Path integrals for actions that are not quadratic in their time derivatives

TL;DR

This paper develops a method to construct path integrals for actions with non-quadratic time derivatives, where traditional Legendre transformation approaches are intractable, focusing on scalar fields.

Contribution

It introduces a novel approach to path integral formulation for complex actions without relying on the Legendre transformation, expanding applicability beyond quadratic cases.

Findings

Provides a new construction method for non-quadratic actions

Focuses on scalar fields with complex time derivative structures

Circumvents the intractability of Legendre transformation in certain cases

Abstract

The standard way to construct a path integral is to use a Legendre transformation to find the hamiltonian, to repeatedly insert complete sets of states into the time-evolution operator, and then to integrate over the momenta. This procedure is simple when the action is quadratic in its time derivatives, but in most other cases Legendre's transformation is intractable, and the hamiltonian is unknown. This paper shows how to construct path integrals when one can't find the hamiltonian because the first time derivatives of the fields occur in ways that make a Legendre transformation intractable; it focuses on scalar fields and does not discuss higher-derivative theories or those in which some fields lack time derivatives.