Talk:Slavnov-Taylor identities

I wish to thank again both referees for helpful comments. I tried to take ino account this comments, in particular enlarging the Introduction andthe first section. I also thank the editor for the help. A.S.

Editor remarks

• To referees, label any new comment with a new number, so that Slavnov can refer to them. Thanks.

Slavnov's answers

I wish to thank the referee for careful reading of my article and useful comments. I agree with most of his comments, however I do not think that the article dealing with rather special problem of gauge theories may and should replace the articles dedicated to the fundamental problems of quantum theory. Instead in my opinion the editors should consider a possibility of including to Scholarpedia corresponding general articles. In particular I have in mind such topics as Green functions in quantum field theory, renormalization and path integrals. Surely these notions should be discussed in Scholarpedia, because without some knowledge of them it is impossible to read papers dealing with quantum field theory. So I share the concern of the referee about this problem , but I do not think that the short comments concerning the Green functions e.t.c. really improve the situation.The people who are not familiar with these questions will not learn much from these comments. In particular, concerning the Green functions in QFT I wish to mention that their importance is not restricted to their connection to scattering amplitudes. Without the Green functions or some analogous off-shell objects it is impossible to formulate such a fundamental notion as causality in quantum field theory, to say nothing about the renormalization procedure. I am also convinced that some basics of the path integrals must be discussed in one of the principle articles on QFT, because at present it is one of the main tools in theoretical studies of QFT, quantum mechanics and quantum statistics. I am sure that the modern development of gauge theories, which form nowadays the theoretical basis of elementary particle physics would be impossible without path integrals. Having this in mind I think a special discussion of these and perhaps some other important questions should be included into Scholarpedia and I see no need to repeat this discussion in the articles dealing with special problems of QFT, like the article presented by me. Meanwhile these terms may be indicated as requiring some explanations which will be given later, as it is done with some other terms.

Below I discuss the referee's comments.

Comment 1.

1. I do not object against insertion of the term (principle of gauge invariance).

2.I agree that the composite operators should be also mentioned, but I do not think that this point should be elaborated mentioning Wilson lines e.t.c. These objects have little to do with ST-identities and hardly a reader is well acquainted with them. I propose the formulate the following modification " and constraints the structure of composite operators corresponding to physical observables. All the operators corresponding to physical observables must be gauge invariant."

Comment 2.

I have already explained my attitude to the discussion of important general concepts of QFT in special articles. In my opinion they should be introduced and discussed in a separate article, so I would not include this modification to the present article.

Comment 3.

I agree with the referee that this chapter of my article is too short and requires some additional explanations. However I would do it in a different way.

The first remark concerns a possibility to write ST-identities for the Green functions avoiding using of path integrals. It certainly can be done, but in my opinion for the reader who does not know what is the Green function in QFT and is not familiar with the technique used in gauge theories, the eq(1) of the referee's comment is hardly more transparent than the eq.(1) of my article. Moreover, in my opinion for anybody who wants to learn more than the very general idea of gauge invariance some basics of path integrals is compulsory. I have already mentioned a necessity of either special article about path integrals, or discussion of this topic in one of the general articles on QFT.

I also do not think that the remark, proposed by the referee "Furthermore they require that the observable operators be gauge invariant functionals of \(A\)" makes much sense. The first sentence of the comment "These relations guarantee independence of observable quantities from the choice of the gauge" already assumes that that observables are gauge invariant functionals.Otherwise their gauge independence is not guaranteed by the eq(1).

The next section of the comment where a simple example (Abelian case) is considered in my opinion is very technical and hardly adds much to understanding of the physical meaning of ST-identities.

In fact the shortness of the section 2 is mainly explained by my desire to avoid as much as possible technical details, which complicate the reading for nonspecialists and hardly are acceptable in Scholarpedia. For this reason I dropped an attempt to prove the identities as any derivation requires some technique, and may seem simple for specialists, but very complicated for nonspecialists (students). So I propose a modification of the section 2, which as I hope clarifies the origin and meaning of ST-identities but avoids any technical derivations.

