Abstract

Superluminal radiation emitted by electrons orbiting in strong magnetic fields is investigated. We show that electrons gyrating in the surface fields of rotation-powered neutron stars can radiate superluminal quanta (tachyons) via the synchrotron mechanism. The tachyonic luminosity of γ-ray pulsars is inferred from COMPTEL and EGRET observations, and so is the magnetospheric electron population generating this radiation. In the surface fields, electromagnetic synchrotron radiation in the γ-ray band is suppressed by a quantum cutoff, but not so tachyonic γ-radiation. This provides an exceptional opportunity to search for tachyon radiation, unspoiled by electromagnetic emission. Estimates of the superluminal power radiated and the tachyonic count rates are obtained for each of the seven established γ-ray pulsars, the Crab and Vela pulsars, as well as PSR B1706−44, Geminga, PSR B1055−52, B1951+32, and B1509−58. Detection mechanisms such as tachyonic ionization and Compton scattering are analyzed with regard to superluminal γ-rays.

1. Introduction

When considering superluminal quanta, we may try a wave theory or a particle picture as starting point. The latter has been studied for quite some time, but did not result in viable interactions with matter [1], [2], [3], [4], [5] and [6]. Here, tachyons will be modeled as wave fields with negative mass-square, coupled by minimal substitution to subluminal particles. Interaction with matter is indeed the crucial point, we will maintain the best established interaction mechanism, minimal substitution, by treating tachyons like photons with negative mass square, a real Proca field minimally coupled to subluminal matter [7] and [8].

We will work out a specific example, tachyonic synchrotron emission from electrons gyrating in the surface fields of γ-ray pulsars. The detectors used to collect the γ-ray fluxes operated in adjacent or partly overlapping energy bands with very different mechanisms, ionization [9], Compton scattering [10], and electron-positron pair creation [11], and we will discuss the respective cross-sections with regard to tachyonic γ-rays. We will calculate the energy stored in the electron population gyrating in the surface magnetic fields, and relate the slope of the electronic power-law distribution to the frequency scaling of the tachyon flux. We will conclude from the break frequencies of the observed fluxes that the gyration energies can reach the low TeV range, and comment on cosmic ray acceleration in the magnetospheres.

In Section 2, we study superluminal synchrotron radiation from electrons subjected to high magnetic field strengths. We will introduce the tachyonic counterparts of certain quantities familiar from electromagnetic synchrotron radiation, such as critical frequencies, break frequencies and critical field strengths, and explain how they relate to the spectral densities, the tachyon mass, and the electronic Lorentz factor. In Section 3, we study the synchrotron power radiated in tachyonic γ-rays by the magnetospheric electron populations of γ-ray pulsars. We calculate the synchrotron power in the high-magnetic-field limit as well as the tachyonic count rates and average the tachyonic spectral densities with electronic power-law distributions. The tachyonic γ-ray luminosity is inferred from the measured fluxes and all that goes with it, such as the electronic power-law indices and the energy range of the gyrating electrons. In Section 4, we present our conclusions.

2. Tachyonic synchrotron radiation in strong magnetic fields

The superluminal spectral densities can be readily assembled from the general formalism of tachyonic radiation theory [12] and [13], which will not be repeated here, but we restate the densities in Figs. (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), (2.7) and (2.8). There is a low-and a high-frequency regime, separated by the break frequency ω_b = γm_tc, where γ  1 is the Lorentz factor of the radiating ultra-relativistic source, typically a circularly orbiting electron, and m_t is the tachyon mass. The latter has the dimension of an inverse length, being a shortcut for m_tc/. The subsequent spectral densities only apply in the range ω  ω_b. There is also high-frequency radiation beyond this break frequency, but this needs second quantization, as the high magnetic field strengths considered here lead to an exponential cutoff of the classical radiation. In the low-frequency regime, however, quantum effects are perturbative, cf. after (2.10).

