From Generalized Dirac Equations to a Candidate for Dark Energy

Abstract

We consider extensions of the Dirac equation with mass terms 𝑚₁+i𝛾⁵𝑚₂ and i𝑚₁+𝛾⁵𝑚₂. The corresponding Hamiltonians are Hermitian and pseudo-Hermitian (𝛾⁵ Hermitian), respectively. The fundamental spinor solutions for all generalized Dirac equations are found in the helicity basis and brought into concise analytic form. We postulate that the time-ordered product of field operators should yield the Feynman propagator (i𝜖 prescription), and we also postulate that the tardyonic as well as tachyonic Dirac equations should have a smooth massless limit. These postulates lead to sum rules that connect the form of the fundamental field anticommutators with the tensor sums of the fundamental plane-wave eigenspinors and the projectors over positive-energy and negative-energy states. In the massless case, the sum rules are fulfilled by two egregiously simple, distinguished functional forms. The first sum rule remains valid in the case of a tardyonic theory and leads to the canonical massive Dirac field. The second sum rule is valid for a tachyonic mass term and leads to a natural suppression of the right-handed helicity states for tachyonic particles and left-handed helicity states for tachyonic spin-1/2 antiparticles. When applied to neutrinos, the theory contains a free tachyonic mass parameter. Tachyons are known to be repulsed by gravity. We discuss a possible role of a tachyonic neutrino as a contribution to the accelerated expansion of the Universe “dark energy.”

1. Introduction

1.1. Generalized Dirac Equations: Mass Terms and Dispersion Relations

Dirac is often quoted saying in some of his talks that the equation that carries his name [1, 2] is “more intelligent than its inventor.” Of course, it needs to be added that it was Dirac himself who found most of the additional insight. Here, we are concerned with extensions of the Dirac equation which contain both tardyonic and tachyonic mass terms. Tardyonic (subluminal) mass terms lead to dispersion relations of the form 𝐸=^→𝑝²+𝑚², whereas tachyonic mass terms lead to superluminal dispersion relations of the form 𝐸=^→𝑝²−𝑚², where 𝐸 is the energy and ^→𝑝 is the momentum. The generalized, matrix-valued mass term 𝑀 enters the Dirac equation in the form (i𝛾^𝜇𝜕_𝜇−𝑀)𝜓(𝑥)=0. The 𝛾^𝜇 are 4×4 matrices that fulfill the relations {𝛾^𝜇,𝛾^𝜈}=2𝑔^𝜇𝜈, where we choose the space-time metric as 𝑔^𝜇𝜈=diag(1,−1,−1,−1). The 𝜕_𝜇 denotes the partial derivative 𝜕/𝜕𝑥^𝜇 with respect to the space-time coordinate 𝑥^𝜇=(𝑡,^→𝑥). It is quite surprising that a systematic presentation of the solutions of the generalized Dirac equations (i𝛾^𝜇𝜕_𝜇−𝑀)𝜓(𝑥)=0, in the helicity basis [3], has not been recorded in the literature to the best of our knowledge. While the following discussion is somewhat technical, we believe that it will be beneficial to give their explicit form, in order to fix ideas for the following discussion.

We set ℏ=𝑐=𝜖₀=1 and use the Dirac matrices in the standard representation 𝛾⁰𝟙=𝛽=^2×200−𝟙_2×2,^→0𝛾=^→𝜎−^→,𝛾𝜎0₅=0𝟙_2×2𝟙_2×20, (1) and define ^→𝛼=𝛾^0→𝛾. For the ordinary Dirac theory, one has 𝑀=𝑚₁ (one should say more precisely 𝑀=𝑚₁𝟙_4×4) with a real mass 𝑚₁, i𝛾^𝜇𝜕_𝜇−𝑚₁𝜓(𝑥)=0. (2) The dispersion relation is 𝐸=^→𝑝²+𝑚²₁. The corresponding Dirac Hamiltonian reads 𝐻⁽¹⁾=^→𝛼⋅^→𝑝+𝛽𝑚₁. (3) Extensions of the Dirac equation with pseudoscalar mass terms that contain the fifth current have been introduced in the literature. In [4], it is shown that for a mass term of the form 𝑀=𝑚₁+i𝛾⁵𝑚₂, the fermion propagator may obtain nontrivial gradient corrections already at the first order in derivative expansion, for a position-dependent mass. In that case, the fermion self-energy may contribute to a conceivable explanation for 𝒞𝒫-violation during electroweak baryogenesis, as pointed out in [4]. We thus study the following generalized form of the tardyonic (subluminal) Dirac equation: i𝛾^𝜇𝜕_𝜇−𝑚₁−i𝛾⁵𝑚₂𝜓(𝑥)=0. (4) The dispersion relation is 𝐸=^→𝑝²+𝑚²₁+𝑚²₂. The Hermitian tardyonic Hamiltonian operator reads as 𝐻^(𝑡)=^→𝛼⋅^→𝑝+𝛽𝑚₁+i𝛽𝛾⁵𝑚₂. (5) We may indicate a further motivation for our study; namely, the unitarity of the 𝑆 matrix implies the existence of useful relations [5] for the even powers (𝑚₂)^2𝑛 obtained upon expanding a one-loop amplitude, formulated with a mass term 𝑚₁+i𝛾⁵𝑚₂, in powers of 𝑚₂. This implies that a better understanding of the tardyonic equation with two mass terms could be of much more general interest.

It has not escaped our attention that the chiral transformation 𝜇expi𝛾⁵𝜃=𝜇cos𝜃+i𝛾⁵𝜇sin𝜃=𝑚₁+i𝛾⁵𝑚₂ (6) connects the two Hamiltonians 𝐻⁽¹⁾ and 𝐻^(𝑡) for 𝑚₁=𝜇cos𝜃 and 𝑚₂=𝜇sin𝜃, but it is computationally easier and more instructive to consider the real and imaginary parts of the mass term separately.

Within a systematic approach to generalized Dirac equations with pseudoscalar mass terms, we also consider tachyonic (superluminal) mass terms of the form 𝑀=𝛾⁵𝑚 which induce a superluminal dispersion relation 𝐸=^→𝑝²−𝑚². The corresponding generalized Dirac equation has been named the “tachyonic Dirac equation” and reads as follows [6–10]: i𝛾^𝜇𝜕_𝜇−𝛾⁵𝑚𝜓(𝑥)=0. (7) The corresponding Hamiltonian reads 𝐻₅=^→𝛼⋅^→𝑝+𝛽𝛾⁵𝑚. (8) The relation 𝐻₅=𝛾⁰𝐻⁺₅(−^→𝑟)𝛾⁰ has been given in [9, 10]. However, it is much more instructive to observe that 𝐻₅ is 𝛾⁵ Hermitian; that is, 𝐻₅=𝛾⁵𝐻⁺₅𝛾⁵. The concept of 𝛾⁵ Hermiticity is known in lattice theory [11, 12] and is otherwise called pseudo-Hermiticity [13–23].

An obvious generalization of the tachyonic case contains an imaginary mass and a 𝛾⁵ mass term, i𝛾^𝜇𝜕_𝜇−i𝑚₁−𝛾⁵𝑚₂𝜓(𝑥)=0. (9) The dispersion relation is 𝐸=^→𝑝²−𝑚²₁−𝑚²₂. The corresponding Hamiltonian reads 𝐻^′=^→𝛼⋅^→𝑝+i𝛽𝑚₁+𝛽𝛾⁵𝑚₂ (10) and is 𝛾⁵ Hermitian, 𝐻^′=𝛾⁵𝐻^′+𝛾⁵. For 𝑚₂=0, (9) has been discussed in [24, 25].

It is our goal here to present the fundamental eigenspinors corresponding to the plane-wave solutions of (2), (4), (7), and (9) in a unified and systematic manner. Furthermore, we discuss the second-quantized versions of the fermionic theories described by the generalized Dirac equations. Anticipating part of the results, we may point out that the massless Dirac equation “interpolates” between the tardyonic equations (2) and (4) and the tachyonic equations (7) and (9). For zero mass, helicity and chirality are equal. Helicity and chirality “depart” from each other in very specific directions, when the tardyonic and tachyonic mass terms are “switched on,” as we shall discuss in the following.

