1 Introduction↩︎

The fundamental advancement of quantum field theory in curved space-time has been significantly influenced by the concept of how a system’s state, which associates an operator with its expectation value in a linear manner, depends on reference frames. A noteworthy application of this theory arises when we restrict to a specific wedge, denoted as R, within Minkowski space-time: R:= \(\{ x \in \mathbb{R}^{1,3} | x_1 > |x_0| \}\). This wedge, referred to as Rindler spacetime is entangled with the complementary left wedge and their union constitutes a globally hyperbolic space-time. In Minkowski spacetime, the existence of a unique Poincaré-invariant state, known as the Minkowski vacuum, is firmly established. Various global families of inertial observers in Minkowski spacetime converge on a common understanding of particle content within any given field state [1]. However, the one-parameter group of Lorentz boost isometries can be used to construct a Rindler spacetime. In the literature, Fulling [2], [3] first gave the Bogolyubov transformation of creation and annihilation operators from the inertial frame to the Rindler frame. The interpretation of particle content obtained by Fulling using Bogolyubov transformation as a black body spectrum was later given by Davies [4]. However, the correct interpretation of these results was later developed by Unruh [5]. It was shown that the vacuum state of an inertial observer appears as a thermal bath for the Rindler observer. Subsequent advancements led to the development of various versions of thermalization theorems [6]. In the literature, the behavior of thermal bath of Unruh particles in the inertial frame as seen by a Rindler observer [7], [8] has also been discussed. There are also attempts to understand the Rindler vacuum [9]. However, since the Rindler basis sees a monochromatic wave of Minkowski modes as a non-chromatic superposition of all frequencies, the literature beyond the single/double Minkowski particle state is scarce in the Rindler frame context. Many physical situations, such as the evolution of the Higgs field in the early universe, models of galaxy formation, fields around astrophysical objects,.. etc, are not in a vacuum or in a thermal equilibrium state. Therefore, it’s important to consider a deviation from the equilibrium or vacuum. The Unruh effect is thought to be the most spectacular manifestation of the vacuum entanglement. Recent advancements in our understanding of quantum entanglement offer a unique avenue for scrutinizing the quantum nature of gravity, as indicated by a study [10]. Furthermore, in the context of stochastic quantum field theory in gravity, entanglement can be conveyed through the local Unruh effect without necessitating the presence of gravitons [11]. This, in turn, encourages us to delve deeper into the intricacies of the Unruh effect.

In many situations, it is possible to associate a unique density matrix with any state of a system. The knowledge of density matrix can be used to compute different correlations. It can also be used to find out entropy and, hence, thermodynamic properties of the system. There are many techniques in quantum optics that have been used in quantum field theory in curved space [12], [13]. It has also been used in the transformation of Wigner distribution ( the phase space representation of density matrix) of a scalar field in a vacuum state from inertial to the Rindler frame [14].

In quantum field theory in curved space, the vacuum state is characterized by being both pure and Gaussian. The vacuum for a given observer depends upon the eigenfunctions/ eigenmodes of the field operators defined with respect to the proper time of that observer. Given the direct interaction between an accelerated detector and Rindler modes, these modes become the more suitable choice for quantizing the field from the perspective of accelerated observers. Moreover, for inertial observers the Minkowski modes are a natural choice for quantizing the field [7]. Nevertheless, one can also design an interaction Hamiltonian in a way that it couples with Unruh modes. It’s important to highlight that many of the conclusions drawn from the Unruh Fock space can be extended to the original Minkowski plane wave Fock space, albeit requiring a basis transformation from plane waves to wave functions defined in log x space. This dual approach encourages us to leverage both Minkowski and Unruh modes in the comprehensive exploration of a wide range of states.

In section 2, we provide an overview of the transformation of the Wigner distribution of a massless scalar field in Minkowski spacetime to that of the Rindler spacetime. This transformation is applicable to a specific subset of states, which draws motivation from Gaussian states but also involves a slight departure from them in the phase space of eigenvalues of the annihilation operator. We accomplish this using both Minkowski and Unruh Minkowski (UM) modes. We employ the Wigner distribution formalism to derive a general form of the reduced Wigner functionals in Rindler spacetime by tracing out the degrees of freedom beyond the Rindler horizon of a real massless scalar field in (1+1) dimensions focusing on a special subset of states in Minkowski spacetime more formally defined in subsection 2.2. The detailed calculation of the transformation is given in Appendix A. Additionally we highlight several features of the reduced Wigner functional for these states. We also present a general expression for the expectation value of the number density operator for each of these Wigner functionals in subsection 2.4. We verify the general result using known results from known states and then proceed discuss several other unknown distributions belonging to the selected subset of states in section 3 and conclude our findings in section 4. Use is made of natural units, namely \(\hbar\)=c= \(k_B\)=1, throughout the paper.

2 Vacuum and non-vacuum Minkowski states in Rindler frame↩︎

2.1 Wigner Functional -Setup↩︎

One can describe physical systems in terms of density function operator \(\rho\), which is a non-negative hermitian operator of trace unity acting on a Hilbert space \(\mathcal{H}\) [15]. The components of \(\rho\) expanded in a complete set of basis eigenstates constitute the density matrix. There exist numerous scenarios, one such example being coherent states of photons, where the uncertainty relation is minimized. It makes the coherent state to be concentrated along classical trajectories. In this paper, we use the basis of coherent states to describe the quantum states. The phase-space representation of the density matrix, which is a pseudo-probability distribution, is known as the Wigner distribution [16]. One uses c-number correspondence to get a c-number function corresponding to any operator [16]. The Wigner functional approach serves the closest analog to classical physics, having roots in quantum theory[17]. It can be said that this feature makes the classical field limit more transparent compared to other formulations. For a quantum harmonic system, one can express the Wigner distribution in the following form for the \(n^{th}\) number state of the harmonic oscillator [14]. \[\label{eq:1} W^{(n)}(\alpha) = \langle n| 2 e^{-2 (\hat{a}^ {\dagger}- \alpha ^*);(\hat{a} - \alpha)} |n \rangle = \langle 0| (\hat{a})^n 2 e^{-2 (\hat{a}^ {\dagger}- \alpha ^*);(\hat{a} - \alpha)} (\hat{a} ^ {\dagger} )^n |0\rangle\tag{1}\] Here \(`;\)’ denotes Schwinger operator ordering, \(\hat{a}\) and \(\hat{a} ^\dagger\) denote the annihilation and creation operators, and \(\alpha, \alpha ^*\) \(\in \mathbf{C}\) with \(\alpha\) being eigenvalue of annihilation operator. The above expression can be formally thought of as a derivative acting on vacuum Wigner distribution. One can also think of the Wigner distribution as the expectation value of the displaced parity operator.

We consider a system of massless real scalar field in the background Minkowski spacetime and apply the Wigner functional approach to analyze how the functional transforms between inertial and Rindler bases states. It is, in general, known how to transform Wigner distribution from one set of complete bases to a new one, corresponding to different reference frames of different observers. We will be dealing with the Wigner distribution based on coherent states [14] because it practically simplifies the relevant calculations. In general, one could adopt the Wigner functional approach to explain the dynamics of almost all quantum field systems where the notion of a density matrix can be suitably defined [17]. We begin by expressing the massless real scalar field in terms of the basis of the Minkowski plane wave mode solution of the Klein-Gordon equation, as follows:
\[\label{eq:2} \hat{\phi}(t,\mathbf{x}) = \iiint _V \frac{d^3 \mathbf{k} }{(2 \pi)^{3/2}\sqrt{2 \omega _{\mathbf{k}}}} [\hat{a}_{\mathbf{k}} e^{ikx} + \hat{a}^\dagger _{\mathbf{k}} e^{-ikx} ].\tag{2}\]
Here \(\hat{a}_{\mathbf{k}}\), \(\hat{a}^\dagger _{\mathbf{k}}\) are standard annihilation and creation operators corresponding to the Minkowski mode \(\mathbf{k}\), and \(\{t, \mathbf{x} \}\) are standard Minkowski coordinates. The above expansion is not formally convergent and lacks the notion of being an observable associated with a spacetime point; however, one can interpret of it as an operator-valued distribution [18]. In the momentum representation, the quantum field is essentially a collection of simple harmonic oscillators, which allows one to apply the Wigner function formalism as described in Eq.@eq:eq:1 for each mode \(\mathbf{k}\). For the sake of simplicity and analytical tractability, we work in (1+1) dimensions for the rest of the paper. We begin by choosing a particular form for the Minkowski-Wigner distribution function, which represents a wide class of initial quantum states. The motivation for such a choice comes from the fact that the Wigner distribution of a thermal bath and Gaussian quantum states, which play a significant role in several areas of theoretical and experimental physics, form a subset of the form chosen. We consider the form of the Wigner distribution in the Minkowski basis to be
\[\label{eq:3} W_M = \bar{N} \exp(-2 \int _{-\infty} ^ {+\infty} \int _{-\infty} ^ {+\infty}\frac{dk dk'}{4 \pi^2} a_{k'} ^* a_{k} f(k,k')),\tag{3}\]
where the subscript \(M\) represents an inertial frame, \(\bar{N}\) is the normalization factor, \(a_{k'} ^*\) and \(a_k\) are c-numbers corresponding to creation and annihilation operators in an inertial frame, obtained by Wigner-Weyl correspondence [14]. One can refer to [17] for a discussion of the Wigner functional for thermal equilibrium states in the field representation. The function \(f(k,k')\) is a two-point function, i.e., we allow the Wignar functional to have off-diagonal elements in the k-k’ space. This allows us to consider Minkowski states beyond the popular \(f(k,k')\) = \(\delta(k-k') f(k)\) class of states usually considered in the literature. The Minkowski vacuum state and the thermal bath both fall in the latter class of states. In general, the two-point function \(f(k,k')\) can be any Hermitian positive two-point function of Schwarz space, or it should have compact support.

