Anomaly meltdown

by C. Gajdzinski and R. F. Streater,

J. of Mathematical Physics, 32, pp 1981-1983, 1991.

ABSTRACT

It is shown that at non-zero temperature it is possible that anomalies in representations of symmetry groups and gauge groups, present at zero temperature, disappear. This does not mean that they are zero, but that they are coboundaries, and can be gauged away without changing the dynamics of the observables. Several examples are given. Thus the idea that anomalies in baryon currents might have caused the baryon imbalance in the early universe needs reconsideration.

REMARKS

Go to an earlier paper for the theory behind this work.

This paper does not mention the well-known work of Jackiw and Dolan, in which an anomaly is calculated at non-zero temperature, and shown to be the same as at zero temperature. This does not mean that there is a contradiction between our result and that of Jackiw and Dolan; their method, perturbation theory, is not suitable for the answering the question as to whether the anomaly, which is a cocycle, is a coboundary. Moreover, the cohomology used in our paper is a severe form of Wignerism: two different states that coincide on the observables are regarded as the same, even though they are not related by an overall phase. The point is that in a temperature state (a KMS state, which stands for Kubo, Martin and Schwinger) the commutant of the observables is a type III von Neumann algebra, and any unitary operator in the commutant can be applied to a vector without altering its physics, as defined by the observable algebra.

After this paper had been accepted, I received a letter from Prof Jackiw, pointing out that we did not quote his afore-said paper with Dolan; his letter continued ..."I find it amazing that you did not consult me before submitting this paper...". I replied that it is not amazing at all, since "...after all, you did not consult Mr. Gajdzinski before submitting your papers on the 3-coboundaries that are not cocycles."

In a paper entitled "On Spin Chains, Charges and Anomalies", H. Grosse, W. Maderner and C. Reitberger show that, by choosing a different implementer for the translation operator, the anomaly returns; that is, it is not zero. That it is a coboundary is implied since there is no change in the automorphisms of either the time-evolution or the space-translation by choosing a different implementer. Indeed, these authors suggest another way to cancel the anomaly, which has uses in other models. They also show that the form of the anomaly depends on the temperature, contrary to the folk-lore established by Dolan and Jackiw. This just reflects the ambiguity in choosing the implementation of symmetries in a reducible representation of the observables. The difference in the physical meaning of the translation operator chosen in this paper, with that of Grosse et al. is suggested in a later work.

Go to papers99 for the references to related works.