Symmetry, Integrability and Geometry: Methods and Applications The Unruh Effect in General Boundary Quantum Field Theory

In the framework of the general boundary formulation (GBF) of scalar quantum field theory we obtain a coincidence of expectation values of local observables in the Minkowski vacuum and in a particular state in Rindler space. This coincidence could be seen as a consequence of the identification of the Minkowski vacuum as a thermal state in Rindler space usually associated with the Unruh effect. However, we underline the difficulty in making this identification in the GBF. Beside the Feynman quantization prescription for observables that we use to derive the coincidence of expectation values, we investigate an alternative quantization prescription called Berezin-Toeplitz quantization prescription, and we find that the coincidence of expectation values does not exist for the latter.


Introduction
In his seminal paper [31] Oeckl proposed an axiomatic framework for the quantum theory that allows to formulate quantum field theory on general spacetime regions with general boundaries. This new formulation, named the general boundary formulation (GBF), assumes as an important ingredient (a particular version of) the mathematical framework of topological quantum field theory [30][31][32][33]. In particular the set of axioms that defines the GBF implements an assignment of algebraic structures to geometrical structures and guaranties the consistency of such an assignment. The physical interpretation relies on a generalization of the Born's rule to extract probabilities from the algebraic structures (in particular amplitudes and observable amplitudes discussed below).
The main motivation at the basis of the development of the GBF is the desire to render the formulation of quantum theory compatible with the symmetries of general relativity, in view of a possible future formulation of a quantum theory of gravity. From such perspective the GBF appears to be particularly advantageous with respect to the standard formulation of quantum theory since it does not require a spacetime metric for its formulation. Indeed the axioms of the GBF necessitate only a topological structure, and not a metric one. Moreover, it is worth noting that no (space)time notion enters in the definition of the generalized Born's rule. See [29] and [10] for more details on the relevance of the GBF for the problem of quantum gravity.
Although, as said, the spacetime background metric does not play any fundamental role in the GBF, evidently a general boundary quantum theory can be implemented for studying the dynamics of fields defined on a spacetime with a definite metric background. In that case the versatility of the GBF makes it possible to consider not only initial and final data on Cauchy surfaces as in the standard approach to quantum field theory but also the dynamics that take place in more general regions; the main interest will be represented by compact spacetime regions, namely regions whose boundaries have spacelike as well as timelike parts. A certain number of results, [7-9, 13-15, 37, 39], have been obtained by applying the GBF for fields in Minkowski space and curved spaces, among which we cite a new perspective on properties of the standard S-matrix (in particular the crossing symmetry, that becomes a derived property within the GBF) and the proposals of new quantization schemes that allow a generalization of the standard Smatrix. In particular in anti-de Sitter space, where the lack of temporal asymptotic regions obstructs the application of the traditional S-matrix techniques, involving temporal asymptotic in and out states, spatial asymptotic states have been rigorously defined within the GBF of a scalar quantum field theory and the corresponding amplitude that has been computed for these states can then be interpreted as a generalization of the standard S-matrix [12].
In this article, we will investigate a certain relation between the QFT in Minkowski space and the QFT in Rindler space in the framework of the GBF. The investigation is inspired by the Unruh effect which is understood as a particular relation between the notions of particle state in Minkowski and in Rindler spacetime. Rindler space is the spacetime naturally associated with uniformly accelerated observers and is isomorphic to a submanifold of Minkowski spacetime called the Rindler wedge. Then, the Unruh effect can be stated as follows: linearly uniformly accelerated observers perceive the Minkowski vacuum state (i.e. the no-particle state of inertial observers) as a mixed particle state described by a density matrix at temperature 1 T = a/(2πk B ), a being the constant acceleration of the observer and k B the Boltzmann constant. This effect was proposed by Unruh in 1976 [43] and has received a considerably amount of attention in the literature because of its relation to other effects, like the particle creation from black holes (the so called Hawking effect) and cosmological horizons. Although, there are some proposals around aiming at an experimental detection of the Unruh effect [16] it was not discovered, yet. There exist only experimental results that can be interpreted as hints to the Unruh effect [4,25].
The Unruh effect must be distinguished from the result that a uniformly accelerated Unruh-DeWitt detector responds as if submersed in a thermal bath when interacting with a quantum field in the Minkowski vacuum state [43]. Instead, the foundation of the Unruh effect is the statement that the Minkowski vacuum state can be interpreted as a thermal state in Rindler space when restricted to the right Rindler wedge. The derivation of this identity in the standard formulation of QFT is done by first, identifying formally the vacuum state in Minkowski with an entangled state containing linear combinations of products of n-particle states of the field defined in the left and right Rindler wedges where the left Rindler wedge is the point reflection of the right Rindler wedge at the origin of Minkowski space. Then tracing out the degrees of freedom in the left Rindler wedge leads to a density operator in the right Rindler wedge describing a mixed thermal state at the Unruh temperature [5,16,23,40,[42][43][44].
