论文标题
通过高斯测度和分析延续中的非标准维度的空间积分
Spatial integrals in non-standard dimensions via Gaussian measure and analytic continuation
论文作者
论文摘要
通过尺寸正则化，非全能维度在量子场理论（QFT）中很普遍。特别是这会影响涉及点产物的角度计算。这些结构从普遍接受的公理中升起，即高斯积分可以写成单个维高斯积分的$ d $维乘积。该结果可以直接扩展，以使任何上述零维的“表面”涉及任何上述结果，但是在考虑尺寸的负值时，融合存在一些明显的歧义。可以通过适当的正则化策略来回答这个障碍，从而导致可接受的分析延续。此外，我们建议一种通过应用Eulerγ函数的对称性来对称角度计算回到正尺寸变体的方法。通过这种方法，[0,1] $中的区域$ d \被认为与剩余的半轴有根本不同。通过将该区域设置为角度生成维度的适当限制，我们充分建立了生成方法的迭代使用规则和最大数字积分角度。
Non-integer dimensions are commonplace in quantum field theories (QFTs) through dimensional regularization. In particular this affects angular calculations involving dot products. The structure of these rises from the generally accepted axiom that Gaussian integrals can be written as a $d$-dimensional product of a single dimensional Gaussian integral. This result can be extended in a straightforward manner to involve any above zero-dimensional "surfaces", but there is somewhat clear ambiguity with convergence when considering negative values of dimensions. This obstacle can be answered with proper regularization strategy, which leads to an acceptable analytic continuation. Furthermore, we suggest a method of symmetrizing the angular calculations back to positive dimensional variants by applying the symmetries of Euler gamma functions. Through this method, the region $d \in [0,1]$ is recognized to be fundamentally different from either remaining half-axis of dimensionality. By setting this region as the proper limit for the angle generating dimension, we fully establish the rules of iterative use of the generating method and the maximal number integration angles.