After "J.C.Taylor" I propose to write: " For the case of the pure Yang-Mills field they may be written in terms of the path integral as follows: " Then follows the eq.(1) of my article and the subsequent text with the change of the sign \(+ \rightarrow -\) in the definition of the covariant derivative and adding after the words "ghost fields " the sentence "\(t^{abc}\) are the structure constants of the gauge group" as suggested by the referee. Then after the words "is the classical external source". I suggest the following insertion:

"To understand the meaning of this equation it is instructive to consider the scattering matrix in the Yang-Mills theory. In terms of the path integral the scattering matrix in the Yang-Mills theory may be written as follows

<math 3> S= \int \exp \{i \int L_{eff}(x)dx \}\Pi_x dA_{\mu}(x)d \bar{c}(x)dc(x) </math> Here the effective Lagrangian is given by the eq(2) and depends explicitely on the gauge parameter \(\alpha\). Nevertheless the value of the integral which defines the scattering amplitudes for the Yang-Mills field is gauge independent. This is a consequence of the relativity principle in the internal space according to which the fields related by the gauge transformation describe the same physical situation, and therefore in the path integral for the scattering matrix one should integrate over the classes of the gauge equivalent fields. In practice that means that one has to select a single representative in the gauge equivalent class by imposing some gauge condition and normalizing correctly the integration measure by adding the integration over Faddeev-Popov ghost fields. The result does not depend on the particular choice of the gauge because the transformation from one gauge to another may be acomplished by a simple change of variables in the path integral.

Contrary to the scattering matrix the Green functions are not directly observable and do depend on the gauge. Any Green function may be presented as a variational derivative of the Green function generating functional given by the following path integral: <math 4> Z(\eta)= \int \exp\{i \int[L_{eff}(x)+J_{\mu}^a(x)A_{\mu}^a(x)]dx \}dA_{\mu}d \bar{c}dc </math>

One sees that the integrand in the formula for the Green function generating functional differs of the corresponding integrand for the scattering matrix by the presence of the non gauge invariant term \( \int J_{\mu}^a(x)A_{\mu}^a(x)dx\) in the exponent. When one passes from one gauge to another by the change of variables this functional is changed. Its variation is given by the left hand side of the eq.(1). As a change of variables does not influence the value of integral this variation is equal to zero. Hence the Slavnov-Taylor identities (1) describe the response of the Green function generating functional to the change of gauge conditions." Then follows the remaining part of the section 2 of my article. I would be very much obliged to the referee for the comments concerning this suggestion. In fact I do not know a simpler way to explain the equation (1). In my opinion the derivation with the help of BRST transformations is not simpler, because to understand it one must be well acquainted with the BRST formalism, and the connection of this symmetry with the relativity principle in internal space, which in my opinion is a physical (and geometric) origin of gauge independence is not straightforward. A simpler possibility would be just to write that the ST-identity describe the response of the Green functions to the gauge transformations.

Comment 4.

I agree with the referee that the notion of renormalization requires some explanation, but as I already mentioned in my opinion these explanations should be given in a separate paper. As for particular suggestions of the referee, I agree with them, but it seems to me that the only essential addition to my article is the explicite form of the field transformations, which may be included into the article by adding after the words "of the classical Lagrangian" the formulae proposed by the referee "and multiplicative transformation of the fields entering the effective Lagrangian \( A_{\mu}^a \rightarrow \sqrt{z_2} A_{\mu}^a, \quad \bar{c}^a \rightarrow \sqrt{\tilde{z}_2}, \quad c^a \rightarrow \sqrt{\tilde{z}_2}c^a\).

Comment 5.

I agree to replace the words "BRST current" by "BRST charge".

Concerning the second remark I agree with the referee that it would be interesting to put into evidence the relation between ST-identities and BRST symmetry, however I think it is much more appropriate to do in the article dedicated to the BRST symmetry, which is planned to be published in the Scholarpedia. First of all it would be a good illustration of the use of BRST symmetry, and secondly I think it may be mentioned in the discussion of the history of BRST symmetry, which was discovered (at least according to R.Stora) by analyzing the ST-identities in the Yang-Mills theory. A.S.

Reviewer A

Introduction

I find the article very stimulating and remarkably clear in its synthetic form. Still in my opinion some points could be improved. I have listed here the major comments. I have inserted few trivial corrections directly in the article.

