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arXiv 2309.03330hep-lathep-th
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使用复朗之万动力学的量子场论非微扰模拟

Non-Perturbative Simulations of Quantum Field Theories using Complex Langevin Dynamics

Arpith Kumar

备注：
PhD thesis, 158 pages. Based on arXiv:1908.04153, arXiv:2011.08107, arXiv:2201.12001, and arXiv:2209.10494
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AI总结：

本论文采用复朗之万方法克服复作用量导致的符号问题，成功模拟了量子场论的非微扰特性，包括超对称场论的自发破缺及IKKT矩阵模型中SO(10)对称性的自发破缺。

摘要：

场论的非微扰表述对于捕捉引人入胜的物理现象至关重要，包括QCD中的禁闭、自发超对称破缺和超弦中的动力学紧致化。格点正规化提供了一个稳健的框架，通过欧几里得路径积分来研究这些非微扰特性。通常，路径积分使用蒙特卡洛方法进行数值评估，其中玻尔兹曼因子被解释为概率权重。然而，各种物理系统中的复作用量使玻尔兹曼因子变复，导致符号问题。复朗之万方法克服了符号问题，可用于评估复积分。本论文采用复朗之万方法研究具有复作用量的场论系统的各种非微扰方面。我们探讨了在超对称场论的最简单实现中发生自发超对称破缺的可能性。这些系统通常具有由费米子算符的复行列式产生的复作用量。我们研究了各类有趣的复势，包括展现PT对称性的势。另一个探索的令人兴奋的方面是超弦理论中额外维度的动力学紧致化。IKKT矩阵模型在大N极限下是10D IIB型弦理论的一种猜想表述。我们采用复朗之万方法研究该矩阵模型的欧几里得版本，该模型具有固有的复Pfaffian，以探讨SO(10)对称性的自发破缺。本论文进行的研究表明，复朗之万方法能够通过解决相关的符号问题，成功模拟量子场论的非微扰特性。

英文摘要：

Non-perturbative formulations of field theories are essential to capture intriguing physical phenomena, including confinement in QCD, spontaneous supersymmetry breaking, and dynamical compactification in superstrings. Lattice regularization provides a robust framework to study these non-perturbative features through Euclidean path integrals. Conventionally, path integrals are numerically evaluated using Monte Carlo methods, where the Boltzmann factor is interpreted as a probability weight. However, complex actions in various physical systems render the Boltzmann factor complex, leading to the sign problem. The complex Langevin method overcomes the sign problem and can be used to evaluate complex integrals. This thesis employs the complex Langevin method to investigate various non-perturbative aspects of field-theoretic systems with complex actions. We probe the possibility of spontaneous supersymmetry breaking in the simplest realizations of supersymmetric field theories. These systems generally have complex actions arising from a complex determinant of the fermion operator. We studied various interesting classes of complex potentials, including those exhibiting PT-symmetry. Another exciting aspect explored is the dynamical compactification of extra dimensions in superstring theory. The IKKT matrix model, in the large-N limit, is a conjectured formulation for the 10D type IIB string theory. We employ the complex Langevin method to investigate the Euclidean version of this matrix model, which has an inherent complex Pfaffian, to probe the spontaneous breaking of SO(10) symmetry. The investigations performed in this thesis suggest that the complex Langevin method can successfully simulate non-perturbative aspects of quantum field theories by taming the associated sign problem.

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