Test of Transitivity in Quantum Field theory using Rindler spacetime

We consider a massless scalar field in Minkowski spacetime $\cal{M} $ in its vacuum state, and consider two Rindler wedges $R_1$ and $R_2$ in this space. $R_2$ is shifted to the right of $R_1$ by a distance $\Delta$. We therefore have $R_2\subset R_1 \subset \cal{M}$ with the symbol $\subset$ implying a quantum subsystem. We find the reduced state in $R_2$ using two independent ways: a) by evaluation of the reduced state from vacuum state in $\cal{M}$ which yields a thermal density matrix, b) by first evaluating the reduced state in $R_1$ from $\cal{M} $ yielding a thermal state in $R_1$, and subsequently evaluate the reduced state in $R_2$ in that order of sequence. In this article we attempt to address the question whether both these independent ways yield the same reduced state in $R_2$. To that end, we devise a method which involves cleaving the Rindler wedge $R_1$ into two domains such that they form a thermofield double. One of the domains aligns itself along the wedge $R_2$ while the other is a diamond shaped construction between the boundaries of $R_1$ and $R_2$. We conclude that both these independent methods yield two different answers, and discuss the possible implications of our result in the context of quantum states outside a non-extremal black hole formed by collapsing matter.

Introduction: Recent astrophysical observations have propelled the study of black holes to the forefront of cutting edge research, both from a theoretical standpoint as well as in the context of observational cosmology. However, there remain quite a number of results regarding black hole physics that still await experimental confirmation -Hawking radiation, information retrieval are yet to receive experimental validation which might come sooner than expected. Additionally from a theoretical perspective, extensive work is being carried out in the fields of black hole entropy, entanglement, quantum information as reviewed in [1][2][3][4][5]. Unsurprisingly, Rindler spacetime, which happens to be the near horizon limit of non-extremal black holes has been extensively used to understand various properties of the event horizon. Entanglement entropy, Hawking radiation, and horizon properties can be investigated in basic models of this space time before being treated in more realistic scenarios like black hole spacetimes [6]. It is well established that if we define a quantum field theory (say a massless scalar field) in Minkowski space and let the quantum state of the field be in vacuum, then a uniformly accelerated observer perceives the same vacuum to be filled with particles that are thermally populated.
The Rindler spacetime is viewed as a quantum subsystem of the Minkowski spacetime and therefore the reduced state of Minkowski vacuum in Rindler spacetime is thermal. In this letter, we point out an important caveat that * sashideep@hyderabad.bits-pilani.ac.in † p20200473@hyderabad.bits-pilani.ac.in ‡ prasant.samantray@hyderabad.bits-pilani.ac.in is missed out in many calculations involving quantum fields in a subsystem or quantum field theory in curved background., that can lead to results that are not generic and are valid as special cases. In general terms the problem we address is this: In a quantum field theoretic setting we consider a system M. A quantum subsystem of M is R 1 and a quantum subsystem of R 1 is R 2 . R 2 is therfore a subsystem of M also. To determine the reduced state in a subsystem from the quantum state of the full system, we trace over the unobserved degrees of freedom. We can arrive at the reduced state in R 2 via two independent methods: 1) we interpret R 2 as a subsystem of M and trace over unobserved degrees of freedom of R 2 in M. 2) we interpret R 2 as a subsystem of R 1 which is in turn a subsystem of M. So R 2 is a sub-subsystem of M. We find reduced state of R 2 from the reduced state of R1. We therefore trace out degrees of freedom outside R 1 first and then trace out degrees of freedom outside R 2 but belongs to R 1 . We justify in the letter that both the methods need not yield the same answer implying that specific criteria regarding entanglement between the subsystems might be needed for the answers to match. The reduced state may indeed depend on the sequence of how the tracing is carried out. In the model we consider below we show explicitly that both the methods yield two different answers.

