Physica A: Statistical Mechanics and its Applications
Volume 307, Issues 3-4, 1 May 2002, Pages 375-404




Quantum statistics of superluminal radiation

Roman TomaschitzCorresponding Author Contact Information, E-mail The Corresponding Author

Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi- Hiroshima 739-8526, Japan

Received 9 July 2001; 
revised 24 October 2001. 
Available online 3 January 2002.

Abstract

A statistical quantization of superluminal (tachyon) radiation is introduced. The tiny tachyonic fine structure constant suggests to depart from the usual quantum field theoretic expansions, and to use more elementary methods such as detailed equilibrium balancing of emission and absorption rates. Instead of commencing with an operator interpretation of the wave function, we quantize the time-averaged energy functional and the energy-balance equation. This allows to use different statistics for different types of modes. Transversal superluminal modes are quantized in Bose statistics, longitudinal ones are turned into fermions, resulting in a positive definite Hamiltonian for the radiation field. We discuss the absorptive space structure underlying superluminal quanta and the energy dissipation related to it. This dissipation leads to an adiabatic time variation of the temperature in the bosonic and fermionic spectral functions, gray-body quasi-equilibrium distributions with a dispersion relation adapted to the negative mass-square of the tachyonic modes. The superluminal radiation field couples by minimal substitution to subluminal matter. Adiabatically damped Einstein coefficients are obtained by detailed balancing, as well as emission and absorption rates for tachyon radiation in hydrogenic systems, in particular the possibility of spontaneous emission of superluminal fermionic quanta is pointed out, and time scales for the approach to equilibrium are derived.

Author Keywords: Tachyonic gray-body radiation; Longitudinal background radiation; Superluminal quantum ensembles; Ether; Superluminal energy dissipation; Quantum tachyons; Spontaneous tachyon emission

PACS classification codes: 05.30.Ch; 42.25.Bs; 32.70.Cs; 98.70.Vc

Article Outline

1. Introduction
2. Tachyons in a permeable spacetime
3. Energy balance for superluminal radiation in a refractive and absorptive space–time
4. Transversal tachyons in Bose-Einstein statistics
5. Longitudinal tachyons in Fermi–Dirac statistics
6. Conclusion
Acknowledgements
Appendix A. Equilibrium mechanics of superluminal fermionic modes
References

1. Introduction

Superluminal radiation fields have never been properly quantized, despite of various attempts, most notably in [1, 2 and 3]. Here we carry out a purely statistical quantization, suggested by the very weak coupling of superluminal radiation to subluminal matter, some eleven orders below the electric fine structure constant. An elementary approach, avoiding propagators and field theoretical expansions, is quite sufficient to describe superluminal quantum effects such as the spontaneous emission of tachyons. The interaction of tachyonic quanta with matter will be settled by detailed balancing of emission and absorption rates, along Einstein's arguments, and the statistics of the superluminal modes will be determined by positivity requirements on energy.

The field theoretical second quantization of superluminal radiation has failed for two reasons. In a relativistic context, there is an insurmountable causality problem [4, 5, 6, 7, 8, 9, 10, 11 and 12], and one also quickly runs into a positivity problem with regard to energy, accompanied by a variety of inconsistencies, such as the violation of unitarity and unstable ground states. In this paper, we focus on the statistics of modes rather than fields, on spectral distributions rather than propagators, and interactions are settled by equilibrium balancing rather than renormalization. Instead of prescribing commutators or anticommutators for the wave function, our starting point will be the operator interpretation of the time-averaged energy functional. The algebraic relations between the Fourier modes are chosen in a way to turn the indefinite classical energy functional into a positive definite Hamiltonian. This is achieved by using commutators for transversal modes, and anticommutators for longitudinal ones. The transversal modes resemble photons with a negative mass-square, but the third degree of freedom is subjected to the exclusion principle, the longitudinal modes being fermions.

The crucial point is to use Bose as well as Fermi statistics for one and the same field. This is in fact very unusual, as second quantization in three space dimensions is customarily set up by an operator interpretation of the classical wave function. The statistics, either Bose or Fermi, is then determined by the spin-statistics theorem [13], which, however, only applies to subluminal fields. The quantization here carried out turns the superluminal radiation field into a mixture of transversal bosons and longitudinal fermions. The tachyonic spectral functions are gray-body Fermi and Bose distributions in quasi-equilibrium (due to the energy dissipation discussed below), with a negative mass-square in the dispersion relation. For equilibrium to be reached, one has to admit emission and absorption processes, so that the particle numbers cannot be prescribed, and accordingly there is no chemical potential, even not for the fermionic modes. Superluminal quanta couple by minimal substitution to subluminal matter, and transition rates will be obtained by detailed balancing.

Tachyon radiation cannot be understood in a relativistic context, because of causality violation [12], it requires an absolute space, tantamount to a medium of wave propagation. Hence, when discussing statistical distributions of superluminal quanta, we have to reckon with the substance of space, the ether [15]. The absorptivity of the ether results in energy dissipation affecting the spectral functions. Superluminal radiation is thus gray- rather than black-body, provided the dissipation is sufficiently adiabatic for equilibrium mechanics to be applicable. This depends on the permeability of the ether and can be made quite quantitative. The derivation of the energy distributions of the bosonic transversal modes and the longitudinal fermions is centered at the time-averaged energy-balance equation, a Poynting theorem in Fourier space, relating the field energy and the energy flux to the dissipated energy. The indefinite classical energy functionals (of field energy and dissipated energy) become positive definite operators by quantizing transversal and longitudinal modes in different statistics as mentioned. In the spectral distributions, the refractivity of the ether enters by the dispersion relation, and its absorptivity generates an adiabatic time variation of the temperature. As an example, we will discuss the cosmic tachyon radiation, in particular how equilibrium can be reached by interactions with subluminal matter.

In Section 2, the tachyonic counterpart to Maxwell's equations is introduced and all that goes with it, such as tachyonic field strengths, inductions, permeabilities and material equations, and the time-averaging of the superluminal energy flux is discussed. In Section 3, we study the energy-balance and the dissipation effected by the absorptivity of the ether. The statistical interpretation of the energy-balance equation is given in Section 4 for transversal and in Section 5 for longitudinal modes. We derive the spectral energy densities, the Einstein coefficients, the emission and absorption rates for bosonic and fermionic tachyons, as well as tachyonic ionization cross sections. In Section 6, we present our conclusions. In the appendix, we calculate the thermodynamic variables and the equations of state for the fermionic modes.