The transversal radiation can be split into two linear polarization components,

The lower plus-sign in (2.2) refers to the -polarization, and q is the tachyonic charge of the electron. The tachyon mass, m_t ≈ 2.15 keV/c², and the ratio q²/e² ≈ 1.4 × 10⁻¹¹ of tachyonic and electric fine structure constants can be inferred from Lamb shifts in hydrogenic systems [7]. The spectral functions can be traced back to Airy functions,

where the argument ξ is a shortcut for

and ω_b = eB/(γmc) is the gyrofrequency, cf. after (2.9). As mentioned, we restrict to frequencies ω  ω_b. The parameter κ will turn out to be the expansion parameter for the power radiated, cf. Section 3. We will consider the κ → 0 asymptotics, attained in high magnetic fields. The spectral density of the longitudinal radiation reads,

with the spectral function L₀(ξ) defined in Figs. (2.3) and (2.4). The ascending series and asymptotic limits of F₀, G₀ and L₀ can be found in Ref. [13].

We split the densities Figs. (2.2) and (2.5) into a linear and a curvature component,

where the densities p^T,lin and p^L,lin stand for the tachyon radiation generated by a charge in linear uniform motion (in the ultra-relativistic limit),

The curvature radiation subtracted in (2.6) can be read off from Figs. (2.2) and (2.5),

where the upper minus-sign refers to . The Lorentz factor of the radiating charge enters the transversal linear density p^T,lin via ω_b. It also enters in p^L,lin, again by ω_b, which is the cutoff frequency for uniform motion; a uniformly moving ultra-relativistic charge can only radiate frequencies ω  ω_b, cf. Ref. [12]. The three densities in (2.6) are safely positive definite, but not so the terms and p^L,curv in (2.8), which are generated by the orbital curvature and oscillate for large ξ.

We consider κ  1, cf. (2.4), but still large enough that κγ²  1. Since γ  1 in the ultra-relativistic limit, we can assume , so that the mass term in the denominator of the spectral densities can be dropped for frequencies . Moreover, 0  ξ(ω)  1 in the interval , cf. (2.4), so that and ξ(ω_b) = 0. Within this interval, the spectral functions ξ^2/3F₀, ξ^2/3G₀ and L₀ are of O(1) . Outside this range, we find ξ  1 for , so that the spectral functions can be replaced by their asymptotic limits. There are three frequencies helpful to understand the qualitative behavior of the spectral densities,

where the three proportionality constants pertinent to the indicated choices of ω are of the same order. The densities decay ∝ ω for ω  m_tc. The longitudinal density attains its maximum at ω ≈ m_tc. The transversal density also has its maximum at ω ≈ m_tc, followed by a minimum at about . The spectral functions (2.3) are composed of (anti-) derivatives of Airy functions, and admit straightforward analytic continuation [13]. The argument ξ(ω) in the spectral functions can still be used in the high-frequency regime, ω  ω_b, if we replace the parentheses in (2.4) by an absolute value. We then find ξ(ω_b/κ) = 1, where the ratio ω_b/κ is independent of the tachyon mass and coincides with the critical photon frequency, ω_c = (3/2)ω_Bγ³. At about this frequency, the analytically continued transversal spectral density attains a second maximum, like the electromagnetic counterpart, before the exponential decay sets in. In actual fact, the classical densities are exponentially cut off before this high-frequency maximum is reached. This is a quantum effect arising in the κ  1 limit, due to the high magnetic field strength.

We shortly summarize the notation and the constants. Gyroradius and gyrofrequency relate as R ≈ c/ω_B, where ω_B = eB/(γmc), and 1G · e ≈ 2.998 × 10⁻⁷ GeV cm⁻¹. We write γ = E/mc², so that eB ≈ E/R, where E and m denote energy and mass of the gyrating electron or positron, other possible source particles like protons and heavier nuclei will not be considered. We use the Heaviside–Lorentz system, so that e²/(4πc) =: α_e ≈ 1/137 and q²/(4πc) =: α_q ≈ 1.0 × 10⁻¹³ are the electric and tachyonic fine structure constants. We restore the mass unit in the above densities, m_t → m_tc/ ≈ 1.09 × 10⁸ cm⁻¹, so that the break frequency reads ω_b = γm_tc²/. Finally, α_q/α_e ≈ 1.4 × 10⁻¹¹ and m_t ≈ m/238 ≈ 2.15 keV/c².