1.2. Tachyonic Dirac Equation and Neutrinos: Possible Connections

The tachyonic generalized Dirac equations (7) and (9) describe the motion of superluminal particles, which may either be important for astrophysical studies (neutrinos) or for artificially generated environments such as honeycomb photonic lattices in which pertinent dispersion relations become practically important [26]. The existence of superluminal particles would not falsify Einstein’s theory of special relativity [27], which according to common wisdom is based on the following postulates. (i) The principle of relativity states that the laws of physics are the same for all observers in uniform motion relative to one another. (ii) The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light. Predictions of relativity theory regarding the relativity of simultaneity, time dilation, and length contraction would not change if superluminal particles did exist. Furthermore, as shown by Sudarshan et al. [28–31] and Feinberg [32, 33], the existence of tachyons, which are superluminal particles fulfilling a Lorentz-invariant dispersion relation 𝐸²=^→𝑝²−𝑚²_𝜈, is fully compatible with special relativity and Lorentz invariance. According to special relativity, it is forbidden to accelerate a particle “through” the light barrier (because 𝐸=𝑚/√1−𝑣²→∞ for 𝑣→1), but a genuinely superluminal particle remains superluminal upon Lorentz transformation. Significant problems are encountered when one attempts to quantize the tachyonic theories, but again, as shown in [9], these problems may not be as serious as previously thought. In particular, the so-called reinterpretation of solutions propagating into the past according to the Feynman prescription [31] is a cornerstone of modern field theory. Furthermore, it has been shown in [9] that tachyonic particles can be localized and equal-time anticommutators of the spin-1/2 tachyonic field involve an unfiltered Dirac-𝛿 (see equation (37) of [9]).

Despite these arguments, we can say that, from the point of view of fundamental symmetries, accepting a superluminal neutrino would be equivalent to “an ugly duckling.” Adding to the difficulties, we notice that recent experimental claims regarding the conceivable observation of highly superluminal neutrinos have turned out to be false. One may point out that a relative deviation 𝑣=𝑐(1+𝛿) with 𝛿∼10⁻⁵ at 𝐸∼30GeV, as claimed by some recent experimental collaborations, would correspond to a negative neutrino mass square in the order of ∼−(100MeV)², if one assumes a Lorentz-invariant dispersion relation [34]. Still, there is at present no conclusive answer regarding the conceivable superluminality of at least one neutrino flavor [35–39], and it is intriguing that all available direct measurements of the neutrino mass square have resulted in negative expectation values, still compatible with zero within experimental uncertainty, whereas published experimental best estimates for the neutrino speed [40–43] have been superluminal, again still compatible with the speed of light within experimental error. The recent ICARUS result [43] is consistent with this trend [44–50]; the best estimate for the neutrino velocity is superluminal, but the deviation from 𝑣_𝜈=𝑐 is statistically insignificant. The OPERA collaboration [51] has indicated a preliminary, revised result of (𝑣−𝑐)/𝑐=(2.7±3.1(stat)^+3.8_−2.8(sys.))×10⁻⁶. Neither subluminal nor superluminal propagation velocities are excluded based on the available experimental data. The “ugly duckling of a superluminal neutrino” is not beautiful; if we are to consider accepting it, then we should be able to hope that the emergence of at least one “added insight” should be the result of this operation.

Before we discuss the possible emergence of these benefits, let us include some historic remarks. According to reliable sources (Professor M. Fink from the University of Austin (Texas) was engaged in discussions with Professor J. A. Wheeler, who found the notion of a nonvanishing neutrino mass so unappealing that he discouraged experimentalists from undertaking any effort to measure the neutrino mass, based on arguments described in Section 1 (M. Fink, private communication, 2012.)), Professor J. A. Wheeler, in his later years at the University of Austin (Texas), used to argue that the neutrino has to be massless, necessarily, and that in his opinion, it could only be a massless Weyl particle with definite helicity (and chirality). Recently presented arguments [52] regarding the possibility of overtaking a subluminal, left-handed neutrino, looking back and seeing a right-handed neutrino, were supposedly already used by Wheeler in order to dispel the conceivable existence of a neutrino mass term. If we assume that right-handed neutrinos are much more massive than their left-handed counterparts, then this leads to a paradox unless one assumes that all light, right-handed neutrinos are sterile. (The problem with a right-handed sterile massive neutrino is that for massive neutrinos, chirality and helicity are different; hence a 𝑉−𝐴 coupling of the form 𝛾^𝜇(1−𝛾⁵) no longer vanishes for massive Dirac neutrinos if one uses the canonical eigenstates of the massive Dirac equation. One therefore has to invoke additional exotic mechanisms in order to ensure the “sterility” of the right-handed Dirac neutrinos.) Wheeler also disliked (Professor M. Fink from the University of Austin (Texas) was engaged in discussions with Professor J. A. Wheeler, who found the notion of a nonvanishing neutrino mass so unappealing that he discouraged experimentalists from undertaking any effort to measure the neutrino mass, based on arguments described in Section  1 (M. Fink, private communication, 2012.)) the notion of a Majorana neutrino, arguing that the charge conjugation invariance condition imposed on the Majorana particle precludes the existence of plane-wave solutions to the Majorana equation and maximally violates lepton number. Again, these arguments (We are grateful to Professor M. Fink for the clarification and confirmation.) are in full agreement with those recently given in [52].

The original standard model thus called for manifestly massless neutrinos. The commonly accepted observation of neutrino oscillations precludes the possibility that all three generations of neutrino mass eigenstates are massless. Lepton number conservation is based on the global gauge symmetry 𝜓→𝜓exp(iΛ), applied simultaneously to all lepton fields. A Majorana neutrino would destroy lepton number as a global symmetry but solve the “autobahn paradox,” because a Majorana neutrino would be equal to its own antiparticle and thus, looking back, the right-handed neutrino state would consist of the same particle = antiparticle.

On the other hand, if we assume that the neutrino is described by the tachyonic Dirac equation, then the following statements are valid.(i)Statement 1: we can properly assign lepton number and use plane-wave eigenstates for incoming and outgoing particles, while allowing for nonvanishing mass terms and thus mass square differences among the neutrino mass (not flavor) eigenstates. (ii)Statement 2: there is a natural resolution for the “autobahn paradox” because a left-handed spacelike neutrino always remains spacelike upon Lorentz transformation and cannot be overtaken. (iii)Statement 3: the right-handed particle and left-handed antiparticle states are suppressed due to negative Fock-space norm.(iv)Statement 4: at least qualitatively, tachyonic neutrinos could yield an explanation for a repulsive force on intergalactic distance scales as they are repulsed, like all tachyons, by gravitational interactions (dark energy).

Pauli [53] postulated the existence of neutrinos, on the basis of the conservation of angular momentum and energy, and also introduced pseudo-Hermitian operators [13]. Here, we describe conceivable connections of neutrino physics and pseudo-Hermitian operators. Final clarification can only come from experiment. When in 1956, Reines and Cowan [54] discovered the electron neutrino, two and a half years before Pauli’s death, Pauli replied [55] by telegram “Thanks for message. Everything comes to him who knows how to wait. Pauli.” In defense of the tachyonic hypothesis, we would like to stress that a tachyonic Dirac neutrino would allow us to retain lepton number conservation as a symmetry of nature. We would thus like to write up these thoughts in the current paper, with attention to detail. We should point out that our approach fully conserves Lorentz invariance, in contrast to the extensions of the Standard Model based on Lorentz-violating terms which can otherwise lead to superluminal propagation (see Tables  11 and 13 of [56]).

Units with ℏ=𝑐=𝜖₀=1 are used throughout the paper. The organization is as follows: in Section 2, we discuss massless and tardyonic theories. Tachyonic extensions of the Dirac equation are discussed in Section 3. Connections of the fundamental tensor sums over the eigenspinors with the derivation of the time-ordered propagator are analyzed in Section 4. A candidate for dark energy is presented in Section 5. Conclusions are reserved for Section 6.