To transform the Wigner functional in terms of the basis of charts of the accelerated frame of reference, we first define the Rindler transformation by considering an observer moving with a uniform linear acceleration ‘a’, along the x-axis with trajectory taken as \((\frac{1}{a} \sinh{a \tau}, \frac{1}{a} \cosh{a \tau} )\). Here, \(\tau\) represents the observer’s proper time, and the motion is restricted to the t-x plane. The wedge part of the entire Minkowski spacetime accessible to the uniformly accelerated observer defined above constitutes a globally hyperbolic spacetime along with the union with its complementary left wedge, though not geodesically complete, is called the right Rindler wedge/patch or the Rindler spacetime. The global hyperbolicity allows one to quantize the field and define modes restricted to the right Rindler wedge. However, the vacuum of an inertial observer, i.e., the Minkowski vacuum, is defined over the full Minkowski spacetime. The real massless scalar field can be expanded in terms of the Rindler plane wave basis mode solutions of the Klein-Gordon equation as \[\label{eq:4} \hat{\phi}(T,X) = \int _{-\infty} ^{+\infty} \frac{d K }{(2 \pi)^{1/2}\sqrt{2 \omega _{K}}} [\hat{b}_{{K,R}} e^{i K_\mu X^\mu} + \hat{b}^\dagger _{K,R} e^{-i K_\mu X^\mu} ] + [R \leftrightarrow L ],\tag{4}\] where \(|K|\) is the Rindler frequency, \(T, X\) are Rindler coordinates and \(\{ \hat{b}_{{K, R}}, \hat{b}^\dagger _{K, R},\hat{b}_{{K, L}}, \hat{b}^\dagger _{K, L} \}\) are creation and annihilation operators corresponding to left and right Rindler wedges. This set of creation and annihilation operators is related to the Minkowski creation and annihilation operators by Bogolyubov transformations.

2.2 Reduced Wigner Functional↩︎

One would like to transform the Minkowski Wigner functional defined in Eq.(3 ) with the help of Bogoliubov transformations. However, it is well known that for a general two-point function \(f(k,k')\), the procedure is analytically untractable due to the mode mixing property of the transformations. Hence, we shall choose a particular subset of two-point function \(f(k,k')\) such that the density functional in terms of the right Rindler basis modes becomes diagonal in the \(K-K'\) space. This is realized if we consider those Hermitian positive definite two-point functions whose 2D Fourier transform with respect to \((K/a, K'/a)\) after a redefinition of the modes as \(k=e^t, k'=e^{t'}\) and weighted by \(e^{(t+t')/2}\) is diagonal for some continuous variable \(K\) and positive nonzero constant ‘a’. The same can be considered as a 2D Mellin transform which is required to be diagonalized. Such a requirement on \(f(k,k')\) can be formally written for all 4 quadrants of the \(k-k'\) space in a matrix sense as \[\label{eq:5} \int _{0} ^{+\infty} \int _{0} ^{+\infty} \frac{dk dk'}{\sqrt{|k||k'|}} f(k,k') |k'/a|^{-iK'/a} |k/a|^{+iK/a} = g_1(K,K') \delta(K-K')\tag{5}\]

\[\label{eq:6} \int _{0} ^{+\infty} \int _{0} ^{+\infty} \frac{dk dk'}{\sqrt{|k||k'|}} f(k,-k') |k'/a|^{-iK'/a} |k/a|^{+iK/a} = g_2(K,K') \delta(K-K')\tag{6}\]

\[\label{eq:7} \int _{0} ^{+\infty} \int _{0} ^{+\infty} \frac{dk dk'}{\sqrt{|k||k'|}} f(-k,k') |k'/a|^{-iK'/a} |k/a|^{+iK/a} = g_3(K,K') \delta(K-K')\tag{7}\]

\[\label{eq:8} \int _{0} ^{+\infty} \int _{0} ^{+\infty} \frac{dk dk'}{\sqrt{|k||k'|}} f(-k,-k') |k'/a|^{-iK'/a} |k/a|^{+iK/a} = g_4(K,K') \delta(K-K')\tag{8}\] for some real two point functions \(g_1(K,K')\), \(g_2(K,K')\), \(g_3(K,K')\) and \(g_4(K,K')\). We denote the diagonal elements \(g_s\)(K,K) as \(g_s\)(K). In principle, the above equations can also be inverted using the inverse Fourier transform to find f(k,k’) for a given set of \(g_i(K, K')\)’s. The inversion property allows us to first choose a Rindler density functional and then obtain a corresponding Minkowski density functional, as we shall demonstrate in the next section. However, one should note that the Minkowski state obtained in this way is not unique because the degrees of freedom in the left wedge have been traced out to get the reduced Wigner functional in the right Rindler wedge. There can be other Minkowski states with a weight function \(f(k,k')\) that does not belong to the subset of functions satisfying properties Eqs.(5 -8 ) and yet may yield the same Rindler state. This behavior could be expected because each Minkowski mode has been mixed up with modes of all frequencies during the Rindler transformation. However, if we have a Minkowski space Wigner functional that belongs to the above mentioned subset, then the Rindler space Wigner functional will be unique. Given the Wigner distribution, physical quantities of interest, such as expectation value of number density, entropy, correlation, .. etc., can be easily determined. The degrees of freedom in the Wigner functional in both inertial and accelerated frames are related by Bogoliubov transformation [19]. The degrees of freedom in the left and right Rindler wedges are entangled. Since the left Rindler wedge is inaccessible to the uniformly accelerated observer in the right wedge, we need to trace over unobserved degrees of freedom after applying the Bogoliubov transformation, which is the standard procedure to obtain the reduced Wigner functional for the right Rindler observer. We provide a detailed derivation of the reduced Wigner distribution in Appendix A. The reduced Wigner distribution corresponding to the Minkowski Wigner functional in Eq.(3 ) and for \(f(k,k')\) satisfying Eqs. (5 -8 ) is obtained to be \[\begin{gather} \label{eq:9} W_{R_{\text{reduced}}} = N \exp \left[ \frac{-1}{8 \pi ^4 a ^2} \int _0 ^{\infty} dK \, \left| K \right| \left| \Gamma \left( \frac{iK}{a} \right) \right| ^2 \left( J(K) \left| b_R (K) \right| ^2 + R_1(K) b_R ^* (K) b_R ^* (-K) \right. \right. \\ \left. \left. + R_2 (K) b_R(K) b_R (-K) + L \left| b_R (-K) \right|^2 \right) \right], \end{gather}\tag{9}\] which is diagonal in \(K-K'\) space. The explicit expressions for weight functions J(K), \(R_1 (K)\), \(R_2(K)\), and L(K) are given in Eqs. 50 to 52 of Appendix A. These weight functions and hence \(W_{R_{\text{reduced}}}\) satisfy some interesting features that we highlight below.