When calculating the power radiated, cf. Section 3, we will use κ in (2.4) as expansion parameter,

In the surface magnetic fields of γ-ray pulsars, κ  1 as well as κγ²  1 usually apply, cf. Table 2. The opposite limit, κ  1, is realized in planetary magnetospheres and supernova remnants [13]. The surface fields are quite comparable to the critical field, B_c = m²c³/(e) ≈ 4.414 × 10¹³ G, cf. Table 1. The gyrofrequency and the ratio of critical and break energy relate to the magnetic field ratio as

This notation is kept close to electromagnetic synchrotron radiation [14], [15], [16] and [17], as is the whole formalism, of course. However, the spectral densities (2.2) applicable below the break frequency do not have an electromagnetic counterpart, only the upper frequency range has, but the high-frequency radiation requires second quantization in strong fields. Quantum corrections [18] are negligible for frequencies much smaller than the electron energy, which is the case below the break frequency. In the opposite limit, ω/E  1, the classical spectral peak near the critical frequency is wiped out by an exponential cutoff, as is the electromagnetic synchrotron radiation [19]. Quantum effects also lead to an attenuation of the classical radiation in the cyclotron limit, for slowly gyrating source particles. The quantization of tachyonic cyclotron and synchrotron radiation will be discussed elsewhere.

To get an overview regarding orders of magnitude, we define the shortcuts E₀ = E[GeV] and B₀ = B[G], and write the preceding formulas as scaling relations in these dimensionless numbers. For instance, and ω_B[GHz] ≈ 29.98/R[cm]. Lorentz factor and expansion parameter scale as γ ≈ 1957E₀ and , respectively. We also note and κγ² ≈ 2.42 × 10¹⁴E₀/B₀. The energies attached to break and critical frequency and to the transversal spectral minimum as defined after (2.9), ε_b,c,min = ω_b,c,min, scale as

Some other numerical relations are needed to connect the foregoing to synchrotron fluxes from pulsars, cf. Section 3. The surface magnetic fields are inferred from the period and period derivative via . The light cylinder field is obtained by a rescaling of the surface field, B_lc = (R_ns/R_lc)³B. Here, R_ns ≈ 10⁶ cm and R_lc = cP/(2π) are the neutron star and light cylinder radii, respectively, so that R_lc/R_ns ≈ 4.77P [ms]. Some of these quantities are listed in Table 1 and Table 2. There is a voltage gap, ΔU = B_lcR_lc, or ΔU[V] ≈ 1.43 × 10⁹B_lc[G]P[ms], between surface and light cylinder. This potential drop is an order of magnitude estimate, essentially on dimensional grounds (1G ≈ 299.8 V/cm), which may even be substantially reduced by pair production. ΔU is of the order of 10¹⁴–10¹⁶ V for the known γ-ray pulsars [20], [21] and [22]. This gap is not related to synchrotron radiation, but there may be an important implication for cosmic ray acceleration [23]. Electrons cycling in the surface fields can acquire energies in the low TeV range, cf. Section 3. When subjected to a voltage of the indicated magnitude or even a few orders less, they are accelerated into the 10²¹ eV region, unless slowed down by electromagnetic radiation loss. In the surface field, electromagnetic synchrotron radiation is suppressed by a quantum cutoff, which annihilates the classical radiation peak [19]. However, tachyonic synchrotron radiation below the break frequency (vanishing in the zero-mass limit) is not affected by this cutoff. When the synchrotron electrons or any other charged particles spiral toward the light cylinder, driven by the electric voltage, the synchrotron radii increase, but the electromagnetic radiation loss may still be much smaller than the classical estimates on curvature radiation along the magnetospheric electric field lines would suggest. There is no radiation damping by tachyon radiation, as the Green function outside the light cone is time symmetric and the radiated energy is drained from the absorber field [8] and [12].

3. Tachyonic γ-rays from pulsar magnetospheres

We will study the tachyonic synchrotron power radiated by electrons gyrating in the surface magnetic fields of γ-ray pulsars, based on the spectral densities Figs. (2.1) and (2.5). The parameters of the individual pulsars are listed in Table 1 and Table 2. The radiated power and the tachyonic count rates calculated in Figs. (3.1), (3.2), (3.3), (3.4), (3.5), (3.6), (3.7), (3.8), (3.9) and (3.10) are stated in Table 3 for each of the known γ-ray pulsars, parametrized by the gyration energy. In Figs. (3.11), (3.12) and (3.13), we will average the tachyonic spectral densities with an electronic power-law distribution, and then proceed with a phenomenological discussion of the observed γ-ray fluxes, including the detection mechanisms of the γ-ray counters.

