Hamiltonian Formalism and the Canonical Commutation Relations in Quantum Field Theory

Abstract

Cyclic representations of the canonical commutation relations and their connection with the Hamiltonian formalism are studied. The vacuum expectation functional E(f)=(Ψ0,ei[open phi](f)Ψ0) turns out to be a very convenient tool for the discussion. The uniqueness of a translationally invariant state (vacuum) is proved under the assumption of the cluster decomposition property for E(f). The existence and near uniqueness of the Hamiltonian in cyclic representations of the canonical commutation relations are established. The conditions for the relativistic invariance of the theory are stated in terms of vacuum expectation values at a fixed time. It is shown that E(f) is the Fourier transform of a quasi-invariant nonnegative measure on the space of all linear functionals of the test functions.