Complex Angular Momentum in General Quantum Field Theory

TL;DR

This paper proves that four-point Green's functions in scalar Quantum Field Theory admit a Laplace-type transform in complex angular momentum, which is holomorphic, invertible, and extends partial-wave coefficients to complex values across all space-time dimensions.

Contribution

It establishes the existence and properties of a Laplace-type transform in complex angular momentum for scalar QFT four-point functions across all dimensions.

Findings

Transform is holomorphic in a half-plane of complex angular momentum.

Transform is invertible and corresponds to absorptive parts in crossed channels.

It extends partial-wave coefficients to complex angular momentum in all space-time dimensions.

Abstract

It is proven that for each given two-field channel - called the ``t-channel''- with (off-shell) ``scattering angle'' $\Theta_t$, the four-point Green's function of any scalar Quantum Fields satisfying the basic principles of locality, spectral condition together with temperateness admits a Laplace-type transform in the corresponding complex angular momentum variable $\lambda_t$, dual to $\Theta_t$. This transform enjoys the following properties: a) it is holomorphic in a half-plane of the form $Re \lambda_t > m$, where m is a certain ``degree of temperateness'' of the fields considered, b) it is in one-to-one (invertible) correspondence with the (off-shell) ``absorptive parts'' in the crossed two-field channels, c) it extrapolates in a canonical way to complex values of the angular momentum the coefficients of the (off-shell) t-channel partial-wave expansion of the Euclidean four-point function of the fields. These properties are established for all space-time dimensions d+1 with $d \ge 2$.