Inequivalent Weyl Representations of Canonical Commutation Relations in an Abstract Bose Field Theory (Mathematical Aspects of Quantum Fields and Related Topics)

Abstract

Considered is a family of irreducible Weyl representations of canonical commutation relations with infinite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are reportedcdot. As a simple application, the well known inequivalence of the time-zero field and conjugate momentum of different masses in a quantum scalar field theory is rederived with space dimension dgeq 1 arbitrary. Also a generalization of representations of the time-zero field and conjugate momentum is presented. Comparison is made with a quantum scalar field on a bounded space of mathbb{R}^{d}. In the case of a bounded space with d=1, 2, 3, the representations of different masses turn out to be mutually equivalent.