We start with the integration of the transversal spectral density (2.1). The power radiated by a single gyrating electron can be split into linear polarization components,

Alternatively, we may decompose P^T into a linear and a curvature component according to Figs. (2.6), (2.7) and (2.8),



so that . The power stemming from the linear transversal density (2.7) is readily calculated,

which is the leading order in the ultra-relativistic 1/γ-expansion of the first integral in (3.2), cf. Ref. [12]. We have restored the natural units, m_t → m_tc/; the tachyonic fine structure constant α_q is defined before (2.10). The second integral in (3.2) gives the transversal curvature radiation,

The lower plus-sign in (3.4) refers to the -polarization, and P^T,curv is the total transversally polarized curvature radiation, cf. (3.2). The expansion parameter κ, the ratio of break and critical frequency, is defined in (2.10), and γ_E ≈ 0.5772.

The longitudinal radiant power can likewise be decomposed into a linear and a curvature term, cf. (2.6),

An elementary integration gives the linear power,

up to O(γ⁻¹) like in (3.3). The κ-expansion of the longitudinal curvature radiation reads,

The total power radiated is P = P^T+P^L, with the polarized powers P^T,L, cf. Figs. (3.2) and (3.5).

The count rates are composed like in Figs. (3.2) and (3.5), with the spectral densities divided by ω; we write the transversal count as N^T = N^T,lin − N^T,curv, where . The linear count is constant in leading order, in the ultra-relativistic regime that is,

The contribution of the orbital curvature to the transversal count is



where we used α_qm_tc²/ ≈ 3.27 × 10⁵ s⁻¹, cf. before (2.10). The longitudinal count rate, N^L = N^L,lin − N^L,curv, is composed of the linear count, N^L,lin  N^T,lin, and the curvature term



In Table 2, the expansion parameter κ is listed for the individual pulsars, parametrized by the electronic gyration energy. In Table 3, we list the above expansions parametrized accordingly, which gives an overview as to how the magnetic field strengths and electron energies impact the weight of the linear and curvature terms. The range of electron energies (k-interval) for these expansions to apply can be read off from Table 2 for each pulsar.

To connect to γ-ray fluxes from magnetospheric electron populations, we need to know the tachyonic spectral densities generated by a non-singular electron distribution. To this end, we average the densities Figs. (2.2) and (2.5) with an electronic power-law, dn(γ) = Aγ^−s dγ, of index s > 1. The ultra-relativistic electronic Lorentz factors range in a finite interval, γ₁  γ  γ₂, with γ₁  1; the normalization constant A relates to the electron count in this interval as stated in (3.12). The transversal density is averaged as

and the longitudinal average p^L(ω)_s is defined in the same way. We will focus on the number densities and n^L (ω)_s = p^L(ω)_s/(ω), since scaling exponents for γ-rays are usually defined with regard to differential number counts. We restore the natural units, m_t → m_tc/, and use the fine structure constant α_q = q²/(4πc) and the scaling variable , to find



This frequency scaling, the leading order asymptotics of the integrals (3.11), is valid in the range and for electron indices s > 1. The amplitude A of the electronic power-law depends on the electronic source number n_e. The product κγ of expansion parameter and Lorentz factor is independent of γ, cf. Figs. (2.4) and (2.10); thus we may take both factors at γ₁ and conclude that the transversal number density overpowers the longitudinal one, since . In fact, the ratio of the longitudinal and transversal densities (3.12) reads,

with s > 1. The constant κγ is listed in Table 2 for the respective pulsars, and the tachyon energy, E_t = ω, is meant in MeV units. In the frequency bands studied in the subsequent examples of γ-ray pulsars, this ratio is well below 10⁻², so that we will focus on the transversal radiation only. The transversal densities in (3.12) also apply to electron indices 1/3 < s < 1, as one may expect, but this requires a further restriction on the scaling range, a limit on the upper cutoff of the electron density, γ₂  γ₁/κ(γ₁). For these indices, the radiation is likewise overwhelmingly transversal.