The derivation of the Unruh effect in algebraic QFT is much more sophisticated but works primarily along the same line of argument [41]: It is proven that the restriction of the Minkowski vacuum state to the right Rindler wedge is identical to a certain abstract thermal state ψ in Rindler space defined as a state fulfilling the KMS-condition formally given as the identity of expectation values A(τ )B ψ = BA(τ + iβ) ψ , for all observables A, B where A(τ ) is the time translation of A [41]. Sometimes, the particle content of the KMS-state is investigated by expressing it as a density matrix in Rindler space which is well known to be only approximately possible (see [47,Section 5.1] and [17, Section 6]) 2 .
The mathematical foundation of the derivation in the standard formulation of QFT was criticized by Narozhnyi et al. in [2,3,19,26,27] which led to an answer by Fulling and Unruh in [22] and a reply by Narozhnyi et al. in [28]. The central point of the criticism by Narozhnyi et al. is that in the derivation of the Unruh effect in the standard formulation of QFT a certain term in the mode expansion of the scalar field is neglected which is equivalent to the requirement of an additional boundary condition at the origin of Minkowski space leading to a topological different spacetime.
We will find that also in the GBF a relation between the Minkowski vacuum and a state in Rindler space cannot be derived in a mathematical rigorous way using global mode expansions. Moreover, since a clear and well defined notion of KMS states has not yet been implemented within the GBF, we will not explore the relation between the Minkowski vacuum state and a thermal state in Rindler. Our strategy will be to compute the expextation value of local observables for the QFTs in Minkowski and Rindler. In particular we will derive the coincidence of expectation values of local observables (i.e. observables with compact spacetime support) obtained in two different settings: in the first setting expectation values are computed on the vacuum state in Minkowski spacetime and in the second setting they are computed on a state in Rindler spacetime that corresponds to the thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory [16].
The approach used in this paper is similar in some technical aspects to the one of Unruh and Weiss [45], and also presents some similarity with the derivation of the Unruh effect in algebraic quantum field theory which is based on the restriction of the algebra of observables to the interior of the Rindler wedge [41]. We emphasize that from a physical perspective the restriction of the set of observables seen by the Rindler observer to the interior of the Rindler wedge is a reasonable condition since no observer should be able to measure at its spacelike infinity.
The coincidence of expectation values we will derive in this article is a purely mathematical result in the framework of the GBF. Such a result suggests a relation between the quantum theory in Minkowski space and the one in Rindler space which could be interpreted as a manifestation of the Unruh effect within the GBF. For the derivation the coincidence of expectation values, we will use a particular quantization prescription for observables known as Feynman quantization prescription. It was introduced in the GBF framework in [36]. For another quantization prescription called Berezin-Toeplitz quantization prescription, we will find no such coincidence which can be seen as a shortcoming of this particular quantization prescription.
Another point that must be considered in the context of the Unruh effect is the issue of the definition of temperature. An up-to-date review on this issue can found in [6] where also conclusions for the interpretation of the Unruh effect are drawn. We shall elaborate on the notion of temperature in the context of the GBF in a future article.
The paper is structured as follows: In Section 2 we present a compact review of the GBF by specifying the two different representations so far implemented within the GBF, namely the Schrödinger representation in which the quantum states of the field are wave functionals of field configurations and the holomorphic representation where the states are holomorphic functions on germs of solutions to the field equations. In Sections 3.1 and 3.2 we formulate the general boundary quantum field theory on Minkowski and Rindler spacetimes respectively for a massive Klein-Gordon field both in the Schrödinger and holomorphic representations. In Section 4 we show that for the massive Klein-Gordon field the GBF on Minkowski space and the one on Rindler space are inherently different and cannot be compared directly using global mode expansions. In Section 5 we show that for observables quantized according to the Feynman quantization prescription that are just defined on the interior of the right Rindler wedge expectation values in the Minkowski vacuum coincide with those calculated for a state of the quantum theory in Rindler space that corresponds to the thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory. In the same section we find that this is not the case if we use the Berezin-Toeplitz quantization prescription. Finally, we summarize our conclusions and give an outlook in Section 6.

The general boundary formulation of quantum f ield theory
This section presents a short review of the two representations in which the general axioms of [31] were implemented following the much more elaborate introduction given in [38]. These are the Schrödinger-Feynman representation [31] and the holomorphic representation [33]. We introduce the main structures that will be used in the rest of the paper, such as state spaces and amplitude maps for both representations.
As usual, we start from an action S[φ] = M d N x L(φ, ∂φ, x) which is considered to describe a linear real scalar field theory in a spacetime region M of an N -dimensional Lorentzian manifold (M, g). Denoting the boundary 3 of the region M with Σ, we associate with this hypersurface the space L Σ of solutions of the Euler-Lagrange equations (derived from the action S[φ]) defined in a neighborhood of Σ. 4 The symplectic potential on Σ results to be where n µ is the unit normal vector to Σ. For every two elements of the space L Σ there is the bilinear map [·, ·] Σ : L Σ × L Σ → R defined such that [ξ, η] Σ := (θ Σ ) ξ (η) and the symplectic structure, that is the anti-symmetric bilinear map ω Σ : The last ingredient for the quantum theory we need to specify is a compatible complex structure J Σ represented by the linear map J Σ : L Σ → L Σ such that J 2 Σ = −id, ω Σ (J Σ ·, J Σ ·) = ω Σ (·, ·) and ω Σ (·, J Σ ·) is a positive definite bi-linear map. Remark, that all ingredients but the complex structure J Σ are classical data uniquely defined by specifying the action.