2. Generalized Dirac Equations: Massless and Tardyonic Theories

2.1. Massless Dirac Theory

The massless Dirac equation and the massless Dirac Hamiltonian read as i𝛾^𝜇𝜕_𝜇𝜓(𝑥)=0,𝐻₀=^→𝛼⋅^→𝑝. (11) We note that 𝐻₀ is both Hermitian as well as 𝛾⁵ Hermitian; that is, 𝐻₀=𝛾⁵𝐻⁺₀𝛾⁵. The dispersion relation is 𝐸=|^→𝑘|. With 𝑘^𝜇=(𝐸,^→𝑘), we seek positive-energy and negative-energy solutions of the form 𝜓(𝑥)=𝑢_𝜎^→𝑘exp(−i𝑘⋅𝑥),𝜙(𝑥)=𝑣_𝜎^→𝑘exp(i𝑘⋅𝑥), (12) where 𝜎=± denotes a quantum number which is equal to the helicity for positive-energy states and equal to the negative of the helicity for negative-energy states. With /^{𝑘=𝛾𝜇𝑘}_𝜇, we have /^𝑘𝑢_±^(𝑘)=/^𝑘𝑣_±^(𝑘)=0. In the massless limit, the solutions to the Dirac equation are given as (see Chapter 2 of [57])𝑢₊^→𝑘=1√2𝑎₊^→𝑘𝑎₊^→𝑘,𝑢₋^→𝑘=1√2𝑎₋^→𝑘−𝑎₋^→𝑘𝑣_+→,𝑘=−𝑢₊^→𝑘,𝑣₋^→𝑘=−𝑢₋^→𝑘. (13a)(13b)The well-known helicity spinors are recalled as 𝑎₊^→𝑘=𝜃cos2𝜃sin2e^i𝜑,𝑎₋^→𝑘=𝜃−sin2e^−i𝜑𝜃cos2. (14) These fulfill the fundamental relations (^→𝜎⋅^→𝑘)𝑎_𝜎(^→𝑘)=𝜎𝑎_𝜎(^→𝑘), as well as ∑_𝜎𝑎_𝜎(^→𝑘)⊗𝑎⁺_𝜎(^→𝑘)=𝟙₂ and ∑_𝜎𝜎𝑎_𝜎(^→𝑘)⊗𝑎⁺_𝜎(^→𝑘)=^→𝜎⋅^→𝑘, where ^→𝑘=^→𝑘/|^→𝑘| and the sum over 𝜎 is over the values ±1. The sums over the fundamental bispinors 𝑢 and 𝑣 fulfill the following sum rules: ^SumruleI:_𝜎^2→𝑘𝑢_𝜎^{→𝑘⊗𝑢}_𝜎^→𝑘=/^𝑘, (15) as well as ^SumruleII:_𝜎^{2→𝑘)𝑢(−𝜎}_𝜎^{→𝑘⊗𝑢}_𝜎^{→𝑘𝛾5=}/^𝑘, (16) Sum rule I can be obtained by a quick explicit calculation, and sum rule II holds because in the massless limit, helicity equals ± chirality (positive sign for positive energy, negative sign for negative energy). We denote the Dirac adjoint by 𝑢_𝜎(^→𝑘)=𝑢⁺_𝜎(^→𝑘)𝛾⁰. One can easily check by an explicit calculation that 𝑢_𝜎(^→𝑘)𝛾⁵=(𝛾⁵𝛾⁰𝑢_𝜎(^→𝑘))⁺=(−𝜎)𝑢_𝜎(^→𝑘) and 𝑣_𝜎(^→𝑘)𝛾⁵=(𝛾⁵𝛾⁰𝑣_𝜎(^→𝑘))⁺=(−𝜎)𝑣_𝜎(^→𝑘). We can thus introduce a factor (−𝜎)²=1 under the summation over spins in (15) and replace one of the factors (−𝜎) by a multiplication of the Dirac adjoint spinor from the right by the fifth current. The Lorentz-invariant normalization of the massless solutions vanishes; that is, 𝑢_𝜎(^→𝑘)𝑢_𝜎(^→𝑘)=𝑣_𝜎(^→𝑘)𝑣_𝜎(^→𝑘)=0.

Eigenstates of the massless Hamiltonian 𝐻₀=^→𝛼⋅^→𝑝 have to be eigenstates of the chirality operator 𝛾⁵ because the chirality commutes with the Hamiltonian, in the sense that [𝛾⁵,𝐻₀]=0. Furthermore, 𝐻₀=^→𝛼⋅^→𝑝=^→𝑝𝛾^5→Σ⋅^→𝑝_→𝑝, (17) where ^→Σ=𝛾^5→𝛼 is the vector of 4×4 spin matrices, and the helicity operator is identified as ^→Σ⋅^→𝑝/|^→𝑝|. Let 𝜆₁ be the eigenvalue of chirality and let 𝜆₂ be the eigenvalue of the helicity operator. Then, the eigenvalue of the Hamiltonian is 𝐸₀=|^→𝑝|𝜆₁𝜆₂. Since 𝜆₁=±1 and 𝜆₂=±1, we easily recover the known fact that helicity equals chirality for positive energy, whereas the relation is reversed for negative-energy states (see also Chapter 2.4 of [57]). We are aware of the fact that the considerations reported in the current section partly refer to the literature, but we give them in some detail because they are essential for the following considerations.

2.2. Massive Dirac Theory

We start from the ordinary Dirac equation given in (2), which reads (i𝛾^𝜇𝜕_𝜇−𝑚₁)𝜓(𝑥)=0. In the helicity basis, the fundamental spinor solutions read as 𝜓(𝑥)=𝑈_±^(1)→𝑘(),𝜙(𝑥)exp−i𝑘⋅𝑥=𝑉_±^(1)→𝑘exp(i𝑘⋅𝑥). (18) The algebraic relations that have to be fulfilled by the bispinor amplitudes 𝑈_±⁽¹⁾(^→𝑘) and 𝑉_±⁽¹⁾(^→𝑘) read as follows: /^𝑘−𝑚₁^𝑈_±^{(1)→𝑘=0,}/^𝑘+𝑚₁^𝑉_±^{(1)→𝑘=0.} (19) The dispersion relation is 𝐸⁽¹⁾=^→𝑘²+𝑚²₁. In the helicity basis, the solutions of (19) with a tardyonic 𝑚₁ mass term are easily written down, using the identity ⁽/^𝑘−𝑚₁⁾⁽/^𝑘+𝑚₁^{)=𝑘2−𝑚2}₁⁼⁰. With an appropriate normalization factor and after some algebraic simplification, the positive-energy solutions read as follows:_{𝑈+(1)→𝑘=}^𝑘+𝑚₁^𝑢₊^→𝑘_𝐸/_{(1)−→𝑘2+𝑚21=𝐸(1)+𝑚12𝐸(1)𝑎+→𝑘𝐸(1)−𝑚12𝐸(1)𝑎+→𝑘,𝑈−(1)→𝑘=}^𝑘/^+𝑚₁^𝑢₋^→𝑘_{𝐸(1)−→𝑘2+𝑚21=𝐸(1)+𝑚12𝐸(1)𝑎−→𝑘−𝐸(1)−𝑚12𝐸(1)𝑎−→𝑘.} (20a)(20b)The negative-energy eigenstates of the tardyonic equations in the helicity basis are given as_{𝑉+(1)→𝑘=}^𝑚₁^−𝑘𝑣₊^→𝑘_{𝐸(1)−→𝑘2+𝑚21=−𝐸(1)−𝑚12}/_{𝐸(1)𝑎+→𝑘−𝐸(1)+𝑚12𝐸(1)𝑎+→𝑘,𝑉−(1)→𝑘=𝑚1−}^𝑘𝑣₋^→𝑘_{𝐸(1)−→𝑘2+𝑚2}/_{1=−𝐸(1)−𝑚12𝐸(1)𝑎−→𝑘𝐸(1)+𝑚12𝐸(1)𝑎−→𝑘.} (21a)(21b)These solutions are consistent with those given in [3] and in Chapter  23 of [58], and the normalizations are (𝜎=±) 𝑈_𝜎^(1)+→𝑘𝑈_𝜎^(1)→𝑘=𝑉_𝜎^(1)+→𝑘𝑉_𝜎^(1)→𝑘=1. (22) One can change the normalization according to 𝒰_𝜎^(1)→𝑘=𝐸⁽¹⁾𝑚₁^1/2𝑈_𝜎^(1)→𝑘,𝒱_𝜎^(1)→𝑘=𝐸⁽¹⁾𝑚₁^1/2𝑉_𝜎^(1)→𝑘. (23) The Lorentz-invariant normalization is equal to one for the fundamental positive-energy bispinors and equal to minus one for the fundamental negative-energy bispinors, 𝒰_𝜎^(1)→𝑘𝒰_𝜎^(1)→𝑘=1,𝒱_𝜎^(1)→𝑘𝒱_𝜎^(1)→𝑘=−1. (24) A little algebra is sufficient to reproduce the following known sums over bispinors: _{𝜎𝒰𝜎(1)}^→_{𝑘⊗𝒰𝜎}^(1)→_𝑘=^𝑘+𝑚_{12𝑚1,𝜎𝒱𝜎(1)→𝑘⊗}/_{𝒱𝜎(1)→𝑘=}/^𝑘−𝑚_12𝑚1. (25) In accordance with general wisdom of the tardyonic case, these do not involve a helicity-dependent prefactor. The sum rule (25) is of type I (see (15)).

