Spatial integrals in non-standard dimensions via Gaussian measure and analytic continuation

TL;DR

This paper develops a method for defining and computing spatial integrals in non-integer and negative dimensions using Gaussian measures and analytic continuation, addressing convergence issues and symmetrizing angular calculations.

Contribution

It introduces a regularization strategy and symmetry-based approach to extend Gaussian integrals to non-standard dimensions, including negative and fractional ones.

Findings

Established rules for iterative angular integrations in non-integer dimensions.

Identified the region d in [0,1] as fundamental for angle generation.

Provided a method to handle convergence and symmetrization in non-standard dimensions.

Abstract

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.