Affine Symmetry and the Group-Theoretic Basis of the Unruh Effect

TL;DR

This paper explores the affine symmetry underlying the Unruh effect, showing how group-theoretic methods involving the affine group and Mellin transforms can derive the thermal spectrum observed by accelerated observers.

Contribution

It provides a novel group-theoretic framework based on affine symmetry to understand the Unruh effect and vacuum thermal phenomena in quantum field theory.

Findings

Derivation of the Unruh thermal spectrum using affine group representations.

Connection between inertial and accelerated observer states via Mellin transform.

Potential applicability of the framework to other quantum systems with translation and dilation symmetries.

Abstract

A massless scalar field in two spacetime dimensions splits into two independent sectors of left and right-moving modes on the light cone. At the quantum level, these two sectors carry a representation of the group of affine transformations of the real line, with translations corresponding to transformations generated by light-cone momenta and dilations given by light-cone Rindler momenta formed by a linear combination of generators of boosts and dilations. One-particle states for inertial observers are eigenvectors of translation generators belonging to irreducible representations of the affine group. Rindler one-particle states are related to eigenfunctions of the generator of dilations. We show that simple manipulations connecting these two representations involving the Mellin transform can be used to derive the thermal spectrum of Rindler particles observed by an accelerated observer. Beyond providing a representation-theoretic basis for vacuum thermal effects, our results suggest that analogous phenomena may arise in any quantum system admitting realizations of translation and dilation eigenstates.