Abstract

Polylogarithmic fugacity expansions of the partition function, the caloric and thermal equations of state, and the specific heat of fermionic power-law distributions are derived in the nearly degenerate low-temperature/high-density quantum regime. The spectral functions of an ultra-relativistic electron plasma are obtained by averaging the tachyonic radiation densities of inertial electrons with Fermi power-laws, whose entropy is shown to be extensive and stable. The averaged radiation densities are put to test by performing tachyonic cascade fits to the γ-ray spectrum of the TeV blazar Markarian 421 in a low and high emission state. Estimates of the thermal electron plasma in this active galactic nucleus are extracted from the spectral fits, such as temperature, number count, and internal energy. The tachyonic cascades reproduce the quiescent as well as a burst spectrum of the blazar obtained with imaging atmospheric Cherenkov detectors. Double-logarithmic plots of the differential tachyon flux exhibit intrinsic spectral curvature, caused by the Boltzmann factor of the electron gas.

1. Introduction

The tachyonic radiation field is a real Proca field with negative mass-square, , subject to the Lorentz condition , where m_t is the mass of the superluminal Proca field A_μ, and q the tachyonic charge carried by the subluminal electron current [3]. In the Proca equation, the mass term is added with a positive sign, and the sign convention for the d’Alembertian is ∂^ν∂_ν=Δ-∂²/∂t², so that is the negative mass-square of the radiation field. The negative mass-square refers to the radiation field rather than the current, in contrast to traditional theories based on superluminal source particles emitting electromagnetic radiation [4], [5], [6] and [7]. Estimates of the tachyon–electron mass ratio and the tachyonic fine structure constant are m_t/m≈1/238 and q²/(4πc)≈1.0×10^-13, obtained from hydrogenic Lamb shifts [8].

Tachyonic radiation implies superluminal signal transfer, the energy quanta propagating faster than light in vacuum, due to their negative mass-square, in contrast to rotating superluminal light sources emitting vacuum Cherenkov radiation [9], [10], [11], [12], [13] and [14]. This superluminal energy propagation by tachyonic vacuum modes is also to be distinguished from superluminal group velocities arising in photonic crystals, optical fibers, or metamaterials [15], [16], [17], [18] and [19]. In contrast to tachyonic quanta, the actual signal speed defined by the electromagnetic energy flow in these media is always subluminal and occasionally even opposite to the group velocity [20].

In Section 2, we derive the fugacity expansion of the internal energy and the partition function in the low-temperature/high-density regime. The thermodynamic variables of nearly degenerate power-law distributions are calculated in Section 3, such as the thermal equation of state, entropy, and free energy. We check the positivity of the isochoric heat capacity and the isothermal compressibility for arbitrary power-law index, demonstrating thermodynamic stability in the quantum regime. A generalization of non-relativistic Fermi–Dirac distributions by way of modified dispersion relations has also been suggested in Ref. [21]. Here, we consider ultra-relativistic multi-component plasmas in the collisionless regime [22], in stationary non-equilibrium described by power-law densities [1], [23] and [24].

In Section 4, we study tachyonic radiation densities averaged over electronic power-law distributions, calculate the spectral functions in the nearly degenerate quantum regime, and discuss the range of applicability of the fugacity expansion, including the crossover into the quasiclassical regime. We perform cascade fits to the γ-ray flux of the Markarian galaxy Mkn 421 (at redshift z≈0.031) in a quiescent state [25] and to a burst spectrum [26], and compare with the tachyonic spectral maps of other BL Lacertae objects. The spectral curvature is reproduced by the tachyonic cascades without resort to intergalactic attenuation mechanisms.

In Section 5, we present our conclusions. In Appendix A, the Sommerfeld asymptotics of incomplete Fermi integrals is derived, which is the basis of the fugacity expansion of the thermodynamic functions in Section 3 and the spectral functions in Section 4. This involves polylogarithms discussed in Appendix B.

2. Ultra-relativistic Fermi power laws

We start with the partition function

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

The μ→∞ asymptotics of the above variables is assembled from the fugacity expansion of the Fermi integrals in Appendix A. The ascending 1/μ series of the particle number reads, cf. Eqs. (A.35) and (A.37),

(2.6)

(2.7)

(2.8)

m/μ1

m/(βμ)1

(2.9)

3. Thermodynamic variables of nearly degenerate power-law distributions

The thermodynamic functions are obtained by iteratively solving (2.6) for the chemical potential μ, which is then substituted into the asymptotic series (2.7) and (2.8) of the internal energy and the partition function. Defining the Fermi momentum as p_F:=(3π²N/V)^1/3, we invert Eq. (2.6) in ascending powers of 1/p_F:

(3.1)

m/p_F1

x1

(3.2)

(3.3)