These basic ingredients can now be used in different ways to specify the Hilbert spaces associated with the boundary hypersurface Σ. In the following subsection, we introduce the two representations developed so far within the GBF, namely the Schrödinger representation, usually associated with the Feynman path integral quantization prescription, and the holomorphic representation.

The Schrödinger-Feynman representation
In this representation, quantum states are represented by wave functionals of field configurations. For its implementation, it is convenient to introduce subspaces of the space L Σ of solutions in a neighborhood of the hypersurface Σ. We start be defining what plays the role of the "space of momentum", denoted by M Σ ⊂ L Σ , It can be shown that M Σ is a Lagrangian subspace of L Σ . 5 Next, we consider the quotient space Q Σ := L Σ /M Σ which corresponds the space of all field configurations on Σ. We denote the 3 Notice that whether the boundary hypersurface Σ is a Cauchy surface (or a disjoint union of Cauchy surfaces) has no bearing on the following treatment. 4 More precisely it is the space of germs of solutions at Σ which is the set of all equivalence classes of solutions where two solutions are equivalent if there exists a neighborhood of Σ such that the two solutions coincide in this whole neighborhood. 5 It is this subspace MΣ that defines the Schrödinger polarization of the prequantum Hilbert space constructed from LΣ, see [38] for details. quotient map L Σ → Q Σ by q Σ . The last ingredient needed for the Schrödinger representation is the bilinear map The map Ω Σ is here induced by the complex structure J Σ but can also be considered independently as the single ingredient leading from the classical to the quantum theory as it was done in [36]. However, it was shown in [38] that there is a one-to-one correspondence between bilinear maps Ω Σ appropriate for the Schrödinger representation and complex structures.
Notice that the symplectic potential [·, ·] Σ is equivalently seen as a map from L Σ × Q Σ to the complex numbers. The Hilbert space H S Σ (the superscript S refers to the Schrödinger representation) is now defined as the closure of the set of all coherent states with respect to the inner product where the bar denotes complex conjugation. The vacuum state K S 0 is then defined as the coherent state with ξ = 0.
So far we have defined the kinematical aspects and we now pass to the dynamical ones. Within the GBF the dynamics are encoded in an amplitude map ρ M : H S Σ → C associated with the spacetime region M . In the Schrödinger representation for a state ψ S ∈ H S Σ , the amplitude ρ M is defined in terms of the Feynman path integral prescription formally given by [36] (recall that Σ is the boundary of M ) where L M is the set of all field configurations in M that solve the Euler-Lagrange equations and N M is the normalization constant defined as where L 0 M is the set of all field configurations in M that are zero on Σ. With "Dφ" we have denoted an hypothetical translation-invariant measure on L M . As it is well known, in general, no such measure exists in mathematical rigor. However, using the mathematically well defined holomorphic representation that we will present in the next section Oeckl was able to give perfect mathematical sense to expressions like (2.1) in [38]. It is then possible to apply the generalized Born's rule [31,35] to extract probabilities for the amplitude map ρ M .

The holomorphic representation
From the complex structure J Σ we define the symmetric bilinear form g Σ : and assume that this form is positive definite. Next, we introduce the sesquilinear form The completion of L Σ with the inner product {·, ·} Σ turns it into a complex Hilbert space. The 6 namely the set of square integrable holomorphic functions on L Σ , is the closure of the set of all coherent states [33] where ξ ∈ L Σ and the closure is taken with respect to the inner product where dν Σ can be represented formally as dν with a certain translation invariant measure dµ Σ . The amplitude map for a state ψ h is defined as where LM ⊆ L Σ is the set of all global solutions on M mapped to L Σ by just considering the solutions in a neighborhood of Σ. 7 The measure dνM is a Gaussian probability measure constructed from the metric g Σ [33]. 8 Independent of the representation the amplitude for coherent states turns out to be 9 where ξ R , ξ I ∈ LM and ξ = ξ R + J Σ ξ I .

GBF in Minkowski and Rindler spacetimes
We start with the action for the real massive Klein-Gordon field on (1+1)-dimensional Minkowski spacetime M = (R 2 , η = diag(1, −1)) which is given by The resulting symplectic potential for a spacetime region M with boundary hypersurface Σ is with n µ the normalized hypersurface normal vector field pointing inside the region M and = ±1 if Σ is everywhere spacelike/timelike respectively 10 . The derivative ζ := d dσ Σ(σ) of the embedding function Σ(σ) is normalized as η µν ζ µ ζ ν = 1. 6 To make this mathematically precise one actually has to construct H h Σ = H 2 (LΣ, dνΣ) whereLΣ is a certain extension of LΣ. For more details about the construction ofLΣ and dνΣ we refer the reader to [33]. 7 More precisely, global solutions are mapped to the corresponding germs at Σ. 8 Again, we refer the reader to [33] where the constructions are given that make all the objects used here well defined. Additionally, in cite [38] it was shown that the one-to-one correspondence between maps ΩΣ and complex structures JΣ leads also to mathematically well defined constructions for all the expressions in Section 2.1. 9 See equation (31) of [36] for normalized coherent states and equation (43) in [33] as well as [38]. 10 We stick here to the conventions used in [33] and earlier publications.

