Volume 387, Issue 14, 1 June 2008, Pages 3480-3494
Roman Tomaschitz, a,
Fermionic power-law distributions are derived by the second quantization of classical power-law ensembles, and applied to ultra-relativistic electron populations in the Galactic center. The γ-ray flux from the direction of the compact central source Sagittarius A* is fitted with a superluminal cascade spectrum. In this way, estimates of the radiating electron plasma in the Galactic center region are obtained, such as the power-law index, temperature, particle number, and internal energy. The spectral averaging of the tachyonic radiation densities with Fermi power-laws is explained. Fugacity expansions of the thermodynamic variables (thermal equation of state, entropy, isochoric heat capacity, and isothermal compressibility) are obtained in the quasiclassical high-temperature/low-density regime, where the spectral fit is carried out. The leading quantum correction to these variables is calculated, and its dependence on the electronic power-law index and the thermal wavelength is discussed. Excess counts of cosmic rays from the Galactic center region are related to the plasma temperature inferred from the cascade fit.
Keywords: Superluminal radiation; Tachyonic cascade spectra; Fermi power-law ensembles; Ultra-relativistic electron plasma; Quasiclassical fugacity expansion; Spectral averaging
PACS classification codes: 05.30.Fk; 05.70.Ce; 52.25.Kn; 95.30.Tg
- 1. Introduction
- 2. Thermodynamic variables of Fermi power-law ensembles
- 3. Equations of state, entropy, and heat capacity of fermionic power-law densities in the quasiclassical regime
- 3.1. Low-temperature asymptotics
- 3.2. Fugacity expansion and thermal wavelength in the high-temperature limit
- 4. Superluminal γ-rays from the Galactic center: Tachyonic spectral maps and electronic source distributions
- 5. Conclusion
1. IntroductionElectronic power-law distributions are commonly used in electromagnetic spectral averages to model the synchrotron emission of astrophysical sources, such as the X-ray spectra of supernova remnants . In this article, we quantize Boltzmann power-law densities, exponentially cut power-law distributions , where H=mc2γ is the free electronic Hamiltonian. In Fermi–Dirac statistics, we arrive at densities
γ. δ is the power-law index, and we use the shortcuts and α=−βμ/(mc2), where μ is the chemical potential. Electronic and protonic power-law densities have been invoked to perform electromagnetic and hadronic spectral fits to various TeV γ-ray sources recently discovered with imaging air Cherenkov detectors . Tachyonic spectral fits to the cascade spectra of γ-ray pulsars and microquasars are based on electronic power-law averages as well  and . Power-law indices inferred from these spectral fits or, for that matter, from the magnetospheric radio emission of planets  and , usually range in the interval 0δ4.
Here, we give evidence for superluminal radiation from the Galactic center by fitting recently obtained spectral maps ,  and  with tachyonic cascade spectra. The tachyonic γ-ray wideband consists of two cascades generated by ultra-relativistic electron populations, and the spectral fit allows us to infer the thermodynamic parameters. The observed spectra are clearly distinguishable from electromagnetic synchrotron and inverse-Compton fits, due to the emergence of extended spectral plateaus. We show that the TeV spectral map of the Galactic center admits a tachyonic extension into the GeV region, providing an excellent fit to the spectrum of the unidentified γ-ray source 3EG J1746 − 2851  associated with the Galactic central source Sagittarius A*. The superluminal cascades are generated by an ultra-relativistic electron gas at high temperature and low density, so that we can use distribution (1.1) in the quasiclassical regime to calculate the thermodynamic variables and the spectral averages.
In Section 2, we set up the thermodynamic formalism of fermionic power-law densities (1.1). Starting with the grand partition function, we derive the quasiclassical fugacity expansion of the thermodynamic variables, such as the caloric and thermal equations of state, entropy, specific heat, and compressibility. In Section 3, we discuss the leading quantum correction to the classical thermodynamic functions. In the high-temperature regime, the quantum corrections become more pronounced with increasing electronic power-law index, exhibiting the same temperature scaling as the classical limit usually dominant in the high-temperature/low-density regime. This is illustrated by calculating the mentioned variables for an increasing sequence of electron indices. We study the thermal wavelength of fermionic power-law ensembles in the low- and high-temperature regime, and discuss the range of applicability of the quasiclassical fugacity expansion.
