Fermionic condensate and the mean energy-momentum tensor in the Fulling-Rindler vacuum

We investigate the properties of the fermionic Fulling-Rindler vacuum for a massive Dirac field in a general number of spatial dimensions. As important local characteristics, the fermionic condensate and the expectation value of the energy-momentum tensor are evaluated. The renormalization is reduced to the subtraction of the corresponding expectation values for the Minkowski vacuum. It is shown that the fermion condensate vanishes for a massless field and is negative for nonzero mass. Unlike the case of scalar fields, the fermionic vacuum stresses are isotropic for general case of massive fields. The energy density and the pressures are negative. For a massless field the corresponding spectral distributions exhibit thermal properties with the standard Unruh temperature. However, the density-of-states factor is not Planckian for general number of spatial dimensions. Another interesting feature is that the thermal distribution is of the Bose-Einstein type in even number of spatial dimensions. This feature has been observed previously in the response of a particle detector uniformly accelerating through the Minkowski vacuum. In an even number of space dimensions the fermion condensate and the mean energy-momentum tensor coincide for the fields realizing two inequivalent irreducible representations of the Clifford algebra. In the massless case, we consider also the vacuum energy-momentum tensor for Dirac fields in the conformal vacuum of the Milne universe, in static open universe and in the hyperbolic vacuum of de Sitter spacetime.


Introduction
The observer dependence of the vacuum and particle notions is among the important lessons from quantum field theory on curved spacetimes.The crucial point in the quantization procedure is the choice of the complete set of mode functions for a given field.The expansion of the field operator over those modes determines the annihilation and creation operators.The construction of the Fock space of states starts from the definition of the vacuum state that is nullified by the action of the annihilation operator.The particle is a state of quantum field obtained acting by the creation operator on the vacuum state.It carries a set of quantum numbers that determines the mode functions of the classical field equations.From this construction it follows that the vacuum and particle states, in general, depend on the choice of the mode functions.The annihilation and creation operators for different sets of mode functions are related by the Bogoliubov transformations.If those transformations mix the annihilation and creation operators, the Bogoliubov β-coefficient is different from zero and the two vacuum states based on two sets of modes are not equivalent: the vacuum state corresponding to one set of modes contains particles of the other set of modes.
The natural mode functions used in the expansion of the field operator may differ for different observers, giving rise to different vacuum states.This may take place already in flat spacetime.The classical example of two inequivalent vacuum states in the Minkowski spacetime are the Minkowski and Fulling-Rindler vacua.They are vacuum states for inertial and uniformly accelerating observers, respectively.The quantization of fields in Rindler coordinates, that are the natural coordinates for uniformly accelerating observers, has been widely discussed in the literature (see [1,2,3] and references therein).The interest is motivated by several reasons.First of all, it comes from principal questions of quantization of fields in geometries having horizons.The latter can be either observer dependent (like Rindler or de Sitter (dS) horizons) or determined by the matter distribution (examples are the black hole horizons).The Rindler geometry is simple enough to allow to find exact solutions in different problems of quantum field theory.This may shed light on the respective problems in more complicated geometries where the exact solutions are not available or they are complicated.Next, the Rindler metric approximates the black hole geometry in the near horizon limit and the roots of a number of quantum field theoretical phenomena around black holes can be found in the Rindler physics.An example is the relation between the Unruh effect and Hawking radiation.The Unruh effect [4,5,6] states a kind of equivalence between thermal fluctuations of a quantum field observed by an inertial observer and fluctuations in the inertial (Minkowski) vacuum of the same field recorded by a uniformly accelerating observer.The temperature of the thermal bath, the Unruh temperature, is proportional to the acceleration of the Rindler observer.The thermal nature of the distribution of Rindler particles is closely related to the presence of the horizon for uniformly accelerating observers.The events outside the horizon are not accessible for those observers and they are traced out, resulting in the information loss and thermal state.Being a background with horizons, the Rindler geometry is an interesting arena to investigate the phenomena of quantum entanglement (see, e.g., Refs.[7,8,9,10]).From the point of view of a uniformly accelerating observer the Minkowski vacuum appears as an entangled state between the states in the right and left wedges of the Rindler decomposition in the Minkowski spacetime.The considerations of specific examples have shown that the Unruh effect can either reduce or enhance entanglement.Though the experimental observation of the Unruh effect requires huge accelerations, different schemes for simulating the effect in laboratory have been discussed in the literature (see, for example, [2,11,12] and references therein).
The investigations in the Rindler physics were carried out in two main directions.The first one considers the properties of the Minkowski vacuum and of the related particle states seen by Rindler observers.In particular, a large number of papers are devoted to the study of the response of various types of particle detectors in an accelerated motion through the inertial vacuum (for different types of particle detectors interacting with quantum fields see, for example, [3,13,14]).The main subject in the second class of investigations are the properties of the Fulling-Rindler vacuum and of the Rindler particle states from the viewpoints of both Rindler and inertial observers.For free quantum fields, among the important local characteristics of the vacuum state are the expectation values of bilinear products of the field operator, like the field squared and the energy-momentum tensor.The present paper deals with those characteristics of the fermionic Fulling-Rindler vacuum state for a massive Dirac field in general number of spatial dimensions.Various aspects of quantum fermionic fields in Rindler spacetime have been considered in the literature.The references [8]- [29] include an incomplete list of some of them.Our consideration of general number of spatial dimension is motivated by possible applications in high energy models of fundamental physics such as string theories, supergravities, Kaluza-Klein type theories and braneworld models.An interesting application of fermionic models in two-dimensional space comes from condensed matter physics.In the long wavelength approximation the excitations of the electronic subsystem in so-called Dirac materials are well described by the Dirac model in two dimensions where the speed of light is replaced by the Fermi velocity (see reviews [30,31]).The latter is smaller than the speed of light by orders of magnitude and the Dirac materials provide a unique possibility for investigations of relativistic effects at smaller velocities.An example of such a condensed matter system is graphene.The graphene based structures, like carbon nanotubes and nanoloops, also provide an opportunity to study the effects of nontrivial spatial topology on the properties of the fermionic vacuum in quantum field theory [32]- [36].Effects of compactification of spatial dimensions in the Fulling-Rindler vacuum for a scalar field have been recently discussed in [37].