2. Tachyons in a permeable spacetime

Tachyons emerge as an extension of the photon concept, a sort of photons with negative mass-square [15 and 16], satisfying the Maxwell equations

divB=0, rotE+c−1B/∂t=0 ,


(2.1)
divD=ρ−c−1mt2C0, rotHc−1D/∂t=c−1j+mt2C .
Tachyonic E and B-fields relate to the vector potential by E=c−1(backward differenceA0−∂A/t) and B=rotA. The inductive potential (C0,C) enters via the tachyon mass and connects to the vector potential by

(2.2)
Image
This complements the familiar material relations

(2.3)
Image
where var epsilon (t) and μ (t) denote the dielectric and magnetic permeabilities, respectively. The tachyon mass mt has the dimension of an inverse length, and is meant as a shortcut for mtc/planck constant over two pi. The signs of the mass terms in (2.1) are chosen in a way that mt2>0 is the negative mass-square. We find mt/me≈1/238, estimated from Lamb shifts in hydrogenic systems [17 and 18]. The tachyon field couples by minimal substitution to subluminal matter. In the case of a classical subluminal particle, the charge density and the current in (2.1) read ρ=qδ(xx(t)) and j=qvδ (xx (t)), where q is the tachyonic charge carried by the particle; quantized interactions will be discussed in 4 and 5. Tachyonic and electric fine structure constants relate as q2/(4πplanck constant over two pic)≈1.0×10−13≈0.66α6, again an estimate from Lamb shifts. We will not consider electromagnetic fields, so that we can use the notation so suggestive in electrodynamics without the risk of confusion. The mass term breaks the gauge invariance, and the tachyon potential (A0,A) becomes observable, resulting in a new set of material equations and an induction (C0,C). In ((2.2) and (2.3)), one may substitute var epsilon (t)=δ (t)+κ (t) (δ stands for the Dirac function) and μ (t)=δ (t)+χ (t), the susceptibilities κ (t) and χ (t) are required to vanish identically for negative t, as the response of the medium cannot happen prior to its exposure to the field.

Defining the Poynting vector as S=cE×H+mt2A0C, we obtain from the Maxwell equations and the material relations

(2.4)
Image
If var epsilon and μ are constant and j=0, we may write divS+∂ρE/∂t=0, with the indefinite density

(2.5)
Image

The energy of the transversal and longitudinal modes can be extracted from (2.4) by time averaging. To this end, we turn to the plane wave decomposition of the spatial component of the vector potential,

(2.6)
Image
with k colon, equalsn/L. The summation is over integer lattice points n in R3, corresponding to periodic boundary conditions, so that the L−3/2 exp (ikx) are orthogonal and complete in a box of size L. The damping of plane waves is described by a complex frequency ω = ωR+iωI, with ωIgreater-or-equal, slanted0, and ω* denotes the complex conjugate. The amplitudes Image (which are not really Fourier due to the damping factor, though we will name them so) are composed with arbitrary real unit vectors var epsilonk,1 and var epsilonk,2 (linear polarization vectors) orthogonal to var epsilonk,3 colon, equalsk0=k/|k|, so that the var epsilonk constitute an orthonormal triad for every n. The amplitudes â(k,λ) are complex numbers. In (2.6) we depart from the standard electromagnetic formalism in a permeable medium, assuming a real wave vector k and an exponential damping factor exp (−ωIt), generated by a complex frequency. The amplitudes of the time component of the real vector potential are defined as in (2.6) with Image replaced by (A0(x,t),Â0(k)), and the same holds for the field strengths and inductions.

The Maxwell equations (2.1) (with ρ=0, j=0) read in Fourier space

Image


(2.7)
Image
The Fourier coefficients of the field strengths and the vector potential relate as and Image . The permeabilities read in Fourier space

(2.8)
Image
and the same for Image and μ(t). Clearly, for Image to be a solution of the field equations (2.7), the dispersion relation,

(2.9)
Image
(with real k colon, equals |k|) as well as the Lorentz condition,

(2.10)
Image
must hold, which is also sufficient. As k2 in (2.9) is real, ω*(k)=ωR−iωI satisfies

(2.11)
Image
Image denotes the complex conjugate of Image . Equation (2.9) is to be solved for (complex) ω(k), and Eq. (2.10) defines the Fourier coefficients Â0(k) of the time component, once the Image are chosen. As the solution ω (k) is not unique, a further summation in (2.6) may be necessary, so that the coefficients Image and â (k,λ,i) depend on a further index i labeling the branches ωi(k). However, we will consider permeabilities where this branching does not occur. In any case, the coefficients Image can arbitrarily be prescribed for each branch, that is, the amplitudes â (k,λ) in (2.6), and then the Â0(k) are determined by (2.10) and (2.9). The Fourier coefficients of the inductions D, H, C and C0 are obtained from the material relations ((2.2) and (2.3)),

(2.12)
Image
the argument in the permeabilities is ω*, unless stated otherwise.

The vector potential can be split into a transversal and longitudinal component, cf. ((2.6) and (2.10)),

(2.13)
Image
this decomposition is unique, as there is no gauge freedom. (We will frequently drop the argument k and/or the polarization index λ in the Fourier coefficients; the same holds for the argument ω* in the permeabilities.) The field strengths read accordingly

(2.14)
Image
where we used the dispersion relation (2.9). The inductions are then obtained from (2.12). Both the transversal and longitudinal components independently satisfy the field equations (2.7), with arbitrarily prescribed amplitudes â (k,λ), λ=1,2,3.

We consider time averages over a period of 2π/ωR; on this scale the variation of exp (−ωIt) is assumed adiabatic. It is also understood that ωR+iωI=ω(k) solves the dispersion relation (2.9). We denote by Ψ(x,t) and Φ(x,t) any of the fields A,A0,E, B, or the inductions (2.12), all defined by series of type (2.6). Image and Image are the corresponding Fourier coefficients, such as Image , etc. The time average of the product Image is readily calculated as

The integral sign refers to the spatial integration, and the averaging has been carried out mode by mode in the product series of ∫ΨΦ dx, cf. (2.6), and has as effect that terms containing Image and Image drop out. The damping factor exp (−ωIt) is regarded as constant within the averaging period 2π/ωR. For instance,

(2.15)
Image


(2.16)
Image
The same relations (2.15) hold for the averaged square of B (BL=0) and the inductions, cf. ((2.12), (2.13) and (2.14)), with

(2.17)
Image
Relations of type (2.15) also apply to the vector potential and the induced potential, to their spatial components as well as A0 and C0, with (2.16) replaced by

Image


(2.18)
Image
The argument in the permeabilities is ω*R−iωI, and ω is a solution of the dispersion relation (2.9), so that (2.11) holds. As for the Poynting vector defined before (2.4), we find the spatially integrated and time averaged flux

(2.19)
Image
The crucial point here is that the interference term of the transversal and longitudinal field components, cEL×H+mt2A0CT, has vanished in the averaging procedure, and we arrive at

(2.20)
Image
The flow components can be made more explicit by ((2.13) and (2.14)),

Image


(2.21)
Image
The meaning of this decomposition into a transversal and longitudinal flux will get apparent in the next section, when we discuss the corresponding splitting of the energy density and the energy dissipation in the ether.

