We review the theory and applications of complex stochastic quantization to the quantum many-body problem. Along the way, we present a brief overview of a number of ideas that either ameliorate or in some cases altogether solve the sign problem, including the classic reweighting method, alternative Hubbard-Stratonovich transformations, dual variables (for bosons and fermions), Majorana fermions, density-of-states methods, imaginary asymmetry approaches, and Lefschetz thimbles. We discuss some aspects of the mathematical underpinnings of conventional stochastic quantization, provide a few pedagogical examples, and summarize open challenges and practical solutions for the complex case. Finally, we review the recent applications of complex Langevin to quantum field theory in relativistic and nonrelativistic quantum matter, with an emphasis on the nonrelativistic case.

中文翻译：

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复朗之万和其他解决量子多体物理中符号问题的方法

我们回顾了复杂随机量化在量子多体问题中的理论和应用。在此过程中，我们简要概述了一些可以改善或在某些情况下完全解决符号问题的想法，包括经典的重新加权方法、替代的 Hubbard-Stratonovich 变换、对偶变量（对于玻色子和费米子）、马约拉纳费米子、态密度方法、假想不对称方法和 Lefschetz 顶针。我们讨论了传统随机量化的数学基础的某些方面，提供了一些教学示例，并总结了复杂案例的开放挑战和实际解决方案。最后，我们回顾了最近复朗之万在相对论和非相对论量子物质中的量子场论中的应用，