Path integrals for awkward actions

TL;DR

This paper introduces the Atlantic City method, a novel approach to evaluate path integrals for theories with non-quadratic time derivatives, enabling Monte Carlo simulations even with complex or negative weight functions, potentially impacting dark energy research.

Contribution

It presents a new technique to perform path integrals without requiring the Hamiltonian and addresses the sign problem in complex path integrals using combined Monte Carlo and numerical integration methods.

Findings

The Atlantic City method estimates energy densities in complex theories.

It enables path integral calculations without explicit Hamiltonians.

The approach can handle theories with finite energy densities and complex weights.

Abstract

Time derivatives of scalar fields occur quadratically in textbook actions. A simple Legendre transformation turns the lagrangian into a hamiltonian that is quadratic in the momenta. The path integral over the momenta is gaussian. Mean values of operators are euclidian path integrals of their classical counterparts with positive weight functions. Monte Carlo simulations can estimate such mean values. This familiar framework falls apart when the time derivatives do not occur quadratically. The Legendre transformation becomes difficult or so intractable that one can't find the hamiltonian. Even if one finds the hamiltonian, it usually is so complicated that one can't path-integrate over the momenta and get a euclidian path integral with a positive weight function. Monte Carlo simulations don't work when the weight function assumes negative or complex values. This paper solves both problems. It shows how to make path integrals without knowing the hamiltonian. It also shows how to estimate complex path integrals by combining the Monte Carlo method with parallel numerical integration and a look-up table. This "Atlantic City method" lets one estimate the energy densities of theories that, unlike those with quadratic time derivatives, may have finite energy densities. It may lead to a theory of dark energy. The approximation of multiple integrals over weight functions that assume negative or complex values is the long-standing sign problem. The Atlantic City method solves it for problems in which numerical integration leads to a positive weight function.