3. Energy balance for superluminal radiation in a refractive and absorptive space–time

To identify the energy density of the transversal and longitudinal modes as well as the dissipated energy, we derive at first the time average of the conservation law (2.4), with the external current dropped. We start with the E·∂D/∂t term in (2.4), write

(3.1)
Image
and use a standard expansion [19] in the imaginary part of the frequency ω* colon, equals ωR−iωI,

(3.2)
Image
which implies adiabatic damping compared to the harmonic time variation. We so find, by means of the time averaging defined after (2.14),

(3.3)
Image
This also holds with the replacements Image , Image and Image , respectively, and thus we may write the time average of (2.4) as

(3.4)
Image
with the averaged energy density

(3.5)
Image
and the same for the dissipated energy per unit time, left angle bracketIdisright-pointing angle bracket, but with Image and Image replaced by Image and Image , respectively. We will identify left angle bracket∫ρEright-pointing angle bracket with the field energy, and left angle bracketIdisright-pointing angle bracket with the dissipated energy, but to do so we still have to disentangle the transversal and longitudinal components like in (2.21).

The preceding time averages as well as the flux components (2.21) can be further simplified by performing the ωI-expansion in the dispersion relation (2.9). To this end we assume Image , so that

(3.6)
Image
and the same expansions hold for Image . By substituting this into (2.9), we easily obtain

(3.7)
Image


(3.8)
Image
In (3.8) the argument of the permeabilities is ωR, calculated via (3.7) as a function of k. The solution ω (k)=ωR+iωI of ((3.7) and (3.8)) also solves the dispersion relation (2.9) up to terms of O(ωI2), and we see from (3.8) that our assumption Image is self-consistent. We may now write in (2.21)

(3.9)
Image
up to O(ωI2). If not otherwise indicated, Image , Image , Image , and the same for Image ; the prime just means ordinary differentiation. If we write Image and Image without explicit indication of its real or imaginary part, we always mean Image and Image , as in Section 2. Analyticity properties, Kramers–Kronig relations, and the fluctuation–dissipation theorem for tachyonic field strengths and inductions will not be discussed in this paper. It is also understood that ωR(k) and ωI(k) depend on the wave vector, being solutions of ((3.7) and (3.8)).

In the averages left angle bracket∫ρEright-pointing angle bracket and left angle bracketIdisright-pointing angle bracket, cf. (3.5), we rewrite the products of the Fourier components by means of ((2.16), (2.17), (2.9) and (2.11)), and then carry out the ωI-expansion as done in ((3.6), (3.7), (3.8) and (3.9)),

Image


(3.10)
Image
The λ-summations stem from the transversal components Image and Image , respectively, and the â(3)â*(3) term can be identified with the longitudinal components Image and Image . In this way we can unambiguously separate the contributions of the transversal and longitudinal modes, left angle bracket∫ρEright-pointing angle bracket=left angle bracket∫ρETright-pointing angle bracket+left angle bracket∫ρELright-pointing angle bracket and left angle bracketIdisright-pointing angle bracket=left angle bracketIdisTright-pointing angle bracket+left angle bracketIdisLright-pointing angle bracket, where the transversal averages are

Image


(3.11)
Image
and the same holds for left angle bracket∫ρELright-pointing angle bracket and left angle bracketIdisLright-pointing angle bracket, but with ωR2/c2 replaced by Image , and the λ-sum over the transversal polarizations is replaced by â(3) â*(3). This is valid up to terms of O (ωI2), like the corresponding decomposition (3.9) of the flow, and the argument in the permeabilities is ωR. Both ωR and ωI depend on the wave vector, being solutions of ((3.7) and (3.8)). Moreover, left angle bracket∫ρET,Lright-pointing angle bracket=O (1), left angle bracketST,Lright-pointing angle bracket=O (1), and left angle bracketIdisT,Lright-pointing angle bracket=O (ωI).

The positivity of the energy averages left angle bracket∫ρETright-pointing angle bracket and left angle bracketIdisTright-pointing angle bracket is ensured by requiring

(3.12)
Image
and the same with Image and Image interchanged. Under the same conditions, the longitudinal components are negative definite. In Section 5, we will quantize them in Fermi statistics, effecting an overall sign change of left angle bracket∫ρELright-pointing angle bracket, left angle bracketIdisLright-pointing angle bracket, and left angle bracketSLright-pointing angle bracket. The transversal modes will be quantized in Bose statistics, which does not affect the positivity of the transversal energy. In this way we will obtain a positive definite Hamilton operator for the transversal as well as longitudinal modes, so that we can identify left angle bracket∫ρEright-pointing angle bracket as field energy, and left angle bracketIdisright-pointing angle bracket stands for the energy per unit time dissipated into the ether. The conservation law (3.4) holds for the transversal and longitudinal components individually.

As a consistency check, we consider a single mode k in the series ((3.9) and (3.11)), and find, with vgr colon, equals k0R/dk,

(3.13)
Image
where the group velocity is determined by the dispersion relation (3.7).

To prepare the second quantization in 4 and 5, we introduce rescaled Fourier coefficients a (k,λ) in the preceding time averages, so that â (k,λ)=:αTa (k,λ) for λ=1,2, and â (k,3)=:αLa (k,3), with the normalization factors

(3.14)
Image
The square of the transversal normalization factor relates to the group velocity in (3.13) via Image . We introduce the adiabatically varying frequency

(3.15)
Image
so that the energy density in (3.11) gets a familiar shape,

(3.16)
Image
The frequency Image is determined by ((3.7) and (3.8)); conditions (3.12) also leave ωI positive (or zero, in the limit of real permeabilities). In the normalization (3.14), the averaged energy flux reads, cf. ((3.9) and (3.13)),

(3.17)
Image
valid up to terms of O (ωI2) like (3.16). The energy per unit time dissipated by transversal and longitudinal modes is found as, cf. ((3.11) and (3.8)),

(3.18)
Image
again up to O (ωI2).

As for the microscopic structure of the ether, we consider a classical oscillator model [15], which gives Image and

(3.19)
Image
with α0 colon, equals N0q02m0−1 and γ0>0. Here, ω0 is the free oscillator frequency, γ0 the damping constant, N0 the number density, m0 the mass, and q0 the tachyonic charge of the uniformly distributed oscillators constituting the ether. We assume a narrow line breadth, γ0→0, and real ω. The maximum of Image is Image , located at ωmaxnot, vert, similarω0. In the vicinity of the resonance, we find the Lorentzian

(3.20)
Image
The real part of Image has apparently a zero at ω0 and one or two extrema,

(3.21)
Image
The Drude formula (3.19) is meant as a power series expansion in α0. As mentioned, Image, and thus the positivity conditions (3.12) are satisfied. For ω→∞, we find
Image
 and Image . If ω→0,
Image
 and Image , so that in the respective limits, cf. (3.8),

(3.22)
Image
Accordingly, ωIR) is uniformly α0-small in the whole frequency range. We will return to these limits when discussing tachyonic gray-body radiation, cf. (4.7), ωI determines the adiabatic time variation of the temperature in the spectral distributions.

4. Transversal tachyons in Bose-Einstein statistics

We quantize in occupation number representation. The Fourier coefficients a (k,λ) in the time averaged transversal energy density (3.16) are replaced by operators ai, and their complex conjugates a*(k,λ) by the adjoints ai+. Bose statistics is defined by the commutation relations [ai,aj+]=δij, [ai,aj]=0 and [ai+,aj+]=0; the indices i and j stand for the modes (k,λ), k=2πn/L, nset membership, variantZ3, λ=1,2, as defined after (2.6). Orthogonal transversal states do not affect each other, and the longitudinal degree of freedom can likewise be treated independently, cf. Section 5, as operators in different orthogonal subspaces are supposed to commute. Therefore, to save notation, we will drop the polarization index λ, that is, quantize modes of a given linear polarization, λ=1, say.

