Geometric integration on Lie groups using the Cayley transform with focus on Lattice QCD

Abstract

This work deals with geometric numerical integration on a Lie group using the Cayley transformation. We investigate a coupled system of differential equations in a Lie group setting that occurs in Lattice Quantum Chromodynamics. To simulate elementary particles, expectation values of some operators are computed using the Hybrid Monte Carlo method. In this context, Hamiltonian equations of motion in a non-Abelian setting are solved with a time-reversible and volume-preserving integration method. Usually, the exponential function is used in the integration method to map the Lie algebra to the Lie group. In this paper, the focus is on geometric numerical integration using the Cayley transformation instead of the exponential function. The geometric properties of the method are shown for the example of the Störmer–Verlet method. Moreover, the advantages and disadvantages of both mappings are discussed.