(3.4)

U/N3p_F/4

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

m/y1

m/(βy)1

(3.15)

(3.16)

(3.17)

In the fully degenerate case, at zero temperature, the power-law exponent δ drops out in all thermodynamic variables, and at finite β it does not enter in leading order. The regime below β_F:=m/(kT_F) is not accessible with the fugacity expansions derived here, which require both β_F1 and β_F/β1. If β1/β_F, the first-order correction proportional to δ is overpowered by the second order, which is independent of δ in this limit. Therefore, the Chandrasekhar mass limit of white dwarfs is not affected by the power-law exponent, as it assumes total degeneracy [28]. Order-of-magnitude estimates of cooling times derived from homology relations are not affected either, as they are based on the leading-order temperature and density scaling of pressure and specific heat [29].

We have put =c=1. To restore the dimensions, we rescale β=mc²/(kT) and p_F=(3π²N/V)^1/3 so that kT_Fμcp_F, and note . The expansion parameter x=mc/(βp_F) is chosen to be dimensionless, and the dimension of y=(12π²³c³P)^1/4 to be that of energy. The expansion parameter mc²/y in Eq. (3.13) is thus dimensionless as well, and p_Fy/c. Hence

(3.18)

4. Tachyonic cascade spectra

The spectral averaging of tachyonic radiation densities with electronic power-law distributions (2.2) has already been explained in Ref. [30], where we mainly focused on the quasiclassical regime, but also derived the general formalism applicable in the nearly degenerate case. In Eqs. (4.1), (4.2), (4.3), (4.4) and (4.5), we summarize the averaged radiation densities:

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

The units =c=1 can easily be restored. We use the Heaviside–Lorentz system, so that α_q=q²/(4πc)≈1.0×10^-13. The tachyon mass is , and the tachyon–electron mass ratio m_t/m≈1/238. These estimates are obtained from hydrogenic Lamb shifts [8]. The particle number reads , where γ₁ is the lower edge of Lorentz factors in the source population. The exponential cutoff in the spectral weights (4.3) is related to the electron temperature by β=mc²/(kT) and the chemical potential by μ=-mα/β. The normalization factor A_F is dimensionless via m→mc/; the volume factor in the thermodynamic functions in Section 3 is thus found as , where is the reduced electronic Compton wavelength.

The weight factors (4.3) are related to the Fermi integral F(a, b) in Eq. (A.1) (with a=k-1 and b=0) and the Sommerfeld decomposition (A.3) by

(4.7)

(4.8)

βγ₁z1

γ₁z1

The amplitude A_F and the fugacity e^-α are two independent fitting parameters in the source density dρ_F(γ), cf. Eqs. (2.2) and (4.3). This is in contrast to the classical limit, , where the factors of the amplitude A_Fe^-α cannot independently be determined from the spectral fit. This amplitude differs from a classical Boltzmann power-law distribution due to the fermionic multiplicity factor, the Boltzmann normalization being A_Fe^-α/2, cf. Eq. (3.7) in Ref. [24]. In the ultra-relativistic limit, γ1, the factor γ^-δ can formally be generated by analytic continuation in the momentum space dimension, since pmγ [33]. In the case of a genuine fermionic power-law distribution in the nearly degenerate quantum regime, one can determine the volume as well as the particle number from the spectral fit, cf. after Eq. (4.5).

We parametrize the spectral weights f_k(γ₁) with the chemical potential via z(μ) in Eq. (4.8). The independent fitting parameters in f_k(γ₁) are thus temperature β, threshold Lorentz factor γ₁, chemical potential μ, and the volume factor A_F of the Fermi distribution in Eq. (4.3). We may fix γ₁ at the lowest possible threshold, γ₁=μ_t, cf. Eq. (4.4) (and put γ₁=1 in the thermodynamic functions (2.3), (2.4) and (2.5)), so that the averaged spectral densities (4.1) simplify to

(4.9)

βγ₁z1

(4.10)

(4.11)

The classical limit of the fermionic spectral functions F^T,L(ω,γ₁) is the Boltzmann average B^T,L(ω,γ₁), obtained by dropping all terms containing m_t/m factors in Eq. (4.2), and replacing dρ_F in the spectral weights (4.3) by the classical density in Eq. (4.11). The classical spectral weights reduce to incomplete gamma functions. Terms containing m_t/m factors in Eqs. (4.4) and (4.5) are dropped as well, and the polarization coefficients reduce to and , cf. after Eq. (4.3). The classical limit of the averaged spectral densities p^T,L(ω)_F in Eq. (4.1) thus reads

(4.12)

The spectral fits of the BL Lacertae object (BL Lac) Markarian 421 in Fig. 1 and Fig. 2 are based on the E²-rescaled flux densities [34]

(4.13)