In Section 4, we average the quantized superluminal radiation densities with the power-law distributions (1.1), and derive the fugacity expansion of the spectral averages. We perform a tachyonic cascade fit to the γ-ray spectrum of the Galactic center, and obtain estimates of the thermodynamic parameters of the electron plasma generating the superluminal radiation. The cutoff energy of the high-energy cascade fitting the TeV spectrum can be related to anisotropies in the cosmic ray spectrum detected by the AGASA and SUGAR air shower arrays  and . If the TeV cascade is generated by a protonic source population, this requires a thermal proton density at . This cutoff temperature very closely matches the upper energy edge of both the AGASA and SUGAR excess counts from the Galactic center region, suggesting that the compact radio source Sagittarius A* is capable of accelerating protons into the 1018 eV region. In Section 5, we present our conclusions.
2. Thermodynamic variables of Fermi power-law ensembles
We start with the fermionic partition function,
H=mγ is the free Hamiltonian. The electronic Lorentz factors γ=(1−υ2)−1/2 range in an interval γ1γ<∞. H1=mγ1 is the lower threshold energy, and γ1≥1 the lower edge of Lorentz factors of the electron distribution. The momentum parametrization is , so that we can substitute , where the angular integration has already been performed, as there is no angular dependence in the integrand. The Heaviside step function θ restricts the integration in (2.1) to γ≥γ1. The exponent α defines the fugacity , and is related to the chemical potential by μ=−mα/β, cf. after (2.26). δ is the electronic power-law exponent, and the cutoff parameter in the Boltzmann factor, so that the relativistic Fermi–Dirac distribution is recovered with δ=0 and γ1=1. Here, we study power-law ensembles of arbitrary real power-law index δ. We use c=ħ=1 for most of this article; the units can easily be restored, e.g., , where the ratio of electron rest energy and Boltzmann constant is .
The grand partition function (2.1) is
obtained via a standard trace calculation in fermionic occupation
We employ box quantization, discretizing the wave vector as .
The following summations are taken over integer lattice points
corresponding to periodic boundary conditions on a box of size L3.
By making use of ,
we may write trace (2.3) over the
basis states |n
L→∞ amounts to replacing the summation over the lattice wave vectors by the integration ; the factor of two accounts for the spin degeneracy. As we have put ħ=c=1, we can identify , so that . Performing these substitutions in (2.5), we arrive at integral representation (2.1) of the partition function, since the Heaviside function in the integrand (2.2) is equivalent to the restriction p≥k1 of the momentum integration (2.1).
The internal energy,
in (2.1) with the Lorentz factor as explained after (2.2). The integral representations (2.6), (2.7) and (2.8) are the key ingredients of the Legendre formalism employed below; in particular, logZ(δ,β,α,V) in (2.8) is the starting point for the quasiclassical fugacity expansion (2.10) (at high temperature and low density) of the thermodynamic functions. The opposite asymptotic limit, the nearly degenerate quantum regime (low temperature, high density), is based on the representation
(2.8) has been removed by a partial integration. That is, we write the integrand in (2.8) as f(γ)g′(γ)/3, with and g=(γ2−1)3/2, and integrate by parts to arrive at (2.9).
For the remainder of this section, we derive the
quasiclassical fugacity expansions of the thermodynamic variables. In
Section 3, we will
discuss the low- and high-temperature limits of these expansions and
conditions for their applicability. We start by expanding partition
function (2.8) in
ascending powers of ,
K1=K and Kn=Kn,0. Differentiation with respect to β is denoted by a prime, . The fugacity expansions of the partition function, internal energy, and particle number thus read
. On substituting (2.16) into (2.13) and (2.14), we find the fugacity expansion of the partition function in Helmholtz parametrization
Ucl is the classical limit, the internal energy of a Boltzmann power-law density , and the first two series coefficients in (2.19) read
-series (2.18), (2.19), and, cf. (2.16),
u1,2 are defined in (2.20), and
 and (4.13), with a term added owing to the multiplicity factor in (2.1).
The thermal equation of state is derived from the Helmholtz
μ=∂F/∂N=−mα/β is found by substituting the series expansion (2.22) of α.