The remainder of the paper is structured as follows.In the next section we present the positive and negative energy Rindler modes for a massive Dirac field in general number of spatial dimensions.By using those modes, in Section 3 the renormalized fermion condensate is evaluated.The renormalization is based on the subtraction of the corresponding vacuum expectation value (VEV) for the Minkowski vacuum state.The corresponding considerations for the vacuum expectation value of the energymomentum tensor are presented in Section 4. We show that, as expected, the mean energy-momentum tensor obeys the covariant conservation equation and the trace relation.Alternative representations for the fermion condensate and vacuum energy-momentum tensor, well adapted for numerical evaluations, are derived in Section 5.In particular, they show that the vacuum stresses are isotropic for the Dirac field.Numerical results are presented as well.Section 7 concludes the main results of the paper.In Appendix A, integral representations are provided for products of the modified Bessel functions with imaginary order.Those representations have been used to simplify the expressions for the VEVs.In Appendix B we give an integral representation for the Hadamard function of the Dirac field in the Minkowski vacuum that is used in the subtraction procedure for renormalized VEVs.An integral representation for the difference of the traces for the Hadamard functions in the Fulling-Rindler and Minkowski vacua is provided in Appendix C.

Fermionic modes in Rindler spacetime
The complete set of fermionic modes are required in the canonical quantization procedure.In the literature they are mainly considered in spatial dimensions D = 1 and D = 3 (for a recent discussion of solutions to the Dirac equation in 4-dimensional Rindler spacetime with analytic continuations to all quadrants of Minkowski spacetime see [9,29] and references therein).Here, we generalize the approach of Ref. [15] for a massive field and for general number of spatial dimension D and present the complete set of Dirac modes in the form that does not depend on the special representation of the Dirac matrices.They are well adapted for investigations of the VEVs of physical observables.
The dynamics of a fermionic field ψ(x), x = (x 0 , x 1 , . . ., x D ), in a (D + 1)-dimensional spacetime with the metric tensor g µν (x) is described by the Dirac equation where γ µ = e µ (b) γ (b) are the curved spacetime gamma matrices, ∇ µ = ∂ µ +Γ µ is the covariant derivative for Dirac fields and Γ µ is the spin connection.The expression for the latter in terms of the flat spacetime matrices γ (a) , a = 0, 1, . . ., D, and (D + 1)-bein fields e µ (a) reads with the semicolon meaning the standard covariant derivative of vector fields.We will consider a fermionic field realizing the irreducible representation of the Clifford algebra {γ µ , γ ν } = 2g µν with N × N Dirac matrices.Here, N = 2 [(D+1)/2] and [z] is the integer part of z (for the Dirac matrices in an arbitrary number of the spacetime dimension see, for example, [38,39]).Up to a similarity transformation, the irreducible representation is unique for odd D. In the case of even D there are two inequivalent irreducible representations.The background geometry under consideration is described by the Rindler line element where x = x 2 , x 3 , . . ., x D , −∞ < τ < +∞, and 0 ≤ ρ < ∞.By the coordinate transformation the line element takes the Minkowskian form ds 2 M = dt 2 − dz 2 , z = (x 1 , x), and the geometry is flat.The worldline with fixed spatial coordinates (ρ, x) corresponds to a uniformly accelerating observer moving along the line parallel to the x 1 axis with proper acceleration 1/ρ.The proper time of that observer is measured by τ p = ρτ .We note that the Rindler coordinates divide the subspace (t, x 1 ) of the Minkowski spacetime into four regions (wedges) depicted in Figure 1.The transformation (2.4) corresponds to the right wedge (R region) with x 1 > |t|.The transformation presenting the left wedge (L region), with x 1 < −|t|, is obtained from (2.4) adding the minus sign in the right-hand side for the expression of x 1 .The coordinate transformations for the remaining regions, future (F) and past (P) regions with t > |x 1 | and t < −|x 1 |, respectively, are given by t = ±ρ cosh τ and x 1 = ρ sinh τ .Here, the upper and lower signs correspond to the F and P regions.In the discussion below we will consider the VEVs in the R region.The same expressions for the VEVs are obtained in the L region.The subspace (t, x 1 ) of Minkowski spacetime covered by Rindler coordinates (τ, ρ).The four wedges, the R, L, F, and P regions, are separated by the Rindler horizons t = ±x 1 (dashed lines).
We want to find a complete set of solutions to equation (2.1) for the geometry (2.3).The corresponding metric tensor is given by (2.5) The (D + 1)-bein fields can be chosen as e µ (0) = δ µ 0 /ρ and e µ (b) = δ µ b for b = 1, 2, . ... The related spin connection is expressed as We present the solution of the equation (2.1) in the form with a new field ϕ(x).For the latter the following equation is obtained with the same operator ∇ µ as in (2.1).By taking into account (2.6), for the geometry at hand the field equation (2.8) is reduced to The geometry possesses a Killing vector ∂ τ and the solution of Eq. (2.9) corresponding to the positive energy fermionic modes with respect to this vector can be presented in the form where , and 0 ≤ ω < ∞.Plugging this in (2.9), we get the equation for the function u(ρ, ω, k): with k = |k| and λ = k 2 + m 2 . (2.12) By taking into account that the solution of (2.11), finite in the limit ρ → ∞, is expressed as where K ν (z) is the modified Bessel function of the second kind (Macdonald function) and χ(k) is a constant spinor.Note that for a function f (x) with the argument x = γ (0) γ (1) we have (see also [15]) Having the function ϕ(x), for the solution corresponding to the Dirac spinor ψ(x) one finds where we have introduced the notation with γ k = D i=2 γ (i) k i .In deriving (2.16) the relations and γ kK ν (λρ) = K ν (λρ) γ k have been used with ν = iω − γ (0) γ (1) /2.