Comment 1

The text puts in due evidence the different nature of the concepts of gauge invariance (principle relativity in internal space) and independence of the quantization procedure from the gauge choice. Still I suggest to insert at the very beginning of the INTRODUCTION:

“Principle relativity in internal space (principle of gauge invariance)…”

since this is the most commonly used term. I should also put into evidence the fact that the above principle is an important constraint for the observables of the theory. One could e.g. insert after:

“determines uniquely…….Lagrangian in gauge theories, “”and constrains the structure of composite operators corresponding to physical observables. These composite operators are given, up to renormalization corrections, as functionals of the fields, which may be local, as e. g. the energy momentum tensor, or non local, as e.g. Wilson lines. All the physical operators must be gauge invariant ””. To quantize the…….”

Comment 2

Introducing the Green functions we encounter one of the major problems that one finds creating an hypertext encyclopedia. The example of Green functions is typical. The natural location of the term Green function is in a mathematical framework. The Green functions of interest in the present situation should be possibly presented in the framework of one of the principal articles on QFT, that is e.g. that on relativistic scattering theory (LSZ reduction). In the present situation one finds no possible reference, I should say even in Wikipedia, thus, I think, one should briefly insert the concept of Green function in the framework of QFT. Mentioning their connection with scattering amplitudes and their relations to the better known Feynman diagrams. Therefore I suggest to replace the paragraph:

“In quantum theory………….ordered field products.”

With:

“”The requirement of independence from the choice of gauge of observable quantities, like the scattering amplitudes and the matrix elements of observable operators, leads in quantum field theory to relations between Green functions. The Green functions are the vacuum expectation values of chronologically ordered field and operator products. In quantum electrodynamics, for example, the physically relevant Green functions involve the vector potential \(A_{\mu}\), the electron spinor fields \(\psi\), \(\bar\psi\) and, possibly, products of composite operators \( O\). Such Green functions have the form\[\langle T \bar\psi(x_1)\cdots\psi(x_m)\, A_{\mu_1}(y_{1})\cdots A_{\mu_n}(y_n)\,O\rangle\]. Green functions are related to scattering amplitudes through the LSZ reduction formulae. Their Fourier transforms evaluated on the physical values of particle momenta, are proportional to the scattering amplitudes and, when composite operators are present, to their matrix elements. In perturbation theory Green functions are expanded in series of terms corresponding to Feynman diagrams.”

Comment 3

It seem to me that the chapter: “Necessary and sufficient…:" is exceedingly short and that the recourse to the Feynman functional integral formalism is not necessary. As a matter of fact it appears as a way of writing the identities and not a way of proving them. Now, in order to present the identities, the already introduced, Green function formalism is largely sufficient. In the same time a comparison of S-T with Ward identities could be usefull. Thus I suggest to substitute the whole chapter with something like:

“The relations between the Green functions in the quantum Yang-Mills theory, which are necessary and sufficient conditions of the gauge independence of the observable quantities, were obtained by A.A.Slavnov and J.C.Taylor.

We shall limit our discussion to the pure Yang-Mills case in which the Faddeev -Popov Lagrangian is given by the equation

\[ L=-{1\over4}F^{\mu\nu a}F^a_{\mu\nu}+{1\over 2\alpha}(\partial_\mu A_\mu^a)^2 -\partial_\mu{\bar c}^a(D^\mu c)^a\]

and the observable operators are functionals of \(A_\mu\) only. In this case the Slavnov-Taylor relations have the form

<math mylabel> {1\over\alpha}\langle T\partial_\mu A^{\mu, a}(y)\prod_{i=1}^nA^{\nu_i, b_i}(x_i)\,O\rangle=-i \sum_{j=1}^n\langle T{\bar c}^{a}(y)D^{\nu_j} c^{b_j}(x_j)\prod_{i=1, i\not= j}^nA^{\nu_i, b_i}(x_i)\,O\rangle</math>

Here \( A^a_\mu \) is the Yang-Mills field, \({\bar c}^a\), \(c^b\) are the Faddeev-Popov ghosts , \(F^a_{\mu\nu}=\partial_\mu A^a_\mu-\partial_\nu A^a_\mu +gt^{abc}A^b_\mu A^c_\nu\) is the Yang-Mills field tensor and \(D_\mu c^a=\partial_\mu c^a +gt^{abc}A_\mu^b c^c\) is the covariant derivative of the ghost field, \(t^{abc}\) are the structure constants of the gauge group. \(O\) is a generic product of observable operators.

