On the PCT-theorem in the theory of local observables

Abstract

We review the PCT-theorem and problems connected with its demonstration. We add a new proof of the PCT-theorem in the theory of local observables which is similar to that one of Jost in Wightman quantum field theory. We also look at consequences in case the PCTsymmetry is given on the algebraic level. At the end we present some examples which answer general questions and throw some light on open problems.

PCT Theorem for the Operator Product Expansion in Curved Spacetime

“…Inspection of the proof of this statement given in [17] shows that in analytic spacetimes, it only relies on (a) and (b) above, but not on (c) and (d). Furthermore, one easily sees that the arguments given in [17] will still be valid when properties (a) and (b) are replaced by the essentially identical properties (L) and (A) assumed for our distributions (14). (In fact, the precise form of our conditions (L) and (A) has been chosen precisely so that the arguments of [17] are still valid.)…”

“…(23) rather than only equivalence under x ∼, or alternatively, why we do not impose that relation (25) holds for all y j in N , rather than some neighborhood of the point x. The reason for this is that we typically expect the coefficients (14) to contain expressions like the geodesic distance, s M (y 1 , y 2 ), between two points in M near x. Now the geodesic distance between two points is not a quantity that is locally constructed out of the metric, since the geodesic distance between two points in a spacetime N (even if it can be defined unambiguously) can be made shorter by embedding N into a suitably chosen larger spacetime M. Therefore, it is not true that χ * s M = s N for the geodesic distance.…”

The CPT and Bisognano-Wichmann Theorems for Anyons and Plektons in d = 2 + 1

“…In the four-dimensional Poincaré covariant case the Bisognano-Wichmann theorem has been shown by the author [39] to hold under physically transparent conditions, namely for massive theories with asymptotic completeness. (Conditions of more technical nature have been found by several authors [5,7,30,35,54], see [7] for a review of these results.) In three-dimensional spacetime, however, there may be charged sectors with braid group statistics [19,24] containing particles whose spin is neither integer nor half-integer, which are called Plektons [23] or, if the statistics is described by an Abelian representation of the braid group, Anyons [59].…”

Borchers’ Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions

“…In fact, pursuing this strategy, we shall find the following result. Let G be the subgroup of the proper Poincaré group generated by the translations, the boosts λ t and the reflection j. Recalling that the representation U ρ may be shifted to a representation e ik·x U ρ (x) whose spectrum has a Lorentz invariant lower boundary [6], 7 we show under the above-mentioned assumptions:…”

“…For example in two-dimensional theories it means that the modular objects generate a representation of the proper Poincaré group, under which the observables behave covariant, and implies the CPT theorem. In higher dimensions, it is a crucial step towards the Bisognano-Wichmann theorem in the general context of local quantum physics [4,7,9,23,25,30,33]. This theorem asserts that a certain class of Poincaré covariant theories enjoys the property of modular covariance, namely that the mentioned unitary modular group coincides with the representers of the boosts, and that the modular conjugation is a CPT operator (where 'PT' means the reflection about the edge of the wedge).…”