Invalidity of TCP-Theorem for Infinite-Component Fields - Project Euclid
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Invalidity of TCP-Theorem for Infinite-Component Fields - Project Euclid
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Commun. math. Phys. 11, 125—130 (1968)
Invalidity of TCP-Theorem for Infinite-Component Fields A. I . OKSAK a n d I . T. TODOROV
Joint Institute for Nuclear Research, Moscow Received June 27, 1968 Abstract. Necessary conditions are given for the TCP invariance of a local field with usual energy-momentum spectrum. Examples are constructed of both Fermi and Bose local free infinite-component fields which violate these conditions and consequently do not allow a TCP symmetry. 1. Introduction
It has been shown recently [1—4] that infinite-component local fields may violate the right connection between spin and statistics. The only example of TCP noninvariant infinite-component field known up to now, the Majorana field [5, 3, 4], is incompatible with usual spectrum conditions. Here we give examples of local infinite-component fields transforming under irreducible unitary representations of SL(2, C) which satisfy usual spectrum conditions and violate TCP. These examples do not contradict Epstein's proof [6] of TCP in a theory of local observables, because his assumption of no mass degeneracy (with respect to spin) is not fulfilled in our case. First of all, in Sect. 2, we derive some general properties of the twopoint function, valid also for tempered local infinite-component fields. The main tool in this investigation is the Bogolubov-Vladimirov theorem [7]. We give necessary conditions on the two-point function for the validity of TCP. We construct in Sect. 3 examples of free (FERMI and BOSE) fields of mass m which violate these conditions. All our examples use the Majorana representations1 [0, 1/2] and [1/2, 0] (but not Majorana equations). 2. Two-Point Functions in Local and TCP-Invariant Theories
We shall deal with the quantum field theory in which all Wightman axioms [9] except Lorentz invariance hold. We impose in particular (weak) locality and translation invariance including normal spectrum 1
For a detailed description of the Majorana representations the reader is
referred to [4]. We are using here the notations p 0 , l{\ of GEL'FAND and NAIMARK
[8] for the irreducible representations of 8L(2, C).
