Action with Acceleration I: Euclidean Hamiltonian and Path Integral

TL;DR

This paper investigates a quantum system with acceleration terms, deriving its Euclidean Hamiltonian and path integral formulation, revealing a non-Hermitian Hamiltonian that can be transformed into a Hermitian form, with a critical coupling where this fails.

Contribution

It introduces a Euclidean path integral and Hamiltonian framework for systems with acceleration, including the explicit transformation to Hermitian form and analysis of its limitations.

Findings

Euclidean Hamiltonian reproduces acceleration Lagrangian

Path integral with correct boundary conditions established

Transformation to Hermitian Hamiltonian fails at critical coupling

Abstract

An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity, and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of this unbounded transformation is explicitly evaluated. The mapping fails for a critical value of the coupling constants.