Abstract
Normalizing flows have recently been applied to the problem of accelerating Markov chains in lattice field theory. We propose a generalization of normalizing flows that allows them to applied to theories with a sign problem. These complex normalizing flows are closely related to contour deformations (i.e. the generalized Lefschetz thimble method), which been applied to sign problems in the past. We discuss the question of the existence of normalizing flows: they do not exist in the most general case, but we argue that exact normalizing flows are likely to exist for many physically interesting problems, including cases where the Lefschetz thimble decomposition has an intractable sign problem. Finally, normalizing flows can be constructed in perturbation theory. We give numerical results on their effectiveness across a range of couplings for the Schwinger-Keldysh sign problem associated to a real scalar field in $0+1$ dimensions.
Highlights
Monte Carlo methods, applied to lattice quantum field theory, are unique in providing nonperturbative access to observables in QCD and other field theories
Feynman’s path integral becomes a finite- dimensional integral. This procedure results in a probability distribution over field configurations, which can be importance sampled with Markov chain Monte Carlo methods
Conditioned on a mild conjecture regarding the dependence of locally perfect manifolds on the parameters of the action, we show that globally perfect manifolds exist for a broad class of physical systems, including the Schwinger-Keldysh sign problem
Summary
Monte Carlo methods, applied to lattice quantum field theory, are unique in providing nonperturbative access to observables in QCD and other field theories. When applied to theories with a finite density of relativistic fermions, or to observables involving real-time evolution, lattice Monte Carlo methods are afflicted by the so-called sign problem. Feynman’s path integral becomes a finite- (but large-) dimensional integral For many theories, this procedure results in a probability distribution over field configurations, which can be importance sampled with Markov chain Monte Carlo methods. A perturbative view of normalizing flows gives rise to a method of computing lattice expectation values by solving a certain high-dimensional first-order partial differential equation. We demonstrate this method on lattice scalar field theory.
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