论文标题
用洛伦兹背景指标与Zeta登记的晶格场理论
Zeta-regularized Lattice Field Theory with Lorentzian background metrics
论文作者
论文摘要
晶格场理论是一种非常强大的工具，可以非扰动地研究Feynman的路径积分。但是，通常要求欧几里得背景指标必须明确定义。另一方面，基于傅立叶积分运算符$ζ$符合的最近开发的正规化方案可以在Lorentzian背景指标中对Feynman的路径积分进行非访问。在本文中，我们正式使用Lorentzian背景正式化晶格理论，并确定适用的傅立叶积分运算符$ζ$ function的条件。此外，我们表明，$ζ$调查理论的经典限制与正则化无关。最后，我们将谐波振荡器视为一个明确的例子。我们讨论了正规化的多种选择，并在分析上表明它们都在晶格上和连续限制中重现正确的基态能量。此外，我们以数值方式在Minkowski背景的晶格上求解了谐波振荡器。
Lattice field theory is a very powerful tool to study Feynman's path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator $ζ$-functions can treat Feynman's path integral non-pertubatively in Lorentzian background metrics. In this article, we formally $ζ$-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator $ζ$-function regularization to be applicable. Furthermore, we show that the classical limit of the $ζ$-regularized theory is independent of the regularization. Finally, we consider the harmonic oscillator as an explicit example. We discuss multiple options for the regularization and analytically show that they all reproduce the correct ground state energy on the lattice and in the continuum limit. Additionally, we solve the harmonic oscillator on the lattice in Minkowski background numerically.
