it is known that all the Fcynman diagrams except for one-loop diagrams arc made finite if one uses the higherderivative regularization in gauge theory. Consequently, one can evaluate all the quantum corrections without encountering divergences if one can make the oneloop diagrams finite. In the actual applications of gauge theory, the dimensional regularization. in which one avoids divergences by performing calculations in space-time dimensions less than four, is commonly used. However, this regularization spoils the algebraic consistency of the chiral symmetry which is defined in terms of Dirac's 7,5 ; no consistent definition of 75 in dimensions slightly away from four is known. In the analysis of quantum anomalies, this property of the dimensional regularization is inconvenient since one cannot decide if the anomalous behavior is coining from this insufficient definition of 7-5 or from a more fundamental origin. In the analysis of quantum anomalies, the gauge invariance (or covariancc) of current operators is essential. It is thus important to find a regularization. which docs not destroy the gauge invariance of currents when one incorporates the quantum effects corresponding to one-loop Feynman diagrams. The covariant regularization to be discussed below gives a regularization of general one-loop Feynman diagrams associated with current operators without spoiling gauge invariance. By this regularization, one can readily judge if a relevant symmetry other than the gauge symmetry is broken by the quantum effects or not. One might then wonder what happens if the quantum anomaly appears in gauge symmetry itself; even in such a case, as is explained later, one can judge if the quantum anomaly exists in gauge symmetry or not by combining a consideration of Bose symmetry.
52
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
We first note that the electric current (or in general simply called the current) to which the photon Afl couples in the Lagrangian density is given by
This current is rewritten as
where 1J) = 7 M (9 M — ieA^) and the trace stands for the sum over Dirac indices. In writing this relation, we used the anti-commuting property of ijj(x) and th propagator for the fcrmion field inside the background c-number electromagnetic field Atl.(x) in Euclidean theory, which is given by
The derivation of this propagator proceeds in the path integral with a proper normalization factor (2.137) as
where we performed partial integration with respect to ift in the last line. We then obtain the desired relation (4.24) by multiplying by (i$> - m)^ 1 on both sides. We next expand the denominator in eqri (4.23) in powers of the electromagnetic field eAp as
If one uses this expansion in the formula for the current (4.23) and retains only the terms linear in Av(x) (the zeroth order term in Av vanishes in eqii (4.23)), one obtains
SELF-ENERGY OF THE PHOTON
53
FIG. 4.2. Photon self-energy correction
In this expression, the derivative 9M acts on all the x-variables appearing to the right of it. If one functionally differentiates the last expression in eqn (4.27) with respect to eAv (z) . one finally obtains
We here used the following representation of the (5-function
and the notation $ = j^k^. The quantity Il^^g) (4.28) obtained here contains two fermion propagators in eqn (A.20) and it is shown that it coincides with the one-loop photon self-energy given by Fig. 4.2 on the basis of the conventional Feynman rules. Uliv(q) gives a basic quantity which is called the vacuum polarization tensor. The correlation functions of more than one-current tj^ip operators, which are usually calculated by expanding in powers of eA^ in perturbation theory, are obtained by differentiating eqn (4.23) with respect to eAp suitably many times. Those correlation functions correspond to Feynman amplitudes with one fermion loop. The important point to be noted here is that we take the limit y = x before we perform the momentum integration in eqn (4.28). Consequently,
54
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
the present calculational method is completely different from the point-splitting method, which regularizes the current correlation functions by considering the current operator ip(y}^i/)(x) which is defined for y ^ x. The vacuum polarization tensor Ii^v(q) (4.28) is quadratically divergent as is confirmed by counting the powers of momentum variables. The regularization of the gauge field propagator in eqn (4.21) does not improve the present divergence since the gauge field does not appear inside the present loop diagrams. We want to regularize this divergence. Our basic strategy is to use an arbitrary smooth function f ( x ) , which vanishes rapidly at large values of x,
and regularize the current (4.23)_as
If one takes the limit A —>• oo before performing any calculation, one recovers the original expression (4.23) by noting /(O) = 1. The parameter A provides the momentum space cut-off, as will be seen later. We also used the operator
which is hermitian in Euclidean theory. This operator ]/) $> is gauge invariant (or to be precise, gauge covariant), and the gauge covariance is automatically ensured if one considers this combination of the differential operator 9M and the gauge field eAf,. In practical calculations
for example, is convenient and the regularized current is given by
Just as in the case of unregularized current, one may expand eqn (4.34) in powers of eA^ and retain the terms linear in eA^* 8 Those who are not familiar with the calculations of Feynman amplitudes may tentatively proceed directly to eqn (4.39) and may examine the calculations later.
SELF-ENERGY OF THE PHOTON
55
(4.35) In this calculation, we used the relation
which is derived by using the properties of 7'' matrices {7^,7"} = 2<7A*!/ and the definition of the electromagnetic field [D M ,jD y ] = —ieF^ in terms of the covariaut derivative Dfl = dIJt — ieA^. Here F^ = d^Av — dvA^. In the above expression, the differential operator <9M acts only on the field immediately after it in (d^A11) and Fltv. but otherwise <9M acts on all the x variables appearing on the right-hand side of it. By considering the functional derivative of this expression with respect to eAv(z), one obtains the regularized vacuum polarization tensor
If one uses the Fourier representation of the ^-function in eqn (4.29). the final momentum representation of the vacuum polarization tensor is obtained:
56
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
In this final formula (4.38), the first term stands for the regularization in terms of a naive form factor [A 2 /(—/c 2 + A 2 )] 2 , which by itself breaks gauge invariancc as is known from the early days of renormalization theory. The remaining two terms in the present regularization recover this lost gauge irivariance. After performing the trace over the Dirac matrices and the momentum integral over kIJ' by using Feynman's parametric representation,9 the result of the lowest-order perturbation theory is obtained in the following gauge invariant form
in the limit of large A. This final result is proportional to q^q" — g^q'2 and satisfies the gauge irivariance condition q^q^q" — g^q'2) = 0 in Fig. 4.1 and equ (4.15). The mass of the photon thus remains 0. If IP"(g) contains an extra term proportional to #'''", the condition (4.15) is not satisfied. In such a case, the Fourier transformation of II'"'(g) back to the coordinate representation would contain a term corresponding to the photon mass (4.4) as a result of quantum corrections. The present covariant regularization gives rise to a gauge invariant result without introducing any divergence in the intermediate stages of calculation. This clearly shows that the gauge invariance in quantum electrodynamics is not spoiled by quantum effects. It is important to notice that we worked in strictly d = 4-dimensional space-time at all stages of the calculation, and this property becomes crucial in the later discussions of quantum anomalies. It is shown that the coefficient of the term containing In A 2 in eqn (4.39) is related to the /3-function in the renormalization group, which is explained in Chapter 7, and it is independent of the choice of the regulator /(&). For example, one can confirm that the choice 9For example,
QUANTUM BREAKING OF CHIRAL SYMMETRY
57
gives the identical coefficient of In A 2 . This result is also understood from the fact that the coefficient of the term In A2 is related to the quantum anomaly (Wcyl anomaly) to be discussed in Chapter 7 and that the Weyl anomaly is independent of the choice of the regulator f(x). In contrast, the constant ter — 1/3 depends on the choice of the regulating function. However, this does not lead to any difficulty, since this constant term together with In A2 is absorbed by the wave function renormalization factor and the final value is uniquely fixed by the renormalization convention. The present covariant regularization is easily extended to the one-loop calculations of multi-point functions, and it is also applied to calculations in the chiral theory such as the Weinberg-Salam theory, which breaks parity. The covariant regularization is easily extended to the cases with Higgs couplings, which mix left and right components of fermions. 4.3
Quantum breaking of chiral symmetry
On the basis of the simplest gauge theory, quantum electrodynamics, we perform a calculation of the quantum breaking of chiral symmetry (or chiral anomaly) associated with an axial- vector current, which was first encountered by Fukuda, Miyamoto, Tomonaga, and Steinberger. The Euclidean path integral (4.20) for quantum electrodynamics is given by
where we set the source term at £./ = 0. We also defined
in the path integral measure. Namely, the measure [PA^] includes all the terms related to gauge fixing. We consider the following change of path integral variables
which is called the chirai transformation. Here, because of the anti-commuting property of 75 and 7°, we have the expression as above. We defined the chirai matrix 75 which is hermitian by We have an identity under this change of variables
58
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
namely, we have the statement that the definite integral does not depend on the naming of path integral variables, which is a generalization of the ordinary integral j ' d x f ( x ) = f dy f(y). We look at parts which depend on fermionic variables in this identity. Firstly, the variation of the action in the exponential factor is given for an infinitesimal
if one notes 75 7M + 7^75 = 0. If one further assumes that the path integral measure does not change under the above change of variables, one has
The above identity is written after those calculations as
when one retains only the terms linear in a(x) by expanding the action in the exponential factor in powers of a(x). If one considers a(x) which has a (^-functional peak in the neighborhood of x, one obtains after partial integration the "naive" chiral identity The current tp(x)^IJ-^i!}(x) which contains 75 is called the axial- vector current. In the case of the vanishing fermion mass m = 0, the axial-vector current is conserved and such a theory is called chiral invariant. The puzzling property, which Fukuda and Miyamoto encountered in 1949 and which was analyzed in greater detail by Bell and Jackiw and by Adler in 1969, was the fact that the naive chiral identity (4.49) does not hold in Lorcntz invariant perturbation theory. The actual perturbative calculation is very complicated and one has to deal with subtle momentum integrals which include a linear
QUANTUM BREAKING OF CHIRAL SYMMETRY
59
FIG. 4.3. Feynman diagrams which give rise to the triangle anomaly divergence. However, if one uses the covariant regularization we introduced, the calculation becomes very simple, and one can show the deviation from the naive chiral identity without relying on perturbative calculations. A perturbative check of the chiral identity proceeds, just like the calculation of the vacuum polarization tensor in eqn (4.23), bv expanding the denominator factor ]/) = in
in powers of eA^. In the present case, the term linear in eA^ vanishes due to charge conjugation properties and the quadratic term in eA^ gives the lowestorder term. In terms of the language of Feynman diagrams, one evaluates the triangular diagrams in Fig. 4.3 where the current appearing on the left-hand side of eqn (4.49) is denoted by j£. For this reason, the chiral anomaly is also called the triangle anomaly. The degree of divergence d of this triangle diagram is d = 1 and the diagram diverges linearly. If one applies the covariant regularization to this calculation, one deals with
with a regulator function f(x) and the trace is taken over the freedom of Dira matrices. A direct evaluation of this quantity is possible, but what we want to know is the deviation from eqn (4.49) and thus it is sufficient to evaluate the derivative of this regularized current
This calculation is simply performed in the following manner. We first define a complete set of eigerifurictioris of the hermitian operator ]/)
60
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
and rewrite eqn (4.51) as
The summation here converges if the function /(A^/A2) rapidly apj for large A?,/A 2 . We now take the derivative of this expression
where we used the fact that 7M is anti-hermitian ( 7 M )^ = — 7^ and 0 and that ^<^n = A n yj re . This formula is further rewritten as
This final expression shows that the gauge covariant regularization gave rise to an extra last term which can break chiral symmetry, in contrast to the naive chiral identity (4.49). The chiral identity is modified from a naive form by quantum effects if the above extra term does not vanish. This last term is evaluated as follows: We first assume that the operator /(^* 2 /A 2 ) is sufficiently convergent, and rewrite the sum over the four-component functions ipn(x) to an integral over a complete set of plane waves
QUANTUM BREAKING OF CHIRAL SYMMETRY
61
where the trace is over the Dirac indices which take four values. This calculation is performed as
where we used the following relation
by noting {7^,7"} = 2 Afc M . We next expand the contents of the function f ( x ) around the value x = —k^k^ = \k2\ (note that x > 0 in our Euclidean metric convention)
and use the fact that only the terms of order I/A 4 or larger survive in the limit A —l oo in cqn (4.58). We also note that the trace with the factor 75 is non-vanishing only when the trace contains four or more 7-matrices, The terms satisfying these non-vanishing conditions are the third term in eqri (4.60) which contains In this wav we have
62
REGULARIZATION OF FIELD THEORY AND GHIRAL ANOMALIES
after calculating the 7-matrix trace
where the completely anti-symmetric symbol e^"0"3 is normalized by the convention e1234 = 1 in the Euclidean metric. We also performed the integral
by noting d4k = Tr'2d\k'2\ \k'2\ = -K^dxx. The basic property of the regulator function (4.30) including lim^^o x f ' ( x ) = 0 and lim^-^co xf (x) = 0 was used. We shall come back to this calculation when we analyze the Jacobian in the next chapter. We thus derived the chiral identity with the anomalous term (the last term)
without relying on perturbation theory. The last extra term does not rely on the details of the regulator f ( x ) . We recognize that the anomalous term, which breaks the naive WT identity (4.49) associated with chiral 75 symmetry, is finite, and thus the divergence and the quantum breaking of symmetry (quantum anomaly) are basically independent notions. A more detailed expression of the identity which includes the effects of the source terms for il>(x), i/)(x) is given by
by taking into account the variation of the source terms under the chiral change of fermionic variables in eqn (4.20). The identity for general Green's functions10 is obtained by functionally differentiating this form of identity with respect to source functions suitably many times and then setting all the source terms to beO. 4.4
Adler— Bardeen theorem
From our derivation of the identity (4.64) which does not depend on perturbation theory, it is regarded as an exact bare identity valid for the regularized theory (4.1). It is in fact known that the identity (4.64) is exact and not modified by higher-order quantum effects even in the large regulator mass M limit 10 The result for the Minkowski metric is given by choosing the convention f1230 = 1 and removing the imaginary factor i in front of the anomaly term.
ADLER-BARDEEN THEOREM
63
in cqn (4.1) after the ordinary renorrnalization operation. This fact is known as the Adler-Bardccn theorem. The Adler-Bardeeri theorem is fundamental, but for technical reasons we do riot discuss the details of the renormalization procedure in this book. We thus explain the essence of this theorem only briefly.11 We first notice that the operator 2im'tjj(x)-j5ib(x) is the only gauge invariant pseudo-scalar operator with mass dimension 3 in the present theory. The operator when inserted into a Green's function thus induces only divergences proportional to itself but it becomes finite after wave-function and mass renorrnalization in the present renormalizable theory where N[^>(x)"/^(x)] is called the normal product and it gives rise to a finite result when inserted into any Green's function. Similarly, i/j(x}^'y5i/}(x') is th only gauge invariant (to be precise, BRST invariant) axial-vector quantity with mass dimension 3 in the present theory, and thus it induces only divergences proportional to itself: To be precise, the divergences arising from the diagrams which do not include the triangle diagrams in Fig. 4.3 are absorbed into the wave-function renormalization factor of \l>. We thus write
with a divergent constant Z and a finite operator N[^}(x)^^c,'ip(x)}. Finally, the anomalous term '2ie2/(32ir'2)fVlAvFa&(x)\. = 2d proportional to itself and also divergences proportional to the divergence of the axial-vector current <9MJV[?/>(x)7A*75'!/'(a;)], but it does not induce divergences proportional to the pseudo-scalar quantity N(v>(xY^(x}}. Actually, a careful analysis shows that it does not induce divergences proportional to itself, and if one uses the conventional renormalization eA^ = erArtl in quantum electrodynamics, one can write
with a divergent constant Z' and a finite operator By using these relations in eqn (4.64). one obtains
11 Those who are not familiar with the details of renormalization theory may skip this section. The main purpose of this book is to formulate the quantum anomaly in a non-perturbative manner as much as possible.
64
REGULARIZATION OF FIELD THEORY AND CHIRAL ANOMALIES
which implies the equality between the divergent coefficients Z = Z' and those divergent terms cancel on both sides of the identity. We thus obtain the exact operator identity for renormalized operators
and the anomaly term does not receive higher-order corrections except for the replacement
• 5
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES In this chapter, a general path integral formulation of the quantum anomaly for chiral symmetry is given. The quantum anomaly is identified as the Jacobian arising from the symmetry transformation of path integral variables. The correct WT identities with an anomalous term arc thus defined from the beginning. To realize this idea, it is essential to have a suitable regulator which controls divergences by preserving as much symmetry as possible. In the path integral formulation, the gauge invariant mode cut-off is shown to be useful. As topics related to quantum anomalies, a brief account of iristanton solutions and the Atiyah Singer index theorem is given. As applications of these notions, we explain that the (naive) unitary transformation to the interaction picture does not exist. We also discuss the general absence of Nambu-Goldstone particles associated with spontaneous symmetry breaking when the relevant symmetry is spoiled by the quantum anomaly and instariton effects. 5.1 The chiral Jacobian in quantum electrodynamics We start with the Euclidean path integral of quantum electrodynamics
where the 7^ matrices are anti-hermitian
and The chiral matrix is hermitian
and it satisfies (75 )2 = 1 and
The covariant derivative is defined by
by using the electromagnetic potential A^, and the action Suax stands for the 2 Maxwell action SMSX = — (1/4) We include (0^.4^ a suitable - d vA^) . /d4z gauge fixing term into the path integral measure of gauge fields [DA^].
65
66
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES To analyze the Jacobian for the chiral transformations
in a reliable way, we expand the Dirac fields
in terms of the eigerif unctions of the hermitian
We used the notation (x\n) = fn(x) following Dirac's transformation theory. This basis set formally diagonalizes the Dirac action
To derive an exact relation such as a WT identity, the basis set which diagonalizes the action is essential. The path integral measure for fermions is then written as
by noting that ip, $, an and an are all Grassmann numbers which aiiti-commute with each other and that the integral over Grassmann numbers is defined by the (left) derivative. This gives a manageable definition of the path integral measure and the path integral (5.1) itself in. our applications.
THE CHIRAL JACOBIAN IN QUANTUM ELECTRODYNAMICS
67
We now consider the evaluation of the Jacobian for coordinate-dependent infinitesimal chiral transformations
To work in the above definition of the path integral measure, we expand the field variables into the complete set
When one multiplies this expression by yj, (x) on the left and integrates over the space-time coordinates, one obtains
Similarly, one may multiply by tpn(x) on the right to give
and integrate over coordinates to obtain
From these expansions, the transformation rules of the path integral measure arc given by
where TV is taken to be infinity later. Here we used the fact that the integration variables an and &„ are both Grassmann numbers and the integral is defined
a by the (left) derivative. Consequently, the Jacobian appears as an inverse of the case for conventional numbers. We next use the relation
valid for an infinitesimal a(x), which is derived by using the relation det M = exp tr In M valid for a general matrix M and the relation
valid for an infinitesimal a(x). If one writes the measure in terms of the variables ifj and if>, the Jacobian for infinitesimal chiral transformations is given by
To evaluate the Jacobian explicitly, we consider
where f ( x ) is an arbitrary smooth function which rapidly approaches 0 at x = oo /v with normalization /(O) = 1. We thus replace the mode cut-off limjv_j.00 z^n=i by a cut-off in terms of eigenvalues limjM_>oo /((An)2/M2). If one chooses f(x) suitably, it is expected that one can approximate the mode cut-off arbitrarily closely by this procedure. See Fig. 5.1. This cut-off in terms of the eigenvalues preserves gauge invariance and thus defines a desirable regulator. In some basic applications we explicitly show that the final results arc independent of the choice of the regulator function f(x) if it satisfies a minimum requirement. In most applications the simple choice f ( x ) = erx is convenient, and it is related to the
THE CHIRAL JACOBIAN IN QUANTUM ELECTRODYNAMICS
69
FlG. 5.1. A smooth regulator function f(x) which approximates the mode cut-off depicted by the broken line heat kernel or ("-function regularization. The present regularization is sometimes called the gauge invariant mode cut-off regularization. Since the operator j$ f (ft'2 / M2) is well regularized, we perform a unitary transformation of basis vectors from {tpn(x}} to plane waves {elkx}, and by doing so we can extract the gauge field dependence of the Jacobian explicitly. This transformation to plane waves is related to the unitary transformation from the Hcisenberg picture to the interaction picture in the operator formalism. The Jacobian is thus given by
where tr on the left-hand side stands for the trace except for the space-time integral which is included in IV. The remaining trace in the last expression stands for the trace over Dirac indices. We used the relation
hghjgjhghgjhgjhgjhgjhgjhgjhghghhghgg k/j, after moving etkx through
70
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES We here recall the trace properties of Dirac matrices
with e1234 = 1. For example, tr7 5 = tr 757^7" = 0 is shown by noting the relations such as
We used the fact, that different 7M matrices anti-commute with each other together with (T^1)2 = -1 and the cyclic property of the trace tr 7 1 7 2 = tr-y 2 ^/ 1 . By using these properties and noting that the terms with powers 1/M5 or higher vanish in the expansion in powers of 1/M in eqn (5.22), we obtain
by noting e1234 = 1. We also used
where x = —k^k^ > 0 in our metric convention. The final result of the Jacobian docs not depend on the regulating function f ( z ) , which was introduced to make the intermediate stages of the calculation more reliable, as long as it satisfies
The regularization by f(x) preserves gauge in variance at all the intermediate stages of the calculation. In the above integral, one recognizes that the contributions from any finite domain —L < fcM < L vanish in terms of the original momentum variables before one applied the scale transformation fc,t —» Mfe^. In this sense, large momentum regions or short distances in space-time determine the value of the Jacobian. Precisely for this reason, the Jacobian (quantum anomaly) is closely related to divergences in the interaction picture perturbation theory, though the Jacobian itself is finite.
WARD-TAKAHASHI IDENTITIES IN QUANTUM ELECTRODYNAMICS
71
We summarize our result so far as
The Jacobian thus evaluated agrees with the chiral anomaly discussed in Section 3 of Chapter 4, as will be shown in the next section. The fact that the Jacobian does not depend on the detailed properties of the regulator f ( x ) is consistent with the perturbative analysis of Adler, who showed that the chiral anomaly for the triangle diagrams is uniquely determined independently of the divergences if one imposes gauge in variance. The regulator function f ( x ) we used here is the same as the function used to regularize the Nother current (and the fermion propagator) in Chapter 4. One may pause here and ask why the quantum anomaly can be evaluated as the Jacobian? Some reasons one can think of at this stage are: The quantum anomaly is finite and independent of regularization as long as gauge irivariance is preserved, and in this sense it is universal. Note that the use of the basis set {<£„}. which diagonalizes the action, and our replacement of the mode cut-off by the eigenvalue cut-off in eqn (5.21) uniquely fixes the regulator f($>2/M'2), which is the essence of the gauge invariant mode cut-off regulari/ation. Also, the quantum anomaly does not depend on the detailed properties of Feynmaii diagrams in perturbation theory, namely, the Feynman diagrams contain more information than the quantum anomaly itself. We can thus evaluate the anomaly without knowing Fcynman diagrams. A further supporting argument for the present prescription to evaluate the Jacobian will be given later by using the notion of index. 5.2
Ward— Takahashi identities in quantum electrodynamics
We have explained how to evaluate the Jacobian associated with a change of path integral variables. We here review the formulation of identities in quantum electrodynamics by using the result. If one considers the infinitesimal localized chiral transformation of path integral variables
in the path integral
where 5jyax stands for the Maxwell action and the source terms are given by
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
we have the identity
This identity is a statement that the definite integral does not depend on the naming of integration variables, namely, it is a generalization of f d x f ( x ) = j dy /(?/). In this identity, the Jacobian is given by, as was shown in the previou section
and the change of the action is given by
The change of the source terms is given by
If one combines these relations in the above identity (5.33) and keeps terms linear in a(x), one obtains the WT identity
If one chooses the parameter a (a;) such that it has a ^-functional peak around the space-time point x, one obtains the more familiar form of identity
which agrees with eqn (4.65) in the previous chapter. We have thus shown that the Jacobian associated with the change of variables represents the quantum
WARD TAKAHASHI IDENTITIES IN QUANTUM ELECTRODYNAMICS
73
breaking of chiral symmetry (namely, the chiral anomaly). This derivation of the chiral identity does not depend on perturbation theory, arid in this sense it gives a non-perturbative definition of the identity. A major difference between the present derivation of eqn (5.37) and the derivation of the identity (4.65) in Chapter 4 is a conceptual one: In the derivation of eqn (5.37), we have the anomaly coming from the Jacobian from the beginning and thus "anticipate" the appearance of the anomaly when one performs a careful evaluation of the divergence of the axial- vector current. In contrast, in the derivation of eqn (4.65) one "discovers" the anomaly only when one performs a careful evaluation of the divergence of the axial- vector current. We have identified the origin of the quantum anomaly and its reliable evaluation, and now we apply the technique to analyze the issues related to gauge invariance in quantum electrodynamics, which were once studied by Tomonaga and Schwingcr. For this analysis we consider the following change of path integral variables
The Jacobian is then evaluated in the same way as for chiral transformations as
which gives a trivial Jacobian independently of the choice of basis vectors ipn (x). On the other hand, the change of the action is given by
We thus obtain the WT identity for gauge transformations following the same procedure as for the chiral identity as (by suppressing source terms)
which leads to the identity in operator language
This identity shows that the gauge symmetry (current conservation) is not broken by the quantum anomaly, and thus it suggests that there exists a calculational scheme which ensures the vanishing photon mass even to non-pcrturbative accuracy. This conclusion is consistent with the analysis in Chapter 4.
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
5.3
Chiral anomaly in QCD-type theory
We start with an explanation of a straightforward generalization of quantum electrodynamics, which is called Abelian gauge theory, to a nori-Abelian gauge theory which has the same structure as QCD (quantum chromodynamics). The Euclidean path integral for this theory is given by
Dirac's 7-matrix convention is the same as for QED and 7'' is anti-hermitian
and the chiral matrix
75
defined by
is hermitian. The covariant derivative Dtl is defined by
by using the generator Ta of a rion- Abelian gauge group. The non- Abelian gauge field carries the same number of components as the generators of the group. Hereafter, we use the convention that we take a summation over the indices which appear twice in the same expression, except when stated otherwise. We often use the notation AjJL = A^Ta as in the last expression above. The field strength tensor F£v of the gauge field, which is a generalization of the electric and magnetic fields E and £>, is defined by
is an action for the non-Abelian gauge field, which is called the Yang-Mills field,
and it gives a generalisation of Maxwell's action of the electromagnetic field. The path integral measure for the gauge field [£>AM] in eqn (5.43) contains a suitable gauge fixing term also, and the details of the gauge fixing are not important for the analysis in this chapter. We give a brief explanation of non-Abelian groups. A general compact group is written as a direct product of simple (non- Abelian) groups and Abelian groups.
CHIRAL ANOMALY IN QCD+-TYPE THEORY
75
The generators of a simple non-Abelian group are given by hcrmitian matrices and satisfy the commutation relation
The constant fabc which is completely anti-symmetric with respect to indices a, b, c is called a structure constant. We normalize the generators by
As an explicit example, the generators of the group SU(2), which is familiar in the analysis of angular momentum, are written in terms of three Pauli matrices as
and they satisfy the relation
where tabc stands for a completely anti-symmetric symbol with respect to the three indices with e12r! = f. The generators of the group SU(3) are written in terms of eight 3 x 3 matrices {A0}, a = 1 ~ 8, which arc called Gell-Mann matrices, as In this book, we discuss only the group SU(n). which is generated by n2 — 1 generators. Coming back to the above path integral, the Dirac field appearing there is written for the group SU(2) in a precise notation as
and it contains two ordinary four-component Dirac fields ib\ (x) and ip% (x). Consequently. Dirac matrices should also be written as 8 x 8 matrices by arranging two conventional 4 x 4 7 matrices as diagonal components. It is however com mon to use the simplified notation as above in eqn (5.43), since this does not induce any confusion. The actual QCD, which describes the strong interaction, is based on the gauge group SU(3), and thus we deal with a three-component field consisting of three conventional Dirac fields. In the case of the gauge group SU(2), for example, an arbitrary function which belongs to SU(2) is written as
76
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
by using three real functions ua(x) and the generators Ta of SU(2), which are given in terms of the Pauli matrices. The non-Abelian (local) gauge transformation is then defined by the replacement of variables
The gauge transformation for matter fields thus corresponds to a rotation in internal space specified by a local g(x) at each space-time point; for the gauge field Ay^ the transformation is given by a combination of rotation (the first term) and translation (the second term). The covariant derivative D^ is the transformed under the gauge transformation as
and consequently, the combination
is transformed in the same manner as the field variable \i>(x] itself. For this reason, D^ is called the covariant derivative. The field strength tensor of the gauge field, which is expressed in terms of the covariant derivative, is transformed as
If one recalls g(x)g^(x) — 1, the action appearing in the above path integral formula (5.43) remains invariant
under the gauge transformation, namely, the action is gauge invariant. In the above expression, we used the convention irTaTb = (l/2)Sab. The function g(x) itself is also often called a gauge transformation. We examine the localized (space-time dependent) infinitesimal chiral transformations
CHIRAL ANOMALY IN QCD-TYPE THEORY
77
for the non-Abelian gauge theory. To analyze the Jacobian for the chiral transformations, we expand the fermioiiic variables
as in the case of QED in terms of the eigeiifunctions of the hermitian operator
The action for the fermioii is formally diagonalized by this expansion
The path integral measure for this case is written as
and it leads to the following Jacobian for an infinitesimal chiral transformation
as in the case of QED in Section 5.1. The actual evaluation of this Jacobian proceeds by replacing the mode cut-off by the eigenvalue cut-off as
where f ( x ) is an arbitrary function of x which approaches 0 rapidly at x = oo with a normalization /(O) = 1. See Fig. 5.1. The condition /(O) = 1 means that one recovers the original expression for M —>• oo before one performs any manipulation, and any regulator function should satisfy this condition. The operator
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
is a well-regularized operator and one may use a complete set of plane waves to extract the gauge field dependence as
where the parameter a(x) and the associated integral over the space-time coordinates arc omitted. The remaining trace here stands for the trace over the indices of Dirac's 7 matrices and the indices of the matrices of gauge group generators. We also used
and performed the scale transformation of integration variables kp —> Mk^ after moving eikx through /(^> 2 /M 2 ). If one expands eqn (5.68) in powers of 1/M. one obtains
by noting tr 7 5 = tr 75 [7^,7"] = 0. We used the normalization of the antisymmetric symbol e1234 = I and
In the present metric convention, a; = —k^k1' > 0. In this momentum integral, an arbitrary finite domain — L < k^ < L for the momentum variables before one performed, the scale transformation k^ —?> Mk^ gives a vanishing contribution. In this sense, the large momenta or short distances determine the quantum anomaly.
INSTANTONS
79
This integral is also independent of the function /(:/;), which was introduced to avoid ambiguity in the intermediate stages of the calculation, as long as it satisfies
as in the case of quantum electrodynamics. The Jacobian for the chiral transformation is thus evaluated as
where the remaining trace is over the indices of matrix generators of the Yang Mills field. Consequently, the chiral identity for the general case with source functions rj(x), fj(x) discussed in Section 5.2 is generalized to
following the same procedure as for quantum electrodynamics. The calculation (5.70) can be readily generalized to any d = In even-dimensional space-time arid the result is written as13
5.4
Instantons
The Euclidean non-Abelian gauge theory, unlike Abelian theory, accommodates a classical solution which is called an instanton. This instanton is fundamental since it describes tunneling from one vacuum in field theory to another. This classical solution when combined with the chiral anomaly leads to an index theorem, which relates the number of zero modes of the Euclidean Dirac operator with the instanton number (or Pontryagin number). We first briefly describe the instanton solution. In Euclidean theory, not only the coordinates but also the field variables are replaced by
and the resulting A4 and x4 are regarded as real quantities. Consequently, the metric of Euclidean space-time becomes g^v = —5^v and, for example, 12 To obtain the result in the Minkowski metric, one may remove the imaginary factor i in front of the anomaly factor with the convention 13 Tho convention in d = 'In dimensions is
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
The gauge action then satisfies
where we included the coupling constant g into the field variables as
We use an imaginary time to describe tunneling in field theory just as in quantum mechanics. In particular, the classical solution of Euclidean theory gives the stationary point of the action and thus describes the path where the tunneling probability becomes a maximum. The instanton solution precisely gives such a stationary point of the action. 5.4.1 The instanton solution We choose the gauge group SU(2) to explain this solution.14 An arbitrary element of the gauge transformation which has a value in SU (2) is written as
by using four real functions (a(x), b(x)) and three Pauli matrices f. If we require that the Euclidean action SE for the gauge field is finite, the field strength tensor is required to satisfy F^v(x) —>• 0 at space-time infinity \x —>• oo. If one remembers that the action SE is invariant under the gauge transformation, the gauge field A^x) itself should approach the configuration which is gauge equivalent to the vacuum
The instanton solution which actually satisfies this condition is explicitly given by
The parameter p is a real constant and we defined
14 In this book we actually use only the fact that there exists a solution which satisfies cqn (5.90).
INSTANTONS
81
where x expresses the spatial coordinates of space-time. The instanton solution Ay for r —> co shows the behavior
and the leading first term has a magnitude of the order 1/r. namely, in the present notation O(l/r). Because of the condition a(x)'2 + [b(x)]2 = 1, one can regard the element g(x) of the gauge group SU(2) as describing a unit hypersurface (which is written as S?>) in a four-dimensional space whose coordinates are given by (a, b). On the other hand, the function g(x) = x4- + ixf which is used to construct the instanton solution shows that a point on the unit hypersurface in four-dimensional Euclidean space-time described by (x4,x), which is also an 53, and a point on the hypersurface described by g(x) of the gauge group SU(2), are in 1 : 1 cor respondence. Namely, when the coordinates (x4,x) of space-time cover S3 once, the element g(x) covers the hypersurface S3 in the gauge space once. The quantity which describes this topological property is called the winding number. The winding number v for the specific g(x) above is given by
by using the totally anti-symmetric symbol f^"aP normalized by g1234 = 1. The integral here stands for a surface integral over the hypersurface S3 located at infinity of four-dimensional space-time, and dS^ stands for the surface element which is orthogonal to the /u-axis. In fact, near the point x4 = oo, namely, near the point 3 x = 0, for example, the integrand of the integral to define v is written as
which shows that the integrand at any point of the hypersurface is given by the surface clement for a unit hypersurface divided by 2?r 2 . If one recalls that the surface area of the unit hypersurface (which is a volume in the conventional sense) is given by 2?r2, the above integral over the entire hypersurface in fact gives the winding number v = 1. If one uses
instead of g ( x ) in the formula defining the winding number //, it is shown that the winding number // = n is obtained. Also, the anti-instantoii with v = — 1 is obtained by setting n = —1 in this formula, or equivalently by
82
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES The winding number v is written by using the instantori solution as
The integrand here is written as a total divergence, and the integral is written as a surface integral at space-time infinity by using the Gauss law10
where we used the behavior of the instanton solution at infinity in the last expression. Namely, this expression agrees with the definition of the windingnumber (5.85). Finally, we would like to show that the instanton discussed so far in fact satisfies the Yang-Mills equation. We define FIJtv = (l/2)e /i)ya ^F a 0 and use the following Schwarz inequality
If one uses the formula for the winding number on the right-hand side of this relation, we have
and the equality holds only for the case
where ± corresponds to the signature of the winding number v. If one uses the we have + signature in this relation (5.93) and uses
which is in fact confirmed to be satisfied by /(r 2 ) = r 2 /(r 2 +p 2 ) used to construct the instanton in eqn (5.82). This shows that the instanton gives the winding number v = 1 and at the same time it gives the minimum of the action for v = 1. The field configuration 16
This property is shown by using the Jacobi identity
The quantity appearing in eqn (5.90). K^ = l/(87r 2 ) the Chcrn-Simons form a,nd is very fundamental.
is called
INSTANTONS
83
which gives the stationary point of the action is a solution of the field equation derived from the action, and thus the instanton is in fact the solution of the Euclidean Yang-Mills field equation. The instanton solution
approaches the configuration ig(x)d fj/<j^(x) at infinity, which is gauge equivalen to the vacuum. The gauge function g ( x ) , however, cannot be expressed as a superposition of conventional "small" gauge transformations which satisfy the boundary condition 5(00) = 1. Physically, this gauge function g(x) is interpreted as describing tunneling starting from one vacuum at x4 = — oo to another vacuum at x4 = oo following the Euclidean imaginary time. (For this reason, this solution is called an "instant-on" , indicating that it appears and then disappears instantly in time unlike the ordinary soliton.) The value of the action for this solution SE = —8n2/g2 describes the height of the tunneling barrier, and in the path integral the factor with this action in the exponential
is understood as giving the leading term of the tunneling probability. The vacuum in non-Abelian gauge theory thus has a structure similar to the motion of the electron moving in a periodic (sharp) potential, and it is described by a wave function similar to the Bloch wave. The path integral of Euclidean gauge theory is shown to be written as
though we here forgo the details of this derivation. In this path integral, the sum over v runs over all the integers, and the term v — 1 corresponds to the contribution from the instanton solution we have discussed so far. The general v corresponds to multi-instanton solutions. The real parameter 0 in this formula is an arbitrary constant, arid the vacuum state \9, ±oc) is called the 9 vacuum. The gauge field configuration in the path integral measure described df by v is Afj,(x) = A, L(x}(v) + a^(x) by using the instanton solution A^(x)^ with instanton number (i.e., the winding number) v and the fluctuation a M (x) around the solution.16 In analogy with the Bloch wave, the exponential factor exp(i0f) has the following meaning: v corresponds to the (difference) of the positions of the electron and the real parameter B corresponds to the Bloch momentum. 16
The instanton solution with v = 1 contains eight deformation parameters corresponding to four parameters describing the position of the center and one parameter p describing the size of the instanton and three other gauge parameters. In the path integral, one needs to perform the integral over these parameters also. For general simple groups, it is known that the instanton solution is constructed for each SU(2) sub-group, and no other solutions.