(a) \(g_2(K,K) \leftrightarrow g_3(K,K) \Rightarrow R_1(K) \leftrightarrow R_2(K)\) where the doublearrow represents an interchange operation
(b) If \(g_2(K,K) = g_3(K,K) = 0\) then \(R_1(K)=R_2 (K)=0\)
(c) L(-K) = L(K) , J(-K) = J(K) , \(R_1(-K)=R_2 (K)\), and \(R_2(-K)=R_1 (K)\)
(d) If \(g_2(K,K) = g_3(K,K)\) = \(g_2(-K,-K) = g_3(-K,-K)\) and \(g_1(K,K) = g_4(K,K)\) = \(g_1(-K,-K) = g_4(-K,-K)\) \(\Rightarrow J(K)=L(K)\)

The first property suggests that if the weight function \(f(k, k')\) of Minkowski Wigner functional satisfies \(f(k, -k')\) = \(f(-k, k')\), then the weights \(g_2(K, K')\) and \(g_3(K, K')\) are also equal. As a result weights \(R_1(K)\) and \(R_2(K)\) in the reduced Rindler Wigner functional, which are linked to the cross terms of positive and negative frequencies, are the same. In those cases where they are not equal, swapping them results in a corresponding interchange of weights, i.e., \(R_1(K) \leftrightarrow R_2(K)\) within the Rindler wedge. The second property asserts that if \(g_2(K, K)\) and \(g_3(K, K)\) vanish, then the corresponding cross term \(R_1(K)\) and \(R_2(K)\) in the Rindler frame also vanish. As a consequence, the anomalous averages such as \(\langle b_R^* (K) b_R^* (-K) \rangle\) in Eq.(9 ) vanish. Non-zero anomalous averages play an important role in several phenomena, such as the BCS theory of superconductors [20]–[23]. The final property elucidates that if the weight of Minkowski Wigner functional exhibits reflection symmetry about the origin, i.e., \(f(k,-k')\) = \(f(-k,k')\) and \(f(k,k')\) = \(f(-k,-k')\) then from Eq. 5 to Eq.@eq:eq:8 we have, \(g_2(K,K) = g_3(K,K)\) = \(g_2(-K,-K) = g_3(-K,-K)\) and \(g_1(K,K) = g_4(K,K)\) = \(g_1(-K,-K) = g_4(-K,-K)\) - which after substituting in Eq.@eq:eq:14 give an equal number of particles and anti-particles in the Rindler frame.

2.3 Wigner Functional for UM modes↩︎

While the concept of particles is not a prerequisite for comparing the relationship between the Fock basis and Bogoliubov transformations in two frames, specific sets of modes and particle states naturally manifest as a consequence of the symmetries inherent in the spacetime under consideration. One such set of modes is the Unruh Minkowski (UM) modes which is a linear superposition of positive frequency Minkowski modes. Its ground state coincides with the Minkowski vacuum. However, other excited states are different. It turns out that a subclass of the reduced Rindler frame Wigner distribution in Eq.@eq:eq:9 , when \(R_1(K)\) = \(R_2(K)\) = 0 and J(K) = L(K), can be obtained by a very specific class of inertial frame Wigner functionals in terms of UM modes. This can be seen as follows. It is known that corresponding to each Rindler frequency, there are two UM modes of positive frequency. One can express the Wigner distribution in terms of modes corresponding to these UM particles as \[\label{eq:10} W_{UM} = \prod_{\xi >0} N_\xi e^{- p(\xi) |\alpha_{\xi}|^2 - q(\xi) |\alpha_{-\xi}|^2 - r(\xi) \alpha_{\xi} \alpha_{-\xi} - s(\xi) \alpha_{\xi} ^* \alpha_{-\xi} ^* }\tag{10}\] Here, \(\alpha\) and \(\alpha^*\) are c- number representations of annihilation and creation operators for Unruh-Minkowski particles. The functions \(p(\xi)\) and \(q(\xi)\), which are positive, nonzero, real, and smooth, dictate the weights linked to \(\alpha_{\xi}\) and \(\alpha_{-\xi}\). Similarly, \(r(\xi)\) and \(s(\xi)\) are real smooth functions. An additional constraint arises from the requirement for the expectation value of number density to be finite, expressed as \(pq \neq rs\). One can also consider a more general form of the weight functions that depend on two arguments, for example, p(\(\xi\), \(\xi '\)), such that the density functional is off-diagonal in \(\xi-\xi'\) space as in Eq.@eq:eq:3 . Here, we consider only the diagonal form as given in Eq.@eq:eq:10 since it suffices to explore different cases in the next section. To obtain the reduced Wigner distribution in the right Rindler frame, we proceed in a manner similar to that in the above section to obtain \[\label{eq:11} W_{R_{reduced}} = \prod_{K >0} N_K \exp \left\{- \frac{ 2 ( p q - r s) \sinh (\frac{\pi |K|}{a}) }{ p e^{-\pi |K|/a } + q e^{\pi |K|/a } - r - s } (|b_{R}(K)|^2 + |b_{R}(- K)|^2 )\right\}\tag{11}\]
where, \(p \equiv p(K)\) , \(q \equiv p(K)\) , \(r \equiv r(K)\) and \(s \equiv s(K)\) are the same weight functions \(p(\xi)\), \(q(\xi)\), \(r(\xi)\) and \(s(\xi)\) in Eq.@eq:eq:10 with the arguments replaced by \(\xi\)= \(\frac{2 \pi |K|}{a}\).

2.4 Number Density in momentum space↩︎

Having obtained the Wigner distribution in two distinct frames, one can now proceed to compute the expectation values of any operator in both frames, especially the expectation value of the number density operator. In this subsection, we provide an expression for the expectation value of the number density associated with various established mode sets and defined by \[\label{eq:12} \langle n \rangle \equiv \int d^2 \alpha (\alpha ^* \alpha -1/2) W\tag{12}\] where integration runs over the respective \(\alpha _s\) in their own phase space corresponding to each W\((\alpha,\alpha^*)\). Substituting our Wigner distributions (\(W_M, W_{UM}\)) and performing the Gaussian integration, we get the following expressions for the particle number density expectation value for Minkowski modes as (\(\langle n_M(k) \rangle\)) and for the UM modes as (\(\langle n_{\alpha_{\xi}} \rangle, \langle n_{\alpha_{-\xi}} \rangle\)) where \[\begin{align} \label{eq:13} \langle n_M(k) \rangle _i = \frac{1}{2} \left[\left(\frac{f(k,k')}{2 \pi}\right)^{-1} _{ii} - 1 \right] \end{align}\tag{13}\]

\[\begin{align} \label{eq:14} \langle n_{\alpha_{\xi}} \rangle = \bigg[\bigg(p(\xi) - \frac{r(\xi) s(\xi)}{q(\xi)} \bigg)^{-1} - \frac{1}{2} \bigg]; \langle n_{\alpha_{-\xi}} \rangle = \bigg[ \bigg(q(\xi) - \frac{r(\xi) s(\xi)}{p(\xi)} \bigg)^{-1} - \frac{1}{2} \bigg] \end{align}\tag{14}\]
Here, the subscript ii in the expression of \(\langle n_M(k) \rangle_i\) refers to the \(ii^{th}\) diagonal element in the inverse matrix. We note from the expectation value of UM particles expression Eq.@eq:eq:14 that if any of the cross terms r or s is 0, then \(\langle n_{\alpha_{\xi}} \rangle\) will not depend upon either \(r(\xi)\), \(s(\xi)\) or \(q(\xi)\). An analogous statement holds true for \(\langle n_{\alpha_{-\xi}}\rangle\). One gets the following expectation value \(\langle n_R (K) \rangle\) in the Rindler frame \[\label{eq:15} \langle n_R(K) \rangle = [(L_1(K,K) - L_2(K,K) L_4 (K,K) ^{-1} L_3 (K,K))^{-1} -\frac{1}{2}],\tag{15}\] where functions \(L_1,L_2,L_3\) and \(L_4\) are related to the weight functions in Eq.@eq:eq:9 as

\(L_1\) = \(\frac{J(K,K)}{8\pi^2 a \sinh (\pi |K|/a)}\); \(L_2\) = \(\frac{R_1(K,K)}{8\pi^2 a \sinh (\pi |K|/a)}\); \(L_3\) = \(\frac{R_2(K,K)}{8\pi^2 a \sinh (\pi |K|/a)}\); \(L_4\) = \(\frac{L(K,K)}{8\pi^2 a \sinh(\pi |K|/a)}.\)
In the present section, we have thus obtained a general expression for the reduced Wigner functional and the corresponding number operator expectation value. In the next section, we shall consider and analyze several explicit examples of interest.

3 Special cases of Wigner distributions↩︎

There are a variety of interesting Minkowski state distributions that belong to the subset of distributions considered in the previous section. We shall investigate a few known cases to demonstrate the consistency of the Wigner functional approach with those taken previously in the literature and then proceed to discuss new ones and highlight their interesting features.

