Talk:Becchi-Rouet-Stora-Tyutin symmetry

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Reviewer A

2008-10-19

Thank you for the authors' quick and serious reply to my comments, among which I accept C) and D) without any problem.

Concerning the remark A), it is unnecessary to stress such a point as "the space of physical states is {\it not} a subspace of the full (indefinite-metric) state space", because the quotient space structure is simply due to the non-negativity of the inner product in the BRS-cocycle space and to the presence of a left ideal with vanishing inner products. If this point is not emphasized in the text, however, we need not dispute on it any more.

Concerning [[B) Regarding Comment 2) of the reviewer. We think that mentioning the superselection rules associated with global U(1) symmetry would be extremely misleading in our context. The global U(1) superselection sectors refer to the space of physical states and thus are totally unrelated to the BRST or Gupta-Bleurer superselection sectors. We do not understand therefore the relevance of this comment to our theme.]], only one comment is in order:

I think that mentioning "a Superselection Sector" would be extremely misleading in your context, because this notion is totally irrelevant to the present context of the space of physical states. See the definition of a superselection sector: http://en.wikipedia.org/wiki/Superselection_sector, where you find such an explanation as follows: [[Suppose A is a unital *-algebra and O is a unital *-subalgebra whose self-adjoint elements correspond to observables. A unitary representation of O may be decomposed as the direct sum of irreducible unitary representations of O. Each isotypic component in this decomposition is called a superselection sector. Observables preserve the superselection sectors.]]

On which ground are you sure of the validity of such a statement in your context that "Each isotypic component in this decomposition is called a superselection sector"?? The superselection rule associated with global U(1) symmetry proved by Strocchi and Wightman gives a counter-example to it, doesn't it?

2008-10-13

Concerning p.5: (the paragraph before "Analysis of the Yang-Mills model") The appearance of negative norm states contrasts with the usual probabilistic interpretation of the inner product (http://mathworld.wolfram.com/InnerProduct.html)in Quantum Mechanics (http://plato.stanford.edu/entries/qm/). The Gupta-Bleuler way out of this paradox consists in identifying the physical state vector space with the subset of the Fock space annihilated by . This choice is justified by the fact that is a free field and hence the prescription selects a Superselection Sector (http://en.wikipedia.org/wiki/Superselection_sector) of the Fock space.

Comment 1) It is important to note that the physical state vector space is specified by a subsidiary condition to guarantee essentially the gauge invariance of physical states in such a form as the (so-called) Gupta-Bleuler (and/or Nakanishi-Lautrup) condition.

Comment 2) In the case of unbroken (global) U(1) gauge symmetry, the superselection sectors associated to this U(1) symmetry are eventually realized (as shown by Strocchi-Wightman, 1974). However, it is logically strange to identify the selected physical state vector space with "a Superselection Sector" (which is repeated later also on the 2nd and 4th lines of p.8, on the second last line of p.13).

Concerning p.7: There is an alternative possibility. One can ask for to be Pseudo-Hermitian. With this choice the full state vector space is an indefinite norm space and hence not compatible with the conventional probabilistic interpretations of the scalar (pseudo-inner) product /Indefinite Metric and BRST Cohomology. A physically consistent interpretation is still recoverable by limiting oneself to a physical subspace of the indefinite norm Fock space ...

Comment 3) While the expression like "negative-norm states" is tolerable, such an expressions as "an indefinite norm space" and as "indefinite norm Fock space" sound mathemtically painful; it is certainly better to replace the word "norm" with "inner(-)product". The same use of the word "norm" appears also p.8.

Comment 4) Similarly to the above 1), the "physical subspace" on the 2nd last line should be qualified by the "Kugo-Ojima subsidiary condition" which has a clear-cut cohomological meaning of the BRS(T) cocycle condition. It is also desirable to mention that it can reproduce the Gupta-Bleuler condition in the abelian situations.

Concerning p.8: The choice of different states in belonging kerQ to the same equivalence class (http://en.wikipedia.org/wiki/Quotient_space)

Comment 5) It is desirable to explicitly mention the meaning of "the same equivalence class" in the form of "im Q".

Concerning p.12: The rightmost part of the equation to define the operator X(c) is unreadable!

Concerning p.13: In the middle, a sentence contains "equation (2) in which ..."; where was the equation (2)?

