Shadow Hamiltonians, Poisson Brackets, and Gauge Theories

TL;DR

This paper explains how symplectic integrators and shadow Hamiltonians can be used to optimize lattice gauge theory simulations, especially with large volumes, by extending Hamiltonian mechanics to include Poisson brackets.

Contribution

It introduces a method to compute forces and Poisson brackets for gauge theories using shadow Hamiltonians and extends Hamiltonian mechanics on Lie groups.

Findings

Symplectic integrators conserve shadow Hamiltonians, improving Monte Carlo efficiency.

Poisson brackets can be expanded to optimize integrator performance.

A practical method for computing forces and Poisson brackets in gauge theories is provided.

Abstract

Numerical lattice gauge theory computations to generate gauge field configurations including the effects of dynamical fermions are usually carried out using algorithms that require the molecular dynamics evolution of gauge fields using symplectic integrators. Sophisticated integrators are in common use but are hard to optimise, and force-gradient integrators show promise especially for large lattice volumes. We explain why symplectic integrators lead to very efficient Monte Carlo algorithms because they exactly conserve a shadow Hamiltonian. The shadow Hamiltonian may be expanded in terms of Poisson brackets, and can be used to optimize the integrators. We show how this may be done for gauge theories by extending the formulation of Hamiltonian mechanics on Lie groups to include Poisson brackets and shadows, and by giving a general method for the practical computation of forces, force-gradients, and Poisson brackets for gauge theories.