3.1 Minkowski Vacuum - Unruh effect↩︎

We first consider the vacuum state of a massless scalar field in the Minkowski spacetime for an inertial observer. This is described by the well-known standard Unruh Minkowski Wigner distribution [14].
\[\label{eq:16} W_{UM} = \prod_{\xi} N_\xi e^{-2 |\alpha_{\xi}|^2 - 2 |\alpha_{-\xi}|^2 }\tag{16}\]

The same can also be expressed in terms of Minkowski modes in Eq.@eq:eq:3 by substituting the two-point function \(f(k,k’)\) = 2 \(\pi \delta (k-k’)\). Both these represent the same Minkowski vacuum state. However, as explained in subsection (2.3), the transformation to a uniformly accelerated frame mixes positive and negative frequencies, and as a consequence, the uniformly accelerated observer finds the inertial frame ground state to be populated with particles. By comparing the distribution in Eq.@eq:eq:16 with the form of the Wigner distribution in Unruh Minkowski form in Eq.@eq:eq:10 , one can read off \(p(\xi)\) = \(q(\xi)\) = 2 and \(r(\xi)\) = \(s(\xi)\) = 0. Then, Eq.@eq:eq:11 for the reduced Wigner functional in the right Rindler wedge can be expressed as

\[\label{eq:17} W_{R_{reduced}} = \prod_{K} N_K e^{-2 |b_{R}(K)|^2 \tanh (\frac{\pi |K|}{a})}\tag{17}\] This represents a thermal bath of Rindler particles with temperature T = a/\(2 \pi\) and is the standard Unruh effect. Alternatively, one could also obtain Eq.@eq:eq:17 starting from Eq.@eq:eq:8 for the Minkowski Wigner functional. One gets the form of the weight functions \(g_1(K), g_2(K), g_3 (K)\) and \(g_4 (K)\) by substituting \(f(k,k’)\) = 2 \(\pi \delta (k-k’)\) in Eqs.@eq:eq:4 to 7 . We have checked that it finally leads to the same reduced Wigner functional as in Eq.@eq:eq:17 using the expressions for different \(L_s\) in Eq.@eq:eq:15 . The expectation value of the number density using Eq.@eq:eq:15 then leads to the expected Planckian as
\[\nonumber \langle n_R(K) \rangle = \frac{1}{e^{\frac{2 \pi |K|}{a}}-1} .\] It can be shown that there are no hidden correlations in observed particles; therefore, the above \(\langle n_R(K)\rangle\) represents a true thermal bath [24], [25]. One can follow [26] for an elegant review of the Unruh effect.

3.2 Rindler vacuum↩︎

The next interesting distribution we consider is the Rindler vacuum. This is represented by a state for which the expectation value of number density in the Rindler frame is zero. The Rindler vacuum distribution can be characterized by the following reduced Wigner distribution in the Rindler frame, where the acceleration has a value greater than zero.
\[\label{eq:18} W_{R_{reduced}} = \prod_{K>0} N_K e^{-2 (|b_{R}(K)|^2 + |b_{R}(-K)|^2 )}\tag{18}\]

To obtain what the inertial observer observes in the inertial frame, we compare the above form in Eq.@eq:eq:18 with Eq.@eq:eq:8 for the reduced Wigner distribution in the Rindler frame and read off weight functions \(g_i\) (K) and then using Eqs.@eq:eq:5 -8 one can obtain the corresponding \(f(k,k')\). The \(g_i\) (K) are found to be of the following form:\(g_1\) (K) = \(g_4\) (K) = 4 \(\pi^2 a \coth{\frac{\pi |K|}{a}}\), \(g_2\) (K) = \(g_3\) (K) = 0. Using the inverse Fourier transform, the form of the Wigner distribution in the Minkowski frame is then found to be \[\begin{gather} \label{eq:19} W_M = N \exp\bigg\{-2 \int _{-\infty} ^ {+\infty} \int _{-\infty} ^ {+\infty}\frac{dk dk'}{4 \pi^2} a_k ^* a_k \bigg[2\pi \delta (k'-k) - \frac{\theta (kk')}{\pi \sqrt{|k||k'|}} [2 \gamma _c + \psi^0(1-i \frac{\ln(k'/k)}{2\pi})+ \\ \psi^0(1+i \frac{\ln(k'/k)}{2\pi}) ] \bigg] \bigg\} \end{gather}\tag{19}\] where \(\gamma _c\) is the Euler-Mascheroni constant. The Eq.@eq:eq:19 represents a Minkowski state that leads to the Rindler vacuum. Surprisingly, the distribution is independent of the acceleration parameter ‘a’, even though the Eq.@eq:eq:5 -8 , which relates f\((k,k')\) and g\((k,k')\), depends on the parameter ‘a’. The only input regarding ‘a’ that has gone in this calculation is that ‘a’ must be nonzero. It can be seen from the above expression that the two-point function \(f(k,k')\), which gives the expectation value of Minkowski particles, has an infrared divergence. This could be handled by the standard procedure of discretizing by considering the system in a finite box. The above expression in Eq.@eq:eq:19 appears complicated; however, the distribution simplifies and is more useful in terms of Unruh-Minkowski modes. This can be achieved by choosing \(p(K)\) = \(q(K)\) = 2 \(\coth{\frac{\pi |K|}{a}}\) and \(r(K)\) = \(s(K)\) = 2 \(\mathop{\mathrm{cosech}}{\frac{\pi |K|}{a}}\) in Eq.@eq:eq:10 which leads to \(W_{R_{reduced}}\) in Eq.(11 ) to take the form of the Rindler vacuum. Note that this choice corresponds to one of the many possible ways to choose p,q,r,s such that the coefficients in front of \(|b_R(K)|^2\) and \(|b_R(- K)|^2\) in Eq.@eq:eq:11 are unity - which is required for the distribution to be Rindler vacuum. With these chosen expressions for p,q,r,s, one can now write the inertial distribution in terms of Unruh-Minkowski modes using Eq.@eq:eq:10 to obtain the following form, \[\label{eq:20} W_{UM} = \prod_{\xi} N_\xi \exp \Bigg[ -2 (|\alpha _\xi |^2 + |\alpha _{- \xi} |^2 ) \coth{(\xi/2)} - 2 \frac{\alpha_\xi \alpha _{-\xi} + \alpha_\xi ^* \alpha _{-\xi} ^*}{\sinh{(\xi/2)}} \Bigg]\tag{20}\] Interestingly, such a choice leads to a form of Wigner distribution, with the corresponding number density of particles turning out to be Planckian, as can be seen by using Eq. 13 , \[\langle n_{\alpha _{\xi}} \rangle = \langle n_{\alpha _{-\xi}} \rangle = \frac{1}{e^{\xi}-1}\] Due to the Planckian form, one may be tempted to associate thermality with the distribution. However, one can check that it cannot be derived from a density operator of the form \(e^{- \beta \hat{H}}\) where \(\hat{H}\) is the Hamiltonian of the Unruh-Minkowski particles. The precise form of the anomalous averages such as \(\langle \alpha _{\xi}^* \alpha _{-\xi}^* \rangle\) breaks this equivalence. For each Rindler frequency, there are two right-moving positive frequencies in Unruh modes. Thus, as mentioned above, there are several other choices of Unruh distributions that lead to the Rindler vacuum. A comparison of the Rindler reduced Wigner distribution in Eq.@eq:eq:18 with the form in Eq.@eq:eq:10 provides a general relation between \(p(\xi)\), \(q(\xi)\), \(r(\xi)\) and \(s(\xi)\) that leads to the Rindler vacuum, as \[\label{eq:21} (p q - r s ) \sinh{(\xi /2)} = p e^{- \xi /2} + q e^{ \xi /2} - r - s\tag{21}\] The choice of r = s = 0 in the above Eq.@eq:eq:21 –which corresponds to zero anomalous averages – yields the following relationship between p and q: \[\label{eq:22} p = \frac{q e^{\xi/2}}{q \sinh{\xi/2} - e^{-\xi/2}}\tag{22}\] By substituting the above relationship, denoted as Eq.@eq:eq:22 , into Eq.@eq:eq:14 , one can obtain the expression: \[\label{eq:23} \langle n_{\alpha _\xi} \rangle = - ( \langle n_{\alpha _{-\xi}} \rangle + 1) e^{-\xi}\tag{23}\] Although this relation, denoted as Eq.@eq:eq:23 , may initially appear to imply the Rindler vacuum to be unphysical, when r=s=0, due to the negative sign, it is indeed physically valid since one only needs the expectation value of the energy density to be bounded from below. One can refer [27]–[29] for the discussion of the negative energy density in the context of the Rindler observer. We provide two particular examples displayed in Fig.1. It can be observed in both examples that \(\langle n_{\alpha _\xi} \rangle\) and \(\langle n_{\alpha _{-\xi}} \rangle\) are bounded from below. Specifically, the minimum of \(\langle n_{\alpha _\xi} \rangle\) and \(\langle n_{\alpha _{-\xi}} \rangle\) in the plot in the left panel of Fig.1 is \(-1/2\), while the minimum of \(\langle n_{\alpha_{\xi}} \rangle\) is \(-1/4\) and \(\langle n_{\alpha_{-\xi}} \rangle\) remains positive for the right panel. Hence, the corresponding energy density obtained by relation E = (\(\langle n \rangle\) + 1/2 ) \(\omega\) is positive. The lower energy density required for the Rindler vacuum, compared to the standard Minkowski vacuum, can be attributed to the relatively weaker constraint implied by quantum inequalities on the world lines of accelerated observers, in contrast to inertial observers, as described in[27], [28].

