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arXiv:2512.22648

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

Michele Arzano, Alessandra D'Alise, Simone del Rosso, Domenico Frattulillo

hep-thgr-qcquant-phSubmitted Dec 27, 2025• Announced Dec 30, 2025• Revised Jan 20, 2026 (v2)
v1 · Dec 27, 2025v2 · Jan 20, 2026
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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.

Comments
12 pages, 3 figures. v2 minor changes, typos corrected

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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.
We develop a group-theoretic interpretation of the Unruh effect based on affine symmetry on a light ray and relate it to modular theory. For a massless scalar field in two spacetime dimensions inertial and uniformly accelerated observers select two different flows within the same chiral one-particle structure, respectively, null translations and dilations. Minkowski modes are adapted to translations, while Rindler modes are adapted to dilations, with the Mellin transform providing the natural bridge between them. When a Minkowski positive-frequency mode is restricted to a single Rindler wedge, its comparison with Rindler modes is non-unitary within the positive-frequency sector. Modular theory gives the corresponding operator-algebraic interpretation: on the horizon the modular flow of the half-line algebra is implemented by dilations, and the restricted vacuum satisfies the KMS condition. The affine group thus appears as the minimal symmetry structure underlying thermality on the Rindler horizon.
For a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half-sided translations. The modular flows corresponding to the algebras of the forward light cone and a space-like wedge admit a simple geometric description in two-dimensional models that factorize in light-cone coordinates. At large distances from the domain boundary compared to the inverse temperature, the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. For each domain there is also a one-parameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh effect.
A uniformly accelerated observer perceives the Minkowski vacuum as a thermal bath, so acceleration acts as a noise channel on correlations shared with an inertial partner. For a Dirac field in the standard treatment, the resulting map is a one-sided amplitude-damping channel, and entanglement and quantum discord survive at infinite acceleration. We introduce a symmetric Unruh-like map in which the vacuum and the one-particle state both exchange amplitude with the inaccessible Rindler region; for a Bell state it yields a bit-flip channel whose flip probability is the Fermi-Dirac occupation at the Unruh temperature. Using negativity, mutual information, and all six pairwise quantum discords, we find that the entanglement and discord between the inertial observer and the accessible mode vanish together at infinite acceleration, leaving only one bit of classical correlation. These results show that the fate of quantum correlations in noninertial frames depends on the structure of the Minkowski-Rindler map.
While the Unruh effect has traditionally been studied under the assumption of uniform acceleration, a simplification motivated by experimental considerations, it is not necessarily true for all non-inertial motions. We propose a novel approach for the indirect detection of the Unruh effect without relying on the former restriction. Previous studies have shown that probing the decoherence of an Unruh-DeWitt detector can significantly reduce the acceleration required for observing the effect by several orders of magnitude compared to earlier proposals. Building on this idea, we develop a theoretical framework describing a non-inertial observer equipped with a detector undergoing non-uniform, time-dependent acceleration. We show that, in a non-uniformly accelerated Rindler spacetime, the particle distribution perceived in the Minkowski vacuum acquires a time-dependent modification of the standard Unruh spectrum. Furthermore, we demonstrate that the inclusion of quantum entanglement leads to a deformation of the Minkowski vacuum into squeezed states.
We present a simple argument to derive the transformation of the quantum stochastic calculus formalism between inertial observers and derive the quantum open system dynamics for a system moving in a vacuum (or, more generally, a coherent) quantum field under the usual Markov approximation. We argue, however, that, for uniformly accelerated open systems, the formalism must break down as we move from a Fock representation over the algebra of field observables over all of Minkowski space to the restriction regarding the algebra of observables over a Rindler wedge. This leads to quantum noise having a unitarily inequivalent non-Fock representation: in particular, the latter is a thermal representation at the Unruh temperature. The unitary inequivalence is ultimately a consequence of the underlying flat noise spectrum approximation for the fundamental quantum stochastic processes. We derive the quantum stochastic limit for a uniformly accelerated (two-level) detector and establish an open system description of the relaxation to thermal equilibrium at the Unruh temperature.
In this paper we attempt to understand Lorentzian tensor networks, as a preparation for constructing tensor networks that can describe more exotic backgrounds such as black holes. To define notions of reference frames and switching of reference frames on a tensor network, we will borrow ideas from the algebraic quantum field theory literature. With these definitions, we construct simple examples of Lorentzian tensor networks, based on Gaussian models of fermions, and solve the spectrum for a choice of ``inertial frame.'' In particular, the tensor network can be viewed as a periodically driven Floquet system that bypasses the ``doubling problem'' and gives rise to fermions with exactly linear dispersion relations. We will find that a boost operator connecting different inertial frames and notions of ``Rindler observers'' can be defined and that important physics in Lorentz invariant quantum field theory, such as the Unruh effect, can be captured by such a skeleton of spacetime. We find interesting subtleties when the same approach is directly applied to bosons---the operator algebra contains commutators that take the wrong sign---resembling bosons behind horizons.