Lattice simulations of real-time quantum fields

TL;DR

This paper explores lattice simulations of real-time quantum fields, demonstrating the use of stochastic processes in Langevin-time for gauge theories, analyzing convergence, stability, and the relation to Dyson-Schwinger identities.

Contribution

It introduces a stochastic Langevin approach for Minkowski space-time lattice simulations of gauge fields and analyzes the stability and convergence of fixed points in this framework.

Findings

Converging results are obtained with small Langevin steps and tilted contours.

Fixed points satisfy Dyson-Schwinger identities but are not unique.

Thermal equilibrium is approached at intermediate Langevin-times, improved by Euclidean deformation.

Abstract

We investigate lattice simulations of scalar and nonabelian gauge fields in Minkowski space-time. For SU(2) gauge-theory expectation values of link variables in 3+1 dimensions are constructed by a stochastic process in an additional (5th) ``Langevin-time''. A sufficiently small Langevin step size and the use of a tilted real-time contour leads to converging results in general. All fixed point solutions are shown to fulfil the infinite hierarchy of Dyson-Schwinger identities, however, they are not unique without further constraints. For the nonabelian gauge theory the thermal equilibrium fixed point is only approached at intermediate Langevin-times. It becomes more stable if the complex time path is deformed towards Euclidean space-time. We analyze this behavior further using the real-time evolution of a quantum anharmonic oscillator, which is alternatively solved by diagonalizing its Hamiltonian. Without further optimization stochastic quantization can give accurate descriptions if the real-time extend of the lattice is small on the scale of the inverse temperature.