Construction of dynamics and time-ordered exponential for unbounded non-symmetric Hamiltonians

Abstract

We prove under certain assumptions that there exists a solution of the Schrödinger or the Heisenberg equation of motion generated by a linear operator H acting in some complex Hilbert space H, which may be unbounded, not symmetric, or not normal. We also prove that, under the same assumptions, there exists a time evolution operator in the interaction picture and that the evolution operator enjoys a useful series expansion formula. This expansion is considered to be one of the mathematically rigorous realizations of so called "time-ordered exponential", which is familiar in the physics literature. We apply the general theory to prove the existence of dynamics for the mathematical model of Quantum Electrodynamics (QED) quantized in the Lorenz gauge, the interaction Hamiltonian of which is not even symmetric or normal.

Dynamical Properties of Nanostructured Josephson‐Junction Based on the Schwinger Quantum Action Analysis

“…[31] This procedure is a good mathematical tool for describing the interaction picture via the time evolution operator in the realm of quantum mechanics. [32,33] Time-ordering procedure is also at the heart of unfolding quantum theory of dynamical systems using the Schwinger action principle. In order to find the time-ordered Hamiltonian of the system, let us consider the Heisenberg equations for the canonical variables:…”

Eliminating unphysical photon components from Dirac–Maxwell Hamiltonian quantized in the Lorenz gauge

“…Here, we take a slightly different formulation from that of Ref. [17] so that the generalization to n-th differentiability is easier.…”

“…The time evolution operator generated by unbounded, non-self-adjoint Hamiltonians has been constructed in Ref. [17]. In this paper, we further develop the theory in several aspects.…”

“…In Section 2, after the brief review of the results obtained in [17], we discuss the abstract theory of time evolution generated by non-self-adjoint Hamiltonians and give a definition of general free field to which the positive frequency part is defined. In Section 3, we intoduce the Difinition of the model and how to mathematically deal with an indefinite metric space.…”

“…Secondly, is it possible to identify the positive frequency part of the operator satisfying Klein-Gordon equation even in an indefinite metric space? The first problem is solved by the general construction method of time evolution operator generated by a non-self-adjoint operator given by the authors [17], and as to the second one, the general definition of "positive frequency part" of a quantum field satisfying Klein-Gordon equation is given in Ref. [18].…”