Problem of Transitivity:
Here we define the transitivity problem in quantum field theory within the framework of Rindler spacetime. We consider twodimensional Minkowski space M with coordinates T M , X M . We consider a Rindler wedge (R 1 ) with standard interpretation in the figure (1), with τ 1 , ξ 1 corresponding to Rindler coordinates. This wedge is such that the bifurcation point is at the origin (O in figure (1)) of the Minkowski spacetime. A second Rindler wedge (R 2 ) is also considered, where the bifurcation point (O 1 ) is moved to the right by ∆, as shown in the figure (1. Let the quantum state of a scalar field in Minkowski space be in vacuum |0 M . The wedge R 1 , with the scalar field constitute a quantum subsystem of Minkowski space, and the reduced quantum state in R 1 has a thermal spectrum with a density matrix ρ R1M . As for the wedge R 2 , we observe that it is a Rindler wedge shifted to the right by a ∆. Once again the quantum state in R 2 is a thermal state with density matrix ρ R2M . Here the subscript 2 indicates the wedge (R 2 ) state, and the second subscript M indicates the initial state, representing the Minkowski state. As R 2 is a quantum subsystem of R 1 , the reduced quantum state of R 2 can be estimated from ρ R1M and is denoted as ρ R2R1 . We take the thermal density matrix in R 1 given by ρ R1 and estimate the reduced density matrix in R 2 . The question is whether the quantum state deduced this way be the same as that which is obtained as a reduced state from Minkowski vacuum, i.e is ρ R2M = ρ R2R1 ? In other words, is the quantum state transitive? The reduced state ρ R2M involves tracing over the degrees of freedom on the T M = 0 line from X M = −∞ to X M = 0 whereas, the reduced state ρ R2R1 is obtained in two steps -first tracing over the degrees of freedom on T M = 0 line from X M = −∞ to X M = 0 to obtain ρ R1 from Minkowski vacuum and then tracing from X M = 0 to X M = ∆ (as shown in the figure (1)) to obtain ρ R2R1 . This letter aims to check whether the reduced quantum state is independent of the tracing details. This question is therefore of fundamental importance in understanding the behavior of quantum fields and quantum information content (when one estimates the Von Neuman entropy of the quantum state in R 2 as S R2M = −tr(ρ R2M ln(ρ R2M )) as compared to S R2R1 = −tr(ρ R2R1 ln(ρ R2R1 ))). This also has implications for understanding final quantum state in exterior of a black hole. This is elaborated in the last section.
Quantum Mechanical situation: We first present a few remarks in the quantum mechanical scenario for completeness. We consider a quantum mechanical system consisting of N coupled oscillators in their ground state along the lines of analysis done in [7], [8]. If we denote the coordinates of the oscillators by (x 1 , x 2 ...x N ) and consider the reduced density matrix of the system obtained by tracing out first 'M' out of the N coupled oscillators, then the final reduced state will depend only on (x M+1 , x M+2 ...x N ). It was explicitly shown in [7] that we obtain a mixed state. If we now ask the question whether the mixed state depends on the sequence of the tracing over the coordinates (x 1 ...x M ), i.e. if we first trace out K oscillators and then L oscillators such that K + L = M for various positive values of K and L. Does the reduced state depends on the order in which the tracing is performed? It is easy to verify using simple calculation that the reduced state of N − M quantum system is independent of the order in which the tracing operation is done. The density matrix obtained by tracing out (x 1 ...x M ) is therefore invariant. However, the situation in quantum field theory is more subtle.
The set-up: Consider a massless scalar field in Minkowski spacetime M, which is in its vacuum state. Based on the standard Bogoliubov method, it is trivial to compute the reduced state in R 2 from the vacuum state in M. The challenging part is estimating the reduced state in R 2 from the thermal state in R 1 .
Prelude: The preprint [9] discusses the question about how a vacuum state in R 1 appears in R 2 . Below, we summarize the relevant coordinates for formulating our question. We use coordinates (T M , R M ) for two dimensional Minkowski spacetime M; (τ 1 , ξ 1 ) for the coordinates in the wedge R 1 ; and (τ 2 , ξ 2 ) for the coordinates in the wedge R 2 . Listed below are the relations between the various coordinates, T M = e aξ1 sinh(aτ 1 )/a = e aξ2 sinh(aτ 2 )/a and X M = e aξ1 cosh(aτ 1 )/a = e aξ2 cosh(aτ 2 )/a + ∆ . Where 'a' indicates the acceleration parameter in each of the wedges R 1 and R 2 and ∆ indicates the shift of the wedge R 2 from R 1 along the common X M − axis as shown in the figure (1). The metric in Minkowski spacetime is . Similarly for R 2 the metric is given by ds 2 = e 2aξ2 (−dτ 2 2 + dξ 2 2 ). Using the above, one can easily deduce the horizon structure (causal boundaries) of R 1 and R 2 as illustrated in the diagram 1. Separate acceleration parameters a 1 and a 2 for the first and second wedges can be specified, but this does not change the qualitative aspects of the results. So for simplicity, we assume both acceleration parameters to be the same.