We denote the energy of the tachyonic γ-rays by E_t as in (3.13). The tachyon flux, F = dN/dE_t, is derived from the normalized differential number count dN = (4πD²)⁻¹n^T(ω)_s dω, where stands for the total (unpolarized) transversal radiation, cf. (3.12), and D is the distance to the pulsar, cf. Table 2. Concerning units, we choose , where E_t is in MeV, and the measured proportionality factor and scaling exponent will be specified below for the enumerated pulsars. These units will be used throughout, mostly without mentioning. The scaling exponent relates to the electron index as α = 1/3 + s, cf. (3.12), and the dimensionless number density can be retrieved from the flux and the distance, n^T(ω)_s = 4πD²F; we will infer the electronic source count n_e by equating in (3.12) with

In the following case study of the seven established γ-ray pulsars, the observational input will be the amplitude and the scaling exponent of the measured flux, , the energy band for a scaling exponent to apply, as well as the spectral breaks where different power-laws join. We will determine the electron distributions generating the radiation in the respective frequency bands, in particular the electronic source count, that is, the number of gyrating electrons generating the tachyon flux or a power-law component of it. When inferring the electronic source number from the measured fluxes, we have to take into account that only a fraction α_q/α_e (ratio of tachyonic and photonic fine structure constants, cf. before (2.10)) of the tachyon flux passing through the detectors is absorbed. This requires a rescaling of the electron count n_e by α_e/α_q ≈ 7.3 × 10¹⁰, where n_e is based on the tachyon count in the detectors. That is, the electronic source count n_e is calculated, via Figs. (3.14) and (3.12), from the observed flux F, but the actual tachyon flux is by a factor of α_e/α_q larger than the observed one, so that the actual number of synchrotron electrons in the surface magnetic field is N_e ≈ n_eα_e/α_q.

The three telescopes employed in measuring the fluxes in the various energy bands cited below operated with very different detection mechanisms, but the rescaling of the observed fluxes withα_e/α_q applies to all of them. In the OSSE detector [9], the basic interaction was ionization generating scintillations in NaI(Tl) crystals. COMPTEL [10] was a counter based on Compton scattering, and the EGRET telescope [11] counted γ-rays by conversion into electron-positron pairs. The crucial point here is, that the cross-sections for ultra-relativistic photoionization, photonic Compton scattering and photonic pair production also apply to transversal tachyons, with the mentioned rescaling by α_q/α_e. The tachyon mass drops out in the dispersion relation, k² = ω²/c² + (m_tc/)², at γ-ray energies, which only makes an overall rescaling of the respective photonic cross-sections [24] necessary. This rescaling also shows in the non-relativistic limit, derived for tachyonic ionization in Ref. [25], where we also studied the effect of the tachyon mass in the low-energy (soft X-ray) regime. The same rescaling of the Compton cross-section (Klein–Nishina formula) can be traced back to classical Thomson scattering, that is, to the acceleration of an electron by the incoming tachyonic wave field, which triggers electromagnetic radiation [8]. Despite the fact that the photonic cross-sections of the three mentioned processes scale with higher powers of the electric fine structure constant, the tachyonic counterpart is obtained in all three cases by an α_q/α_e-rescaling, for transversal γ-rays at least; this is further discussed in the Conclusion.

Apart from the electronic source count and the total gyration energy, , stored in these electrons, we will determine their tachyonic γ-ray luminosity, the power transversally radiated in a given frequency band,

where the interval boundaries E_t,1,2 are in MeV units. The conversion factors used in the sequel are 1 TeV ≈ 1.602 erg and 1 kpc ≈ 3.086 × 10²¹ cm. The tachyon energy E_t is given in MeV and the electron energy E in GeV, parametrized by 10^k like in the tables, and the flux F is in units of cm⁻² s⁻¹ MeV⁻¹, cf. (3.14).

