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arXiv:hep-lat/0508030 · Physical Review Letters

Simulating Nonequilibrium Quantum Fields with Stochastic Quantization Techniques

Jürgen Berges, Ion Olimpiu Stamatescu

Published Nov 11, 2005• 92 citations
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Abstract

We present lattice simulations of nonequilibrium quantum fields in Minkowskian space-time. Starting from a nonthermal initial state, the real-time quantum ensemble in (3 + 1) dimensions is constructed by a stochastic process in an additional (5th) "Langevin-time." For the example of a self-interacting scalar field, we show how to resolve apparent unstable Langevin dynamics and compare our quantum results with those obtained in classical field theory. Such a direct simulation method is crucial for our understanding of collision experiments of heavy nuclei or other nonequilibrium phenomena in strongly coupled quantum many-body systems.

DOI
10.1103/physrevlett.95.202003

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We present lattice simulations of nonequilibrium quantum fields in Minkowskian space-time. Starting from a nonthermal initial state, the real-time quantum ensemble in (3 + 1) dimensions is constructed by a stochastic process in an additional (5th) "Langevin-time." For the example of a self-interacting scalar field, we show how to resolve apparent unstable Langevin dynamics and compare our quantum results with those obtained in classical field theory. Such a direct simulation method is crucial for our understanding of collision experiments of heavy nuclei or other nonequilibrium phenomena in strongly coupled quantum many-body systems.
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We present simulations of nonequilibrium dynamics of quantum field theories on digital quantum computers. As a representative example, we consider the Schwinger model, a (1+11+11+1)-dimensional U(1) gauge theory, coupled through a Yukawa-type interaction to a thermal environment described by a scalar field theory. We use the Hamiltonian formulation of the Schwinger model discretized on a spatial lattice. With the thermal scalar fields traced out, the Schwinger model can be treated as an open quantum system and its real-time dynamics are governed by a Lindblad equation in the Markovian limit. The interaction with the environment ultimately drives the system to thermal equilibrium. In the quantum Brownian motion limit, the Lindblad equation is related to a field theoretical Caldeira-Leggett equation. By using the Stinespring dilation theorem with ancillary qubits, we perform studies of both the nonequilibrium dynamics and the preparation of a thermal state in the Schwinger model using IBM's simulator and quantum devices. The real-time dynamics of field theories as open quantum systems and the thermal state preparation studied here are relevant for a variety of applications in nuclear and particle physics, quantum information and cosmology.
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We consider two strategies for stochastic quantization. With the first, one posits an additional time dimension (fictitious time) and describes the evolution of classical fields by means of the Langevin equation. One then evaluates stochastic averages of the field functions. In the limit that the fictitious time goes to infinity, these approach the time ordered correlation functions of canonically quantized field theory. We conclude that, while this strategy successfully describes QED and other quantum field theories, it is contrived and probably lacks deep physical significance. With the second strategy for field quantization, one begins with a classical action in either one or two extra dimensions coupled to an external random source. We review a method of quantizing bosonic fields which uses this strategy. Furthermore, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss applications to quantum mechanics and stochastically quantize the simple harmonic oscillator.