Embedding Space Approach to Lorentzian CFT Amplitudes and Causal Spherical Functions

TL;DR

This paper develops an embedding space framework for Lorentzian conformal field theories, emphasizing causality constraints and constructing causal spherical functions that relate to Minkowski conformal blocks.

Contribution

It introduces a novel embedding space approach to Lorentzian CFT amplitudes, incorporating causality via semigroups and connecting to causal spherical functions and Minkowski conformal blocks.

Findings

Causality constraints are implemented through semigroups.

Four-point amplitudes are shown to be equivalent to H bi-invariance.

Construction of causal zonal spherical functions linked to Minkowski conformal blocks.

Abstract

Conformal Field Theory in a Minkowski setting is discussed in an embedding space approach, paying special attention to causality constraints for four-point amplitudes. The physics of dilatation and Lorentz boost is emphasized in specifying the non-compact Maximal Abelian subgroup (MASG) of $SO(d,2)$. Reduction of a Conformal Field Theory (CFT) four-point amplitudes as functions of cross ratios is shown to be equivalent to enforcing $H$ bi-invariance, i.e., $F(hgh')=F(g)$, with $g\in SO(d,2)$ and $H$ an appropriate subgroup. Causality is imposed by introducing appropriate semigroups. Causal zonal spherical functions are constructed, making contact with Minkowski conformal blocks introduced previously.