The number operators Ni colon, equalsai+ai are Hermitian and commute. We consider the basis vectors |n1,…,ni,…,nright-pointing angle bracket (shortcut |nright-pointing angle bracket, e.g. |0right-pointing angle bracket for the vacuum state). The occupation numbers ni are non-negative integers indicating the number of particles in state i. In each basis vector, only a finite number of the ni are non-zero. A scalar product is defined by left angle bracketn|nright-pointing angle bracketn1,n1…δn,n. Operators satisfying the above commutation relations are readily found,

Image


(4.1)
Image
and ai|nright-pointing angle bracket=0 (zero-vector) if ni=0. Hence, left angle bracketai+n|nright-pointing angle bracket=left angle bracketn|ainright-pointing angle bracket, as well as Ni|nright-pointing angle bracket=ni|nright-pointing angle bracket. We so find the Hamilton operator of the var epsilonk,1-polarized modes in (3.16) as

(4.2)
Image
and the same for the var epsilonk,2-modes. The partition function and the internal energy of modes of a given transversal polarization is calculated in the usual way,

Image


(4.3)
Image


(4.4)
Image
In the thermodynamic limit [20], ∑kL3/(2π)3∫dk, we find

(4.5)
Image
where Image , cf. (3.15), and k0 colon, equalsmtc/planck constant over two pi is the smallest possible value of k:=|k|, attained for ω=0 according to the dispersion relation (3.7). ωI depends on k via ωR(k), cf. (3.8). We will frequently write ω for ωR, and replace in (3.7) mt by mtc/planck constant over two pi, see after (2.3). To account for two transversal degrees, we have to multiply log Z as well as U by a factor of two, so that the spectral and internal energy densities of transversal tachyonic gray-body radiation read

(4.6)
Image
k(ω) is defined in (3.7), and k′(ω)=dk/dω coincides with (3.13). We have here expanded (i.e., dropped) the damping factor exp (−2ωI(k)t) in the nominator of the integrand in (4.5), and in the denominator we have scaled this factor into the temperature variable, appealing to small ωI and adiabatic variation on time scales tmuch less-than1/(2ωI), otherwise the use of an equilibrium distribution would not be justified. As pointed out in (3.22), one may assume ωI(k) small in the whole frequency range. The cosmic tachyon background also requires a conformal time scaling of the temperature with the cosmic expansion factor, β=a(t)/(kT).

We derive some limit cases of the density (4.6), with permeabilities as in (3.19). k(ω) and k′(ω) are explicit in ((3.7) and (3.13)). In the high frequency limit, we find knot, vert, similarω/c. For ω→0, we may approximate knot, vert, similarmt and knot, vert, similarωmt−1c−2(1+α002), see after (3.21). Hence, for high frequencies and with permeabilities as suggested in ((3.19), (3.20), (3.21) and (3.22)), ρT (ω) converges to the Wien limit of the photonic Planck distribution, and in the Rayleigh–Jeans limit we find a linear frequency scaling,

(4.7)
Image
Here we defined Image , with the limits ωI(ω→∞,0) substituted, cf. (3.22). In the limit Image , that is α0→0 in (3.22), we recover from (4.6) the spectral energy density of tachyonic black-body radiation [14 and 18],

(4.8)
Image

We turn to the coupling of the tachyon field to matter, and study a subluminal, non-relativistic and spinless quantum particle carrying tachyonic charge q. We start with the Schrödinger equation for an attractive Coulomb potential V, coupled by minimal substitution to the tachyon field,

Image


(4.9)
tA colon, equalst−iq/(planck constant over two pic)A0, backward differenceA colon, equals backward difference−iq/(planck constant over two pic)A ,
so that the tachyonic charge density in (2.1) reads ρ=qψ*ψ. If Aμ=0, we find the discrete hydrogen-like spectrum En=(1−1/n2)E0. The ionization energy E0 has been inserted in (4.9) to define a zero ground state energy E1. In this section we consider transversal fields AT, so that A0=0, cf. (2.13). The interaction Hamiltonian can be readily extracted from (4.9),

(4.10)
Image
where we dropped terms quadratic in AT. The bound states of the unperturbed Coulomb problem are expanded as ψ=∑nbnune−iEnt/planck constant over two pi, with time-separated normalized eigenfunctions, ∫umun* dxnm, resulting in a bound state energy functional H(ψ)=∑nEnbn*bn. As above, we substitute for the Fourier amplitudes bn statistical operators, writing bn+ for the adjoint (instead of bn*). These operators may satisfy Bose statistics as above, or, equally well, the anticommutation relations of Fermi statistics, [bi,bj+]+ij, [bi,bj]+=0 and [bi+,bj+]+=0. We will consider a single subluminal particle, and so the statistics does not matter. The particle number operators Ni colon, equalsbi+bi are in either case Hermitian and commute. As for the fermionic occupation number representation, the ni are restricted to 0 and 1, and the Fermi operators are defined by

bi|n1,…,ni,…,nright-pointing angle bracket=(−)n<ini|n1,…,1−ni,…,nright-pointing angle bracket ,


(4.11)
bi+|n1,…,ni,…,nright-pointing angle bracket=(−)n<i(1−ni)|n1,…,1−ni,…,nright-pointing angle bracket ,
with n<i colon, equalsk=1i−1 nk, so that the anticommutation relations are satisfied and Ni|nright-pointing angle bracket=ni|nright-pointing angle bracket. It is assumed that the statistical operators ak of the tachyon field and their adjoints commute with the bi(+). The interaction Hamiltonian (4.10) can be represented in the tensorial product of the vector spaces used in ((4.1) and (4.11)). The statistical operators are extended to the product space by bn(+)circle times operatorid and idcircle times operatoran(+), so that these two sets commute.

We study induced tachyon radiation and spontaneous emission of tachyons, based on the interaction (4.10). The initial and final states for absorption read

|iright-pointing angle bracketabs=|0,…,1i,…,0j,…right-pointing angle bracketcircle times operator|n1,…,nk,…,nright-pointing angle bracket ,


(4.12)
|fright-pointing angle bracketabs=|0,…,0i,…,1j,…right-pointing angle bracketcircle times operator|n1,…,nk−1,…,nright-pointing angle bracket .
The first factor represents a single subluminal particle, and the second a set of tachyons distributed over some energy range. As for emission,

|iright-pointing angle bracketem=|0,…,0i,…,1j,…right-pointing angle bracketcircle times operator|n1,…,nk,…,nright-pointing angle bracket ,


(4.13)
|fright-pointing angle bracketem=|0,…,1i,…,0j,…right-pointing angle bracketcircle times operator|n1,…,nk+1,…,nright-pointing angle bracket .
The first factor in ((4.12) and (4.13)) is a basis vector for Bose or Fermi statistics. In the second factor (bosonic), k=(k,λ) labels the tachyonic occupation number diminished or augmented when passing from the initial to the final state. As at the beginning of this section, we drop the index λ and consider a linearly polarized tachyon field, so that we can identify k with k in the subsequent summations.