84
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES When one performs a change of the integration variables
with an infinitesimal constant parameter a, the calculation of the Jacobian (5.73) shows that the path integral measure is transformed as
and thus the parameter 0 is changed as
At the same time, the Euclidean action changes as
These properties show that the 0 parameter can be freely changed in the theory with a massless fermion, and the parameter 0 has no physical meaning in such a theory. (In the presence of the source terms / dx (fjiji + iprj), one needs to perform the simultaneous re-definition of sources t] —>• /7exp(—id/2) and 77 —S> cxp(—i6/2)r to completely eliminate the 9 dependence from the generating functional.) The 9 term is written as
and ^""PFpvFap is a generalization of EB in QED and thus breaks CP symmetry. This breaking of CP symmetry for 0 ^ 0 in QCD is known as the strong CP problem. 5.5
Atiyah Singer index theorem
Coming back to the analysis of the quantum anomaly, we re-examine the calculation of the Jacobian factor for the chiral transformation
If one considers a global limit of this relation by choosing a as a constant, both sides of this relation are written as
ATIYAH-SINGER INDEX THEOREM
85
If one combines this relation with the relations valid for the eigenfunctions with
which is derived from ^75 + J5$> = 0, one obtains
by rioting /(O) = 1. In this relation, n± respectively represent the number of eigenstates with vanishing eigenvalue for the Dirac operator T/) with positive or negative chirality eigenvalues
and the right-hand side of eqri (5.106) shows the instanton number (or Pontryagin number). For $>
17
This index theorem was formulated in the late 1960s, about the same time as the modern
recognition of the quantum anomaly in physics.
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
and noting
which arises from the relation 75^+^75 = 0. We further note that (1 ± 7s)/2 is the projection operator satisfying [(1 ± 7s)/2] 2 = (1 ± 7s)/2. The index theorem is then written as
for M H> large. The Atiyah-Singer index theorem written in this form has the same form as the index theorem (3.80) we discussed in Chapter 3 in connection with the photon phase operator. This index theorem provides another motivation and justification for identifying the Jacobian with the chiral anomaly. The Jacobian for the global chiral transformation with a constant a is given by
independently of the large N behavior, and the expression for the Pontryagin index v = n+ — n_ in eqn (5.88) shows that the Jacobian in fact carries the proper information about the chiral anomaly. We have thus provided three major supporting arguments for the identification of the Jacobian with the quantum anomalies and for the gauge invariant mode cut-off regularization of the Jacobian: The first is the gauge invariant regularization of the current operators in Section 4.3: the second is the diagonalizatior of the basic action in the path integral; and the third is the Atiyah-Singer index theorem. In subsequent chapters, we shall show that the gauge invariant mode cut-off of the Jacobian gives rise to all the known local anomalies. 5.6
Nambu—Goldstone theorem
We discuss two implications of the instanton solutions and the index theorem. The first is the general aspect of field theory, and the other is concerned with the Nambu-Goldstorie theorem. 5.6.1
Unitary transformation to the interaction picture
We first discuss the implication on the so-called interaction picture in field theory. In our formulation of quantum anomalies, we expand the fermionic variables as
NAMBU GOLDSTONE THEOREM
87
in terms of the eigenfunctions of the hermitian operator
This expansion formally diagonalizes the fermionic action
and in this sense the above expansion may be regarded as corresponding to the Heisenberg representation in the operator formalism. We also obtained the Jacobiari for an infinitesimal chiral transformation
which carries the information about the gauge field for a = constant in eqn (5.111). On the other hand, one could in principle expand the fermionic variables as
in terms of a complete set of eigenfunctions of the hermitian operator
This expansion diagonalizes the free Dirac operator
and in this sense this expansion is regarded as corresponding to the interaction representation in the operator formalism. In this expansion, the Jacobian for the chiral transformation is given by
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
which carries no information about the gauge field for a = constant unlike eqn (5.115), and thus we cannot assign a physical significance to this expression. If one uses the index theorem
but
one learns that a smooth unitary transformation from non-zero integer to 0 does not exist.18 The naive unitary transformation from the exact basis set {
which is also the basic procedure in the proof of the index theorem in mathematics by using the heat kernel. In the expression with the free field basis in eqn (5.122), the effect of the index is recovered from high-frequency sectors, which explains why the quantum anomaly was closely related to the treatment of ultraviolet divergences in perturbation theory. WTe emphasize that the quantum breaking of s}'mmetry (namely, the quantum anomaly) itself is perfectly finite and independent of divergences. 5.6.2
Nambu-Goldstone theorem
We next discuss an issue related to the quantum anomaly and the NambuGoldstone theorem. The main purpose of this analysis is to show that the Nambu-Goldstone bosons do not generally appear for a spontaneously broken symmetry if the relevant global symmetry is broken by the effects of the anomaly 18
To define a unitary transformation, one needs to truncate the ultraviolet components of the action in a gauge invariant way.
NAMBU-GOLDSTONE THEOREM
89
and iiistantons. We discuss this problem by using the Lagrangian of QCD. The path integral of QCD is given by
and the gauge group involved is SU(3). We use the 9 vacuum \0) which incorporates the instanton effects. In the actual QCD we need to think of at least three quarks u, d, s, but the essence of the U(l) problem (or the rf problem) related to the competition between the index and the spontaneous symmetry breakdown is analyzed by a theory with a single flavor of quark, and thus we discuss this problem on the basis of the simplified model. If one considers the special case with the vanishing quark mass m = 0, the action is invariant under the global chiral transformation with a constant parameter a To derive a chiral identity based on this change of variables in a reliable way, we generalize the parameter to be space-time dependent a(x) and use the following identity (namely, the integral itself is independent of the naming of path integral variables)
We wrote the path integral for the operator insertion for the convenience of the following discussion. If we treat a(x) as an infinitesimal parameter in this identity
we then have the changes of the action and the path integral measure
If one collects the terms linear in a(x) and functionally differentiates the resulting expression with respect to a(x), one obtains the identity (we here simplify the notation \d,-oo) and (#,+oo by \9) and (9\, respectively)
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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
We also used
in terms of the Chern-Simons form K^. We next analyze the spontaneous symmetry breakdown of chiral symmetry. We start with a global chiral transformation by setting a(x) to be a constant in the transformation law oitp(yYfflti}(y) in eqn (5.126) by assuming for the moment that, there is no anomaly. The invariance of the action and the measure then leads to the relation
The chiral invariance of the vacuum implies that the left-hand side and the first term on the right-hand side are equal, as is confirmed by introducing a formal unitary transformation, ijj' = U(a)-ij}U*< (a), ib' = V"(a}^!}^(a), and U^(a)\6) — \0). Consequently, the second term on the right-hand side vanishes. On the other hand, the spontaneous breakdown of chiral symmetry implies the rion-invariance of the vacuum and thus where we used the translational invaria,nce of the vacuum. The spontaneous symmetry breakdown corresponds to a second-order phase transition in condensed matter theory, and it is generally characterized by an order parameter. The order parameter of the spontaneous symmetry breakdown of chiral symmetry is thus given by (
and the Fourier transform of this expression
should hold even for p^ —i 0. We thus conclude that the amplitude on the lefthand side of eqn (5.133) contains a pole 1/p2, which fact is expressed in symbolic notation by
with a constant fv. The axial-vector current •07A'75^) thus contains a massless (pseudo-)scalar particle represented by the pole 1/p2- This particle is called a
NAMBU-GOLDSTONE THEOREM
91
Nambu Goldstone particle (the Nambu Goldstone theorem). The Nambu Goldstone particle arises from the mismatch between the global symmetry and the global limit of the localized WT identity. The localized identity holds regardless of the presence or absence of the spontaneous symmetry breakdown. On the other hand, if one integrates both sides of the actual chiral identity with the chiral anomaly (5.128) by multiplying by clp^""y^ and takes the limit pu —> 0, one obtains
by using the translational invariance. We assume that the non-vanishing order parameter persists with or without the anomaly. For the non-vanishing instanton number, the second term on the left-hand side does not vanish and thus the presence of a rnassless particle in the operator (V'T^Ts^/OCp) 'IS n°t generally concluded. If one defines a modified axial-vector current by
which satisfi.es in the operator notation
it appears that the Nambu-Goldstone theorem for the current JA' formally holds. However this current J^ is not gauge invariant since the Chem-Sirrioiis form K1'' is not invariant under the SU(3) gauge transformation of QCD and thus a gauge invariant physical particle for JM is not defined. One thus concludes that the Nambu—Goldstone theorem does not hold in general when the global chiral symmetry is broken by the effects of the chiral anomaly and instantons. This is regarded as a reason why we do not observe a light pseudo-scalar meson rf in actual QCD.
6
QUANTUM BREAKING OF GAUGE SYMMETRY In this chapter we discuss the quantum anomalies which appear in gauge symmetry itself. This class of anomalies generally appear in gauge theory where both of the vector and axial-vector couplings of gauge fields co-exist. As a special case, the chiral gauge theory where parity is broken is important in physical applications. The quantum breaking of gauge symmetry implies that the conservation of probability (unitarity) fails and one cannot calculate higher-order quantum corrections in a consistent manner. The absence of the quantum anomaly in gauge symmetry is the basic requirement of consistent gauge theory, and the requirement imposes a stringent condition on the allowed lepton and quark multiplets in the Weiiiberg-Salarri theory. This class of anomalies also has important applications to the study of. for example, non-linear a models. In this connection, an interesting constraint called the integrability (or the Wess--Zumi.no consistency) condition appears. The actual evaluation of the covariant form of anomalies, which do not satisfy the integrability condition but have a gauge covariant form, is much easier. Consequently, a conversion prescription of the covariant form of anomalies to the integrable form of anomalies becomes important. In the context of the current algebra, it is explained that the quantum anomalies generally lead to anomalous commutation relations. 6.1
Gauge theory with axial-vector gauge fields
We analyze the Euclidean path integral
where the Dirac matrix convention is the same as in QED and QCD, and 7^ is chosen to be anti-hermitian
The 75 matrix which satisfies
is herrnitiaii. The covariant derivative D^ is defined by
92
GAUGE THEORY WITH AXIAL-VECTOR GAUGE FIELDS
93
by using the generators Ta of a non-Abelian gauge group
The non-Abelian gauge field has the same number of components as the generators of the gauge group. To simplify the notation, we include the coupling constant into the gauge fields, for example, gV£ —>• V®. We adopt the convention to sum over the indices which appear twice in the same expression. We also often write Vf, = V£Ta and .4M = A°Ta as in eqn (6.4) above. In this chapter, the vector-like gauge field, which has been discussed so far in this book, is denoted by V£ and the axial-vector gauge field, which appears together with Dirac's 75 matrix, is denoted by A"^. The field strength tensors V£v and A.^v, which are generalizations of the electric and magnetic fields E and B, are defined by
The fcrmion action is invariant under the following gauge transformations
The last equation is also written as
where we defined
We have the gauge parameter with the 75 matrix in addition to the conventional gauge transformation in these equations, arid thus a generalization of gauge transformations. When one considers the Dirac operator in Euclidean theory, the operator with both of the vector and axial-vector fields V'M and A^
does not define a hermitian operator with respect to the following inner product
94
QUANTUM BREAKING OF GAUGE SYMMETRY
since the signature of the axial-vector field AIL changes. If one rotates the axialvector field AH to an imaginary field as
the operator becomes hermitian However, as a price for making the axial-vector field A^ pure imaginary, the axial part of the gauge transformation is spoiled. We define a complete set of basis functions by using the hermitian operator ]j) (6.13), and expand the fermionic fields as
The path integral is then written in the form
which can be exactly (though formally) integrated. The Jacobiaii for the gauge transformations (6.7) is then given by
GAUGE THEORY WITH AXIAL- VECTOR GAUGE FIELDS
95
and the regularized Jacobian (namely, the quantum anomaly) is given by the master formula
This formula of the Jacobian shows that the quantum anomaly associated with the conventional gauge transformations parametrized by a(x) vanishes in Che present method of calculation, and the gauge symmetry for the axial-vector gauge field parametrized by /3(x)"/s is spoiled by the quantum anomaly. The explicit evaluation of the Jacobian (6.20) proceeds by using the complete set of plane waves just as in the case of the previous chapter. The result after a somewhat lengthy calculation is given by (with the convention e 1234 = I)19
by rotating back to the original field A^ —» —iA^ after performing the calculation. In the actual calculation of the Jacobian, one encounters the term
multiplied by the cut-off parameter Af 2 , but this term is cancelled by a gauge transformation of a local counter-term M2Aa^A^ with a suitable coefficient Af 2 . If one adds this counter-term to the original Lagrangian in eqn (6.1)
the variation of the local counter-term under eqn (6.7) cancels the part of the Jacobian (6.22). In the analysis of the quantum anomaly, those terms in the Jacobiaii which, can be cancelled by suitable local counter-terms in the Lagrangian are generally regarded not to be a genuine anomaly. This definition of the quantum anomaly is important in the analysis of quantum anomalies associated with gauge symmetry.20 19
One may use a simple regulator function f(x) — e x. Unfortunately, this calculation is not systematic and is very complicated compared to the calculations in Chapter 5. We shall later present a simpler calculational method which gives rise to the covariant form of anomalies. We shall then give a method to convert the covariant form of anomalies into the anomalies equivalent to eqn (6.21). 20 To be precise, one says that the theory is anomaly-free if the main part of the anomaly factor in cqn (6.21) is absent. In the presence of the main part of the anomaly, it is more general to retain the extra term (6.22) in the Jacobian (6.21) with an arbitrary finite coefficient as part of the anomaly by adjusting M2 suitably.
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QUANTUM BREAKING OF GAUGE SYMMETRY
6.2 Pauli—Villars regularization The calculatiorial scheme we described above corresponds to perturbative calculations by using the Pauli-Villars regularization. When the quantum anomaly appears in the gauge symmetry itself, the gauge theory becomes inconsistent and thus no non-perturbativc calculation exists. The Pauli Villars regularization is defined in the path integral by
where the mass M of the regulator field (x), which was introduced to make the calculation well defined, is set to infinity after the calculation. The field
the same Jacobian factors appear in the denominator and numerator corresponding, respectively, to the field variables ib(x) and (f>(x), and thus the Jacobians completely cancel each other. We here recall that the Jacobian appears in the denominator for the Grassmann number for which the integral is defined by the (left) derivative. The regularized path integral thus does not produce the quantum anomaly manifested as a Jacobian. However, the action for the Pauli-Villars regulator field
contains a mass term M, and the large limit of this mass term defines a physical theory. In the present gauge theory, the mass term of the regulator field M<j>(x)(p(x) is not invariant under the axial gauge transformation cxp[i/?a (x)Ta^\ and thus the gauge symmetry is explicitly broken. This gauge symmetry breaking in the limit M —?> oc induces a strong breaking of gauge symmetry, which is a definition of the quantum anomaly in the context of the Pauli-Villars regularization. From the viewpoint of Ward-Takahashi identities, the change of variables in the path integral with a = 0 in the gauge transformations (6.25) gives rise to the identity (which is a generalization of the fact that the definite integral does not depend on the naming of integration variables f dx f(x) = f dy f(y))
PAULI-VILLARS REGULARIZATION
97
In the last line we used the fact that no Jacobian appears in the path integral measure. The action in the last expression is explicitly written for an infinitesimal as
When one uses this expression in the above identity (6.27) arid collects only the terms linear in j3(x), the identity is written as
The covariant derivative here is defined by
and also we define
for the general composite operator O(x). The left-hand side of the identity (6.29) defines a regularized axial-vector current, and it is confirmed that the mass term on the right-hand side gives the quantum anomaly in the limit M —> oo (with the convention e1234 = I) 21
which reproduces the anomalous identity derived in the previous section. Since the regularization is introduced simply to avoid ambiguities in the actual perturbativc calculations, it is natural that the final result agrees with the 21
One calculates all the one-loop diagrams which contain the operator 2iMip(x)^sTa(x') at one of the vertices and the gauge fields at the remaining vertices. The diagrams which do not vanish in the limit M —> oo give the result in cqn (6.32). This calculation was first performed bv W. Bardeen.
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QUANTUM BREAKING OP GAUGE SYMMETRY
one obtained by a different method. To see what is going on in the Pauli-Villars regularization in a more intuitive way, we write an identity which is based on the transformation of only the path integral variables ip(x) and ijj(x)
and an identity based on the transformation of only the path integral variables and
When one adds these two identities, one obtains the result of the Pauli-Villars regularization (6.29), which avoids ambiguities in the intermediate stages of perturbative calculations associated with the axial-vector current. Here we wrote the anomaly which arises from the Jacobian in a symbolic notation In J. The change of the signature of In J between the two equations above is a result of the fact that the Jacobian appears in the denominator and the numerator for the Grassmann number and the ordinary number, respectively. Note that both sides of eqn (6.34) approach zero in the limit M —>• oo. The path integral method discussed in Section 6.1 used a regularization which cuts off the eigenvalues of the operator Tjb (6.13), and as we explained there this regularization generally spoils axial gauge symmetry though it preserves vector gauge symmetry. This property is shared with the Pauli-Villars regularization. 6.3
Chiral gauge theory and the quantum anomaly
The quantum anomaly discussed so far may be called V-A-type, but from the viewpoint of gauge theory it is natural to separate the left component ipL (x) = [(I - ~f5)/2]il;(x) arid the right component WR(X) = [(I + 7 5 )/2]ii(x) and rewrite the covariant derivative as
and the action as
Since we have and thus (1 ± 7s)/2 are projection operators, one can treat the left and right components of the gauge field as independent components.
CHIRAL GAUGE THEORY AND THE QUANTUM ANOMALY
99
In this separation, the theory with only the left components, for example,
is called a chiral gauge theory, and the action is invariant under the gauge transformations
The path integral measure is not invariant under these transformations. A way to understand this non-invariance is to introduce a dummy fermionic component ij)H into the path integral (6.38), which does not interact with the gauge field,
and one recognizes that this theory is derived from the V-A-type theory we discussed already by setting V^ = —A^ — L^/2 in eqn (6.4). Consequently, the quantum anomalies are also derived by the same replacement in eqn (6.21). If one considers the gauge transformation of the fermionic variables with gauge field kept fixed, one obtains the anomalous identity
where we defined
The theory where the gauge field couples only to the right-handed fermion tpn is treated in a similar way by setting Vlt = A^ — R^/2 in the V-A-typc theory in eqn (6.21), and one obtains
100
6.3.1
QUANTUM BREAKING OF GAUGE SYMMETRY
Quantum theory of chiral gauge theory
We study the quantum theory of chiral gauge theory defined by
to understand the physical implications of quantum anomalies obtained above for the chiral theory. As a gauge condition we set the time components of all the gauge fields to be 0 (which is also called the Weyl gauge condition)
From the equation of motion for L^(x) we obtain the constraint
related to the Gauss law
where a non-Abeliaii generalization of the electric field in Maxwell theory is denned by The indices k which run over 1 ~ 3 stand for the spatial components. The action with the condition Lg (x) = 0 still contains a gauge freedom which does not depend on time
and the Gauss law operator is a generator (Nothcr current) of such a gauge transformation. This fact is understood by considering a time-dependent parameter u } a ( t , x ) instead of the time-independent gauge parameter uja in eqn (6.49) in the action which shows that Ga(x) is the Neither current for the time-independent gauge transformation. When one considers the above time-dependent gauge transformation uja(t, x) in the path integral (by using the fact that the path integral measure for the gauge field is gauge invariant)
CHIRAL GAUGE THEORY AND THE QUANTUM ANOMALY
101
one obtains an identity by including the quantum anomaly (6.41) arising from the Jacobian
where we used e1230 = 1 in the Minkowski metric. This equation is written in an operator notation by using the Hamiltoriiari H derived from the Lagrangian density C in eqri (6.44) as
for the theory with the gauge anomaly. Namely, the Gauss operator Ga(x) no longer commutes with the Hamiltonian H due to the effects of the quantum anomaly, and the Hamiltonian H is no longer gauge invariant. This fact shows that the basic two equations in the Schrodinger functional representation of quantum field theory
are no longer compatible with each other since [H,Ga(x]] ^ 0. Even if one starts with a physical state which satisfies G°* = 0 at t — 0, the physical condition Ga^> = 0 does not hold after a time development to t / 0. The Gauss law operator Ga, which generates the time-independent gauge transformation, eliminates the unphysical longitudinal component of the gauge field Lak(x) when applied as a constraint on the wave functional G°*(i;L^(x),^i(x)) = 0. The failure of the Gauss constraint by the quantum anomaly means that a consistent unitary theory with only the transverse physical components for the quantized gauge field is not constructed. When one adopts a general Lorentz covariaut gauge condition, the constraint on the state vector (in the Fock representation)
becomes fundamental. For example, a general covariant gauge fixing of eqri (6.44)
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QUANTUM BREAKING OF GAUGE SYMMETRY
is invariant under the BRST symmetry (sec cqn (3.51))
Here ca(x) and ca(x) stand for the non-Abelian generalization of the FaddeevPopov ghost fields, which were briefly explained in Section 3.2, and A is a constant Grassmann number; i\ca(x) replaces the non-Abelian gauge parameter. The BRST current is derived by a localized transformation with \(x) and the variation of the action 5 J d^x £eff = / d^x 9/i A(a;) jgRST (x). The BRST charge is then delined by QBRST = / d3x JBRST^^)- ^n a theory with the quantum anomaly in gauge symmetry, it is shown by considering a localized BR.ST transformation that we have
and thus the BRST charge QBRST ceases to be a conserved quantity, and the constraint on the state vector QBR.ST|X!'} = 0 becomes incompatible with the time development. We thus conclude that the quantum anomalies in gauge symmetry need to be cancelled among fermionic variables appearing in the theory to define a consistent theory of quantized chiral gauge theory. This gives a stringent constraint on the allowed fermionic contents, such as the number of fields or the representations of the gauge group, in the Weinberg-Salam theory which breaks parity symmetry. The cancellation of the quantum anomaly in the actual Standard Model, which incorporates the Weinberg Salam theory, is explained later. 6.4 Covariant anomaly Among the methods to evaluate quantum anomalies in gauge symmetry explicitly, in addition to the method already explained which is related to the PauliVillars regularization, a method which gives a gauge covariant form of anomalies is known and the anomalies thus obtained are known as "covariant anomalies." We explain the method for chiral gauge theory
We first note that the Dirac operator Jfii = ^(d,,, - «£ M )(1 - 7o)/2 appearing in the Euclidean path integral is not hermitian
namely, the left-handed system characterized by (1 - 75)72 changes to a righthanded system. In this case, the operator
COVARJANT ANOMALY
103
is confirmed to be hermitian and positive semi-definite. Consequently, one can define a complete orthonormal system
where A n is a real non-negative number. The 1 : 1 correspondence of vanishing eigenvalues of $>L and JPL^>L is confirmed as follows
Similarly, one can define a complete system by using the hermitian ]/)L^L
and confirm the 1 : 1 correspondence of vanishing eigenvalues of If)L and ^*L^| • From the relation
one can define a correctly normalized eigenfuncliori ipn by
by choosing Xn > 0 for \n / 0. This shows that the two operators $>L$>L and J/>r,J/>L share exactly the same number of non-vanishing eigenvalues. By using the basis sets {ipn(x)} and {ipn(x)} thus defined, one may expand the fermionic variables as22
and the fcrmion action is formally diagonalized as 22 In the framework of the covariant anomaly, one can show that the anomalous Jacobian is independent of the choice of path integral variables, namely, either T/J£ and V'r. or y;L and 1(1 *L. In Lhc latter choice, i/ij, is defined by the operator ("tof)L)^"fof>L = W^L and 1/4 is denned by the operator 70-Pr,(70^1,)^ = 7o^L-PJ,7o- One can then confirm that the identical (regularized) Jacobian is obtained by either choice of path integral variables.
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QUANTUM BREAKING OF GAUGE SYMMETRY
6.4.1 Fermion number anomaly As an application of the present definition of the path integral, we first consider the phase transformation related to the fermion number by using a local parameter a(x) (we thus discuss the non-gauge freedom for the moment) in eqn (6.59)
and then the path integral measure is transformed as
A more detailed account of the appearance of this Jacobian is given soon. This Jacobian is evaluated by replacing the mode cut-off by the cut-off in terms of the eigenvalues as
where we used the property of the projection operators (I ± 7s)/2
The explicit evaluation of the Jacobian (6.71) is the same as the calculations in Chapter 5, and one finally obtains
COVARIANT ANOMALY
105
This result when combined with the Nother current for eqn (6.69) gives
which shows that the fermion number operator contains a quantum anomaly. This was first shown by G. 't Hooft. To understand the meaning of this anomaly, we write the path integral measure more explicitly as2s
Here we consider the general case where $>L^L and $\$L, respectively, have n0 and mo vanishing eigenvalues. This property is generally expressed as
where dimker^i, for example, expresses the number (dimension) of the normalizable 0 eigcnmodes (kernel) of the operator J/>i. and the above difference of zero modes defines an important notion of index. The index relation here has the same form as the index relation we discussed in connection with the photon phase operator in Chapter 3, if one identifies the creation operator of the photon with a^ = Jfii. We discuss the meaning of the index by noting 75$) + ^75 = 0. In general (we analyze the eigenvalues of J/) = 7M[c^ — igL^(x}])
and thus the 0 eigenmodes of ]}) can be chosen as the simultaneous eigenstates of 75, [(l±7 B )/2]^ n (.T). Also
and the index (6.76) is expressed by using the basis set $>'il>n(x) = Xn^->n(x) as 23 If one denotes 4>n(x) — {x[
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QUANTUM BREAKING OF GAUGE SYMMETRY
where we defined HO = n+, m® = H_. As was explained in Section 5.5, this index agrees with the Pontryagin index carried by the gauge field (by using the notation including the coupling constant into the field variables)
which is known as the Atiyah-Singer index theorem. Coming back to the path integral of chiral gauge theory, the path integral measure is transformed for a global constant fermion number a in eqn (6.70) as
where we have a difference of zero modes, since {
which shows that np number of tyi and rno number of 'tpi are absorbed into the vacuum in the presence of the general iustanton solution. Since the integral over the Grassmanri numbers is defined by the (left) derivative and the Grassmann coefficients of the expression in eqn (6.67) corresponding to the zero modes do not appear in the action of the path integral, we need to insert at least the above number of ipL and 'I/JL into the integrand to obtain the non-vanishing path integral. If one inserts more I/JL and -I/JL, one obtains a generalization of conventional Green's functions.
COVARIANT ANOMALY 6.4.2
107
Covariant anomaly in gauge transformations
We next examine the gauge transformation of fenriioii variables defined by
The path integral measure is transformed by using eqn (6.67) as
with a(x) = aa(x)Ta, and the Jacobian is evaluated just as in eqn (6.71)
This quantum anomaly for the gauge transformation is called the covariant anomaly because of its gauge covariant appearance. We show that this covariant anomaly is associated with a specific regularization of the source current of the gauge field. A regularized current which gives rise to this covariant anomaly is defined by using the complete set
and the construction of covariantly regularized current discussed in Chapter 4 as
When we have \n = 0 we need to control the infrared divergence by using, for example, the Higgs mechanism to be discussed in the next section without changing the quantum anomaly. The quantum anomaly itself is free of infrared divergences as is shown shortly. The gauge transformation is written as
with oj(x.) = (jja(x)Ta. By noting the gauge invariance of the eigenvalues in eqn (6.86), we have a covariant form of transformation law of the current
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QUANTUM BREAKING OF GAUGE SYMMETRY
for an infinitesimal w. The quantum anomaly is given by using e1234 = 1
which agrees with eqn (6.85). In fact this identity is derived by considering the transformations (6.83) in the path integral of the chiral gauge theory (6.59) and by using the Jacobian (6.85). This shows that the calculational method of the Jacobian for the covariant anomaly is understood as a covariant regularization of the current. This result (6.90) shows that gauge symmetry is broken by the quantum effects. Conversely, if all the quantum anomalies (6.85) thus evaluated are cancelled, the chiral gauge theory is consistent even in the quantum level. If one recalls that the quantum anomaly (6.85) is symmetric under the replacement of F^vTb and F^T0, a formulation of quantized chiral theory is consistent if the following relation holds24
It is known that the orthogonal gauge group SO(ri) with n > 2 and n ^ 6, for example, leads to anomaly-free gauge theory. As an explicit example, if one chooses the group SO(3) or locally equivalent SU(2), it is confirmed that all 24 From the viewpoint of Feynman diagrams, Ihc covariant anomaly corresponds to a calculational scheme where one collects all the effects of the anomaly to a specific current vertex while imposing the ga,uge invariance on all the remaining vertices coupled to the gauge field. I is thus clear thai chiral gauge theory is consistent if the covariant form of anomalies all cancel. In the case of the gauge group SU(2), one needa to include the analysis of the globa.1 anomaly, which is briefly mentioned in Chapter 11.
ANOMALY CANCELLATION IN WEINBERG-SALAM THEORY
109
the anomalies cancel by using the properties of the Pauli matrices which satisfy eqn (6.91). A simple way to understand the absence of anomalies in the realistic Weinberg Salain theory may be to use the fact that all the quarks and leptons are precisely accommodated in a representation of the anomaly-free gauge group SO (10) in a grand unified model. In the general case where the gauge field couples to the left-handed fermion field with a representation T£ and the right-handed fermion field with a representation Tft, the above anomaly cancellation condition is written as
For example, the vector-like theory, where the gauge field couples to the lefthanded and right-handed fermion fields with an identical representation, defines a consistent theory since the quantum anomaly is automatically cancelled. The quantum electrodynamics and QCD belong to vector-like theory. 6.5
Anomaly cancellation in Weinberg—Salain theory
We examine the anomaly cancellation in the Weinberg Salam theory which is based on the gauge group SU(2)^ x C7(l)y. When we denote the generators of the gauge group SU(2)i x U(l)y by (I\. T2, T3) and Y, respectively, the electric charge of fermions satisfies the Gell-Mann-Nishijima-type relation
When we write a doublet of quarks as
the Lagrangian for the quarks is given by
If one recalls that Ta = (l/2)r a is written in terms of Pauli matrices, the generator of U(l) is written as
by remembering that the electric charges of quarks (u,d) are (2/3, —1/3). Consequently, the formula (6.90) for the quantum anomalies gives the anomaly for SU(2) gauge transformations (by using the notation W^ = W^Ta)
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QUANTUM BREAKING OF GAUGE SYMMETRY
arid for U(l) gauge transformations
We multiplied all the expressions by the factor 3 since three quarks appear for each case due to the color degrees of freedom. Similarly, for the doublet of leptons
(by writing the right-handed component of the neutrino explicitly, though it does not couple to gauge fields), the U(l) charge is given by
by recalling the electric charge (0. —1) of leptons (v,e). The Lagrangian for the leptons is written as
Consequently, the leptons induce the quantum anomalies for the SU(2) gauge transformations
and for the U(l) gauge transformations
The gauge field belonging to SU(2) docs not contribute to the anomaly for the SU(2) gauge transformations, since SU(2) is an anomaly-free group, arid the properties trT" — 0 and 3F/ + Y'L — 0. which are confirmed by using the explicit expressions in eqn (6.96) and eqn (6.100), ensure the absence of the anomaly for all the SU(2) gauge transformations. The remaining anomaly for the U(l) gauge transformations cancels if one uses
The Weinberg-Salam theory thus defines a consistent quantum theory if one considers quarks and leptons together.