We start with the Crab pulsar. The flux in the 30–4000 MeV interval scales as , cf. Refs. [27] and [28], suggesting an electron index s ≈ 1.72. As pointed out after (3.14), the observed flux has still to be rescaled by a factor of α_e/α_q ≈ 7.3 × 10¹⁰, as most tachyons pass unnoticed through the detectors. The cited interval boundaries correspond to electron energies of 10^0.85 and 10^3.0 GeV, respectively. There is a spectral break at 30 MeV, followed by a second power-law, , in the 0.12–30 MeV band, cf. Refs. [29] and [30]. This flux component is generated by electrons gyrating with energies from 10^−1.55 to 10^0.85 GeV, with power-law index s ≈ 1.92. The tachyonic spectral densities above and below the spectral break relate to the amplitudes of the respective electronic power-laws as and , cf. (3.12). The normalization constants of the electron distributions in the adjacent intervals above and below the break energy (of 10^0.85 GeV) read A^α=2.05 ≈ 690n_e and A^α=2.25 ≈ 36.9n_e, respectively. We find n_e ≈ 1.4 × 10³⁵ in the upper interval (where ) and n_e ≈ 2.0 × 10³⁷ below the break energy (where ). These electron counts are calculated by equating flux and number density, cf. Figs. (3.14) and (3.12), and they still have to be renormalized; the actual count in the 10^0.85−10^3.0 GeV range reads , and is the count below the break energy, in the 10^−1.55−10^0.85 GeV interval. The total electronic energy in the surface magnetic field is calculated as indicated before (3.15); we find above the break energy, and below. The respective electron populations produce the tachyonic luminosity in the 30–4000 MeV band, and in the 0.12–30 MeV interval, cf. (3.15). These gyration energies and luminosities are based on the observed flux F; the actual tachyon flux is by a factor of α_e/α_q higher, so that we have to replace the by the renormalized counts .

Remarks. There is no superluminal radiation damping that could slow down the gyration, as the tachyonic Green function is time symmetric. The energy radiated stems from the oscillators of the cosmic absorber [25] and [31]. The absorber field breaks the time symmetry and renders the interaction causal and non-local [8]. The latter has implications on energy conservation, as the radiated energy is drained from the absorber. For this reason, the search for missing negative mass-squares suggested in Ref. [4] fails, where bubble-chamber events were reanalyzed, assuming a local but otherwise unspecified interaction of tachyons with matter, discounting the time symmetry of the Green function outside the light cone. Had there been detection in these experiments, in the relativistic framework in which they were interpreted, this would have been tantamount to causality violation. The radiation mechanism scrutinized here, based on a Proca field with negative mass-square minimally coupled to subluminal matter, is non-relativistic. It implies the absolute spacetime defined by the absorber [5] and [6], even though the Lagrangians are still covariant. Finally, electromagnetic synchrotron radiation would lead to radiation damping, but is suppressed by a quantum cutoff, cf. after (2.11).

There is a further spectral break at 0.12 MeV, where the spectrum flattens to , cf. Ref. [29], and then continues as an unbroken power-law even into the soft X-ray band [32] and [33]. The radiation below 0.12 MeV is not accessible in the ultra-relativistic limit because of the moderate electronic Lorentz factors; the spectral densities would have to be recalculated, with the Nicholson asymptotics replaced by Debye’s approximation of large-order Bessel functions [34]. At the opposite end, above 4000 MeV, there is some indication that the radiation terminates in a steep spectral slope, and no pulsed emission is detected in the TeV region [28]. The fluxes used here and in the subsequent examples refer to the pulsed emission, and they are phase averages. Phase-resolved spectra exist for the Crab and Vela pulsars and Geminga. As for the Crab pulsar, the phase-dependent exponents α range from 1.42 to 2.65 in the 30–4000 MeV band [27], and from 2.16 to 2.41 in the 0.12–30 MeV interval, cf. Ref. [29]. The respective phase-resolved electron indices, s = α − 1/3, are usually larger than 1, and they never fall below 1/3, cf. after (3.13).

The Vela pulsar radiates a flux of in the 30–2000 MeV interval, cf. Refs. [27] and [35], generated by electrons in the 10^0.85–10^2.7 GeV range with power-law index s ≈ 1.29. The normalization of the electron distribution in this range is A ≈ 6.32n_e, and the electronic source count reads n_e ≈ 5.9 × 10³³ or N_e ≈ 4.3 × 10⁴⁴ if renormalized. These electrons store an energy of E_e[erg] ≈ 0.125N_e and generate the number density , which results in a tachyonic γ-ray luminosity of L_t[erg/s] ≈ 10.4N_e in the 30–2000 MeV band. The phase-resolved exponents of this pulsar range from 1.38 to 2.21, cf. Ref. [27]. Pulsed TeV radiation is not detectable [36].