We substitute the tachyonic and subluminal statistical operators into the interaction Hamiltonian (4.10). The transversal tachyonic wave operator of linear polarization λ is defined by ((2.6) and (3.14)),

(4.14)
Image
with ak(+) as in (4.1). The bound state wave operator ψ is defined after (4.10). Using the hermiticity of ibackward difference in (4.10), we find the interaction operator as

(4.15)
Image
The summation index k stands for k, and the polarization only enters via var epsilonk. [In ((4.12) and (4.13)), the indices i, j and k are fixed and should not be confused with summation indices.] We so find, cf. ((4.1), (4.11), (4.12), (4.13) and (4.15)),

(4.16)
Image


(4.17)
Image
These matrix elements are independent of the statistics used in defining bi(+), since the subluminal factors in (4.12) and (4.13) are one-particle states. nk is a tachyonic occupation number in Bose statistics.

The transition rate for induced absorption, with a given linear polarization λ, is obtained by a standard procedure,

(4.18)
Image


(4.19)
ωE colon, equalsωj−ωi−ωR, ωk=:ωR(k)+iωI(k) .
(The same formula also applies to transitions effected by longitudinal fermionic tachyons, cf. Section 5, so we do not indicate here the T-superscripts in wabsT,ind and TabsT.) For arbitrary real numbers ωE,I,

(4.20)
Image


(4.21)
Image
and we assume ωIgreater-or-equal, slanted0, tmuch greater-than1, ωItmuch less-than1, so that FEI) is strongly peaked as a function of ωE, with maximum FE≈0,ωI)≈t. Due to dissipation, cf. Section 3, energy conservation is only approximate, Re ωk≈ωj−ωi, hence ωE≈0.

Like in ((4.4) and (4.5)), we replace the box-summation in (4.18) by the continuum limit

(4.22)
Image
with dΩ=sin θ dθ dphi as solid angle element. Explicit formulas for kR) and its derivative are given in (3.7) and (3.13). The ωR-integration in (4.18) is then carried out by steepest descent, at ωji colon, equalsωj−ωi,

(4.23)
Image
ωI is by virtue of (3.8) a function of ωR, positive and uniformly bounded, cf. the end of Section 3, so that the restrictions mentioned after (4.21) are satisfied over the whole integration range. The steepest descent procedure boils down to (4.21); ωI is taken at ωRji, likewise the remaining ωR-dependent factors in (4.18), and Image , cf. (3.15). The direction of the tachyonic wave vector k is specified by the angular variables in dΩ, and its magnitude by kji), cf. (3.7). The coefficient αkT in (3.14) is likewise taken at ωRji. The spectral density ρT is defined in (4.6). In short, wabsT,ind in (4.23) is the transition probability per unit time for an electron to be moved from an initial state ωi to an excited state ωj by the absorption of a transversal tachyon (k,λ).

The transition rate for emission is

(4.24)
Image
with ωE,I defined in (4.19). (ωj is now the excited initial state and ωi the final state.) Like in (4.18), we have dropped here the T-superscripts of wemT and TemT, cf. (4.17). The time integration gives the same result as for absorption, cf. (4.20), and thus the ωR-integration in the continuum limit (4.22) is the same as in (4.21). The crucial difference to absorption is the nk+1 factor in |left angle bracketTemTright-pointing angle bracket|2, contrary to the nk-proportionality of |left angle bracketTabsTright-pointing angle bracket|2, cf. ((4.17) and (4.16)). We split wemT colon, equalswemT,ind+wemT,sp, where wemT,ind denotes the contribution of the nk-proportional terms in (4.24), responsible for induced emission. A procedure completely analogous to the foregoing gives

(4.25)
Image
where all factors outside the integral are taken at ωRjij−ωi as in (4.23), and αkT2 relates to dωR/dk as indicated after (3.14). Applying Green's formula, we recover the symmetry of Einstein's B-coefficients, BjiT(k,λ)=BijT(−k,λ). The transversal spontaneous emission rate wemT,sp is obtained by dropping the nk+1-factor of |left angle bracketTemTright-pointing angle bracket|2 in (4.24). The k-summation in (4.24) is replaced by the integration indicated after (4.4), and we find, via ((4.20) and (4.21)),

(4.26)
Image
The A- and B-coefficients for transversal tachyon radiation thus read

(4.27)
Image
A consistency check of ((4.27) and (4.6)) is provided by the equilibrium balancing of emission and absorption events,

(4.28)
Image
with occupation numbers related by their weight factors as Image , where we used Image and Image , cf. (4.6). This can also be regarded as a derivation of the spectral density ρT(ω) alternative to ((4.2), (4.3), (4.4), (4.5) and (4.6)), when combined with (4.27). The factors of one-half in ((4.23), (4.25) and (4.28)) just indicate that we consider a single transversal degree, that is, modes of a fixed polarization, to compare better to the longitudinal radiation discussed in Section 5. The transition rates for unpolarized radiation are obtained by replacing var epsilonk·backward difference in ((4.23), (4.25) and (4.27)) by the transversal component of the gradient, backward differenceT colon, equals backward differencek0(k0·backward difference).

In dipole approximation, we may drop the exponential e±ikx in the preceding integrals and use the identity

(4.29)
Image
which is also valid for longitudinal polarization, i.e., for var epsilonk,3, as well as for unpolarized transversal radiation via the substitutions var epsilonk·backward differencebackward differenceT and var epsilonk·djidjik0(k0·dji). As for the latter, the angular integration in ((4.23), (4.25) and (4.26)) can easily be carried out with dji as polar axis and ∫sin2θ dΩ=8π/3. We so find the unpolarized total transversal transition rates as

(4.30)
Image


(4.31)
Image
The detailed balancing condition (4.28) apparently also applies to A(T,d) and B(T,d), the Einstein coefficients in dipole approximation for unpolarized transversal radiation in all directions.

Finally we shortly discuss the tachyonic analog to the photoelectric effect, the transfer of a bound electron into the continuum by the absorption of a tachyon. The Coulomb wave functions of the continuum are approximated by plane waves, that is, we put uj=L−3/2eikex in the matrix element (4.16) (Born approximation) to maintain the discrete states assumed in ((4.1) and (4.11)), and we write u0 for the initial bound state (not necessarily the ground state). In formula (4.18) for the absorptive transition rate wabsind, we take the k-summation over the electronic (rather than tachyonic) wave vectors. (There is still only one electron, so that the statistics used is irrelevant.) In the continuum limit, we may replace this summation by

(4.32)
L3/(2π)3∫dke, dke=(m/planck constant over two pi)keee, ωe=planck constant over two pike2/(2m) ;
the solid angle is now defined with the angular variables of the electronic wave vector. We also write, instead of (4.19), ωE colon, equalsωe−ω0−ωR(k). Here, ω0,e are the frequencies of u0 and uj, respectively, and k and ωkR+iωI refer to the tachyons. The dωe-integration is carried out by steepest descent around ωe0R, by making use of (4.21). Energy conservation, ωE≈0, is again only approximate, unless the dissipation generating ωI vanishes. We so find

(4.33)
Image
The integral can be replaced by var epsilonk·ke∫ei(k−ke)xu0 dx, the electronic ke is taken at ωe≈ωR, cf. (4.32), as ωRmuch greater-thanω0 is required by the Born approximation. The tachyonic kR) and ωIR) are defined in ((3.7) and (3.8)), and the amplitude αkT2 in (3.14). Dividing wabsT,ind by the incoming tachyonic flux density, |vgr|nk/L3, cf. (3.13), we find the differential cross section for polarized transversal radiation,

(4.34)
Image
The unpolarized transversal section is obtained by replacing |var epsilonk·ke|2 by Image , or by Image with k as polar axis. A phenomenological discussion of (4.34), with regard to the tachyonic ionization of Rydberg atoms, is given in [14], where (4.34) was derived semiclassically in the limit ωI=0, neglecting dissipation and the resulting adiabatic damping.