ANOMALY CANCELLATION IN WEINBERG SALAM THEORY 6.5.1
111
Effects of Higgs particles
As an application of the covariant anomalies, we show that the Higgs (or scalar) particles do not contribute to the quantum anomalies. We analyze this issue by using the following Lagrangian
where 4>(x] stands for the Higgs field (or in general, a scalar field). Note that the operator 0r, = 7''(<9;j ~ i/^i), for example, docs not contain the projection operator (1 — 7s)/2. By noting that the left- and right-handed fermions are coupled to each other mediated by the Higgs field, we use the following general operator
In Euclidean theory we expand the field ?/; in terms of the complete set of eigen-
functions of 'D~'D and the field ' in terms of the complete set of 'D'D~ in the calculational scheme of covariant anomalies. The quantum anomaly for the gauge transformations of left-handed components, for example.
is given by the master formula
We now use the following relation
The terms linear in the Higgs field, for example.
112
QUANTUM BREAKING OF GAUGE SYMMETRY
show that the differential operator acts on the Higgs field only and does not act on fields coming to the right of it. As a result, in the expression with the plane wave basis of the master formula
the terms with the Higgs field is always expanded in powers of 1/M2, when one moves c.'lkx through the differential operators in, for example, TjD\ and then rescales as k^ —> Mk^ and expands all the terms in the exponential in powers of 1/M. Also, the terms linear in the Higgs field convert the right-handed components to left-handed ones and vice versa, and thus only the even powers in these terms contribute to the master formula.2''1 For this reason, the identical form of terms appear in both of the anomaly factors coming from ip and ip in eqn (6.108), and consequently we take the trace over these terms together with 75. For example, in d = 4 space-time, we have
which vanishes since it contains only two 7^ matrices. The master formula eqn (6.108), after eliminating the terms odd in powers of the Higgs field in eqn (6.109), thus gives
One can confirm that those terms with the Higgs field in eqn (6.113), which do not contain Dime's 7 matrices, do not contribute to the anomaly if one performs the same calculation as in the previous section. 6.6
The Wess Zuinino integrability condition
We examine the following path integral by setting A/t(x) = A®(x)Ta
The covariant current defined in eqn (6.87) with f) = 7^(9^ — iA^)
as was already explained, contains the covariant form of anomaly 25 This fact is seen by writing the trace over the 7^ matrices in the first term in eqn (6J 11), for example, as tr[(l - 7o)/2] cxp(-£>t?3/Af 2 )[(l - 75)72].
WESS-ZUMINO INTEGRABILITY CONDITION
113
In the applications of quantum anomalies, the anomalies for a finite transformation often become important. We thus have to superpose the Jacobian for an infinitesimal transformation to realize a finite transformation. In this analysis, we make the following correspondence between the Jacobian and the transformation of the fermionic variable for an infinitesimal parameter a(x) — aa(x)Ta as
The composition law eta^x'e^x'e ia(x>e IP(X> = exp(—[a,/?]) of group theory then imposes the following integrability condition on the Jacobian
The first term on the left-hand side is given by the Jacobian evaluated for the parameter /? by using the gauge field, which is a result of the gauge transformation parametrized by —a, and the second term is given by the Jacobian obtained by a reversed order of calculation. For the above covariant anomaly (6.116), however, because of the gauge covariance of F^ we obtain twice
on the left-hand side, namely, the integrability condition is spoiled by a factor of 2. On the other hand, it is shown that the anomaly related to the Pauli Villars regularization discussed in Sections 2 and 3 of this chapter satisfies the integrability condition (the Wess-Zumino condition), after a somewhat lengthy calculation. The Wess-Zumino condition becomes important when the gauge theory is not consistent due to the presence of quantum anomalies in gauge symmetry, and thus only the perturbative calculation with one-loop fermion diagrams becomes relevant. The Wess-Zumino condition is then shown to correspond to the Bose symmetry with respect to the vertices of the one-loop Feynman diagrams. The covariant anomaly, which collects the effects of the anomaly to only one of the vertices, does not satisfy the Bose symmetry and thus the integrability condition. In an actual explicit calculation of anomalies in arbitrary space-time dimensions d (which in general includes the gravitational field) . one can calculate only the covariant form of anomalies. It is therefore important to give a prescription 'which converts the covariant anomalies to intcgrable anomalies which satisfy the Wess -Zumino condition.26 There are several ways to do this conversion, and we give here a method which uses a current to define a result of the path integral (or the partition function) in the presence of the background gauge field. 26 This anomaly is also called a consistent anomaly, but it is clear that the covariant anomaly is not "inconsistent."
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QUANTUM BREAKING OF GAUGE SYMMETRY
If one replaces the gauge field as A^ —> sA^ by introducing the parameter s in cqn (6.114), the quantity defined by W(sAp,) = lnZ(sA/t) satisfies
We thus define the regularized path integral by means of the covariantly regularized current J°''(x-) (6.115) by27
The current Jap'(x} is a finite quantity and thus W(Ali} is also finite. If one regards the finite W(A^) as a functional of Ap and expands it in powers of AH, the expansion naturally satisfies the Bose symmetry. Consequently, the current jav(x] defined by
satisfies the Wess Zurnino condition. From its definition, we have
The comparison to eqn (6.121) shows that W thus defined docs not, depend on the class of currents we use to define it. To be more explicit, the current ja"(x) is given by adopting the notation
Ja»(x.s) = Ja>l(x,sA
which combined with the fact that the current Jal*(x,s) depends on s only
through the combination sA
finally gives
The last term in this expression is a manifestation of the fact that the current J afi (x) does not satisfy the integrability condition (if it is written in a form 27l ['he imaginary part, of eqn (6.121) contains interesting information about the quantity called the '/-/-invariant.
WESS-ZUMINO INTEGRABILITY CONDITION
115
Ja^(x) = 5W/6A^(x), the last term vanishes) arid at the same time it directly depends on the quantum anomaly, since in the absence of the anomaly, these two currents should coincide.28 In the following, we would like to explain the evaluation of the last term in eqn (6.126) above,29 by using the regulator function f ( x ) = e~x in eqn (6.115) as an example. We first consider the case s = 1. If one defines
the "curvature" 5Jb"(y]/5A^1(x} combination
— (a -B- b, p.• O v, x •<-?• y) is obtained from the
by identifying the subscripts as 1 —> (a,fj,,x) above expression, we used the identity
and 2 —> (b,v,y). In deriving the
valid for a general operator X. By noting the relation
and the cyclic property of the trace Tr, it can be confirmed that all the non-local terms in eqn (6.128) which contain 1/0 cancel each other after integration byparts with respect to a. Thus we have
28
In the absence of gauge anomalies, one can impose the gauge invariance on all the vertices in the one-loop Feynman diagram without spoiling the Rose symmetry. In fact, this is the only sensible calculation and thus these two currents coincide. 29 Rcadcrs who are interested in the final formula may proceed directly to eqn (6.135).
116
QUANTUM BREAKING OF GAUGE SYMMETRY
where we have noted that the terms which do not contain 75 vanish under the anti-symmetrization of 1 4-» 2. From this expression, we have (by recalling that
In writing the last expression, we discarded the commutator {j",e ^ a^ /M"] and the anti-commutator {7^,7"} = 2
by noting eqn (6.129). It can be shown that this relation holds for a general regulator function /(x). 31 If one uses SJbv(y)/5A°(x) - 5Ja»(x)l5Abv(y) thus obtained in eqn (6.126) (the index s indicating that sA™ is used) one obtains
where Str indicates that we take a trace of the expression which is arranged to be symmetric with respect to the generators of the gauge group. The quantum anomaly is then given by the covariant divergence of this expression'32 30
Powers in 1/M do not change but the number of 7 matrices is reduced by two in these operations. 31 To show this, we introduce the Laplace transformation
and the result for a general regulator function f(x) is obtained by multiplying liniM->oo f£° dpg(p) by the above evaluation in eqn (6.132) with the scaled regulator mass M2 —5- M 2 /p. The terms which survive in the large M limit are independent of M and p, and we obtain the result identical to the case f(x) = e x, because f£° dp g(p) = /(O) = 1'. 32 Quantities inside brackets { and } are treated as a block, namely, they are not subject to svmnietrization under Str.
WESS-ZUMINO INTEGRABILITY CONDITION
117
which gives a formula to derive the integrable anomaly (6.41) starting with the covariant anomaly (6.116). In this derivation we used the Jacob! identity ^vafi\^^^Aa,Ao^ = 0 and it is important to recognize that the symmetrized trace Str is used. As a special example, we obtain
for Abclian gauge theory, and the factor 1/3 is the Bose symmetrization factor of the triangle Feynman diagram. It is clear that these formulas are generalized to arbitrary even d = 2n dimensional space-time, and by using the covariant, anomaly, which is derived in a manner identical to the U(l)-type anomaly (5.75) in Chapter 5,
one can write the current
and the anomaly factor
As is clear from the above derivation, -4"ons = 0 is automatically satisfied if -4£ov = 0. 6.6.1
The Wess-Zumino term
We define the Wess-Zumino term I\vz by33 33
Under an infinitesimal gauge transformation parametrized by a, SL^ = l)^a, SR^ — 0, 5U = iaU, the left-hand side in eqn (6.141) is invariant and the second factor on the right-
118
QUANTUM BREAKING OF GAUGE SYMMETRY
by considering the background fields L^(x)T" = L^ and R^(x)Ta = R^ belonging to the group, for example, SU(n). The factor U(x) is an element of the group SU(n), and it is given explicitly by
We define the path integral measure in eqri (6.141) by the V-A prescription in Section 6.1. In the defining path integral of I\yz, one may perform a change of variable and then the U(x) dependence is completely removed if the Jacobian is ignored. Consequently, I\vz contains only the effects of the anomaly. The path integral can also be written as
if one performs the vector-like transformation ib —>• U^ip. which does not contain the anomaly as was shown in Section 6.1. To evaluate FWZ explicitly, we introduce a real parameter s and consider the gauge field which, after a certain amount of gauge transformation, becomes
Starting with this configuration, one performs a further infinitesimal chiral transformation and an anomaly-free vector-like transformation
alternatingly. The result of this operation is expressed as an integral over the variable s of the anomaly appearing for the chiral transformation in eqn (6.146) as hand side produces the gauge anomaly by considering combined chiral and vector-like gauge transformations of the fermionic variables. In this way, the Wess Zumino term which turns out to be a local functional reproduces the anomaly under the same gauge transformation,
WESS-ZUMINO INTEGRABILITY CONDITION
119
where A(L/Jl(x),Rlj(x)) stands for the integrable form of anomaly34 discussed in Section 6.1
with the replacements
When one adds this FWZ (6.148) evaluated for the group SU(3) to the nonlinear
one obtains the effective interaction among the neutral pion and electromagnetic fields after multiplying by the color factor Nc = 3 as
Here v^ stands for the electromagnetic field and FIW = 9Miv - dvv^. To obtain the same result from a more fundamental consideration, one may consider two flavors of quarks (u,d) with the electric charges (2e/3, —e/3). respectively, and the Lagrangian
The axial-vector current corresponding to the third component of isospin symmetry 34 Equation (6.148) is written in the Minkowski metric with e123n = 1. The imaginary factors i arising from the Minkowski convention of e 12 ^ 0 — l and the volume element d^x compensate, and the expression (6.149) still gives the correct result.
120
QUANTUM BREAKING OF GAUGE SYMMETRY
is shown to contain the following chiral anomaly (by including the color degrees of freedom Nc = 3 which is not written explicitly in the above Lagrangian)
The Nambu-Goldstone theorem in the presence of the spontaneous breakdown of chiral symmetry states
When one combines the anomaly relation eqn (6.155) with this theorem (at p2 ~ 0), one has
which agrees with the effective Lagrangian given by the Wess-Zumino term (6.152). This analysis of eqii (6.157) was performed by Bell and Jackiw, and Adler in 1969, and it is known that the result describes well the experimental result. As the second application, we consider the case where all the external gauge fields vanish (L M (x) = R/j.(x) = 0) and the case of group SU(3)
where we introduced the new notation
with U(x,s) = exp{2i[7r(x)// ff ](l - s)}. When one uses <9 M t/t = -U^(d^U)U^ which arises from U(x, s)U^(x, s) = 1 and defines a five-dimensional completely anti-symmetric tensor by e12305 = 1 by regarding the variable s formally as the fifth coordinate of space-time, one can write FWZ(^) in a symmetric way as
The integration domain D (which means a disc) shows 0 < s < 1. When one expands this Tw7,(U) in powers of TT(.T), one can describe the new interactions among the eight Nambu-Goldstone bosons induced by the quantum anomaly.
QUANTUM ANOMALIES AND ANOMALOUS COMMUTATORS 6.7
121
Quantum anomalies and anomalous commutators
Anomalous commutation relations generally appear in the theory which contains the quantum anomalies. This is based on the fact that a quantum field theory is defined by the commutation relations of various operators and their representations in the operator formalism, and thus if all the commutation relations are normal we have no space to accommodate the quantum anomalies. An example of the anomalous commutator already appeared in cqn (6.53) in Section 6.3, and further examples arc discussed in detail in Chapter 10 later in connection with the Kac-Moody and Virasoro algebras. In this section, we would like to briefly explain the difference between the Goto-Imamura— Schwinger term, which appears in the absence of the quantum anomalies, and the anomalous commutators associated with the quantum anomalies. In quantum electrodynamics, the correlation function of the current j ^ ( x ) = ip(x)^ij}(x) is given by the calculation of the vacuum polarization tensor as (in Minkowski metric)
which was explained in eqn (4.39) of Chapter 4.35 In the analysis of canonical commutation relations, one implicitly assumes a theory which is defined by a cut-off in momentum space (instead of renormalized quantities) and thus we write the bare correlation function with an explicit cut-off parameter A. We now apply the BJL (Bjorken-Johnson-Low) prescription which states the rule: If the right-hand side of the above Fourier transformation approaches 0 for go —>• oo, one can replace the Lorentz covariant T* product by the canonical T product. If the Fourier transformation does not approach 0, one defines the T product after subtracting the non-vanishing quantity, which is expressed by a polynomial of q0, from both sides. A physical reason for this prescription is that in the canonical formulation which is based on the equal-time commutation relations the correlation functions of general currents should be well defined and smooth in x^ — j/M around x° = y°- The limit
To be precise, the result in eqn (4.39) is written for the case A 2 ;§> |
122
QUANTUM BREAKING OF GAUGE SYMMETRY
since one obtains the S function when the derivative acts on the T product and the term with derivative operation inside the T product vanishes in the limit (?o —> oo by the definition of the T product.36 One then concludes by using eqn (6.162) for the correlation functions in eqii (6.161)
where C — i/(27T 2 )(A 2 — m 2 ) is a divergent c-number in the present example. The term which contains C is called the Goto-Imamura-Schwinger term. On the other hand, if one applies the same analysis to the correlation functions containing the Gauss law operator for the gauge condition AQ = 0 (a non-Abelian version of the Gauss operator is given in eqn (6.47), while we use here an Abelian version)
where Ak (x) is the time derivative of Ak (x), one obtains
after applying the BJL prescription. The basic difference between [ G ( x ) , j k ( y ) ] x 6(x° - y°) and [j°(x), j k ( y ) ] S ( x ° - y°) is that the contribution from the term where the gauge field contained in G(x) couples to the current jk(y) via the photon propagator cancels the C term in eqn (6.163). This shows that the Gauss law operator defines a proper generator of the gauge transformation in anomalyfree theory and the current j ^ ( y ) is a gauge invariant operator. On the other hand, in a, theory with the anomalies in gauge symmetry, the commutators containing the Gauss law operators themselves have anomalous terms. See. for example, eqn (6.53) in Section 6.3. A detailed analysis of the commutation relations among the Gauss law operators in non-Abelian gauge theory is rather technical, and we forgo the discussion here.
36
This effectively gives a prescription to separate the equal-time commutator from the correlation function.
7
THE WEYL ANOMALY AND RENORMALIZATION GROUP In this chapter we discuss the quantum anomaly associated with the scale transformation of space-time coordinates or the transformation generally called the Weyl transformation. In flat space-time, this anomaly is related to the renormalization group and the calculation of the ft function in the renormalization group equation is related to the calculation of the Weyl anomaly. In other words, the renormalization group equation is regarded as an expression of the Weyl anomaly in terms of Green's functions. We illustrate the calculation of the one-loop ft functions in QED and QCD by means of the Jacobians for the Weyl symmetry. The Weyl anomalies in curved space-time arc briefly explained. We also mention an improved finite energy-momentum tensor in renormalizable theory on the basis of an analysis of the Weyl anomaly. 7.1
Scale transformation in field theory
We explain the scale transformation in field theory by taking QED as an example. The starting action is given by
We consider the following transformation of coordinates and field variables parametrized by a constant a, which is called the scale transformation,
If one writes the action in terms of the variable x', one has
If one uses the above scale transformation laws and the relations such as & = exp(a)9jj. d4x' = d 4 xexp(—4a), one obtains 123
124
THE WEYL ANOMALY AND RENORMALIZATION GROUP
which shows that the action is invariant except for the mass term, which spoils the scale transformation. In this sense, the field theory for massless particles is generally scale invariant. The characteristic constants appearing in the scale transformation such as 3/2 for i/j and 1 for AM are called the canonical mass dimensions. As is well known in renormalization theory, the above naive scale symmetry is broken since a new mass parameter is introduced in the renorrnalization procedure even into a massless theory. The purpose of the present chapter is to study these properties of field theory from the viewpoint of quantum anomalies. It is known that QED is free of anomalies associated with general coordinate transformations. Consequently, it would be strange if one has an anomaly for the coordinate transformation x1-1 —> exp(—a)x^. It is also shown that the Jacobian for the scale transformation with naive canonical dimensions does not give rise to the correct anomalies. Those apparently paradoxical properties are all resolved if one analyzes the above scale transformation as a combination of coordinate and Weyl transformations in a slightly curved space-time. In fact all the anomalies of the scale transformations arise from the Weyl transformations when one defines a general coordinate invariant measure, and the coordinate transformation docs not give rise to the anomalies in QED in agreement with the general expectation. 7.2
Identities for the Weyl transformation arid Weyl anomalies
We formulate the Ward-Takahashi identity associated with the Weyl transformation in the flat limit of curved space-time. The Wick rotation to the Euclidean metric in curved space-time is defined by e°(x) -> —ie^(x), and consequently we have \f^g = dete* —»• — zdete* = —i^/g. To derive an identity associated with the general coordinate transformation in Euclidean theory, we first recall that the action for QED in curved space-time is written as (see Appendix B) 37
. 37In Euclidean theory, one needs to rotate the auxiliary field B(x) —> iB(x) (and thus c >. ic) when performing the path integral,
THE WEYL TRANSFORMATION AND WEYL ANOMALIES
125
where F^v = d^A^—d^A^ and we used the Landau gauge explained in Chapter 3. As is explained in Appendix B. this theory is invariant under the local Wcyl transformation
except for the mass term. It is important to notice that ordinary gauge fields (including Yang-Mills fields) are not transformed under the Weyl transformation, which arc sometimes called the Weyl scalars, and thus the ghost field c(x), which appears in gauge fixing and is associated with the gauge field, is not transformed under the Wcyl transformation. We now consider the path integral
where the path integral measure is given by the general coordinate invariant measure in eqn (B.4f)
The Weyl transformation property of each variable in eqn (7.8) is given by
It should be noted that the naive Weyl transformation laws in eqn (7.6) and the Weyl transformation laws of those variables in eqn (7.9) which define the coordinate invariant measure arc quite different. In particular, the anti-ghost and the auxiliary field B are no longer transformed under the Weyl transformation. Incidentally, we have the relation 'DBtt = VBVc.
126
THE WEYL ANOMALY AND RENORMALIZATION GROUP
To write the identity related to the Weyl transformation in the present Euclidean path integral in a compact manner, we define a generic variable for QED by The general coordinate invariant measure is then written as
and the general Green's function is defined by
The path integral (7.7) and the Weyl transformation laws (7.9) with an infinitesimal a(x) give the following identity
where the variables with primes represent the variables after the Weyl transformation, in particular, e'^'(x) = exp[a(z)]e^(x). The first equality in this relation shows the fact that the path integral itself is independent of the naming of path integral variables. The transition from the first expression to the last expression is based on the fact that the action is invariant under the Weyl transformation except for the mass term together with the evaluation of the Jacobian J(a) for the transformation of variables. The last two expressions in eqri (7.13) show that the change of the action for an infinitesimal Weyl transformation of the vierbeins gives the same effects as the Jacobian and the explicit Weyl symmetry breaking by the mass term together with the transformation of variables in the integrand. The variation of the action with respect to the vierbeins defines the energy-momentum tensor, and in particular, the variation of the vierbein by the Weyl transformation defines the trace of the energy-momentum tensor. We use the definition
and T%(z) = fkk(x) = (e%(x)/^/g)5S/6e%(x) with the variables $, c, c and B kept fixed.
IDENTITIES RELATED TO COORDINATE TRANSFORMATIONS
127
We thus obtain the following identity (in the flat space-time limit)
where wn stands for the weight factors in eqn (7.9) for the Weyl transformation, and to be explicit, wn = -1/2 for the fermion and wn = -1 for the gauge field. If one integrates this identity over the coordinates x (or equivalently setting a = 1 in eqn (7.13)). one obtains the identity for the global Weyl transformation
In this equation Nf stands for the number of fermion fields appearing in the Green's function and NA stands for the number of gauge fields. If one can evaluate the Jacobian exactly in this expression (7.16), one obtains the exact Weyl identity. In practice, no exact non-perturbative evaluation of the Jacobian for the Weyl transformation is known, and in this sense this identity is a formal one. However, we can still extract some interesting physical results from eqns (7.15) and (7.16), as is illustrated in the following sections. 7.3
Identities related to coordinate transformations
To derive an identity related to the general coordinate transformation, we use the following variables of QED-type theory defined in eqn (7.10)
The general coordinate invariant measure is then given by cqn (7.11), and a general Green's function is defined by eqri (7.12). The general coordinate transformation law for the vierbeins is given by
and all the path integration variables transform as
This transformation law of $(x) is determined by the weight factors in eqn (7.9). Note that we do not integrate over the vierbeins e^k (x) .
128
THE WEYL ANOMALY AND RENORMALIZATION GROUP
One can understand that the measure (7.11) is free of anomalies under the general coordinate transformation (7.19) by expanding the variable Aa(x) as
and then the Jacobian vanishes
irrespective of the choice of the basis sets {4>an(x}}- For the complex fermioiis ij) and -il>, one expands
and the Jacobians for these two variables added together vanish
irrespective of the choice of the basis sets {ipn(x)}. (In a chiral theory which could contain a genuine gravitational anomaly, this simple argument fails since one generally expands ip and ?/) in terms of different complete sets in such a theory if one follows the calculational scheme of the covariant anomalies.) The identity for the above general coordinate transformation (7.19) is given
by
The first equality in this expression shows that the value of the integral is independent of the naming of integration variables, and the transition from the first line to the last line expresses the fact that the path integral measure is anomaly-free and the action is invariant under the coordinate transformation. We first note the relation
IDENTITIES RELATED TO COORDINATE TRANSFORMATIONS
129
by using the metric condition in eqn (B.I7) and by noting that the tensor T^v is not symmetric in general. When one combines this relation with $'(:/;) = $(:/;) + £"c^$(a;) + (l/2)(d^)$(a;) in the last two lines of the identity (7.24), one obtains the identity
We here collected only the terms linear in the infinitesimal parameter £"(x) and then differentiated those terms with respect to £"(x). If one takes the flat space-time limit in the identity (7.26) and uses the renormalized fields $(#) —>• \/~Z$ r(x) with the wave function renormalization factor Z in the case of renormalizablc theory, one obtains an identity for the renormali/ed Green's functions
where we used the fact that the spin connection A%® (x) vanishes in the flat spacetime limit. Since the right-hand side of this relation is finite in renormalizable theory, we learn that the energy-momentum tensor we defined gives a finite operator. A concrete expression of this finite T^ will be given later in cqn (7.35). When one multiplies by xv on both sides of the above renormalized identity (7.27) and integrates over x. one obtains
which shows that the trace of the energy-momentum tensor generates a scale transformation of space-time coordinates. The renormalized Green's function
130
THE WEYL ANOMALY AND RENORMALIZATION GROUP
satisfies the following relation, which is based on a dimensional analysis in renormalizable theory.
where n stands for a new mass parameter specifying the renormalization point. m stands for the renorrrialized fermion mass, and N/ and NA, respectively, show the numbers of fermion fields and gauge fields appearing in the Green's function with N = Nf + NA- One thus concludes
The identity (7.28) for the renormalized Green's function is then written as
The renormalization group equation for the renormalized Green's function is derived from the fact that the bare Green's function does not depend on the renormalization point [i
and it is written by using the renormalized coupling constant g as
By using the relations (7.33), the identity (7.31) is also written in the form
The function I3(g) which appears inside DN in eqn (7.33) is a fundamental quantity in field theory and it is called the /? function. We emphasize that eqri (7.34) is an exact relation valid up to any finite order in perturbation theory.
IDENTITIES RELATED TO COORDINATE TRANSFORMATIONS
131
7.3.1 Improved energy-momentum tensor We here give an explicit form of the improved energy-momentum tensor for a QED-type theory in eqn (7.5), which is finite in reiiormalizable theory as we explained in eqn (7.27), in the flat space-time limit
where !),„ is the conventional naive energy-momentum tensor. The difference between 2)tJ, and T^v arises from the fact that we defined the tensor Tltv(x) by differentiating the action S with respect to the vierbeins by keeping the variables $(z) with the weight, factors fixed. One recognizes that the difference is proportional to the equations of motion. When one writes the identity for the change of the variable <&'(x) — $(x) + a(x)$(x), one obtains
where J(a) is the Jacobian for the change of the variable, and the difference of two tensors is thus basically the difference of (Weyl) anomalies. A characteristic feature of this improved tensor in eqn (7.35) is that, the mass term does not appear explicitly. If one recalls the local Lorentz transformation of the vierbeins parametrized by an infinitesimal anti-symmetric parameter ujmn(x)
and uses the definition of the energy-momentum tensor in eqn (7.14) (see also Appendix B), one can see that the anti-symmetric part of the energy-momentum
132
THE WEYL ANOMALY AND RENORMALIZATION GROUP
tensor generates the Lorentz rotation.38 Consequently, if one separates the symmetric and anti-symmetric components as
the symmetric part TS^V satisfies the identity
which explicitly separates the Lorentz rotation of $(xn) generated by S1'1 „(«,) from the identity (7.27).
7.4 Weyl anomalies and j3 functions in QED and QCD In Section 7.2 the identity containing / d ^ x T I J - ^ ( x ) was derived from the identity for the Weyl symmetry which contains the Weyl anomaly, and the identity which contains the same object / d 4 x T ^ ^ ( x ) was also derived from the identity for general coordinate transformations in Section 7.3 which is anomaly-free. By combining these two identities (7.16) and (7.34) we obtain
where the symbol N[ip^](x) defines a quantity called the normal product, which gives a finite result when inserted in any Green's function without inducing any divergence. The relation moil}ip(x) = mN[ipii>](x) is known to be valid in QED and QCD-type theories. The index c in [J(l) + J d'ixmoipip(x)]c indicates a connected component, namely, one retains only those components connected by the propagator to other variables when inserted in any Green's function. To express the right-hand side of the equation (7.41) in terms of operators, we write the action in terms of renormalized variables (by including the coupling 38
If one transforms the variables <S>(x) as *'(xj) = e.xp[^Smn(l)ujmn(x}l^(xi) as eqn (7.37), one obtains the identity
simultaneously
where TA^^X) stands for the anti-symmetric part of the energy-momentum tensor. We used the fact that the localized Lorentz transformation is anomaly-free in the present model; the Lorentz anomaly for a chiral theory will be explained in a later chapter. This Lorentz identity combined with eqn (7.27) leads to eqn (7.40).
WEYL ANOMALIES AND /3 FUNCTIONS IN QED AND QCD
133
constant into field variables as g0A^(x) -*• A/_l(x) and rewriting the gauge fixing term anew in terms of A(x)
where we omitted the subscript "r" to field variables. By using Schwinger's action principle (in a renormalized form) we can operate the derivatives with respect to g and m in the identity (7.41), which was derived by using T^, directly on the action in the path integral arid we obtain
This relation, which was derived here in a heuristic manner, is known to be justified up to any finite order in perturbation theory by a more detailed analysis in the framework known as the normal product algorithm.39 Accepting the result of this analysis, we can identify the Weyl anomaly in eqn (7.41) with the operators containing field variables as
We derived this relation for the QED action in eqn (7.42), but the same analysis is known to be valid for QCD-type theories also. The relation (7.44) shows that the part of the Wcyl Jacobian J(l) a that depends on the gauge field gives rise to a term containing the j3 function, and the part of the Jacobian that depends on the ferinioiiic field gives the term with the mass renormalization factor S(g), although we cannot evaluate Weyl anomalies exactly. In the following we illustrate the oneloop level evaluation of the 0 function by calculating the Wreyl anomaly in QED and QCD-type theories. 39
See S.L. Adlcr, J.C. Collins and A. Duncan, Phys. Rev. D 15 (1977) 1712; N.K. Nielsen, Nucl. Phys. B 120 (1977) 212.
134 7.4.1
THE WEYL ANOMALY AND RENORMALIZATION GROUP Lowest-order 13 function in QED
The Weyl anomaly to one-loop accuracy is evaluated by the background field method by separating the field variables into the background c-number parts and the quantum fluctuations around the c-number parts. In the following we evaluate only those terms which depend on the background gauge field and thus we obtain the one-loop j3 function.'10 We start with the analysis of QED. The term which contains the field variables higher than quadratic order appears only in the fermionic action in QED, and thus the lowest-order 8 function is given by the evaluation of the Weyl anomaly generated by fermions in the presence of the background gauge field. This calculation of the Weyl anomaly proceeds just like the calculation of the chiral anomaly in Section 5. We first expand the path integral variables in terms of Grassmann numbers as (in the Euclidean metric)
with the notation goA^(x) —>• Atl(x), and we define the path integral measure by
The Jacobian for the Weyl transformation in the flat space-time limit 41
for an infinitesimal a(x) is given by the gauge invariant mode cut-off regularization (the variables tb and tb give the same Jacobian, and an extra minus sign arises from the fact that these variables are Grassmann numbers) 40
The evaluation of the Jacobian with the c-number fermionic parts requires an evaluation of the "supertrace." Such a calculation can be performed systematically in the superfield technique in the theory with supersymmetry. We are here content with the calculation only with the gauge field background, which is sufficient to give rise to the one-loop /? function in QED and QCD. 11 The identity associated with the Weyl transformation of the fermionic variables can be written in a form similar to the chiral identity as | (1/117'*D^ 1/1 — (Df,.ip)i"t^ip) — (l/247T2)-FA,l, where Ihe right-hand side stands for the Weyl anomaly to be evaluated below.
WEYL ANOMALIES AND fl FUNCTIONS IN QED AND QCD
135
A more detailed account of this calculation is given later in cqn (7.54) again. Consequently, the connected components which contain the field variables are given by (using the result of eqn (7.56) below)
and the lowest-order 0 function in QED is obtained by using the relation as
which reproduces the well-known result.
7.4.2
Lowest-order 3 function in QCD
We next analyze QCD-type theories with the notation g^A^x) —?• A^(x)
where we denoted the bare coupling constant go explicitly by a suffix 0. The gauge fixing procedure is important in the present application and it is explained later. The calculation of fermion contributions to the lowest-order 13 function in QCD type theory is evaluated in the presence of the background gauge field, and the actual calculation is almost identical to the calculation in QED except for the appearance of the trace over the color freedom in the Jacobian.
136
THE WEYL ANOMALY AND RENORMALIZATION GROUP
We here examine the calculation of the Jacobian for the Weyl transformation of fermionic variables (7.48) in more detail in the actual case of QCD-type theories, so that we can apply the calculational technique to the calculation of gauge field contributions later. We start with the formula
where we rcscalcd /% ->• Mk^ in the last expression after passing the factor e%kx through the regulator arid tr here stands for the trace over the Dirac 7^ matrices and color indices. Only the terms of order 1/M4 or larger survive in the expansion in terms of 1/M. If one defines
with the generators Ta of the gauge group SU(n) normalized by tiTaTb — \8ab (in actual QCD we have SU(3)), one obtains
where we used the fact that terms odd in fcM vanish and that only F contains the
Dirac matrices and tr F = 0. The notation f^(x) stands for the n-th derivative of/(x). We next note that we can replace
when one performs a symmetric integration for the terms with k^. Also, by noting k2 < 0 in the present metric convention we have
WEYL ANOMALIES AND 13 FUNCTIONS IN QED AND QCD
137
We thus have
and the connected component, which is finite, is finally given by using [D/^, Da] = -iFlllr as
The important property in this calculation is that the finite connected component does not depend on the details of the regulator function f ( x ) except for the fact that f ( x ) approaches 0 rapidly for large x and /(O) = 1 together with (see Fig. 5.1)
In QCD-type theories the gauge field also contributes to the lowest-order /? function. The Yang-Mills Lagrangian is given by
and we split the field variables as
138
THE WEYL ANOMALY AND RENORMALIZATION GROUP
to apply the background field method. Here jE?° stands for the background field, and a™ for the quantum fluctuation around it. The field strength tensor is then given by where the covariant derivative is defined by
The gauge transformation of the gauge field parametrized by uia(x) is defined by
by recalling that a^ is the dynamical variable. If one uses the above field strength tensor in the action and assumes that the variable B1^ formally satisfies the field equation (this is the basic ingredient of the background field method), the action is written as
since the terms linear in a^ disappear. To simplify the notation and also the evaluation of the Jacobian later, we write the covariant derivative as
and regard the gauge field a M (x) as a vector with n 2 — 1 components for the group SU(n). The operation of the covariant derivative in eqn (7.63) is then written in the simple form
As for the gauge fixing term and the compensating Faddeev-Popov term, we employ the background Feynman gauge with £ = 1 (the BRST transformation of the gauge field atl(x) is determined by eqn (7.64) by the replacement <jja(x) ->• i\ca(x))
where we used vector notation for the ghost variables c and c just as for a M , and eliminated the auxiliary field B by integrating over it in the path integral.