These relations guarantee independence of the observable quantities from the choice of the gauge, and hence of physical equivalence of different gauge conditions. Furthermore they require that the observable operators be gauge invariant functionals of \(A\).

A simple example is the Abelian case in which \(A\) only couples to charged fields while \(c\) and \(\bar c\) are free fields. In this situation the Green function in the left-hand side of Eq. (1) factorizes into products of contributions corresponding to two different kinds of Feynman diagrams, those connecting pairs of \(A\) fields and those in which \(A\) fields are connected to observable operators. If the operators are gauge invariant the Feynman diagrams of the second kind do not contribute to the right-hand side of Eq. (1) while those of the first kind contribute through the derivative of a scalar field propagator, that is \(\langle T\bar c(y)\partial^\nu c(x_j)\rangle\).

If one introduces charged, e.g. spinor, fields, further terms appear in the right-hand side of Eq. (1). In the abelian case these terms reproduce exactly the charged field contributions to the Ward identities. In this sense the Slavnov-Taylor identities extend the Ward identities to the non-abelian case.”

Comment 4

Concerning the chapter on “S-T identities and renormalization” I think one can repeat the comment on the Green functions, again there is no article which we can refer to, yet. Thus in particular the central point, concerning the renormalization constants, needs some introductory words. This could be done quite simply adding a sentence after: …and is equivalent to a redefinition of the parameters (..) of the classical Lagrangian. The new sentence could be:

“This redefinition consists, either in the addition of further contributions to some parameters, as in the case of masses, or in a multiplicative transformation of fields and coupling constants. In particular the original (bare) Faddeev-Popov Lagrangian gets renormalized and transformed into an effective Lagrangian after the following transformations of the fields \( A^{a}_{\mu}\to \sqrt{z_2} A^{a}_{\mu}\, \ ,\, c^{a}\to \sqrt{\tilde z_2} c^{a}\,\ ,\, {\bar c}^{a}\to \sqrt{\tilde z_2}{\bar c}^{a}\) and analogous substitutions of the coefficients. The factors \( z\) are called renormalization constants.”

Comment 5

Finally concerning the last chapter I have two comments. the first is:

“I think that the concept of BRST current should be avoided, as a matter of fact the conservation, time-independence, of the BRST charge is the necessary and sufficient condition for physical consistency of the theory, therefore I suggest to replace BRST current with BRST charge in this section.”

The second one is: after the reference to the BRST symmetry it could be interesting to put into evidence the relations between S-T identities and BRST symmetry. This could be done by translating the S-T identities written in the second chapter in the BRST form based on the charge Q. One could insert immediately after the reference to BRST symmetry and before the sentence; Similar identities….. the paragraph:

“BRST symmetry appears rather clearly from the structure of the Faddeev-Popov Lagrangian. Indeed it is a trivial exercise to verify that this Lagrangian commutes with the BRST charge \(Q\) whose action on the fields is: \[ [ Q, A^{a}_{\mu}]=-i D_{\mu} c^{a}\,\ , \,\{Q,\bar c^{b}\}=-{i\over\alpha} \partial_{\nu} A^{\nu, b}\,\ ,\, \{ Q,c^{a}\}={ig\over2}f^{abc}c^{b}c^{c}\] and which commutes with observable operators. Then it is also apparent that Eq.(1) is equivalent to the equation \[i\langle T\{Q,\bar c^{a}(y)\prod_{i=1}^nA^{\nu_i, b_i}(x_i)\,O\}\rangle=0 \] which is a direct consequence of the \(Q\)-invariance of the vacuum state.”

Reviewer B

Comment B1

I agree with the first referee that some parts of the article should somewhat expanded to make them comprehensible to a wider audience, in particular, the introduction and the first section. Indeed, it would be most useful to give some clearer idea of the origin and the meaning of the ST identities. On the other hand, I agree also with the author's reply that such specialized articles should be supplemented with article of more general nature explaining gauge invariance, path integrals...

Comment B2

I agree that this new version is much improved and in my opinion, considering the complexity of the question,is acceptable as such. Just a few misprints should be corrected. Also, I hope that in due time other more general articles will provide some additional background to Pr. Slavnov's contribution.