5. Longitudinal tachyons in Fermi–Dirac statistics

To render the longitudinal energy density in (3.16) positive, we quantize the modes in Fermi statistics, cf. (4.11), otherwise the reasoning is the same as in Section 4. Like in (4.2), we replace in the longitudinal density the Fourier coefficients by statistical operators, and make use of the anticommutation relation [ai,aj+]+ij, which effects the sign change turning the energy density into a positive definite operator,

(5.1)
Image
The a(+) admit the representation (4.11), so that the ni are restricted to zero and one. Partition function and internal energy read, cf. ((4.3) and (4.4)),

(5.2)
Image


(5.3)
Image
In the thermodynamic limit, we find the spectral energy density and the internal energy of the longitudinal modes as, cf. ((4.5) and (4.6)),

(5.4)
Image
Apparently, ρL differs from the transversal density ρT only by a sign change in the denominator and a factor of 1/2. (There is only one longitudinal degree of freedom.) As for the number density of the longitudinal modes, this is of course nL(ω) colon, equals ρL(ω)/(planck constant over two piω), so that Image , which will be made more explicit in the black-body limit at the end of this section. In the high frequency regime, we find ρLnot, vert, similarρT/2, and the Rayleigh–Jeans limit is quadratic in the frequency and temperature independent, Image . (The respective limits (4.7) of the transversal density are to be inserted.) In the black-body limit Image ,

(5.5)
Image

Next, we calculate the Einstein coefficients for longitudinal radiation. The interaction Hamiltonian is readily found, cf. ((4.9) and (4.10)),

(5.6)
Image
The spatial component of the longitudinal wave operator reads, cf. ((2.6) and(3.14)),

(5.7)
Image
and the time component follows from (2.13) via Â0(k)=α0LαLa(k,3) and the expansion ((3.6), (3.7) and (3.8)),

(5.8)
Image
We also restore mtmtc/planck constant over two pi in k and αL, cf. ((3.7) and (3.14)). Hence we may write the interaction as

(5.9)
Image
The operators a(+) are assumed in the representation (4.11). The b(+) of the subluminal particle may satisfy Bose or Fermi statistics, and we use for them the representations (4.1) or (4.11). The b(+) and a(+) commute. Thus we may take the initial and final states defined in ((4.12) and (4.13)), but the particle numbers ni in the second tensorial factor are now restricted to zero and one.

As for the absorption of a longitudinal fermionic tachyon, with initial and final states defined in (4.12), we find

left angle bracketf|HintL|iright-pointing angle bracketabs=left angle bracketTabsLright-pointing angle bracketei(ωj−ωi−ωk*)t ,


(5.10)
Image
and the matrix element for emission is likewise easily assembled from ((5.9), (4.13) and (4.11)),

left angle bracketf|HintL|iright-pointing angle bracketem=left angle bracketTemLright-pointing angle brackete−i(ωj−ωi−ωk)t ,


(5.11)
Image
In either case, the nk only admit the values zero and one, and it is instructive to compare to the transversal bosonic radiation ((4.16) and (4.17)).

The transition rates can be compiled as in Section 4. Equations ((4.18), (4.19), (4.20) and (4.21)) remain unaltered, as well as (4.22), apart from an obvious sign change in the denominator. We so find the induced absorption rate for longitudinal fermionic quanta, cf. (5.10),

(5.12)
Image
which is to be compared to (4.23). The factors outside the integrals are again taken at ωRji, and |k|=kji) in the exponents. The symmetry BjiL(k)=BijL(−k) follows from the Green formula and the condition planck constant over two piωRmuch less-thanmc2, as the subluminal particle is non-relativistic.

Concerning emission, the transition rate is determined by (4.24) with left angle bracketTemLright-pointing angle bracket defined in (5.11). To keep the formal analogy to the derivation following (4.24), we write Image , where −wemL,ind stands for the contribution of the nk-proportional terms in (5.11), and Image for the remaining terms independent of nk. The actual spontaneous emission rate wemL,sp will be identified in (5.19). By proceeding as after (4.24), we find

(5.13)
Image


(5.14)
Image
We restore the mass unit in αkL, that is, replace mt in (3.14) by mtc/planck constant over two pi, and so obtain

(5.15)
Image
In this way we recover, via the balancing condition

(5.16)
Image
the equilibrium distribution (5.4). In (5.22), we will write this balance in a more comprehensible form, with the proper A-coefficient.

The dipole approximation of the transition rates ((5.12), (5.13) and (5.14)) is obtained by dropping the exponentials in the integrals, so that the second integral in ((5.12), (5.13) and (5.15)) vanishes, and the first integral is settled by Ehrenfest's theorem (4.29). We so find, in the same notation as in ((4.29), (4.30) and (4.31)), the total (that is angular-integrated, ∫cos2 θ dΩ=4π/3) longitudinal transition rates

(5.17)
Image


(5.18)
Image
with dji as defined in (4.29). Clearly, Bji(L,d)=Bij(L,d), and the first equality in (5.15) as well as the balancing conditions (5.16) are likewise evident in dipole approximation.

The spontaneous emission rate wemL,sp and the Einstein A-coefficient attached to it are readily identified,

(5.19)
Image


(5.20)
Image


(5.21)
Image


(5.22)
Image
It is clear from ((5.14) and (5.19)) that wemL,sp>0. Relations ((5.19), (5.20), (5.21) and (5.22)) also hold in dipole approximation, and we may compare to the transversal rates, cf. ((4.30) and (4.31)),

(5.23)
Image
The transition rates derived in this section for longitudinal radiation in Fermi statistics are evidently quite similar to those for transversal radiation in Bose statistics discussed in Section 4; had we quantized the transversal tachyonic modes in Fermi–Dirac statistics, this would have only affected the transversal absorption rates by a factor Image .