Minkowski spacetime
We want to investigate the GBF in a region M ⊂ M bounded by the disjoint union of two spacelike hypersurfaces represented by two equal time hyperplanes (this corresponds to the standard setting), which we denote as Σ 1,2 : . Then the boundary of the region M corresponds to the disjoint union Σ := Σ 1 ∪ Σ 2 (the bar denotes the inverted orientation). The set of solutions in the neighborhood of Σ decomposes in a direct sum as L Σ = L Σ 1 ⊕ L Σ 2 where L Σ 1 and L Σ 2 are the sets of solutions in the neighborhood of Σ 1 and Σ 2 respectively each equipped with the corresponding symplectic form ω Σ 1 respectively ω Σ 2 and a complex structure J Σ 1 respectively J Σ 2 . The inversion of the orientation is implemented by The corresponding Hilbert space associated with Σ is given by the tensor product H Σ = H Σ 1 ⊗ H Σ 2 , where H Σ 1 and H Σ 2 are the Hilbert spaces associated with the hypersurface Σ 1 and Σ 2 respectively and the inversion of the orientation translates to the level of the Hilbert spaces by the map ι : H Σ 2 → H Σ 2 , ψ → ψ as can be seen from the definition of the coherent states in Section 2.
In order to provide an explicit expression to the structures introduced in the previous section we expand the scalar field in a complete basis of solutions of the equation of motion, where ψ p (x, t) are chosen to be the eigenfunctions of the boost generator, namely the boost modes 11 where we have introduced the operator ω = −∂ 2 x + m 2 . These modes are normalized as The Hilbert space H i of the quantum theory, associated to a hyperplane Σ i , i = 1, 2, is defined by the vacuum state written in the Schrödinger representation as N being a normalization constant and ϕ i ∈ Q Σ i are the boundary field configurations on the hypersurface Σ i , namely ϕ i (x) = φ(x, t) t=t i . This vacuum state corresponds to the standard Minkowski vacuum state 12 , whose GBF expression has been given in [14], and it is uniquely defined by the complex structure [1] which defines a unitary complex structure on L Σ in the sense that it is compatible with the dynamics of the field. The boost modes (3.3) are eigenfunctions of this complex structure, i.e.
The structures introduced in the previous section, namely the symplectic form ω Σ i (·, ·), the metric g Σ i (·, ·) and the inner product {·, ·} Σ i , evaluated for two solutions φ, φ ∈ L Σ i , i = 1, 2, take the form The dense subset of the Hilbert space associate to Σ i , defined by the coherent states, as well as the amplitude map associated to the region M are implementable in terms of the above quantities.

Rindler spacetime
For the quantization of the scalar field in Rindler spacetime we consider again the action in equation (3.1) but restricted to the right wedge of Minkowski space, namely R : , which is covered by the Rindler coordinates (ρ, η) such that ρ ∈ R + and η ∈ R. The relation between the Cartesian coordinates (x, t) and the Rindler ones is t = ρ sinh η and x = ρ sinh η, and the metric of Rindler space results to be ds 2 = ρ 2 dη 2 −dρ 2 . We consider the region R ⊂ R bounded by the disjoint union of two equal-Rindler-time hyperplanes In order to repeat the construct of the quantum theory implemented in Minkowski spacetime, we start by expanding the field in a complete set of solutions of the equation of motion, where the Fulling modes K ip is the modified Bessel function of the second kind, also known as Macdonald function [24]. The modes (3.9) are normalized as The Hilbert space associated with the hypersurface Σ R i , i = 1, 2, is characterized by the following vacuum state in the Schrödinger representation, expressed in terms of the boundary field configuration ϕ i , where ω now denotes the operator ω = (ρ∂ ρ ) 2 − m 2 and N is a normalization factor. This vacuum state is in correspondence with the following complex structure, defined by the derivative with respect to the Rindler time coordinate η, 10) and the Fulling modes (3.9) are eigenfunctions of this complex structure: The algebraic structures defined on the hypersurface Σ R i , considered for two solutions φ R , ψ R ∈ L Σ R i result to be It is important to notice that in order for the quantum theory to be well defined the following condition must be imposed on the field in Rindler space: φ R (ρ = 0, η) = 0. Indeed the complex structure (3.10) is well defined except in the origin of Minkowski spacetime as can be seen by expressing (3.10) in terms of the Cartesian coordinates (x, t), The relevance of such a condition in the derivation of the Unruh effect has been emphasized, and discussed both in the canonical and algebraic approach to quantum field theory, by Belinskii et al. in [2,3,19,26,27]. This condition plays indeed a fundamental role in the attempt to compare the quantum theories in Minkowski and Rindler spacetimes, as will be discussed in the next section.