WEYL ANOMALIES AND /3 FUNCTIONS IN QED AND QCD
139
The Lagrangian for the quadratic parts in the dynamical variables related to the Yang-Mills field and ghost variables is obtained from cqns (7.65) and (7.68) as
where we used the notation
by using eqn (7.66). We now evaluate the Weyl anomaly by using this quadratic Lagraiigian. The Weyl transformation laws of integration variables are fixed as was explained in eqn (7.9)
in the flat space-time limit. Consequently, the connected components of the Jacobian for the Weyl transformation are given by the general formula (7.58), which was explained for fermion variables, by suitably adjusting the transformation coefficient and the operator appearing in the regulator. For the gauge field we have
where we used the notation (F)^ — —1iF^v(B) and replaced $92 —>• -DT>;, + F inside the regulator, which formally has the same form as in eqn (7.52). In the third line in this calculation we evaluated the trace with respect to the Minkowski indices of a^ and then evaluated the trace over the adjoint representation of the gauge group by using
For the group SU(n) we have C2(G) = n.
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THE WEYL ANOMALY AND RENORMALIZATTON GROUP
Similarly, the Weyl anomaly for the ghost variables is evaluated as (by replacing f)2 -t D^DH inside the regulator in eqn (7.58))
where we used the fact that the ghosts are described by Grassmann variables and thus there is an extra minus sign in the Jacobian. One may replace the background field B^ (x) by the full variable A/( (x) in the Jacobians evaluated so far to the accuracy of the present one-loop approximation. If one adds those results of the Wcyl anomalies in eqns (7.58). (7.72) and (7.74), the /3 function for a QCD-type theory with / fermions is given by using eqn (7.44) as
namely, one recovers the well-known result of the lowest-order 3 function
where 62 (G) = n for the gauge group SU(n). Note that J3(g) < 0 and thus the theory is asymptotically free for / < 33/2 for SU(3). 7.5
The Weyl anomaly in curved space-time
We illustrate a typical example of the Weyl anomaly, which appears in the background gravitational field, by taking QED as an example. Actual calculations of the Wcyl anomaly rely on the technically involved regularization schemes such as the heat kernel method and the C function regularization. We here simply state the final results. We employ the action in the Euclidean metric defined by the Feynman gauge
We first analyze the Weyl anomaly generated by quantized fermion fields whose Weyl transformation rules are given by
THE WEYL ANOMALY IN CURVED SPACE-TIME
141
We expand the fermionic variables as
and the Jacobian for the Weyl transformation is given by (by writing a sum of '(/.' and -0 contributions, which are Grassmann numbers)
where we used the gauge invariant mode cut-off regularization, and the final expression uses the common notation in the heat kernel regularization. This calculation is performed by using a formula which appears in the intermediate stage of the C function regularization. and the Weyl anomaly is obtained as42
In this expression we write only the gravitational contributions. The fermions couple to the gauge field and thus the Weyl anomaly actually contains the contributions from the gauge field which were discussed in the previous section. The quartic divergent term ^/gM4 . which corresponds to the cosmological constant, could be subtracted by a suitable local counter-term. It is known that 42 The calculational scheme by converting the complete set to plane waves is not convenient here since we have to take into account the effects of the curved space-time. The heat kernel or £ function regularization provides a more systematic method to evaluate eqn (7.80).
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the quadratically divergent term ^/gRM2 disappears in the final result of the C function regularization. We next note that the action for the gauge field and Faddeev-Popov ghosts in eqn (7.77) is written as
where Aa(x) — e^(x)AIJi(x) and the covariant derivative acting on the gauge field is defined by
as explained in Appendix B. The Weyl transformation rules of the variables which define the general coordinate invariant measure are given by (see eqn (7.9))
and we expand the variables as
arid
The Jacobiaii for the Weyl transformation of the gauge field is
THE WEYL ANOMALY IN CURVED SPACE-TIME
143
\uJA(a)
where tr stands for the trace over the vector indices of the gauge field, and the Jacobian for the ghosts is (by recalling the fact that the ghost fields are Grassmann numbers)
These results are based on the formulas which appear in the intermediate stage of the £ function regularization. We omitted the cosmological term ^/TjM4. which could be subtracted by a suitable counter-term, and the term ^/gRM2 which is eliminated in the final result of the £ function regularization. The final contributions of the gauge field and the ghost fields to the Jacobian are thus given by
144
THE WEYL ANOMALY AND RENORMALIZATION GROUP
It is known that, the Wcyl anomalies (7.81) and (7.89) agree with the results of other calculational methods. 7.6
The Weyl anomaly in two-dimensional space-time
The Weyl anomaly in Euclidean two-dimensional curved space-time is important in connection with conformal field theory. In this section we discuss the Weyl anomaly generated by massless scalar and fermion fields. We employ the conformal gauge for the metric variables, which is convenient for practical calculations. This gauge condition is specific to two-dimensional space-time and is defined by
We define the Euclidean metric by a Wick rotation from the Lorentz metric as Gmn = (-!,-!) = ??M,,. In this gauge the action for the scalar field X(x) is written as
and the action for the Dirac field is written as (by using the definition of the spin connection A^/, in Appendix B)
These expressions of the action show that the scalar theory is invariant under the Weyl transformation
THE WEYL ANOMALY IN TWO-DIMENSIONAL SPACE-TIME
145
and the Dirac theory is invariant under the Weyl transformation
These classical symmetries are broken by the quantum anomaly if one uses the path integral measure which preserves general coordinate invariance. If one defines the path integral measure, which is invariant under the general coordinate transformation, by using the weight 1/2 variables X(x) = (g)l^X(x) = ^fpX(x) and il! (x) = (g)1^4"4>(x) = \fp'4!(x] the path integrals are respectively written as
The complete orthonormal sets which define the path integral measure in the gauge invariant mode cut-off are then respectively given by
The Jacobians (quantum anomaly) for the Weyl transformations
arc, respectively, given by
and
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THE WEYL ANOMALY AND RENORMALIZATION GROUP
where the symbol tr stands for the trace over the (two-component) spinor freedom. Since one can write #0/2) = (If p * - / * ) f i ( l / p l / z ) $ ( l / p 1 / * ) in eqn (7.96), the Jacobians for the scalar and Dirac fields are evaluated bv
if one sets n = 0 or n = -1/2, respectively, after the calculation and adjusting the factor 2 arising from the trace over 7'* matrices. The evaluations of Weyl anomalies are performed by
where the overall factor 2 arises from the trace over the spinor freedom which should be omitted for the scalar field. The actual calculation of the Jacobian proceeds as follows: We define
and we expand
OTHER APPLICATIONS OF WEYL ANOMALIES
147
in powers of 1/M to the second order by imitating the calculation in interaction picture perturbation theory. In this calculation
is obtained from Hj in cqn (7.102) by the replacement
When one uses the above expansion in
and uses the relations (by noting k2 < 0 in our metric convention)
together with the fact that terms odd in k^ vanish, one obtains the final result (7.101). We thus obtain the Weyl anomalies for massless fields in two-dimensional space-time from eqns (7.98). (7.99) and (7.101) as
where we used ^fg = p and ^/gR = —d^d^lnp with R standing for the scalar curvature. The term ^/gM'2 could be subtracted (or made finite) if one adds a suitable cosmological constant to the starting action as a counter-term. 7.7
Other applications of Weyl anomalies
Other important applications of the calculational method of Weyl anomalies appear in theories with supersymmetry (boson fermion symmetry), though we do not discuss supersymmetry in this book. The first example is called the Konishi anomaly, which is regarded as a supersymmetric generalization of the Weyl anomaly. (The Konishi anomaly is also regarded as a supersymmetric generalization of the chiral anomaly.) The other example is a generalization of the calculation of the ,3 function discussed in this book to supersymmetric theories. A characteristic feature of supersymmetric theories is that renormalization properties are greatly improved compared to ordinary theory (the non-renormalization
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THE WEYL ANOMALY AND RENORMALIZATION GROUP
theorem and its generalization). As a result one can evaluate quantum corrections up to all orders in perturbation theory in some special cases. It is known that the present calculational method is generalized to supcrsymmetric theories defined in terms of superfields. which ensures manifest supersyrnmetry, and interesting results have been obtained. For details of these calculations, readers are referred to the references at the end of the present book.
8
TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION In this chapter we discuss the quantum anomalies in two-dimensional field theory. Two-dimensional field theory is important in connections with conformal field theory and its applications to string theory and condensed matter theory. We first summarize essentially all the steps in evaluating the chiral anomalies in the path integral formulation so that a reader who starts with this chapter can understand the subject without excessive reference to earlier chapters. We then formulate the description of fermionic theory in terms of bosonic theory, namely, the bosonization in the path integral formulation. An issue related to a local counter-term is clarified. We next explain that the central extensions in KacMoody and Virasoro algebras are the algebraic representations of chiral and general coordinate anomalies, respectively. We also explain the connection of the identities written in terms of the operator product expansion in conformal field theory with the identities in conventional field theory. Finally, we discuss the calculations! method of Wcyl anomalies in string theory and its implications. The ghost number anomaly in the first quantization of string theory is related to the Riemann-Roch theorem. 8.1
Chiral anomalies in two-dimensional theory
We start with the fermionic path integral of a theory which contains Abelian U(l) vector V^(x) and axial-vector Afl(x) gauge fields
where we regard Vtl(x) and A^x) as the background c-number fields for the moment. We adopt natural units by setting H = c — 1. Our Minkowski metric conventions are
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
where ak stands for the Pauli matrices arid r]IJ/v = rf" — (1,— 1). We adopt the convention of summing over the indices which appear twice in the same expression. The actual path integral is performed in the Euclidean metric convention with imaginary time. The Euclidean theory is defined by the Wick rotation
and x2 is treated as a real number. The Euclidean inner product is then defined bv and the metric tensor is now given by 'r\^v ~ r^iv = (—1, —1). The 7-matrices are defined in Euclidean theory by the replacement 7° —>• —/'7 2 .
and they are aiiti-hermitian (7''')^ = ~7 // but 7? = 75. We also defined the Wick rotation of the gauge fields
so that Tf) retains the same form as in the Minkowski metric. The Dirac conjugate of il>(x) is defined by ip(x) = t/J^(x)j°, but in the path integral these two quantities are treated as independent variables. In Euclidean theory, ip(x) is transformed in the same manner as the spinor ['(z)]t under the Euclidean rotation group SO (2). The Dirac operator in Euclidean theory f) = 7^(3,, — iVp. — iApjs) is not hermitian in the Euclidean inner product
To define a hermitian operator, which renders the path integral manipulation more reliable, we rotate all the components of the axial gauge field in Euclidean theory into pure imaginary variables
and rotate them back to the original field A^ —?> —iA^ after the calculation. The operator after the rotation of A/L
CHIRAL ANOMALIES IN TWO-DIMENSIONAL THEORY
151
becomes hermitian ffi = $ in the Euclidean sense, and one can define a complete orthonormal set and the expansion of fermionic variables
where {an} and {&„} are the Grassmann numbers. The path integral measure is then written as
and we define the path integral by
This path integral is performed exactly though in a formal sense. The integration with respect to Grassmann numbers is defined by the left derivative. We now examine the Jacobian under gauge transformations. Under an infinitesimal vector-like transformation
we have by remembering the expansion (8.10)
The Jacobian for the transformation of {an}, for example, is evaluated by
for an infinitesimal a(x) by remembering that the integral over the Grassmann numbers is defined by the left derivative and thus the inverse of the ordinary Jacobian appears. Similarly, one has
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
We thus define the total Jacobian by
which vanishes identically, and thus we have no anomaly for the vector-like transformation in the present formulation. On the other hand, for the infinitesimal chiral transformation43 generated by 7s,
we obtain the Jacobian following a procedure similar to the vector-like transformation as
This Jacobian is evaluated by the gauge invariant mode cut-off,44 namely, by replacing the cut-off in N by the eigenvalue cut-off with a simple regulator / (x) = e.~\
where in the last line we transformed {(pn(x)} to the plane wave basis for the well-defined operator 75 exp(—$> 2 /M 2 ) to extract the gauge field dependence, and the trace is taken over the freedom of Dirac indices. 43
In accord with eqn (8.8) one may also set a(x) —} io(x) in the chiral transformation, but we forgo this refinement. 1/1 Actually, we preserve only the vector-like gauge symmetry for the present case.
CHIRAL ANOMALIES IN TWO-DIMENSIONAL THEORY
153
This Jacobian is evaluated as follows: When one defines D,,, = d^ — iV^, as a covariant derivative with only the vector-like gauge field (by noting if = (—1, —1) in Euclidean metric) we have
where the differential operator in (d^A^) acts only on A^ and F^ = d^v—dvV^,. If one recalls the definitions of 75 = cr3, 71 = iu'2 and 72 = ial in terms of Pauli matrices in Euclidean theory and the relation
one finds that the last two terms in eqn (8.21) do not contain the 7 matrices and thus do not contribute to the Jacobian (8.20) in d = 2 dimensions. Consequently, one can use
in the calculation of the Jacobian (8.20). In this way one can evaluate the integrable form of anomaly
by applying the scale transformation k^ -» Mfe,( after passing e*fc:£ through the regulator as in four-dimensional theory and then retaining the terms to order 1/M2 in the expansion of the exponential factor in powers of 1/M. We also used
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
by recalling that fc2 < 0 in our metric convention. When one first goes back to the original variable A^ -» —iA^ in eqn (8.5) in the Euclidean metric, and then if one Wick rotates back to the Minkowski metric by noting the convention of eM" in the Minkowski metric
the Jacobian (8.24) is finally written as
for the theory (8.1) defined in the Minkowski metric. We here also used
with YH = V®Ta and A/( = A^Ta, the Jacobian (8.27) is generalized to
where tr stands for the trace over the generators T° of the group SU(n), for example. The generators are defined by
We also defined
in eqn (8.29). The calculation of eqn (8.29) is essentially the same as eqn (8.27), and the expression (8.23) in Abelian theory is replaced by
in non-Abelian theory with the definition D^ = d^ — iV*Ta. The trace in eqn (8.20) is then taken over the freedom of the non-Abclian generators arid Dirac matrices by using a(x) = aa(x)Ta.
CHIRAL ANOMALIES IN TWO-DIMENSIONAL THEORY 8.1.1
155
Local counter-term in gauge theory
Prom the viewpoint of the general analysis of the anomaly in gauge theory, the first term of the Jacobian In J(a), namely, the term (d^A^) in Abelian theory (8.27) or (D^A^) in non-Abelian theory (8.29) could be cancelled if one adds a suitable mass term A^A1* for the gauge field A^ in the starting Lagrangian. We illustrate this property for the Abcliari theory (8.1) by adding a mass term to the Lagrangian
In fact we have
where the second equality states that the path integral is independent of the naming of path integral variables, arid we set tj)'(x) = exp[ia(x)75]'i/>(2;) and tp'(x) = i,b(x) exp[ia(a;)75] for an infinitesimal a(x). The Jacobian for this transformation is given by our result in eqn (8.27) and thus the first term in the Jacobian is in fact cancelled by the local counter-term. It is customary not to assign a physical significance to these terms cancelled by a local counter-term in the ordinary treatment of the anomaly. However, those terms do not diverge in two-dimensional theory and they play a basic role by giving the kinetic terms for boson fields in the path integral bosonization to be discussed below. In fact, we shall later show that those terms are not cancelled by a local counter-term in the context of the path integral bosonization. The usual argument of the local counter-term thus needs to be carefully examined depending on the physical context one is studying. Another important issue in the path integral bosonizatiori is the symmetry relation with the anti-symmetric symbol e1*", eM" — —(v>i and e10 = 1,
156
TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
which is valid in strictly d — 2 dimensional Minkowski space-time. This relation is used in an essential way inside the action of the path integral. One thus has to preserve the corresponding relation in the current level
for the sake of consistency. This relation is generally satisfied if one regularizes the currents in terms of the regularized correlation function (T*i^(x)ip(y))ieg in d = 2 dimensions. The representative regularization schemes of the Jacobian in the path integral, namely, the V-A-type regularization presented in Section 8.1 and the covariant regularization discussed in Section 6.4, both satisfy the above condition (8.36). As for the covariant current associated with the covariant form of anomalies, we already explained that it is defined in terms of the regularized correlation functions in Section 4.3 and also in eqn (6.87). As for the V-A-type regularization in eqns (8.17) and (8.20), it is related to the regularized currents in the Euclidean metric defined by
and similarly
where we used the complete set { i f n } in eqn (8.10). The divergence of the current (8.37) gives
ABELIAN BOSONIZATION OF FERMIONS
157
where we used the fact that 7^ is anti-herniitian (7^)^ = —7 M and 7^75 + 757'* = 0 and that tf)ipn = Xn
We can thus correctly generate the anomalies identical to the Jacobians from the regularized currents. For the regularized currents defined in eqn (8.37) and eqn (8.38), the relation (8.36) or
in the Euclidean metric is trivially satisfied. In the path integral formulation, the V-A-type regularization satisfies the Wess-Zumino integrability (or consistency) condition45 and also the relation (8.36). We thus adopt the V-A scheme for the path integral bosonization. Besides, the V-A-typc regularization preserves the fermion number symmetry as is seen in eqn (8.17), which is essential in the bosonization since the fermion number should be well defined by definition when one talks about the bosonization of a fermion theory. 8.2
Abelian bosonization of fermions
We diseuss the bosonization related to Abelian gauge transformations. We can learn the basic idea of bosonization by the analyses of these simple models. There are various versions of bosonization, and we here discuss three representative examples. 4 °The Wcss-Zumino condition is related to Bose symmetry in the Feynman diagram language. The Bosc symmetry in this sense is not sufficient for the path integral bosonization. For example, it is shown that the currents related to the consistent anomaly defined in Section 6.6 do not satisfy the relation (8.36).
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
8.2.1 Bosonization of free fermions We study the path integral
where the path integral measure is specified by the regularization in Section 8.1. We next define a generating functional of connected Green's functions W(v^) by
by regarding v^x) as a source function. In this setting any local counter-term should be expressed as a local polynomial in v^(x). Now we observe that the vector field v^x) in two-dimensional space-time contains two independent components and it can be decomposed into two arbitrary real functions n and £ as
After this decomposition, we can write cqn (8.43) as
where we used eqn (8.35), £^7''' = 71/75, which is valid in a strictly d = 2 Miiikowski metric. This path integral has formally the same form as eqn (8.42) if one denotes Vp, = d^r) and A/,, = <9^£. We extract the functions r; and £ as integrated Jacobians associated with the transformations of integration variables ip and il>. Using the results of Section 8.1 for the infinitesimal transformations
we have the Jacobians in eqn
where Vp, = d^rj and AIL = d^. Using these Jacobians, the r\ and £, dependences in the action arc extracted as
where F(w |U ) stands for the integrated Jacobian (or anomaly), and it has the form in the Minkowski metric
ABELIAN BOSONIZATION OF FERMIONS
159
The derivation of this integrated Jacobian proceeds as follows: One first eliminates the component r)(x) by a vector-like transformation, which is anomaly-free and thus without any Jacobian. One then performs an infinitesimal transformation parametrized by dn£(x) in the intermediate result where the axial-vector component d^(x] is partially extracted and given by AIL = (1 - s)9 /t ^(x) arid Vp — 0 in the general formula of the Jacobian (8.47). One then obtains the above formula r(i^) by integrating over the parameter s of infinitesimal transformations from 0 to 1. We can also rewrite eqn (8.48), where the fermionic path integral gives a numerical constant, as
since the absolute normalization of the path integral does not matter in the definition of W(v^. We next shift the variable
In writing the second line, we used the relation
and in the last line we rescaled if -> ^/TT^ so that the kinetic term of
By differentiating the generating functional with respect to the source field Vp,(x) twice and then setting v/Ji(x) — 0, we obtain
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
which shows the equivalence of the correlation functions described by a Dirac fermion and a real boson in d = 2 dimensional space-time. . The part of the Jacobian d^A^ in eqn (8.27). which could be subtracted by a local counter-term l/(27r)^ in the context of gauge theory in eqn (8.33), plays a central role in the above bosonization. In fact it gives the kinetic term of the bosonized field. We examine this connter-term in more detail. The term i/(2ir)A^l, which is local in terms of the axial vector field A^ appearing in eqn (8.33), is actually not local in the context of eqn (8.43). The local counter-term in the.context of eqn (8.43) should be expressed as a local polynomial of the source field v^(x). as we have emphasized there. The would-be counter-term (in the context of gauge theory) is written in terms of v^ as
which is not local. It is thus not allowed to add this term as a counter-term to the definition of the original partition function (8.43). This term, if added, modifies the physical contents of the original fermionic theory even in the conventional understanding of local counter-terms. 8.2.2
The massless Thirring model
We study a theory of interacting fermions, which is called the massless Thirring model,
where the first representation is obtained from the second one when one eliminates the auxiliary field B^ by using the equation of motion. The path integral formula is given by
ABELIAN BOSONIZATION OF FERMIONS
161
where we used the decomposition generally valid in two-dimensional space-time
by using the relation (8.35). We also used
which is a generalization of the derivation of the volume element ^fgtf'x from the formula for the length (line element) ds2 = g^lv dx^dxv, and discarded the constant factor det(—d^d^). We first eliminate the coupling between p and $ in eqn (8.57) by a change of variables corresponding to a vector-like gauge transformation
which do not give a non-trivial Jacobian. We next perform the change of variables corresponding to a chiral gauge transformation
which eliminates the coupling between the fcrmion and <j>, but we obtain a nontrivial Jacobiaii
The evaluation of this Jacobian proceeds just as the calculation (8.49) in the preceding subsection. In this way, the path integral is written as
which shows that the interacting theory of the i'ermion *& is rewritten as a sum of the theories of a free Dirac fermion tjj and two real scalar particles p and d>. But the variable rp corresponds to a theory with negative metric.
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
By using the above result we can, for example, write the propagator (or correlation function) for the original variable in eqn (8.56)
as
The Green's function which satisfies
in the minkowski metric is given by
where p, is an arbitrary parameter with a unit mass dimension. This expression when combined with
gives a correlation function of a free fermion (by using
The calculation for the field <j> by adding a source function J(x)
gives (by choosing J ( x ) with a (^-functional peak at x and y such that J dx J ( z ) x (j)(z) = 4>(x) — o(y) and by noting that the combinations x and y appear twice)
ABELIAN BOSONIZATION OF FERMIONS
163
where z is a cut-off dependent constant arising from the propagator at the coincident point. Similarly we obtain (by a replacement 1 + g2 fir ->• -1 in the exponential factor for the case of 6(x) in eqn (8.71))
where z' is a cut-off dependent constant. We can thus derive the exact correlation function for the interacting *
after absorbing the cut-off dependent factor (zz1)2 into the left-hand side. Note that the ju-deperidence on the right-hand side of eqn (8.73) carries information about the anomalous dimension of ^S'. More general 2n-point functions are similarly evaluated exactly. 8.2.3
The massive Thirring 'model and the. sine-Gordon model
We now discuss the equivalence between the massive Thirring model and the so-called sine-Gordon model. This analysis is a combination of the analyses in Subsections 8.2.1 and 8.2.2. The massive Thirring model is defined by the Lagrangian
and the sine-Gordon model is defined by
where ao, /3 and 70 are real parameters. We start with the generating functional of connected Green's functions for the massive Thirring model
where we added the source term v/^ (x). We now use the decomposition
where we used eqn (8.35). The path integral (8.76) is then written in the form of eqn (8.57) as
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TWO-DIMENSIONAL FIELD THEORY AND BOSOMZATION
We first apply a vector-like transformation to the fermion variables
which eliminates the couplings of p(x) and r/(x) to the fermion without generating a non-trivial Jacobiari. We next perform a chiral transformation
which eliminates the couplings of 4>(x) and £(x) to the fermion in the kinetic term but generates a Jacobian
The derivation of this integrated Jacobian proceeds just as in eqri (8.49). In this way, the generating functional of connected Green's functions is written as
We next define and rewrite the path integral as
by noting that T>[
ABELIAN BOSONIZATION OF FERMIONS
165
This formula is further modified to
where we used / d2xd^d11'^ = - f d2xvfte^"dIJif of p since it gives a constant. Finally, we define
and dropped the path integral
and write the path integral as
which gives a basis of our analysis. We now fix the variable (p and perform the path integral over the remaining variables. We first observe that the mass term can be written as
with I!JR,L = [(1 ± 7s)/2]V ; > an(l expand the path integral (8.87) in powers in the mass term.46 Since the massless fermion is chiral invariant and also the Lagrangian for the massless 0 is invariant under the translation in <•/>, only the following terms survive in the expansion
where the numerical coefficient l/(fc!) 2 appears from the binomial coefficient
We show below that the expression (8.89) is equal to 46 This perturbative treatment of the mass term is consistent -with our calculation of the Jacobian (8.27), which treats the fermion action without the mass term in eqn (8.74) as the basic action. Tf one treats the mass term non-perturbatively, one generally needs to include the scalar and pseudo-scalar fields in the basic operator (8.20) for the present V-A-type evaluation of the Jacobian.
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
where A stands for the ultraviolet cut-off to define the propagator for the scalar field 0 at the coincident point. The generating functional is thus written by following the reverse steps to the above and by noting that the free field part of (f is invariant under the translation ip —>
where we added an arbitrary constant 70- If one defines the variable (p with the standard normalization by
we have the generating functional expressed in terms of the sine-Gordon model
where the parameter 0 is given by
and ao/fi2 = Am 0 /7r. By comparing eqn (8.94) to eqn (8.76). we have the identification We thus recover the well-known bosonization rules. Note that the coupling constant g in some of the literature corresponds to our g2. We now briefly explain the transformation of eqn (8.89) to eqn (8.91). The correlation function of the scalar field <j> is given by
ABELIAN BOSONIZATION OF FERMIONS
167
for generic points \Xi} and {yj}- The constant A stands for the ultraviolet cut-off in the propagator for the coincident point. This result is obtained by a generalization of the calculation (8.70) with suitable adjustments of numerical coefficients. The free fernaion propagator is evaluated by the same procedure as in the path integral in Appendix A (A.22) (by using the correct normalization Z(0, 0) = 1)
where we used the explicit form of iSp(x - y ) = (T*if)(x)ip(y)) in eqri (8.69). To simplify the following calculation, we write the two components of the fermions explicitly
In this notation the result of the above path integral is written as
where we used the light-cone coordinates x± = XQ ± %i to be denned in eqns (8.106)-(8.108) below. We also used the 7-matrix convention in the Minkowski metric in eqn .(8.2). in particular, 75 = a^. In the present notation the source terms in eqn (8.98) are written as rjiV-'i + ^2^2 + Wi7?! + 'ta'te, an(i thus the correlation functions of component fields arc written as
In this notation we also have
We then evaluate
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
where the determinant is defined for the k x k matrix specified by the indices i and j. We next apply the identity (Cauchy's lemma)47
to the two determinant factors in eqn (8.103). We thus obtain
where we have used the relation x+X- = x2 and introduced an arbitrary mass parameter //. By combining eqn (8.97) and cqn (8.105) in eqii (8.89), we finally establish the desired form of cqn (8.91). 8.3
Non-Abelian bosonization of fermion theory
To analyze the non-Abelian bosonization, we first briefly summarize the notation. We use the light-cone coordinates in Minkowski space-time defined by
47 This identity may be proved by examining the residues of poles of both sides and invoking the Liouville theorem. See. for example, M. Stone, Bosonization. World Scientific, Singapore (1994).
NON-ABELIAN BOSONIZATION OF FERMION THEORY
169
where our convention is e10 = 1. The following relations hold in these coordinates
and similarly
8.3.1
Non-Abelian bosonization of free fermions
We define the generating functional of Green's functions by
where the fermions belong to a representation of the group SU(n) and the source field v_(x) is Lie-algebra valued, v~(x) = vaL(x)Ta. In two-dimensional spacetime, the two independent components of WM can be written as
where h(x) = ha(x)Ta and l(x) = la(x)Ta. By using the above relations, the generating functional is written as
Formally, this path integral can be regarded as a non-Abelian chiral gauge theory in which vector and axial gauge fields are given by
In eqn (8.111), we. repeatedly apply an infinitesimal transformation of integration variables,
and integrate the resulting Jacobian by using eqn (8.29) with respect to the parameter s. In the intermediate steps, the gauge fields are given by
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
where This operation then extracts the v- (x) dependence from the action of the path integral. The result is
where
Various factors in this expression are given explicitly as
Thus we can also write
By using
the first term in eqn (8.119) is written as
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171
where only the value at a = 0 contributes. The second term in eqn (8.119) is written as
where we used d^U(h,s)^ = -U(h,,rfd,iU(h,s)U(h,s)^ which follows from dn(U*U) = 0. The last equation can be written in a symmetric way by introducing a three-dimensional notation
for eqn (8.115) and by using a totally anti-symmetric symbol eM"A with the normalization e102 = 1 as
where the integration domain D = disc indicates 0 < x2 = 1 — s < 1. We thus finally obtain where the Wcss-Zumino-Witteii action FWZW(^) is given by
It can be verified that Fwzw(£0 defined above satisfies the composition law
for general U = U(£). To show this relation we start with
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
where £/(£) = exp[-i£a (x)Ta]. We now consider the change of variables
and we obtain
Here iT = lnJ stands for an integrated Jacobian, and it is given by
The difference of zT(u_, f/(£)) from the previously evaluated iT(U(h)) (8.117) is that besides the replacement U(h) —> U(£) we ha,vc
instead of
We thus obtain from eqn
(8.119)
where the second term is independent of v- . The first term in this expression is written as
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173
where only the value at s = 0 contributes. On the other hand we have
which is a result of the Wess-Zumino integrability condition. We thus have
which gives the desired result (8.127) when written in terms of FWZWWe have shown in eqn (8.116) and eqn (8.125) that
since the absolute normalization of the path integral does not matter in the definition of W(v-). We can rewrite this relation by multiplying by the constant J 'DU exp[jTwzw(£0] on the right side as
We then change the integration variable as U —> U(h)U. By using the gauge invariance of the integration measure, T>(U(h)U) = DU, and the composition law in eqn (8.127), we recover the non-Abelian bosonization formula
Comparing eqn (8.139) to eqn (8.109) we obtain the bosonization rule of the free Dirac fermion theory
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
and the free fermion theory is translated into a non-linear a model with a WessZumino term. This bosonization of a fermion theory plays a fundamental role in conformal field theory in d = 2 dimensions. The first term of r\yzw(^ r (7 1 )) in eqn (8.126) comes from the first term in the anomaly in eqn (8.29). which could be subtracted by a local counter-term
in the context of gauge theory. The would-be counter-term (in the context of gauge theory) is not local in the context of the present bosonization. In fact, by noting the relation, which arises from the fact that U(h)^d^U(h) is a pure gauge and thus FIJ/V = 0,
the would-be counter-term is written in terms of V- as (by recalling A^ = (i/2)U(h^d,U(h))
which is non-local with regard to V-(x). The kinetic term in I\vzw(k r ) cannot be subtracted by a local counter-term. 8.4
Kac—Moody algebra and Virasoro algebra
We discuss some of the basic subjects in conformal field theory in d = 2 Euclidean space-time. We show that the central extensions of the Kac-Moody and Virasoro algebras are the algebraic representations of quantum anomalies. We also explain the connection between the identity expressed in terms of the operator product expansion, which is commonly used in conformal field theory, and the more conventional identity in field theory. 8.4.1
Kac-Moody Algebra
The starting Lagrangian of a fermion theory in two-dimensional Euclidean spacetime is given by
in the path integral we consider the chiral transformation for an infinitesimal aa (x)
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175
We then have the identity
where the first equality is the statement that the path integral itself is independent of the naming of the path integral variables, and in the last line we evaluate the Jacobian and the variation of the action. By noting tiTaTb = (l/2)<5a{,, we then obtain
where we used the result of the anomaly (8.29) with AM = 0 (written in Euclidean space-time). Similarly, the vector-like transformation
which is free of anomaly as was explained in Section 8.1, gives rise to the identity
We normalized e12 = 1 and defined the Nother currents
As a special property of two-dimensional theory, the vector and axial-vector currents (in the Euclidean metric) are related by
in the present path integral formulation. See eqn (8.41). When one functionally differentiates both sides of the identity (8.148), which contains the axial-vector current, with respect to V^(y) and then sets V°' = 0, one obtains
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
where jbv(y) appeared from the action in the exponential of the path integral. Note the definition
Similarly, one obtains from the identity with vector currents (8.150)
We now recall the Bjorken-Johnson-Low (BJL) prescription. This prescription tells us how to convert the time ordering in Lorentz covariarit calculations, which is denoted by T*, to the conventional time ordering operation T, which is not necessarily Lorentz covariant. To be explicit, if the relation
holds for arbitrary operators A(x) and B(y) (by taking into account the translatiorial invariance), one can replace it by
Intuitively, the canonical formulation, which is based on the equal-time commutator, should give rise to a smooth x2 (= ix°) dependence of the correlation function near x'2 ~ 0, and thus its Fourier transform vanishes for large ki (recall the Riernann-Lcbesgue lemma). If the Fourier transform does not vanish in the large momentum limit, we define the time ordering T by subtracting it
The subtraction term is generally a polynomial in k%. When we apply the BJL prescription to eqns (8.153) and (8.155), we obtain
The derivation of the second relation in eqn (8.159), for example, proceeds as follows: We have a time derivative of the (J-function on the right-hand side of
but such a term does not appear in the time derivative of the left-hand side if it is defined in terms of the T product, as can be confirmed by considering
KAOMOODY ALGEBRA AND VIRASORO ALGEBRA
177
the Fourier transform of both sides. Such a term on the right-hand side is thus eliminated when one converts the left-hand side to the T product. In contrast, the spatial derivative of a <5-function on the right-hand side does not contradict the T product on the left-hand side and thus it remains on the right-hand side of the first relation of eqn (8.159). The relation with the term dtl(Tja>1(x)jbl(y)} is in general inflicted with the Goto-Imamura- Schwinger term we mentioned in Chapter 6 and thus its analysis is involved, but a careful analysis shows that, it gives a relation equivalent to the first relation in eqn (8.159). Namely, the chiral anomaly and the Goto-Imamura-Scrrwinger term gives the same effect in the current algebra in d = 2 dimensions. The identities (8.159) are summarized in terms of operator language as48
where we used j b i ( y ) = ijf (y) and J 5 cl (z) = i j c 2 ( x ) , and d^ = 8^ = 0 for free fermions. We also used the fact that the time derivative acting on the T product symbol converts it into a 5-function, for example,
by noting When we define
eqri (8.161) gives rise to equal-time commutation relations
If one understands that these relations arc written for a theory defined on the surface of a cylinder
and if one defines for an integer n 48 Thc relations (8.159) are valid for the path integral with source terms for fermions added, namely, for general malrix elements. They are thus converted to the operator statements (8.161).