Next we derive the longitudinal ionization cross section, cf. ((4.32), (4.33) and (4.34)). The same procedures as outlined after (4.31) lead to, cf. ((4.33) and (5.12)),

(5.24)
Image
Dividing this by the flux density, |vgr|nk/L3, cf. (4.34), we find

(5.25)
Image
The second term in the parentheses should be taken into account, despite ωR≈ωemuch less-thanmc2/planck constant over two pi, since ke/kmuch greater-than1 is possible and ωRmuch greater-thanω0 is needed for the plane wave approximation. Also compare in this regard the discussion of the maxima of the tansversal cross section in [14].

Next we discuss the equilibrium mechanics of longitudinal tachyonic black-body radiation, Image , ωI=0; the thermodynamic formalism for the superluminal transversal modes has already been studied in [14 and 18]. The peak of the fermionic spectral density ρL(ω) in (5.5) depends on the tachyon mass. Defining x colon, equals βhν and γ colon, equals βmtc2, we find the location of this peak by solving

(5.26)
Image
As for the cosmic tachyon background, γ≈9.1×106 and βh≈1.76×10−11 s, based on mt≈2.15 keV/c2 and kT≈2.35×10−4 eV. Since x(γ→∞)≈2.218, the longitudinal radiation has a frequency peak at νtL≈126 GHz [5.2×10−4 eV or hνtL/(mtc2)≈2.4×10−7], which is rather close to the peak of the photon density at νph≈160 GHz (6.6×10−4 eV). The transversal tachyon spectrum is peaked at νtT≈90.6 GHz or 3.7×10−4 eV, cf. [18]. In dipole approximation, cf. (5.23),

(5.27)
Image
where αte≈1.4×10−11 is the ratio of tachyonic and electric fine structure constants. Accordingly, at T≈2.725 K, we find wL/wTtL)≈1.45×10−14 and wT/wphtL)≈5.8×10−5, and comparable ratios for νtT and νph. Apparently, νtL also lies well within the core of the photon and transversal tachyon distributions. In the high-temperature limit, we find from (5.26) x(γ→0)≈3.131. The solution of (5.26) varies only moderately over the whole temperature range, and the same holds for the maximum of the transversal bosonic distribution, cf. [18]. A frequency defined by βhν0(T)=x≈2.218 lies for any temperature in the bulk of the enumerated distributions. Thus the location of the bulk of the spectral densities scales linearly with temperature, and Wien's displacement law is more or less recovered.

With these preparations, we can readily show that both the longitudinal and transversal quanta of the cosmic tachyon background have reached equilibrium at a rather early stage. Primordial nucleosynthesis requires a photonic equilibrium distribution at kT≈1 MeV, corresponding to a cosmic age of 1 s. At this temperature, we find hν0≈2.2 MeV, and the ratios wL/wT0)≈2.6×105 and wT/wph0)≈1.4×10−11. With increasing temperature, wL/wT gets quickly larger and overpowers wT/wph in their product. Thus one can assume that the longitudinal tachyon background has reached equilibrium within the first second, even before the photon background did, due to its stronger interaction with subluminal matter. As for the transversal tachyon radiation, this happened at a much later stage, but well within 1011 s (assuming a linear space expansion in this early epoch), the more so as wT/wph increases in time. In the present low-temperature regime, the ratios indicated after (5.27) hold, which makes the longitudinal fermionic radiation much harder to observe than the transversal boson background.

We turn to atoms in equilibrium with tachyon radiation. The dipole approximation (5.27) is the same for induced and spontaneous emission and absorption. We identify the Ly-α lines of hydrogen (10.2 eV) with the core frequency ν0 as defined above, corresponding to a temperature of kT0)≈4.6 eV. We so find from (5.27) wL/wT0)≈5.6×10−6 and wT/wph0)≈3.0×10−9. The Ly-α1 transition in hydrogenic uranium (ν0≈0.23 MeV) results in a bulk temperature of kT0)≈0.1 MeV. (At this temperature, the frequency ν0 lies in the core of the photonic and the two tachyonic spectral distributions.) In this case we find wL/wT0)≈2.9×103, but a very small wT/wph0)≈1.4×10−11. The chances to detect transversal tachyons improve in the low frequency fringe, due to the different frequency scaling of the photon and transversal tachyon distributions, cf. [18].

Finally we assemble the equations of state and the various thermodynamic variables for the longitudinal fermionic quanta. The subsequent high- and low-temperature expansions are derived in the Appendix, cf. ((A.27) and (A.28)). For T→0, we find in lowest order,

Image


Image


(5.28)
Image
(We have dropped the subscript L.) The caloric and thermal equations get independent of the tachyon mass in this limit. The high-temperature limit reads in leading order

Image


Image


(5.29)
Image
The tachyon mass does not enter in lowest order. The corresponding expansions for transversal tachyon radiation are very similarly structured, with moderately modified numerical constants [14], despite the different frequency scaling of the spectral densities in the Rayleigh–Jeans limit.

6. Conclusion

A new quantum statistics for superluminal radiation has been suggested, resulting in a positive definite Hamiltonian and a stable ground state. We have departed from the customary field theoretic quantization, and chosen more elementary statistical procedures adapted to the extremely small tachyonic fine structure constant, q2/(4πplanck constant over two pic)≈1.0×10−13. The tiny coupling constant is key to the quantization of superluminal radiation fields, this has been overlooked hitherto, as there have not been quantitative estimates on the interaction strength of superluminal radiation with matter. The weak interaction renders systematic quantum field theoretic expansions academic, which are anyway marred by negative energies and unstable vacua [1, 2 and 3]. In this paper we have used detailed equilibrium balancing to describe the tiny interaction of tachyons with subluminal matter.

The statistical quantization developed here is completely self-consistent, and we have shown that it works, being capable of quantitative predictions based on realistic interactions, extremely weak but not out of reach. In the following I summarize the main features of superluminal quantum statistics. Tachyonic quantum ensembles are neither bosonic nor fermionic, they are a mixture of both. Statistics does not apply to the field, but to its modes, and in the case of superluminal radiation, we are not bound by the spin-statistics theorem, which requires the same algebraic relations for all modes. This freedom in the choice of the commutation relations for the Fourier amplitudes is used to render the energy density positive definite, ensuring a stable ground state. The superluminal quanta, transversal bosons and longitudinal fermions, admit gray-body spectra in quasi-equilibrium, with an adiabatic time variation of the temperature due to energy dissipation.

This energy dissipation relates to the underlying space structure. A consistent statistics of superluminal quanta cannot be achieved in a relativistic spacetime, due to causality violation manifested in advanced wave modes. Superluminal quanta need a very different context, a permeable space, the ether [15]. They interact with the refractive and absorptive ether, and this results in adiabatic energy dissipation. The starting point for quantization is the time averaged energy-balance equation, relating the superluminal energy flux to the field energy and the dissipated energy, cf. Section 3. This energy-balance is quantized by defining algebraic relations for the Fourier amplitudes. The statistics is chosen in a way to turn both the field energy and the dissipated energy into positive definite hermitian operators, transversal modes are bosons, longitudinal ones satisfy Fermi statistics, cf. 4 and 5. Once a positive definite Hamiltonian is established for the free superluminal modes, it is straightforward to balance emission and absorption rates, in this way arriving at Einstein coefficients which are adiabatically damped by the energy dissipation.