Null coordinates provide us with greater insight into our problem. Here the null rays are defined as for the wedges R 1 and R 2 respectively. It is easy to deduce the relationship between them, as shown below, The coordinate range for all the null rays are −∞ < U i , V i < ∞ (where i takes values M, 1 and 2). This range makes it easy to see that the horizon for R 1 is given by ( . Similarly for the wedge R 2 , the horizons can be found to be given by (u 2 = ∞, v 2 = −∞) which in Minkowski null coordinates is given by (U M = −∆,V M = ∆) and in terms of R 1 null coordinates, the horizons of R 2 map to (u 1 = − ln(a∆)/a, v 1 = ln(a∆)/a). Hereafter we choose the value of ∆ = 1/a so that the horizons of the wedge R 2 pass through the origin of R 1 , (τ 1 = 0, ξ 1 = 0). We evaluate the relationship between (u 1 , v 1 ) and (u 2 , v 2 ) in the near horizon limit for the wedge R 2 . In the limit of u 2 → −∞, in which case, from equation 1, we get u 1 = u 2 . In the limit v 2 → −∞, from 2 we get e av1 = e av2 + 1. Taking logarithm on both sides and power expanding ln(1 + x) for small x, we get v 1 = e av2 /a (these limits also apply to early time behavior of (u 2 , v 2 )). Similarly, v 1 = v 2 and u 1 = −e −au2 /a represents the late time behavior. The above relations between null rays of R 1 and R 2 can be summarized as follows: u 1 = −e −au2 /a, v 1 = e av2 /a. This observation is reminiscent of the relation between Kruskal coordinates and Schwarzschild light cone coordinates or between Minkowski and the Rindler coordinates. Later, we will demonstrate that this near-horizon relation plays a crucial role in particle content.
Cleaving of Rindler chart R 1 . The rich substructure of Rindler spacetime: We now take the following detour that lets us estimate the reduced state based on the relation between the null coordinates near the horizon of R 2 . The strategy is to cleave the Rindler spacetime R 1 into two parts such that they form a thermofield double of each other. This split regions play the same role in the context of Rindler spacetime R 1 as Rindler Left and Right wedges play in the context of Minkowski spacetime. In [10], the authors have defined an interesting spacetime called Rindler-Rindler (RR) spacetime. We use these coordinate to carry out the cleaving of R 1 spacetime. The RR coordinates (τ rrr , ξ rrr ) are defined as τ 1 = e aξrrr sinh(aτ rrr )/a and ξ 1 = e aξrrr cosh(aτ rrr )/a. We have used the subscript 'rrr' for Rindler-Rindler-Right since we show later how we can define Rindler-Rindler-Left coordinates. In [10], the authors have constructed quantum field theory in Rindler-Rindler-Right (RR − R) spacetime and showed that the vacuum state in R 1 appears as a thermally populated state in RR − R spacetime. The metric in these coordinates is conformal and is given by ds 2 = e 2a(ξ1+ξrrr ) (−dτ 2 rrr + dξ 2 rrr ). We show below that the RR − R spacetime and R 2 wedge share the same horizons and therefore are two different coordinate systems for the same region of spacetime. In the figure  (1), we indicate that R 2 and RR − R occupy the same wedge region. To demonstrate this, we define null coordinates of RR − R spacetime and find the map between null coordinates of R 1 and RR − R. From the definition of coordinates, we easily derive that u 1 = − e −aurrr a and v 1 = e avrrr a where (u rrr = τ rrr − ξ rrr ) and (v rrr = τ rrr + ξ rrr ). Considering the range of null rays −∞ < u rrr , v rrr < ∞, we arrive at the fact that the relevant boundary/horizon of RR − R spacetime is given by (u rrr = ∞, v rrr = −∞). This corresponds to (u 1 = 0, v 1 = 0), showing that the RR − R spacetime and R 2 spacetime share the same boundary. In fact this relation between Rindler wedges and Rindler-Rindler spacetimes can be made more general by choosing different acceleration parameters 'a' for the wedges given a shift ∆. This implies that given a shifted wedge, the cleaving of the R 1 spacetime can be done by choosing an appropriate acceleration parameter 'a' such that RR − R coincides with the shifted wedge. Our goal is to estimate the features of particle content in R 2 due to the thermal state in R 1 which is in turn a reduced state from pure vacuum state of the scalar field in Minkowski spacetime. We observe that in the near horizon (of R 2 ) behavior of the two coordinate systems (u 2 , v 2 ) and (u rrr , v rrr ) are equal due to the fact that u 1 = − e −au 2 a = − e −aurrr a , v 1 = e av 2 a = e avrrr a . Close to the horizon, the coordinates of R 2 and RR − R converge. We can also arrive at the same conclusion from the expression for the metric in RR − R spacetime. We can write the metric in terms of null coordinates as ds 2 = e 2a(ξ1+ξrrr ) (−dτ 2 rrr + dξ 2 rrr ) = e a(v1−u1+vrrr −urrr ) (−du rrr dv rrr ).