2.3. Two Tardyonic Mass Terms

Inspired by the discussion in Section 1, we consider an equation with two tardyonic mass terms, which has already been indicated in (2) and reads (i𝛾^𝜇𝜕_𝜇−𝑚₁−i𝛾⁵𝑚₂)𝜓^(𝑡)(𝑥)=0. For the corresponding bispinors in the fundamental plane-wave solutions, this implies that^𝑘/^−𝑚₁^{−i𝛾5𝑚}₂^𝑈_±^{(𝑡)→𝑘}₋^=0,/^𝑘−𝑚₁^{−i𝛾5𝑚}₂^𝑉_±^{(𝑡)→𝑘=0.} (26a)(26b)The dispersion relation is 𝐸^(𝑡)=^→𝑘²+𝑚²₁+𝑚²₂. The fundamental positive-energy bispinors read as follows:_{𝑈+(𝑡)→𝑘=}^𝑘+𝑚₁^{−i𝛾5𝑚}/₂^𝑢₊^→𝑘_{𝐸(𝑡)−→𝑘2+𝑚21+𝑚22=𝑚1−i𝑚2+𝐸(𝑡)−→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎+→𝑘√2𝑚1−i𝑚2−𝐸(𝑡)+→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎+→𝑘√2,𝑈−(𝑡)→𝑘=}^𝑘/^+𝑚₁^{−i𝛾5𝑚}₂^𝑢₋^→𝑘_{𝐸(𝑡)−→𝑘2+𝑚21+𝑚22=𝑚1+i𝑚2+𝐸(𝑡)−→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎−→𝑘√2−𝑚1−i𝑚2+𝐸(𝑡)−→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎−→𝑘√2.} (27a)(27b)The negative-energy eigenstates of the equation with two tardyonic mass terms are given as_{𝑉+(𝑡)→𝑘=}^−𝑘/^{−i𝛾5𝑚}₂^+𝑚₁^𝑣₊^→𝑘_{𝐸(𝑡)−→𝑘2+𝑚21+𝑚22=−𝑚1+i𝑚2+𝐸(𝑡)−→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎+→𝑘√2−𝑚1+i𝑚2−𝐸(𝑡)+→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎+→𝑘√2,} (28a) for negative helicity (positive chirality in the massless limit) and _{𝑉−(𝑡)→𝑘=}^−𝑘/^{−i𝛾5𝑚}₂^+𝑚₁^𝑣₋^→𝑘_{𝐸(𝑡)−→𝑘2+𝑚21+𝑚22=−𝑚1−i𝑚2+𝐸(𝑡)−→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎−→𝑘√2𝑚1+i𝑚2+𝐸(𝑡)−→𝑘𝐸(𝑡)−→𝑘2+𝑚21+𝑚22𝑎−→𝑘√2,} (28b)for positive helicity (negative chirality in the massless limit). In the massless limit (first 𝐸^(𝑡)→|^→𝑘|, then 𝑚₂→0, and then 𝑚₁→0), we again reproduce the massless solutions 𝑈₊^(𝑡)(^→𝑘)→𝑢₊(^→𝑘), 𝑈₋^(𝑡)(^→𝑘)→𝑢₋(^→𝑘), 𝑉₊^(𝑡)(^→𝑘)→𝑣₊(^→𝑘) and 𝑉₋^(𝑡)(^→𝑘)→𝑣₋(^→𝑘). Of course, in the limit 𝑚₁→0, one also has to expand the normalization denominators in powers of 𝑚₁. For 𝑚₂=0 the solutions (27a), (27b), (28a), and (28b) reduce to the solutions of the ordinary Dirac equation in (20a), (20b), (21a), and (21b) and can be expanded in 𝑚₂ to yield corrections to the ordinary Dirac equation for small 𝑚₂; that is, 𝑚₁≫𝑚₂. The states are normalized with respect to the condition 𝑈_𝜎^(𝑡)+→𝑘𝑈_𝜎^(𝑡)→𝑘=𝑉_𝜎^(𝑡)+→𝑘𝑉_𝜎^(𝑡)→𝑘=1. (29) In the normalization 𝒰_𝜎^(𝑡)→𝑘=𝐸^(𝑡)𝑚₁^1/2𝑈_𝜎^(𝑡)→𝑘,𝒱_𝜎^(𝑡)→𝑘=𝐸^(𝑡)𝑚₁^1/2𝑉_𝜎^(𝑡)→𝑘, (30) the positive-energy solutions acquire a “positive Lorentz-invariant norm,” whereas the negative-energy solutions have “negative Lorentz-invariant norm,” 𝒰_𝜎^(𝑡)→𝑘𝒰_𝜎^(𝑡)→𝑘=1,𝒱_𝜎^(𝑡)→𝑘𝒱_𝜎^(𝑡)→𝑘=−1. (31) After some algebra, one can derive the following sums over bispinors: _{𝜎𝑈𝜎(𝑡)}^→_{𝑘⊗𝑈𝜎}^(𝑡)→_𝑘=^𝑘+𝑚₁^{−i𝛾5𝑚}_{22𝑚1,𝜎𝑉𝜎}/_{(𝑡)→𝑘⊗𝑉𝜎(𝑡)→𝑘=}/^𝑘−𝑚₁^+i𝛾5𝑚_22𝑚1. (32) These are easily identified as the positive- and negative-energy projectors. The sum rules do not involve helicity-dependent prefactor and are of type I (see (15)).

The solutions (27a), (27b), (28a), and (28b) approach the massless solutions if one replaces 𝑚₂→0 first and then let 𝑚₁→0. They are thus useful for systems where the 𝑚₁ mass is greater than 𝑚₂. For 𝑚₂≫𝑚₁, one would like to calculate solutions that approach the massless case for the sequence 𝑚₁→0, then 𝑚₂→0. These read as follows: 𝑈_𝜎^′(𝑡)→𝑘=i𝜎𝑈_𝜎^(𝑡)→𝑘,𝑉_𝜎^′(𝑡)→𝑘=i𝜎𝑉_𝜎^(𝑡)→𝑘. (33) In comparison to the solutions (27a), (27b), (28a), and (28b), they acquire a nontrivial phase factor.

3. Generalized Dirac Equations: Tachyonic Mass Terms

3.1. Tachyonic Dirac Equation

The tachyonic Dirac equation is given in (7) and reads (i𝛾^𝜇𝜕_𝜇−𝛾⁵𝑚)𝜓(𝑥)=0. The fundamental bispinors entering the equations fulfill the equations /^{𝑘−𝛾5𝑚𝑈}_±^→𝑘=0,/^{𝑘+𝛾5𝑚𝑉}_±^→𝑘=0. (34) Using ⁽/^{𝑘−𝛾5𝑚)(}/^{𝑘−𝛾5𝑚)=𝑘2+𝑚2} and some algebra, the prefactors in the fundamental bispinors (for positive energy) take a very simple form,_{𝑈+→𝑘=}^{𝛾5𝑚−𝑘𝑢}₊^→𝑘_𝐸−/_{→𝑘2+𝑚2=}^{→𝑘+𝑚2→}_𝑘^𝑎+→𝑘_{→𝑘−𝑚2→𝑘𝑎+→𝑘𝑈,−→𝑘=}/^{𝑘−𝛾5𝑚𝑢}₋^→𝑘_{𝐸−→𝑘2+𝑚2=}^{→𝑘−𝑚2→}_𝑘^{𝑎−→𝑘}_{−→𝑘+𝑚2→𝑘𝑎−→𝑘.} (35a)(35b)For negative energy, the solutions read as follows: _{𝑉+→𝑘=}^{𝛾5𝑚+𝑘𝑣}₊^→𝑘_𝐸−→/_𝑘2+𝑚2=^{−→𝑘−𝑚2→}_𝑘^𝑎+→𝑘_{−→𝑘+𝑚2→𝑘𝑎+→𝑘𝑉,−→𝑘=−}/^{𝑘−𝛾5𝑚𝑣}₋^→𝑘_{𝐸−→𝑘2+𝑚2=}^{−→𝑘+𝑚2→}_𝑘^𝑎+→𝑘_{→𝑘−𝑚2→𝑘𝑎+→𝑘.} (36a)(36b)The normalization condition is (𝜎=±) 𝑈⁺_𝜎^→𝑘𝑈_𝜎^→𝑘=𝑉⁺_𝜎^→𝑘𝑉_𝜎^→𝑘=1. (37) One can change to Lorentz-invariant normalization by a multiplication with (|^→𝑘|/𝑚)^1/2, 𝒰_𝜎^→𝑘=^→𝑘𝑚^1/2𝑈_𝜎^→𝑘,𝒱_𝜎^→𝑘=^→𝑘𝑚^1/2𝑉_𝜎^→𝑘. (38) The “calligraphic” spinors fulfill the following helicity-dependent normalizations: 𝒰_𝜎^→𝑘𝒰_𝜎^→𝑘=𝜎,𝒱_𝜎^→𝑘𝒱_𝜎^→𝑘=−𝜎, (39) where we observe that 𝜎 is a good quantum number because the helicity operator commutes with the Hamiltonian (8). The sum rule fulfilled by the fundamental plane-wave spinors is of type II (see (16)). For the positive-energy spinors, we have_{𝜎()𝒰−𝜎𝜎}^→_{𝑘⊗𝒰𝜎}^→_𝑘𝛾5=/^{𝑘−𝛾5𝑚}_,2𝑚 (40a) where for the negative-energy spinors, the sum rule reads _{𝜎()𝒱−𝜎𝜎}^→_{𝑘⊗𝒱𝜎}^→_𝑘𝛾5=/^{𝑘+𝛾5𝑚}_.2𝑚 (40b)The expressions on the right-hand sides are the positive- and negative-energy projectors.

