JAMP  Vol.8 No.7 , July 2020
Lefschetz Thimbles and Wigner Functions
Abstract: The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.
Cite this paper: Filinov, V. and Larkin, A. (2020) Lefschetz Thimbles and Wigner Functions. Journal of Applied Mathematics and Physics, 8, 1278-1290. doi: 10.4236/jamp.2020.87098.

[1]   Fedoryuk, M.V. (1976) The Asymptotics of the Fourier Transform of the Exponential Function of a Polynomial. Doklady Akademii Nauk, 227, 580-583.

[2]   Fedoryuk, M. (1989) Asymptotic Methods in Analysis. In: Gamkrelidze, R.V., Ed., Analysis I, Springer, Berlin, 83-191.

[3]   Witten, E. (2011) Analytic Continuation of Chern-Simons Theory. AMS/IP Studies in Advanced Mathematics, 50, 347.

[4]   Alexandru A., Başar, G. and Bedaque, P. (2016) Monte Carlo Algorithm for Simulating Fermions on Lefschetz Thimbles. Physical Review D, 93, Article ID: 014504.

[5]   Alexandru, A., Başar, G., Bedaque, P.F. and Ridgway, G. (2017) Schwinger-Keldysh Formalism on the Lattice: A Faster Algorithm and Its Application to Field Theory. Physical Review D, 95, Article ID: 114501.

[6]   Parisi, G., Klauder, J., Petersen, W., Ambjorn, J., Yang, S., Karsch, F. and Wyld, H. (1988) On Complex Probabilities. Stochastic Quantization, 131, 381.

[7]   Klauder, J.R. (1983) A Langevin Approach to Fermion and Quantum Spin Correlation Functions. Journal of Physics A: Mathematical and General, 16, L317.

[8]   Klauder, J.R. (1984) Coherent-State Langevin Equations for Canonical Quantum Systems with Applications to the Quantized Hall Effect. Physical Review A, 29, 2036.

[9]   Fujii, H., Honda, D., Kato, M., Kikukawa, Y., Komatsu, S. and Sano, T. (2013) Hybrid Monte Carlo on Lefschetz Thimbles—A Study of the Residual Sign Problem. Journal of High Energy Physics, 2013, Article No. 147.

[10]   Mukherjee, A., Cristoforetti, M. and Scorzato, L. (2013) Metropolis Monte Carlo Integration on the Lefschetz Thimble: Application to a One-Plaquette Model. Physical Review D, 88, Article ID: 051502.

[11]   Di Renzo, F. and Eruzzi, G. (2015) Thimble Regularization at Work: From Toy Models to Chiral Random Matrix Theories. Physical Review D, 92, Article ID: 085030.

[12]   Cristoforetti, M., Di Renzo, F., Scorzato, L., Collaboration, A., et al. (2012) New Approach to the Sign Problem in Quantum Field Theories: High Density QCD on a Lefschetz Thimble. Physical Review D, 86, Article ID: 074506.

[13]   Weinberg, S. (1995) The Quantum Theory of Fields. Vol. 1: Foundations. Cambridge University Press, Cambridge.

[14]   Mou, Z.-G., Saffin, P.M., Tranberg, A. and Woodward, S. (2019) Real-Time Quantum Dynamics, Path Integrals and the Method of Thimbles. Journal of High Energy Physics, 2019, Article No. 94.

[15]   Berges, J., Borsanyi, S., Sexty, D. and Stamatescu, I.-O. (2007) Lattice Simulations of Real-Time Quantum Fields. Physical Review D, 75, Article ID: 045007.

[16]   Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953) Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics, 21, 1087.

[17]   Hastings, W.K. (1970) Monte Carlo Sampling Methods Using Markov Chains and Their Applications. Biometrika, 57, 97-109.