γ-Radiation from PSR B1706−44 has been detected from 50 MeV up to 20 GeV, cf. Ref. [37]. A flux is observed in the 50–1000 MeV interval, corresponding to electron energies from 10^1.1 to 10^2.4 GeV. This is the only example where the (phase-averaged) electron index drops below 1, s ≈ 0.94. The electronic power-law normalization in the indicated GeV range is A^α=1.27 ≈ 0.160n_e, and the tachyonic number density scales as in the 50–1000 MeV interval. There is a spectral break at 1000 MeV, followed by a second flux component, , cf. Ref. [37], which applies up to 20 GeV, suggesting electron energies up to 10^3.7 GeV in the surface field. This flux is generated by an electronic power-law of index s ≈ 1.92 and amplitude A^α=2.25 ≈ 1.54 × 10⁵n_e in the 10^2.4–10^3.7 GeV range. The tachyonic number density in the 1–20 GeV interval reads . By the way, the frequency scaling is obtained by substituting ω/1 MeV for E_t. Both electronic power-laws admit the same source number, n_e ≈ 9.2 × 10³³, valid for s ≈ 0.94 as well as s ≈ 1.92; the renormalized source count for each of these slopes is accordingly N_e ≈ 6.7 × 10⁴⁴. These electron populations below and above the break energy of 10^2.4 GeV contain a gyration energy of and , respectively; they produce a tachyonic luminosity of in the 50–1000 MeV band, and of above the spectral break.

Geminga generates the flux in the 30–2000 MeV band, corresponding to electron energies between 10^0.85 and 10^2.7 GeV, cf. Refs. [27], [38] and [39]. This flux allows us to infer the electron index, s ≈ 1.09, and the tachyonic number density, . The electronic power-law normalization in the mentioned GeV interval is A ≈ 0.649n_e, with source count n_e ≈ 7.9 × 10³². We find the renormalized electron count, N_e ≈ 5.8 × 10⁴³, the total energy of these electrons, E_e[erg] ≈ 0.16N_e, as well as their tachyonic γ-ray luminosity in the 30–2000 MeV interval, L_t[erg/s] ≈ 9.6N_e. The power-law indices of the phase-resolved flux range from 1.27 to 1.89, cf. Ref. [27].

PSR B1055−52 radiates the flux in the 70–1000 MeV band [40], generated by electrons cycling with energies from 10^1.2 to 10^2.4 GeV. In this range, we infer an electronic power-law distribution of index s ≈ 1.25, normalization A^α=1.58 ≈ 6.65n_e, and source number n_e ≈ 4.4 × 10³³, which produces the tachyonic number density in the 70–1000 MeV interval. Above the spectral break at 1000 MeV, the flux scales as up to at least 4000 MeV, cf. Ref. [40]. This means electron energies up to 10^3.0 GeV in the surface field, an electronic power-law index s ≈ 1.71 above the electronic break energy of 10^2.4 GeV, and a tachyonic number density of in the 1000–4000 MeV interval. The electronic power-law normalization in the 10^2.4−10^3.0 GeV range reads A^α=2.04 ≈ 1.15 × 10⁴n_e, with n_e ≈ 1.5 × 10³³. Below the electronic break energy, we count electrons, which store a total energy of and radiate a power of in tachyonic γ-rays in the 70–1000 MeV band. Above the break energy, there are electrons gyrating in the 10^2.4–10^3.0 GeV range, which add up to an electronic energy of and produce a tachyonic luminosity of in the 1000–4000 MeV interval.

γ-radiation from PSR B1951+32 has been detected in a wide band, from 10 MeV to 20 GeV, cf. Refs. [41] and [42], generated by electrons gyrating in the 10^0.38–10^3.7 GeV range. The flux scales as , cf. Ref. [41], seemingly without a spectral break in this band. We find the electron index s ≈ 1.41, the electronic power-law normalization A ≈ 13.2n_e, and the tachyonic number density , from which we infer the electronic source count n_e ≈ 6.2 × 10³⁴ or N_e ≈ 4.5 × 10⁴⁵ when renormalized. The gyration energy of these electrons is E_e[erg] ≈ 0.25N_e, and their tachyonic radiation power in the 10 MeV–20 GeV interval amounts to L_t[erg/s] ≈ 2.9N_e. A spectral break must occur below 10 MeV, according to the upper flux limits derived in Refs. [43] and [44]; no pulsed TeV radiation has been found [45].

