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.
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.
Consider a massless scalar field in Minkowski spacetime
M in its vacuum state and two Rindler wedges
R1 and
R2. Here
R2 is shifted to the right of
R1 by a distance $\mathrm{\ensuremath{\Delta}}$. We, therefore, have ${R}_{2}\ensuremath{\subset}{R}_{1}\ensuremath{\subset}\mathcal{M}$ with the symbol $\ensuremath{\subset}$ implying a quantum subsystem. We find the reduced state in
R2 using two independent ways: (a) Evaluation of the reduced state of
R2 from vacuum state in
M, which yields a thermal density matrix. (b) First, evaluating the reduced state in
R1 from
M yielding a thermal state in
R1, and subsequently evaluating the reduced state in
R2 from
R1. In this article, we attempt to address the question of whether both these independent paths yield the same reduced state in
R2. To that end, we devise a method that involves cleaving the Rindler wedge
R1 into two domains such that they form a thermofield double. One of the domains aligns itself along the wedge
R2 while the other is a diamond-shaped construction between the boundaries of
R1 and
R2. We conclude that both these independent methods yield two different answers, indicating that there are systems in which the reduced quantum state of a subsystem can be path dependent and hence not unique in a quantum field theoretic setting. We discuss the possible implications of our result in the context of quantum states outside a nonextremal black hole formed by collapsing matter.
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.
Abstract The Minkowski vacuum
∣0⟩M | 0 ⟩ M , which for an inertial observer is devoid of particles, is perceived as a thermal bath by Rindler observers living in a single Rindler wedge (Unruh in Phys Rev D 14:870, 1976) as a result of the discrepancy in the definition of positive frequency between the two classes of observers and a strong entanglement between degrees of freedom in the left and right Rindler wedges. We revisit the problem of quantification of the correlations in a two-mode state of a free neutral scalar field which is observed by an inertial observer Alice and left/right Rindler observers Rob/AntiRob, a problem that pertains to the field of relativistic quantum information and has been previously studied in Martin-Martinez et al. (Phys Rev D 82:064006, 2010) and Datta (Phys Rev A 80:052304, 2009). We focus on the analysis of informational quantities like the locally accessible and locally inaccessible information (Koashi and Winter in Phys Rev A 69:022309, 2004; Fanchini et al. in Phys Rev A 84:012313, 2011; Fanchini et al. in New J Phys 14:013027, 2012) and a closely associated entanglement measure, the entanglement of formation. We conclude that, with respect to the correlation structure probed by inertial observers alone, the introduction of a Rindler observer gives rise to a correlation redistribution which can be quantified by the entanglement of formation. Given the informational meaning of the derived correlations, we discuss on the capacity of a quantum channel to communicate classical information between accelerated parties.
Abstract The Minkowski vacuum is often presented in textbooks and reviews as a thermofield double (TFD) state, an entangled state of field modes in the left and right Rindler wedges. This picture is widely used to explain the Unruh effect, motivate entanglement entropy calculations, and connect quantum field theory to black hole thermodynamics and AdS/CFT. However, we show that this interpretation, while elegant, is not exact. We explicitly compute two-point functions and their derivatives for a massless scalar field in two-dimensional Minkowski space, comparing results obtained from canonical quantization with those obtained by assuming a TFD form of the vacuum. Mixed-derivative correlators agree perfectly, but higher-derivative correlators show systematic mismatches that persist even for points well away from horizons and are not removed by infrared regularization. To further test this picture, we construct an alternate coordinate system that divides Minkowski spacetime into two disconnected regions, apply the same derivation that leads to the standard TFD expression, and obtain a new ‘entangled-state’ representation of the vacuum that is not thermal. This demonstrates that the appearance of a TFD structure is a feature of the derivation method rather than a fundamental property of the vacuum. Our results clarify the limits of interpreting the Minkowski vacuum as a literal TFD state, emphasizing that while it captures key thermal features, it should be viewed as a powerful calculational tool rather than a precise statement about Hilbert space structure.
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.