Feynman's Path Integral to Ostrogradsky's Hamiltonian for Lagrangians with second derivatives

TL;DR

This paper demonstrates how Feynman's path integral naturally leads to Ostrogradsky's Hamiltonian for nonsingular Lagrangians with second derivatives, providing a new derivation and insights into the classical-quantum transition.

Contribution

It introduces a novel derivation connecting Feynman's path integral with Ostrogradsky's Hamiltonian for higher-derivative Lagrangians, extending previous methods to more general cases.

Findings

Path integral implies Ostrogradsky's Hamiltonian for second derivatives

New derivation of Legendre transformation for classical Lagrangians

Feynman's method starts with classical Lagrangian and ends with classical Hamiltonian

Abstract

A calculation is presented that shows that Feynman's path integral implies Ostrogradsky's Hamiltonian for nonsingular Lagrangians with second derivatives. The procedure employs the stationary phase approximation to obtain the limiting change of the wave function per unit time. By way of introduction, the method is applied anew to the case of nonsingular Lagrangians with only first derivatives, but not necessarily quadratic in the velocities. A byproduct of the calculation is an alternate derivation of the Legendre transformation of taking general classical Lagrangians into Hamiltonians. In both the first and second derivative cases, the outcome contains precisely the classical Hamiltonian, which represents the so-called "symbol" of a (not necessarily Hermitean) pseudodifferential operator acting on the wave function at an instant of time. The derivation herein argues for a claim that Feynman's method starts with a classical Lagrangian and ends with a classical Hamiltonian---nonclassical operator-ordering prescriptions in the passage from classical to quantum Hamiltonians require external input, and are generally not inherent in Feynman's path integral formalism.