3.3 A near-Minkowski vacuum functional↩︎

Next, we consider a Minkowski Wigner functional, which represents a state ‘close’ to the Minkowski vacuum but not exactly the vacuum. To construct such a distribution, we choose a weight function \(f(k,k')\) to be highly peaked in the momentum space and parameterized by a non-negative parameter \(\gamma\). The motivation behind choosing such a two-point function \(f(k,k')\) is that, in an appropriate limit, the weight function \(f(k,k')\) will reduce to a Dirac Delta distribution which leads to the Minkowski vacuum state functional in subsection 3.1. Hence, by introducing a non-negative parameter \(\gamma\) such that \(f(k,k')\) is highly peaked, one can construct a Minkowski functional, which slightly deviates from the vacuum state as shown below. We choose \(f(k,k')\) to be \[\label{eq:24} f(k,k') = \sqrt{\frac{4\pi}{\gamma}} \frac{\theta(kk')}{\sqrt{|k||k'|}} e^{-(\ln(k'/k))^2/\gamma} .\tag{24}\] One can note that this choice corresponds to an approximation of the Dirac Delta distribution and in the limit \(\gamma \rightarrow 0\) reduces to the Dirac Delta distribution. The Wigner distribution in the inertial frame is then \[\label{eq:25} W_M = N \exp\left\{-2 \int _{-\infty} ^ {+\infty} \int _{-\infty} ^ {+\infty}\frac{dk dk'}{4 \pi^2} a_k ^* a_k \sqrt{\frac{4\pi}{\gamma}} \frac{\theta(kk')}{\sqrt{|k||k'|}} e^{-(\ln(k'/k))^2/\gamma} \right\}\tag{25}\]

As a consistency check, the above expression reduces to the vacuum state for an inertial observer in the limit \(\gamma \rightarrow\) 0. To obtain the effect of a small perturbation of \(\gamma\) away from 0, we look at the expectation value of number density( \(<n_M(k)>\)). To proceed, one needs to obtain the inverse of \(f(k,k')\) to evaluate Eq.@eq:eq:13 . However, the form chosen for \(f(k,k')\) is non-diagonal. Hence, we employ an approximation for the inversion, proceeding with a Taylor series expansion of \(f(k, k'; \gamma)\) around \(\gamma = 0\). The detailed calculation of \(\langle n_M(k) \rangle\) is presented in Appendix B. In the first-order approximation with respect to \(\gamma\), the expression for \(\langle n_M(k) \rangle\) is given by \(-\frac{\gamma \delta (0)}{8} + \mathcal{O} (\gamma ^2)\). Here, \(\langle n_M(k) \rangle\) may seem independent of \(k\). However, one must note that such a form is contingent not only on the limit \(\gamma \rightarrow 0\) but also on the condition imposed on the frequency to be not sufficiently large to render \(\gamma k k'\) significant. Here \(\delta\)(0) is a consequence of the continuum limit of modes. One can divide both sides by the volume of the box to get rid of this divergent term -viz. the expectation value of particles per unit frequency per unit volume is finite. To comprehend the negative sign, one can multiply \(\langle n_M(k) \rangle\) by the frequency and then add \(\omega /2\) to obtain the energy density in natural units –as explained in the last subsection. In particular, as \(\gamma\) \(\rightarrow\) 0, the resulting energy density is slightly less than \(\omega\)/2 but remains positive.

To investigate the high-frequency behavior, we turn to numerical techniques to find the inverse of \(f(k,k')\) using its discrete form with a finite box size in Mathematica. We display the numerical results in Fig2. It can be seen from the first three plots in Fig.2 that the expectation value of number density in an inertial frame is negative at small frequencies–consistent with what we get by the analytical approximation –and diverges at high frequencies for a finite \(\gamma\). By choosing smaller values of \(\gamma\), we find that the expectation value of number density per unit volume per unit frequency range starts diverging at frequencies higher than for the corresponding case with a comparatively higher \(\gamma\). In the strict limit \(\gamma \rightarrow 0\), \(\langle n_M(k) \rangle\) approaches zero for all finite frequencies –as expected for the vacuum state, since the divergence pattern as seen in the Fig.2 will be strictly at infinite value of the frequency.

To obtain the expectation values of observables for the functional in Eq.(25 ) in the Rindler frame, we compute the reduced Wigner functional for a nonzero acceleration parameter \(a\). By substituting the two-point function \(f(k,k')\) from Eq. 22 into Eq. 4 to 7 , one obtain the following \(g_s\): \(g_1(K) = g_4(K) = 4 \pi ^2 a \exp{(-\gamma K^2/4 a^2)}\), and \(g_2(K) = g_3(K) = 0\). These lead to the reduced Wigner functional in 8 to take the following form \[\label{eq:26} W_{R_{reduced}} = \prod_{K>0} N_K \exp \left\{-2 (|b_{R}(K)|^2 + |b_{R}(-K)|^2 ) e^{-\gamma K^2/4a^2} \tanh{\frac{\pi |K|}{a}} \right\}\tag{26}\] Using Eq.@eq:eq:12 we get the following expectation value of the number density in the Rindler frame:

\[\label{eq:27} \langle n_R(K) \rangle = \frac{1}{2} \bigg( e^{\frac{\gamma K^2}{4a^2}} \coth{\frac{\pi |K|}{a}} -1 \bigg).\tag{27}\] As expected, in the limit \(\gamma \rightarrow 0\), the reduced Rindler Wigner functional turns out to be thermal, corresponding to the Minkowski vacuum. But for a nonzero \(\gamma\), the expectation value of the number density (see Eq. 12 ) diverges at high frequencies for ( see the right panel of Fig.2) \(\gamma\) comparable or greater than \(a/K\). This divergence can be regulated by choosing a frequency cutoff. However, the corresponding number density distribution may or may not be regulated by the same cutoff in the inertial frame as well as in a different frame. The form of divergence may depend on the state as well as the frame of reference, and further, a small perturbation around a given state can change the regularization significantly in different frames.
In the same context, one can analyze the large frequency behavior of the number density expectation value in terms of the Unruh Minkowski modes. To re-express the Minkowski functional in Eq.@eq:eq:25 in terms of the UM mode functional in an inertial frame we compare Eq.@eq:eq:10 with Eq.@eq:eq:24 which suggests identifying \(p(\xi) = q(\xi)\) = 2 \(e^{-\gamma \xi ^2 /16 \pi ^2}\) under a choice r = s = 0. Thus, the weight functions in the Wigner functional in terms of the Unruh Minkowski degrees of freedom also have a Gaussian form. Using Eq.@eq:eq:12 , the expectation value of the number density of Unruh-Minkowski particles for smaller frequencies, to the linear order in \(\gamma\), is found to be \(\gamma \xi ^2 / 32 \pi ^2 + \mathcal{O}(\gamma ^2)\). However, for larger frequencies, it grows rapidly. One concludes from here that in terms of Unruh particles, the \(\langle n_{\alpha_\xi} \rangle\) and \(\langle n_{\alpha_{-\xi}} \rangle\) both diverge at high frequencies for a \(\gamma\) greater than \(32 \pi^2/\xi ^2\). The functional form of the divergence associated with number density is sensitive to the form of \(f(k,k')\) chosen. Even with a choice of \(f(k,k')\), which slightly deviates from the corresponding form of the vacuum, the \(\langle n(k) \rangle\) may be affected in a nontrivial way. For example, let us consider the following form \[\label{eq:28} f(k,k')=\frac{\pi}{c} \frac{\theta(kk')}{\sqrt{|k||k'|}} e^{-|c \ln(k'/k))|}\tag{28}\] for some nonzero constant ‘c’. The corresponding Rindler frame occupation number \(\langle n_R(K) \rangle\), is found to be such that its second derivative becomes constant at high frequencies and goes to \(0\) at lower frequencies.