Introducing a new constant spinor χ (+) η (k), the positive energy fermionic modes are presented in the form where σ = (ω, k, η) presents the set of quantum numbers specifying the modes.Here, η enumerates the spinorial degrees of freedom.For the operator (2.17) the properties can be easily checked.We specify the spinors χ η (k) by the relation by the orthonormality condition and by the completeness relation where α and β are spinor indices.The conditions (2.21)-(2.23)are the generalizations of the respective relations in [15] for a massless field in 4-dimensional spacetime.Note that in [15] the Majorana representation was used in which the components of the spinor ψ(x) and the Dirac matrices are real.
In our consideration the representation is not fixed.With the condition (2.21), the following final expression is obtained for the positive energy fermionic normal modes: Note that the energy measured by an observer with (ρ, x) = const is given by ε ρ = ω/ρ.The constant N σ in (2.24) is determined from the normalization condition where g is the determinant of the metric tensor and the Dirac conjugate ψ(x) for the field ψ(x) is defined as ψ(x) = ψ † (x)γ (0) .Here, δ σσ ′ is understood as Dirac delta function for continuous quantum numbers and as Kronecker symbol for discrete ones.By taking into account that γ 0 = γ (0) /ρ and |g| = ρ, the orthonormality condition for the modes (2.24) is reduced to For the evaluation of the integral in the right-hand side we use the result [15] ∞ The contribution of the second term in the square brackets of (2.26) to the normalization integral in (2.25) contains the combination χ By taking into account χ it can be seen that χ Hence, the second term in the square brackets of (2.26) does not contribute to the normalization integral in (2.25).As a consequence, for the normalization coefficient we get where λ is given by (2.12).The negative energy fermionic modes are found in a similar way: with the same normalization coefficient.The constant spinors χ (−) η (k) obey the same relations (2.21)-(2.23)with the N × N matrix P (k) from (2.17).It can be checked that the positive and negative energy modes are orthogonal: The integral is evaluated with the help of (2.26), where now the replacement ω ′ → −ω ′ should be made.By using the relations χ η (k) = 0.The remaining part contains the delta function δ (ω ′ + ω).From the relation (2.23) and similar relation for χ for j = +, −.In the discussion below this relation will be used in the evaluation of the mode sums for the fermion condensate and the VEV of the energy-momentum tensor.
We have described the fermionic modes in the R region of the Minkowski spacetime.The modes in the L region have the same structure, whereas the modes in the F and P regions are obtained by respective analytic continuations (for the analytic continuation of the modes in 4-dimensional Rindler spacetime see, e.g., [8,9]).Note that in those regions the line element reads ds 2 F,P = dρ 2 − ρ 2 dτ 2 − dx 2 and the roles of the coordinates τ and ρ are reversed: they appear as spatial and time coordinates, respectively.Here the situation is similar to that for Schwarzschild time and radial coordinates near black hole horizons.In the F and P regions the special case of the Kasner metric is realized with flat spacetime geometry.Alternatively, by the coordinate transformation we can present the line element in the F and P regions in the form corresponding to the Milne universe.In that representation the spacetime is foliated by constant negative curvature spatial sections (see Section 6).
Note that, by using the procedure described above, we can also construct a set of fermionic modes in problems where an additional uniformly accelerating boundaries (mirrors) are present, for example, in the R region.For a single boundary and reflecting conditions on the fermionic field, the R region is divided into two subregions 0 < ρ < ρ 0 and ρ 0 < ρ < ∞, where 1/ρ 0 is the proper acceleration of a planar mirror.In the region between the Rindler horizon and mirror the dependence on the coordinate ρ is expressed in terms of the linear combination of the modified Bessel functions I ν (z) and K ν (z) and the relative coefficient in that combination is determined by the boundary condition.In the region ρ 0 < ρ < ∞ the mode functions should have the form (2.24) and (2.30), but now the boundary condition on the mirror will lead to discrete spectrum of the energy ω.The problems for polarization of the Fulling-Rindler vacuum by uniformly accelerating boundaries in the cases of scalar and electromagnetic fields have been discussed in Refs.[40]- [45].

Fermionic condensate
In this section we investigate the fermionic condensate (FC) by using the mode functions from the previous section.It is an important physical characteristic in models of symmetry breaking and dynamical mass generation and in the studies of phase transitions.In the most popular model of the Unruh-DeWitt detector interacting with fermionic fields the interaction Hamiltonian is proportional to the FC operator evaluated at the location of the detector (see, e.g., [3,13,14,18]) and the FC is the central quantity in the interpretation of the response for that type of detectors.