Spontaneous emission is in no way hampered by the exclusion principle, it applies to longitudinal fermionic quanta as well. The semiclassical arguments used in [14] are quite efficient to describe bosonic superluminal modes, but they also have their limits. The bosonic transition rates derived in second quantization ( Section 4) coincide with those of [14], provided we drop the adiabatic damping factors in the Einstein coefficients; the semiclassical derivation given in [14] does not account for the absorptivity of the ether and the resulting energy dissipation. Also, the exclusion principle is beyond semiclassical mechanics, and so we didn't attempt to study the longitudinal fermionic modes in [14]. The black-body limit of the fermionic equations of state is derived in the Appendix; the complete set of fermionic equilibrium variables is given in ((5.28) and (5.29)), complementing the variables for the transversal bosonic modes listed in (5.36) of [14].

In this paper we studied the equilibrium mechanics of superluminal quanta. We have not discussed applications, apart from some estimates on the cosmic tachyon background. But there are a variety of systems where one can try to spot quantum tachyons by means of the transition rates calculated in 4 and 5. Estimates of tachyonic ionization cross sections of Rydberg atoms are given in [14], the effect of tachyon radiation on Lamb shifts in hydrogenic systems and on hyperfine intervals is studied in [18]. Superluminal cyclotron radiation in planetary magnetospheres, and tachyonic synchrotron radiation and inverse Compton scattering in the magnetic fields of supernova remnants will be discussed elsewhere.

Acknowledgements

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, the Institute of Mathematical Sciences, Madras, and the Tata Institute of Fundamental Research, Bombay, are likewise gratefully acknowledged. I would like to thank Nandor Balazs and George Contopoulos for exciting discussions.

References

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4. Th. Des Coudres Arch. Néerland. Sci. II 5 (1900), p. 652.

5. A. Sommerfeld Proc. Konink. Akad. Wet. (Sec. Sci.) 7 (1904), p. 346.

6. Ya.P. Terletsky Sov. Phys. Dokl. 5 (1961), p. 782.

7. R. Newton Science 167 (1970), p. 1569. 

8. E. Chaliasos Physica A 144 (1987), p. 390. 

9. D. Wycoff and N.L. Balazs Physica A 146 (1987), p. 175. 

10. R. Tomaschitz Chaos Solitons Fract. 7 (1996), p. 753. 

11. R. Anderson, I. Vetharaniam and G.E. Stedman Phys. Rep. 295 (1998), p. 93. 

12. R. Tomaschitz Celest. Mech. Dyn. Astron. 77 (2000), p. 107. 

13. R.F. Streater and A.S. Wightman PCT, Spin and Statistics, and All That, Benjamin, New York (1964).

14. R. Tomaschitz Physica A 293 (2001), p. 247. 

15. R. Tomaschitz Class. Quant. Grav. 18 (2001), p. 4395. 

16. A.S. Goldhaber and M.M. Nieto Rev. Mod. Phys. 43 (1971), p. 277. 

17. R. Tomaschitz Int. J. Mod. Phys. A 14 (1999), p. 5137. 

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Appendix A. Equilibrium mechanics of superluminal fermionic modes

We derive the high- and low-temperature expansions of the internal and free energy as well as the number density, based on the Fermi–Dirac spectral function (5.5) of the longitudinal modes in the black-body limit. The thermodynamic formalism for the transversal modes in Bose–Einstein statistics has already been worked out in [14], and the longitudinal modes do not really require new calculations but some reassembling.

We start with the internal energy, cf. (5.4) (with Image , ωI=0),

(A.1)
Image
where α colon, equals mtc2/(kT). The asymptotic low-temperature limit is easily calculated via Watson's Lemma, by replacing the root in (A.1) by its series expansion,

(A.2)
Image
In the high-temperature limit, a systematic convergent expansion of Û(α) is obtained by means of the Euler series (x<π)

(A.3)
Image
We split the integral (A.1) into Û(α)=Û0,

(A.4)
Image
and choose α<δ<π. Series (A.3) converges absolutely in [0,δ]; we substitute it into Û0 and interchange summation and integration, arriving so at

(A.5)
Image
The integrals in (A.5) represent hypergeometric functions [14], that is,

(A.6)
Image
Finally we insert (A.6) into (A.5), and interchange summations, Û0(2)0(2)odd0(2)even,

(A.7)
Image


(A.8)
Image
The series in ((A.4), (A.5), (A.6), (A.7) and (A.8)) constitute the high-temperature expansion of the internal energy,

(A.9)
Û(α)=Û0(1)0(2)odd0(2)even .
The coefficients in this expansion must be independent of δ, which can easily be checked by means of (A.4) and the integral representation of the cn(δ),

(A.10)
Image
and similarly for n>2, with more terms subtracted. The series in ((A.4) and (A.7)) converge for α<π, which defines the convergence radius of the high-temperature expansion. I also remark, as an addendum to Eq. (4.19) of [14], where the internal energy of the transversal bosonic modes was studied, that Image . This constant is defined in Eq. (4.17) of [14] by an integral which can be evaluated in closed form as indicated.

The free energy can be handled quite similarly,

(A.11)
Image
so that the low-temperature expansion reads

(A.12)
Image
The high-temperature expansion is again obtained by splitting the integral and expanding either nominator or denominator, Image ,

(A.13)
Image
Image is calculated by means of the Euler expansion (A.3), Image ,

(A.14)
Image
The integrals in (A.14) admit the expansion [14]

(A.15)
Image
and by combining ((A.14) and (A.15)), we obtain Image ,

(A.16)
Image
The coefficients cn(δ) are the same as in (A.8) and (A.10). [Apart from an integration constant, Image follows from Û(α) via term by term integration, Image ] The high-temperature expansion of the free energy reads, cf. ((A.13), (A.14) and (A.16)),

(A.17)
Image

We turn to the fermionic number density,

(A.18)
Image
which is not an independent variable, as the chemical potential vanishes. The low-temperature expansion reads

(A.19)
Image
and the high-temperature limit is likewise found completely analogously to the foregoing. At first we write Image ,

(A.20)
Image
and Image is then further split into Image ,

(A.21)
Image
The series expansion of the integrals in (A.21) is

(A.22)
Image
ψ is the logarithmic derivative of the gamma function, elementary for (half-)integers. It enters here by a limit procedure, kk+var epsilon, needed to circumvent the poles in the hypergeometric functions defined by the integrals, cf. [14]. Thus, Image ,

Image


(A.23)
Image


(A.24)
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The coefficients (A.24) admit integral representations such as, cf. (A.10),

(A.25)
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The high-temperature expansion of the particle density is thus, cf. ((A.20), (A.21) and (A.23)),

(A.26)
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To summarize, we list the first two terms of the series expansions derived in this Appendix. In the low-temperature limit, we find from ((A.2), (A.12) and (A.19)),

(A.27)
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with ζ(3)≈1.202 and ζ(5)≈1.037. The first two terms of the high-temperature expansions follow from ((A.9), (A.17) and (A.26)),

(A.28)
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All other fermionic variables, cf. ((5.28) and (5.29)), can be assembled from ((A.1), (A.11) and (A.18)).

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