3.4 An almost uniform momentum distribution↩︎

In the earlier subsections, we discussed the vacuum state Wigner functionals with its weight function, \(f(k,k')\), sharply peaked in the Dirac delta distribution sense about the origin in momentum space. We also analyzed a case of Wigner functional, which slightly deviates from the vacuum state functional by considering its corresponding weight function \(f(k,k')\) to have a small width around a peak value in momentum space. Here, we consider a particular Wigner functional such that the expectation value of the corresponding Minkowski number density in momentum space is sharply peaked instead of the sharply peaked two-point weight function \(f(k,k')\). One can, therefore, call it an almost uniform momentum distribution. We choose the following form for the inertial frame Wigner distribution: \[\label{eq:29} W_M = N \exp(-2 \int _{-\infty} ^ {+\infty} \int _{-\infty} ^ {+\infty}\frac{dk dk'}{4 \pi^2} a_k ^* a_k \frac{2 \theta(kk')}{\gamma \sqrt{kk'} \left( (k/k')^{1/2\gamma} + (k'/k)^{1/2\gamma} \right)} ) ,\tag{29}\] where \(\gamma\) is a non-negative parameter. Such a choice leads the expectation value of the Minkowski number density operator \(\langle n_M(k) \rangle\) to be peaked in the momentum space as described below. We follow the procedure outlined in the previous subsection to numerically obtain the plots of inertial frame number density \(\langle n_M(k) \rangle\) and Rindler frame number density \(\langle n_R(K) \rangle\) as shown in Fig.3. The peak of number density \(\langle n_M(k)\) in frequency space shifts with a change in the value of the parameter \(\gamma\). From the plots, one can observe that the position of the peak shifts rightward, that is, towards a higher frequency with a decrease in \(\gamma\). Further, the absolute value of the peak of \(\langle n_M \rangle\) also decreases with a decrease of \(\gamma\). Thus, as \(\gamma \rightarrow\) 0, the peak of the number density shifts to infinity; however, the absolute value of the peak also approaches zero. This behavior is consistent with the expected limit wherein the \(\langle n_M(k)\) approaches the vacuum state \(<n_M>\), when \(\gamma \rightarrow 0\). Using Eqs. 5 -9 , one obtains the form of \(g_i(K)s\) to be \(g_1(K) = g_4(K) = 4 \pi^2 a \mathop{\mathrm{sech}}{(\pi \gamma |K|/a)}\) and \(g_2(K) = g_3(K) = 0\). The corresponding Rindler frame reduced Wigner distribution for a nonzero ‘a’ is found to be \[\label{eq:30} W_{R_{reduced}} = \prod_{K>0} N_K \exp \left\{-2 (|b_{R}(K)|^2 + |b_{R}(-K)|^2 ) \mathop{\mathrm{sech}}{\frac{\pi |K| \gamma}{a}} \tanh{\frac{\pi |K|}{a}} \right\}.\tag{30}\] The corresponding expectation value of the number density in the Rindler frame is obtained to be

\[\label{eq:31} \langle n_R(K)\rangle = \frac{1}{2} (\cosh(\frac{\pi \gamma |K|}{a}) \coth(\frac{\pi |K|}{a}) -1).\tag{31}\] For \(\gamma = 0\), \(\langle n_R(K) \rangle\) reduces to the standard Planckian form with the Unruh temperature T = a/\(2\pi\), as is expected since \(\gamma \rightarrow 0\) represents the vacuum Minkowski-particle state. The bottom panel in Fig.3 shows the expectation value of Rindler number density \(\langle n_R(K) \rangle\) given in, Eq.@eq:eq:31 . One can see that it diverges at higher frequencies. The divergence becomes sharper as \(\gamma\) increases from 1/7 to 1. Whereas at low-frequencies the behavior of \(\langle n_R(K) \rangle\) is finite and approaches zero for the cases of \(\gamma\) = 1/3 and 1, while for \(\gamma\) = 1/7 there is a rise in \(\langle n_R(K) \rangle\) when the Rindler frequency approaches zero. These observations suggest that there is no infrared divergence in \(\langle n_R(K)\rangle\) for the field in this state when \(\gamma \rightarrow > 1/7\). Alternatively, one can conclude that as \(\gamma \rightarrow 0\), \(\langle n_R(K) \rangle\) diverges at lower frequencies and decays exponentially at high frequencies, while the behavior is opposite for a relatively high \(\gamma\).

3.5 Indistinguishability of thermal and quantum fluctuations↩︎

Previous works in the literature have attempted to explore the relationship between quantum and gravitational phenomenon by exploring the relation or connection between quantum and statistical fluctuations [30]. There are also proposals to describe inertia as a consequence of the second law of thermodynamics [31]. In [8], the indistinguishability of thermal and quantum fluctuations was demonstrated in the context of the Unruh thermal bath. Using the density matrix formalism, the corresponding reduced density matrix in the Rindler frame was obtained starting from a thermal bath of one set of Unruh Minkowski particles. The resultant \(\rho_{red}\) in [8], [32] was symmetric in the temperature \(T_{UM}\) of the thermal bath considered and the Unruh acceleration temperature T= a/2 \(\pi\). Here, we explore further this relationship in the Wigner functional formalism. We consider a Wigner functional that represents a thermal bath of Unruh-Minkowski particles, where both \(\langle n_{\alpha_\xi} \rangle\) and \(\langle n_{\alpha_{-\xi}} \rangle\) follow a thermal distribution. In [8], \(\langle n_{\alpha_{-\xi}} \rangle\) is taken to be zero, that is, thermality is considered only for \(\langle n_{\alpha_{\xi}} \rangle\). We consider the following Unruh Minkowski Wigner distribution:

\[\label{eq:32} W_{UM} = \prod_{\xi > 0} N_\xi e^{-2 |\alpha_{\xi}|^2 \tanh{\frac{a \xi}{2\gamma}} - 2 |\alpha_{-\xi}|^2 \tanh{ \frac{a \xi}{2\gamma}}} = \prod_{K} N_K e^{-2 |\alpha_{K}|^2 \tanh{\frac{\pi |K|}{\gamma}} - 2 |\alpha_{-K}|^2 \tanh{ \frac{\pi |K|}{\gamma}}} ,\tag{32}\] where \(\gamma\) is some nonnegative real parameter. It is known that different quantum states can lead to the same expectation value of, say, the number density operator, which is a two-point correlator, while the corresponding expectation values of higher order correlations of the same operator, which are essentially fluctuations, can be different. In subsection 3.2, we found a Planckian expectation value of Unruh Minkowski particles \(\langle n_{UM}(\xi)\rangle\) for the state functional in Eq.@eq:eq:20 that leads to the corresponding reduced Wigner functional in the Rindler frame to be that of the Rindler vacuum. In contrast, the Wigner functional in Eq.@eq:eq:32 represents a true thermal bath, with both \(\langle n_{\alpha_\xi} \rangle\) and \(\langle n_{\alpha_{-\xi}} \rangle\) describing a thermal bath at the same temperature, characterized by \(\gamma\). By substituting \(p'(K)=q'(K)= \tanh{\pi |K|/\gamma}\) and \(r = s= 0\) in equation 11 , we obtain the following reduced Wigner distribution in the Rindler frame for a nonzero acceleration parameter \(a\): \[\label{eq:33} W_{R_{\text{reduced}}} = \prod_{K>0} N_K \exp \left\{-2 (|b_{R}(K)|^2 + |b_{R}(-K)|^2 ) \tanh{\frac{\pi |K|}{\gamma}}\tanh{\frac{\pi |K|}{a}} \right\}\tag{33}\] which, using Eq.@eq:eq:12 , yields the following expectation value of number density in the Rindler frame: \[\label{eq:34} \langle n_R(K) \rangle = \frac{1}{(e^{2\pi |K|/\gamma}-1)(1-e^{-2\pi |K|/a)}} + \frac{1}{(e^{2\pi |K|/a}- 1)(1-e^{-2\pi |K|/\gamma})}.\tag{34}\] We first note, that the above expression is different from the expectation value \(\langle n_R (K) \rangle\) for the single thermal bath of \(\langle n_{\alpha_\xi}\rangle\) considered in [8]. However, the expression above, Eq. 34 , is the same as the expression for the expectation value of the number density given in Eqs. (28) and (40) of [33], with the identification \(\beta = 2 \pi / \gamma\) and ‘g’ = ‘a’. In [33], along with the Bogoliuobov calculations, the authors have also illustrated the response rate of a Unruh-DeWitt detector for a Rindler Rindler trajectory in the Minkowski vacuum. One can interpret the result in Eq.@eq:eq:34 as the sum of expectation values of the number density of two gray bodies with the gray body factor equal to the partition function of the other one. One can also note that Eq.@eq:eq:33 is invariant under the exchange of two temperatures \(T_1 = a/2\pi\) and \(T_2 = \gamma /2\pi\). In other words, the reduced Wigner functional with acceleration parameter ‘a’ for a thermal bath at temperature \(\gamma /2\pi\) is identical to the reduced Wigner functional with acceleration parameter corresponding to Unruh-Davies temperature \(\gamma /2\pi\) for a thermal bath of UM particles at temperature a/\(2\pi\). This property supports the idea of indistinguishability of thermal and quantum fluctuations [8], which tells that within the domain of thermodynamic experiments, the quantum fluctuations and statistical fluctuations are indistinguishable.

