New Representation Spaces of the Poincaré Group and Functional Quantum Theory

Abstract

Group theory and the structure of linear metrical spaces used in quantum theory are in close connection. This is established by the theorem that a linear selfadjoint operator being forminvariant with respect to a symmetry group has eigenstates which must be base states of the corresponding representations of this group. Since the quantum observables have to be represented by selfadjoint operators and since the infinitesimal generators of a symmetry group are selfadjoint, it follows that they have to be themselves quantum observables. From the set of symmetry observables a subset of complete compatible observables can be chosen which fixes the structure of the corresponding representation space. Thus, all results and calculation methods of quantum theory depend strongly upon the representation spaces of symmetry groups under consideration. For example, the representation spaces of the Poincaré group which were first investigated by Wigner and Barg-mann [1, 2] and which are used in ordinary quantum field theory lead in connection with the principle of microcausality and locality to divergent results. Out of the various attempts to remove these defects we discuss those which are closely connected with the introduction of new representation spaces of the Poincaré group.