The FC is defined as the VEV 0| ψψ |0 = ψψ , where |0 stands for the vacuum state (the Fulling-Rindler vacuum in the consideration at hand) and ψ = ψ † γ (0) .The FC is expressed in terms of the fermionic Hadamard function S (1) (x, x ′ ) with the spinorial components where α and β are spinorial indices.For the FC one has Expanding the field operator in terms of the complete set of fermionic modes {ψ σ }, the following mode sum is obtained for the trace of the Hadamard function: where Substituting the mode functions (2.19) and (2.30), by using Eq.(2.15) and the relations for the products of the normal modes we get where ∆τ = τ − τ ′ and ∆x = x − x ′ .Now, by using the relation (2.15) for the modified Bessel functions, the trace is presented as The summation over η is done by using the result (2.32): where the relations Tr γ (i 1 ) γ (i 2 ) • • • γ (i 2n+1 ) = 0 and Tr γ (a) γ (b) = (D + 1)η ab have been used.This leads to the result (3.9) The limit in the right-hand side of (3.2) is divergent and the renormalization of the FC is required.The background spacetime is flat and the divergences are the same as those for the Minkowski vacuum.Renormalizing the FC in the Minkowski vacuum to zero, for the renormalized FC in the Fulling-Rindler vacuum we get where S (1) MR (x, x ′ ) is the Hadamard function for the Minkowski vacuum transformed to the Rindler coordinates.The expression for the latter, adapted for the subtraction in (3.10), is provided in Appendix B (see (B.11)).Plugging the representations (3.9) and (B.11) (the latter multiplied by cosh (∆τ /2)) and integrating over the angular coordinates of k, for the renormalized FC in the Fulling-Rindler vacuum one finds where D ≥ 2. In the special case D = 1, with N = 2 for the irreducible representation of the Clifford algebra, the corresponding expression has the form (3.12) For a massless field the FC vanishes.Note that we could write a formal expression obtained from (3.9) taking the coincidence limit in the integrand: The divergence comes from the term e πω /2 in the definition of the hyperbolic cosine function.Comparing with (3.12) we see that the subtraction of the Minkowskian VEV is equivalent to the subtraction of the part with that term.
The ratio ψψ FR /m D is a function of a single variable mρ.Let us consider the behavior of that function in asymptotic regions for D ≥ 2. For mρ ≫ 1 the argument of the modified Bessel function is large and we use the leading order estimate This gives and the FC is exponentially small.In particular, for an observer with a given acceleration the FC is exponentially small for large values of the mass.
In the opposite limit mρ ≪ 1 we introduce a new integration variable u = λρ and put in the integral mρ = 0 for the leading order term: The integral is evaluated by using the formula with ν = D − 2, and we get Note that for the gamma function in the integrand we have where for n ≥ 2 and B 0 = B 1 = 1.In (3.20), {n/2} is the fractional part of n/2 and we have defined Note that, for a given value of the mass the condition mρ ≪ 1 corresponds to points near the Rindler horizon ρ = 0. From (3.18) we conclude that near the Rindler horizon the FC behaves like 1/ρ D−1 .

VEV of the energy-momentum tensor
We turn to the study of another important local characteristic of the fermionic vacuum, namely, the expectation value of the energy-momentum tensor.In addition to describing the local properties of the vacuum state, it appears as a source of gravity in the semiclassical Einstein equations and determines the back reaction of quantum effects on the spacetime geometry.The possibility to measure the VEV of the energy-momentum tensor for a scalar field in non-gravitational way by using particle detectors has been discussed in [46].Similar to the case of the FC, the point splitting procedure will be used for regularization.The corresponding mode sum is expressed as where the brackets denote symmetrization with respect to the enclosed indices and the action of the covariant derivative operator on the Dirac conjugate spinor is given by σ Γ µ with the spin connection from (2.6).The summation σ is understood in the sense (3.4).
The evaluation procedure for the energy-momentum tensor is similar to that for the FC and we will omit the details.Substituting the mode functions (2.24), (2.30) and using the relation (2.15), for the diagonal components we get (no summation over µ) with the functions where in the last expression l = 2, . . ., D. The off-diagonal components vanish.Again, the limit in the right-hand side of (4.2) is divergent and in order to obtain the renormalized VEVs we need to subtract the corresponding expectation values for the Minkowski vacuum.Similar to the case of the FC it can be seen that the subtraction is equivalent to omitting the term e πω /2 in the expression for the hyperbolic cosine function cosh (πω).In this way, for the renormalized VEV, where D ≥ 2 and the functions for separate components are defined as with l = 2, . . ., D. In the special case D = 1 one obtains The off-diagonal components of the renormalized mean energy-momentum tensor are zero.Alternative representations of the FC and mean energy-momentum tensor will be provided below in Section 5.In particular, they show that (no summation over l) T l l FR = T 1 1 FR for l = 2, 3, . . ., D, and the vacuum stresses are isotropic.Note that this property does not follow from the symmetry of the problem.For example, in the case of a scalar field the stresses T 1 1 FR and T 2 2 FR , in general, differ (see [41]).They coincide for a conformally coupled massless field only (see below).In analogy of the perfect fluid, we can interpret the quantities (no summation over l) − T l l FR as effective pressures along the respective directions.
The energy density corresponding to Eq. (4.4) is always negative, T µ µ FR < 0. The representations given in Section 5 explicitly show that the vacuum pressures − T l l FR are negative as well.We expect that the renormalized VEVs will obey the covariant conservation equation ∇ µ T µ ν FR = 0.For the geometry at hand it is reduced to the relation T 0 0 FR = ∂ ρ ρ T 1 1 FR .By using the differential equation for the modified Bessel functions it is easy to check that this relation indeed takes place.Writing the VEV (4.4) in the form T ν µ M − T ν µ = − T ν µ FR , we can interpret the tensor − T ν µ FR as the energy-momentum tensor of the Minkowski vacuum with respect to the Fulling-Rindler vacuum state.In this interpretation the respective energy density is positive.
For a massless field the integrals over λ in (4.4) for µ = 1 are evaluated by the formula (3.17).The integral for µ = 1 is obtained from the general formula in [47] for the product of modified Bessel functions: For the renormalized VEV the following result is obtained The corresponding stresses are isotropic and the energy-momentum tensor is traceless.We could expect the latter property on the basis of conformal invariance of a massless fermionic field and flatness of the background geometry (zero trace anomaly).