3.2. Two Tachyonic Mass Terms

We study the equation (i𝛾^𝜇𝜕_𝜇−i𝑚₁−𝛾⁵𝑚₂)𝜓(𝑥)=0, as given in (9). The fundamental spinors, which we denote 𝑈^′_±(^→𝑘) and 𝑉^′_±(^→𝑘), fulfill the following equations:/^𝑘−i𝑚₁^−𝛾5𝑚₂^𝑈′_±^→𝑘=0,/^𝑘+i𝑚₁^+𝛾5𝑚₂^𝑉′_±^→𝑘=0. (41a)(41b)

The positive-energy solutions are obtained using the identity ⁽/^𝑘−i𝑚₁^−𝛾5𝑚₂⁾⁽/^𝑘+i𝑚₁^−𝛾5𝑚₂^{)=𝑘2+𝑚2}₁^+𝑚2₂. With 𝐸^′=^→𝑘²−𝑚²₁−𝑚²₂, they read as follows:𝑈^′₊^→𝑘=i𝑚₁+𝑚₂−𝐸^′+^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₊^→𝑘√2i𝑚₁+𝑚₂+𝐸^′−^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₊^→𝑘√2𝑈_′−→,𝑘=i𝑚₁+𝑚₂+𝐸^′−^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₋^→𝑘√2−i𝑚₁−𝑚₂+𝐸^′−^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₋^→𝑘√2. (42a)(42b)

The negative-energy solutions for the tachyonic equation with two mass terms are given as𝑉^′₊^→𝑘=i𝑚₁−𝑚₂−𝐸^′+^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₊^→𝑘√2i𝑚₁−𝑚₂−𝐸^′−^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₊^→𝑘√2𝑉_′−→,𝑘=−i𝑚₁−𝑚₂+𝐸^′−^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₋^→𝑘√2i𝑚₁+𝑚₂+𝐸^′−^→𝑘𝐸^′−^→𝑘²+𝑚²₁+𝑚²₂𝑎₋^→𝑘√2. (43a)(43b)The normalization condition is 𝑈_𝜎^′+(^→𝑘)𝑈^′_𝜎(^→𝑘)=𝑉_𝜎^′+(^→𝑘)𝑉^′_𝜎(^→𝑘)=1. We use a definition of the “calligraphic” spinors analogous to (38), 𝒰^′_𝜎^→𝑘=^→𝑘𝑚₂^1/2𝑈^′_𝜎^→𝑘,𝒱^′_𝜎^→𝑘=^→𝑘𝑚₂^1/2𝑉^′_𝜎^→𝑘. (44) In analogy to (40a) and (40b), a sum rule of type II (see (16)) is fulfilled by the fundamental plane-wave spinors, _{𝜎()𝒰−𝜎′𝜎}^→_𝑘⊗𝒰^′_𝜎^→_𝑘𝛾5=^𝑘+i𝑚₁^−𝛾5𝑚_{22𝑚2,𝜎()𝒱−}/_{𝜎′𝜎→𝑘⊗𝒱′𝜎→𝑘𝛾5=}/^𝑘−i𝑚₁^+𝛾5𝑚_22𝑚2. (45) We thus obtain that the desired projectors onto positive- and negative-energy solutions for the Dirac equation with two tachyonic mass terms (9). The generalized equation (i𝛾^𝜇𝜕_𝜇−i𝑚₁−𝛾⁵𝑚₂)𝜓(𝑥)=0 is fully compatible with the Clifford-algebra-based approach recently described in [59].

4. Theorems for Generalized Dirac Fields

4.1. Spinor Sums and Time-Ordered Propagator

Our central postulate regarding the quantized fermionic theory is that the time-ordered vacuum expectation value of the field operators should yield the time-ordered (Feynman) propagator, which, in the momentum representation, is equal to the inverse of the Hamiltonian (upon multiplication with 𝛾⁰). This postulate implies that under rather general assumptions regarding the mathematical form of the elementary field anticommutators, sum rules have to be fulfilled by the tensor sums over the fundamental spinor solutions. It is perhaps not surprising that these sum rules are precisely of the form investigated in Sections 2 and 3 of this paper.

For definiteness, we consider the solution of the tachyonic Dirac equation (Section 3.1). The generalization to other generalized Dirac equations is straightforward. We start from the field operator [9] 𝜓(𝑥)=d³𝑘(2𝜋)³𝑚𝐸×_𝜎=±𝑏_𝜎(𝑘)𝒰_𝜎^→𝑘e^{−i𝑘⋅𝑥}+𝑑⁺_𝜎(𝑘)𝒱_𝜎^→𝑘e^i𝑘⋅𝑥,𝑘=𝐸,^→𝑘,𝐸=𝐸^→_𝑘=^→𝑘²−𝑚²−i𝜖, (46) where 𝑏_𝜎 annihilates particles and 𝑑⁺_𝜎 creates antiparticles. The following anticommutators vanish:𝑏_𝜎(𝑘),𝑏_𝜌𝑘^′=𝑏⁺_𝜎(𝑘),𝑏⁺_𝜌𝑘^′𝑑=0,_𝜎(𝑘),𝑑_𝜌𝑘^′=𝑑⁺_𝜎(𝑘),𝑑⁺_𝜌𝑘^′=0. (47a)(47b)We assume the following general form for the nonvanishing anticommutators,𝑏_𝜎(𝑘),𝑏⁺_𝜌𝑘^′=𝑓𝜎,^→𝑘(2𝜋)³𝐸𝑚𝛿^3→𝑘−^→𝑘^′𝛿_𝜎𝜌,𝑑_𝜎(𝑘),𝑑⁺_𝜌𝑘^′=𝑔𝜎,^→𝑘(2𝜋)³𝐸𝑚𝛿^3→𝑘−^→𝑘^′𝛿_𝜎𝜌, (48a)(48b)with arbitrary 𝑓 and 𝑔 functions of the quantum numbers 𝜎 and ^→𝑘. One might argue that since 𝑓 and 𝑔 must be dimensionless, they can depend only on the dimensionless arguments 𝜎 and ^→𝑘/𝑚, but that is a detail of the discussion which we do not pursue any further. Our only assumption concerns the fact that the field anticommutators should be diagonal in the helicity and wave vector quantum numbers, leading to the corresponding Kronecker and Dirac-𝛿’s.

We assume that the spin-matrix Γ either constitutes a Lorentz scalar or a pseudoscalar quantity, which is a scalar under the proper orthochronous Lorentz group. The time-ordered product of field operators reads as 0𝑇𝜓(𝑥)𝜓𝑦Γ0=d³𝑘(2𝜋)³𝑚𝐸×Θ𝑥⁰−𝑦⁰e^{−i𝑘⋅(𝑥−𝑦)}×_𝜎=±𝑓𝜎,^→𝑘𝒰_𝜎^→𝑘⊗𝒰_𝜎^→𝑘Γ−Θ𝑦⁰−𝑥⁰e^{i𝑘⋅(𝑥−𝑦)}×_𝜎=±𝑔𝜎,^→𝑘𝒱_𝜎^→𝑘⊗𝒱_𝜎^→𝑘Γ. (49) This equation contains the same coefficient functions 𝑓 and 𝑔 that enter into (48a) and (48b). In order to proceed with the derivation of the propagator, we must postulate that the following sum rules hold: _{𝜎𝑓𝜎,}^→_𝑘𝒰𝜎^→_{𝑘⊗𝒰𝜎}^→_𝑘Γ=^{𝑘−𝛾5𝑚}_{,2𝑚𝜎𝑔𝜎,→𝑘𝒱𝜎→}/_{𝑘⊗𝒱𝜎→𝑘Γ=}/^{𝑘+𝛾5𝑚}_.2𝑚 (50) The sum rule (50) is crucial for the further steps in the derivation of the time-ordered propagator. Introducing a suitable complex integral representation for the step function, one obtains from (49), using (50), after a few steps which we do not discuss in further detail (see also equations (3.169) and (3.170) of [57]) 0𝑇𝜓(𝑥)𝜓𝑦Γ0d=i³𝑘(2𝜋)³𝑚𝐸d𝑘₀𝑒2𝜋^{−i𝑘0⋅(𝑥0−𝑦0)+i→𝑘⋅(→𝑥−→𝑦)}×𝛾⁰𝑘₀−^→𝛾⋅^→𝑘−𝛾⁵𝑚2𝑚𝑘₀d−𝐸+i𝜖+i³𝑘(2𝜋)³𝑚𝐸d𝑘₀e2𝜋^{−i𝑘0⋅(𝑥0−𝑦0)+i→𝑘⋅(→𝑥−→𝑦)}.×−𝛾⁰𝑘₀+^→𝛾⋅^→𝑘+𝛾⁵𝑚2𝑚𝑘₀.+𝐸−i𝜖 (51) The convention is that in any integrals ∫d³𝑘, the component 𝑘₀ is set equal to 𝐸=^→𝑘²−𝑚² in the integrand when it occurs in scalar products of the form 𝑘⋅(𝑥−𝑦), and so forth, but if the integral is over the full d⁴𝑘, then the integration interval is the full 𝑘₀∈(−∞,∞). With the convention which is adopted in many quantum field theoretical textbooks, including [57, 60], we finally obtain the result _{0(𝑥)𝑇𝜓𝜓𝑦Γ0}^d_=i^4𝑘_{()2𝜋4e−i𝑘⋅(𝑥−𝑦)}^𝑘−/^𝛾5𝑚_{𝑘2+𝑚2.+i𝜖} (52) The tachyonic propagator 𝑆_𝑇 is identified, under the integral sign, as ^{𝑆𝑇(𝑘)=}/^{1𝑘−𝛾5()=𝑚+i𝜖𝑘−}/^𝛾5𝑚_{𝑘2+𝑚2.+i𝜖} (53) The sum rules (40a) and (40b) imply that the derivation is valid for the choice 𝑓𝜎,^→𝑘=𝑔𝜎,^→𝑘=−𝜎,Γ=𝛾⁵, (54) in which case the relations given in (50) are fulfilled. Note that this observation does not imply that the choice (54) necessarily is the only one for which we are able to fulfill the postulates given in (50), but it is the only structurally simple choice that we have found.