We now proceed further to obtain an inertial frame functional in terms of Minkowski modes that yields the reduced Wigner functional Eq.@eq:eq:33 . We first read off \(g_1(K) = g_4(K) = 4 \pi^2 a \tanh{(\pi |K|/\gamma)}\) and \(g_2(K)=g_3(K) = 0\) by comparing Eq.@eq:eq:33 with Eq.@eq:eq:9 . Next, we use the inverse Fourier transform in Eq.(5 - 8 ) to determine the weight function f\((k,k')\). Substituting the resulting weight function into Eq.@eq:eq:3 yields \[\begin{gather} \label{eq:35} W_M = N \exp\bigg[-2 \int _{-\infty} ^ {+\infty} \int _{-\infty} ^ {+\infty}\frac{dk dk'}{4 \pi^2} a_k ^* a_k [2 \pi \delta(k-k') + \frac{\gamma' \theta (kk')}{2 \pi \sqrt{|k||k'|}} [-\psi^0(1-i \frac{\gamma' \ln(k'/k)}{4\pi}) \\ -\psi^0(1+i \frac{\gamma' \ln(k'/k)}{4\pi}) + \psi^0(\frac{1}{2}-i \frac{\gamma' \ln(k'/k)}{4\pi})+ \psi^0(\frac{1}{2}+ i \frac{\gamma' \ln(k'/k)}{4\pi})] ]\bigg] \end{gather}\tag{35}\] where, \(\gamma' =\gamma/a\). The range of the integration over k and \(k'\) in the above expression is - \(\infty\) to \(\infty\). However, one can split the integration and use the \(\theta (kk')\) term to change the integration limit from 0 to \(\infty\). This yields some another weight, say h, where the integration runs from 0 to \(\infty\). The expectation value of the number density of Minkowski particles can be determined using Eq.@eq:eq:12 . To obtain the inverse of \(h(k,k')\) we use the definition \(\int h(k,k')h^{-1} (k',k'') dk'\) = \(\delta(k-k'')\) and substitute the form of \(h(k,k')\) from Eq.@eq:eq:35 to identify the resultant expression as a Fredholm integral equation of the second kind [34]. Using the standard solution of Fredholm integral equation into Eq.@eq:eq:13 yields the following series –describing the expectation value of the number density of the Minkowski particles –which is convergent as \(\gamma' \rightarrow 0\). \[\begin{gather} \label{eq:36} \langle n_M(k) \rangle = \frac{1}{2} \bigg[ - \frac{\gamma'}{2\pi} h(k,k) + \frac{\gamma'^2}{4\pi ^2} \int _{0^+} ^\infty h(k,k') h(k',k) dk' - \\ \frac{\gamma'^3}{8\pi ^3} \int _{0^+} ^\infty h(k,k') \int _{0^+} ^\infty h(k',k'')h(k'',k) dk' dk'' +.....\bigg] \end{gather}\tag{36}\] where \(h(k,k')\) represents the term, apart from \(\theta (kk')\), in the weight function \(f(k,k')\) that appears alongside the Dirac delta function, given by
\[\begin{gather} \label{eq:37} h(k,k') = \frac{1}{2 \pi \sqrt{|k||k'|}}\bigg[-\psi^0(1-i \frac{\gamma' \ln(k'/k)}{4\pi}) -\psi^0(1+i \frac{\gamma' \ln(k'/k)}{4\pi}) + \psi^0(\frac{1}{2}-i \frac{\gamma' \ln(k'/k)}{4\pi})+ \\ \psi^0(\frac{1}{2}+ i \frac{\gamma' \ln(k'/k)}{4\pi})\bigg] . \end{gather}\tag{37}\] Additionally, one can confine the system to a finite box which will ensure the lower limit of integration to be \(0^+\). For \(\gamma'\) = 0 the \(\langle n_M(k) \rangle\) in Eq. 37 , yields the vacuum state in Minkowski and the \(\langle n_R(K) \rangle\) in Eq.@eq:eq:34 reduces to Planckian distribution in the Rindler frame, as expected. Another interesting aspect would be to explore the effect of the extra term \(h(k,k')\) to first order in \(\gamma'\). Using \(\lim_{x \to 0} [-\psi^0(1-i \frac{\gamma' x}{4\pi}) -\psi^0(1+i \frac{\gamma' x}{4\pi}) + \psi^0(\frac{1}{2}-i \frac{\gamma' x}{4\pi})+ \psi^0(\frac{1}{2}+ i \frac{\gamma' x}{4\pi})]\) = -2.77259 and retaining terms linear in \(\gamma'\) in Eq.@eq:eq:36 we obtain the following expression for the number density of Minkowski particles: \[\label{eqn:37} \langle n_M(k) \rangle = 0.035115 \frac{\gamma'}{|k|} .\tag{38}\] Interestingly, for small values of \(|k|\), the above expression of \(\langle n_M(k) \rangle\) exhibits a dependence on |k| akin to that of a bath with Planckian distribution.

Further, it is worth mentioning that in the case when we choose \(p'(K) = \tanh(\beta |K| / 2)\) and \(q'(K) = 1\) in Eq.@eq:eq:10 , the corresponding configuration represents a thermal bath of Unruh-Minkowski (UM) particles in \(\langle n_{\alpha _{\xi}}\rangle\), and with a zero average particle number density for \(\langle n_{\alpha _{-\xi}} \rangle\), which is the specific scenario discussed in reference [8]. Remarkably, this case is also consistent with detector response rate calculations [35]. However, it’s crucial to recognize that, since a particle detector also gets extra contributions apart from particle-like excitations, in more general cases, detector response may lead to varying conclusions, as demonstrated in different scenarios in reference [36].

3.6 Fermionization of a bosonic field↩︎

In curved spacetime, a relationship between statistics and spin naturally arises from the underlying spacetime dynamics— a connection that is absent for inertial observers in the flat geometry of Minkowski spacetime. [37]–[39]. However, the nontrivial Bogoliubov coefficients for an accelerated observer lead to the emergence of the spin-statistics connection for accelerated observers even in Minkowski spacetime [39]. Additionally, the equivalence principle, which posits a local equivalence between acceleration and gravitational effects, also suggests that this connection persists in Rindler space. The response rate calculation in [6] for a detector moving on a uniformly accelerated trajectory shows that the statistics of the Minkowski vacuum of a massless scalar field are Fermionic in an odd number of dimensions and are bosonic for even number of dimensions. However, the Bogoliubov calculation yields a Bosonic distribution in any dimension for the number density expectation in terms of the uniformly accelerated observer operators in a Minkowski vacuum [40]. In this subsection, we consider a Minkowski Wigner functional which leads to a mixture of both Fermionic and bosonic distributions in the Rindler frame, as described below. We define a Minkowski state by modifying the weight function \(f(k,k')\) discussed in the preceding subsection in Eq.@eq:eq:35 and introducing a change in sign of all the terms other than the Dirac delta in the weight function \(f(k,k')\). The objective is to assess the influence of the corresponding remaining term, h\((k,k')\). More precisely, we take the Wigner distribution for the inertial frame to be of the following form: \[\begin{gather} \label{eq:38} W_M = N \exp \bigg[-2 \int _{-\infty} ^ {+\infty} \int _{-\infty} ^ {+\infty}\frac{dk dk'}{4 \pi^2} a_k ^* a_k [2 \pi \delta(k-k') - \frac{\gamma' \theta (kk')}{2 \pi \sqrt{|k||k'|}} [-\psi^0(1-i \frac{\gamma' \ln(k'/k)}{4\pi}) \\ -\psi^0(1+i \frac{\gamma' \ln(k'/k)}{4\pi}) + \psi^0(\frac{1}{2}-i \frac{\gamma' \ln(k'/k)}{4\pi})+ \psi^0(\frac{1}{2}+ i \frac{\gamma' \ln(k'/k)}{4\pi})] ] \bigg] . \end{gather}\tag{39}\]
The expectation value of the number density in the inertial frame can be obtained by the similar procedure as in the preceding subsections. It yields a series that differs from Eq.@eq:eq:36 just by a negative sign, and it is convergent for \(\gamma \rightarrow 0\), if one considers the system to be in a box. However, in the present scenario, the expectation value of Minkowski particles up to the first order in \(\gamma\) turns out to be \(\langle n_M(k) \rangle = - 0.035115 \gamma/|k|\). We note that this value is negative of the \(\langle n_M(k) \rangle\) in Eq.@eq:eqn:37 . Once again, we can multiply \(\langle n_M(k) \rangle\) by frequency and add the zero point energy to get the positive and finite answer. One can get the weight function corresponding to the inertial frame Wigner functional displayed in Eq.@eq:eq:38 by substituting \(g_1(K) = g_4(K) = 4 \pi^2 a (2-\tanh{(\pi |K|/\gamma)}), g_2(K)=g_3(K) = 0\) in Eq.@eq:eq:4 -8 and using the inverse Fourier transform. The substitution of these \(g_s\) in Eq.@eq:eq:9 yields the following reduced Wigner distribution in the Rindler frame for a nonzero ‘a’:
\[\label{eq:39} W_{R_{reduced}} = \prod_{K>0} N_K \exp \left\{-2 (|b_{R}(K)|^2 + |b_{R}(-K)|^2 ) (2-\tanh{\frac{\pi |K|}{\gamma }}) \tanh{\frac{\pi |K|}{a}} \right\}.\tag{40}\] Using 12 for the expectation value of number density one obtain \[\begin{align} \label{eq:40} \langle n_R(K) \rangle = \frac{1}{-1 + e^{2 \pi |K|/a}} - \frac{1}{3 + e^{2 \pi |K|/\gamma }} - \frac{2 }{(-1 + e^{\frac{2 \pi |K|}{a}})(3 + e^{\frac{2 \pi |K|}{\gamma}})}, \end{align}\tag{41}\] where \(\gamma =\gamma' a\). In the above expression, the second term exhibits the Fermionic statistics. The overall structure of the expression, with the exception of the Fermionic form instead of bosonic component is similar to, Eq.(9) of [8], where the authors found the expectation value of the number density in the Rindler frame corresponding to a Unruh thermal bath in the Minkowski frame, to be the sum of two bosonic distributions plus an additional term that is the product of the first two terms. One can interpret Eq.@eq:eq:40 in terms of the spontaneous and stimulated emission of Rindler particles by writing as \(\langle n^\beta \rangle\) + \(\langle n^{\beta '} \rangle\) + 2 \(\langle n^\beta \rangle\) \(\langle n^{\beta '} \rangle\). It’s important to note that the Fermionic nature of Rindler noise was previously known in the case of odd dimensions for the Unruh Dewitt detector response [6]. Here, a Fermionic component arises in even dimensions in spite of using the Bogoliuobov calculation, which goes into calculating the Rindler reduced Wigner functional from the Minkowski functional.