An equivalent representation of the mean energy-momentum tensor for a massless field is obtained by using the relation (3.19).This gives where Here, B D (ω) is defined by (3.20) for D ≥ 2 and B 0 = B 1 = 1.In the special case D = 3 we get which coincides with the result from [15].We can write the integral in (4.11) in terms of the proper energy ε ρ = ω/ρ, measured by an observer with proper acceleration 1/ρ.The respective factor e 2πρερ + 1 −1 is interpreted as an indication of the thermal nature of inertial vacuum with respect to a uniformly accelerating observer [4]- [15].The corresponding temperature (Unruh temperature) is given by T = 1/(2πρ).An interesting point to be mentioned is that in even number of spatial dimensions the thermal factor for Dirac field is of bosonic type, (e 2πρερ − 1) −1 .Similar features in the response of a uniformly accelerating Unruh-DeWitt detector interacting with Dirac field prepared in the Minkowski vacuum have been observed in [17,18].Note that for a scalar field in general number of spatial dimension the thermal factor is in the form [e 2πω + (−1) D ] −1 (see [18,41,48]).The appearance of the Fermi-Dirac type factor for scalar fields in even number of spatial dimensions has been further discussed in [49,50] (see also [51] for more recent consideration and references).The calculation of Rindler noise on the basis of the fluctuation-dissipation theorem, that gives a hint to the origin of inversion of statistics, is presented in [52].The coefficient A = [π 2 ζ(3) + 3ζ(5)]/ 16π 7 .In Table 1 we give the numerical values of the coefficient for D = 2, . . ., 10.
As seen from (4.4), for a massive field the ratio T ν µ FR /m D+1 depends on the mass and coordinate ρ in the form of dimensionless combination mρ.Let us consider the behavior of the VEV in asymptotic regions of that combination.For mρ ≫ 1 the argument of the modified Bessel functions in (4.4), (4.5) is large and the dominant contribution to the integral over λ comes from the region near the lower limit of the integration.By using the asymptotic (3.14) and  1: The values of the coefficient in the expression (4.9) for the vacuum energy-momentum tensor in different spatial dimensions.
in the leading order we get (no summation over l) for l = 1, 2, . . ., D. In particular, for large masses of the field quanta the vacuum energy-momentum tensor is exponentially small.An interesting feature seen from the asymptotic estimate (4.13) is that the absolute value of the energy density is much smaller than the absolute value of the pressure, | and in the non relativistic limit |T 1 1 | ≪ T 0 0 .In the opposite limit mρ ≪ 1, the leading term in the expansion over mρ coincides with the VEV for a massless field given by (4.9).In particular, for a given mass, near the Rindler horizon the diagonal components of the energy-momentum tensor behave as 1/ρ D+1 .
It is of interest to compare the VEV (4.9) for a massless Dirac field with the respective result for a massless scalar field with curvature coupling parameter ξ.The corresponding mean energy-momentum tensor is given by the formula (no summation over µ) [41] with the functions for l = 2, 3, . . ., D. Here, ξ D = (D − 1)/(4D) and the factor B (sc) D (ω) is the scalar analog of (3.20), defined as for D ≥ 3 and As it has been already mentioned, for scalar field, in general, the stresses are anisotropic.For a conformally coupled field ξ = ξ D and the formula (4.14) is reduced to with the notation This result generalizes the relation between the VEVs for massless scalar and Dirac fields, conjectured in [53] (see also [1,15]).
For even values of spatial dimension D the Clifford algebra of Dirac matrices has two inequivalent irreducible representations.These two representations can be constructed by using an irreducible representation γ (a) , a = 0, 1, . . ., D − 1, for (D − 1)-dimensional space and adding the matrix γ (D) (s) = i 2{D/4} sγ (D) , where γ (D) = γ (0) γ (1) • • • γ (D−1) and s = ±1 (see, e.g., [38]).Two inequivalent representations for even D are realized by two sets γ The consideration we have presented above does not depend on the representation of Dirac matrices.Hence, we conclude that the FC and the mean energy-momentum tensor coincide for Dirac fields ψ (s) realizing two inequivalent representations of the Clifford algebra in even numbers of spatial dimensions.Note that for even D the mass term in the Lagrangian density for a field ψ (s) with given s is not invariant under the parity (P) transformation.Additionally, the mass term is not invariant under the charge conjugation (C) for {D/4} = 0 and under the time reversal (T) for {D/4} = 1/2.Fermionic models invariant under those transformations can be constructed combining two fields ψ (+1) and ψ (−1) with the same masses.In those models the VEVs are obtained from the expressions presented above with an additional coefficient 2.

Alternative representations and numerical analysis
In this section, equivalent representations are provided for the VEVs, more convenient in numerical evaluations.We start with the FC given by (3.11).From the integral representation (A.2) for the product of the modified Bessel functions it follows that Substituting this in (3.11), the integral over λ is evaluated by using the formula [47] ∞ where we have introduced the function The integration over ω is elementary and we get This shows that FC is always negative.Note that this formula could also be directly obtained from the representation (C.9) for the trace of the Hadamard function derived in Appendix C. The FC is monotonically decreasing function of the proper acceleration 1/ρ.It is exponentially suppressed by the factor e −2mρ for large values of ρ and behaves like 1/ρ D−1 for small ρ (near the Rindler horizon).In Figure 2 we have plotted the quantity ρ D ψψ FR as a function of dimensionless combination mρ for different values of the spatial dimension.For a massless field the FC vanishes.
For the transformation of the energy density we use the representation which is a direct consequence of Eq. (A.4) with ρ ′ = ρ.Inserting in (4.4) with µ = 0 and evaluating the integral with the help of (5.2) one finds (2mρ cosh u) cosh u.
