In recent work by M.H.Lynch E.Cohen Y.Hadad and I.Kaminer LCHK a modified model of the UnruhDeWitt quantum detector coupled to a 4vector current has been proposed to examine the radiation emitted by high energy positrons channeled into silicon crystal samples. Inspired by their ideas we analyze theoretical aspects of such a model its internal consistency and ignore all questions related to experiments. The twopotential correlation functions for the quantized electromagnetic field in a vacuum state and the corresponding detector radiation power DRP considered in proper time formalism are used as the basis for investigating the radiation observed at an accelerating point detector. The quantum detector is assumed to be moving through an electromagnetic vacuum along a classical hyperbolic trajectory with a constant proper acceleration. The DRP is obtained for three possible cases. First the DRP is found in a Lorentzinvariant manner. It contains both transverse and nonphysical longitudinal polarization modes and is a divergent quantity. Second the radiation power holds only physical transverse modes but it is nonrelativistic and also depends on the detector proper time which contradicts the fact that there is no preferred time for hyperbolic detector motion. Third in the case considered by LCHK for zero detector proper time when its velocity in the lab inertial system is zero the radiation power with transverse modes shows some signs of thermality which could be associated with a detector acceleration but different from the BoseEinstein statistics expected for the photon field. If the detector energy gap is zero then in complete contradiction with what LCHK claim there is no radiation and no thermalized Larmor formula. Based on our analysis we do not believe that the LCHKs model can be used to support the idea about thermal effects of uniform acceleration.
['General Relativity and Quantum Cosmology', 'Physics - Atomic Physics']
Some Theoretical Aspects of Observation of Acceleration Induced Thermality
{'Abstract': "In recent work by M.H.Lynch, E.Cohen, Y.Hadad and I.Kaminer (LCHK), a modified model of the Unruh-DeWitt two-level quantum detector, coupled to a semi-classical 4-vector current, has been proposed to examine the radiation emitted by high energy positrons channeled into silicon crystal samples. Inspired by their ideas, we analyze theoretical aspects of such a model, its internal consistency, and ignore all questions related to experiments. The two-potential correlation functions for the quantized electromagnetic field in a vacuum state and the corresponding detector radiation power (DRP), considered in proper time formalism, are used as the basis for investigating the radiation observed at an accelerating point detector. The quantum detector is assumed to be moving through an electromagnetic vacuum along a classical hyperbolic trajectory with a constant proper acceleration. The DRP is obtained for three possible cases. First, the DRP is found in a Lorentz-invariant manner. It contains both transverse and non-physical longitudinal polarization modes and is a divergent quantity. Second, the radiation power holds only physical transverse modes but it is non-relativistic and also depends on the detector proper time, which contradicts the fact that there is no preferred time for hyperbolic detector motion. Third, in the particular case considered by LCHK, for zero detector proper time when its velocity in the lab inertial system is zero, the radiation power with transverse modes shows some signs of thermality which could be associated with a detector acceleration but different from the Bose-Einstein statistics expected for the photon field. If the detector energy gap is zero then, in complete contradiction with what LCHK claim, there is no radiation and no 'thermalized Larmor formula'. Based on our analysis we do not believe that the LCHK's model can be used to support the idea about thermal effects of uniform acceleration.", '1 Introduction': "The theoretical prediction of a warmed vacuum observed by an accelerated observer ([1],[2],[3]) was a by-product of the extension of quantum theory to curved spaces. P.W. Milonni found this result  \nso remarkable that much later, in 1994, he wrote in his book ([4]p 64): 'It took half a century after the birth of the quantum theory of radiation for the thermal effect of uniform acceleration to be discovered.' He provides an elegant description of the Unruh-Davis effect in a scalar field done in terms of the correlation function. The concept of an accelerating observer is given there in general terms, specifying only its classical trajectory and without indicating its internal structure and its interaction with the vacuum. However, in any experimental situation you cannot avoid this issue. This situation is clearly formulated by B.S. DeWitt ([5]): 'These and many examples discovered in the last few years have brought about major changes in our ways of thinking about particles and in our ways of defining them and their associated vacuum states. As has happened before in relativity theory, and in the quantum theory, one has had to fall back on operational definitions 1 . Here, for example, one must ask: How would a given particle detector respond in a given situation.' Today, when there is a great interest in an experimental proof of the effect, this remark becomes relevant again. What theoretical implementation and what kind of a detector model can be used to explain experiments?  \nRecently, Cozzella et al.([6]) made a proposal for observing the Unruh-Davis effect, which outlined a method of measuring the Fulling-Davis-Unruh (FDU) temperature directly from a data set. There is an opinion ([7],[8]) that experimental evidence already exists. This opinion is based on both experimental data and a theoretical model for their interpretation. An essential part of their analysis is a slightly changed Unruh-DeWitt detector model. The latter was proposed for an observation of a scalar field.  \nFor the case of an electromagnetic field, LCHK use the Unruh-DeWitt detector, coupled to the semi-classical vector current:  \nˆ j µ = u µ ˆ q ( τ ) δ 3 ( x -x tr ) (1)  \nand the current interaction for QED  \nˆ S I = ∫ d 4 x ˆ j µ ( x ) ˆ A µ ( x ) , (2)  \nwhere u µ is the 4-velocity of the detector, the monopole operator ˆ q ( τ ) is Heisenberg evolved via ˆ q ( τ ) = e i ˆ Hτ ˆ q (0) e -i ˆ Hτ , with ˆ q (0) defined as ˆ q (0) | E i > = | E f > and with E i and E f the initial and final energy of a two energy state detector moving along the trajectory, x tr ( τ ).  \nInspired by LCHK's ideas, we analyze theoretical aspects of such a model, its internal consistency,  \nand ignore all questions related to experiments. For this purpose we slightly modify their model and explicitly use proper time formalism. Our model is described in details in Section 2.  \nThe main tool of our investigation is a calculation of a two-potential correlation function of the potentials at the locations of the detector on a hyperbolic trajectory at two proper times, τ 1 and τ 2 , for the vacuum state of an electromagnetic field determined at the lab inertial system in the Minkowski space-time. This is done in a relativistic invariant manner in Subsection 3.1. Calculated this way, the correlation function can be split, as shown in Subsection 3.2, into two parts corresponding respectively to transverse polarization and scalar/longitudinal modes. In Subsection 3.3, the transverse part is obtained by explicitly using the supplementary Lorentz condition.  \nTo be an observable in a real experiment, the detector radiation power (DRP) is a more convenient variable than just a correlation function. In Section 4 we consider 3 cases. First, relativistic DRP is discussed in Subsection 4.1. Second, DPR consisting of transverse polarization modes is obtained and discussed in Subsection 4.2. Third, derivation and discussion of LCHK's DRP can be found in Subsection 4.3.  \nThe author intends to focus only on theoretical validation of LCHK' model, leaving aside other work in this area. As a result, the bibliography contains only references which can be associated with LCHK's work.", '2 Detector Model': "A point-like quantum detector, which can be in one of two quantum states, is uniformly accelerated along a classical hyperbolic trajectory in a quantized electromagnetic field which is in a vacuum state. Our goal is to find the probability of detector transition from one of its states to the other and its radiation power, induced by the interaction of the detector with the vacuum and accompanied by all possible one-photon states of radiation.  \nIn our analysis we used a fixed inertial frame I M in Minkowski 4-space and a set of inertial frames I τ [10],[11] defined by the condition that the detector is instantaneously at rest at I τ at the proper time τ measured by a clock at the detector position. In each I τ -frame the detector labeled by ( t τ , x τ , y τ , z τ ) = ( τ, x τ , 0 , 0) has the same acceleration, d 2 x τ /dt 2 τ = a = const .  \nIts trajectory in the Minkowski space is 2  \nx ( t ) = ( 1 a )(1 + a 2 t 2 ) 1 / 2 (3)  \nor in terms of proper time τ  \nt ( τ ) = 1 a sinh( aτ ) , x ( τ ) = 1 a cosh( aτ ) , y ( τ ) = z ( τ ) = 0 , v ( τ ) = tanh( aτ ) , d 2 x/dt 2 = γ -3 τ a, (4)  \nwhere γ τ is the Lorentz boost of a local Lorentz transformation between I M and I τ  \nγ ( τ ) = 1 (1 -v 2 ( τ )) 1 / 2 = cosh( aτ ) , (5)  \nThe Unruh-DeWitt detector is coupled to the vector current operator, according to our definition, in I τ frame 3 :  \nˆ j τ,µ ( τ, x τ , 0 , 0) = u τ,µ ( τ, 0 , 0 , 0)ˆ q ( τ ) , µ = 0 , 1 , 2 , 3 , u τ,µ ≡ ( u τ 0 , u τ 1 , u τ 2 , u τ 3 ) = (1 , 0 , 0 , 0) (6)  \nHere u τ,µ is 4-velocity of the detector in I τ .  \nThe Hamiltonian of the detector's interaction with an electromagnetic field in I τ is  \nˆ H i ( τ ) = ˆ j τ,µ ( τ ) ˆ A τ,µ ( τ, 0) = ˆ j τ, 0 ( τ ) ˆ A τ, 0 ( τ, 0) = ˆ q ( τ ) ˆ A τ, 0 ( τ, 0) . (7)  \nIt depends on the zeroth component of the 4-potential, ˆ A τ, 0 ( τ, 0), defined at the location of the detector in I τ at the proper time τ . The amplitude of the transition from | E i , 0 > to | E f , ⃗ kλ > in the first order of perturbation is 4  \nA E f , ⃗ k,λ ; E i , 0 = -i < E f , ⃗ kλ | ∫ ∞ -∞ dτ ˆ q ( τ ) ˆ A τ, 0 ( τ, 0) | E i , 0 > = (8)  \n= -i < E f | ˆ q (0) | E i > ∫ ∞ -∞ dτ e i ( E f -E i ) τ < ⃗ kλ | ˆ A τ, 0 ( τ, 0) | 0 > . (9)  \nHere | E i , 0 > is a detector state with energy E i , the electromagnetic field in a vacuum state, and | E f , ⃗ kλ > is the state with energy E f and the electromagnetic field in a one-photon state with momentum ⃗ k and polarization mode λ . The states | E i > , | E F > and the states | ⃗ k, λ > , | 0 > are defined  \nin different reference systems, I τ and I M respectively.  \nThe total probability of the detector transition from the state with energy E i to the state with energy E f , after summation over final one-photon states with all possible ⃗ k and polarizations λ , is  \n| A | 2 ≡ q 2 ∫ dτ 1 dτ 2 exp[ -i ∆ E ( τ 2 -τ 1 )] ∫ d ⃗ k λ =3 ∑ λ =0 < 0 | ( ˆ A τ, 0 ) † ( τ 2 , 0) | ⃗ kλ >< ⃗ kλ | ˆ A τ, 0 ( τ 1 , 0) | 0 > ≡ ≡ q 2 ∫ dτ 1 dτ 2 exp[ -i ∆ E ( τ 2 -τ 1 )] < 0 | ( ˆ A τ, 0 ) † ( τ 2 , 0) ˆ A τ, 0 ( τ 1 , 0) | 0 >  \n(10)  \nor 5 in new variables  \n| A | 2 = q 2 ∫ dσdτ exp[ -i ∆ E τ ] × < 0 | ( ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 >, (11)  \nwhere  \nq 2 = | < E f | ˆ q (0) | E i > | 2 , ∆ E = E f -E i , σ = ( τ 2 + τ 1 ) / 2 , τ = τ 2 -τ 1 (12)  \nSummation over λ in the previous expression should include both transverse modes, with λ = 1 , 2, and scalar/longitudinal ones, with λ = 0 , 3. Separation of them is possible in a fixed reference frame only but in proper time formalism a reference frame can not be fixed. We will see in the next section that calculation of the correlation function involves three reference systems, I τ 1 , I τ 2 and I M .", '3.1 The Correlation Function in Terms of Variables Determined in the Minkowski Space Time.': 'The vacuum state, | 0 > , and one-photon states, | ⃗ kλ > , are defined in I M but ˆ A τ, 0 ( σ + τ/ 2 , 0) and ˆ A τ, 0 ( σ -τ/ 2 , 0) are defined at detector locations in two different instantaneous inertial reference frames. Therefore we have to represent ˆ A τ, 0 ( σ ± τ/ 2 , 0), the scalar component of 4-potential, in terms of 4potential operator components in I M using Lorentz transformations with boosts γ σ + τ/ 2 and γ σ -τ/ 2 (', '5) respectively:': 'ˆ A τ, 0 ( σ ± τ/ 2 , 0) = ˆ A 0 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0] γ σ ± τ/ 2 -ˆ A 1 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0]( v σ ± τ/ 2 /c ) γ σ ± τ/ 2 = ˆ A 0 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0] cosh[ a/c ( σ ± τ/ 2)] -ˆ A 1 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0] sinh[ a/c ( σ ± τ/ 2)] (13)  \nUsing the expansion of A µ ( t, x, 0 , 0) in plain waves in I M (60), properties of annihilation/creation operators (61) and trajectory equations (4) this correlation function can be reduced to the form:  \nˆ † ˆ  \n< 0 | A τ, 0 ) ( σ + τ/ 2 , 0) A τ, 0 ( σ -τ/ 2 , 0) | 0 > = -∫ d 3 k (2 π ) 3 1 2 ω k exp {-i [ ω k ( t σ + τ/ 2 -t σ -τ/ 2 ) -k x ( x σ + τ/ 2 -x σ -τ/ 2 )] } × ×{ +cosh( a ( σ + τ/ 2)) cosh( a ( σ -τ/ 2)) 3 ∑ λ =0 [ e 0 ( ⃗ k, λ )] 2 g λλ + -cosh( a ( σ + τ/ 2)) sinh( a ( σ -τ/ 2)) 3 ∑ λ =0 [ e 0 ( ⃗ k, λ )][ e 1 ( ⃗ k, λ )] g λλ --sinh( a ( σ + τ/ 2)) cosh( a ( σ -τ/ 2)) 3 ∑ λ =0 [ e 1 ( ⃗ k, λ )][ e 0 ( ⃗ k, λ )] g λλ + +sinh( a ( σ + τ/ 2)) sinh( a ( σ -τ/ 2)) 3 ∑ λ =0 [ e 1 ( ⃗ k, λ )] 2 g λλ } (14)  \nFor our further analysis, it is useful to consider the contribution to the correlation function from scalar/longitudinal and transverse polarization modes of the detector radiation. It is done in the next subsection.', '3.2 Scalar/Longitudinal and Transverse Polarization Modes': "With the help of the properties of polarization vectors (62) and hyperbolic function (63), provided in Appendix A, the correlation function can be split into two parts corresponding to scalar/longitudinal (sl) and transverse (tr) polarization modes  \n< 0 | ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 > = -∫ d 3 k (2 π ) 3 1 2 ω k exp {-i [ ω k ( t σ + τ/ 2 -t σ -τ/ 2 ) -k x ( x σ + τ/ 2 -x σ -τ/ 2 )] } × { () sl +() tr } , (15)  \nwhere  \n() sl ≡ [ 1 2 cosh(2 aσ ) + 1 2 cosh(2 aτ ) ] [ e 0 ( ⃗ k, λ = 0)] 2 -[ 1 2 cosh(2 aσ ) -1 2 cosh(2 aτ ) ] [ e 1 ( ⃗ k, λ = 3)] 2 = = 1 2 cosh(2 aσ ) sin 2 θ + 1 2 cosh( aτ )(1 + cos 2 θ ) , (16)  \n() tr ≡ [ 1 2 cosh(2 aσ ) -1 2 cosh(2 aτ ) ] ([ e 1 ( ⃗ k, λ = 1)] 2 +[ e 1 ( ⃗ k, λ = 2)] 2 ) = = -1 2 cosh(2 aσ ) sin 2 θ + 1 2 cosh( aτ ) sin 2 θ (17)  \nand θ is an angle between the wave vector ⃗ k and the detector motion direction along axis x.  \nObviously the splitting of the correlation function into sl- and tr-parts is not Lorentz invariant and depends on σ . The σ -term in the correlation function contains contributions from both scalar/longitudinal and transverse modes, which are mutually compensated in the sum of () sl and () tr  \n() sl +() tr = cosh( aτ ) (18)  \nand  \n< 0 | ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 > = -∫ d 3 k (2 π ) 3 1 2 ω k exp {-i [ ω k ( t σ + τ/ 2 -t σ -τ/ 2 ) -k x ( x σ + τ/ 2 -x σ -τ/ 2 )] } × cosh( aτ ) (19)  \nThe factor cosh( a c ) τ does not depend on σ . We should expect that the correlation function does not depend on σ as well because there is no preferred time for hyperbolic motion. The independence on σ can be exhibited explicitly by changing the variables of integration ⃗ k to ⃗ k ' , where  \nω = ω ' cosh( aσ ) + k ' x sinh( aσ ) , k x = k ' x cosh( aσ ) + ω ' sinh( aσ ) , k y = k ' y , k z = k ' z , ∫ d ⃗ k (2 π ) 3 1 ω = ∫ d ⃗ k ' (2 π ) 3 1 ω ' , (20)  \nwhich corresponds exactly to a Lorentz transformation from the unprimed laboratory frame I M over to the primed inertial frame in which the accelerating detector is instantaneously at rest at proper  \ntime σ . Then we obtain  \n< 0 | ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 > ≡ < 0 | ˆ A τ, 0 ) † ( τ/ 2 , 0) ˆ A τ, 0 ( τ/ 2 , 0) | 0 > = = -∫ d 3 k ' (2 π ) 3 1 2 ω ' exp {-i 2 a ω ' sinh( a 2 τ ) } cosh( aτ ) . (21)  \nThis result is obtained in a relativistically invariant manner but it contains nonphysical , scalar/longitudinal, polarization modes. Now we will find the correlation function without unphysical modes.", '3.3 Supplementary Condition': "Until now we considered all polarization modes of the quantized electromagnetic field on an equal basis even though nonphysical scalar and longitudinal modes should be excluded in the quantized field. Now we explicitly introduce the supplementary condition to exclude nonphysical modes. For this purpose we will use the gauge transformation [15](9.41) in Gupta-Bleyer formalism  \na † µ ( ⃗ k ) → ˜ a † µ ( ⃗ k ) = a † µ ( ⃗ k ) -k µ k 0 a † 0 ( ⃗ k ) , a µ ( ⃗ k ) → ˜ a µ ( ⃗ k ) = a µ ( ⃗ k ) -k µ k 0 a 0 ( ⃗ k ) (22)  \na µ ( ⃗ k ) ≡ 3 ∑ λ =0 a ( ⃗ k, λ ) e µ ( ⃗ k, λ ) , a † µ ( ⃗ k ) ≡ 3 ∑ λ =0 a † ( ⃗ k, λ ) e µ ( ⃗ k, λ ) (23)  \nof the 4-vector potential operator defined in I M  \nˆ A µ ( t, ⃗x ) = ∫ d 3 k (2 π ) 3 1 2 ω k [ a µ ( ⃗ k ) e -kx + a † µ ( ⃗ k ) e kx ] (24)  \nto  \n˜ ˆ A µ ( t, ⃗x ) = ∫ d 3 k (2 π ) 3 1 2 ω k [˜ a µ ( ⃗ k ) e -kx +˜ a † µ ( ⃗ k ) e kx ] . (25)  \nThen, after this gauge transformation performed in I M , instead of (13) and (14) we have respectively  \nˆ A τ, 0 ( σ ± τ/ 2 , 0) = ˜ ˆ A 0 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0] γ σ ± τ/ 2 -˜ ˆ A 1 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0]( v σ ± τ/ 2 /c ) γ σ ± τ/ 2 = ˜ ˆ A 0 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0] cosh[ a/c ( σ ± τ/ 2)] -˜ ˆ A 1 [ t ( σ ± τ/ 2) , x ( σ ± τ/ 2) , 0 , 0] sinh[ a/c ( σ ± τ/ 2)]  \n(26)  \nand  \n< 0 | ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 > = ∫ d 3 k (2 π ) 3 1 2 k 0 × d 3 k ' (2 π ) 3 1 2 k ' 0 × × exp {-i [ k ' 0 t σ + τ/ 2 -k ' 1 x σ + τ/ 2 ] + i [ k 0 t σ -τ/ 2 -k 1 x σ -τ/ 2 ] } × ×{ +cosh( a ( σ + τ/ 2)) cosh( a ( σ -τ/ 2)) < 0 | ˜ a 0 ( ⃗ k ' )˜ a † 0 ( ⃗ k ) | 0 > --cosh( a ( σ + τ/ 2)) sinh( a ( σ -τ/ 2)) < 0 | ˜ a 0 ( ⃗ k ' )˜ a † 1 ( ⃗ k ) | 0 > --sinh( a ( σ + τ/ 2)) cosh( a ( σ -τ/ 2)) < 0 | ˜ a 1 ( ⃗ k ' )˜ a † 0 ( ⃗ k ) | 0 > + +sinh( a ( σ + τ/ 2)) sinh( a ( σ -τ/ 1)) < 0 | ˜ a 1 ( ⃗ k ' )˜ a † 1 ( ⃗ k ) | 0 > . } (27)  \nTaking into account (22) and (61) we have  \n[ a µ ( ⃗ k ) , a † ν ( ⃗ k ' )] = -(2 π ) 3 2 k 0 g µν δ 3 ( ⃗ k -⃗ k ' ) (28)  \nand  \n< 0 | ˜ a 0 ( ⃗ k ' )˜ a † 0 ( ⃗ k ) | 0 > = < 0 | ˜ a 0 ( ⃗ k ' )˜ a † 1 ( ⃗ k ) | 0 > = < 0 | ˜ a 1 ( ⃗ k ' )˜ a † 0 ( ⃗ k ) | 0 > = 0 , < 0 | ˜ a 1 ( ⃗ k ' )˜ a † 1 ( ⃗ k ) | 0 > = (2 π ) 3 2 k 0 (1 -k 2 1 k 2 0 ) δ 3 ( ⃗ k -⃗ k ' ) (29)  \nThen the correlation function takes the form  \n< 0 | ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 > = -1 (2 π ) 3 1 4 [2 cosh 2 ( aσ ) -1 -cosh( aτ )] ∫ d 3 k (2 π ) 3 1 k 0 (1 -k 2 1 k 2 0 ) × × exp {-i [ k 0 ( t σ + τ/ 2 -t σ -τ/ 2 ) -k 1 ( x σ + τ/ 2 -x σ -τ/ 2 )] } (30)  \nor  \n< 0 | ˆ A τ, 0 ) † ( σ + τ/ 2 , 0) ˆ A τ, 0 ( σ -τ/ 2 , 0) | 0 > = 1 4 [2 cosh 2 ( aσ ) -1 -cosh( aτ )] ∫ d 3 k (2 π ) 3 1 k 0 sin 2 θ × × exp {-i [ k 0 ( t σ + τ/ 2 -t σ -τ/ 2 ) -k 1 ( x σ + τ/ 2 -x σ -τ/ 2 )] } , (31)  \nwhere θ is an angle between ⃗ k and the direction of the detector motion.  \nIt is obvious that the correlation function is exactly the transverse part of the correlation function obtained in (15) and (17), as it is supposed to be.", '4 Detector Radition Power': 'Detector radiation power for correlation function (21) contains nonphysical polarization modes (that do not correspond to a real electromagnetic field) but it is relativistic and will be useful for interpretation of the results obtained under the supplementary condition when unphysical modes are annihilated. Therefore we consider both cases: for correlation functions (21) with unphysical states, and the function defined by (15) and (17) without unphysical states. Then we discuss the approach adapted by LCHK to arrive at their expression for DRP.', '4.1 Relativistic DRP': "From equation (21) and (11) we have  \nd | A | 2 dσ = -q 2 ∫ + ∞ -∞ dτ exp( -i ∆ Eτ ) ∫ d 3 k ' (2 π ) 3 1 2 ω ' × exp {-i 2 a ω ' sinh( a 2 τ ) } × cosh( aτ ) (32)  \nThis is the probability of the detector transition from | E i > to | E f > per unit proper time, σ , accompanied by one-photon radiation of all possible ⃗ k ' and polarizations λ = 0 , 1 , 2 , 3. The corresponding electromagnetic energy radiation, W, per unit proper time, involves an additional factor ω ' in the last integrand  \nS ≡ dW dσ = -q 2 ∫ + ∞ -∞ dτ exp( -i ∆ Eτ ) ∫ d 3 k ' (2 π ) 3 1 2 ω ' × ω ' exp {-i 2 a ω ' sinh( a 2 τ ) } × cosh( aτ ) (33)  \nand after integration over ⃗ k ' , it is  \nS = -i 4 π αa 3 ∫ + ∞ -∞ dτ e -i ∆ Eτ cosh( aτ ) sinh 3 ( aτ 2 ) ≡ -i 4 π αa 3 ( I 1 + I 2 ) , α = q 2 4 π , (34)  \nwhere  \nI 1 ≡ ∫ + ∞ -∞ dτ cos(∆ Eτ ) cosh( aτ ) sinh 3 ( aτ 2 ) , I 2 ≡ ∫ + ∞ -∞ dτ -i sin(∆ Eτ ) cosh( aτ ) sinh 3 ( aτ 2 ) (35)  \nThis expression is Lorentz invariant and does not depend on σ . It means that any observer which is at rest at the location of the detector measures the same radiation power. This good feature of the model is obscured by two facts.  \nThe first is that S is divergent. Indeed, the integral I 1 = 0 because its integrand is an odd function but I 2 is divergent (Appendix D).  \nThe second fact is that we have not taken into consideration the supplementary condition to get rid of unphysical polarization modes. This will be done in the next section.", '4.2 Transverse DRP': "In this section we consider the correlation function obtained with the use of the supplementary condition. It contains only the transverse polarization modes.  \nFrom ( 11) and (15), (17) we have  \nd | A | 2 tr dσ = q 2 1 4(2 π ) 3 ∫ dτe -i ∆ Eτ [cosh(2 aσ ) -cosh( aτ )] ∫ d 3 k 1 ω k sin 2 θ × × exp {-i [ ω k ( t σ + τ/ 2 -t σ -τ/ 2 ) -k x ( x σ + τ/ 2 -x σ -τ/ 2 ) ] } (36) ω k = k 0  \nand, after multiplication of the second integrand by ω k , the detector radiation power due to transverse polarization modes is  \nS tr ( σ ) ≡ dW tr dσ ( σ ) = q 2 1 4(2 π ) 3 ∫ dτe -i ∆ Eτ [cosh(2 aσ ) -cosh( aτ )] ∫ d 3 k sin 2 θ × × exp {-i [ ω k ( t σ + τ/ 2 -t σ -τ/ 2 ) -k x ( x σ + τ/ 2 -x σ -τ/ 2 ) ] } . (37)  \nVariables t σ ± τ/ 2 and x σ ± τ/ 2 are defined in (4).  \nObviously, this expression depends on proper time σ , and transformation (20) does not help to get rid of this dependence because of the factor sin 2 θ in its second integral. So observers which are at rest at different points of the detector in the hyperbolic trajectory observe completely different radiation powers, although there is no preferred time for such movement. So the case σ = 0 considered in [7] is a very special one and does not represent the properties of a uniformly accelerating detector. Nevertheless, it is useful to consider this case because all theoretical conclusions and experimental considerations done by LCHK are associated with this case.  \nFor σ = 0 we have from 6 (37) and (4)  \nS tr ( σ = 0) = q 2 1 4 ∫ + ∞ -∞ dτ exp( -i ∆ Eτ ) ∫ d 3 k (2 π ) 3 × exp {-i 2 a ω sinh( aτ ) } × sin 2 θ (1 -cosh aτ ) (38)  \nIt can be reduced to  \nS tr ( σ = 0) = 2 3 α i π a 2 2 ∫ + ∞ 0 du u -i ∆ E/a ( u -1) 2 u -1 / 2 . (1 -u ) 3 (39)  \nΓ = q 2 1 (2 π ) 3 1 4 ∫ dξ (1 -cosh aξ ) ∫ d 3 k ω sin 2 θe -i (∆ Eξ + ω ∆ t )  \nthe same way as it is done in [7] (S14)-(S20) This expression coincides with [7](S20) but despite this after integration we come to (Appendix C)  \nS tr ( σ = 0) = 1 3 αa 2 e 2 π ∆ E/a -1 e 2 π ∆ E/a +1 (40)  \nwhile LCHK, after integrating the same expression, obtained a different formula [7](S21)  \nS LCHK ( η = 0) = 2 3 αa 2 1 1 + e 2 π ∆ E/a . (41)  \n̸  \nThe LCHK result is obviously wrong when considering a boundary condition. From (39) we can see that its principle value integral is zero when ∆ E = 0, and S tr ( σ = 0 , ∆ E = 0) = 0. The expression (40) satisfies this condition, but the latter one does not: S LFCHK ( η = 0 , ∆ E = 0) = 0. More details are provided in Appendix B.  \nTherefore, LCHK's statement, based in particular on (41), that 'we discover a Larmor formula and power spectrum that are both thermalized by acceleration' is not supported by their calculations.  \nWe must add that expression (38) and folowing from it (39), which we have obtained in our approach, are obtained by LCHJK by 'ad hoc', after some miscalculations. Following their approach you can not arrive at that result. This leaves hope to preserve their main claim about acceleration induced thermality by making the necessary adjustments in their approach to the problem. We discuss this issue in detail in Section 4.3.", "4.3 LCHK's Derivation of Detector Radiation Expression": "LCHK start with the equation [7] (S7)  \nP = ∫ d 4 xd 4 x ' | < E f | ˆ j µ | E i > | 2 < 0 | ˆ A † µ ( x ' ) ˆ A ν ( x ) | 0 >, (42)  \nwhere integrals are taken over ( x ) = ( t, x, y, z ) and ( x ' ) = ( t ' , x ' , y ' , z ' ) in the laboratory inertial reference frame, and immediately move on to the next equation  \nΓ = dP dη = q 2 ∫ dξe -i ∆ Eξ U µν ( x ' , x ) G µν ( x ' , x ) , (43)  \n( ξ = τ ' -τ , η = ( τ ' + τ ) / 2, and τ , τ ' are proper times of the accelerating detector)  \nmaking some implicit assumptions. Discussion of these assumptions is the subject of this section. To  \nexpose this we need to go into some details of LCHK's calculations.  \nLCHK use the matrix element [7](S5)  \n| < E f | ˆ j µ ( x ) | E i > | 2 = q 2 u µ ( x ' ) u ν ( x ) δ 3 ( x ' -x ' tr ( τ ' )) δ 3 ( x -x tr ( τ )) e -i ∆ E ( τ ' -τ ) ⃗x tr ( τ ) = (0 , 0 , 1 a cosh( aτ )) , u µ = (cosh( aτ ) , 0 , 0 , sinh( aτ )) , (44)  \nwhich is a function of proper times τ and τ ' , in (42) to integrate it over ⃗x , ⃗x ' , t and t ' in the lab system. This creates some ambiguity. More uncertainty also comes from semi-classical character of the model. The detector is endowed with two contradictory properties. It is both a quantum and a classical object. Its quantum properties are defined in a proper reference frame and presented by the factor e -i ∆ E ( τ ' -τ ) . Classical properties are defined by space-time coordinates ( t, x 1 , x 2 , x 3 ) in the lab system in which the quantum nature of the detector is ignored and the detector is considered as a point-like object. This approach is accepted in the literature, see for example [1], and we employ it to further explore the capabilities of the proposed LCHK model. To support their idea we have to represent the matrix element in a different form  \n| < E f | ˆ j µ ( x ) | E i > | 2 = q 2 u µ ( x ' ) u ν ( x ) δ 3 ( x ' -x ' tr ( t ' )) δ 3 ( x -x tr ( t )) e -i ∆ E ( τ ' -τ ) (45)  \n⃗x tr ( t ) = (0 , 0 , (1 /a )(1 + a 2 t 2 ) 1 / 2 ) , (46)  \nwhere all 'classical' quantities related to the detector motion in 4-space-time must be considered as functions of ( t, ⃗x ) rather than τ . Then after integration over ⃗x and ⃗x ' we come to the equation 7  \nP cor = ∫ dt ' dt q 2 u µ ( t ' , 0 , 0 , x ' tr, 3 ( t ' )) u ν ( t, 0 , 0 , x tr, 3 ( t )) e -i ∆ E ( τ ' -τ ) < 0 | ˆ A † µ ( t ' , 0 , 0 , x ' tr, 3 ( t ' )) ˆ A ν ( t, 0 , 0 , x tr, 3 ( t )) | 0 > (47)  \nwhere all variables are defined on trajectory (46) as functions of t. Particularly  \nu µ = ((1 -v 2 ) -1 / 2 , ⃗v (1 -v 2 ) -1 / 2 ) , ⃗v = (0 , 0 , at (1 + a 2 t 2 ) -1 / 2 ) (48)  \nIn this expression τ and τ ' should be considered as parameters associated with quantum character of the detector and which are not connected with a specific classical trajectory of the detector. Our next step is to connect both classical and quantum features of the model and consider ( t, t ' ) as functions  \nof ( τ, τ ' ). However, it is more convenient to work with proper time variables ( η, ξ ):  \nτ ' = η + ξ/ 2 , τ = η -ξ/ 2 , t ' = (1 /a ) sinh( a ( η + ξ/ 2)) , t = (1 /a ) sinh( a ( η -ξ/ 2)) . (49)  \nThen  \nx ' tr, 3 ( t ' ) = (1 /a ) sinh( a ( η + ξ/ 2)) , x tr, 3 ( t ) = (1 /a ) sinh( a ( η -ξ/ 2)) , u µ ( t ' , 0 , 0 , x ' tr, 3 ( t ' )) = (cosh( a ( η + ξ/ 2)) , 0 , 0 , sinh( a ( η + ξ/ 2))) u µ ( t, 0 , 0 , x tr, 3 ( t )) = (cosh( a ( η -ξ/ 2)) , 0 , 0 , sinh( a ( η -ξ/ 2))) (50)  \nThe Jacobian of a transformation from ( t, t ' ) to ( η -ξ/ 2 , η + ξ/ 2) is  \nJ ( t, t ' ; η, ξ ) = cosh( a ( η + ξ/ 2)) cosh( a ( η -ξ/ 2)) (51)  \nIn the Coulomb gauge adopted in [7], (S11), the zero component of the operator ˆ A µ is ignored 8 , and equation (47), taking into account (51), becomes  \nP cor = ∫ dηdξ q 2 cosh( a ( η + ξ/ 2)) cosh( a ( η -ξ/ 2)) sinh( a ( η + ξ/ 2)) sinh( a ( η -ξ/ 2)) × × e -i ∆ Eξ < 0 | ˆ A † 3 ( t ' , 0 , 0 , x ' tr, 3 ( η + ξ/ 2)) ˆ A 3 ( t, 0 , 0 , x tr, 3 ( η -ξ/ 2)) | 0 > (52)  \nor using the expansion of ˆ A 3 ( t, 0 , 0 , x 3 ( τ )) (60) in plane waves and properties of polarization vectors (62)  \nP cor = ∫ dηdξ q 2 cosh( a ( η + ξ/ 2)) cosh( a ( η -ξ/ 2)) sinh( a ( η + ξ/ 2)) sinh( a ( η -ξ/ 2)) e -i ∆ E ξ × × ∫ d 3 k (2 π ) 3 1 2 ω k e -iω k ( t ' -t )+ ik 3 ( x ' tr, 3 ( t ' ) -x tr, 3 ( t )) × sin 2 θ. (53)  \nwhere variables t, t ' , x ' tr, 3 ( t ' ), and x tr, 3 ( t ) are defined in (49) and 50) as functions of ( η + ξ/ 2) and ( η -ξ/ 2), and θ is an angle between wave vector ⃗ k and the direction of the detector motion along axis x 3 .  \nThis corrected expression is different from the corresponding expression, [7] (S13), by an additional factor, Jacobian J ( t, t ' ; η, ξ ). Because this step was skipped by LCHK, they lost this factor. Then  \nΓ cor = dP cor dη = q 2 1 (2 π ) 3 1 16 ∫ dξ [2 cosh 2 (2 aη ) -1 -cosh(2 aξ )] × × ∫ d 3 k ω k sin 2 θe -i ∆ E ξ -iω k ( t ' -t )+ ik 3 ( x ' 3 ( t ' ) -x 3 ( t )) . (54)  \nFollowing LCHK, we set η = 0, even though this assumption contradicts to the feature of the hyperbolic motion that it has no preferred time. Then  \nΓ cor ( η = 0) = q 2 1 (2 π ) 3 1 16 ∫ dξ [1 -cosh(2 aξ )] × ∫ d 3 k ω k sin 2 θe -i ∆ E ξ -iω k ( t ' -t ) = = 1 4 ( 2 3 α 1 2 π ) ∫ dξ (1 -cosh 2 aξ ) e -i ( δE ) ξ ∫ dωωe -i 2 ω a sinh( aξ/ 2) , α ≡ q 2 4 π (55)  \nand finally, weighting the frequency integral with an additional factor of frequency,  \nS cor = 1 4 ( 2 3 α 1 2 π ) ∫ dξ (1 -cosh 2 aξ ) e -i (∆ E ) ξ ∫ dωω 2 e -i 2 ω a sinh( aξ/ 2) , α ≡ q 2 4 π (56)  \nThis result is significantly different from [7](S16)  \nS LCHK = 2 3 α 1 2 π ∫ dξe -i ∆ Eξ (1 -cosh aξ ) ∫ dω ω 2 e -iω ∆ t , ∆ t = 2 a sinh( aξ/ 2) (57)  \nIt has cosh(2 aξ ) instead of cosh( aξ ) and an additional factor, 1 4 .  \nHaving integrated (56) we come to  \nS cor = S cor, 1 + S cor, 2 = ( 2 3 αa 2 ) 1 4 e 2 π ∆ E/a -1 e 2 π ∆ E/a +1 -( 2 3 αa 2 ) i 4 π ∫ + ∞ 0 dw w -1 / 2 -i ∆ E/a ( w -1) 2 (58)  \nThis expression has two parts. The first one, convergent S cor, 1 , is significantly different from the so the called 'thermalized Larmor formula'  \nS LCHK (∆ E ) = 2 3 αa 2 × 1 1 + e 2 π ∆ E/a (59)  \nThe second one, S cor, 2 , is divergent and it is absent in LCHK's calculations. This demonstrates that even analysis carried out as closely as possible to the approach accepted by LCHK leads us to a totally different result from their 'thermalized Larmor formula' [7](S21).", '5 Discussion': "The transition between theoretical predictions and experimental observations made by LCHK is based on their model of the Unruh-DeWitt detector, coupled to a semi-classical 4-vector current. We would expect this model to satisfy at least two conditions. It should predict and/or explain possible results of experiments and it must correctly represent the theoretical concept to be proved. In the foregoing work we have considered the LCHK model only from the theoretical point of view.  \nOur analysis was focused on two issues: first, whether LCHK's claim that they 'discover[ed] a  \nLarmor formula and power spectrum that are both thermalized by the acceleration' is supported by the calculations and second, whether this model is consistent and has no internal contradictions in the frame of quantum electrodynamics.  \nFor this purpose we examined possible expressions of a detector radiation power, its derivations and interpretations, and associated two-potential correlation functions for a quantized electromagnetic field observed at the location of an accelerating detector. We have considered two cases in the frame of the Lorentz gauge, with and without use of the Lorentz complementary condition, and analyzed in great detail LCHK's approach made in the Coulomb gauge.  \nRadiation power expression (34), corresponding to correlation function (21), is easy to interpret. It is the same for any observer at a location of the detector on a hyperbolic trajectory, an expected feature of hyperbolic motion. But, as our calculations show, it includes non-physical states and is divergent. So, (34) is not a good representation of the thermal effects phenomenon.  \nExpression (37) for DRP, obtained in the Lorentz gauge frame and under the complementary condition, includes only physical, transverse, polarization modes. However, it depends on detector proper time. This contradicts the fact that there is no preferred time for hyperbolic motion. Therefore its calculation in a general form does not make sense.  \nNevertheless, calculation of (37) for a very special case, when proper time and detector lab velocity is zero, can easily be done and leads us to (39). This expression was also considered by LCHK, though as we have indicated in Subsection 4.3 was obtained ad hoc, and after its integration our result (40) is totally different from the mistaken LCHK result (41). Radiation power (40) shows some signs of thermality which could be associated with a detector acceleration but they are different from the Bose-Einstein statistics expected for the photon field. We have shown that if the detector energy gap is zero, ∆ E = 0, then there is no radiation, S tr ( σ = 0), and, in complete contradiction of the LCHK claim, no 'thermalized Larmor formula'.  \nThere remains one more feature of the model to be discussed. To separate transverse modes from scalar/longitudinal ones we use a gauge transformation. This can only be done in one certain reference frame and it is impossible to do it in all three reference frames I M , I τ 1 and I τ 2 , involved in the calculations of the correlation functions, simultaneously. For example, after a gauge transformation (22), 4-potentials (25) have only transverse polarization modes. They are defined in the inertial lab system I M . However, these potentials being considered in I τ 1 and I τ 2 by means of Lorentz transformations  \nfrom I M to I τ 1 and I τ 2 with different boosts, correspondingly γ τ 1 and γ τ 2 , would have contained both transverse and scalar/longitudinal polarization modes. Thus the theory of this detector cannot be constructed based on transverse polarizations only, and non-physical modes are also involved.  \nThe only very special case (38) for the LCHK model with a finite valued result (40) was obtained. It has the following features: 1. there is no certain thermality, 2. the DRP expression is not Lorentz-invariant and represents only one point of the classical hyperbolic trajectory, 3. it involves both transverse and non physical longitudinal/scalar modes, in contradiction with the features of a quantized electromagnetic field, 4. a detector, representing a quantum device, moves along a classical trajectory. This is in contradiction with the quantum uncertainty principle.  \nThese features dictate how well the model answers DeWitt's question mentioned in the introduction: 'How would a given particle detector respond in a given situation?' Based on our analysis, we must conclude that LCHK's model cannot provide a theoretical justification of a quantum electrodynamic phenomenon being discussed.", 'A Vector Potentaials, Polarization vectors and Hyperbolic Functions': "In the reference frame I M , the expansion of the operator of a 4-potential of the field in plain waves is [4],(10.166):  \nˆ A µ ( t, ⃗x ) = ∫ d 3 k (2 π ) 3 1 2 ω k 3 ∑ λ =0 [ a ( ⃗ k, λ ) e -kx + a + ( ⃗ k, λ ) e kx ] e µ ( ⃗ k, λ ) , (60)  \nwhere annihilation/creation operators obey the commutation relationship  \n[ a ( ⃗ k, λ ) , a + ( ⃗ k ' , λ ' ))] = -(2 π ) 3 2 ω k g λλ ' δ ( ⃗ k -⃗ k ' ) , g λλ ' = (+1 , -1 , -1 , -1) . (61)  \nand polarization vectors have the following properties  \n3 ∑ λ =0 e µ ( ⃗ k, λ ) e ν ( ⃗ k, λ ) g λλ = -g µν , { g µν } = { g λλ ' } = (1 , -1 , -1 , -1) , e µ ( ⃗ k, λ = 0) = (1 , 0 , 0 , 0) , e µ ( ⃗ k, λ = 1) = (0 , ⃗e ( ⃗ k, λ = 1)) , e µ ( ⃗ k, λ = 2) = (0 , ⃗e ( ⃗ k, λ = 2)) , e µ ( ⃗ k, λ = 3) = (0 , ⃗ k/k ) , ⃗e ( ⃗ k, λ ) ⃗e ( ⃗ k, λ ' ) = δ λλ ' , λ, λ ' = 1 , 2 , 3 ( ⃗ k, 2) = ⃗e ( ⃗ k, 3) , ⃗e ( ⃗ k, 2) × ⃗e ( ⃗ k, 3) = ⃗e ( ⃗ k, 1) , ⃗e ( ⃗ k, 3) × ⃗e ( ⃗ k, 1) = ⃗e ( ⃗ k, 2) (62)  \n⃗e ( ⃗ k, 1) × ⃗e  \nHere are two useful formulas for hyperbolic functions used in this paper:  \ncosh[ a ( σ + 1 2 τ )] × cosh[ a ( σ -1 2 τ )] = 1 2 cosh(2 aσ ) + 1 2 cosh( aτ ) , sinh[ a ( σ + 1 2 τ )] × sinh[ a ( σ -1 2 τ )] = 1 2 cosh(2 aσ ) -1 2 cosh( aτ ) (63)", "B Error in LCHK' calculation of integral (39)": 'Integral in [7](S20)  \nS LCHK = 2 3 α i π ( a 2 ) 3 8 a ∫ ∞ 0 dw w 1 / 2 -i ∆ E/a -1 2 w 3 / 2 -i ∆ E/a -1 2 w -1 / 2 -i ∆ E/a ( w -1) 3 (64)  \nat ∆ E = 0 becomes  \nI (∆ E = 0) = ( -1 / 2) ∫ ∞ 0 dw w -1 / 2 w -1 . (65)  \nAfter variable change w = 1 /u we get the same expression with an opposite sign  \nI (∆ E = 0) = (+1 / 2) ∫ ∞ 0 du u -1 / 2 u -1 . (66)  \nSo I (∆ E = 0) = 0, and S LCHK defined in (64) should be 0 at ∆ E = 0. But their final formula [7](S21)  \nS LCHK = (2 / 3) αa 2 1 1 + e 2 π ∆ E/a (67)  \ndoes not satisfy this boundary condition.  \nThis result was used in Section (4.2)', 'C Evaluation of the Integral in (39).': 'The principle value integral in (39)  \n<!-- image -->  \nI ≡ ∫ + ∞ 0 u -i ∆ E a -1 2 1 -u du (68)  \nafter a variable change has a form  \nI = ∫ + ∞ -∞ e t (1 / 2 -ib ) dt 1 -e t , u = e t , b = ∆ E a (69)  \nFor integration purposes we use a complex variable function f ( z ) = e z (1 / 2 -ib ) 1 -e z and integration contour γ = L 0 -L 1 + γ + -γ --C 0 -C 1  \nwhere R → + ∞ in Fig 1. By the residue theorem we have  \n0 = ∫ γ f ( z ) dz = ( ∫ L 0 -∫ L 1 + ∫ γ + -∫ γ --∫ C 0 -∫ C 1 ) f ( z ) dz,  \nwhere  \n∫ L 0 f ( z ) dz → I, ∫ L 1 f ( z ) dz → ∫ + ∞ -∞ e ( t +2 πi )(1 / 2 -ib ) dt 1 -e t = -Ie 2 πb , ∫ γ + f ( z ) dz → 0 , ∫ γ + f ( z ) dz → 0 , ∫ C 0 f ( z ) dz → iπRes ( f ( z ) , 0) = -iπ, ∫ C 0 f ( z ) → iπRes ( f ( z ) , 0) = iπ e 2 πb  \nThen we obtain  \nI = iπ ( e 2 π ∆ E/a -1) e 2 π ∆ E/a +1 (70)', 'D Divergence of I 2 defined in (35)': "Integral I 2 in (35) has a singular point at τ = 0. Its principle value, if it exists, should be defined as a sum of two integrals  \nI 2 = lim ϵ → 0 ∫ -ϵ -∞ dτ -i sin(∆ Eτ ) cosh( aτ ) sinh 3 ( aτ 2 ) + lim ϵ → 0 ∫ + ∞ + ϵ dτ -i sin(∆ Eτ ) cosh( aτ ) sinh 3 ( aτ 2 ) ≡ I 21 + I 22 (71)  \nBut none of these limits exists. More exactly, both limits are positive infinite, and the integral is divergent.  \nFor example, integral I 21 would have existed if for any small positive δ there exists such positive η that  \n| ∫ -ϵ '' -ϵ ' dτ -i sin(∆ Eτ ) cosh( aτ ) sinh 3 ( aτ 2 ) | < δ, ϵ '' < ϵ ' < η (72)  \nBut it is easy to see that it is impossible. The integral tends to infinity when ϵ ' and ϵ '' become small. Indeed,  \n| ∫ -ϵ '' -ϵ ' dτ -i sin(∆ Eτ ) cosh( aτ ) sinh 3 ( aτ 2 ) | = | ∫ -ϵ '' -ϵ ' dτ { 2 sin(∆ Eτ ) sinh( aτ 2 ) + sin(∆ Eτ ) sinh 3 ( aτ 2 ) }| = | 2∆ E a/ 2 ( -ϵ '' + ϵ ' ) + ∆ E ( a/ 2) 3 ( 1 ϵ '' -1 ϵ ' ) | → + ∞ (73)", 'E Acknolegments': 'I would like to thank prof. A. Tumanov for helpful discussion of some mathematical issues in this paper.', 'References': "- [1] Davis, P.C.W., Scalar Particle Production in Schwarzschild and Rindler Metrics, J. Phys.A8, 609 (1975)\n- [2] W.G. Unruh, 'Notes on Black-Hole Evaporation', Phys. Rev.D14, 870 (1976)\n- [3] Birrell, N.D.,Davis, P.C.W. 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