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
FIG. 8.1. Time arid space coordinates in the complex plane
in the first relation of eqn (8.164). for example, one obtains
which is called the Kac Moody algebra. The last c-rmrnber term which arises from the chiral anomaly gives a central extension of the algebra. 8.4.2
Method in conformal field theory
To compare the above result of the Kac-Moody algebra with the derivation in conformal field theory, we define the complex coordinates
in Euclidean theory. We also understand that, the above coordinates are in fact obtained from the coordinates (£1, C 2 ) on the cylinder, —TT < C1 < if and —oo < C2 < oo, by the projection
on the entire two-dimensional plane. Sec Fig. 8.1. The point £2 = — oo is then mapped to the origin and the point £2 = oo is mapped to z = oo, and the time development is described by the clock which proceeds along the radial direction starting from the origin (denoted by T in Fig. 8.1). The spatial freedom is then described by a circle with center at the origin (denoted by x in Fig. 8.1). This is called radial quantization. As a result, the field variable i/j(x), for example, is obtained from the variable on the cylinder by a projection, and thus its definition in general differs from the variable naively defined in two-dimensional theory.
KAC-MOODY ALGEBRA AND VIRASORO ALGEBRA
179
For general vector quantities, we define
When one remembers the definitions of Dirac matrices
the currents for fermions are written as (by noting ip = ^7°)
where i/>i, 02 are respectively the first and second components of the spinor ip(x). See also eqn (8.163). In conformal theory ip and ^ are treated as independent variables unlike the covariaiit formulation with the variables tp and ip so far in this chapter. As a special property of two-dimensional theory, the axial-vector and vector currents are directly related to each other by (see eqn (8.152))
In terms of the present notation, the identities (8.153) and (8.155) for j^ and jatt are written as (by using tzz = —*/2)
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
By considering the sum and difference of these "two identities we obtain
When one applies the BJL prescription to the present problem, the identities (8.175) are written as
In the first identity we regard z as the time variable, and we directly obtained the T product since there are no terms which diverge when the momentum conjugate to the coordinate z becomes large. In the second identity we regard z as the time variable, and the time derivative of the 5-function on the right-hand side is absorbed by the left-hand side when defining the T product. To relate those identities (8.176) in conventional field theory to the operator product expansion (OPE) in conformal field theory, we use the relation
which is explained in eqn (8.183) later and we write the first relation in eqn (8.176) as
where we used the fact that d=jcz(x) = dzjcz(x) — 0 holds for free fermions, as is seen in eqn (8.181) below. In general the relation dsF(z,z) = 0 implies that F(z.z) does not depend on z and F(z,z) has no poles in z. Consequently, we have the relation from eqn (8.178) which shows that both sides have the same pole structure
and similarly In conformal field theory we consider the Euclidean Lagrangian
KAC-MOODY ALGEBRA AND VIRASORO ALGEBRA
181
The equations of motion then imply dzipi = 0 and Q^i = 0. If one uses the complex coordinates z = (xl + ix2)/2, w — (yl + % 2 )/2 in the definition of the Green's function in Euclidean theory
one obtains the Green's function in two-dimensional theory without boundaries
When one combines eqns (8.181) and (8.183), one obtains
which is consistent with the Minkowski relation y)+] in eqn (8.100) if one remembers t/»i = (V-^7°)i = V4- If orie considers a contour integral with respect to z along an infinitesimal circle around, w in eqn (8.184), one obtains (since the T* product in eqn (8.184) is replaced by the T product by the BJL prescription)
One can convert this integral to a difference of two contour integrals along the circles \z = w ± f. around the origin z = 0 if one recalls that the correlation function has a singularity only at z = w. Since the Grassmann variables anticommute, one thus obtains in the limit c —¥ 0
which gives (a part of) the canonical quantization condition of the fermion. This relation (8.186) is generalized for two arbitrary operators A(x) and B(y) as
where ± corresponds to Fermi or Bose statistics. The function f ( z ) is a test function with possible singularities only at z = 0 and z = oo. One can thus
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
recover all the information about equal-time commutation relations once one knows the pole structure of the operator product expansion of two operators. We now come back to the current algebra for the current
By using (T^2(z)ipl(w)} = i/[4ir(z — w)] and counting all the possible combinations (or contractions) of tp and ip^, one obtains the operator product expansion
which agrees with the identity (8.179) in conventional field theory including the c-rmmber term. The symbol ~ implies that the pole structures of both sides are equal. A contour integral after multiplying the test function f ( z ) gives
if one defines
by choosing f ( z ) = zn, one recovers the Kac-Moody algebra
after multiplying eqri (8.190) by wm and contour integrating around the origin. 8.4.3 Virasoro algebra We show that the central extension of the Virasoro algebra is a manifestation of the quantum anomaly. The analysis is somewhat lengthy and involved, but our main purpose is to show that the central extension of the Virasoro algebra arises from the general coordinate anomaly when one defines a traceless energy-momentum tensor. We consider an action for a scalar field X(x) in twodimensional Euclidean curved space-time
which is invariant under the Weyl transformation
KAC-MOODY ALGEBRA AND VIRASORO ALGEBRA
183
Since X(x) is invariant under the Weyl transformation, the Weyl invariant path integral measure is denned in terms of X (x) and the path integral is given by
To specify the measure more definitely we define
and expand
The measure is then given by
When one defines (pn(x) = (g)l^tpn(x)
one has
The identity for general coordinate transformations
is given just as for other identities we discussed so far by
The first equality shows the fact that the path integral is independent of the naming of integration variables, and the last expression is obtained from the first expression by noting the invailance of the action and evaluating the Jacobian. The identity is thus obtained by comparing the last two expressions
where the energy-momentum tensor T^v is defined by keeping the integration variable X(x) fixed as
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
The tensor T^ thus defined satisfies the relation
identically in d = 2. The Jacobian is evaluated as
In the conformal gauge which is specified by g^ = p(x)r]IJ/v, we have
and the actual calculation of the Jacobian becomes identical to that of the Weyl anomaly for the scalar field in Section 7.6. For the general case with d scalar fields Xa(x), the anomaly is multiplied by d. In this way the identity is written as
where the covariant derivative is given by using the affine connection F^ given in Appendix B
We next derive the correlation functions of !),„. For this purpose we define an infinitesimal e^"(x) by
where ^"(x) is assumed to be symmetric and satisfies r/ MJ /e MZ/ (a;) = 0. (In the present discussion of the Virasoro algebra, we use e^v(x) to denote the infinitesimal deviation of the metric ffv from the flat metric. Elsewhere in this book e'i!/
KAC MOODY ALGEBRA AND VIRASORO ALGEBRA
185
is used for an anti-symmetric symbol.) In this case we have det g = 1 to accuracy linear in t^ (x]. To the same accuracy linear in ^ (x) we have
and we can write the energy-momentum tensor (8.203) as
where Tji,, is zeroth order in e MI/ (x). We can then write
From this equation we obtain (by writing T^J on the right-hand side simply by Tpv for notational simplicity after setting e^i,(x) = 0)
In the following equations, all TMl, stand for quantities zcroth order in efH/(.7;). By using the definitions of the Ricmanri-Christoffcl curvature tensor and the quantities Rpv = Rapal, and R = g^R^v in terms of the affine connection F$ (see Appendix B), we have
We next functionally differentiate both sides of the identity (8.207) with respect to eai3(y} and use the fact that another Tctfj(y) appears from the functional differentiation of the action on the exponent in the path integral. By considering the flat space-time limit e^v (x) = 0 after the functional differentiation we have
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
We here recall that the second-rank tensors are generally written in complex coordinates z = (xl + ix2')/2, z = (xl - ix2)/2 as
In complex coordinates, the tracelcss property of T^ is expressed by
and the non-vanishing components are given by Tzz = 8ZXOZX and Tzz — dzXdsX, which satisfy in flat space-time
The above identity (8.215) which contains the correlation functions of Tp,v is written in complex coordinates by setting v = z,a = j3 = was
In this expression, if one notes dzTzz = ~(1/2)8ZTZZ and interprets z as time and z as space coordinates, the time derivative does not appear on the right-hand side and the correct T product is defined. If one uses
in the identity (8.219) and writes the expression after one removes the operator d, from both sides, one obtains
KAC-MOODY ALGEBRA AND VIRASORO ALGEBRA
187
which expresses the fact that the pole structure of both sides is identical. In this last expression we give a result valid for a theory with d scalar fields Xa(x). This relation has been derived from the identity with the anomaly, and it gives a relation corresponding to the operator product expansion in conformal field theory. It is important to recognize that the last c-number term arises from the anomaly associated with general coordinate transformations when one imposes the Wcyl invariance. On the other hand, the path integral in conformal field theory is defined by
and one obtains the correlation function by taking cqn (8.183) into account
and thus
which is also valid for a T product. The traceless energy-momentum tensor TzZ = 0 is given by (see eqn (8.217))
or TZz = d'gXdzX, and the operator product expansion of the energy-momentum tensor is calculated as
This relation including the last c-numbcr term agrees with the identity (8.221) with d = I derived in conventional field theory.
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
By multiplying the test function £,z(x) by the identity (8.221) or (8.226) derived by the operator product expansion and contour integrating with respect to the variable z around an infinitesimal circle around w, one obtains
If one defines
by choosing £ 2 (x) = zn+1, and if one contour integrates (8.227) by multiplying wm+\ one ob|;ains the Virasoro alebra
where the last term stands for the central extension. Note that the central extension in eqn (8.221) arises from the general coordinate anomaly. The essential point in the present consideration is that the central extensions of the Kac-Moody and Virasoro algebras give the algebraic manifestation of chiral and general coordinate anomalies. In this connection, the conserved energy-momentum tensor T^. which is defined by a variational principle with the variable X = (g^^X kept fixed (see Appendix B), is related to the traceless tensor TIW by
and the conservation condition D^f^ = 0 gives £"%„ = -(l/48n)dvR (8.207). On the other hand the operator 7),,, which is conserved contains the trace anomaly T^ = J?/(24?r). 8.5
Quantum theory of strings and Liouville action
In this section we discuss the basic aspects of the first quantization of a bosonic string. The Polyakov-type Lagrangiau of a string is given by
where Xa(x'), a — 1 ~ d, stand for the coordinates of a bosonic string in d-dimcnsional Euclidean space-time. The two-dimensional internal coordinates (x 1 ,^ 2 ) = (cr, r) stand for the parameter cr, which describes the spatial extension of the string, and the parameter r, which describes the extension in the time direction (or time development). See Fig. 8.2.
QUANTUM THEORY OF STRINGS AND LIOUVILLE ACTION
189
FIG. 8.2. Free motion of a closed string A naive path integral which defines the first quantization of a string is given by
We present a precise definition of this path integral below. Formally, this path integral is equivalent to a theory of d scalar fields Xa(x) defined in a curved two-dimensional space. A characteristic property of string theory is that the two-dimensional space takes various topological shapes. A two-dimensional space with a cylindrical shape describes the free motion of a closed string, and a twodimensional space with a doughnut in the middle describes the motion of a string with higher-order (one-loop) quantum corrections. To perform the path integral on the space with holes (or handles) is an important but in practice very difficult problem in string theory. Here we discuss the quantum effects (namely, quantum anomalies) instead of those topological issues. The BRST invariant path integral measure with respect to general coordinate transformations in curved two-dimensional space is given by Sec cqn (B.41) in Appendix B. The actual calculation is performed in the conformal coordinate defined by
where 77^ is the flat metric. In this coordinate the variables with weight factors become
The conformal coordinate conditions are written in terms of metric variables as
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
and thus the gauge conditions are fixed by the Lagrangian
When one uses the equations of motion for BI and B%. one obtains the gauge conditions (8.236) written in terms of variables with weight factors in eqn (8.235). When one defines the new anti-ghost variables
and similarly iBzz = BI +iB^ and —iBzS = BI —iB^, the BRST invariant gauge fixing Lagrangian which contains the Faddeev-Popov ghost fields is given by
In deriving £g, we used the supcrfield notation derived from eqn (B.34) in Appendix B
If one sets /j• — v = z in this expression, for example, the first term gives gzz in the gauge fixing term, and the term proportional to 0 gives the last term in eqn (8.239) proportional to bzz. After the integration over the auxiliary fields Bzz and Bzz in Cg, we have gzz = g^z = 0 and gzz — -2^/p. If one uses these relations in the Faddeev-Popov ghost parts, the gauge fixing Lagrangian is simplified as
although the BRST symmetry is no longer manifest.
QUANTUM THEORY OF STRINGS AND LIOUVILLE ACTION
191
In this way the path integral for a bosonic string is given after the integrations over Bzz, Bsz and then over gzz, g^ by
An important property of this path integral formula is that the Weyl freedom p(x) explicitly appears in the action as a result of the choice of the BRST invariant path integral measure (8.233) with respect to general coordinate transformations. When one makes the change of variables (the Weyl transformation)
where the Weyl freedom p(x) is extracted from other variables in the action in eqn (8.242). To evaluate the Jacobian for this transformation, we fix the hermitiaii operators, which define the basis vectors to expand the integration variables, by (using the calculational method of covariant anomalies)
The operators for the variables bzz(x) and cz(x) are given by replacing z by z in HI, arid HC, respectively, and it is shown that the Weyl anomaly does not change under this replacement. All these operators in eqn (8.245) are obtained from the general operator
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
by choosing the parameter n at n = 0, -2, 1, respectively. Consequently, the general calculation of the Weyl anomaly for an infinitesimal parameter a(x) is given by
where we defined
and used a generalization or the gauge invariant mode cut-off. This calculation agrees with the one already performed in eqn (7.101), and its result is given by
where R is the scalar curvature and we used ^/g = p and ^/gR = —d^d^ \np = dzdzlnp. From the viewpoint of the Weyl anomaly, the term ^/gM2 could be removed by adding a suitable local-counter term (the cosrnological term) to the starting Lagrangian. The integrated Jacobian by extracting the freedom p from the action is evaluated as follows: We start with the intermediate stage after some Weyl transformation where p = cxp[(l — s)a], and we apply a further infinitesimal Weyl transformation ds a and integrate the resulting anomaly factor (by taking into account the weight factors in cqn (8.243) and also the fact that b and c arc Grassmann numbers) as
The path integral of the bosonic string (8.242) is then given by
QUANTUM THEORY OF STRINGS AND LIOUVILLE ACTION
193
where the Liouvillc action SL is defined by
In the last term of SL, which corresponds to the cosmological term, we give the result after adjusting the coefficient to be a finite constant \i? by adding a suitable counter-term to the original string Lagrangian. From eqn (8.250) one might think that the main part of the Liouville action is subtracted by a local counter-term. If one uses the gauge condition other than the conformal gauge, the main part of the Liouville action is written as
which is not local, and thus it is not subtracted away by a local counter-term. A specific quantum bosonic string at d = 26, which is defined without considering the Liouville freedom p ( x ) , is called a critical string and d = 26 is called the critical dimension. (When one considers the superstring, which contains the same number of bosonic and fermionic freedoms, the critical dimension is given by d = 10.) On the other hand, the string theory for d < 26 is called a non-critical string and a BRST invariant quantum string theory is defined only when one includes the Liouville freedom p ( x ) . In the quantum theory of strings, it is convenient to consider the motion of string coordinates Xa(x) on a two-dimensional parameter space, which is called the world sheet, and this parameter space is treated as a Riemaim surface. One can naturally treat the topological notion (genus) if one introduces a background c-numbcr metric for this Ricmann surface. In the calculations so far we used the specific metric specified by the conformal gauge g^ix) = p(x)r]ll,v^ but it is more natural to treat a more general situation
where p'(x) carries the dynamical freedom and the background gfu, (x) carries the topological information. The variables Xa(x), bzz(x), cz(x) are thus influenced by the presence of the background g^' (x). Even for the critical string for d = 26 where the dynamical freedom p'(x) is discarded, the topological effects of the background metric remain. For the non-critical string for d < 26. the freedom p' (x) behaves like an extra scalar freedom in the path integral. It is thus expected that one unit of Liouville action is induced by p ' ( x ) , and as a result, the coefficient in front of the Liouville action (8.252) is modified to (25 - d)/(487r). It is in fact known that this modified coefficient leads to a more natural quantum theory of Liouville theory and thus to a quantum theory of non-critical strings.
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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
8.6
Ghost number anomaly and the Riemann—Roch theorem
We now come back to the analysis of the quantum anomaly. When one considers an infinitesimal ghost number transformation in eqri (8.242)
The associated Jacobian, which is obtained from the above calculation of the Weyl anomaly, leads to an identity (by recalling that bzz and cz are Grassmann numbers)
and similarly
and the sum of the right-hand sides of these two relations (3/^Tr)^/gR gives the ghost number anomaly. When one integrates the sum of these two identities over a closed twodimensional surface and uses the Gauss-Bonnet theorem, one obtains
where x = 2(1 — g) is the Euler number with g standing for the genus; g = 0 for a sphere and g = I for a torus, for example. We thus obtain a two-dimensional ghost number identity whose integral has a form analogous to the chiral anomaly in the presence of the iristanton in four-dimensional non-Abelian gauge theory. The index theorem related to the ghost number anomaly is written by denning HcLpn = \^
where we used the relation in eqri (8.247) with a = constant and the 1 : 1 correspondence of (pn and * for non-vanishing An for the operators Hc and H£ defined in eqri (8.245), and similarly for H* and HI,. If one defines the operators
GHOST NUMBER ANOMALY AND THE RIEMANN-ROCH THEOREM
195
which appear in eqn (8.242) the above index relation is written as
which has the same form as in eqn (3.77) if one introduces the two sets of operators. This index relation is known to correspond to the Riemann-Roch theorem in the theory of Riemann surfaces. The Riemann-Roch theorem is thus understood as the ghost number anomaly in string theory. The contents of this index relation and the Riemann-Roch theorem are understood from.eqn (8.241) as follows. If one uses the zero modes for the ghosts (f or cz in the BRST symmetry, the conformal gauge conditions
are not modified. Thus the ghost zero modes correspond to the conformal Killing vectors. On the other hand, the zero modes of the anti-ghosts bzz or bzz are orthogonal to the variations of gzz or gzz induced by coordinate transformations. The zero modes of the anti-ghosts bzz or bzz thus generate the variations of the metric which are not generated by coordinate transformations. This freedom of the metric variations other than the coordinate transformations is called the modulus. The number of zero modes of bzz and bzz thus corresponds to the number of the moduli parameters of the Riemann surface. The index relation (8.259) then expresses the Riemann-Roch theorem.
9 INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES Recently there has been important progress in the treatment of Dirac fields in lattice gauge theory and we can now discuss the chiral anomaly on the lattice in a manner similar to that in continuum theory. In particular, we can discuss the index theorem on the lattice. The analysis of the index theorem on the discrete lattice itself has certain subtle aspects, but lattice theory deals with completely regularized quantities and thus some of the subtle aspects in continuum theory are now given a more rigorous basis. We explain that all the results of chiral anomalies in continuum theory are reproduced in a suitable continuum limit of lattice gauge theory, and thus we can give a uniform and consistent treatment of both continuum and lattice theories. 9.1
Lattice gauge theory
We first briefly summarize the basic aspects of lattice gauge theory. We consider a hypercubic lattice with a lattice spacing a in four-dimensional Euclidean space-time. The gauge field (connection) is defined on the link in the /i direction connecting the points h and n + ap, as in Fig. 9.1
and it takes a value in a gauge group, for example, SU(n) in our treatment. Here fl stands for a unit vector in the fj, direction, jj, = 1 ~ 4. and Tb stands for a generator of SU(n)
FIG. 9.1. Smallest plaquette specified by the fiv plane 196
LATTICE GAUGE THEORY
197
with the normalization tiTaTb = (l/2)<5 at , and g is the gauge coupling constant. We also define and the gauge transformation rule is given by (9.4)
where U(u}(n)) takes a value in the group SU(n). In Fig. 9.1 we show the smallest plaquette P^ lying on the plane specified by // and v. The Wilson action of the lattice gauge theory is defined by
where tr stands for the trace with respect to the gauge freedom. We consider a product of gauge fields defined on each link of a plaquette P^v in Fig. 9.1 by taking into account the direction of the link, and we define the action by summing over all the possible P/j,v. It is confirmed that the action thus defined is invariant under the gauge transformation (9.4) on the lattice. In the limit of small lattice spacing a, we have
and thus the trace in the action gives
Consequently, the above lattice action in the naive, continuum limit a —>• 0 becomes
and the Yang-Mills action in the continuum is realized. Namely, the lattice gauge theory defines a regularization by approximating the action of continuum YangMills theory, which has an infinite number of degrees of freedom, by a theory with a finite number of degrees of freedom on the link. This regularization clearly preserves the gauge symmetry, but the rotational symmetry of the four-dimensional Euclidean space-time is spoiled for a finite a. The hypercubic symmetry defined by a 90-degree rotation is preserved, and it is believed that the hypercubic symmetry is sufficient to recover the Lorentz symmetry in the continuum limit.
198
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES The path integral quantization of lattice gauge theory is defined by
where the integration is performed for all the link variables U/j,, and the path integral measure is denned by a Haar invariant measure specified by the followin rules
The function f(Ufl) is an arbitrary function of U^ and UQ £ SU(n) is an arbitrary group element. Since Ut,U^ = I, the volume of the entire gauge field can be normalized to be unity by j T>U = 1. Namely, the present lattice gauge theory defines a compact gauge theory. For this reason, the path integral in lattice gauge theory is finite even without gauge fixing and the quantization can be defined without gauge fixing. This integration rule gives for SU(3), for example.
The vanishing of the first integral is understood by noting that the value of the integral does not change when one applies independent gauge transformations, which belong to the representations 3 and 3* of SU(3), to the indices i, j of (Up)ij, respectively. One can understand the second integral intuitively by setting j = k and summing over 9.2
Lattice Dirac fields and species doubling
The Dirac field is defined to be located on each lattice point and the action is defined by
where the sum over x runs over all the lattice points. Also a1' = aft and the sum is taken from fj, = 1 to /j, = 4. The last term in the action with the coefficient r
LATTICE DIRAC FIELDS AND SPECIES DOUBLING
199
is called the Wilson term, and its meaning will be explained later. The metric convention is chosen as
This action is gauge invariant if one performs the transformation
simultaneously with the gauge transformation of the gauge field (9.4). When a is small we have
and the above Dirac action in the naive continuum limit a —> 0 becomes
where the covariant derivative is defined by
In the naive continuum limit a —S> 0, the Wilson term with the coefficient r vanishes. To understand the meaning of the Wilson term, we go back to the action (9.12) with a finite a and consider the free Dirac field obtained by setting r = 0 and II^ = 1. By performing the Fourier transformation
we rewrite the free fcrmioii part of the action (9.12) as
The propagator for the Dirac particle is given by the inverse of the factor between •0(fe) and V-'(fc) in the momentum representation of the Lagrangian
200
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
As is well known in condensed matter theory which deals with the matters of lattice structure, the momentum in lattice theory is defined in the fundamental Brillouin zone
for each component specified by /K with yti = 1 ~ 4. When one considers the limit a —> 0 in the above propagator (9.20) of the Dirac particle with all the momentum components k/j, kept fixed within —n/(2a) < kfl < 7r/(2a). one obtains the ordinary propagator in continuum theory with a mass mo
In addition to the conventional propagator, however, if one considers the limit a. —T> 0 by choosing the first component k\ of the momentum at fci = TV /a + k[ with fixed k\ and the remaining components of kIL being kept fixed, one obtains
Since the components of the momentum run between — oo and oo in the continuum limit, this second propagator (9.23) also describes the propagation of a conventional Dirac particle with massTOO. Namely, the lattice Dirac action which was introduced to describe one Dirac particle in fact describes two physical Dirac particles in each momentum direction when rotated back to the Minkowski metric. In four-dimensional space-time there appear 24 = 16 Dirac particles altogether. This phenomenon is called species doubling. On the other hand if one keeps the Wilson term with r ^ 0 the Dirac propagator in lattice theory is given by
and this propagator describes the ordinary Dirac propagator with mass m0 only when one takes the limit a —>• 0 with all the momentum components fc,, kept fixed within — ?r/(2o) < kp, < 7r/(2a). The propagators for other momentum domains such as cqn (9.23) describe particles with mass of the order m<) + r/a and thus with an infinite mass in the continuum limit. The extra particles are thus eliminated from the physical Hilbert space in the continuum limit and the problem associated with the species doubling is resolved.
LATTICE DIRAC FIELDS AND SPECIES DOUBLING
201
The species doubling is understood from the viewpoint of symmetry as follows: The lattice Dirac action without the Wilson term r = 0
is invariant under the global chiral transformation with a constant a
if one sets m,Q = 0. The above action (9.25) is also confirmed to be invariant under the transformations of the Dirac field (even with mo 7^ 0)
Here Tn represents one of the following 16 operators
and
We define T^ for
by
which satisfies the following relation
This algebraic property is the same as that of the four ordinary Dirac matrices 17^, and thus we can construct the above 16 independent operators from TM. For the anti-hermitian 7^ in the present Euclidean theory, we have
We denote those 16 operators by Tn, n = 0 ~ 15, and define in particular TO = 1- If one notices that the operator TM adds momentum -K/O, to ?/>(fc M ) in eqn (9.18), one can cover the entire fundamental Brillouin zone by starting with the momentum domain which is called the physical domain, if one applies the operation (9.27) by using Tn.
202
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
The important property of Tn is that eight operators in cqn (9.28) commute with 75 and eight operators in eqn (9.29) anti-comrnute with 75 and thus change the signature of the chiral 75 charge. Because of this property, eight species douhlers give the chiral anomaly with a positive signature and the remaining eight species doublers give the chiral anomaly with a negative signature when evaluated in the theory with r = 0. In this way the chiral anomaly vanishes in the naive latticized Dirac theory. In other words, the mechanism which produces the chiral anomaly disappears in a regularized theory with only finite degrees of freedom such as the lattice theory. To be consistent with the fact that the chiral anomaly is inevitable, the theory adjusts by itself such that it is anomaly-free. Namely, the 15 species doublers are generated so that the sum of all the chiral charges vanish
This suggests that it is generally difficult to regularize a continuum theory and preserve exact chiral symmetry by simply replacing the differential operator by the difference operator in lattice theory. Namely one cannot avoid species doubling by a naive discretization. This fact is known as the Nielsen-Ninomiya theorem. From this viewpoint, the ordinary chiral symmetry is broken by the Wilson term in the lattice Dirac action with r ^ 0. The Wilson term is understood as a generalized (infinite) mass term which breaks chiral symmetry, analogous to the mass term in the Pauli-Villars regularization. It is known that the chiral anomaly corresponding to one Dirac fermion is correctly reproduced in the continuum limit by the lattice Dirac action with the Wilson term. 9.3
Representation of the Ginsparg—Wilson algebra
We write the action for a Dirac particle on the lattice in the general form
where D(x,y) generally contains the gauge field and the sum runs over all the lattice points x. The Dirac operator D(x,y) is not ultra-local in general, namely, the operator has a non-vanishing value even at points far apart on the lattice but we assume that the operator D(x, y) decreases exponential!}' for a large value of
where ga with a constant Q is the localization range of the Dirac operator. For a hennltiaii 75, we assume the following hermiticity property when we regard D(x,y) as a matrix specified by the raw indices x arid column indices y. In lattice theory the operator D contains both kinetic and (generalized)
REPRESENTATION OF THE GINSPARG-WILSON ALGEBRA
203
mass terms, and the operator can be hermitian only after being multiplied by 75; this property is shared with the continuum Euclidean Dirac theory with a mass term. The recent progress in the treatment of the lattice Dirac operator is based on the operator which satisfies the algebraic relation
which is called the Ginsparg-Wilson relation19 where k stands for a non-negative integer. This relation shows that the continuum chiral symmetry is modified by the term on the right-hand side which depends on the lattice spacing a. In the naive continuum limit a — ¥ 0. this relation is reduced to the ordinary continuum relation for a massless Dirac field. An explicit construction of the lattice Dirac operator which satisfies this algebraic relation will be discussed later. Here we discuss the general characteristic properties of the operator D, which are implied by the algebraic relation (9.38). We analyze all the possible finitedimensional representations of the algebraic relation (9.38) by considering a hermitian operator defined on a finite lattice. As the boundary condition we employ the periodic boundary condition. We first define the operator
which is hermitian by its construction, and the Ginsparg-Wilson relation (9.38) is written as This relation implies if
then where the positive definite inner product of eigenfunctions is defined by
49 The relation corresponding to k — 0 is commonly called the Ginsparg-Wilson relation. We can discuss the algebraic properties and chiral anomaly for all these operators in a uniform manner.
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INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
with the summation running over all the lattice points. If Xn = 0. these relations imply that H(j)n = 0 and thus H(j56n) = 0. namely, one has
We can thus choose the eigenstates with An = 0 as the simultaneous eigenstates of We next consider the case \n ^ 0. If \n ^ 0 and (F5<;6n, F B
Since the eigenvalue An of the hermitian operator H is real, the state <j>n satisfies the "highest state" condition familiar in the representation of the angular momentum only if 1 — (a\n)2 = 0. Here we used the fact that the inner product in Euclidean theory is positive definite. We thus understand that the states tf>n with \n = ±l/o do not give rise to paired states by the operation T^(j>n in general. We also see from eqn (9.49) that these states are the simultaneous eigenstates of 7,5. By using the relation valid for the states with
which is derived by sandwiching the Ginsparg-Wilson relation (9.38) by <£>,], and 4>n, one can show that these eigenvalues Xn = ±l/o are the largest or the smallest eigenvalues of j&D. Namely.
Finally, the states with 0 < |An < I/a always appear pairwise as and satisfy the relation
where <£>_-„ stands for the state with eigenvalue —Xn, and we note the relation FsCFsaC) = [1 - (aXn)2(2k+ind).T, oc <£„. Since 0 < |aAJ < 1, these states cannot be the eigenstates of
REPRESENTATION OF THE GINSPARG WILSON ALGEBRA
205
On the other hand, the relation tr~/5 = 0 generally holds on a finite lattice, and it implies
where in the last line we used the fact that all the states with Xn ^ 0 except for those with \n = ±l/a appear pairwise with positive and negative eigenvalues ±|A n | and thus cancel in the summation. Here N± respectively stands for the number of states §n with eigenvalues \n = ±l/a and 75
This relation shows that the chiral asymmetry n+ — n_ in the eigcnstates with zero eigenvalue is always balanced by the chiral asymmetry N+ — N- in the states with the largest eigenvalues |A n | = I/a for the operator satisfying the Ginsparg-Wilson relation. The analysis in this section is valid both for Abelian as well as for non-Abeliau theories. As for the boundary condition, we assumed the periodic condition for simplicity, but the analysis is valid for more general boundary conditions whenever there appear non-trivial zero eigenstates. The analysis of the index (i.e., the difference n+ — ri_ of eigenstates with vanishing eigenvalue) in this section is formal, since it is shown that the index or the chiral anomaly is not uniquely specified by the Ginsparg-Wilsoii relation alone. We shall show this property later in the present chapter. To summarize the analysis of this section, the normalized eigcnstates of 7oD = H/a on a finite lattice arc classified into the following three categories: (i) ri± states;
(ii) N-J- states (highest states,
respectively. (iii) The remaining states 0 < |A n | < I/a (paired states):
206
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
and the chirality sum rule n+ + N+ = n_ + N- holds. All the states n± and N± are also the eigenstates of the operator D. and they satisfy D(f>n = 0 and D
Atiyah Singer index theorem on the lattice and the chiral anomaly
The starting path integral for the Dirac field in lattice theory is given by
where the operator D satisfies the Ginsparg Wilson relation (9.38) and the gauge field is treated as the c-number background field. The action in the exponent is invariant under the generalized chiral transformation with an infinitesimal constant parameter
if one writes the Ginsparg-Wilson relation (9.38) in the form with which is shown to satisfy 75 = 1 by noting ^H2 - H2^5 = 0. Under the transformation (9.59) one obtains the following Jacobian from the path integral measure (by remembering that the fermionic variables are described by Grassmarm numbers) where FS is defined in eqn (9.41). We have the following relation for the trace appearing in the Jacobian
by using the representation of the Ginsparg-W7ilson algebra in the preceding section. Here n± stands for the number of zero eigenvalue solutions
ATIYAH-SINGER INDEX THEOREM AND ANOMALY
207
of the hcrmitian operator 75!? with 75^»n = ±(f>n, respectively, and n+ - n_ is called the index of the operator 75!). We also used eqn (9.52). In lattice theory the inner product is denned by a sum over all the lattice points (®n,&n} = Y^xa4'
where f ( x ) is an arbitrary smooth function with /(O) = 1 and rapidly approaching 0 at infinity.50 By using the result of the. representation of the GinspargWilson algebra in the preceding section, one can confirm that the index n+ — n_ (9.63) is not modified for any smooth function f ( x ) with /(O) = 1. In our analysis of the index, the hcrmitian operator 7,5!? plays a special role. We now analyze a local version of the index
by omitting the sum (or integral) over the coordinates in the formula for the index. The trace tr here stands for the sum over the Dirac and Yang-Mills indices. Since the local version of the index is insensitive to the precise boundary condition, we consider the infinite lattice limit L — Na —> oo in the following analysis of the index. (The limit N —> oo is in any case required in the continuum limit.) We now consider the continuum limit a —> 0 in eqn (9.65), which corresponds to a "naive" continuum limit in lattice gauge theory. We first note that
approaches 0. Because of the presence of the regulator function /(x), which rapidly approaches 0 for large x (see Fig. 5.1), the eigenvalues of 75!) are cut-off s
°Wc here present this calculational scheme to show that the lattice calculation in the continuum limit is reduced to the continuum calculational schemes discussed in preceding chapters. A more direct evaluation of the Jacobian (9.62) by the lattice formulation will be given later.