(5.6) Yet another representation is obtained by using in (5.6) and integrating by parts.By taking into account the relations (5.8) that gives (2mρ cosh u) . (5.9) Again, this shows that the energy density is negative.Figure 3 presents the dependence of the dimensionless product ρ D+1 T 0 0 FR on mρ in spatial dimensions D = 2, 3, 4, 5.For the vacuum stresses we can proceed in a similar way.For the stress T 1  1 FR we use the representation obtained from (A.5).The following steps are based on (5.2) and are similar to those for the FC and energy density.The final result reads (2mρ cosh u) . (5.11) For the stresses T l l FR , l = 2, . . ., D, we employ the representation (5.1) and the subsequent evaluation shows that (no summation over l) (5.12) Hence, we conclude that the vacuum stresses are isotropic for massive fields as well.From (5.11) it follows that they are positive and, hence, the corresponding vacuum pressure is negative.By using the representations obtained, we can prove the trace relation T µ µ FR = m ψψ FR .This serves as an additional check for the calculations provided above.In Figure 4 we display the function ρ D+1 T 1 1 FR versus mρ.The numbers near the curves are the values of the spatial dimension D. The representations given in this section are further simplified for a massless field.By taking into account that f ν (z) ∼ 2 ν−1 Γ (ν) z −2ν for z → 0, we get and FC vanishes.The integral in (5.13) is strongly convergent and this representation is well adapted for numerical evaluation.For large values of mρ, by using f ν (x) ≈ π/2x −ν−1/2 e −x for x ≫ 1, the dominant contribution to the integrals in (5.9) and (5.11) comes from the region near the lower limit and it can be seen that, to the leading order, the asymptotics (4.13) are obtained.
6 VEVs in backgrounds conformally related to Rindler spacetime Having the VEVs for a massless Dirac field in the Fulling-Rindler vacuum we can generate the corresponding VEVs in the problems where the background geometry is conformally related to the Rindler spacetime.Some examples for a conformally coupled massless scalar field in 4-dimensional spacetime are summarized in [1,54].In Ref. [55] the brane induced vacuum energy-momentum tensor is investigated in dS spacetime by using the conformal relation with the problem of a planar boundary uniformly accelerated through the Fulling-Rindler vacuum and the results from [41].
Let ḡµν be the metric tensor for a spacetime conformally related to the Rindler metric, ds 2 = ḡµν dx µ dx ν = Ω 2 (x)ds 2 R .The following relation takes place for the respective mean energy-momentum tensors [1]: where g and ḡ are the determinants of the corresponding metric tensors and T ν µ G is the geometrical part that is completely determined by the geometrical characteristics of the metric tensor ḡµν .The expectation value in the left-hand side of Eq. (6.1), denoted as • • • CR , is taken for the state of quantum field conformally related to the Fulling-Rindler vacuum in flat spacetime.A similar relation is valid for the state of the field conformal to the Minkowski vacuum.By taking into account that for the latter state the renormalized VEV of the energy-momentum tensor vanishes, we see that T ν µ CM = T ν µ G , where CM stands for the state in the geometry with the metric ḡµν that is conformal to the Minkowski vacuum in flat spacetime.Hence, the relation (6.1) can be written as The trace anomaly comes from the part T ν µ CM of the energy-momentum tensor.In odd-dimensional spacetimes there is no trace anomaly and T ν µ CM = 0.As the first example we consider the conformal relation between the Rindler spacetime and static spacetime with constant negative curvature spatial sections.In order to see the relation between the metric tensors we introduce angular coordinates (θ 1 , θ 2 , . . ., θ D−2 , φ) in accordance with (see also [56]) x l = ρw l sinh r, l = 2, . . ., D, where n=1 sin θ n , and w D = sin φ D−2 n=1 sin θ n , with 0 ≤ θ l ≤ π, l = 1, . . ., D − 2, 0 ≤ φ ≤ 2π.The new time and radial coordinates, η and r, are defined by the relations and the parameter α determines the curvature radius of spatial sections.In terms of the new coordinates, the Rindler line element is presented as ds 2 R = (ρ/α) 2 ds 2 NC , where is the line element for a static spacetime with negative constant curvature spatial foliation (static open universe).Here, dΩ 2 D−1 is the line element on a (D − 1)-dimensional unit sphere.Hence, for the example under consideration Ω(x) = α/ρ and g/ḡ = (ρ/α) D+1 .Now, from (6.1) we conclude that the mean energy-momentum tensor for a massless Dirac field in the spacetime given by the line element (6.4) is given by Geometry is homogeneous and it is natural that the VEV does not depend on the spacetime point.In Ref. [57] it has been shown that (see also Ref. [1]) for a conformally coupled scalar field the renormalized VEV T ν µ NC is zero in spatial dimension D = 3.A similar feature was assumed for spin 1/2 and 1 fields.
The static spacetime with negative constant curvature spatial sections is conformally related to the Milne universe.The latter is flat and the corresponding line element reads ds 2  Milne = e 2η/α ds 2 NC = αe η/α /ρ 2 ds 2 R .For the synchronous time coordinate in the Milne universe one has t = αe η/α and the line element is written as The counterpart of the Fulling-Rindler vacuum in the Milne universe is the conformal vacuum.The latter is different from the adiabatic vacuum being the counterpart of the inertial vacuum in the Minkowski spacetime.Hence, for a massless Dirac field in the conformal vacuum of the Milne universe the expectation value of the energy-momentum tensor reads The corresponding result for a conformally coupled massless scalar field is presented in [56].