Comparison of Minkowski and Rindler quantization using global mode expansions
The Unruh effect can be expressed as the statement that an uniformly accelerated observer perceives the Minkowski vacuum state as a mixed thermal state at a temperature proportional to its acceleration. In many treatments of the Unruh effect this claim relies on a comparison between the quantum theory in Minkowski and the one in the right Rindler wedge which is naturally associated with an accelerated observer. The quantization scheme proposed by Unruh to implement such a comparison rests on the properties of particular linear combinations of the boost modes, known as the Unruh modes: with p > 0, whose normalization is determined by the one of the boost modes (3.4). The key property of the Unruh modes is their behavior when evaluated in the right and left wedge of Minkowski spacetime 13 : the modes R p (x, t) (respectively L p (x, t)) vanish for (x, t) ∈ L (respectively (x, t) ∈ R) and moreover R p (x, t) coincide with the Fulling modes (3.9) in R. Equations (4.1) and (4.2) are then interpreted as Bogolubov transformations connecting the expansion of the field in the basis of the boost modes and the one in the Unruh modes. The existence of such a Bogolubov transformation allows to relate the corresponding annihilation and creation operators defined within the canonical approach for the two quantization schemes and consequently the quantum states defined between the quantum theories. By inverting relations (4.1) and (4.2) and substituting in the expansion of the field in the basis of the boost modes provides an expression of the field in terms of the Unruh modes. Then, the restriction of the field to the right Rindler wedge allows a direct comparison, via the Bogolubov transformation, to the expansion of the field in Rindler spacetime in the basis of the Fulling modes. In their review [16], Crispino et al. provide the following definition of the Unruh effect (end of Section II.B): "The Unruh effect is defined in this review as the fact that the usual vacuum state for QFT in Minkowski spacetime restricted to the right Rindler wedge is a thermal state [. . . ]". The derivation of the Unruh effect in the standard formulation of QFT is to use the Bogolubov transformations derived from (4.1) and (4.2) to calculate the expectation value of the canonical operator corresponding to the Rindler particle number operator evaluated on the Minkowski vacuum state which turns out to be that of a thermal bath of Rindler particles. This derivation relies on the properties of particle states and number particle operator which have a global nature 14 .
The above mentioned derivation of the Unruh effect in the standard formulation of QFT has been criticized [2] in virtue of the existence of the boundary condition mentioned at the end of the preceding section. We will not repeat here the arguments presented in the cited papers but limit ourself to a few considerations. First, we notice that the correct expansion of the field in Minkowski spacetime in terms of the Unruh modes reads which does not coincide with the expansion of the field in the basis of the Fulling modes for the world points located inside the right Rindler wedge, i.e. for (x, t) ∈ R. The difference is due to presence of the last term in the r.h.s. of (4.3): for (x, t) = (0, 0) the integral vanishes in the limit → 0 because the boost modes take finite values 15 , but for (x, t) = (0, 0) the contribution of this term cannot be neglected since the boost modes reduce to a delta function, as can be seen from expression (3.3), i.e. ψ p (0, 0) = δ(p)/ √ 2. The importance of this term is evident when considering the algebraic structures needed for the implementation of the GBF. For simplicity and without loss of generality, we consider the hyperplane Σ 0 at t = 0 and the structures (3.6), (3.7) and (3.8) defined on it. When restricting Σ 0 to the right wedge R, by using (4.3) we obtain (we denote such restriction with the superscript (R)) where Σ R 0 is the semi-hyperplane η = 0 (η being the Rindler time) corresponding to the intersection Σ 0 ∩ R. Analogue expressions are obtained for the restriction of (3.7) and (3.8). From (4.4) we notice that the restriction to R of the symplectic structure defined in Minkowski spacetime coincides with the symplectic structure in Rindler spacetime only if the second term in (4.4) vanishes, and for that we must impose φ(p) = 0 for p = 0. However, requiring such a condition implies imposing the vanishing of the field at the left edge of the right wedge, namely at the 14 In the canonical approach particle states are determined by the action of the creation and annihilation operators whose definition involves the values that the field takes over a (non compact) Cauchy surface. In the GBF treatment, particle states are elements of the Hilbert space associated with the hypersurface under consideration and the construction of such a Hilbert space depends on the field configurations on this hypersurface, i.e. it depends on the global properties of the elements of the space LΣ. 15 We are here assuming that |φ(p)| < ∞ in the limit p → 0.
origin of Minkowski spacetime. While this is a built-in boundary condition the field in Rindler has to satisfy 16 , there is no reason to require the same condition for the field in Minkowski. Indeed imposing such condition translates in the exclusion of the zero boost mode from the set of modes on which the field is expanded, namely in the suppression of the second term in (4.3). But the remaining expression will then represent the expansion of a different field in Minkowski spacetime, namely a field that satisfies a zero boundary condition at (x, t) = (0, 0). The appearance of this condition is a consequence of the fact that the basis of the boost modes is not anymore complete without the zero boost mode. The consequence of this fact at the quantum level manifests in the loss of the translational invariance of the Minkowski vacuum state, see [2] for a detailed discussion on this point. The claim that the Minkowski vacuum state can be written as an entangled state composed by multi particle states defined in the left and right wedges is consequently not acceptable. For example Unruh and Wald [44] provide the following equality where |0 M is the vacuum state in Minkowski space, N j = (1 − exp(−2πω j /a)) 1/2 , and |n j , L , |n j , R represent the state with n j particles in the mode j in the left, right wedge L and R respectively. From (4.5) it is then possible to obtain a reduced density matrix by tracing over the degrees of freedom in the left wedge, This reduced density matrix is interpreted as representing the restriction of the Minkowski vacuum to the region R. As already mentioned, the left and right hand side of formulas (4.5) and (4.6) refer to states that belong to unitarily inequivalent quantum theories and are consequently not mathematically well-defined. In fact, that (4.6) can only be formally true was already observed much earlier than the articles [2,3,19,26,27] started the discussion about the boundary condition at the origin. For example, in [47], we find the statement that the Minkowski vacuum state cannot be expressed as a density matrix in the quantum field theory in Rindler space. This mathematical problem was avoided by showing that the Minkowski vacuum is a KMS-state in Rindler space and using KMS-condition to define thermal states in a way that does not rely on the particle number operator [41]. Since there is no definition of KMS-states in the GBF, yet, we cannot give a mathematically rigorous version of equation (4.6) in this article.