Authors

2008-10-20

We seem to understand that the reviewer is objecting to the usage of the expression super-selection sector to refer to the subspace of BRST-invariant states. It seems to us that this wording is in fact justified, although we realize that it might generate some confusion.

In the notation used by the cited wikipedia article, H is the full space of states, O is the subalgebra of BRST-invariant observables. O sends kerQ to ker Q, therefore kerQ is a sub-representation of O on H. It is not irreducible since O sends imgQ into imgQ but it is not decomposable (since it is not unitary). One must consider kerQ/ImgQ to obtain a (potentially) irreducible representation of algebra of the physical observables, which is O modulo Q-trivial observables.

The difference with respect to the situation described in the wikipedia article is that H is not a unitary representation of O in the BRST context, of course, and therefore it is not fully decomposable. In this sense the question that the reviewers rises

On which ground are you sure of the validity of such a statement in your context that "Each isotypic component in this decomposition is called a superselection sector??

does not apply to our case, since H is not decomposable. Yet, we do have a subrepresentation on kerQ, which we refer to as the BRST-superselection sector. Precisely for this reason we pointed out in our previous response to the referee that the superselection sectors associated to physical symmetries like the U(1) global symmetry are fundamentally different from the one we are considering.

Our response, in conclusion, is the following: we think it would be legitimate to talk of BRST superselection sector, since the situation does fit into the general framework of superselection sectors as described (for example) in the cited wikipedia article. It is true, however, and we understand this is the concern of the reviewer, that the BRST case differs crucially from the specific examples listed in the wikipedia article (which have to do with unitary physical symmetries) because of the lack of full decomposability of the big space H.

Therefore, to avoid any possible confusion, we decided to eliminate all references to (BRST) superselection sector and we introduced the expression physically invariant subspaces, i.e. subspaces invariant under the action of the observables. We believe this should address correctly the reviewer's concern.

2008-10-15

We thank the reviewer for his detailed reports, that we examined with great care. Here are our answers to his comments. We inserted the corresponding corrections into the body of the article.

A) Regarding Comments 1), 4) and 5):

It should be kept in mind that the space of physical states is not a subspace of the full (indefinite-metric) state space. Both in the Gupta-Bleuler and in the BRST formalisms physical states are (gauge) equivalence classes.

As far as Gupta-Bleuler is concerned, this point is made at the end of the relative section (Section 3) of the original text, in a way that appears to us sufficiently clear:

The physical space is spanned by states generated by polynomials of a^+( \vec k,h) which are positive norm states and mixed polynomials of a^+( \vec k,h) and \beta^+( \vec k) which are orthogonal to the rest of the physical space and have zero norm. These states can be freely added to the positive norm physical states without changing their inner products and correspond to generic gauge variations of the physical states.

In the context of the BRST formalism we also explained in the Kugo-Ojima section (Section 5) that the physical states are elements of the quotient Ker Q/ Im Q, which means precisely that they are classes, not vectors, of the original space.

One proves that the quotient space H_{phys}= kerQ/imQ, known as the BRST cohomology space, is a Hilbert space. It is naturally identified with the space of physical states.

We reworded a bit the paragraph which follows the lines above, with respect to the original text, to emphasize the point even more:

States in kerQ which differ by an element of imQ belong to the same equivalence class. They correspond to the same physical state in H_{phys}. The Kugo-Ojima Q-invariance condition reduces to the Gupta-Bleuler condition in the case of electrodynamics and BRST equivalence is the analogue of the gauge equivalence of the Gupta-Bleuler formalism.

This re-wording shoud also take care of Comments 4) and 5) of the reviewer.

B) Regarding Comment 2) of the reviewer.

We think that mentioning the superselection rules associated with global U(1) symmetry would be extremely misleading in our context. The global U(1) superselection sectors refer to the space of physical states and thus are totally unrelated to the BRST or Gupta-Bleurer superselection sectors. We do not understand therefore the relevance of this comment to our theme.

C) Regarding Comment 3) of the reviewer.

We replaced the expression indefinite norm space with indefinite-metric space. Indeed it seems to us that the expression indefinite-metric space is overwhelmingly more common in the relevant physical literature.

D) Regarding the concerns of the reviewer about the readability of the equation for the operator X(c) and the numbering of the equation (2):

In the current version of the wiki-text the equation for X(c) and the number of equation (2) are both readable.