The third example of the conformal relation we are going to study is the dS spacetime foliated by negative curvature spatial sections.The corresponding line element reads Introducing the conformal time η as e η/α = tanh(t/2α), the conformal relations are seen.For this example g/ḡ = [ρ sinh (|η|/α) /α] D+1 .In dS spacetime the conformal counterpart of the Minkowski vacuum is the Bunch-Davies vacuum.The latter is maximally symmetric and the corresponding VEV has the structure T ν µ CM = T ν µ BD = const • δ ν µ (the conformal relation between the dS and Rindler spacetimes has been recently used in [58] to clarify the entanglement structure in the Bunch-Davies vacuum state).For even values of D the trace anomaly is absent and T ν µ BD = 0.For dS spacetime, the conformal counterpart of the FR vacuum is the hyperbolic vacuum.Denoting the respective VEV by T ν µ H , in terms of the synchronous time coordinate, from (6.2) we get The corresponding relation for a massless scalar field has been recently discussed in [56] (for boundary induced effects in the hyperbolic vacuum of dS spacetime for a massive scalar field with general curvature coupling see [59]).We can also find the expectation value of the energy-momentum tensor for a massless Dirac field in the static vacuum of dS spacetime.The latter is the vacuum state for an observer with fixed radial coordinate r s and angular coordinates (θ 1 , θ 2 , . . ., θ D−2 , φ).The respective line element has the form Introducing a new radial coordinate r in accordance with r s /α = tanh (r/α), the following conformal relations are obtained: where, as it is seen from ( 6.3), we have Now, from (6.2) one obtains where T ν µ St is the VEV for the static vacuum in dS spacetime.For D = 3 one has A (ferm) D = 17/(1920π 2 ) and (6.14) is reduced to the result given in [54].Note that in 4-dimensional dS spacetine and for the Bunch-Davies vacuum T ν µ BD = 11δ ν µ / 1920π 2 α 4 .

Conclusion
We have studied the local properties of the Fulling-Rindler vacuum for a massive Dirac field in general number of spatial dimensions.As characteristics of the properties the FC and the expectation value of the energy-momentum tensor are considered.For evaluating the corresponding VEVs the mode summation technique is employed with combination of the point-splitting regularization procedure.As the first step we have generalized the result of Ref. [15] for the complete set of fermionic normal modes in Rindler spacetime for a massive field in (D + 1)-dimensional spacetime.The positive and negative energy mode functions with respect to the Rindler Killing vector ∂ τ are given by expressions (2.24) and (2.30) with the normalization coefficient (2.29).The constant spinors χ (j) η (k), j = +, −, obey the relations (2.21)-(2.23),with (+) in superscript replace by (j), and no specific form of the Dirac matrices has been used in deriving the modes.Both the FC and the VEV of the energy-momentum tensor are presented as coincidence limits of two-point functions and those limits are divergent.The Rindler spacetime is flat and the renormalization is reduced to the subtraction of the corresponding quantities for the Minkowski vacuum, transformed to the Rindler coordinate system.In Appendix B we have presented the Minkowskian VEVs in the form adapted for the subtraction.
The renormalized FC is given by the expression (3.11).An alternative representation is provided by Eq. (5.4).The FC vanishes for massless fields and is negative for massive fields.The dependence of the combination m −D ψψ FR on the mass of the field and on the acceleration 1/ρ appears in the form of a single argument mρ.For large accelerations corresponding to small values of that argument the FC behaves as m/ρ D−1 .This limit corresponds to spacetime points near the Rindler horizon.In the opposite limit of small accelerations the FC is suppressed by the factor (mρ) − D+3 2 e −2mρ .Another important local characteristic of the vacuum state is the expectation value of the energymomentum tensor.It is diagonal for the Fulling-Rindler vacuum and the respective components are given by (4.4) with the functions (4.5) for separate components.Simpler formulas are obtained by using the integral representations of the products for the modified Bessel functions, given in Appendix B. The vacuum energy density is expressed as (5.6) (or equivalently by (5.9)) and is negative.The vacuum stresses are isotropic and are given by (5.11).The respective vacuum pressures are negative.As expected, the VEV of the energy-momentum tensor obeys the covariant conservation equation and the trace relation with the FC.For a massless field the general expression is simplified to the form (4.9) with the coefficient A (ferm) D defined by (4.10).The massless Dirac field is conformally invariant in all spatial dimensions and the respective energy-momentum tensor is traceless.The background geometry is flat and the trace anomaly is absent.Near the Rindler horizon, corresponding to large accelerations, the effect of the mass on the vacuum energy-momentum tensor is weak and the leading term in the asymptotic expansion coincides withe the VEV for a massless field with behavior 1/ρ D+1 .For small accelerations the decay of the vacuum energy-momentum tensor for massive fields is exponential.The suppression factor is the same as that for the FC.It should be noted that the isotropy of the stresses for the Fulling-Rindler vacuum does not follow from the symmetry of the problem.In the case of a scalar field, in general, the stress T 1  1 FR differs from those along the other directions, T l l FR , l = 2, . . ., D. The equality takes place only for a conformally coupled massless field.
One can interpret the tensor − T ν µ FR as the energy-momentum tensor for the Minkowski vacuum seen by a uniformly accelerating observer.For a massless fermionic field the form of that tensor follows from (4.9) and it describes a radiation type source with the equation of state P = ε/D, where (no summation over l) P = T l l FR , l = 1, . . ., D, is the pressure and ε = − T 0 0 FR is the energy density.Both these quantities are positive.By taking into account that the energy of the fermionic mode with a given ω, measured by an observer with proper acceleration 1/ρ, is given by ε ρ = ω/ρ, we can interpret the factor [e 2πρερ − (−1) D ] −1 in the integrand of (4.10) as an indication for the thermal nature of the energy distribution with the Unruh temperature T U = 1/(2πρ).An interesting point to be mentioned is that the thermal factor is of the Fermi-Dirac type only in odd number of spatial dimensions D. In even dimensional spaces the Bose-Einstein type factor appears.For a conformally coupled massless scalar field one has an inverted situation: the thermal factor is of the Bose-Einstein type for odd D and of the Fermi-Dirac type in the case of even D. A similar feature was observed in the response of uniformly accelerating particle detectors.
