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Representations of discrete symmetry operators in quantum field theory Mariwalla, Kishin Hariram

Abstract

The object of the work reported in this thesis was to construct and study the explicit representations of discrete symmetry operators (D.S.O.'s) in quantum field theory. In spite of the considerable importance of the D.S.O.'s in present day physics, not much has been reported in the systematic study of such representations. Furthermore, in the work reported hitherto, only incomplete representations for the operators of space inversion (⊓) particle conjugation (⌐) and time reversal ( T ) have been given. Starting from general considerations on invariance principles and infinitesimal transformations, with the associated conservation laws, a systematic procedure for constructing the representations of the D.S.O.'s has been formulated. The procedure consists in enumerating the bilinears in creation and annihilation operators. It is shown that eight symmetries are the only possible ones. In view of the TCP - theorem and the so called non-conservation of parity in weak interactions, the product operators, such as reflection ( ⋀ = ⊓ ⌐ ) and strong reflection (S = ⊓ ⌐ T), in addition to time reversal, should be considered as the most basic symmetries. Working in linear momentum representation, the unitary operators ⋀, ⊓, ⌐, E ( = identity) and the unitary factors of the antiunitary operators: S, I = ⊓ T, J = T ⌐ and T are constructed for the following free fields: (I) The non-hermitian scalar field representing, for example, kaons. (II) The electromagnetic field. (III) The four-component spinor field. The operators for the scalar field have also been worked out in the angular momentum representation. Using the anti-commutation relations for C.O.’s and A.O.’s, an alternate construction of D.S.O.'s of the Dirac field is exhibited. More than one representation has been given in each case. In addition a two dimensional matrix representation has been given. It is shown that by an appropriate unitary transformation these can be reduced to the ordinary form.

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