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arXiv:2606.01071

Modular theory and affine representations on the Rindler horizon

Michele Arzano, Paolo Palumbo

hep-thgr-qcquant-phSubmitted May 31, 2026• Announced Jun 2, 2026
v1 · May 31, 2026v2 · Sep 28, 2026
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Abstract

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.

Journal reference
Fortschritte der Physik 74, no. 9 (2026): e70150
DOI
10.1002/prop.70150
Comments
17 pages, 1 figure. Invited contribution to the special issue "Quantum Horizons: Emerging Interfaces of Quantum Mechanics, Quantum Information, and Gravitation" of Fortschritte der Physik

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•Sep 1, 2026•0 citations
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.
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.
Padmanabhan and Kolekar had demonstrated the phenomenon of recursive thermality in two dimensional free field theories where the Rindler vacuum appears thermal to an observer accelerated with respect to the Rindler observer. We study the same construction at an algebraic level for conformal field theories. At each step, a parent wedge is conformally mapped to a reference Minkowski spacetime, where a new right Rindler wedge and its left sibling are selected. We illustrate that under the Bisognano--Wichmann property, thermality holds at every generation: the state associated with a parent wedge becomes a KMS state when restricted to its right child. Further, using the modular structure of the wedge and its causal sibling, we identify the sibling algebra with the modular commutant and derive the corresponding recursive modular flow explicitly.
In this article, we explore the answer to an inverse question to the Unruh effect. Given a two-dimensional Rindler wedge R with a thermal distribution of massless scalar particles, we explore the supersets of R , whose reduced state yields the observed particle content in R . If we restrict our analysis to particle content close to the Rindler horizon in R , we show that the answer to the inverse question is not unique. The analysis is done using two methods, the first uses Bogolibov coefficients and the second method uses Virasoro anomaly. We identify four inequivalent paths with various “parent” spacetimes of R from which we can arrive at the given particle content in R . We show that the supersets are Minkowski spacetime, a Rindler wedge in vacuum, a Rindler wedge with a thermal flux of left-moving particles and lastly a Rindler wedge with a thermal flux of right moving flux of particles. We also show an interesting phenomenon; flux to density conversion, wherein a thermal flux of either left-moving or right-moving particles can be converted to thermal density of particles involving both left-moving and right-moving particles. The analysis presents a natural possibility to define the information entropy for a given Rindler wedge. This stems from the fact that a given Rindler wedge with its particle content could have originated from various supersets found in the article. Based on the analysis done in the article, we present qualitative arguments regarding the possibility that the continuous evaporation of a black hole might be punctuated by a series of pauses and bursts of radiation.
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.
We investigate the information-theoretic structure underlying causal horizons through the formalism of modular quantum channels. By analyzing the singular value decomposition of the channel induced by the partial trace over inaccessible regions, we show that the Unruh effect and the Page curve can be understood as manifestations of a universal thermal filtering mechanism governed by the modular Hamiltonian. This structure leads to a Gibbs-weighted hierarchy of information transmission modes, with entanglement entropy corresponding to spectral activation and thermal capacity. We reinterpret the first law of entanglement as a Clausius relation for modular flow and derive Einstein's equations by requiring its local validity across Rindler horizons. In the case of black hole evaporation, we find that the modular channel undergoes an informational phase transition at the Page time, marked by a shift from entanglement preservation to recoverability, quantified through fidelity and channel capacity. Based on these results, we propose the Modular Channels Flow Correspondence (MCFC): a minimal holographic principle whereby the area of a causal screen measures the storage of filtered quantum information. Our framework offers a unified operational account of holography, entropy, and curvature, grounded in the spectral dynamics of modular channels.
This manuscript establishes a mathematically controlled one-parameter modular-time coordinate for the vacuum representation of a free neutral scalar field on the right Rindler wedge. By restricting the state space strictly to the finite-entropy Weyl-coherent sector, the vacuum relative entropy is shown to be exactly quadratic based on rigorous external results by Longo and Casini–Grillo–Pontello. Consequently, the entropy Hessian exists without remainder estimates (O(t^3)), is strictly positive, and remains invariant under the modular group. By applying the Bisognano–Wichmann theorem, the modular parameter directly identifies with the Rindler boost rapidity up to orientation and the standard KMS-Unruh normalization (s = -2\pi\tau). The manuscript explicitly addresses the separate status of unbounded Araki perturbations (\phi(f)) and defines the exact boundary between proven theorems (⁠[RIGOROUS]⁠) and open matching hypotheses (⁠[OPEN]⁠), avoiding any completing-the-square assumptions for the modular Hamiltonian.