In accordance of the equivalence principle, the local effects of the gravitational filed on the properties of a physical system can be modeled by considering the system in non-inertial reference frame with an appropriate acceleration.In this context, the results presented above shed light on the influence of the gravitational field on the local properties of the fermionic vacuum in different numbers of spatial dimensions.The Rindler spacetime also approximates the near-horizon geometry of a large class of black holes and the VEVs described above give the leading terms in the respective expansions over the distance from the horizon.Other applications are based on the conformal relations of the Rindler spacetime with curved geometries in gravitational physics and cosmology.For a massless field the fermionic expectation values in those geometries are obtained by using the conformal relation between the VEVs.Some examples are presented in the discussion above.

B Hadamard function and FC for the Minkowski vacuum
In order to obtain a finite result for the FC (3.9) we subtract the corresponding quantity for the Minkowski vacuum.The background geometry is flat and the divergences are the same as those in the Minkowski vacuum and this subtraction is sufficient.Here we obtain a representation for the Minkowskian FC adapted for the subtraction procedure.The Minkowskian line element is presented as ds 2 M = dt 2 −dz 2 , where z = (x 1 , x).The corresponding Dirac equation reads iγ (µ) ∂ µ − m ψ M (x) = 0. Similar to the case of the Rindler geometry, we present the field in the form where the new function obeys the equation η µν ∂ µ ∂ ν + m 2 ϕ M (x) = 0 with the Minkowski metric tensor η µν = diag(1, −1, . . ., −1).The solution for the latter equation is expressed as ϕ . Now, from the relation (B.1) for the normalized positive and negative energy fermionic modes, realizing the Minkowski vacuum, we get where the constant spinors χ (±) Mη (K) are the eigenspinors of the projection operator with the properties First we consider the positive frequency Wightman function S + M (x, x ′ ) = ψ(x) ψ(x ′ ) M .Substituting the mode functions (B.2) in the corresponding mode sum formula and using the relation (B.4) for the summation over η, we get The respective expression for the negative frequency Wightman function where T stands for the transponation.The formula (B.5) gives the Wightman function for the inertial vacuum in the Minkowskian coordinates.Though the FC ψψ M is a scalar with respect to the transformation to the Rindler coordinates, that is not the case for two-point functions and also for the corresponding traces.The Wightman function transforms as a product of two spinors ψ(x) and ψ(x ′ ).The coordinate transformation (t, x 1 , x) → (τ, ρ, x), given by (2.4), induces a local Lorentz transformation L(x) (for a general discussion in curved spacetime with an arbitrary number of spatial dimensions see, e.g., [39]).Under the latter transformation the spinor transforms as ψ MR = U (L(x))ψ M , where the transformation matrix reads By taking into account that ψMR = ψM U −1 (L(x)), for the Wightman function in the Rindler coordinates one obtains S + MR (x, x ′ ) = U (L(x))S + M (x, x ′ )U −1 (L(x ′ )).Combining this with (B.5), for the trace of the Wightman functions we get For the trace of the Hadamard function S MR (x, x ′ ) = j=+,− jS j MR (x, x ′ ), required in the evaluation of the FC, this gives Tr(S where λ is given by As the next step, we express the Minkowskian coordinates (t, x 1 ) in terms of the Rindler coordinates (τ, ρ) for the replacement where σ(x, x ′ ) = (∆x 1 ) 2 + |∆x| 2 − (∆t) 2 determines the geodesic distance between the points x and x ′ and the function f ν (z) is defined by (5.3).The formal expression of the FC for the Minkowski vacuum is obtained taking the limit x ′ → x in (B.11).

C Representation for the trace of the Hadamard function
In this appendix we will provide representations for the Wightman and Hadamard functions in terms of the respective functions for the Minkowski vacuum.A similar procedure for the Feynman function of a scalar field has been presented in [61].The starting point for the transformation of Tr S (1) (x, x ′ ) will be the formula (3.9).First of all we note that the Hadamard function can be written in terms of the positive and negative energy Wightman functions as S (1) (x, x ′ ) = j=+,− jS j (x, x ′ ), where the respective matrix elements are defined by In this form and for real ∆τ the change of the order of integrations is not allowed.
To make the change admissible, we will temporarily assume that jIm ∆τ < −π.First integrating over ω one gets Recall that this representation is obtained for Im ∆τ < −π for j = + and Im ∆τ > π for j = −.An analytical continuation is required in the limit Im ∆τ → 0. The integrand has poles at u = ∆τ ± iπ and they are located below the integration contour (real axis) for j = + and above the contour for j = −.These locations dictate the deformation of the integration contour in the limit Im ∆τ → 0. The deformed contours are depicted in Figure 5 for j = + (upper contour) and for j = − (lower contour).The integrals along the circles near the points u = ∆τ ± π are expressed in terms of the respective residues and we get Tr S j (x, x ′ ) = j 2m D N (2π) The coincidence limit of the second term in the right-hand side of this formula, multiplied by −1/2, gives the renormalized fermionic condensate in the form (5.4).A similar relation between the Feynman Green functions of the Fulling-Rindler and Minkowski vacua for a scalar field has been discussed in [61,62].In particular, in Ref. [62] it was shown that the relation is a consequence of the fact that the Minkowskian Green function is a periodic sum of the corresponding function for the Fulling-Rindler vacuum (the thermal nature of the Minkowski vacuum for Rindler observers).The generalization of the relation between the Hadamard functions of scalar fields in locally Minkowski and Rindler spacetimes, with a part of spatial dimensions compactified to a torus, is given in [37].

9 )
is expressed in terms of the products of the Euler gamma function and the Riemann zeta function ζ(z).For example, A

Figure 3 :
Figure 3: The vacuum energy density in units of 1/ρ D+1 as a function of mρ for separate values of spatial dimension (the numbers near the curves).

Figure 4 :
Figure 4: The vacuum stress T 1 1 FR in units of 1/ρ D+1 as a function of mρ in different spatial dimensions.