Abstract
This chapter reveals the connection between Feynman path integrals in Euclidean quantum field theory and Markov chain Monte Carlo. We begin with transition amplitudes in quantum field theory and then introduce Feynman path integral in Euclidean spacetime as a way to extract the observables in a quantum field theory. We also look at physics examples such as supersymmetry breaking in a zero-dimensional quantum field theory with a square-well potential, two-point correlation function in a one-dimensional simple harmonic oscillator, and a matrix model with U(N) symmetry, that undergoes a phase transition as the coupling parameter is varied.
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In a Markov chain, the updates can be performed using a fixed scan (such as raster scan) or a random scan, depending on the model and the algorithm. It is best to use random scan if we are using Metropolis sampling. As an example, for the case of two-dimensional Ising model, a Gibbs sampler (which is a variant of Metropolis-Hastings algorithm with the property that the proposed moves are always accepted), with any of these scans would produce an irreducible Markov chain. However, using a fixed scan with Metropolis updates fails to produce an irreducible Markov chain for this model.
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Joseph, A. (2020). MCMC and Feynman Path Integrals. In: Markov Chain Monte Carlo Methods in Quantum Field Theories. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-46044-0_5
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DOI: https://doi.org/10.1007/978-3-030-46044-0_5
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Online ISBN: 978-3-030-46044-0
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