208
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
at ~ M. Consequently, the above expression (9.66) approaches 0 in the limit a —> 0, or in terms of the integrated index it vanishes as
We thus analyze the small a limit of
where the operator is well-regularized by f ( x ) , and thus one can use the plane waves for the Dirac fields to evaluate this trace and in this way one can extract the gauge field dependence of a local version of the index. The basic momentum domain for the hypercubic lattice is defined by the Brillouin zone (9.21)
We also assume that the operator D blows up as ~ I/a for small a in the momentum domain of species doublets, namely, the operator D is free of species doublcrs. See eqn (9.20) for species doublets. (An explicit construction of D of this property is given later.) Consequently, in the limit a —5- 0 with, for example,
in the momentum domain of the would-be species doublers which corresponds to outside — ?r/(2o) < k^ < 7r/(2a). and it does not contribute to the trace because of the regulator f ( x ) . We are thus allowed to consider only the momentum domain for the physical species
in the evaluation of the above trace. In the limit a —>• 0 we obtain
where we first considered the limit a —>• 0 with the momentum variables k^ fixed in — L < ku < L, and then took the limit L —t oo. This procedure is justified
ATIYAH-SINGER INDEX THEOREM AND ANOMALY
209
if the integral is well convergent.51 In this calculation we also assumed that the (difference) operator D in the limit a -> 0 satisfies
for an arbitrary fixed momentum fc;,,, —vr/(2a) < ktl < ?r/(2a), and for an arbitrary smooth function h(x). The function h(x) in the present calculation corresponds to the gauge potential. The gauge potential is thus assumed to be smooth and vary very little over the distances of the basic lattice spacing a. We shall show that the explicit example of D given by Neuberger (and its generalizations) in fact satisfies all these conditions in eqri (9.114) later. In this way we arrive at the same formula as in Chapter 5. The formula (9.64) combined with eqn (9.72) and eqn (5.104) gives
in the limit M —> oo. The index theorem in the presence of topologically nontrivial gauge fields and the chiral anomaly in the continuum are thus recovered. It is important to recognize that we used only the general properties of the operator D, namely, the existence of a hcrmitiaii 75!) which satisfies the Ginsparg-Wilson relation with a smooth continuum limit a —>• 0 without producing the species doublers. This analysis shows that the chiral Jacobian (9.62) in lattice gauge theory contains the correct chiral anomaly. (We however implicitly assumed that the index defined in lattice theory does not change in the process of taking the continuum limit.) A direct evaluation of the Jacobian for the Neuberger operator will be given later. We would like to add a comment on the present analysis of the index. The relation Tr7s = 0 contains important information about the Hilbert space in lattice gauge theory. In the perturbative calculation of the chiral anomaly in B1
To be precise, we are dealing wi f~^ /(2 dx fa(x) where fa(x) depends on the parameter a. (The generalization to fourdimensional integrals is straightforward.) Consequently, we need to show that both lim a _Kj/£/ a dx fa(x) and lim a _j,o J_^/(2 a ) ^x f&(x) cari ^e made arbitrarily small for large L. The integral we encounter in lattice gauge theory generally has a .structure \ii^^of"ff$a-)dxe~3in2aX/(a2M''i) = ^mr^oo/^^e-^^ 2 and the convergence condition is satisfied for the choice f(x) = e~x. Note that J^/(2a) dxtTsin2 ax/(a2M2) <• J^ a dxe~c x IM for a sufficiently small constant c, and the right-hand side goes to zero for a ->• 0 and then L -> oo. Incidentally, we have Iim0.^0 f^/ffi dx e"1/1"2^1'2'1 = 7r 0; e 1 l <1 M / ) ^ '' -
a_>o(
•* = 0 for the momentum domain of the would-be species doublers be-
cause of our assumption about the property of the operator D, which was used in eqn (9.70).
210
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
lattice gauge theory one usually assumes the relation Tr7g = 0. In this case, the relation holds by remembering F5 = 7.5 - (aj5D)2k+L. Namely, only the states N± in eqn (9.56) contribute to the Jacobian (9.62) if one uses the results in Section 9.3, since other states give either vanishing contributions or cancelling contributions of positive and negative values. When one combines the above expression (9.75) with the chirality sum rule Tr7g = n+ — n_ + Ar+ — N- = 0 in eqn (9.53), the correct index theorem Tr Fg = n+ — n~ is derived. This chirality sum rule also shows that the relation Tr 7.5 = 0 has no well-defined meaning in the continuu limit a —> 0, since one cannot count the number of states N±, which have infinitely large eigenvalues in the continuum limit, in a reliable way. It is in fact shown that the states N± correspond to the heavy topological states related to the would-be species doublers. in contrast to the physical topological states n± which represent the states with the vanishing eigenvalue. Consequently, one has the relation in the physical Hilbert space with the species doublers excluded, which agrees with the continuum analysis. 9.5 The operator D satisfying the Ginsparg—Wilson relation An explicit form of the operator D, which satisfies the Ginsparg-Wilson relation (9.38) with k = 0, has been given by Neuberger and it has the form
It is confirmed that this operator D satisfies the relation by a direct insertion of D, independently of the detailed form of the hermitian operator HW- In the explicit construction of D, one uses DW = 7&Hw which was discussed in Section 9.1 as the Wilson operator (9.12) for the Dirac field on the lattice
but the mass term is chosen as mo —* rn^/a and the signature of the Wilson term is reversed. Namely,
OPERATOR D SATISFYING THE GINSPARG WILSON RELATION
211
It has been proven that the operator D thus constructed satisfies the locality condition in the sense of eqn (9.36) for sufficiently smooth gauge field configurations, though we do not give the proof here. One way to specify the smoothness of gauge fields is to impose the so-called admissibility condition on lattice gauge fields, where \\O\\ is the operator norm and e is a certain small fixed number. This condition divides the space of lattice gauge fields into topological sectors; the index (9.63) takes a well-defined value for each topological sector. The necessity of such a restriction on gauge field configurations is thus suggested by the consideration of the index theorem (9.63) also. The chiral Jacobian (9.62) in the preceding section is given for this operator by The conventional calculation of this Jacobian and the chiral anomaly in lattice gauge theory proceeds as
where we used the relation Tr75 = 0 valid on a finite lattice. The local version of the index (i.e., the chiral anomaly) is given by
where the trace runs over the Dirac and Yang-Mills indices. The factor D\y x rD I D — is a phase factor, and its behavior in the continuum limit is somewhat V w W
subtle. But this calculation illustrates an interesting aspect of the lattice calculation, and we present its details below. 9.5.1 Explicit lattice evaluation of the chiral Jacobian We first note
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INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
where B is the Brillouin zone (scaled by a)
We then move the plane wave Klkx/a through the operator to the left-hand side of the operator. For the Wilson operator, we have the identity
where f(x) is a smooth function, and the operators QM and R are defined by
and
Thus, by using
we have
where we have used our Euclidean metric convention With this form of the ohiral Jacobian, it is straightforward to find the naive continuum limit by noting the following facts. Firstly, the operators Q^ and R are of O(a°) in the naive continuum limit (9.15),
and thus
OPERATOR D SATISFYING THE GINSPARG- WILSON RELATION
213
Secondly, the trace over Dirac indices requires at least four gamma matrices; see eqn (5.24). Also it is sufficient to retain terms to the magnitude O(ai) in the integrand. After some rearrangements, we have
where the equality holds in the limit a —i 0. The coefficient J(mo,r) is given by the integral
By comparison with our previous result (9.74), one expects that I(mo,r) = 1 if the Dirac operator D is free from species doubling, namely, for 0 < mo/r < 1 as will be explained in the next subsection. We now confirm this expectation by evaluating /(mo,r) explicitly. We first change the integration variables from fcM to
and split the Brillouin zone (9.85) into -vr/2 < k,L < K/2 and vr/2
where
The four quantities e^ which assume the values cfl = ±1 specify the 16 domains in the Brillouin zone and we sum the contributions from those 16 domains.
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INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
We can first confirm that I (mo, r) is invariant under an infinitesimal change of parameters mo and r. This follows from the identities
and
By integration by parts, we then have (d/dmQ)I(mo,r) = (d/dr)I(m,0,r) = 0. The surface terms do not contribute because they are independent of e^ and thus vanish due to J^f =±i(n,u £ M) = 0- Once the invariance of I(mo,r) has been established, we may evaluate /(TOO , r) by considering some convenient limiting cases of the parameters. We set a. = mo/r and regard /(TOO,'') as a function of a arid r. After the rescaling s^ —>• rs M , we have
One can see that the integrand becomes singular near s2p ~ 0 only when a = 0, 2. 4, 6 and 8. Therefore, by fixing a avoiding these values, we may consider the r —> 0 limit as a convenient limit of parameters. It is shown that the r —> 0 limit of eqn (9.99) is given by first taking the r —> 0 limit in the integrand and then taking the r —>• 0 limit of the integration domain.52 Therefore, we have 52
To show this, we first divide the integration domain [— 1/r, 1/r] into a four-dimensional cylinder of size L with L < 1/r, C(L) = Ss x [-L, L], and the remaining part R(L). The radius of S'3 is L and the direction of the cylinder is taken along the /^-direction of the numerator of eqn (9.99). If the r —f 0 limit of the integral over R(L) vanishes for L —j- oo, lim^-joo lim,—>o I(cvr,r)\R(L
OPERATOR D SATISFYING THE GINSPARG-WILSON RELATION
215
By examining the last expression, we finally obtain
where 6(x) is the step function. As we expected, when and the present direct calculation of the chiral Jacobian (9.92) coincides with our previous analysis (9.74). 9.5.2
Physical meaning of the operator D
The definition of the Ncuberger operator D (9.77) is somewhat abstract, and to clarify its physical meaning we consider its continuum limit by the followingtwo steps. This analysis also supplements the explicit calculation of the Jacobian presented above. We first take the limit a —5- 0 with
kept fixed, and then consider the continuum limit by letting 2r/a, m 0 /a -> oo. This two-step procedure is known to simplify perturbative calculations in the lattice theory defined by the Wilson operator. and fL ' /T dp f_L dz with the same integrand. After performing the integration over p, these integrals are bounded by
Similarly, the integral over fL dpf^Ldz lim£,_j.i:x; limr_,,o I(ar, r)1 R(!,) = 0 is established.
is bounded by 47T/L.
In this way,
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INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
We represent the fundamental Brillouiri zone in momentum space by starting with the basic domain
and applying the operators Tn, n = 0 ~ 15, defined in eqns (9.28) and (9.29). We first observe that the relation for Dw in cqn (9.79)
holds in the continuum limit a —> 0 for any fixed k/j in the domain kp. < 7r/(2a) and fixed MO = —mo/a and for a sufficiently smooth h(x). We next define
where we used the fact that @ = 7^CM in eqn (9.79) and Tn commute. The second term on the right-hand side of (Dw)n gives the mass term generated by the Wilson term for a -> 0 with &M kept fixed in — 7r/(2a) < &M < 7r/(2a). If one recalls the momentum representation of DW with the vanishing gauge field
the explicit expression of the 16 mass terms Mn is given by MO = —rrio/a and one of the following 15 expressions
for n = 1 ~ 15. We indicate inside the brackets of eqri (9.109) the (multiplicity, chiral charge) of the would-be species doublers for each mass parameter. For example, Mn = 2r/a — mo/a appears four times with the chiral charge —1. These multiplicities and chiral charges appear as the coefficients in cqn (9.103). Consequently, the value of (Dw)n (9.107) in the continuum limit a -» 0 is given by
If one recalls the definition
OPERATOR D SATISFYING THE GINSPARG-WILSON RELATION
217
one obtains in the same limit as eqn (9.110)
When one chooses the mass parameter such that
namely 0 < mo < 2r, one can show that eqn (9.112) gives rise to
for any k^ fixed in —7r/(2a) < /% < 7r/(2a) in the limit a—>• 0 now with IT and mo kept fixed. In the last expression we chose the parameters as 0 < 2mo = 1 < 4r. The property (9.114) provides sufficient information to prove the index theorem (9.74) in the preceding section. These considerations, namely, the explicit evaluation of the Jacobian and the present analysis of the continuum limit show that the Ginsparg-Wilson relation by itself does not fix the coefficient of the chiral anomaly uniquely, since the Ginsparg- Wilson relation (9.38) with k = 0 is satisfied independently of the details of the hermitian operator H\y in eqn (9.77), whereas the chiral anomaly depends on the details of HW- For example, if one chooses mo < 0 then MO > 0 and thus 70~1D'7o ~ I/a, and the chiral anomaly vanishes. See eqn (9.103). When one imposes the condition 0 < mo < 2r which ensures only the physical species in D and the absence of species doublers. the chiral anomaly is uniquely fixed. In passing, we would like to briefly explain the construction of the solution for eqn (9.38) with general k. We first note that the condition (9.38) is equivalent to the two relations
When one defines H^k+i) = H2k+1, one has
which has the same form as eqn (9.38) with the simplest choice k = 0. One can thus construct the solution following eqn (9.77) as
218
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
where we denned
by using eqri (9.79). One may thus define the Dirac operator H by
in the representation where H(2k+i) is diagonal. If one chooses 0 < m 0 < 2r and
one obtains a lattice Dirac operator which is free of species doublers and gives the correct Dirac operator iSfi in the (naive) continuum limit. The general Dirac operator thus constructed gives the correct index theorem and chiral anomaly as was shown in Section 9.4. But the locality condition (9.36) for k / 0 has not been proved though we expect the locality of D for a sufficiently smooth weak background gauge field, since the locality for the vanishing gauge field is proved by an explicit analysis of D in the momentum representation. 9.6
Some characteristic features of lattice chiral theory
The Dirac operator on the basis of the Ginsparg-Wilson relation thus resolved many of the basic issues associated with fcrmions on the lattice. We would like to comment on several important properties of this formulation. First of all, we have noted the existence of the chirality sum rule
where AT± in eqn (9.56) stands for the number of states with eigenvalues with 756n = ±
for the trace Tr constrained to the physical Hilbert space, and that this index relation has a smooth continuum limit a. —>• 0 as was discussed in eqn (9.72). This phenomenon is the same as the behavior of the index discussed for the phase operator of the photon in Section 3.5. Namely, the index vanishes Tr 75 = in the truncated Hilbert space, but a non-trivial index generally appears after the elimination of the cut-off. The Dirac field exists only on the lattice point in a discretized theory such as the lattice theory, and thus it is not obvious if the notion of index in the AtiyahSinger index theorem is rigorously defined. The Atiyah-Singer index theorem is valid for a continuum Dirac equation with non-Abelian gauge fields defined on, for example, the smooth compact four-dimensional space such as the 4-sphere S4. The analysis of the index in this chapter shows that one can formulate the index theorem in the continuum limit of the lattice Dirac operator for sufficiently
SOME CHARACTERISTIC FEATURES OF LATTICE CHIRAL THEORY 219 smooth gauge fields, and that the result is confirmed by detailed perturbative and non-perturbativc calculations. In this sense, the notion of the index theorem is valid on the lattice also. The notion of the index is somewhat subtle on the lattice, but the treatment of the Jacobian in the lattice path integral is based on well-defined finite quantities only. One can thus analyze the chiral Jacobian and the anomaly on the basis of mathematically well-defined quantities. 9.6.1
Chiral gauge theory and fermion number anomaly
When one defines
by using D which satisfies the Ginsparg-Wilson relation (9.38), one can confirm •yj? = 1 by using eqn (9.115). One can thus define the projection operators
with One can also confirm that the operator satisfying the Ginsparg-Wilson relation (9.38) is decomposed into left- and right-handed components as
As a result one can define the right-handed or left-handed Dirac field by
One of the important implications of this decomposition into chiral components is that the pseudo-scalar operators Pa(x) representing the pseudo-scalar particles such as TT, r\ and t]' mesons in QCD are given by
where Xa, a = 1 ~ 9, standing for the Gell-Mann matrices of flavor SU(3) and the unit matrix, which express the flavor freedom of three light quarks. In this construction, the states N± in eqn (9.56) are automatically eliminated by the presence of the operator r 5 .
220
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
The chiral theory which breaks parity such as the Wcinberg-Salam theory is defined by the Lagrangian03 and the path integral on a finite lattice is given by
A lattice non-perturbativc formulation of chiral gauge theory has many potential applications, but its final formulation for non-Abeliari gauge theory has not been given yet. We here briefly sketch the basic idea of how to define the path integral measure in the present scheme. The functional space in our problem is naturally spanned by the eigenfunctions of the basic hermitian operator
However this eigenvalue equation is gauge covariant as are all the quantities in the gauge invariant lattice regularization. To accommodate the gauge noncovariant quantities such as a consistent form of anomaly, one defines the path integral in a specific topological sector specified by M by
where we expanded fermionic variables as
The basis vectors {wn} and {wn}, which satisfy
are suitable linear combinations of {(/)„,} and {4>\,}-, respectively; wn oc P-6n and wn K-<j>llP+. See also eqn (9.52). The measure factor 7?^ stands for the Jacobian for the transformation from the ideal bases {vn} and {vn}, which automatically ensure the Wess- Zumino intcgrability condition, to the bases specified by // and thus <&M crucially depends on the ideal bases.54 In practice, the ideal bases are 53
Note that the basic lattice operator D ( x , y ) is always vector-like and we do not define the generators T° (1 — 75)/2 for SU(2)]J, for example. We define the chiral theory by the projection operators P_ and P+. °4The measure factor is thus chosen to be a constant for the ideal bases.
SOME CHARACTERISTIC FEATURES OF LATTICE CHIRAL THEORY 221 not given explicitly and one determines the measure factor by imposing several physical conditions. The lattice path integral (9.131) corresponds to the formulation of the covariant anomaly in continuum theory in eqn (6.75). When one considers the change of fermionic variables which corresponds to the gaug_e transformations (including the U(l) phase transformation), -0 -» 1/1' and t/5 ->• '• {a'n} and {an} —» {a'n}Since the naming of path integral variables does riot matter, one obtains the identity
In this form of identity, the Jacobian of the path integral measure gives a lattice version of the covariant anomaly and the variation of the action gives the divergence of the covariant current. As an important application of this derivation of the anomaly, one can calculate the gauge invariant ferrnion number anomaly in chiral gauge theory. When one considers the fermion number transformation with a constant a
the Jacobian is given by (by noting that both of (1 — 7s)/2 and (1 + 7s)/2 are projection operators)
which shows that there appears an asymmetry between the number of freedoms Tr(l — 7s)/2 describing the particle ibi, and the number of freedoms Tr(l + 7s)/2 describing the anti-particle 4'L- We also used the index in eqn (9.63). There thus appears an asymmetry between fermions and anti-fermions proportional to the index when one defines chiral fermions on the lattice. As was discussed in Chapter 6, the fermion number is not conserved in chiral gauge theory in the presence of the instantori background. The proton can thus decay in the Weinberg Salam theory. The fermion number anomaly is naturally built-in in the present lattice formulation of chiral gauge theory. If one performs the simultaneous gauge transformation of the link variables U in the above path integral ZM(U) (9.134), the action becomes invariant but one
222
INDEX THEOREM ON THE LATTICE AND CHIRAL ANOMALIES
needs to take into account the variation of the measure factor MM induced by the gauge transformation of U. This variation S'&M converts the covariant anomaly given by the Jacobian to a lattice form of consistent anomaly, which is one of the requirements on the measure factor. In the theory with an anomaly-free gauge group, the measure factor should completely cancel the non-vanishing Jacobian arising from the lattice artifacts. Unlike the continuum theory, the Jacobian in lattice theory does not vanish completely even for an anomaly-free gauge group for a finite lattice spacing a. The current associated to ST^M should be local and satisfy several other requirements: The existence proof of such a measure factor $M amounts to a definition of lattice chiral gauge theory. The construction of the path integral measure with the desired measure factor for chiral non-Abelian gauge theory has not been given yet, and interested readers are referred to the references at the end of this book.
10
GRAVITATIONAL ANOMALIES In this chapter we discuss chiral anomalies associated with the gravitational field. We first discuss a gravitational generalization of chiral U(l) anomalies for ordinary gauge fields. We thus give a method to evaluate the Dirac genus and Chern characters in general 2n-dimensional Euclidean space-time. We next explain that there generally appear quantum anomalies in general coordinate and local Lorentz transformations for chiral Dirac fields in (4n + 2)-dimensional curved space-time, and thus the basic principles of general relativity and quantum theory become incompatible with each other if these anomalies arc not cancelled. 10.1
Chiral U(l) gravitational anomalies
In this section we study the Dirac field in the presence of the background gravitational field and QCD-type SU(JV) gauge fields. We consider an arbitrary 2ndimensional (Euclidean) space-time. The action for the Dirac field is given by (see Appendix B)
This action except for the last mass term is invariant under a global chiral transformation
with a constant parameter a. Here 72™+1 is a generalization of Dirac's 75 in four-dimensional space-time to 2n-dimensional space-time, and it is defined as a product of all 2n Dirac 7 matrices
All the In 7-matriccs are anti-hermitian iri our convention
223
224
GRAVITATIONAL ANOMALIES
and we define 7^ = e%ik, and Gab = (—1, —1, • • • , -1). In this case, the most general Dirac operator (see eqn (B.3))
becomes a hermitiari operator for the Euclidean inner product
(10.6) When this operator acts on the Dirac field itself which carries no Minkowski indices, the generator of general coordinate transformation U® is set to 0. We define a complete orthoiiormal set
to define the path integral measure, and expand the field variables as
If one defines the path integral measure by using weight 1/2 variables such as •tp — (gY^tp-, we have
and the path integral is given by
If one considers a localized chiral transformation in this path integral
one obtains the following identity
The first equality in this identity expresses the fact that the path integral itself is independent of the naming of integration variables, and the transition from the
CHIRAL U(l) GRAVITATIONAL ANOMALIES
225
first line to the last line expresses the result of expanding the action in powers of the infinitesimal parameter a(x) combined with the evaluation of the Jacobian. We also defined
When one uses the notation J(a) = cxp[-2i J dZnx v/ga(x)^45(x)], the chiral identity is written as
We thus need to evaluate A&(x). We obtain the master formula for the quantum anomaly by using the gauge invariant mode cut-off with the simple regulator function
just as the case in flat space-time. The trace here runs over the spinor and Yang -Mills indices. We next notice the following formula55
For an arbitrary spinor ipn (x), we have
Here -ft""lMJ, stands for the Riemann-Christoffel curvature tensor. By using the symmetry properties of the curvature tensor, one can further rewrite BB
7" commutes with Dp. if one uses the metric condition (B.17).
226
GRAVITATIONAL ANOMALIES
The master formula (10.15) is thus written as
where d(x,x') is a generalization of the 6 function to a curved space-time and it is given by the representation
The quantity a(x, x') is called a geodesic biscalar, and it is a generalization of (1/2)(x — x')2 to a curved space-time. We perform the calculation of the Jacobian around a point x$ where the space-time becomes flat. The coordinate conditions are specified by (see Appendix B)
We then set and treat y]Jl as coordinate variables. The actual calculation is performed for an infinitesimal r/ M and y^ is set y^ —>• 0 after the calculation. If one defines
the master formula becomes
a,nd the desired result is obtained by setting j/M to 0 after the calculation. We here replaced 5(x, x') in the neighborhood of XQ by the ^-function in a flat space-time, which is justified in the present calculation of the chiral anomaly. It is possible
CHIRAL U(l) GRAVITATIONAL ANOMALIES
227
to perform the calculation (10.24) directly, but to derive the result for arbitrary space-time dimensions in a simple way we set
and take the large M limit simultaneously as setting y11 to 0. When one expands various variables in powers in y^/M one obtains
where to simplify the calculations we used the fact that the relations such as 9a9ii.v(%o) = 0 and A™n(x0) = 0 hold in the present choice of coordinates and that the final result depends on quantities anti-symmetric with respect to the indices fj, and v (the symmetric components are eliminated by using the gauge freedom). We thus have
If one rescales fc^ —> Mk^ in the master formula (10.24), the chiral anomaly is written as
where we used the fact that only the terms with n-th powers in aab /M2 survive at the end since only those terms which are larger than or equal to (l/M) 2n with at least In 7-matrices survive at the end. In the derivation of this formula, we eliminated various terms at intermediate stages, for example, the terms (1/4M2)J? and (IjM)A^(x^ + y»/M')Ta in eqn (10.27) were omitted in eqn (10.28). These terms do not carry a sufficient number of 7-matrices relative to the powers in
228
GRAVITATIONAL ANOMALIES
1/M. It is confirmed that these eliminated terms do not modify the final result of the chiral anomaly to be given later. When one defines
in the above final formula for the anomaly (10.28), RVfl/M'2 can be treated as quantities commuting with each other; the commutator of two Rvff/M'2's increases the powers in 1/M but it does riot increase the number of 7 matrices. Similarly, RV^/M2 and F/M2 are treated as quantities that commute with each other. Consequently, the master formula is also written as
We perform an explicit evaluation of this formula by using a pa,th integral technique of first quantization in the next subsection. 10.2
Evaluation by a quantum mechanical path integral
We formally define a Hairiiltonian
with y^ as a variable conjugate to p^, and we define the time evolution operator as in Chapter 2. After a Legendre transformation, we define a path integral of the first quantization for a unit time interval
and the normalization constant Ar is an undetermined constant at this moment and it is fixed at the end of the calculation. By recalling that we can treat Rnti(xo)/M2 as a commuting constant we perform the path integral over the momentum variables
We thus confirm that only the anti-symmetric components of Rv(1 contribute. By taking into account the boundary condition for we expand £ M (T) by a complete orthonormal set
EVALUATION BY A QUANTUM MECHANICAL PATH INTEGRAL
229
The path integral measure is then expressed in terms of the expansion coefficients
where we defined
If one performs the path integral over 6^ and then a%, one obtains
where we used the matrix notation
and the determinant in eqn (10.37) is defined with respect to these indices. We also used the orthogonal property56
and an infinite product representation of sinhx. 57 The normalization constant N is fixed by considering the special case J2,,,, = 0
and we obtain the final result
56 Em=i A(n,m)A(l,m) = 4 £)^_ 0 / 0 drcfrr' sin 2mrr cos m?rT sin 2!7rr'cos mirr' = Sn.i where we extend the sum over odd integers m to the sum over all the integers m by using JQ dr sin 2ri7TT cos 2m7rr = 0. B7 siiih x =
230 10.3
GRAVITATIONAL ANOMALIES Cherri character and Dirac genus
By using the result (10.41) of the previous section, the chiral U(l) anomaly is written in a general coordinate system as
although we performed the actual calculation in a specific choice of the coordinates (10.21). The trace in front of the entire expression stands for the trace over the Dirac matrices and Yang-Mills indices, and the trace in the exponent is taken over the Minkowski indices of R,j,a. In this formula, the term with exp(F/M2) is called the Chern character and the determinant with R is called the Dirac genus. The expression (10.42) when combined with the classical solutions of the Dirac equation gives rise to
where n±, respectively, stand for the number of normalizable solutions with the vanishing eigenvalue flipfl(x) — 0 and jzn+i'-Pn = ±¥>n- We here used f d2nx ^fg(p\(x)^n+\(pi(x) = 0 for A; / 0. This relation (10.43) gives a generalization of the Atiyah Singer index theorem. If one recalls the following definitions in the formula (10.42) and if one expands the formula (10.42) in powers of aab/M2 by using
the term with the rt-th power in aab /M2 combined with tr 72,^1 gives the final explicit result of the chiral anomaly in 2n-dimcnsional space-time. From the expansion of sinhx/x one recogni/es that the chiral U(l) anomaly with the gravitational field appears only in d = 4fc (k = 1, 2, • • • ) dimensional space-time. In contrast the chiral anomaly with the gauge field F appears in arbitrary even-dimensional space-time. For example, in four-dimensional spacetime, the contribution of the gravitational field to the identity (10.14) in QEDtype theory is written as
In a general higher even-dimensional space-time the cross-terms of F and R. which are called the mixed anomaly, appear.
