Shadow Hamiltonians, Poisson Brackets, and Gauge Theories

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.

References in corpus (6)

• Accelerating Dynamical Fermion Computations using the Rational Hybrid Monte Carlo (RHMC) Algorithm with Multiple Pseudofermion Fields
• First results from 2+1 dynamical quark flavors on an anisotropic lattice: light-hadron spectroscopy and setting the strange-quark mass
• Accelerating Staggered Fermion Dynamics with the Rational Hybrid Monte Carlo (RHMC) Algorithm
• Dynamical QCD simulations on anisotropic lattices
• Improving dynamical lattice QCD simulations through integrator tuning using Poisson brackets and a force-gradient integrator
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Cited by in corpus (19)

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• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Status and challenges of simulations with dynamical fermions
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents
• Reversibility Violation in the Hybrid Monte Carlo Algorithm
• Hamiltonian Monte Carlo on Symmetric and Homogeneous Spaces via Symplectic Reduction
• Hessian-free force-gradient integrators
• RHMC with Block Solvers and Multiple Pseudofermions
• Comparison of smoothening flows for the topological charge in QCD-like theories
• Tuning the Hybrid Monte Carlo algorithm using molecular dynamics forces' variances
• A Unifying and Canonical Description of Measure-Preserving Diffusions
• Exploiting the hopping parameter expansion in the hybrid Monte Carlo (HMC) simulation of lattice QCD with two degenerate flavours of Wilson fermions
• Generalized HMC using Nambu mechanics for lattice QCD
• Numerical stability of force-gradient integrators and their Hessian-free variants in lattice QCD simulations
• Force-Gradient Nested Multirate Methods for Hamiltonian System
• Extended framework for the hybrid Monte Carlo in lattice gauge theory
• Symmetrically processed splitting integrators for enhanced Hamiltonian Monte Carlo sampling