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

TL;DR

The paper analyzes the Unruh effect in two dimensions from a representation-theoretic perspective by linking Minkowski and Rindler modes to representations of the affine group (the $ax+b$ group). It decomposes the massless scalar field into light-cone sectors, identifies affine symmetries as translations and dilations on the light-cone, and uses the Mellin transform to connect translation eigenstates to dilation eigenstates, yielding a Bogoliubov-like map between inertial and accelerated descriptions. The main result is that Minkowski and Rindler one-particle states reside in inequivalent one-dimensional affine representations, and the Mellin-based construction reproduces the Unruh thermal spectrum with temperature $T={\alpha}/{(2\pi)}$. This work provides a group-theoretic origin for horizon thermality and suggests that similar thermal phenomena may arise in any quantum system admitting translation and dilation eigenstates, with potential experimental implications for affine-symmetric setups.

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.

Figures (3)

• Figure 1: The domains of the four components of $\phi(t,x)$.
• Figure 2: The action of $T(b)$ is transitive on the whole real line; the action of $D(a)$ is transitive only on half-lines.
• Figure 3: The expression (\ref{['psi t pos e neg']}) is analogous to the expansion of $\psi(v)$ (or $\chi(u)$), component of the 2-dimensional massless scalar field in Minkowski spacetime, into $\psi_+(v)$ and $\psi_-(v)$ (\ref{['psi hvs']}) (or $\chi_-(u)$ and $\chi_+(u)$ (\ref{['chi hvs']}), respectively). In this sense, the division of Rindler spacetime into two wedges corresponds to the partition of the real line induced by $D(\lambda)$.