ANOMALY IN GENERAL COORDINATE TRANSFORMATIONS
231
In mathematical literature the notation of differential forms is often used. To convert our formula to the notation of differential forms, one first replaces
in the formula (10.42). and then the chiral anomaly is written as
where the remaining trace is over the Yang-Mills indices. When one extracts 2n-forms from this formula by noting
the result expressed in terms of differential forms is obtained in arbitrary 2ndimcnsioiial space-time. The extraction of the 2ri-forms replaces the trace with
10.4
Anomaly in general coordinate transformations
We have explained in Chapter 6 that the gauge symmetry itself can contain an anomaly in chiral gauge theory which breaks parity. It was pointed out by Alvarez-Gaume arid Witten that Einstein's general coordinate transformation itself can contain an anomaly in chiral theory, which plays a basic role in the quantum theory of superstring theory. In general, the self-dual tensor fields, for example, can contain gravitational anomalies besides the chiral fermion, but we discuss only the simplest chiral fermion in this book. We consider a general 2n-dimensional Euclidean space-time and quantize only the Dirac field by treating the gravitational and gauge fields as classical background fields. The action for chiral theory is given by
The Einstein equation for the quantized Dirac field is written as (see Appendix B)
where (T^v(x)} stands for the energy-momentum tensor generated by the quantized Dirac field. This (Tllv(x)} with a correct normalization is given by
232
GRAVITATIONAL ANOMALIES
We now analyze the symmetry properties of the quantized (TnV(x)}. Since the action S = f d4x ^/g.C is invariant under the local Lorentz and general coordinate transformations (see Appendix B), we have
and thus we obtain the identity
where the first equality expresses the fact that the path integral itself is independent of the naming of integration variables, and the transition from the first expression to the last one depends on the evaluation of the Jacobian and the invariance of the action. When one writes an infinitesimal transformation of the vicrbein as eJA = eA + JeA and retains the lowest-order term in <5e A . one obtains the identitv
It is important to recognize that the variables such as ip instead of ip are kept fixed when one takes the functional derivative with respect to the vierbein. If one uses the variation corresponding to the local Lorentz transformation in the above identity, we obtain
By noting the anti-symmetry ujml = -wj m of the local Lorentz transformation arid the general structure .of the Jacobian
ANOMALY IN GENERAL COORDINATE TRANSFORMATIONS
233
the identity is rewritten as
If the Jacobian (i.e., the quantum anomaly) for the local Lorentz transformation does not vanish, the symmetric property of the left-hand side of the Einstein equation (10.51) and the anti-symmetric components of the energy-momentum tensor generated by the chiral Dirac field on the right-hand side are not compatible with each other. An infinitesimal form of general coordinate transformations is given by but to deal with the energy-momentum tensor, which is not symmetric in general, in a natural manner we consider a more general coordinate transformation where Dv is the covariant derivative and thus satisfies the metric condition D,,e% = 0. See Appendix B. Namely, we consider a combination of the general coordinate transformation and the local Loreritz transformation which contains the spin connection as a parameter For this variation the above identity (10.55) is written as
If one assumes a general form of the Jacobian
one concludes since ^(x) is arbitrary. If one combines this identity with the fact that the anti-symmetric parts of Tvp,(x) satisfy the identity for the Lorentz transformation (10.58), the symmetric components T,swi(x) satisfy which is called the WT identity for general coordinate transformations. If the anomaly on the right-hand side of eqn (10.65) is not 0. the Einstein equation becomes inconsistent for the quantized chiral Dirac field since the left-hand side of the Einstein equation (10.51) satisfies the (Bianchi) identity
234 10.5
GRAVITATIONAL ANOMALIES General properties of gravitational anomalies
In the calculational scheme of a covariant form of anomaly in Chapter 6, we used the complete orthonormal system defined by f)\f>L and ^>i^L to expand the path integral variables ip arid ip, where An equivalent but more direct way of expanding the variables is to start with
and to note the properties
We then expand
In this case, the action in the path integral is formally diagonalizcd
where we ignored the phase factor which appears as the Jacobian for the change of variables from ip and il> to their expansion coefficients. This phase factor, which does not appear in vector-like theory, is important in the present chiral theory when one analyzes the consistent form of anomalies. Since we evaluate the covariant form of anomalies, we can ignore this phase factor in the present analysis. The Jacobian for the local Lorentz transformation
is given by
GENERAL PROPERTIES OF GRAVITATIONAL ANOMALIES
235
where we sum over all the eigenvalues (negative as well as positive and 0 eigenvalues) in the last expression of cqn (10.72). Similarly, the Jacobian for the general coordinate transformation (by denning
is given by
where we sum over all the eigenvalues in the last expression. An important property of this last expression is that a spurious gravitational anomaly which does not contain 72n+i is automatically eliminated as a result of the use of integration variables • = (g)l/4ip with the weight factor (fl) 1//4 . (See Appendix B.) In this way, the quantum anomalies for the local Lorentz and general coordinate transformations are respectively given by
where we choose the simple regulator function e x. These two anomalies are related to each other in the following manner (by omitting to write the explicit regulator e~x^/M and by noting
236
GRAVITATIONAL ANOMALIES
This relation shows that the anomaly for general coordinate transformations for the Dirac field is automatically obtained once one can evaluate the local Lorentz anomaly. It is known that the anomaly for local Lorentz transformations exists only in d = (4tk + 2)-dimcnsional space-time. An intuitive way to understand this fact is to decompose the d = (4k + 2)-dimensional space-time as a product of d = 2-dimensional and d = 4/fc-dimensional space-times
In this case, if one considers Ai2 in eqn (10.75) and notices
the local Lorentz anomaly A12 is decomposed into a product of the two-dimensional Weyl anomaly (generated by 1) and the 2(n — l)-dimcnsional gravitational chiral U(l) anomaly. As was explained in Section 7.6, the Weyl anomaly appears in two-dimensional space-time and the gravitational U(l) anomaly exists only in d = 4&-dimensional space-time as was explained in Section 10.3. Consequently, one can at least intuitively understand that the local Lorentz anomaly exists only in d = (4k + 2)-dimensional space-time. 10,6
Explicit examples of gravitational anomalies
We first consider a chiral U(l) gauge theory in two-dimensional Euclidean curved space-time, though the gravity in this space-time is trivial. The local Lorentz anomaly is given by
EXPLICIT EXAMPLES OF GRAVITATIONAL ANOMALIES
237
and this expression agrees with (—z/4)e m ™ times the Weyl anomaly in twodimensional space-time discussed in Chapter 7. By using the result of the Weyl anomaly in Section 7.6 (the gauge field does not contribute to the Weyl anomaly in two dimensions) and the relation between the Lorentz anomaly and general coordinate anomalies (10.76), we obtain
where we used In the space-time of general dimensions, one can use for example the heat kernel method to obtain the explicit results of gravitational anomalies by a generalization of eqn (10.19) but the actual calculations are very involved. As for the anomaly of general coordinate transformations, a calculational scheme for general space-time dimensions is known. This method is based on the formula (10.74) in the preceding section which is written as
and the following rules based on this expression: The Dirac matrix 72^+1 m this formula is denned in d = 2n = (4.k + 2)-dimensional space-time, but one may formally regard it as 72n+3 multiplied by
in d = 2n + 2 = (4fc + 4)-dimensional space-time which contains two more dimensions. One may then regard ^ as the spin connection associated with these extra two dimensions by formally assigning the two indices £^n. We then use the gravitational chiral U(l) anomaly (10.48) written in differential forms in (4A; + 4)-dimensional space-time, which was derived in Section 3 of this chapter,
In this formula we make the following replacement of
238
GRAVITATIONAL ANOMALIES
or. in terms of the curvature 2-form
and we retain only the terms linear in £^ and multiply the. result by 2-iri. In this way one finally obtains the gravitational anomaly ^A^x) in cqn (10.81). From this rule, one can understand that the anomaly for the general coordinate transformations appear only in d = (4k + 2)-dimensional space-time. (This rule also shows that the mixed anomaly of gravitational and gauge fields appears in more general 2n-dimeiisional space-time.) An intuitive understanding of the rule (10.84) is obtained if one recalls the evaluation of the gravitational chiral U(l) anomaly in eqns (10.19) and (10.30). If one indicates the two extra dimensions by a tilde, one can write the covariant derivative as
where we assumed that there is no spin connection with mixed indices such as Apmn- We also assume that the spinor field and the spin connection depend only on the first 2n — (4fc + 2)-dimensional coordinates. If one considers that Aiann(x) = £/j,mn(%) as an infinitesimal quantity, and expands the formula (10.19) in powers of Al2.7jlfl(x) and retains the term linear in AllT-nn(x), one obtains the formula (10.81) if one takes into account the numerical constant factor properly. This gives an intuitive understanding of the above rule (10.84). The rule (10.84) in fact reproduces the gravitational anomaly in d=4k+2= 2 dimensions, which was obtained in eqn (10.80) by using a different method. For this case, one starts with the gravitational chiral U(l) anomaly in d = 4, and according to eqns (10.83) and (10.45). when
Under the replacement (10.85), the term linear in £ is given by
In two dimensions, the Rierriann-Christoffcl curvature has only one independent component which is given by the scalar curvature {Jua-gvp)- By using this fact, we have
and consequently the replacement rule gives
EXPLICIT EXAMPLES OF GRAVITATIONAL ANOMALIES
239
This is identical to f
11 CONCLUDING REMARKS In this book we have discussed the basic ideas of quantum anomalies and their applications in the path integral formulation of quantum field theory. The first quantum anomaly was discovered in the detailed analysis of triangle Feynman diagrams. In this sense, the triangle diagram has a special meaning in fourdimensional space-time (in d = 2n dimensional space-time, the (n + l)-gon diagrams have a special meaning). We have explained that all the known (local) quantum anomalies are formulated as the non-trivial Jacobians associated with the symmetry transformations of integration variables in the path integral formulation of field theory. Namely, the path integral measure (or the quantization procedure itself) breaks some classical symmetries. In this sense we have given an answer to the question of what is the quantum anomaly. In particular, the recent demonstration that the chiral anomaly in lattice gauge theory is formulated as the Jacobian for chiral transformations has given additional support for this view of quantum anomalies. At the same time we have also explained that the quantum anomalies are characterized by anomalous commutation relations (or central extensions in commutation relations) in the operator formalism. In either case,. one has to go beyond the naive canonical formulation (or its straightforward path integral transcription) to accommodate quantum anomalies in local field theory. We have thus learned that various characterizations of quantum anomalies arc possible. One might still ask if a more intuitive and elementary characterization of quantum anomalies is possible. In the following, we briefly describe such an elementary characterization. We first recall the fact that quantum anomalies are closely related to divergences but a careful analysis shows that they arc perfectly finite and independent of divergences. In this sense the presence of an infinite number of degrees of freedom is essential for the anomaly. For example, in the representation which diagonalizes 75 we have
and the chiral anomaly in a naive interaction picture corresponds to the evaluation of oscillating infinite series
240
CONCLUDING REMARKS
241
The value of this infinite series generally depends on how to sum the series. In the evaluation of chiral anomaly in the present book we evaluated this series as
by using the complete basis set {ipn(x)} for a Dirac operator lf>. The basis set plays a fundamental role to give a definite value to the oscillating scries. We thus understand that the quantum anomaly is located in the border of divergence and convergence. In the case of a finite or absolutely convergent series the sum is uniquely given, and thus there is no freedom to control the sum of the series by a choice of the basis set. This shows that the quantum anomaly is closely related to the presence of infinite degrees of freedom.58 In connection with the phase operator of the photon discussed in Chapter 3, we also explained that the infinite-dimensional space spanned by the creation operator of a bosonic particle gives rise to the notion of index and provides important information. As another characteristic feature of quantum anomalies, we observe the competition of gauge symmetry and chiral symmetry in the chiral anomaly, for example, and we observe the competition of gauge symmetry (including general coordinate transformations) and Weyl symmetry in the Weyl anomaly. This is reminiscent of a certain kind of "uncertainty relation." For example, the fact that we cannot diagonalize 75 and Tf> simultaneously is closely related to the appearance of the chiral anomaly. To diagonalize lf> implies that we impose gauge invariance, and in this case the relation
implies that 75 cannot be simultaneously diagonalizcd. In the sector of the 0 eigenvalue of $>, 75 is simultaneously diagonalized and the notion of the index arises from this sector. This property at first sight might imply that the appearance of the index is independent of the property that 75 and T/) are not diagonalized simultaneously. But as we explained above the infinite degrees of freedom is fundamental to render the notion of index to a square matrix, and the non-commuting property of two operators plays a basic role in the analysis of quantum anomalies. In fact, when one evaluates the average of the right-hand side of the above commutation relation (11.4) by using the single-particle states in Hilbert space (see eqn (4.54)), one obtains °8The presence of tke-notion of index on-the finite lattice does not contradict this view. We have Tr75 = N+ + n+ — N— — n_ = 0 on the finite lattice. We find a non-vanishing index TrFo = n+ — n _ only when we use a modified operator FS which projects out the freedom N±. See Chapter 9.
242
CONCLUDING REMARKS
which gives both of the explicit breaking of chiral symmetry by the mass term and the quantum breaking of chiral symmetry by the anomaly. In the last step of the calculation, however, we need to specify how to sum the series in increasing order of the magnitudes of eigenvalues Xn (as a rather weak form of regularization). Similarly, when one defines the generator of the Weyl transformation formally by w, one has and thus for the Dirac operator $> = 7ae^(a;)[3^ - (i / 2) Af aat> - igA^} defined in a slightly curved space-time, one has
by noting [w,A^] = 0. namely, the gauge fields are Weyl scalars. The average of the right-hand side of this commutation relation in the flat space-time limit as in eqii (11-5) gives
where the last result is obtained as in eqn (7.48) by applying a suitable cut-off in terms of the eigenvalues of ]/) and retaining only the connected components as the operator. We thus correctly evaluate both of the explicit breaking of Weyl symmetry by the mass term arid the quantum breaking of Weyl symmetry by the anomaly. These examples illustrate that we can understand (certain aspects of) the quantum anomalies as a kind of uncertainty relation originating from the fact that two basic operators are not simultaneously diagonalized. We now briefly comment on several other topics related to quantum anomalies which we did not discuss in the body of the book. We have so far concentrated on the explicit evaluation of Jacobians in the path integral. There are elegant
CONCLUDING REMARKS
243
though somewhat abstract constructions of the anomalies related to gauge transformations, which are sometimes called the descent formula. In this construction of the integrable form of gauge anomalies in d = In dimensions, one starts with the chiral U(l)-type anomalies in d = In + 2 dimensions. The differential form is convenient for this construction and we define
We start with the, relation (see also the Chern character in Chapter 10)
where W2n+i (A, F] is a In + 1 form. This relation is derived by a generalization of the derivation of the Chern-Simoiis form in Chapter 5. We next define
where Au stands for the gauge transformation of A
We note that
by recalling the definition of ^/(U, A, F) and the gauge invariance of tr Fn+l. For an infinitesimal a (a;), we have
where Aa(A. F) gives the gauge anomaly in eqn (6.140) (or eqn (6.136) in the case d = 4) written in form notation. The Wess-Zumino term is then written as
where the integration domain D (= disc) is a generalization of the domain in fourdimensioiial theory in cqn (6.160) (or eqn (6.148) with R/j, = 0) for the Euclidean metric. As for a justification of these constructions, readers are referred to the references at the end of this book. We next comment on the non-perturbative (global) SU(2) anomaly which appears in the Euclidean path integral
where L^ = La^(x)Ta stands for the gauge field belonging to SU(2). We impose the boundary condition on the allowed set of gauge transformations U(\x =
244
CONCLUDING REMARKS
oo) = 1. The Euclidean space-time is then regarded as the compact S4. The chiral SU(2) gauge theory is anomaly-free for a gauge transformation which is smoothly connected to the identity, as we already explained. However, it is known that the homotopy from the space-time S4 to the group manifold of SU(2) is given by and thus there are two kinds of "large" gauge transformations: one is smoothly connected to the identity and the other is a non-trivial one which is not smoothly connected to the identity. The non-trivial one applied twice is reduced to the trivial one which is smoothly connected to the identity. Because of 7r/i(SU(3)) = 0, the non-trivial gauge transformation in SU(2) when embedded in SU(3) is now smoothty connected to the identity. We thus formally extend the SU(2) gauge theory to an SU(3) theory by enlarging the SU(2) doublet ib to the SU(3) triplet ip, but the gauge field L0^ still couples to only the generators of SU(2). The non-trivial gauge transformation in SU(2), which we denote by g, is embedded in SU(3) in the form
where g is a 2 x 2 matrix. When one varies the gauge transformation from the identity to g, the gauge transformation assumes values in SU(3) in the intermediate stages. In the definition of the Wess Zumino term, the gauge transformation assumes the specific value g at the boundary of the disc D. We then consider the gauge transformation
and obtain the identity
where we extracted the Wcss-Zumino term as an integrated Jacobian for the gauge transformation inside SU(3), where one can smoothly connect the identity to g. See eqn (6.141). A detailed analysis shows that,
The path integral over all the possible gauge field configurations thus gives rise to the result
CONCLUDING REMARKS
245
where we adopt the path integral prescription to take a sum over the (large) gauge freedom associated with Zi. Note that the path integral is invariant under an ordinary SU(2) gauge transformation which is smoothly connected to the identity, and thus the ordinary gauge transformation is gauge fixed in a conventional manner when one performs the path integral over the gauge field with the Yang-Mills action added. The path integral of a single chiral SU(2) doublet thus has no physical meaning, which was first pointed out by Witten. In the Wcinberg-Salam theory we have an even number (= 4) of SU(2) doublets in each generation by counting lepton and quark multiplets, and thus the above phase factor (11.21) becomes unity and the difficulty associated with the global SU(2) anomaly does not appear. A detailed analysis of the global SU(2) anomaly requires some mathematical background, and we give references and a monograph on this subject at the end of the present book. The anomaly cancellation in superstriiig theory is another important subject which we did not discuss in this book. This issue was analyzed by Green and Schwarz to show the existence of quantum mechanically consistent superstriiig theory. The basic idea and the machinery to analyze this problem are explained in the present book, but a concrete analysis of this problem requires further detailed knowledge of gravitational anomalies. It is also interesting to evaluate the quantum anomalies directly in superstring theory, and interested readers are referred to references given at the end of the present book. We did not discuss supersymmetry and associated quantum anomalies, which are subjects of great interest among particle physicists recently. When one considers supersymmetric theory in terms of its component fields, all the evaluations of quantum anomalies are performed by the methods given in the present book. However, some of quantum anomalies which appear to be independent of each other are often related to each other in the framework of supersymmetric theory. To understand such a relation it is necessary to generalize the calculational schemes in this book to evaluate the quantum anomalies in the superfield formulation. It is in fact known that the anomalies can be evaluated as non-trivial Jacobians in the path integral formulation in superspacc. Interested readers are referred to the references at the end of the present book.
APPENDIX
A
BASICS OF QUANTUM ELECTRODYNAMICS In this appendix we summarize the basic aspects of quantum electrodynamics. Though the knowledge summarized here is not essential to understand the main part of the present book, it helps to understand some of the analyses better. A.I
Quantum electrodynamics
The basic action of quantum electrodynamics is defined as a combination of the Maxwell action and the Dirac action
An important characteristic of quantum electrodynamics is that the electromagnetic field An. couples to the Dirac field -0 only through the covariant derivative
As a result, the action is invariant under the gauge transformation, which is a generalization of the gauge transformation in Maxwell theory,
where u>(x) is an arbitrary function. The co-variant derivative is transformed under this gauge transformation as
The basic principle of gauge theory is formulated, following the reversed way of reasoning, as a requirement that the action is invariant under the phase transformation i}>'(x) — U(x)ip(x) of the matter field tp. Both the gauge field and covariant derivative are then introduced, and the gauge transformation rule of the gauge field (the photon in the present case) is determined. Consequently, the action of the gauge field itself is fixed so that it is invariant under this gauge transformation. Moreover, it is known that the action is uniquely determined if one requires the renormalizability (i.e., the consistent treatment of higher-order 246
QUANTUM ELECTRODYNAMICS
247
quantum effects) of the theory thus constructed. This is the reason why the gauge principle (and gauge theory) is so powerful and useful. In the present example the gauge transformation is composed as
namely, following the rules of addition. The gauge group thus forms an Abelian U(l) group, and for this reason the electromagnetic field A^ is called an Abelian gauge field. In general, one can introduce the gauge field consistently for an arbitrary compact group and the gauge field associated with a non-Abelian group is called the Yang-Mills field. The name of the covariant derivative comes from the fact that D^(x) is transformed
in the same manner as the original field ijj(x) under the gauge transformation. The electromagnetic field strength tensor is generally defined as the "curvature"
by using the covariant derivative. From this definition one can understand that the electromagnetic field strength is gauge invariant
From the Jacob! identity
one can derive two of the four Maxwell's equations
Another characteristic feature of the action of quantum electrodynamics is that the interaction term which is gauge invariant by itself (the Pauli term) gFnVtf}[jlt,jl']^j is not included. This is because the action with the Pauli term spoils the renormalizability of the theory. The action such as that of quantum electrodynamics where the matter fields interact with the gauge field only through the covariant derivative is called a theory with minimal coupling. The path integral quantization of quantum electrodynamics is given by the analyses in Chapters 2 and 3 by
248
BASICS OF QUANTUM ELECTRODYNAMICS
We employ the £ gauge condition (3.52) and also write the Faddeev Popov fields explicitly, though they are not essential in the present Abelian theory. The integral over the Dirac field is defined by the left derivative in terms of Grassmaim numbers. Following Schwinger we introduce the ordinary c-number source Jli(x) for the gauge field A^ and the Grassmann number sources fj(x) and 77(0;) for Dirac fields. When one writes the action appearing in the exponential of the path integral as Sj, the equations of motion in quantized theory arc given by considering the variational derivative of 5j with respect to, for example, A,L and ip as
The matrix elements of those quantum operator equations should vanish. To be precise, if one replaces the operator field variables in the matrix element by derivatives with respect to source functions by using the action principle of Schwinger and if one uses the above path integral representation, the equations of motion written in terms of c-number fields appear in the integrand of the path integral as
and
where we used the "translational invariarice" in functional space of the path integral measure
both for bosons and fermions, namely, the fact that the definite integrals of derivatives vanish. The quantum operator equations are thus ensured in the path integral. The path integral formula thus defines the correct quantum theory for interacting gauge and Dirac fields.
INTERACTION REPRESENTATION AND PERTURBATION FORMULAS 249 A.2 Interaction representation and perturbation formulas We now separate the (effective) Lagrangian density in the above path integral formula (A. 11) as
where £j stands for the free part consisting of only the terms quadratic in field variables and stands for the interaction part. The perturbative calculations are then reduced to the evaluation of
In the last two expressions the interaction is specified by the derivatives with respect to source terms, but when one brings the exponential terms with derivatives inside the path integral the functional derivatives act on the source functions contained in Cj' and produces the corresponding field variables. We thus reproduce the starting formula. The important property of the last formula in eqn (A.18) is that the path integral is performed only for the free fields. This corresponds to the interaction representation in the operator formalism. In the interaction representation all the field variables satisfy the equations of motion of free fields. In fact, if one applies the Schwinger action principle to the path integral denned in terms of Cj the Dirac field ^>, for example, satisfies the free field equation
The above perturbation formula (A.18) thus corresponds to the Dyson formula for the S-matrix. It is interesting that the operation related to the Wick theorem
250
BASICS OF QUANTUM ELECTRODYNAMICS
in operator formalism is realized by the functional derivative with respect to source functions. We can perform the path integral explicitly for free fields. To show this we define the following Feynman propagators
where the derivative <9A acts on the coordinate x. By using those functions we make the following change of variables in the path integral
If one uses the fact that the path integral measure is invariant under the above change of variables, for example, T>ip = 'Dt/j', the path integral for free fields is given by
When one uses the normalization of the path integral in the interaction picture (0. +oo|0, — oo}j_i0 = 1, the path integral for the free fields is given by
When one applies the Schwinger action principle to the above path integral of free fields (A.23) one obtains
INTERACTION REPRESENTATION AND PERTURBATION FORMULAS 251
where we used the representation 6^\x —2'y)?4=exp[— / , ip(x — ?y)] of the delta function. The last formula is obtained by performing the integral over po by deforming the contour to the direction where the exponential function decreases and the result is known to agree with the result in the operator formalism. The Feynman ie prescription dictates that the negative-energy solutions of the Dirac equation propagate only in the negative time direction and, as a result, they represent the anti-particle (the positron in the present context) with positiveenergy propagating in the positive time direction. (The Feynman ic prescription is also understood as imposing the positive energy condition in the path integral formulation, as was mentioned in Chapter 2 in the present book.) Similarly, we obtain
where we give the result of the simplest Feynman gauge with £ = 1 in the last expression. This result is also known to agree with the operator formulation for the Feynman gauge £ = 1, though we do not demonstrate it here. The ie prescription specifies that the negative-energy solutions of the Maxwell equation propagate only in the negative time direction and, as a result, they describe the anti-particle (in the present case of a real field A^, the anti-particle and particle are not distinguished) with positive-energy propagating in the positive time direction.
252
BASICS OF QUANTUM ELECTRODYNAMICS
The h'nal formula for perturbation theory in quantum electrodynamics is thus given by
If one expands this formula in powers of the coupling constant e up to a suitable order and then performs the functional derivative of (0, +oo|0, -oo)j with respect to source functions by suitable times (and setting the remaining source functions to zero), the formula for the Green's function which is expressed as the vertex functions expressed by 67^ connected by the Feynman propagators Dp and SF is obtained. The Feyrirnan diagrams give an intuitive picture of this representation of the Green's function.
APPENDIX B FIELD THEORY IN CURVED SPACE-TIME In this appendix we briefly summarize the basic properties of field theory in curved space-time. B.I Coordinate transformation and energy-momentum tensor The Lagrangian of Einstein's general theory of relativity in the presence of QCDtype gauge theory is defined by
where K = 8?rG with the Newton constant G. The Greek indices which describe the Minkowski coordinates of the curved space and the Roman indices which describe the Lorentz coordinates of the flat space attached to each point of the curved space are related to each other by the vierbein e^(x). The metric of the flat Lorentz frame is denoted as Gmn. The vierbein is connected to the metric by
The raising and lowering of the Minkowski indices p is realized by the metric 9tj,v(x) and the raising and lowering of the flat Lorentz indices m is realized by Gmn. The basic symmetry of Einstein's theory, namely, the general coordinate transformation, does not admit a double-valued representation such as the rotation group for spin 1/2 in flat space, and thus we represent the Dirac field as a double-valued representation of the Lorentz group of the flat space attached to each point of the curved space-time. The covariant derivative in the curved space is generally given by
and contains the generators Smn of the Lorentz group and the generators U& of the general coordinate transformation GL(4, R) in addition to the generators Ta of the ordinary gauge group. These generators are defined by
253
254
FIELD THEORY IN CURVED SPACE-TIME
To be more explicit, the generators for spin 1/2 and 1 are, respectively, given by
On the other hand, the generators U® of the general coordinate transformation are given for the covariant vector Av and the contravariant vector A" respectively by Consequently, the covariant derivatives are defined by
The gauge field A™n appearing in the covariarit derivative is called the spin connection and the gauge field Fjg is called the affine connection, respectively, and these are expressed in terms of e^ and g^ as
The fields A™n and Pg in the present formulation, in which the torsion freedom is ignored, arc also called respectively the Ricci rotation coefficient and the Christoffel symbol. By recalling the basic relation of Riemann which states that the geometrical length is independent of the choice of coordinate systems
the general coordinate transformation laws of the quantities with Minkowski indices are given by
For an infinitesimal transformation
the general coordinate transformation is expressed in terms of the generators as
COORDINATE TRANSFORMATION
255
The transformation law of the metric tensor is derived from the transformation law of the vierbein as
and the transformation law of the affirie connection is derived from its definition as
The infinitesimal local Lorente transformation is given in terms of the parameter ojmn(x) as
by using the generators of the Lorentz group, just as the ordinary gauge transformations. The transformation law of the spin connection in the last expression is obtained from the transformation of the covariant derivative
or from the definition of A™n in terms of e% and the transformation law of e% . Einstein's theory contains two gauge fields F^ and A™n, but both of these fields are expressed in terms of the vierbein e'k. As the gauge freedom, the vierbein contains the localized Poincare transformations with 10 parameters, namely, four localized translations £ M (x) and six localized Lorentz transformations umn(x). In the present formulation, the condition called the metric condition
is automatically satisfied. This metric condition implies that the inner product of two vectors parallel transported from a point x to the point x + dx is defined by the metric g^, (x + dx) . As an application of this metric condition, we can show that the vierbein e% at an arbitrary point, for example, at the origin, is chosen to be
This choice of the vierbein is used in the body of the present book to evaluate quantum anomalies. The proof of this choice of the vierbein proceeds as follows:
256
FIELD THEORY IN CURVED SPACE-TIME
We first apply a coordinate transformation x'v- — a^x" to the vicrbein e'Jk by choosing a suitable constant a^ near the origin such that
We next apply a specific infinitesimal coordinate transformation near the origin
such that F^(0) = 0 without changing e'£(Q)- Finally, we apply a local Lorentz transformation parametrized by 'jjmn(x) - -A"m(Q)x^' so that A'™n(Q) = 0 without changing e^*(0). When one uses the two relations 1^(0) = 0 and A'™n(Q) = 0 in the metric condition (B.I7), we obtain the desired result. The metric condition shows that we can freely commute the covariant derivative with e% and g^v. For example, when one applies the commutator of covariarit derivatives to an arbitrary covariant vector Ap = e™An, one obtains
This relation shows that the Riernann-Christoffel curvature tensor evaluated in terms of the affinc connection Pg^ .
agrees with the curvature tensor Rmn^ evaluated in terms of the spin connection A™n. As an important local symmetry of the matter and gauge fields in gauge theory, we have the Weyl symmetry defined by
where the transformation law of the spin connection A™n is derived from the transformation law of the vierbein e£, since A™n is expressed in terms of e£. The Weyl transformation changes the length as
but the local angle is preserved, and for this reason it is also called a conformal transformation. When one defines g = det.g/u,, it is confirmed that the matter
COORDINATE TRANSFORMATION
257
part of the action / d4x ^f^gC- in terms of the Lagrangian (B.I) is invariant under the Weyl transformation if one sets the fermion mass m = 0. The action for a massless scalar theory is also rendered invariant under the Weyl transformation if one chooses £ = ^g^d/^dup + \R
is obtained. The (naive) energy-momentum tensor T^v(x) generated by the matter fields in this equation is defined by°9
wtu,re we symmetrized the expression with respect to the replacement of ?/> and ijj. We also normalized the completely anti-symmetric symbol e1230 = 1 and used the property of the Dirac matrix {^k,Smn} = ekmnljij5. To study the symmetry requirement on the energy-momentum tensor, we recall the basic property of the Einstein Hilbert action. We first note the following relation
valid for an arbitrary variation 5gp'v(x) of the metric tensor. As a specific variation of the metric, we consider the variation induced by a coordinate transformation
S9
ln deriving this expression, it is convenient to note the relation SA™n — (l/2)cmXen>>(8CXpl, - <5CpV - <5C, iAp ), where 5CXpl, -= e*(DpSek^ - Dp,5ekp). This relation follows from the fact that the difference of two spin connections behaves as a tensor.
258
FIELD THEORY IN CURVED SPACE-TIME
When one recalls that the Einstcin-Hilbert action is invariant under the coordinate transformation and that the covariant derivative commutes with the metric, the above relation (B.29) valid for an arbitrary variation implies
Since £"(#) is arbitrary we conclude
When one combines this relation with the Einstein equation (B.27). one concludes the requirements on the classical energy-momentum tensor
B.2
Path integral measure in gravitational theory
It is necessary to define the path integral measure carefully when one analyzes the quantum theory of gravity and the quantization of fields in curved spacetime in the path integral formulation. In particular, the treatment of general coordinate transformations is subtle and a naive treatment could induce spurious anomalies. Also the Weyl anomaly is uniquely specified only when one uses the general coordinate invariant measure. In the following, we discuss the definition of the path integral measure which does not induce spurious breaking of general coordinate transformations. It is convenient to use the BRST symmetry in this analysis, though the BRST symmetry itself is somewhat technical. The basic idea of the BRST transformation is based on writing the parameter f M (a;) of general coordinate transformations in the form £^(x) = i\dl(x) where A is a constant real Grassmann parameter and c>L (x) is the Grassmann variable called the Faddeev-Popov ghost field. When one uses the BRST supcrfield notation, one writes the original field variable together with its variation under the BRST transformation by using a constant Grassmann parameter 0 as
whore we used the transformation law of tf>, 5tp(x) = ip'(x) — ib(x) = £.pdp'ip(x), and wrote the anti-ghost field c-n(x) together with its variation, the auxiliary field B^ (x).
PATH INTEGRAL MEASURE IN GRAVITATIONAL THEORY
259
The BRST transformations are denned as a translation in the Grassmann parameter 9 ->• 9 + X. Consequently, the first component of the superfield, which does not contain the parameter 8, varies by an amount proportional to the parameter A and the second component, which contains the parameter 0, remains invariant. To be explicit, the transformation law 6\ of the ghost field cll(x) is given by (noting that both CM(O;) and A are Grassmann numbers)
Namely, the, first component receives a transformation proportional to the second component, and if one uses its result the second component becomes invariant. The BRST transformation is thus consistently denned. The treatment of the antighost is exceptional, and its variation is defined by fi\c^(x) = \B(x). 5\B(x) = 0 and. as a result the measure X>cMX>£>M is BRST invariant. When one uses the notation of the BRST superficld, the gauge condition d^g^ = 0. for example, is defined by collecting the terms linear in 9 in the expression
which defines both the gauge fixing and compensating terms simultaneously. In passing we note that the transformation law of the ghost (B.34) does not appear as a transformation law of a contravariant vector quantity, but if one considers the differential of the superfield itself one obtains which corresponds to the replacement. £p (a;) —> i9cp (x) in the general coordinate transformation of the contravariant vector dcIJ-(x). This property plays a fundamental role when we discuss the BEST invariant path integral measure. We now discuss the path integral measure which is invariant under the BEST symmetry associated with general coordinate transformations. We start with the simplest fields which do not carry the Minkowski indices such as the fermions ?/>(x) and $(x). We define the weight 1/2 fields60
and examine the path integral of the mass term of the fermion
In this relation, the action on the left-hand side is invariant under the general coordinate (or associated BRST) transformation and the right-hand side is a 60
The weight 1/2 fields are defined as the field variables (without the Minkowski indices) multiplied by the weight factor (—g) 1 / 4 . and these variables transform under general coordinate transformations as in eqn (7.19).
260
FIELD THEORY IN CURVED SPACE-TIME
constant independent of the metric variable. We thus conclude that the path integral measure on the left-hand side is invariant under the general coordinate (or BRST) transformation. One can in fact confirm that the Jacobian for the BRST transformation becomes trivial, i.e., 1, independently of the choice of basis sets, and thus free of general coordinate anomalies. See eqn (7.21). We now generalize this analysis to the cases such as the vector field Aa(x) and the second-rank tensor field Aap(x). We first convert these fields into fields without the Minkowski indices Aa(x) = e^Aa(x) and Aab(x) = e%e%Aa/3(x) byusing the vierbein. We then consider the weight 1/2 field prescription as above
which define the general coordinate (or associated BRST) invariant path integral measure. When we give two definitions of the measure in eqn (B.40), the latter definition is obtained from the first one by first extracting the vierbein as the Jacobian and then distributing the Jacobian to all the degrees of freedom equally. This second definition may not be said to be precise, but we cannot convert the metric g^v into a quantity without the Minkowski indices by multiplying the vierbein. The second definition of the measure works for the case of the metric tensor also. We also give the definitions for the vector and tensor fields in arbitrary d = n dimensional space-time in eqn (B.40) for the convenience of applications in the present book. In this way the path integral measure invariant under the BRST symmetry associated with the general coordinate transformation is given by
The part containing the metric is defined by d/j,(gal3) = T)[(-g}kgai3] as we explained above. The measure for the ghost variable is determined by c^ = (-gr) 1 / 4 e^c M by using the fact that the differential of the ghost variable is transformed as a contravariant vector under the BRST transformation. In this path integral measure the BRST invariancc of the part dfj^(ga^)Vc>J is not manifest, but its invariance is confirmed by showing that the result after BRST transformation agrees with the result before the BRST transformation as
PATH INTEGRAL MEASURE IN GRAVITATIONAL THEORY
261
The above proof proceeds as follows: The measure for the metric is invariant for a fixed c^(x, 0) we can set 0 = 0. It is important to keep in mind that one needs to consider the metric and the ghost variables always together in the analysis of the gravitational path integral measure.
APPENDIX
C
REFERENCES WITH BRIEF COMMENTS We first give some references which are directly related to the descriptions of the present book with brief comments. We then present some references to further advanced research in path integrals and quantum anomalies. We also briefly comment on the subjects or applications of quantum anomalies which were not covered in the present book. C.I
Genesis of quantum anomalies
The basic references to modern field theory (renormalization theory) are found in: 1. J. Schwinger, Quantum, electrodynamics. Dover (1958). The letter of Tomonaga to Oppenheimer is printed in: 2. S. Tornonaga, Phys. Rev. 74 (1948) 224. The evaluation of the two-photon decay of the neutral TT meson which eventually led to the discovery of quantum anomalies was first performed by: 3. H. Fukuda and Y. Miyamoto, Prog. Theor. Phys. 4 (1949) 49. This calculation was refined by: 4. H. Fukuda, Y. Miyamoto, T. Miyajima, S. Tomonaga, Prog. Theor. Phys. 4 (1949) 385; H. Fukuda. Y. Miyamoto, T. Miyajima, S. Tomonaga, S. Oncda and S. Sasaki, Prog. Theor. Phys. 4 (1949) 477. 5. J. Steinberger, Phys. Rev. 76 (1949) 1180. 6. J. Schwinger. Phys. Rev. 82 (1951) 664. The Pauli-Villars rcgularization was introduced in: 7. W. Pauli and F. Villars. Rev. Mod. Phys. 21 (1949) 434. 8. S.N. Gupta, Proc. Phys. Soc. A 66 (1953) 129. The detailed analyses of the decay of the soft TT meson and the triangle diagrams in spinor electrodynamics led to the modern formulation of quantum anomalies: 9. J.S. Bell and R. Jackiw, Nuovo Cim. A 60 (1969) 47. 10. S.L. Adler, Phys. Rev. 177 (1969) 2426. We here give some well-written readable reviews of quantum anomalies: 11. S.L. Adler, in: Lectures on elementary particles and quantum field theory. S. Deser et al. (eds.), MIT Press (1970). 12. S.B. Treirnan, R. Jackiw, B. Zumino and E. Witten, Current algebra and anomalies. World Scientific (1985). 262
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13. P. van Nieuwenhuizen, Anomalies in quantum field theory: cancellation of anomalies in D = 10 super-gravity. Leuven University Press (1988). 14. R. Bertlemann, Anomalies in quantum field theory. Oxford University Press (1996). In these books and reviews together with a more recent lecture note by J. ZinnJustiii, Chiral anomalies and topology, hep-th/0201220, readers can learn various viewpoints on quantum anomalies complementary to those in the present book. C.2
The Feynman path integral and Schwinger's action principle
The classic paper of Dirac is found in: 15. P.A.M. Dirac, Phys. Zeit. Sow. 3 (1933) 64. The Feynman path integral is given in, for example: 16. R.P. Feynman, Rev. Mod. Phys. 20 (1948) 367; Phys. Rev. 80 (1950) 440. Other references are found in [1]. For Schwinger's action principle, see, for example: 17. J. Schwingcr, Phys. Rev. 91 (1953) 713. Further references are found in [1]. A readable account of the action principle is given in: 18. C.S. Lam. Nuovo Cim. 38 (1965) 1755. The fermioriic path integral is discussed in, for example: 19. F.A. Berezin, The method of second quantization. Academic Press (1966). 20. T. Kashiwa, Y. Ohnuki and M. Suzuki, Path integral methods. Clarendon Press (1997). 21. M.S. Swarison, Path integrals and quantum processes. Academic Press (1997). 22. M. Chaichian and A. Demichcv, Path integrals in physics. Inst. of Phys. Pub. (2001). C.S
Quantum theory of photons and the phase operator
The quantization of the electromagnetic field was clearly formulated in: 23. P.A.M. Dirac, Proc. Roy. Soc. of London A 114 (1927) 243 where the notion of the photon phase operator was introduced. The general treatment of constrained systems is found in: 24. P.A.M. Dirac, Lectures on quantum field theory. Yeshiva University, New York (1966). The path integral formulation of gauge theory was given in: 25. L.D. Faddeev and V.N. Popov, Phys. Lett. B 25 (1967) 29. The BRST symmetry was introduced and its applications were discussed in: 26. C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42 (1975) 127. 27. J. Zinn-Justin, Lecture notes in physics 37 Springer-Verlag (1975).