From these expressions, we can see that near the horizon of R 2 (v 1 − > 0, u 1 − > 0), the two coordinate systems (u 2 , v 2 ) and (u rrr , v rrr ) coincide. We make an assumption that the particle content in the reduced state is crucially dependent on the near horizon behavior of the modes. We therefore estimate the particle content of R 2 in the Rindler coordinates by considering the isometry between both the coordinates in the near horizon limit and the causal structure of RR − R region. We note that ∂ τrrr is not a killing vector and the metric in RR − R coordinates is not stationary. Nevertheless the vector ∂ τrrr , in the near horizon limit as well as late/early times, aligns itself along the Killing vector ∂ τ2 = X M ∂ TM +T M ∂ XM −∆∂ TM due to the coordinates(τ 2 , ξ 2 ) coinciding with (τ rrr , ξ rrr ). We now construct the Rindler-Rindler-Left wedge. Causal Diamond construction for Rindler-Rindler-Left Chart : We define the 'left wedge' RR − L (Rindler-Rindler-Left) using the definition, (τ rrl , ξ rrl ) such that τ 1 = −e aξ rrl sinh(aτ rrl )/a and ξ 1 = −e aξ rrl cosh(aτ rrl )/a. The null coordinates with proper range determine the boundary of this left wedge. This region is like a diamond in the Minkowski spacetime diagram as shown in the figure (1). Surprisingly this region is the thermofield twin of the RR − R region as justified subsequently in this article. As suggested in [10] one can think of a series of Rindler-Rindler-Rindler-Rindler.... frames. Now by constructing RRR − L, RRRR − L... spacetimes we can get these diamonds and arrive at a diamond necklace structure in Minkowski spacetime as seen in the figure (2). In fact by choosing different acceleration parameters 'a', one can vary the size of the diamonds in the necklace without the physicist worrying about the budget! As of now, reduced quantum states and entropy aspects of these diamond regions have not been studied. with ω k = |k|. Similarly the positive frequency modes (defined with respect to −∂ τrr ) with support in RR − L are found to be f RR−L k = e iω k τrr+ikξrr / (4π). These modes, together with their complex conjugates form a complete set in RR − R and RR − L spacetimes respectively. We note that the Cauchy surface for R 1 is τ 1 = 0 (positive X M axis), while τ rrr = 0 is the Cauchy surface for RR − R (X M = 1/a to X M = ∞) and similarly, τ rrl = 0 for RR − L spacetime (from X M = 0 to X M = 1/a). The scalar product for the modes has the standard definition and is given by, where √ h is the square root of the determinant of the induced metric on the spacelike hypersurface (e aξ1 for τ 1 = 0 in R 1 and e a(ξ1+ξ2) for the hypersurface τ RR−R = 0 and τ RR−L = 0 for RR − R, RR − L spacetime) and n µ is the unit normal to the hypersurface (in all the cases discussed in this article, n µ turns out to be non zero only for the time component and has magnitude 1/ √ h). The normalization of the modes is defined with respect to the above norm. |0 RR−L = 0 respectively. In the paper [11], it is shown that if we take a vacuum state for the Rindler spacetime R 1 , the state |0 R1 yields a mixed thermal state in RR − R with the temperature a/2π. This calculation is done by finding out the Bogoliubov coefficients between the modes of R 1 and RR − R. We rederive the same fact by defining equivalent Unruh-modes for the RR spacetime below. By using these Unruh modes we can determine the particle content of RR − R given a thermal state in R 1 .