However, inspired by some results derived within the algebraic approach to quantum field theory 17 , in the next section we present a result, obtained within the GBF, that suggests the existence of a relation between the Minkowski vacuum state and a state of the quantum theory in Rindler spacetime that corresponds to the thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory. To be more precise, we compute the expectation value of a Weyl observable defined on a compact spacetime region in the interior of the right Rindler wedge in two different contexts: first on the vacuum state in Minkowski spacetime and then on a state in Rindler space, whose form represents the analogue of the r.h.s. of (4.6) in the GBF language. It turns out that these two expectation values are equal when the observables are quantized according to the Feynman quantization prescription.

The relation between operator amplitudes on Minkowski and Rindler space
In this section, we derive the coincidence of the expectation values of local observables computed in the Minkowski vacuum and a certain state in Rindler spacetimes which is the central result of this article. The observables we consider have been called Weyl observables in [36] and are given by an exponential of a linear functional of the field, in our case µ(x) is assumed to have compact support in the interior of the right wedge R and (5.1) is consequently a well defined observable in both Minkowski and Rindler spacetime. The interest for looking at the Weyl observables is twofold: first, consistent quantization schemes have been established within the GBF, and second, general results concerning expectation values of these observables have been obtained in [36]. Here we consider the Feynman and Berezin-Toeplitz quantizations of (5.1) and compute for the corresponding quantum observables two different expectation values: one on the vacuum state in Minkowski spacetime and the other on a state in Rindler spacetime that corresponds to the thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory.
In the GBF, as in algebraic quantum field theory, quantum observables O M are associated with a spacetime region M . They are defined by a linear map, called observable map or observable amplitude, from (a dense subspace of) the Hilbert space associated with the boundary of the region to the complex numbers, O M : H Σ → C, Σ being the boundary of the region M . A set of axioms establishes the properties of this map, in particular the spacetime composition of observables. The specific form of the observable map depends on the quantization scheme adopted. In the following two sections, the Feynman and Berezin-Toeplitz quantization schemes combined with the Schrödinger and holomorphic representations are used for the Weyl observable in the settings specified above.
The Feynman quantization prescription is inspired from the purely formal expression for the operator amplitude known from the path integral formulation of quantum field theory. The corresponding observable map associated with an observable O M evaluated on a state ψ S ∈ H S Σ in the Schrödinger representation takes the form We see from this expression that observables are considered as functions on spacetime configuration space in the Feynman quantization prescription. Instead, in the Berezin-Toeplitz quantization prescription, observables are functions on phase space. The corresponding observable map, in the holomorphic representation, for the observable O for a state ψ h ∈ H h Σ is given as where ξ ∈ L Σ and LM is the space of solutions of the equation of motion, defined in a neighborhood of the boundary hypersurface Σ that admit a well defined extension in the interior of the region M . dνM (ξ) is a suitable measure on LM and we refer to [33,34,36] for details concerning the definition of such structures. In [36] Oeckl was able to quantify the difference between the observable maps computed in the Feynman quantization scheme and the one computed in the Berezin-Toeplitz quantization scheme. The result is presented in two propositions, in particular Propositions 4.3 and 4.7 of the cited paper, where the amplitude of a Weyl observable is derived for the two quantization prescriptions. We reproduce in the following formulas the statements of these propositions: For a coherent state K τ we have from the Feynman quantization (whereτ is a complex solution of the equation of motion determined by the coherent state K τ and η D is the unique element of J Σ LM fulfilling the condition D(ξ) = 2ω ∂M (ξ, η D ) for all ξ ∈ LM , Σ being the boundary of the region M ) and from the Berezin-Toeplitz quantization.
In particular the Feynman quantization prescription and the Berezin-Toeplitz quantization prescription differ in a property called composition correspondence [34]: while the application of the observable axioms of the GBF to the product of classical observables with disjoint support quantized via the Feynman quantization prescription leads to another observable such that its expectation value is exactly the product of the expectation values of the original observables (composition correspondence), this is not the case for the Berezin-Toeplitz quantization prescription.