For the egregiously simple choice (54), let us study the transition to the massless limit (16) in some further detail. Indeed, in the limit 𝑚→0, the denominator of the spin sums in (40a) and (40b) vanishes, and a finite limit is obtained after multiplication with 2𝑚,^lim_𝑚→0𝜎^{()𝒰2𝑚−𝜎}_𝜎^{→𝑘⊗𝒰}_𝜎^{→𝑘𝛾5=𝑘,}_{lim𝑚→0𝜎()𝒱2𝑚−𝜎𝜎→𝑘⊗𝒱𝜎→𝑘𝛾}/₅₌/^𝑘. (55a)(55b)In order to compare the normalizations of the fundamental spinors in the massless limit, we calculate the following quantities: 𝒰_𝜎^→𝑘𝛾⁵𝛾⁰𝒰_𝜎^→𝑘𝐸=−𝜎𝑚,𝑢_𝜎^→𝑘𝛾⁵𝛾⁰𝑢_𝜎^→𝑘=−𝜎,𝒱_𝜎^→𝑘𝛾⁵𝛾⁰𝒱_𝜎^→𝑘𝐸=−𝜎𝑚,𝑣_𝜎^→𝑘𝛾⁵𝛾⁰𝑣_𝜎^→𝑘=−𝜎, (56)(57) where the fundamental spinors 𝑢_𝜎 and 𝑣_𝜎 of the massless equation have been given in Section 2.1. Observing that 𝐸=|^→𝑘| in the massless limit, the identifications √𝑚𝒰_𝜎(^→𝑘)→|^→𝑘|𝑢_𝜎(^→𝑘) and √𝑚𝒱_𝜎(^→𝑘)→|^→𝑘|𝑣_𝜎(^→𝑘), as implied by (56) and (57), show that the identities (55a) and (55b) precisely reduce to the sum rule (16) in the massless limit.

4.2. Generalized Field Anticommutators for Tardyonic and Tachyonic Fields

For definiteness, we have considered the case of the tachyonic Dirac field in the above derivation. The decisive observation is that the tachyonic choice, tachyonicchoice:𝑓𝜎,^→𝑘=𝑔𝜎,^→𝑘=−𝜎,Γ=𝛾⁵, (58) is consistent with both massive tachyonic fields discussed in Sections 3.1 and 3.2, whereas the tardyonic choice, tardyonicchoice:𝑓𝜎,^→𝑘=𝑔𝜎,^→𝑘=1,Γ=𝟙_4×4, (59) yields the time-ordered propagator for both massive tardyonic fields discussed in Sections 2.2 and 2.3. The nonvanishing anticommutators for tardyons take the simple form (cf. (48a) and (48b)), 𝑏tardyonicanticommutators:_𝜎(𝑘),𝑏⁺_𝜌𝑘^′=(2𝜋)³𝐸𝑚𝛿^3→𝑘−^→𝑘^′𝛿_𝜎𝜌,𝑑_𝜎(𝑘),𝑑⁺_𝜌𝑘^′=(2𝜋)³𝐸𝑚𝛿^3→𝑘−^→𝑘^′𝛿_𝜎𝜌. (60) Again, compared with (48a) and (48b), the nonvanishing anticommutators for tachyons take the simple form, 𝑏tachyonicanticommutators:_𝜎(𝑘),𝑏⁺_𝜌𝑘^′=(−𝜎)(2𝜋)³𝐸𝑚𝛿^3→𝑘−^→𝑘^′𝛿_𝜎𝜌,𝑑_𝜎(𝑘),𝑑⁺_𝜌𝑘^′=(−𝜎)(2𝜋)³𝐸𝑚𝛿^3→𝑘−^→𝑘^′𝛿_𝜎𝜌. (61) With these universal choices, the theory of the tardyonic and tachyonic spin-1/2 fields can be unified. The time-ordered propagator is given as 0𝑇𝜓(𝑥)𝜓𝑦Γ0,𝑆d=i𝑆𝑥−𝑦𝑥−𝑦=⁴𝑘(2𝜋)⁴e^{−i𝑘⋅(𝑥−𝑦)}𝑆(𝑘). (62) We use the sum rules for the tensor sums over fundamental spinors given in (25) (for the tardyonic Dirac field), (32) (for the tardyonic Dirac field with two mass terms), (40a) and (40b) (for the tachyonic Dirac field), and (45) (for the Dirac field of imaginary mass). Going through the exact same derivation as outlined above in between (49) and (53), we obtain the following results for the time-ordered propagators of tardyonic and tachyonic fields. For the tardyonic Dirac field (Section 2.2), one has ^{𝑆(1)(𝑘)=}/^1𝑘−𝑚₁^{=+i𝜖𝑘+𝑚}/_{1𝑘2−𝑚21.+i𝜖} (63) For the tardyonic field with two mass terms (Section 2.3), the Feynman propagator is easily found as ^{𝑆(𝑡)(𝑘)=1}/^𝑘−𝑚₁^{+i𝜖−i𝛾5𝑚}₂₌^−𝑖𝜂/^𝑘+𝑚₁^{−i𝛾5𝑚}₂/^{𝑘2−𝑚2}₁^−𝑚2₂^,+i𝜖 (64) where 𝜖 and 𝜂 are infinitesimal imaginary parts. Both tardyonic mass terms acquire an infinitesimal negative imaginary part, and the prefactor 𝑚/𝐸 from (46) in the field operator needs to be replaced by 𝑚₁/𝐸^(𝑡), where the tardyonic energy is 𝐸^(𝑡)=^→𝑘²+𝑚²₁+𝑚²₂. For the tachyonic Dirac field (Section  3.1 in [9]), one has the result given in (53). Finally, for the Dirac field with two tachyonic mass terms, we have _{𝑆′(𝑘)=}^𝑘+/^i𝑚₁^−𝛾5𝑚_{2𝑘2+𝑚21+𝑚22.+i𝜖} (65) In the latter case, the prefactor 𝑚/𝐸 from (46) in the field operator needs to be replaced by 𝑚₁/𝐸^′, where the tachyonic energy is 𝐸^′=^→𝑘²−𝑚²₁−𝑚²₂. For both tachyonic fields discussed here, the mass acquires an infinitesimal positive imaginary part, as manifest in the results given in (53) and (65).

4.3. Tachyonic Gordon Identities

It is useful to illustrate the derivation outlined above by exploring its connection to tachyonic Gordon identities. For definiteness, we again concentrate on the tachyonic Dirac equation discussed in Section 3.1. The matrix element of the vector current finds the following Gordon decomposition for positive-energy spinors: 𝒰_±^→𝑘^′𝛾^𝜇𝒰_±^→𝑘^′=12𝑚𝒰_±^→𝑘^′𝛾⁵𝑘^′𝜇−𝑘^𝜇+i𝜎^𝜇𝜈𝑘^′_𝜈+𝑘_𝜈𝒰_±^→𝑘. (66) For negative-energy solutions, the identity reads as 𝒱_±^→𝑘^′𝛾^𝜇𝒱_±^→𝑘1=−2𝑚𝒱_±^→𝑘^′𝛾⁵𝑘^′𝜇−𝑘^𝜇+i𝜎^𝜇𝜈𝑘^′_𝜈+𝑘_𝜈𝒱_±^→𝑘. (67) For 𝑘^′=𝑘, one has𝒰_±^→𝑘𝛾^𝜇𝒰_±^→𝑘=i𝑚𝒰_±^→𝑘𝛾⁵𝜎^𝜇𝜈𝑘_𝜈𝒰_±^→𝑘,𝒱_±^→𝑘𝛾^𝜇𝒱_±^→𝑘i=−𝑚𝒱_±^→𝑘𝛾⁵𝜎^𝜇𝜈𝑘_𝜈𝒱_±^→𝑘. (68a)(68b)The matrix element of the axial current reads𝒰_±^→𝑘^′𝛾⁵𝛾^𝜇𝒰_±^→𝑘1=−2𝑚𝒰_±^→𝑘^′𝑘^′𝜇+𝑘^𝜇+i𝜎^𝜇𝜈𝑘^′_𝜈−𝑘_𝜈𝒰_±^→𝑘, (69a) whereas for negative-energy solutions 𝒱_±^→𝑘^′𝛾⁵𝛾^𝜇𝒱_±^→𝑘=12𝑚𝒱_±^→𝑘^′𝑘^′𝜇+𝑘^𝜇+i𝜎^𝜇𝜈𝑘^′_𝜈−𝑘_𝜈𝒱_±^→𝑘. (69b)For 𝑘^′=𝑘, this simplifies to𝒰_±^→𝑘𝛾⁵𝛾^𝜇𝒰_±^→𝑘1=−𝑚𝒰_±^→𝑘𝑘^𝜇𝒰_±^→𝑘,𝒱_±^→𝑘𝛾⁵𝛾^𝜇𝒱_±^→𝑘=1𝑚𝒱_±^→𝑘𝑘^𝜇𝒱_±^→𝑘. (70a)(70b)The results (69a), (69b), (70a), and (70b) for the tachyonic axial vector current have a similar structure as the Gordon decomposition for the tardyonic vector current obtained with the ordinary Dirac equation (see equation (2.54) of [57]). The role of the Dirac adjoint for the tardyonic case is taken over by the “chiral adjoint” 𝒰_±(^→𝑘)𝛾⁵ for the tachyonic particle. Here, the designation “chiral adjoint” is inspired by the fact that 𝒰_±(^→𝑘)𝛾⁵𝒰_±(^→𝑘) transforms as a pseudoscalar under Lorentz transformations.

