论文标题
量子力学和量子统计的路径积分公式中的符号问题
Sign problems in path integral formulations of quantum mechanics and quantum statistics
论文作者
论文摘要
如今，“标志问题”一词用于识别两个不同的问题。克服Feynman Path积分中强烈振荡复杂的Imtegrand的第一种“符号问题”的想法来自Picard-Lefschetz理论和莫尔斯理论的复杂版本。主要思想是选择Lefschetz Thimbles作为接近路径积分处的临界点的循环，其中复杂作用的虚构部分保持恒定。由于动作的假想部分在每个顶针上都是恒定的，因此符号问题消失了，可以更有效地计算积分。在这里，基于大都市 - 黑斯廷斯算法已经提出了一种强烈振荡整合体积分的新方法。已经进行了一些简单的测试计算和与可用分析结果的比较。 “符号问题”的第二种类型是通过路径积分方法在研究费米系统上产生的，这是由于密度矩阵的真实值矩阵元素的反对称性而引起的。在量子力学的Wigner公式中，已经讨论了相位空间中有效对假苯二电势的显式分析表达。获得的假能力可以考虑费米的统计效应，因为意识到由于相同的费米子之间的排斥而导致的保利（Pauli）阻止了费米子（Fermions），从而阻止了它们占领同一相空间单元。为了测试这种方法，在广泛的动量和退化参数中，已经介绍了费米颗粒理想系统的动量分布函数的计算。
Nowadays the term 'sign problem' is used to identify two different problems. The ideas to overcome the first type of the 'sign problem' of strongly oscillating complex valued imtegrand in the Feynman path integrals comes from Picard-Lefschetz theory and a complex version of Morse theory. The main idea is to select Lefschetz thimbles as the cycle approaching the critical point at the path-integration, where the imaginary part of the complex action stays constant. Since the imaginary part of the action is constant on each thimble, the sign problem disappears and the integral can be calculated much more effectively. Here based on the Metropolis -- Hastings algorithm a new method of calculations of the integral of the strongly oscillating integrands has been prosed. Some simple test calculation and comparison with available analytical results have been carried out. The second type of the 'sign problem' arises at studies Fermi systems by path integral approach and is caused by the requirement of antisymmetrization of the real valued matrix elements of the density matrix. An explicit analytical expression for effective pair pseudopotential in phase space has been discussed in Wigner formulation of quantum mechanics. Obtained pseudopotential allow to account for Fermi statistical effects as realizes the Pauli blocking of fermions due to the repulsion between identical fermions, which prevents their occupation of same phase space cell. To test this approach, calculations of the momentum distribution function of the ideal system of Fermi particles have been presented over a wide range of momentum and degeneracy parameter.