The isochoric specific heat and the isothermal compressibility
CV,cl denotes the classical limit
(2.26) for V (since , cf. (2.17)),
κT,cl=1/P is the classical limit. Thermodynamic stability requires CV≥0 and κT≥0. The leading order of the fugacity expansion of CV in (2.28) and κT in (2.30) is in either case positive, as it coincides with the classical limit based on a Boltzmann power-law density .
3. Equations of state, entropy, and heat capacity of fermionic power-law densities in the quasiclassical regime
3.1. Low-temperature asymptotics
In the quasiclassical regime covered by the fugacity
expansions in Section 2, the
low-temperature limit of the thermodynamic variables is determined by
the β1 asymptotics of integral K(δ,β,γ1)
in (2.11). At γ1=1,
the asymptotic 1/β-series
K2,1/(K1,1K1), with 45/16 replaced by 57/16. As for coefficient u1 in (2.20), we note , up to terms of O(1/β). Finally, K1,1/K11 in leading order. Since , cf. (2.17), these ratios already suffice to obtain the temperature and density scaling of the leading quantum correction to the thermodynamic variables. The low-temperature limit of the internal energy (2.19) is found as
, that is, higher orders in N/V. The leading factor Ucl is the low-temperature expansion of the internal energy of a classical power-law density . The thermal equation of state (2.26) reads
∝N/V stems from the linear -term in (2.26). We may replace β by the thermal wavelength, , to find condition for the quasiclassical fugacity expansion to apply; at low temperatures, the gas has to be sufficiently dilute. The mean kinetic energy per particle in the low-temperature regime is EavU/N−m, so that β∝m/Eav in leading order. We thus recover the nonrelativistic wave-mechanical scaling . (2.28) and (2.29),
κT,cl=1/P, and the quantum correction is found as, cf. (2.30),
(3.3)–(3.7) are in ascending powers of N/V or P. We have only stated the leading quantum correction, that is, the term linear in N/V or P. The indicated temperature scaling of this term is meant in leading order as well, the next-to-leading order being smaller by a factor of O(1/β), cf. (3.3).
3.2. Fugacity expansion and thermal wavelength in the high-temperature limit
Like in Section 3.1, the
quasiclassical high-temperature asymptotics of the thermodynamic
variables are based on the ascending -series
derived in Section 2, namely U
in (2.19), S
in (2.23), the
thermal equation of state (2.26), and CV
in (2.28). As for
the compressibility κT,
we will use the ascending P-series
in (2.30). The
leading quantum correction to these variables is the term linear in
which is in all cases composed of the K-ratios,
cf. (2.11) and (2.12),
(3.9) as yet. The entropy has the usual logarithmic divergence, the heat capacity approaches a finite limit, and the internal energy has a 1/β divergence. All quantum corrections vanish for β→0. At δ=2, we obtain
1/β as compared to the previous cases δ=0,1, which has implications for the thermal wavelength, cf. after (3.16). The quantum correction to the compressibility is of the same order as in the thermal equation, since PmN/(βV).
β→0, owing to a double-logarithmic divergence. The heat capacity approaches zero logarithmically, but stays positive at finite temperature. The quantum corrections vanish logarithmically, except for κT. At δ=4, the expansions read
lE(β) defined in (3.12). In this case, the internal energy diverges logarithmically, and the entropy approaches a finite limit at β=0. (For power-law indices δ>4, the internal energy reaches saturation, attaining a finite limit at β=0 like the entropy, since integral (2.6) stays finite without exponential cutoff.) In (3.14), the quantum correction to the thermal equation is in leading order independent of β, approaching the indicated finite limit linear in N/V. The quantum correction to the specific heat has the same linear β-dependence as the classical term, in leading order that is, and both terms are positive. (At δ=5, the classical term as well as the quantum correction scale ∝β2lE(β) in leading order, with positive proportionality constants.)
In the fugacity expansions (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14), the quantum correction is the term linear in N/V (or P in the case of κTP). Only the leading order in β is indicated, in the classical term as well as the quantum correction, except for the entropy function in (3.14), where the next-to-leading order in β is included in the quantum correction, so that the heat capacity can be recovered by differentiation, cf. (2.27). Otherwise, the omitted β-terms are by at least a factor of O(βlogβ) smaller than the indicated ones. (In the expansion procedure, the logs are treated as constants.) The ellipses in expansions (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14) indicate higher-order quantum corrections in powers of N/V. The units are restored by replacing m by mc2 on the left-hand side of the above equations as well as in β, cf. after (2.2); on the right-hand side, m is replaced by mc/ħ.
