Quantum Systems with Linear Constraints and Quadratic Hamiltonians

TL;DR

This paper explores mathematical frameworks for constrained quantum systems with linear constraints and quadratic Hamiltonians, extending classical stability results to quantum and infinite-dimensional cases.

Contribution

It introduces new mathematical constructions for the Hilbert space of constrained systems and generalizes the Maslov theorem to these systems, including infinite-dimensional cases.

Findings

Properties of Gaussian and quasi-Gaussian wave functions analyzed

Generalization of Maslov theorem to constrained quantum systems

Discussion of constrained Fock space with infinite degrees of freedom

Abstract

Quantum systems with constraints are often considered in modern theoretical physcics. All realistic field models based on the idea of gauge symmetry are of this type. A partial case of constraints being linear in coordinate and momenta operators is very important. Namely, when one applies semiclassical methods to an arbitrary constrained system, the constraints in "general position case" become linear. In this paper, different mathematicals constructions for the Hilbert space space for the constraint system are discussed. Properties of Gaussian and quasi-Gaussian wave functions for these systems are investigated. An analog of the notion of Maslov complex germ is suggested. Properties of Hamiltonians being quadratic with respect to the coordinate and momenta operators are discussed. The Maslov theorem (it says that there exists a Gaussian eigenfunction of the quantum Hamiltonian iff the classical Hamiltonian system is stable) is generalized to the constrained systems. The case of infinite number degrees of freedom (constrained Fock space) is also discussed.