The following investigations will show another difference in the two quantization prescriptions: First, we will show that the expectation value of the local observable F (φ) of the form (5.1) in the state D in Rindler space that corresponds to the thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory coincides with the expectation value of F (φ) in the Minkowski vacuum when F (φ) is localized in the interior of the right Rindler wedge. Second, we will show that this is not true for the Berezin-Toeplitz quantization prescription.

Observable maps from Feynman quantization
Consider the spacetime region M defined in Section 3.1 in Minkowski spacetime. We start by computing the observable amplitude ρ F M : H Σ 1 ⊗ H Σ 2 → C for the Weyl observable (5.1) on the quantum state, in the Schrödinger representation, given by the tensor product of two copies of the vacuum state (3.5), namely K S 0,Σ 1 ⊗ K S 0,Σ 2 . Using the expression in equation (5.3) we arrive at where G M F is the Feynman propagator in Minkowski spacetime, which is evaluated only in the interior of the right Rindler wedge since the field µ(x) has support there. The explicit form of the Feynman propagator can be obtained in terms of the expression of the boost modes (3.3) in the right Rindler wedge, namely [18] ψ k (x, t) where • R denotes the interior of the right Rindler wedge. Then the Feynman propagator reads Now, consider the region R defined in Section 3.2 in Rindler spacetime. The evaluation of the observable map is now performed on the state D ∈ H Σ R 1 ⊗ H Σ R 2 given by expression where ψ n i is the state with n i particles defined in H Σ R i , i = 1, 2, 18 and N i = 1 − exp(−2πk i /a). In particular, the state D corresponds to the mixed thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory [16].
From now on we set a = 1. Since for the observable map evaluated on coherent states we can use the general result in equation (5.3) 19 , it is convenient to express the state in equation (5.7) in terms of coherent states 20 ; the observable map in the region R for the state D then reads where the terms in the second line come from the scalar product of the n i -particle states appearing in (5.7) and the coherent states K S ξ 1 and K S ξ 2 respectively, see Section II.B of [14]. The observable map ρ F R (K S ξ 1 ⊗ K S ξ 2 ) has been shown to satisfy a factorization property, see Proposition 4.3 of [36], which corresponds to the amplitude map of the theory with a source field interaction [11], where we are now using x as global notation for the Rindler coordinates (η, ρ). The first term in the r.h.s. of (5.9) is the free amplitude map (2.1) for the state K S 18 Notice that the factor (2ki) n i appearing in the denominator of (5.7) comes from the normalization of the ni-particle state, where the sum runs over all permutations σ of n elements. 19 See also [11] for the expression of amplitude maps in terms of modes expansion. 20 An important property satisfied by coherent states is the completeness relation expressed by the resolution of the identity operator id which, in a bra ket notation, takes the form ξ(x) is a complex solution of the equation of motion determined by the two coherent states K S where φ R k (ρ, η) are the Fulling modes (3.9). Finally, G R F (x, x ) appearing in the last term of (5.9) is the Feynman propagator in Rindler spacetime and in the region R it reads 21 (5.10) We now have at our disposal all the ingredients to compute the integrals in (5.8). It is convenient to proceed by expressing the powers of the modes ξ 1,2 (k i ) in (5.8) in terms of functional derivatives, Substituting this in (5.8), the integrals are evaluated by the following shift of integration variables: To compute the derivative with respect to α we use Rodrighues' formula, see [24, (8.970.1)], and obtain where L n i is the Laguerre polynomial of order n i . According to formula (8.975.1) of [24], the sum over n i gives . 21 The general expression of the Feynman propagator for fields in (a wide class of) curved spacetimes has been obtained in [11], to which we refer also for details concerning the calculation presented here.
Finally, the substitution of the expression of the quantities µ 1 (k i ) and µ 2 (k i ) leads to Noticing that only the symmetric component of the first term in the square bracket contributes to the integral and using (3.9) and (5.10), a straightforward calculation shows that the sum in the square bracket coincides with the Feynman propagator in Minkowski spacetime evaluated in the right Rindler wedge (5.6); and so do the observable maps (5.5) and (5.11). In the next section we present the same calculation performed according to the Berezin-Toeplitz quantization (5.2) of the Weyl observable.

Observable maps from Berezin-Toeplitz quantization
By examining the expressions (5.3) and (5.4) one can see that the difference between the observable maps of a Weyl observable in the two quantization schemes amounts to an exponential factor. Moreover the exponential factor in (5.3) corresponds to the last exponential factor in (5.9), and consequently we have that Hence, in Minkowski spacetime the observable map (5.4) on the vacuum state evaluated in the Berezin-Toeplitz quantization scheme in the spacetime region M , is given by In Rindler spacetime the observable map (5.4), in the same quantization scheme in the spacetime region R, takes the form As in (5.11) only the symmetric part of the terms in the square bracket contribute to the integral and the situation is similar to the one in the preceding section apart from the factor 1/2 appearing in (5.13). It is precisely this factor that prevents the coincidence of (5.12) and (5.13). We conclude that the Berezin-Toeplitz prescription for the quantization of observables gives no ground for the Unruh effect.