As mentioned after (3.8), high-temperature expansions at a non-integer power-law exponent δ are calculated analogously, by making use of ratios (3.8) and the expansions of K(δ,β,1) in Ref. . High-temperature expansions of thermodynamic variables are differently structured in different δ-intervals, n−1<δ<n, joining the expansions (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14) at integer δ. Ultra-relativistic high-temperature expansions of K(δ,β,γ11) have been obtained in Ref.  as well, to be substituted into the ratios (3.8) in the case of power-law distributions with Lorentz factors exceeding a high-energy threshold, cf. (2.3).λT is chosen in a way to recover the usual definition, , in the low-temperature limit: for β1. The numerical proportionality factors in and λT are a mere convention; λT provides a second length scale to be compared to (V/N)1/3, which sets the scale for the quantum correction, the quasiclassical regime being defined by , cf. after (3.4). The high-temperature scaling of λT at integer power-law indices δ=0,1,…,4 is
lE(β) denotes the logarithmic temperature dependence (3.12). If δ3, we invert the caloric equation of state to find in leading order, cf. (3.9), (3.10), (3.11), (3.12) and (3.13),
Eav stands for the mean particle energy U/N, and β1 is implied. On combining (3.16) and (3.17), we find the high-temperature dispersion relation λT(Eav). At δ=0,1, we recover the ultra-relativistic wave-mechanical scaling λT∝1/Eav. For δ≥4, the thermal wavelength is proportional to the Compton wavelength, λT∝1/m. For intermediate power-law indices, the thermal wavelength is a hybrid of ultra-relativistic and Compton wavelength, e.g., .
4. Superluminal γ-rays from the Galactic center: Tachyonic spectral maps and electronic source distributions
In this section, we average tachyonic radiation densities  with fermionic power-law distributions, and use the spectral averages to perform a cascade fit to the γ-ray broadband of the Galactic central source Sagittarius A*, cf. Fig. 1 and Fig. 2. The thermodynamic parameters of the electron plasma in the Galactic center generating the superluminal cascades are extracted from the spectral fit, cf. Table 1. First we briefly summarize the tachyonic radiation densities, cf. (4.1) and (4.2), then we explain the spectral averaging, in particular the fugacity expansion of the spectral functions and their low- and high-temperature asymptotics, cf. (4.3), (4.4), (4.5), (4.6), (4.7), (4.8), (4.9), (4.10), (4.11), (4.12), (4.13), (4.14), (4.15), (4.16), (4.17), (4.18) and (4.19). The spectral fit in Fig. 1 and Fig. 2 is discussed after (4.20).
Fig. 1. γ-ray wideband of the Galactic center. EGRET data points from Ref. , HESS points from Refs.  and , MAGIC points from Ref. . EGRET points refer to the source 3EG J1746 − 2851, HESS and MAGIC points to HESS J1745 − 290. The solid line T+L depicts the unpolarized differential tachyon flux , obtained by adding the flux densities ρ1,2 of two electron populations and rescaled with E2 for better visibility of the spectral curvature, cf. (4.20). The transversal (T, dot-dashed) and longitudinal (L, double-dot-dashed) flux densities add up to the total flux T+L. The χ2-fit is done with the unpolarized tachyon flux T+L, and subsequently split into transversal and longitudinal components. The exponential decay of the cascades ρ1,2 sets in at about , implying cutoffs at 5.5 TeV for the ρ1 cascade and at 38 GeV for ρ2, which terminate the spectral plateaus. The unpolarized flux T+L is the actual spectral fit, the parameters of the electron densities are recorded in Table 1.
Each ρi stands for a Maxwell–Boltzmann density , cf. (4.13). β is the cutoff parameter in the Boltzmann factor. determines the amplitude of the tachyon flux generated by ρi, from which the electron count is inferred at the indicated distance of 8 kpc, cf. after (4.21). (The subscript 1 in and has been dropped.) is the electron temperature, and U (erg) the internal energy of the thermal densities ρi. The cascades labeled ρ1,2 in Fig. 1 and Fig. 2 are obtained by averaging the tachyonic radiation densities (4.1) with the electron densities ρi, cf. (4.14), (4.15) and (4.16). The parameters β and are extracted from the least-squares fit T+L in Fig. 1.