The structure of (68a), (68b), (70a), and (70b) is somewhat peculiar with regard to parity. In (68a), an apparent vector current on the left-hand side appears to transform into an axial current on the right-hand side, whereas in (70a), an apparent axial vector on the left-hand side of the equation becomes what appears to be a vector on the right-hand side. The reason lies in the more complicated behavior of the tachyonic Dirac equation under parity as investigated in [10]. Namely, the tachyonic Dirac equation (7) contains a term which transforms as a scalar under parity,i𝛾^𝜈𝜕−−_𝜇^𝒫→𝛾⁰i𝛾⁰𝜕₀+i𝛾^𝑖−𝜕_𝑖𝛾⁰=i𝛾^𝜈𝜕_𝜇, (71a) as well as a term which transforms as a pseudoscalar, 𝛾⁵𝑚−−^𝒫→𝛾⁰𝛾⁵𝑚𝛾⁰=−𝛾⁵𝑚. (71b)The mass term in the tachyonic Dirac equation is pseudoscalar and changes sign under parity. Indeed, in [10], the tachyonic Dirac equation has been shown to be separately 𝒞𝒫 invariant, and 𝒯 invariant, but not 𝒫 invariant, due to the change in the mass term.

In order to put this observation into perspective, we recall that the entries of the electromagnetic field strength tensor are composed of axial vector components (magnetic ^→𝐵 field), as well as vector components (electric ^→𝐸 field). The transformation properties of the electromagnetic field strength tensor under the proper orthochronous Lorentz group are nevertheless well defined.

The transformation (71b) can be interpreted as a transformation 𝑚→−𝑚 under parity. Thus, if we interpret the mass 𝑚 as a pseudoscalar quantity, then the right-hand sides of (68a) and (70a) transform as a vector and an axial vector, respectively. It is the parity noninvariance of the mass term in the tachyonic Dirac equation which leads to the somewhat peculiar structure of (68a) and (70a).

4.4. Helicity-Dependence and Gupta-Bleuler Condition

The anticommutator relations for tardyons given in (60) imply that both left-handed and right-handed helicity states, for both particles and antiparticles, have positive norm. However, the anticommutator relations for tachyons given in (61) imply that right-handed particle as well as left-handed antiparticle states acquire negative norm. This is shown in equations (31) and (32) of [9]. Indeed, for one-particle states |1_𝑘,𝜎⟩=𝑏⁺_𝜎(𝑘)|0⟩, 1_𝑘,𝜎1_𝑘,𝜎=0𝑏_𝜎(𝑘)𝑏⁺_𝜎(𝑘0=0𝑏_𝜎)(𝑘),𝑏⁺_𝜎(𝑘0=()𝑉𝐸)−𝜎𝑚, (72) where 𝑉=(2𝜋)³𝛿³(^→0) is the normalization volume in coordinate space. The Fock-space norm ⟨1_𝑘,𝜎|1_𝑘,𝜎⟩ is negative for 𝜎=1.

For clarification, the corresponding Gupta-Bleuler condition should be indicated explicitly. In full analogy to the instructive discussion of the Gupta-Bleuler mechanism for the photon field, as given in full clarity in Chapter  (9b) of Schweber’s textbook [61], we select the positive- and negative-frequency component of the (𝜎=1)-component of the neutrino field operator,𝜓⁽⁺⁾_𝜎=1(𝑥)=d³𝑘(2𝜋)³𝑚𝐸𝑏_𝜎=1(𝑘)𝒰_𝜎=1^→𝑘e^{−i𝑘⋅𝑥},𝜓⁽⁻⁾_𝜎=1(𝑥)=d³𝑘(2𝜋)³𝑚𝐸𝑑⁺_𝜎=1(𝑘)𝒱_𝜎=1^→𝑘e^i𝑘⋅𝑥, (73a)(73b)and postulate that it annihilates any physical Fock state |Ψ⟩ of the tachyonic field, ⟨Ψ|𝜓⁽⁻⁾_𝜎=1(𝑥)=𝜓⁽⁺⁾_𝜎=1(𝑥)|Ψ⟩=0. (74) These relations automatically imply that the Gupta-Bleuler condition also is realized in terms of the expectation value ⟨Ψ|𝜓_𝜎=1(𝑥)|Ψ⟩=⟨Ψ|𝜓⁽⁻⁾_𝜎=1(𝑥)+𝜓⁽⁺⁾_𝜎=1(𝑥)|Ψ⟩=0, (75) but the condition (74) is stronger. We recall that the Gupta-Bleuler condition on the photon field reads ⟨Ψ_𝛾|𝜕^𝜇𝐴_𝜇|Ψ_𝛾⟩=0, where |Ψ_𝛾⟩ is a Fock state of the photon field. As stressed in Schweber’s book [61] on page 246, the condition ⟨Ψ_𝛾|𝜕_𝜇𝐴^𝜇|Ψ_𝛾⟩=0 is not sufficient for the suppression of the longitudinal and scalar photons, but one must postulate that 𝜕^𝜇𝐴_𝜇⁽⁺⁾|Ψ_𝛾⟩=0, where 𝐴_𝜇⁽⁺⁾ is the positive-frequency component of the photon field operator. Because the tachyonic fermion, unlike the photon, is not equal to its own antiparticle, we need two conditions, given in (74).

A crucial question now concerns the possibility of reversing the helicity-dependence, that is, the question of whether or not a different choice for the helicity-dependent factors in (48a) and (48b) exists that would imply negative norm for left-handed particles and right-handed antiparticles. A related question is whether other tachyonic Dirac Hamiltonians exist for which the choice 𝑓(𝜎,^→𝑘)=𝑔(𝜎,^→𝑘)=+𝜎 instead of (−𝜎) would fulfill our general postulate, namely, the sum rule (50). For reasons outlined in the following, we can ascertain that this is not the case; tachyonic spin-1/2 particles should always be left handed.

The arguments supporting this conclusion are as follows. First, if we assume that the tachyonic field fulfills a sum rule of type II which for massless fields is given in (16), then it is impossible to replace (−𝜎) by (+𝜎) in the sum rule because of the necessity to preserve a smooth massless limit. The considerations in the text following (16) imply that if one were to replace (−𝜎) by (+𝜎) in the massless case, then one would violate the sum rule (16). The second argument is obtained by explicit calculation. We have checked that if one replaces 𝑚→−𝑚 in the tachyonic and imaginary-mass Dirac equations (7) and (9), then the sum rules fulfilled by the corresponding fundamental spinors still contain the characteristic factor (−𝜎). For the imaginary-mass Dirac equation, this result is obtained in [25]. Intuitively, we can understand this result as follows: the mass 𝑚 in the denominators of the right-hand sides of (40a), (40b), and (45) is obtained as the modulus √𝑚²=|𝑚| and does not change if we replace 𝑚→−𝑚 in the superluminal Dirac equation. The mass in the numerator of the right-hand sides of (40a), (40b), and (45) changes sign, but this is consistent with the obvious change in the functional form of the positive-energy and negative-energy projectors as we change the sign of the mass term. Again, this consideration supports the conclusion that we cannot invert the helicity-dependence by choosing a different Hamiltonian; the factor (−𝜎) persists.