As mentioned earlier, one can obtain the reduced Wigner functional described in Eq.@eq:eq:39 by using several other inertial frame Unruh Minkowski distributions. We arrive at one such interesting inertial frame Wigner functional, in terms of Unruh-Minkowski modes, that yields Eq.@eq:eq:40 by substituting the following weight functions in Eq. 10 : \[\begin{align} \addtocounter{equation}{1}\label{eq:41} p(\xi) = & q(\xi) = \mathop{\mathrm{sech}}{\left(\frac{\xi}{2}\right)}, \\ r\left( \xi \right) = & \frac{2 e^{\frac{\xi}{2 \gamma}} j(\xi)}{3 + e^{\frac{\xi}{\gamma}}}, \\ s\left( \xi \right) = & -\frac{e^{-\frac{\xi}{2 \gamma}} \, \text{sech}\left(\frac{\xi}{2}\right) \left(2 + e^{\frac{\xi}{\gamma}} \left(-2 + \text{sech}\left(\frac{\xi}{2}\right)\right) + \text{sech}\left(\frac{\xi}{2}\right)\right) \left(-2 + \tanh\left(\frac{\xi}{2 \gamma}\right)\right)}{2 j(\xi)}, \\ j(\xi) = & 2 e^{- \xi /2 \gamma} + \frac{\cosh\left(\frac{\xi}{2 \gamma}\right)}{\left(1 + e^{\frac{\xi}{\gamma}}\right)^{3/2}} \left\{ \left( 3 + 4 e^{\frac{\xi}{\gamma}} + e^{\frac{2 \xi}{\gamma}} + 8 \cosh\left(\frac{\xi}{2}\right) \right) \right. \times \\ & \left. \left( -3 + 2 \cosh\left(\frac{\xi}{2}\right) + e^{\frac{\xi}{\gamma}} \left( -1 + 2 \cosh\left(\frac{\xi}{2}\right) \right) \right) \text{sech}^2\left(\frac{\xi}{2}\right) \right\}^{1/2} . \end{align}\tag{42}\]
Substituting the above weight functions, denoted as Eq.@eq:eq:41 , in Eq.@eq:eq:14 we obtain the expectation value of the Unruh Minkowski number density as \[\begin{align} \label{eq:42} \langle n_{\alpha_{\xi}} \rangle = \langle n_{\alpha_{- \xi}} \rangle = \frac{1}{e^{2 \pi |K| / \gamma} - 1}, \end{align}\tag{43}\] which is nothing but a Planckian distribution. The Bosonic statistics of inertial frame expectation value Eq.@eq:eq:42 suggests that there can be a mixing of statistics by just choosing a different frame. However, as discussed in subsection 3.2, it does not represent a true thermal bath because of the nonzero r\((\xi)\) and s\((\xi)\) that yield nonzero anomalous averages.

4 Discussion↩︎

We employed the Wigner distribution formalism to derive a general form of the reduced Wigner functionals in Rindler spacetime by tracing out the degrees of freedom beyond the Rindler horizon of a real massless scalar field in (1+1) dimensions focusing on a special class of states in Minkowski spacetime more formally defined in subsection 2.2. Additionally we highlighted several features of the reduced Wigner functional for the these states. In conjunction with the Wigner distributions for Minkowski states, we also provided a general form of Wigner distributions expressed in terms of Unruh-Minkowski modes, which simplifies the technical challenges associated with formulating the former in a closed analytical form in certain special cases. The resulting Wigner functionals can be used to compute the expectation value of any relevant physical observables. We presented a general expression for the expectation value of the number density operator for each of these Wigner functionals.

We further explored a few specific distributions within this special class, providing various insights, as summarized below: The first Wigner functional examined describes the Minkowski vacuum which yields the reduced Wigner functional of a thermal bath in the Rindler frame –illustrating the standard Unruh effect. This serves as a consistency check for the more general result in Eq.(9 ). The next special case considered pertains to the Rindler vacuum. Interestingly, the Minkowski space Wigner functional in terms of Minkowski modes, which yields the Rindler vacuum as the reduced Wigner functional in the Rinder frame, is found to be independent of the acceleration parameter. Additionally, we identified a constraint relation on the weight functions \(p,q,r,s\) of inertial frame Wigner functional in terms of Unruh-Minkowski modes, which led to the Rindler vacuum. Subsequently, we examined three examples of UM mode Wigner functionals that satisfy this constraint. Notably, one example resulted in a Planckian distribution of number density \(\langle n_{UM}(K)\rangle\) of the UM modes, albeit with nonzero anomalous averages. The remaining two examples discuss distribution with vanishing anomalous averages

The third Minkowski Wigner functional considered is a near-Minkowski vacuum functional, with a slight deviation characterized by a non-negative parameter \(\gamma\). We find that the expectation value of the number density, \(\langle n_M(k)\rangle\), remains constant at lower frequencies but increases at higher frequencies. This constant value of \(\langle n_M(k)\rangle\) approaches zero across all frequencies as the system approaches the Minkowski vacuum. In the corresponding Rindler space, the number density \(\langle n_R(K)\rangle\) increases sharply at high frequencies for a finite \(\gamma\); however, it aligns with the thermal bath distribution when \(\gamma = 0\).

The next Wigner functional describes a distribution where \(\langle n_M(k)\rangle\) is sharply peaked around a specific Minkowski mode frequency. The peak can be shifted within the \(k\) space by varying the parameter \(\gamma\); however, moving the peak to higher frequencies reduces the overall magnitude of \(\langle n_M(k)\rangle\). Thus when the peak is strictly shifted to infinity, \(\langle n_M(k)\rangle\) becomes zero in the inertial frame, resulting in a thermal distribution in the Rindler frame. We also illustrated the frequency dependent behavior of both \(\langle n_M(k)\rangle\) and \(\langle n_R(K)\rangle\) in the non-vacuum scenario. Notably, we observed that \(\langle n_R(K)\rangle\) does not exhibit infrared divergence for the field in this state when \(\gamma \rightarrow \infty\).

The next intriguing case we discussed involves a bath of Unruh-Minkowski particles in the inertial frame, which exhibits thermal characteristics in both \(\langle n_{\alpha_\xi}\rangle\) and \(\langle n_{\alpha_{-\xi}}\rangle\). The corresponding reduced Wigner functional is found to be invariant under the exchange between the Unruh-Davies temperature and the bath temperature in the inertial frame. This finding supports the notion of the indistinguishability of quantum and statistical fluctuations, as described in [8]. Finally, we derived the corresponding Minkowski Wigner functional in terms of Minkowski modes that produce the same temperature-symmetric reduced Wigner functional and calculated \(\langle n_M(k)\rangle\) to linear order

The final reduced Wigner functional describes a distribution where \(\langle n_R(K)\rangle\) is a superposition of two components, exhibiting Bosonic and Fermionic statistics, respectively. We obtained the corresponding Minkowski Wigner functional and highlighted the similarities and differences with the distribution discussed in subsection 3.5. Lastly, we illustrated an inertial frame distribution corresponding to the reduced Wigner functional in terms of the Unruh-Minkowski modes, with Planckian \(\langle n_{\alpha_\xi}\rangle\) and \(\langle n_{\alpha_{- \xi}}\rangle\).

Indeed, there are several other distributions where the formalism presented can be applied. However, we have restricted our analysis to real massless scalar fields in Minkowski space with (1 + 1) dimensions for a special class of states. Extending to other dimensions with different fields in different states may yield further insight that can be explored in future work.

References↩︎