Fig. 2. Close-up of the HESS spectrum in Fig. 1. The TeV spectral map coincides with the ρ1 cascade, since the ρ2 flux is exponentially cut at 38 GeV. T and L stand for the transversal and longitudinal flux components, and T+L labels the unpolarized flux. The HESS points define a spectral plateau in the high GeV range typical for tachyonic cascade spectra  and . The spectral curvature is intrinsic, being generated by the Boltzmann factor in the electron densities.
The quantized tachyonic radiation densities of a uniformly
moving spinning charge read 
0ωωmax(γ) can be radiated by a uniformly moving charge, the tachyonic spectral densities being cut off at the break frequency ωmax. The units ħ=c=1 can easily be restored. We use the Heaviside–Lorentz system, so that αq=q2/(4πħc)≈1.0×10−13. The tachyon mass is , and the tachyon–electron mass ratio mt/m≈1/238. These estimates are obtained from hydrogenic Lamb shifts . A positive ωmax(γ) requires Lorentz factors exceeding the threshold μt in (4.2), since ωmax(μt)=0. The lower threshold on the speed of the electron for radiation to occur is thus υminmt/(2mμt). The tachyon–electron mass ratio gives υmin/c≈2.1×10−3, cf. Ref. . AFm3V/π2, cf. (2.6), (2.7) and (2.8). The particle number is , where γ1 is the lower edge of Lorentz factors in the source population. The exponential cutoff in (4.3) is related to the electron temperature by . In this section, we write the exponent of the fugacity with a hat, cf. after (2.2), to distinguish it from the electron index customarily defined as α=δ−2, cf. (4.13). The normalization factor AF is dimensionless via m→mc/ħ, cf. (1.1); the volume factor in the quantum corrections discussed in Section 3 is thus found as , where is the reduced electronic Compton wavelength.
The spectral average of the radiation densities (4.1) is carried
(4.4) can be reduced to the fermionic spectral functions
γ1≥μt, cf. after (4.2). The threshold Lorentz factor γ1 defines the break frequency 
, and if ω>ω1. By making use of the spectral functions (4.6), we can write the averaged radiation densities (4.4) as
in (4.5) and ω1 in (4.7). The superscripts T and L denote the transversal and longitudinal radiation components, cf. (4.1). The spectral functions in (4.8) are assembled by substituting the radiation densities (4.1) into the integral representation (4.6),
in (4.3). fk(γ1) in (4.10) reads
is defined by the leading term in series (4.11), the Boltzmann power-law density
. (As mentioned after (4.3), exponent in the normalization factor defines the fugacity, and is not to be confused with the electron index α=δ−2.) The electron count based on the classical density (4.13) is , to be identified with the renormalized electron count obtained from the spectral fit, cf. after (4.21). is the classical spectral density, obtained by dropping all terms containing mt/m factors in (4.1) and (4.2). In particular, and , cf. (4.1). Carrying out the integration in (4.14), we find the classical spectral functions
in (4.9) by dropping the mt/m terms and substituting the leading order of the fugacity expansion of fk(γ1), cf. (4.12). The classical limit of the fermionic spectral average in (4.8) thus reads
is the classical limit of the break frequency (4.7). (3.4) and (3.15). Spectral averaging at low temperature and high density will be discussed elsewhere. The spectral fit discussed below is carried out in the high-temperature regime, βγ11. In the fugacity expansion (4.12) of the reduced spectral functions, we can therefore substitute the ascending series
k−δn is zero or a negative integer, k−δn=−m, m≥0, we use instead of (4.18) the ascending series of the exponential integral Em+1(βγ1),
Fig. 1 and Fig. 2 is based on the E2-rescaled flux densities 
d is the distance to the source and the spectral average (4.16) (with ω=E/ħ). The fit is done with the unpolarized flux density of two electron populations ρi=1,2, thermal Maxwell–Boltzmann distributions (4.13) with α=−2 and γ1=1, cf. Table 1. Each electron density generates a cascade ρi, and the wideband fit is obtained by adding two cascade spectra, labeled ρ1 and ρ2 in Fig. 1. As for the electron count, , we use a rescaled parameter for the fit,
(4.20). Here, implies the tachyon mass in keV units, that is, we put mt≈2.15 in the spectral density (4.1). At γ-ray energies, only a tiny -fraction (the ratio of tachyonic and electric fine structure constants) of the tachyon flux is absorbed by the detector, which requires a rescaling of the electron count n1, so that the actual number of radiating electrons is . We thus find the electron count to be , where defines the tachyonic flux amplitude extracted from the spectral fit . This renormalized count is to be identified with the particle number N in the thermodynamic variables of the electron plasma. The electron temperature and cutoff parameter in the Boltzmann factor are related by , and the energy estimates in Table 1 are based on , cf. (3.9).