Unruh modes:
The discussion follows the standard treatment as given in [12] and [13]. It is easily shown that f RR−R k = (−u 1 ) iω k /a a iω k /a / √ 4πω k in the RR − R region. In the RR − L region we can show that f * RR−L −k = e πω k /a (−u 1 ) iω k /a a iω k /a / √ 4πω k which implies that the modes can be analytically continued into each other over the entire τ 1 = 0 plane. Just as in the case of Rindler spacetime, we can now construct the Unruh modes that are well defined in both the left and right Rindler-Rindler wedges. The normalized Unruh modes take the form similar to the Minkowski Unruh modes, are the annihilation (creation) operators corresponding to the two Unruh-modes. Both the annihilation operators operate on the vacuum state of R 1 to yield d 1 k |0 R1 = 0, d 2 k |0 R1 = 0. To rederive the result in [10] using these Unruh modes, we express the Rindler-Rindler operators as, To find out how Rindler vacuum (R 1 ) appears in RR − R, we evaluate the expression for the expectation of the number operator on R 1 vacuum state and obtain, showing the Planckian distribution with temperature given by a/2π. The same can be shown for RR − L case.
Particle content in near horizon limit of R 2 : With this background, We now attempt to address the central issue raised in this article. We begin by expressing Rindler vacuum in terms of its thermofield double, with A 2 k being the normalization constant. We have started with Minkowski vacuum |0 M . The reduced state in Rindler wedge R 1 is a mixed thermal state with temperature a/(2π). In the papers [11], [14] the reduced state in Rindler spacetime (R 1 ) is estimated when the Minkowski space is in a thermal state with temperature T ′ . The reduced state in R 1 is shown to be non thermal and they derive a nice explicit analytical expression for the particle number density in R 1 . Our analysis closely follows [11], [14] owing to the conformal nature of the metric as well as scalar field considered here being massless. We consider the analysis for one set of Unruh-Rindler particles with modes h 1 k . The density matrix for a thermal state in R 1 is given by with B 2 k normalization constant. Following the notation in [11], [10], and using equations (7), (8) we express d 1 k in terms of RR-R and RR-L creation and annihilation operators as, withQ = e −πω k /a andP = (e πω k /2a )/ 2sinh(πω k /2a). Using the expression for (d †1 k ) m from Appendix in [10], obtained by binomially expanding equation (12), and tracing over the resultant density matrix over the RR-L states, we obtain the reduced density matrix given by (after equating both the temperatures in [10] to a/2π), Here C k is the normalization constant which is calculated to be C k = (1−e −ω k 2π/a ) by imposing the condition that the trace of density matrix equals one. We are interested in the number density in RR − R spacetime and this can be evaluated from the expression This is therefore the particle content in the shifted Rindler spacetime R 2 in the near horizon limit. The above particle spectrum is obviously not Planckian, and is therefore non thermal. If transitivity were to hold in quantum field theory in the presence of horizons, we would have had a thermal spectrum with temperature given by a/2π. But instead our particle estimate yields a non thermal spectrum and therefore the result is indicative of the fact that transitivity is lost in the problem. The conclusion being that the reduced quantum state in a quantum field theoretic setting can depend on the order in which the tracing out is carried out.

Consequences and Discussion:
In situations where transitivity is absent, the reduced state of a subsystem is specific to one particular sequence of tracing out procedure and does not imply a general answer. Failure of transitivity raises many questions. Some of the pertinent questions in the model discussed in the letter are as follows. If we refine the wedges between R 1 and R 2 with N number of wedges, say W 1 , W 2 ..W N such that R 2 ⊂ W N ⊂ ..W i ... ⊂ W 1 ⊂ R 1 , one may ask the question -What is the reduced state in R 2 due to intermediate N wedges calculated from R 1 ? Can one estimate the form of the reduced state after N iterations? Is there an asymptotic expression for the reduced state (in terms of particle content, density matrix ) as N tends to infinity? Can one deduce an asymptotic formula if it exists? How does this Rindler model discussed in the letter have relevance in the real world situation? To that end, we claim that our model using shifted Rindler wedges becomes relevant in the realistic scenario involving dynamical horizons. In [15], it is proved that in the collapse of generic matter, the dynamical horizon is a spacelike hypersurface. This fact is crucial in connecting our shifted Rindler situation to the realistic gravitational collapsing scenario. In the figure 3, we describe a two step process. Two null shells with null coordinates given by V 1 and V 2 (with masses M 1 and M 2 ) collapse to form the black hole. The arbitrary shift between the two Rindler wedges in the Letter now gets naturally determined by the details of the two null shells as can be seen in the