A Generalization of Mackey's Theory of Induced Representations

Abstract

The paper has a technical character in which we extend Mackey theory of induced representations so as to embrace representations of the covering of the Poincaré group in Krein spaces (with Gupta-Bleuler operator) such as the Lopuszański representation encountered in construction of free fields with non trivial gauge freedom, e. g. electromagnetic field. Proposed extension allows us to pursue explicit spectral-operator construction of space-time in the Fock space of free fields -a crucial point in the solution of the representation problem (viz. convergence problem of the formal perturbation series) in the perturbative algebraic QFT (developed mainly by Dütsch, Fredenhagen and Brunetti) which we have proposed in our previous paper. The proposal of our previous paper is in fact an extension of the general program proposed by Fredenhagen, Dütsch and Brunetti whose main point lies in separating the algebraic aspectby interpreting the perturbation series as a formal deformation from the analytic aspect tied to the representation problem. The representation problem cited to above is immediately connected to the general problem in QFT of proving the existence of interacting fields with non trivial gauge freedom. Presented generalization of Mackey theory is given an independent character so as the "subgroup theorem" and "the Kronecker product theorem" are proved for general separable locally compact groups and for their Krein-isometric not necessary bounded representations induced by Krein-unitary (bounded) representations.