Fig. 1 and Fig. 2 show the spectral map of the TeV γ-ray source HESS J1745 − 290 and the GeV source 3EG J1746 − 2851. Both sources are located in the vicinity of the unidentified compact radio source Sagittarius (Sgr) A* at the core of the Galactic center region, which comprises the supernova remnant Sgr A East and the pulsar wind nebula G359.95 − 0.04, as well as molecular cloud complexes such as Sgr B and Sgr C  and .
In Table 1, we list the thermodynamic parameters of the electron populations generating the tachyonic cascade spectra ρ1,2 depicted in the figures. In the case of protonic source densities (4.13), the cutoff energy in the Boltzmann factor has to be multiplied by 1.84×103, the proton/electron mass ratio, resulting in a cutoff energy of 1016.2 eV for a protonic ρ2 population. The protonic high-energy population ρ1 is cut at 1018.4 eV, which is to be compared to the excess fluxes from the Galactic center region observed by AGASA in the 1018–1018.4 eV range , and the SUGAR array in the 1017.9–1018.5 eV interval .
The tachyonic spectral maps are further explained in the figure captions, and can be compared to electromagnetic inverse-Compton fits  or hadronic fits based on proton–proton scattering and pion decay  and . The electronic source count for the Crab Nebula at is 1.9×1050, cf. Ref. , as compared to 3.4×1049 for HESS J1745 − 290 and 3EG J1746 − 2851, at a distance of 8 kpc, cf. Table 1. The internal energy of the electron gas is 7.2×1051 erg, to be compared to hadronic model estimates predicting 5×1045 erg , 1049–1050 erg , and 5.1×1050 erg  for a protonic source population.
We have investigated superluminal radiation from electron populations in the Galactic center region, and found the thermodynamic parameters of the source densities. We demonstrated that the γ-ray wideband of the Galactic center can be fitted with a tachyonic cascade spectrum. In particular, the extended spectral plateau in the MeV–GeV range as well as the spectral curvature apparent in double-logarithmic plots are reproduced by the cascade fit in Fig. 1 and Fig. 2. Estimates of the temperature, the source count, and the internal energy of the electron plasma generating the superluminal cascades are given in Table 1.
In Section 4, we averaged the superluminal radiation densities with fermionic power-law distributions, and derived the quantized spectral functions. The power-law densities in Sections Sections 2 and 3 and the spectral averages were studied mostly in the quasiclassical limit, as γ-ray spectral fits are done in the high-temperature regime. The opposite asymptotic limit of nearly degenerate fermionic power-law ensembles at low temperature and high density will be discussed elsewhere.
We quantized Boltzmann power-law densities in Fermi–Dirac statistics, derived the fermionic partition function, developed the Legendre formalism of fermionic power-law distributions, and calculated the fugacity expansion of the thermodynamic variables, cf. Section 2. The qualitative dependence of the variables on the electronic power-law index, the temperature scaling of the quantum corrections, and the thermal wavelength of power-law ensembles were investigated in Section 3. We have focused on γ-ray spectra, which can be fitted with ultra-relativistic electron densities at high temperature, cf. Section 4. As for tachyonic X-ray spectra obtained from diffraction gratings , the tachyon mass of 2 keV has to be included in the dispersion relation when parametrizing the glancing angle in the Bragg condition with energy, which affects the shape of the spectral maps in the X-ray bands; this will be discussed elsewhere. The spectral fit in Fig. 1 and Fig. 2 is performed with the unpolarized tachyon flux. At γ-ray energies, the speed of tachyons is close to the speed of light, the basic difference to electromagnetic radiation being the longitudinal flux component . The polarization of tachyons can be determined from transversal and longitudinal ionization cross-sections of Rydberg atoms, which peak at different scattering angles .
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.