Expectation values in the holomorphic representation
In this section, we present the computation of the observable maps ( this is a consequence of the equivalence between the Schrödinger and holomorphic quantizations shown in [38]. We now consider the same observable map in Rindler spacetime on the state D h in the holomorphic representation corresponding to the state in equation (5.7) in the Schrödinger representation. For later convenience we write this state in terms of derivatives of coherent states, , (5.14) where K h ξ 1 ∈ H h Σ 1 and K h ξ 2 ∈ H h Σ 2 are the coherent states in the holomorphic representation defined by ξ i ∈ L Σ R i , i = 1, 2. Consequently D h is a state in the whole boundary Hilbert space . The corresponding observable map for the Weyl observable (5.1) reads We now specify the three terms appearing in the expression (5.3) for the observable map ρ F • the free amplitude ρ R (K h ξ 1 ⊗K h ξ 2 ) can be computed using (2.2), where in the present context ξ R = ξ 1 + ξ 2 and ξ I = ξ 1 − ξ 2 , leading to • the Weyl observable evaluated on the complex solutionξ given in this case by 23 • the last term in the r.h.s. of (5.3) coincides with the last term in the r.h.s. of (5.9).
The observable map (5.15) can then be written as We proceed by evaluating the first line in the r.h.s. of (5.16) by applying the general Leibniz rule d n dγ n f (γ)g(γ) = n k=0 n k and using the relation which we proof in the appendix. We obtain Hence, substituting in (5.16) we obtain after some rearrangements which coincides with expression (5.11). Consequently, (5.18) equals the observable map computed in Minkowski spacetime on the vacuum state, and we recover the coincidence of the expectation values we found for the Schrödinger representation also in the holomorphic representation.
As already noticed, the difference between the Berezin-Toeplitz quantization and the Feynman one amounts to the last factors in (5.3) and (5.4). These terms are independent of the representation chosen for the quantum states, and so we are reduced to the same situation as in Section (5.1.2): the coincidence of expectation values that exists for the Feynman quantization prescription does not appear adopting the Berezin-Toeplitz prescription for quantizing local observables.

Conclusions and outlook
We have applied the general boundary formulation of quantum field theory to quantize a massive scalar field in Minkowski and Rindler spacetimes. We showed that the expectation values of Weyl observables with compact spacetime support in the interior of the right Rindler wedge, computed in the Minkowski vacuum state coincide with those calculated in a state in Rindler space that corresponds to the thermal state known from the derivation of the Unruh effect in the standard formulation of quantum field theory if the observables are quantized according to the Feynman quantization prescription. This result could be interpreted as the manifestation of the Unruh effect within the GBF. Furthermore, we showed that the coincidence of the expectation values does not hold in the Berezin-Toeplitz quantization, which is an alternative quantization scheme for observables in the GBF.
The work in this article is of immediate relevance for the GBF program. It represents a concrete application of the quantization of observables and the opportunity to compare the Feynman quantization prescription and Berezin-Toeplitz quantization prescription in a specific context. The observed difference in the two quantization prescriptions adds up to the difference in the existence of a property called composition correspondence [34]: while the application of the observable axioms of the GBF to the product of classical observables with disjoint support quantized via the Feynman quantization prescription leads to another observable such that its expectation value is exactly the product of the expectation values of the original observables, this is not the case for the Berezin-Toeplitz quantization prescription. It will be of interest for the GBF to explore the differences of the two quantization prescriptions in more detail.
Another issue for future work is the definition of KMS states in the GBF. Such a definition could be used to make the relation between the quantum theory in Minkowski and the one in the Rindler wedge more explicit. It would be interesting to explore if the corresponding results of the GBF would differ from those found in algebraic quantum field theory.
It should be noted that the spacetime regions considered for the evaluation of the observable maps are the standard ones bounded by two equal (Minkowski and Rindler) time hyperplanes. Of course the versatility of GBF enables to quantize the field and to compute expectation values in more general regions. Although the main focus is represented by compact spacetime regions, inspired by previous results obtained applying the GBF in Minkowski and curved spacetime, an interesting region is represented by the one bounded by one connected and timelike boundary. In particular, it is possible to apply the GBF for a field defined in a region of Rindler spacetime bounded by one hyperbola of constant Rindler spatial coordinate ρ. The origin of Minkowski spacetime lies outside this region and the comparison of the quantum field theory defined there and the one in Minkowski will then avoid the difficulty inherent with the behavior of the field in (x, t) = (x, 0).
Furthermore, the analysis of the properties of the Minkowski and Rindler quantum theories can be the basis for solving an open question within the GBF. The hyperplane t = 0 in Minkowski spacetime is the union of the two semi-hyperplanes η R = 0 and η L = 0 in the right and left Rindler wedge respectively. However the Hilbert space associated to the hypersurface t = 0 is not the tensor product of the Hilbert spaces associated with η R = 0 and η L = 0, due to the additional boundary condition at the origin. In order to compare the different Hilbert spaces one possibility would be to consider hypersurfaces with boundaries: in the present context the hyperplane t = 0 for x ≥ 0 (x ≤ 0), namely with a boundary in the origin of Minkowski spacetime. However it is still not clear within the GBF which algebraic structure should be associated with an hypersurface with boundaries 24 . The solution of such a question is of paramount importance from the perspective of the Unruh effect in the GBF, as well as for more general contexts. which proves that the ansatz was correct.