PSR B1509−58 is not detected in hard γ-rays, possibly due to a quantum cutoff inflicted by its high magnetic field, but this needs further investigation in second quantization, cf. after (2.11). The observed flux, , extends from 0.2 to 10 MeV, cf. Refs. [46] and [47], and is produced by electrons with energies in the 10^−1.3–10^0.38 GeV range. We infer an electronic power-law of index s ≈ 1.35 and normalization A ≈ 2.25n_e, where n_e ≈ 1.75 × 10³⁶, as well as a tachyonic number density of in the mentioned MeV band. We note the electron count in the surface field, N_e ≈ 1.3 × 10⁴⁷, the energy stored in these electrons, E_e[erg] ≈ 6.5 × 10⁻⁴N_e, and their tachyonic luminosity, L_t[erg/s] ≈ 0.82N_e, in the 0.2–10 MeV interval. Above 10 MeV, there is a steep spectral break [47], and no pulsed TeV radiation is observed [48]. At the lower end, at 0.2 MeV, there is likewise a spectral break, where the spectrum hardens, admitting a spectral index of about α ≈ 1.3 in the hard X-ray band [49] and [50], but this radiation is not accessible in the ultra-relativistic synchrotron limit [34].

4. Conclusion

We have given a quantitative and in part phenomenological discussion of superluminal synchrotron radiation in strong magnetic fields. We will further comment on the interaction of tachyons with matter, on the basis of the three detectors used to infer the magnetospheric electron populations and their tachyonic γ-ray luminosity. The ionization in the scintillation crystals of the OSSE counter [9] happens in the relativistic regime. OSSE operated above a 50 keV threshold, so that the tachyon mass of 2 keV can be neglected in the dispersion relation, given that only the squared energies enter, cf. before (3.15). Therefore, the relativistic photonic ionization cross-section [24] can be used for transversal tachyons, if properly rescaled. The only change necessary for tachyonic X-rays or γ-rays above 50 keV is a rescaling with the ratio of tachyonic and electric fine structure constant. This rescaling can be traced back to the non-relativistic limit, which depends on the tachyon mass and applies in the soft and hard X-ray bands [25]. The longitudinal relativistic cross-section has to be calculated from scratch without reference to a zero-mass limit [51]. The tachyon flux generated by synchrotron radiation in the surface fields of γ-ray pulsars is overwhelmingly transversally polarized, cf. (3.13), in contrast to the low-magnetic-field limit [13], and therefore we have not attempted a quantitative study of the longitudinal fraction beyond some basic estimates.

Compton scattering and pair creation were employed in the COMPTEL and EGRET detectors [10] and [11]. The tachyonic Thomson cross-section, the non-relativistic classical limit of Compton scattering, was derived in Ref. [8], but a quantum mechanical version is still lacking, especially if the incident tachyonic X-rays have energies close to the tachyon mass. In the relativistic regime, for γ-rays above 0.75 MeV (the threshold of the COMPTEL counter), the transversal cross-section is obtained by a rescaling of the Klein–Nishina formula with α_q/α_e, the same rescaling that applies to the Thomson and ionization cross-sections. The EGRET detector was a spark chamber recording electron-positron pairs produced by γ-rays scattered in tantalum foils. The conversion of tachyons into electron-positron pairs has not been studied in any limit and context as yet. However, one can reckon from the foregoing that the cross-section for the conversion of transversal γ-rays is just the Bethe–Heitler formula rescaled with α_q/α_e; the fluxes collected by EGRET were in the 20 MeV–30 GeV range, so that the tachyon mass is negligible in the dispersion relation. This rescaling of the EGRET fluxes is also suggested by cross-calibration with the COMPTEL detector, to avoid discontinuities in the band overlap.

Tachyonic synchrotron radiation above the break frequency has not been discussed here, cf. after (2.9) and Ref. [13]. This requires second quantization in strong magnetic fields, as quantum corrections in the high-frequency regime cannot be treated perturbatively. Above the break frequency, the classical radiation theory is severely modified by a quantum cutoff, which results in exponential decay of the spectral densities at the critical energy, where the classical spectral functions are peaked. This quantum cutoff may well be the reason that no pulsed TeV radiation is detected from the known γ-ray pulsars.

Acknowledgments

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Tata Institute of Fundamental Research, Mumbai, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.