Volume 307, Issues 3-4, 1 May 2002, Pages 375-404
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
- 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
- Appendix A. Equilibrium mechanics of superluminal fermionic modes
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 , 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 , 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 . 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
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(x−x(t)) and j=qvδ (x−x (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/(4πc)≈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 (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
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,
The Maxwell equations (2.1) (with ρ=0, j=0)
read in Fourier space
2.7), the dispersion relation,
2.9) is real, ω*(k)=ωR−iωI satisfies
Eq. (2.10) defines the Fourier coefficients Â0(k) of the time component, once the are chosen. As the solution ω (k) is not unique, a further summation in (2.6) may be necessary, so that the coefficients 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 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)),
2.9). 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). and are the corresponding Fourier coefficients, such as , etc. The time average of the product is readily calculated as2.6), and has as effect that terms containing and drop out. The damping factor exp (−ωIt) is regarded as constant within the averaging period 2π/ωR. For instance,
2.15) hold for the averaged square of B (BL=0) and the inductions, cf. ((2.12), (2.13) and (2.14)), with
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
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
(2.13) and (2.14)),
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
3.8) 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 is self-consistent. We may now write in (2.21)
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 ∫ρE and ∫Idis, 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)),
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, ∫ρET,L=O (1), ∫ST,L=O (1), and ∫IdisT,L=O (ωI).
The positivity of the energy averages ∫ρET and ∫IdisT is
ensured by requiring
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
3.11) gets a familiar shape,
(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)),
3.16). The energy per unit time dissipated by transversal and longitudinal modes is found as, cf. ((3.11) and (3.8)),
As for the microscopic structure of the ether, we consider a
classical oscillator model , which gives
3.19) is meant as a power series expansion in α0. As mentioned, , and thus the positivity conditions (3.12) are satisfied. For ω→∞, we find
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, nZ3, 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
are Hermitian and commute. We consider the basis vectors |n1,…,ni,…,n∞
e.g. |0 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 n|n′=δn1,n′1…δn∞,n′∞.
Operators satisfying the above commutation relations are readily found,
20], ∑k→L3/(2π)3∫dk, we find
3.15), and k0 mtc/ 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/, 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
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 t1/(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 kω/c. For ω→0, we may
approximate kmt and k′ωmt−1c−2(1+α0/ω02),
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,
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,
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) 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(+)id and idan(+), 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
(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)),
(4.12) 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)),
The transition rate for induced absorption, with a given
linear polarization λ, is obtained by a standard procedure,
Section 5, so we do not indicate here the T-superscripts in wabsT,ind and TabsT.) For arbitrary real numbers ωE,I,
Section 3, energy conservation is only approximate, Re ωk≈ωj−ωi, hence ωE≈0. 3.7) and (3.13). The ωR-integration in (4.18) is then carried out by steepest descent, at ωji ωj−ωi,
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 ωR=ωji, likewise the remaining ωR-dependent factors in (4.18), and , cf. (3.15). The direction of the tachyonic wave vector k is specified by the angular variables in dΩ, and its magnitude by k(ωji), cf. (3.7). The coefficient αkT in (3.14) is likewise taken at ωR=ωji. 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
4.23), 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 |TemT|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)),
(4.27) and (4.6)) is provided by the equilibrium balancing of emission and absorption events,
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 k,λ· in ((4.23), (4.25) and (4.27)) by the transversal component of the gradient, T −k0(k0·).
In dipole approximation, we may drop the exponential e±ikx
in the preceding integrals and use the identity
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), as ωRω0 is required by the Born approximation. The tachyonic k(ωR) and ωI(ωR) 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), with regard to the tachyonic ionization of Rydberg atoms, is given in , 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+]+=δij,
which effects the sign change turning the energy density into a
positive definite operator,
(4.5) and (4.6)),
4.7) of the transversal density are to be inserted.) In the black-body limit ,
2.13) via Â0(k)=α0LαLa(k,3) and the expansion ((3.6), (3.7) and (3.8)),
(3.7) and (3.14)). Hence we may write the interaction as
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
(5.9), (4.13) and (4.11)),
(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),
Concerning emission, the transition rate is determined by (4.24) with TemL
defined in (5.11). To keep
the formal analogy to the derivation following (4.24), we write
, where −wemL,ind
stands for the contribution of the nk-proportional
terms in (5.11), and 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
3.14) by mtc/, and so obtain
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
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
(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)),
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 . 4.34), we find
Next we discuss the equilibrium mechanics of longitudinal
tachyonic black-body radiation,
, 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
βhν and γ
we find the location of this peak by solving
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. . 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/wT(ν0)≈2.6×105 and wT/wph(ν0)≈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 kT(ν0)≈4.6 eV. We so find from (5.27) wL/wT(ν0)≈5.6×10−6 and wT/wph(ν0)≈3.0×10−9. The Ly-α1 transition in hydrogenic uranium (ν0≈0.23 MeV) results in a bulk temperature of kT(ν0)≈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/wT(ν0)≈2.9×103, but a very small wT/wph(ν0)≈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. .
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,
14], despite the different frequency scaling of the spectral densities in the Rayleigh–Jeans limit.
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/(4πc)≈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 . 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  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 , provided we drop the adiabatic damping factors in the Einstein coefficients; the semiclassical derivation given in  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 . 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 .
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 , the effect of tachyon radiation on Lamb shifts in hydrogenic systems and on hyperfine intervals is studied in . 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.
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.
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 , and the longitudinal modes do not really require new calculations but some reassembling.
We start with the internal energy, cf. (5.4) (with
A.1) into Û(α)=Û0+Û∞,
A.3) converges absolutely in [0,δ]; we substitute it into Û0 and interchange summation and integration, arriving so at
A.5) represent hypergeometric functions , that is,
A.6) into (A.5), and interchange summations, Û0(2)=Û0(2)odd+Û0(2)even,
(A.4), (A.5), (A.6), (A.7) and (A.8)) constitute the high-temperature expansion of the internal energy,
A.4) and the integral representation of the cn(δ),
(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 , where the internal energy of the transversal bosonic modes was studied, that . This constant is defined in Eq. (4.17) of  by an integral which can be evaluated in closed form as indicated.
The free energy can be handled quite similarly,
A.14) admit the expansion 
(A.14) and (A.15)), we obtain ,
A.8) and (A.10). [Apart from an integration constant, follows from Û(α) via term by term integration, ] The high-temperature expansion of the free energy reads, cf. ((A.13), (A.14) and (A.16)),
We turn to the fermionic number density,
14]. Thus, ,
A.24) admit integral representations such as, cf. (A.10),
(A.20), (A.21) and (A.23)),
(A.9), (A.17) and (A.26)),
(5.28) and (5.29)), can be assembled from ((A.1), (A.11) and (A.18)).