Path-Integral for Quadratic Hamiltonian Systems and Boundary Conditions

TL;DR

This paper reviews a path-integral approach for quadratic Hamiltonian systems, providing explicit formulas for the heat kernel and a general scheme for boundary conditions, applicable to various quantum models.

Contribution

It introduces a general method for incorporating boundary conditions in path-integrals and derives explicit formulas for systems with quadratic constraints.

Findings

Explicit heat kernel formulas for quadratic Hamiltonian systems

A general scheme for boundary conditions in path-integrals

Applicability to diverse quantum models with quadratic constraints

Abstract

A path-integral representation for the kernel of the evolution operator of general Hamiltonian systems is reviewed. We study the models with bosonic and fermionic degrees of freedom. A general scheme for introducing boundary conditions in the path-integral is given. We calculate the path-integral for the systems with quadratic first class constraints and present an explicit formula for the heat kernel in this case. These results may be applied to many quantum systems which can be reduced to the Hamiltonian systems with quadratic constraints (confined quarks, Calogero type models, string and $p$-brain theories, etc.).