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REFERENCES WITH BRIEF COMMENTS
For reviews of the BRST symmetry sec, for example: 28. N. Nakanishi and I. Ojima, Covariant operator formalism, of gauge theories and quantum gravity. World Scientific (1990). 29. J. Gomis. J. Paris and S. Samuel, Phys. Rep. 259 (1995) 1. For an attempt to construct the photon phase operator (including the past references) see: 30. D.T. Pcgg and S.M. Barnett. Phys. Rev., A 39 (1989) 1665. The connection between the phase operator of the photon (and harmonic oscillator) and the index theorem was discussed in: 31. K. Fujikawa, Phys. Rev. A 52 (1995) 3299. C.4
Regularization of field theory and chiral anomalies
A brief summary of U(l) gauge theory (quantum electrodynamics) is given in Appendix A, and we here give several standard textbooks: 32. J.D. Bjorkeii and S.D. Drell, Relativistic quantum fields. McGraw-Hill (1965). 33. K. Nishijima, Fields and particle,?. Benjamin (1969). 34. N.N. Bogoliubov arid D.V. Shirkov. Introduction to the theory of quantized fields (3rd edn). John Wiley (1980). 35. ,T.C. Taylor, Gauge theories of weak interactions. Cambridge University Press (1976). 36. L.D. Faddeev and A.A. Slavnov. Gauge fields. Benjamin/Cummings (1980). 37. P. Ramond, Field theory—A modern primer. Addison-Wesley (1989). 38. M. Kaku, Quantum field theory: a modern introduction. Oxford University Press (1993). 39. M.E. Peskin and D.V. Schroeder, Quantum, field theory. Addison-Wesley (1995). 40. S. Weinberg, The quantum theory of fields, I and II. Cambridge University Press (1995). 41. J. Zinn-Justin, Quantum field theory and critical phenomena (4th edn). Oxford University Press (2002). The notation of the present book follows that of the textbook [32]. As for the contents of Chapter 4, the higher-derivative regularization is explained in detail in [36]. A readable account of the Wick rotation is found in: 42. P. van Nieuwenhuizcn and A. Waldron, Phys. Lett. B 389 (1996) 29. The covariant regularization of the currents discussed in Chapter 4 follows the references: 43. K. Fujikawa, Phys. Rev. D 29 (1984) 285. 44. K. Fujikawa. Nucl. Phys. B 428 (1994) 169. The covariant regularization is closely related to the generalized Pauli-Villars regularization:
THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES
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45. S.A. Frolov and A.A. Slavnov, Phys. Lett. B 309 (1993) 344. 46. R. Narayanan and H. Neuberger, Phys. Lett. B 302 (1993) 62. 47. S. Aoki and Y. Kikukawa, Mod. Phys. Lett. A 8 (1993) 3517. In ordinary perturbative calculations, the dimensional regularization in: 48. G. 't Hooft and M. Veltman, Nud. Phys. B 44 (1972) 189. 49. C.G. Bollini and J.J. Giambiagi, Nuovo Cim. B 12 (1972) 20 is often used, but this regularization is not convenient for the analyses of quantum anomalies. The analyses of triangle Fcynman diagrams in Chapter 1 are explained in detail in [9] [10]. The Adler-Bardeen theorem is given in: 50. S. Adler and W.A. Bardccn, Phys. Rev. 182 (1969) 1517. 51. A. Zee, Phys. Rev. Lett. 29 (1972) 1198. 52. K. Higashijima, K. Nishijima and M. Okawa, Prog. Theor. Phys. 67 (1982) 668. C.5
The Jacobian in path integrals and quantum anomalies
Chapter 5 gives a path integral reformulation of classic Feynman diagrammatic analyses, which directly leads to the chiral identities with anomaly terms present. This path integral method was given in: 53. K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195. 54. K. Fujikawa, Phys. Rev. D 21 (1980) 2848; 22 (1980) 1499(E). The proof of the Atiyah-Singer index theorem, which is based on a technique closely related to the method in physics, is found in: 55. M. Atiyah, R. Bott, and V.K. PatodL Invent. Math. 19 (1973) 279. An explicit analysis of this theorem in the context of flat four-dimensional Euclidean space is given in: 56. R. Jackiw and C. Rebbi, Phys. Rev. D 16 (1977) 1052. Our analysis of the index theorem closely follows these two references. The instanton solution is found in: 57. A.A. Belaviri, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett. B 59 (1975) 85. The physical implications of this solution have been clarified in: 58. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. 59. G. 't Hooft, Phys. Rev. D 14 (1976) 3432. 60. R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. 61. C.G. Callan, R.F. Dashen arid D.J. Gross, Phys. Lett. B 63 (1976) 334. Reference [59] analyzes the so-called U(l) problem related to the 77' meson from the viewpoint of the competition of the Nambu-Goldstone theorem and the quantum anomaly, and references [60] [61] introduced the notion of the 0 vacuum. The U(l) problem, strong CP a,nd axions are reviewed in:
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62. S. Coleman, The uses of instantons. in: Aspects of symmetry. Cambridge University Press (1985). 63. G. 't Hooft, Phys. Re.pt. 142 (1986) 357. 64. J.E. Kirn, Phys. Rc.pt. 150 (1987) 1. The analyses of the Jacobian in a general context have been given in: 65. A. Diaz, W. Troost, P. van Nieuwerihuizen and A. Van Proeyen, Int. J. Mod. Phys. A 4 (1989) 3959. 66. W. Troost. P. van Nieuwerihuizen and A. Van Proeyen, Nud. Phys. B 333 (1990) 727. The regularization of the Jacobian has been discussed in: 67. A.A. Andrianov, L. Bonoraand R. Gamboa-Saravi. Phys. Rev. D 26 (1982) 2821. 68. D.W. McKay and B.-L. Young, Phys. Rev. D 28 (1983) 1039.
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Quantum breaking of gauge symmetry
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QUANTUM BREAKING OF GAUGE SYMMETRY
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84. A.A. Andrianov and L. Bonora, Nud. Phys. B 233 (1984) 232; 233 (1984) 247. Reference [84] discussed the rotation of the gauge field to a pure imaginary value. The Pauli-Villars regulari/ation in the path integral was discussed in [54]. For the anomaly cancellation in gauge theory (in particular, in WeinbergSalam theory), see: 85. C. Bouchiat, J. Iliopoulos and P. Meyer, Phys. Lett. B 38 (1972) 519. 86. D.J. Gross and R. Jackiw, Phys. Rev. D 6 (1972) 477. The analysis of anomalous gauge theory in the present book follows: 87. K. Fujikawa, Phys. Lett. B 171 (1986) 424. The algebraic analyses of the Gauss operators in anomalous theory are found in, for example: 88. L.D. Faddeev, Phys. Lett. B 145 (1984) 81. 89. S.G. Jo, Nud. Phys. B 259 (1985) 616. 90. M. Kobayashi, K. Sco and A. Sugamoto, Nud. Phys. B 273 (1986) 607. There are attempts to give a consistent physical meaning to anomalous gauge theories. See, for example: 91. 92. 93. 94. 95.
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A more mathematical differential geometrical method, such as cohomology and descent formulas, which were only briefly commented on in the present book, are found in [99] and the following references 102. R. Jackiw, in: Relativity, groups and topology II. B.S. DeWitt and R. Stora (eds.), North Holland (1984) p. 221. 103. B. Zumino, in: Relativity, groups and topology II. B.S. DeWitt and R. Stora (eds.), North Holland (1984) p. 1291. 104. R. Stora, in: Progress in gauge field theory. H. Lehman et al. (eds.), NATO ASI Series B, Physics, 115, Plenum (1984) p. 543. 105. B. Zumino. Y.-S. Wu and A. Zee, Nud. Phys. B 239 (1984) 477. 106. L. Alvarez-Gaume and P. Ginsparg, Nud. Phys. B 243 (1984) 449. 107. T. Sumitarii, J. Phys. A 17 (1984) L811. For the analyses of the Wess-Zumino term in chiral effective theory, sec the following references in addition to [98] 108. E. Witten, Nud. Phys. B 223 (1983) 422. 109. E. Witten, Nud. Phys. B 223 (1983) 433. 110. K.-C. Chou, H.-Y. Guo, K. Wu and X.-C. Song, Phys. Lett. B 134 (1984) 67. 111. H. Kawai and S.H.H. Tye, Phys. Lett. B 140 (1984) 403. The Goto-Iinarnura-Schwinger term was introduced in: 112. T. Goto and I. Imamura. Prog. Theor. Phys. 14 (1955) 196. 113. J. Schwinger, Phys. Rev. Lett. 3 (1959) 296. The analysis of the cancellation of the Goto-Imamura-Schwinger term in the Gauss operator for quantum electrodynamics in four-dimensions follows: 114. K. Fujikawa, Phys. Lett. B 188 (1987) 115. In the two-dimensional theory such as in the analysis of the Kac-Moody algebra in Chapter 8, the distinction between the quantum anomaly and the GotoImarmira-Schwinger term becomes less clear. Further analyses of the general aspects of path integrals and anomalies are given in: 115. 116. 117. 118.
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The Weyl anomaly and renormalization group
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260. 261. 262. 263. 264. 265. 266. 267. 268. 269. 270. 271.
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Index theorem on the lattice and chiral anomalies
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291. H. Neuberger, Phys. Rev. D 57 (1998) 5417. 292. Y. Kikukawa and T. Noguchi, hcp-lat/9902022. 293. M.F. Golterman, K. Jansen and D.B. Kaplan, Phys. Lett. B 301 (1993) 219. The overlap formula for fermions was discussed in: 294. R. Narayanan and H. Neuberger, Nucl. Phys. B 412 (1994) 574; Phys. Rev. Lett. 71 (1993) 3251; Nucl. Phys. D 443 (1995) 305. 295. S. Randjbar-Dacmi and J. Strathdee, Phys. Lett. B 348 (1995) 543; Nucl. Phys. B 443 (1995) 386; 466 (1996) 335. The recent developments are based on the algebraic relation noted in: 296. P.H. Ginsparg and K.G. Wilson, Phys. Rev. D 25 (1982) 2649 and the construction of an explicit solution (the overlap operator): 297. H. Neuberger, Phys. Lett. B 417 (1998) 141; 427 (1998) 353 which contains an analysis of the index. See also: 298. P. Hasenfratz, Nucl. Phys. B 525 (1998) 401. The index theorem on the lattice was further clarified in: 299. P. Hasenfratz. V. Laliena and F. Niedermayer, Phys. Lett. B 427 (1998) 125 and the recognition of the chiral anomaly as the Jacobian even for the lattice theory in 300. M. Liischer, Phys. Lett. B 428 (1998) 342. was important. For reviews of these developments, see: 301. F. Niedermayer, Nucl. Phys. (Proc. Suppl.) 73 (1999) 105. 302. H. Neuberger, Ann. Rev. Nucl. Part. Sci. 51 (2001) 23. 303. M. Liischer, Lectures given at the International School of Subnuclear Physics. Erice 2000, hep-th/0102028. 304. M. Creutz, Rev. Mod. Phys. 73 (2001) 119. For the detailed evaluation of the chiral anomaly on the lattice, see, for example: 305. 306. 307. 308. 309.
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A generalized form of the Ginsparg-Wilson algebra and its solutions are rioted
in: 310. K. Fujikawa, Nucl. Phys. B 589 (2000) 487.
GRAVITATIONAL ANOMALIES
277
The proof of the locality of the fermion operator, when a suitable admissibility condition is imposed, exists at this moment only for the simplest operator given in [297]: 311. P. Hernandez, K. Jansen and M. Luscher, Nucl. Phys. B 552 (1999) 363. 312. H. Neubergcr, Phys. Rev. D 61 (2000) 085015. The analysis of the locality for the general case is incomplete in 313. K. Fujikawa and M. Ishibashi, Nucl. Phys. B 605 (2001) 365. 314. T-W. Chiu, Nucl. Phys. (Proc. Suppl.) 94 (2001) 733 but sufficient to analyze the chiral anomaly: 315. K. Fujikawa and M. Ishibashi, Nucl. Phys. B 587 (2000) 419. The meaning of the heavy topological excitations N± appearing in the GinspargWilson operators was clarified in: 316. T-W. Chiu, Phys. Rev. D 58 (1998) 074511. 317. K. Fujikawa, Phys. Rev. D 60 (1999) 074505. Abelian chiral gauge theories were formulated on the basis of the GinspargWilson relation in: 318. M. Luscher, Nucl. Phys. B 549 (1999) 295. For non-Abelian theories, however, the formulation is still in the process of construction. See, for example: 319. H. Neuberger, Phys. Rev. D 59 (1999) 085006. 320. H. Suzuki, Prog. Theor. Phys. 101 (1999) 1147. 321. M. Luscher, Nucl. Phys. B 568 (2000) 162; JEEP 0006 (2000) 028. 322. H. Suzuki, Nucl. Phys. B 585 (2000) 471; JHEP 0010 (2000) 039. 323. Y. Kikukawa and Y. Nakayama, Nucl. Phys. B 597 (2001) 519. 324. Y. Kikukawa, Phys. Rev. D 65 (2002) 074504. A non-commutative technique in lattice theory is noted in: 325. T. Fujiwara, H. Suzuki and K. Wu, Nucl. Phys. B 569 (2000) 643. For some interesting applications of the Ginsparg-Wilson fermions, see: 326. H. So and N. Ukita, Phys. Lett. B 457 (1999) 314. 327. O. Bar and I. Campos, Nucl. Phys. B 581 (2000) 499. 328. W. Bietenholz and J. Nishimura, JHEP 0107 (2001) 015. 329. L. Giusti, G.C. Rossi, M. Testa and G. Veneziano, Nucl. Phys. B 628 (2002) 234. C.10
Gravitational anomalies
The calculation of the chiral U(l) anomaly in the presence of the background gravitational field was performed in: 330. T. Kimura, Prog. Theor. Phys. 42 (1969) 1191. 331. R. Delbourgo and A. Salam, Phys. Lett. B 40 (1972) 381.
278
REFERENCES WITH BRIEF COMMENTS
332. T. Eguchi and P.G.O. Freund, Phys. Rev. Lett. 37 (1976) 1251. The appendix in reference [330] gives a concise account of the basic properties of curved space-time. The generalization to spin 3/2 particles is given in: 333. N.K. Nielsen, M.T. Grisaru, H. Romer and P. van Nicuwenhuizen, Nud. Phys. B 140 (1978) 477. 334. R. Endo and T. Kimura, Prog. Theor. Phys. 63 (1980) 683. 335. R. Erido and M. Takao, Phys. Lett. B 161 (1985) 155. For the analysis of the Atiyah Singer index theorem by using supersymrnetric quantum mechanics, see: 336. L. Alvarez-Gaume, J. Phys. A 16 (1983) 4177. A detailed review of topological aspects of anomalies is found in: 337. T. Eguchi, P.B. Gilkey and A.J. Hanson, Phys. Rep. 66 (1980) 213. The calculation of chiral anomalies in arbitrary dimensions in the present book is based on: 338. K. Fujikawa, S. Ojima and S. Yajima, Phys. Rev. D 34 (1986) 3223 which is a simplified version of the calculation in [336] without using supersymmetry. The quantum breaking of general coordinate and local Lorentz transformations has been found and analyzed in detail in: 339. L. Alvarez-Gaume and E. Witten, Nud. Phys. B 234 (1984) 269. As for related early analyses, see: 340. L. Alvarez-Gaume and P. Ginsparg, Ann. Phys. 161 (1985) 423. 341. L.N. Chang and N.T. Nich, Phys. Rev. Lett. 53 (1984) 21. 342. S. Yajima and T. Kimura, Phys. Lett. B 173 (1986) 154. and references [99] [335]. The simple relation between the general coordinate and local Lorentz anomalies was noted in: 343. K. Fujikawa, M. Tomita arid 0. Yasuda, Z. Phys. C 28 (1986) 289. Further general aspects of gravitational path integrals are analyzed in: 344. H.T. Nieh, Phys. Rev. Lett. 53 (1984) 2219. 345. R. Erido. Prog. Theor. Phys. 80 (1988) 311. 346. M. Hatsuda, P. van Nieuwenhuizen. W. Troost and A. Van Proeyeri, Nud. Phys. B 335 (1990) 166. 347. Z. Bern, E. Mottola, S.K. Blau, Phys. Rev. D 43 (1991) 1212. 348. 0. Chandia and J. Zarielli, Phys. Rev. D 58 (1998) 045014. Various path integral evaluations of higher-dimensional anomalies have been discussed in: 349. T. Matsuki, Phys. Rev. D 28 (1983) 2107. 350. R. Delbourgo and T. Matsuki, J. Math. Phys. 26 (1985) 1334.
CONCLUDING REMARKS
279
351. 352. 353. 354.
R. Endo and M. Takao, Prog. Theor. Phys. 73 (1985) 803. J.M. Gipson, Phys. Rev. D 33 (1986) 1061. R. Endo and M. Takao, Prog. Theor. Phys. 78 (1987) 440. J. de Boer. B. Peeters, K. Skenderis and P. van Nieuwenhuizen, Nucl. Phys. B 459 (1996) 631. 355. S. Yajima, Class. Quant. Grav. 13 (1996) 2423. The anomalies in a space with torsion have been analyzed in: 356. S. Yajima and T. Kimura, Prog. Theor. Phys. 74 (1985) 866. 357. S. Yajima, Prog. Theor. Phys. 79 (1988) 535. 358. 0. Chandia and J. Zanelli, Phys. Rev. D 55 (1997) 7580.
C.ll
Concluding remarks
The argument related to "uncertainty relations" in Chapter 11 is based on [54][160]. The details of the global SU(2) anomaly, which we just briefly mentioned, are found in [108] [109] and 359. E. Witten, Phys. Lett. B 117 (1982) 324. 360. S. Elitzur and V.P. Nair, Nucl. Phys. B 243 (1984) 205. 361. L. Alvarez-Gaume, S. Delia Pietra, V. Delia Pietra, Phys. Lett. B 166 (1986) 177. 362. R.A. Baadhio, Quantum topology and global anomalies. World Scientific (1996). The Green-Schwarz mechanism of the anomaly cancellation in superstring theory is given in: 363. M.G. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. 364. M.G. Green, J.H. Schwarz and P.O. West, Nucl. Phys. B 254 (1985) 327. 365. D.J. Gross, J.A. Harvey, E..J. Martinec and R. Rohm. Nucl. Phys. B 256 (1985) 253. A direct evaluation of the anomaly in superstring theory is found in, for example: 366. H. Suzuki and A. Sugamoto, Phys. Rev. Lett. 57 (1986) 1665. 367. K. Pilch, A.N. Schellekens and N.P. Warner, Nucl. Phys. B 287 (1987) 362. The detailed references to string theory are found in: 368. M.G. Green, J.H. Schwarz and E. Witten. Superstring theory I,II. Cambridge University Press (1987). 369. K. Kikkawa, Quantum theory of strings. Asakura Publishing Co. (1991). 370. J. Polchinski, Superstring theory I.II. Cambridge University Press (1998). There are a vast number of references to anomalies in supersymmctric theories. We just mention: 371. S. Ferrara and B. Zumiiio. Nucl. Phys. B 87 (1975) 207.
280
REFERENCES WITH BRIEF COMMENTS
372. 373. 374. 375. 376. 377. 378. 379. 380.
T.E. Clark, O. Piguct and K. Sibold, Nucl. Phys. B 143 (1978) 445. N.K. Nielsen, Nucl. Phys. B 244 (1984) 499. M.T. Grisaru and P.C. West, Nucl. Phys. B 254 (1985) 249. E. Guadagnini, K. Konishi, M. Miiitchev, Phys. Lett. B 157 (1985) 37. V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Phys. Lett. B 157 (1985) 169. H. Itoyama, V.P. Nair and H.-C. Ren, Nucl. Phys. B 262 (1985) 317. K. Harada and K. Shizuya. Phys. Lett. B 162 (1985) 322. H. Suzuki, Phys. Rev. Lett. 56 (1986) 1534. F. Cooper, A. Khare arid U. Sukhatme, Phys. Kept. 251 (1995) 267.
For recent analysis and related references, see for example: 381. Y. Ohshima, K. Okuyama, H. Suzuki and H. Yasuta, Phys. Lett. B 457 (1999) 291. The standard textbooks on supcrsymmetry are: 382. J. Wess and J. Bagger, Supersymmetry and super gravity. Princeton University Press (1983). 383. S. Wcinberg, The quantum theory of fields, III. Cambridge University Press (1999). Supergravity is reviewed in: 384. P. van Nieuwenhuizen, Phys. Rept. 68 (1981) 189. The anomalies in fuzzy physics have also been discussed in path integrals: 385. 386. 387. 388. 389. 390. 391.
A.P. Balachandran and S. Vaidya, Int. J. Mod. Phys. A 16 (2001) 17. J.M. Gracia-Boridia and C.P. Martin, Phys. Lett. B 479 (2000) 321. E.F. Moreno and F.A. Sdiaposnik, JHEP 0003 (2000) 032. J. Nishimura and M.A. Vazquez-Mozo, JHEP 0108 (2001) 033. T. Nakajima, Phys. Rev. D 66 (2002) 085008. C.P. Martin, Nucl. Phys. B 623 (2002) 150. H. Aoki, S. Iso and K. Nagao, Phys. Rev. D 67 (2003) 065018.
For other applications, the anomaly matching condition by 't Hooft and its implications on the rnassless composite particles are discussed in: 392. G. 't Hooft, in: Recent developments in gauge theories. G.'t Hooft et al. (eds.), Plenum (1980) p. 135. 393. Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz. Nucl. Phys. B 177 (1981) 157. 394. S.R. Coleman and B. Grossman, Nucl. Phys. B 203 (1982) 205. The anomaly in stochastic quantization has been studied in: 395. J.D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B 233 (1984) 61. 396. M. Namiki. I. Ohba, S. Tanaka and D.M. Yanga, Phys. Lett. B 194 (1987) 530
CONCLUDING REMARKS
281
and anomalies in topological field theory in: 397. R.K. Kaul and R. Rajaraman, Phys. Lett. B 249 (1990) 433. 398. D. Birmingham, M. Blau, M. Rakowski and G. Thompson, Phys. Rept. 209 (1991) 129. The anomaly-related phenomena in condensed matter theory are discussed in: 399. K. Ishikawa, Phys. Rev. Lett. 53 (1984) 1615; Phys. Rev. D 31 (1985) 1432. 400. R. Jackiw, Phys. Rev. D 29 (1984) 2375. 401. Z.-B. Sn and B. Sakita, Phys. Rev. Lett. 56 (1986) 780. 402. D. Boyanovsky, E. Dagotto arid E.H. Fradkin, Nucl. Phys. B 285 (1987) 340. 403. C. Mudry, C. Chamon and X.-G. Wen, Nucl. Phys. B 466 (1996) 383. 404. D.H. Kim and P.A. Lee, Annals Phys. 272 (1999) 130. Anomaly (and parity) related phenomena in odd dimensions are discussed in: 405. L. Alvarez-Gaume, S. Delia Pictra and G.W. Moore, Annals Phys. 163 (1985) 288. 406. D. Boyanovsky and R. Blankenbecler, Phys. Rev. D 31 (1985) 3234. 407. S.G. Naculich, Nucl Phys. B 296 (1988) 837. 408. M. Gomes, R.S. Mendes, R.F. Ribeiro, A.J. da Silva, Phys. Rev. D 43 (1991) 3516. 409. C. Fosco, G.L. Rossini arid F.A. Schaposnik, Phys. Rev. Lett. 79 (1997) 1980; 79 (1997) 4296(E). The quantum anomaly in molecular physics was also noted in: 410. H.E. Camblong, L.N. Epele, H. Fanchiotti and C.A. Garcia Canal. Phys. Rev. Lett. 87 (2001) 220402. 411. R. Jackiw, in: M.A.B. Beg memorial volume. A. All and P. Hoodbhoy (eds.), World Scientific, Singapore (1991). 412. B.R. Holstein, Am. J. Phys. 61 (1993) 142.
INDEX Abelian bosonization of fcrmions, 157 Abelian gauge field, 246 Adler. S., 5 Adler-Bardcen theorem, 62 afflne connection, 254 Alvarcz-Gaume, L., 231 anomalous commutator, 121 a priori probability, 23 Atiyah Singer index theorem, 84, 106 on the lattice, 206 axial-vector current, 58, 90 axial-vector gauge field, 93 background field method, 138 Bardeen, W., 97 bare identity, 62 basic principle of gauge theory, 246 basis vectors which diagonalize the Dirac action, 88 Bell, J.S., 5 P function. 130 for a QCD-typc theory. 140 in QED, 135 Bjorken-Johnson-Low (BJL) prescription, 121, 176, 180 Bloch wave, 83 Bose particle. 15 Bose symmetrization factor, 117 Bose symmetry, 113 Bose-like spinor field, 96 bra state, 41 Brillouin zone, 200 BRST charge, 102 BUST invariant path integral measure, 189 BRST transformation, 258 canonical mass dimension. 124 canonical quantization of the electromagnetic field, 31 canonical T product, 121 central extension, 178, 188 Chern character, 230 Chern-Simons form, 82, 90 chiral gauge theory, 98 chiral identity, 62, 225 chiral Jacobian, 05 chiral symmetry spontaneous breakdown of, 90 chiral transformation, 57 identity for, 79 chiral WT identity, 72
chirality sum rule, 206 coherent state. 16 color degrees of freedom, 110 compact gauge theory. 198 compact group, 7-1 completely anti-symmetric symbol, 62 conformal field theory, 144, 178 conformal gauge, 144 connected component, 132 contravariant vector, 259 correlation function, 162 cosmologica.1 term, 143 Coulomb gauge, 32 cova.ria.nt anomaly, 107, 117 covariant derivative, 74, 76 covariant regularizatioii, 51 critical string, 193 current conservation, -19 descent formula, 243 differential form, 231 dimensional regularization, 51 Dira,c equation, 20 Dirac genus, 230 Dirac matrices, 20 Dirac operator, 50 Dirac's transformation theory, 66 doublet of leptons, 110 doublet of quarks, 109 -Dyson. F., 2 Dyson's formula for the S-matrix. 2-19 Einstein equation, 231, 257 electromagnetic tensor, 31 energy-momentum tensor. 126, 183 naive, 257 trace of, 129 equal-time commutation relation, 177 Huclidean theory, 50 in curved space-time, 124 2n-dimcnsional, 223 Killer number, 194 evolution operator, 8, 17 Faddeev Popov formula, 40 Faddeev Popov ghost, 38 fermion number, 104 asymmetry in. 221 ferrnion, 15 Feynman gauge, 138 Feynman path integral, 23 Feynman, R.P., 2 Feynman's it prescription, 21, 251 282
283
INDEX Fcynmaivs parametric representation, 56 Feynman's propagator, 250 field strength tensor, 74 finite operator, 129 first quantization of a bosonic string, 188 flavor freedom, 219 Fock space representation, 12 four-component spinor, 20 Fresnel integral, 37 Fukuda, H., 4 fundamental Brillouin zone, 200 gauge condition, 32 gauge invariancc, 76 gauge transformation. 31 of left-handed component, 111 Gauss operator, 101 Gauss Bonnet theorem, 194 Gell-Mann matrices, 75 Gell-Mann-Nishijima-type relation, 109 general coordinate transformations, 231 anomaly associated with, 187 identities related to, 127 measure invariant under the BRST symmetry associated with, 260 WT identity for, 233 general theory of relativity, 253 generators of a simple nou-Abclian group, 75 genus, 193 geodesic biscalar, 226 ghost number anomaly, 194 Ginsparg Wilson relation, 203 global SU(2) anomaly, 243 Goto-Imamura-Schwinger term, 121, 122, 177 Grassmann number, 15 Green's function, 29 Grecn-Schwarz mechanism, 245 Haar invariant measure, 198 harmonic oscillator, 7 heat kernel method, 237 Ileisenberg, W., 7 Heisenberg's equation of motion, 21 hermitian phase operator, 42 Higgs particles, 111 higher derivative regularization, 47 hypcrcubic lattice, 196 improved energy-momentum tensor, 131 index, 44, 105 infinitesimal chiral transformation, 76 instanton, 79, 80 instanton number. 85 integrability (Wess Zumiiio) condition, 112 integrable anomaly, 117, 153
interaction representation, 249 unitary transformation to, 86 Ja.ckiw, R., 5 Jacobi identity, 82 Jacobian. 68 Kac-Moody algebra, 174 ket state, 41 Kimura, T., 6 Konishi anomaly, 147 Landau gauge, 32 lattice gauge theory, 196 left derivative, 16 Legendre transformation, 228 length in the functional space, 35 light pseudo-scalar meson, 91 local counter-term, 95 local Lorentz transformation. 232 local Weyl transformation, 125 Lorentz covariant T* product, 121 Lorentz indices, 253 LSZ prescription. 29 mass term for the photon, 48 Maxwell's electromagnetic field, 31 metric condition, 255 metric of space-time, 50 minimal coupling, 247 minimum uncertainty relation, 46 Minkowski indices, 253 Miyamoto, Y.. 4 mode cut-off, 68 naive chiral identity, 58 naive continuum limit, 197 naive energy-momentum tensor, 257 naive form factor, 56 Nambu—Goldstohe boson, .119 natural unit, 47 Xeuberger. H., 210 Nielsen—Ninomiya theorem, 202 non-Abelian bosonization, 168 nou-Abelian gauge transformation, 76 non-critical string, 193 non-linear a model, 119, 174 non-renormalization theorem, 148 norm of the wave function, 1 normal product, 63, 132 normalization condition of the path integral, 29 normalization of generators, 75 Mother current, 100 operator product expansion, 180 Oppenheimer, J., 2 orthogonal group, 108 overlap Dirac operator, 210 particle number representation, 13 partition function in statistical mechanics, 23
284 path integral measure, 26 Paul! matrices, 20. 75
Pauli, W,, 15 Pauli-Villars regularization, 96 photon phase operator. 40 physical Hilbert space, 210 plaquette, 197 point-splitting method, 54 Poisson bracket, 42 pole, 180 Polyakov, A.M., 188 Pontryagiu index, 106 Pontryagin number, 85 positive definite inner product, 204 projection operator, 98 proton decay, 221 pseudo-scalar operator, 219 • quantum anomaly associated with general coordinate transformations, 187 cancellation condition, 109 cancellation of, 102 identity, 97 integrable, 117, 153 master formula for, 152, 225 quantum chromodynamics (QCD), 74 f) function, 140 quantum electrodynamics (QED), 246 0 function, 135 perturbation formula, 252 quark mass, 89 radial quantization, 178 rcnormalization group equation, 130 renormalization point, 130 Riemann surface, 193 Riemann-Christoffel curvature tensor, 225 Riemann-Roch theorem, 194 scalar curvature, 147 scalar field, 11 scale invariance, 124 scale transformation, 123 Schrodinger equation, 1 Schrodinger functional representation, 101 Schwarz inequality, 82 Schwinger, J., 2 Schwinger's action principle, 23, 250 Schwinger's source function, 28 short distance, 78 sine-Gordon model, 163 small gauge transformation, 83 SO(10) grand unified model, 109 source field, 25 species doubling, 200 spin connection, 254 squeezing of the photon, 45
INDEX Steinberger. J., 5 structure constant, 75 superstring theory, 239 supersymmetry, 147 supertrace, 19 symmetric component, 132 6 vacuum, 83 Thirring model massive, 163 massless, 160 't Hooft, G., 105 time ordering, 24 Tbmonaga. S., 2 translation invariant measure, 26 triangle anomaly, 59 T* product, 176 tunneling, 83 uncertainty relation, 42 unit hypersurface, 81 vacuum polarization tensor, 53 vector-like gauge field, 93 vierbein, 253 Virasoro algebra, 182, 188 Ward-Takahashi (WT) identity, 47, 71 for general coordinate transformations, 233 wave function renormalization factor, 129 weight 1/2 variables, 224, 259 Weinberg-Salam theory. 109 Wess, J., 112 Wess-Zumino term, 117 Wess Zumino-Witten action, 171 Weyl anomaly, 145 in two-dimensional space-time, 144 Weyl gauge, 32 Weyl transformation, 256 Jacobian for. 139 local, 125 Wick rotation, 23, 28 Wick's theorem, 250 Wilson, K., 197 Wilson term, 199 winding number, 81 VVitten, E.. 231 £-gauge, 40 Yang-Mills field, 74 f function regularization, 140 Zumino, B., 112