The third argument comes from the tachyonic Gordon identities discussed in Section 4.3. We use the Gordon identity (70a) and (70b) and the normalization (39) to calculate the bispinor trace (with a 𝛾⁰ multiplied from the right) of the left-hand side of (40a),tr_𝜎(−𝜎)𝒰_𝜎^→𝑘⊗𝒰_𝜎^→𝑘𝛾⁵𝛾⁰=_𝜎(−𝜎)𝒰_𝜎^→𝑘𝛾⁵𝛾^𝜇=0𝒰_𝜎^→𝑘=_𝜎()−𝑘−𝜎⁰𝑚𝒰_𝜎^→𝑘𝒰_𝜎^→𝑘_=𝜎𝐸=2𝑚. (76a) The bispinor trace of the right-hand side of (40a) is _𝛾tr0/^{𝑘−𝛾5𝑚}_{𝛾2𝑚=tr0}/^𝑘𝐸_2𝑚=4^𝐸_2𝑚=2𝑚, (76b)which shows the consistency of the bispinor sum (40a) with the Gordon decomposition (70a) and (70b). If we were to replace (−𝜎) by (+𝜎), the two sides of the relation (76a) would differ by a minus sign. The bispinor trace of the left-hand side of (40b) istr_𝜎(−𝜎)𝒱_𝜎^→𝑘⊗𝒱_𝜎^→𝑘𝛾⁵𝛾⁰=_𝜎(−𝜎)𝒱_𝜎^→𝑘𝛾⁵𝛾^𝜇=0𝒱_𝜎^→𝑘=_𝜎()𝑘−𝜎⁰𝑚𝒱_𝜎^→𝑘𝒱_𝜎^→𝑘_=−𝜎𝐸=2𝑚. (77a) We have used the tachyonic Gordon decomposition for negative-energy states as given in (70a) and (70b), which differs from the positive-energy Gordon decomposition by a minus sign, but an additional minus sign is obtained from the Lorentz-invariant normalization of the negative-energy fundamental bispinors. From the right-hand side of (40b), we have _𝛾tr0/^{𝑘+𝛾5𝑚}_{𝛾2𝑚=tr0}/^𝑘𝐸_2𝑚=2𝑚, (77b)which again is fully consistent, but only because we have a factor (−𝜎) in the sum rule (40a) and (40b), which combines with the factor (−𝜎) from the Lorentz-invariant normalization of the 𝒱_𝜎(^→𝑘)𝒱_𝜎(^→𝑘), to give 2𝐸/𝑚 as a final result in (77a) and (77b).

5. Physical Interpretation: From Tachyonic Neutrinos to Cosmology

5.1. Arguments for and against Tachyonic Neutrinos

In the absence of conclusive experimental evidence, the hypothesis of tachyonic neutrinos has been controversially discussed in the literature. Three main arguments [62] have been brought forward against tachyonic neutrinos. (1) They would require us to give up the notion of a Lorentz-invariant vacuum state, and even the vacuum would become unstable in the presence of tachyonic fields. (2) Given the tachyonic dispersion relation 𝐸=^→𝑘²−𝑚², the role of states with |^→𝑘|<𝑚 needs to be clarified. (3) The physical (probability!?) interpretation of the conserved Noether current of the free tachyonic Dirac equation has been called into question [62], and it has been argued that no consistent interpretation can be given because certain zero components of the conserved current were conjectured to vanish for all tachyonic momentum eigenstates [62].

A possible answer for question (1) has been proposed in [9]. Summarizing the argument, it has been concluded in [9] that one can solve the problem in two ways. (i) One can Lorentz transform the vacuum state and Lorentz transform all fundamental creation and annihilation operators of the fermion field (some of these will change from annihilators to creators upon transformation, due to the spacelike nature of the tachyons). (ii) One keeps a Lorentz-invariant vacuum state, and only transform the space-time arguments 𝑘^𝜇 and 𝑥^𝜇 of the field operators, keeping all creation operators as creators and annihilation operators as annihilators. The amplitudes, cross sections, and so forth, obtained using approach (ii), then depend on scalar product of four vectors which are equal to the result obtained by first calculating the process in the original Lorentz frame and then performing the Lorentz transformation into the moving frame.

The conjecture regarding an expansion of a “false” vacuum in the presence of tachyons can be traced to the fact that most of the tachyonic theories discussed so far in the literature are scalar [28–33]. Indeed, scalar tachyons have a problem with instability, because of the structure of the mass term in relation to the field Hamiltonians, which changes sign 𝑚²→−𝑚² in a tachyonic theory, suggesting that the field energy can be lowered by creating tachyons. The problem does not occur in tachyonic spin-1/2 theories because a linear, not quadratic, mass term enters the field Lagrangian and Hamiltonian. Provided one reinterprets the spin-1/2 antiparticle solutions in the usual way (negative energy for propagation into the past becomes positive energy for propagation into the future), it then becomes immediately clear that the vacuum energy cannot be lowered upon spin-1/2 tachyon antitachyon pair production.

A solution to problem (2) has also been proposed in [9]. Namely, the energies with 𝐸=±^→𝑘²−𝑚²−i𝜖 (for |^→𝑘|<𝑚) find a natural interpretation in terms of complex resonance and antiresonance energies, which describe unstable states which decay in time. Particle resonances are damped for propagation into the future, and antiparticle antiresonances are damped for propagation into the past, as they should be [9]. The occurrence of momentum eigenstates with real energies, and resonances with complex resonance energies, is a well-known phenomenon all across physics (e.g., in atomic physics, the ground-state energy of the helium atom is strictly real, whereas autoionizing resonances in the three-particle system have a manifestly complex resonance energy).

Another “myth” which should be refuted concerns a conceivable “runaway reaction” where a moving tachyon releases an arbitrarily large amount of energy, as it loses energy and accelerates, given its classical energy-velocity relation 𝐸=𝑚/√𝑣²−1. According to this relation, a tachyon indeed accelerates as its energy is lowered and becomes commensurate with the invariant mass square. (At high energy, a tachyon approaches the light cone, though.) An infinitely fast tachyon takes the role of a tardyon at rest [28]; the explicit eigenstates have been indicated in [9]. However, energy conservation holds, and one cannot mix the notion of energy increase by acceleration, which only holds for tardyonic particles, in order to “convert” a tachyon losing energy by acceleration into a tardyon that gains energy in the same process. The “runaway reaction” is impossible; only a finite amount of energy is released as the tachyon accelerates and the energy goes from 𝐸=𝑚/√𝑣²−1 to zero. Energy conservation holds for tachyons, even if they accelerate, somewhat counterintuitively, when losing energy.

An answer to question (3) has not yet been provided so far in the literature to the best of our knowledge. Here, we aim to provide a possible physical interpretation for the conserved current and scalar product and also show where certain arguments presented originally by Hughes and Stephenson in their research paper [62] entitled “against tachyonic neutrinos” become inconsistent. The Lagrangian density of the tachyonic Dirac particle reads, in first quantization [6], ℒ(𝑥)=𝜓(𝑥)𝛾⁵i𝛾^𝜇𝜕_𝜇−𝛾⁵𝑚𝜓(𝑥). (78) This is equivalent, up to partial integration, to the symmetric form [10] iℒ=2𝜓𝛾⁵𝛾^𝜇𝜕_𝜇𝜓−𝜕_𝜇𝜓𝛾⁵𝛾^𝜇𝜓−𝑚𝜓𝜓, (79) where we suppress the space-time argument 𝑥=(𝑡,^→𝑟). The nonsymmetric form (78) clearly exhibits the presence of the “chiral adjoint” 𝜓(𝑥)𝛾⁵ in the Lagrangian. The conserved current is 𝒥^𝜇(𝑥)=𝜓(𝑥)𝛾⁵𝛾^𝜇𝜓(𝑥),𝜕_𝜇𝒥^𝜇(𝑥)=0. (80) The zero component 𝒥⁰(𝑥)=𝜓(𝑥)𝛾⁵𝛾⁰𝜓(𝑥) (81) assumes the following values in plane-wave eigenstates, according to (56) and (57):𝒰_𝜎^→𝑘𝛾⁵𝛾⁰𝒰_𝜎^→𝑘𝐸=−𝜎𝑚,𝒱_𝜎^→𝑘𝛾⁵𝛾⁰𝒱_𝜎^→𝑘𝐸=−𝜎𝑚. (82a)(82b)We conclude that 𝒥⁰ is positive for left-handed particle states (𝜎=−1) and right-handed for antiparticle states (likewise, 𝜎=−1; the helicity is equal to −𝜎 for antiparticles). For 𝐸=0, the “axial norm” of the momentum eigenstates given in (82a) and (82b) vanishes, but this is a very special case. Namely, for tachyons, 𝐸=0 implies |^→𝑘|=𝑚=𝑚𝑣/√𝑣²−1 for the momentum and thus corresponds to an “infinitely fast” tachyon (𝑣→∞), which remains “infinitely fast” upon Lorentz transformation. It would thus be wrong to conclude, as done in [62] in the text following (54) of [62], that all tachyonic states have zero axial norm, because this would correspond to a forbidden generalization of a result which holds asymptotically, for 𝑣→∞, to all finite values of the tachyonic velocity 𝑣. Our (82a) and (82b) gives the explicit result for any finite value of the energy 𝐸.

The spatial integral of the zero component of the conserved current, d³𝑟𝒥⁰(𝑥)=d³𝑟𝜓(𝑥)𝛾⁵𝛾^𝜇𝜓(𝑥)