(4.14)

(4.15)

dcz/H₀

Fig. 1 shows the cascade fit of the Markarian galaxy Mkn 421 in a quiescent state [25], and Fig. 2 in a high emission state [26]. The redshift of Mkn 421 is z≈0.031, implying a distance of 140 Mpc. TeV γ-ray spectra of active galactic nuclei are usually assumed to be generated by inverse Compton scattering or pp scattering followed by pion decay. Both mechanisms result in a flux of TeV photons, assumed to be partially absorbed by interaction with background photons due to pair creation, so that the intrinsic spectrum has to be reconstructed on the basis of intergalactic absorption models depending on vaguely known cosmological input parameters. In contrast, the extragalactic tachyon flux is not attenuated by interaction with the background light, there is no absorption of tachyonic γ-rays. The curvature of the γ-ray spectra in double-logarithmic plots is caused by the Boltzmann factor of the electron densities generating the tachyon flux, so that the observed spectrum is already the intrinsic one, and no reconstruction is needed. The curvature present in the γ-ray spectra of BL Lacs is not correlated with distance; the spectral curvature does not increase with redshift if we compare the spectral fits in Fig. 1 and Fig. 2 to the spectral maps of other active galactic nuclei, cf. the figure captions.

5. Conclusion

Tachyonic γ-ray spectra of active galactic nuclei are generated by ultra-relativistic electron populations. Tachyons are radiation modes, unrelated to electromagnetic radiation. Electrons radiate tachyons, and these tachyonic quanta produce the observed γ-ray cascades. The tachyonic radiation modes are coupled by minimal substitution to the electron current. This field theory, a real Proca field with negative mass-square, admits a static potential analogous to the Coulomb potential, but oscillating because of the negative mass-square, and much weaker due to the small tachyonic fine structure constant. In the spectral maps, the tachyon–electron mass ratio shows in the cutoff energy of the cascades. The negative mass-square of tachyons implies superluminal velocity and allows longitudinal polarization. The tachyonic radiation field does not couple to electromagnetic fields, nor is it affected by electric charge. Thus, interaction of tachyons with photons can only happen indirectly via matter fields. In contrast to electromagnetic γ-rays, there is no extinction of the extragalactic tachyon flux by the cosmic background light, as tachyons do not interact with infrared photons. The ultra-relativistic electron plasma in the active galactic nucleus produces tachyonic γ-rays propagating unattenuated over intergalactic distances.

The tachyonic radiation density averaged over the electron populations in the galactic nucleus is generated by electrons in uniform motion. In particular, there is no electromagnetic radiation damping, as photons can only be radiated by accelerated charges, in contrast to tachyonic quanta, where the emission rate primarily depends on the electronic Lorentz factor rather than on acceleration [3]. The high plasma temperature inferred from the spectral fits implies ultra-high energy electrons. Such high electron temperatures are also found in Galactic pulsar wind nebulae, production sites of ultra-high energy cosmic rays [36].

Specifically, we fitted a quiescent as well as a flare spectrum of the γ-ray blazar Mkn 421 with tachyonic cascades, and found that the spectral curvature is intrinsic and reproduced by the superluminal spectral densities (4.1) averaged with ultra-relativistic electron distributions. The curvature present in the γ-ray spectra of active galactic nuclei is not correlated with distance, so that absorption of electromagnetic radiation due to interaction with background photons is not a viable explanation for the spectral curvature. By contrast, there is no intergalactic attenuation of the tachyon flux, as tachyons cannot interact with photons. In Table 1, we have given estimates of the temperature, the source count, and the internal energy of the electron populations in the galactic nucleus generating the superluminal γ-ray cascades.

The fugacity expansion of the thermodynamic variables of a nearly degenerate electron plasma was derived in Sections 2 and 3, and fermionic power-law densities were shown to admit a stable and extensive entropy function in the quantum regime, cf. Eqs. (3.5) and (3.6). In Section 4, we averaged tachyonic spectral densities with electronic power-law distributions, and obtained the fugacity expansion of the quantized spectral functions in the nearly degenerate low-temperature/low-frequency/high-density regime. The integral representation (4.3) of the spectral weights covers the crossover into the quasiclassical high-temperature/high-frequency/low-density regime studied in Ref. [2].

Superluminal radiation from ultra-relativistic electrons orbiting in magnetic fields was investigated in Ref. [37]. In the zero-magnetic-field limit, the averaged tachyonic synchrotron densities converge to the spectral densities (4.12). Orbital curvature induces modulations in the spectral slopes, but these ripples are attenuated when performing a pitch-angle average, cf. Figs. 1–3 of Ref. [37]. Thus we can use uniform radiation densities even in the presence of magnetic fields in the galactic nuclei.

Acknowledgments

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, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.