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Hindawi Publishing Corporation ISRN High Energy Physics Volume 2013, Article ID 374612, 21 pages http://dx.doi.org/10.1155/2013/374612

Research Article From Generalized Dirac Equations to a Candidate for Dark Energy

U. D. Jentschura and B. J. Wundt

Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409, USA

Correspondence should be addressed to U. D. Jentschura; [email protected]

Received 12 November 2012; Accepted 1 December 2012

Academic Editors: G. A. Alves, C. A. D. S. Pires, and F.-H. Liu

Copyright © 2013 U. D. Jentschura and B. J. Wundt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

5 5 We consider extensions of the Dirac equation with mass terms 𝑚1 + i𝛾 𝑚2 and i𝑚1 +𝛾𝑚2. The corresponding Hamiltonians 5 are Hermitian and pseudo-Hermitian (𝛾 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 secondsum 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.”

𝜇𝜈 1. Introduction choose the space-time metric as 𝑔 = diag(1,−1,−1,−1). 𝜇 The 𝜕𝜇 denotes the partial derivative 𝜕/𝜕𝑥 with respect to the 𝜇 1.1. Generalized Dirac Equations: Mass Terms and Dispersion space-time coordinate 𝑥 =(𝑡,𝑥)⃗ .Itisquitesurprisingthat Relations. Dirac is often quoted saying in some of his talks a systematic presentation of the solutions of the generalized that the equation that carries his name [1, 2]is“more 𝜇 Dirac equations (i𝛾 𝜕𝜇 − 𝑀)𝜓(𝑥), =0 in the helicity basis intelligent than its inventor.” Of course, it needs to be added [3], has not been recorded in the literature to the best of that it was Dirac himself who found most of the additional our knowledge. While the following discussion is somewhat insight. Here, we are concerned with extensions of the Dirac technical, we believe that it will be beneficial to give their equation which contain both tardyonic and tachyonic mass explicit form, in order to fix ideas for the following discussion. terms. Tardyonic (subluminal) mass terms lead to dispersion We set ℎ=𝑐=𝜖0 =1andusetheDiracmatricesinthe 2 relations of the form 𝐸=√𝑝⃗ +𝑚2, whereas tachyonic mass standard representation terms lead to superluminal dispersion relations of the form 2 0 12×2 0 0 𝜎⃗ 𝐸=√𝑝⃗ −𝑚2,where𝐸 is the energy and 𝑝⃗ is the momentum. 𝛾 =𝛽=( ), 𝛾=(⃗ ), 0−12×2 −𝜎0⃗ The generalized, matrix-valued mass term 𝑀 enters the Dirac ( 𝛾𝜇𝜕 − 𝑀)𝜓(𝑥) =0 𝛾𝜇 4×4 (1) equation in the form i 𝜇 .The are 5 0 12×2 𝜇 𝜈 𝜇𝜈 𝛾 =( ), matrices that fulfill the relations {𝛾 ,𝛾 }=2𝑔 ,wherewe 12×2 0 2 ISRN High Energy Physics

0 0 + 0 and define 𝛼=𝛾⃗ 𝛾⃗. For the ordinary Dirac theory, one has The relation 𝐻5 =𝛾𝐻5 (−𝑟)𝛾⃗ has been given in [9, 10]. 𝑀=𝑚1 (one should say more precisely 𝑀=𝑚114×4)witha However, it is much more instructive to observe that 𝐻5 is 5 5 + 5 5 real mass 𝑚1, 𝛾 Hermitian; that is, 𝐻5 =𝛾𝐻5 𝛾 . The concept of 𝛾 Hermiticity is known in lattice theory [11, 12] and is otherwise 𝜇 (i𝛾 𝜕𝜇 −𝑚1)𝜓(𝑥) =0. (2) called pseudo-Hermiticity [13–23]. An obvious generalization of the tachyonic case contains 5 √ ⃗2 2 an imaginary mass and a 𝛾 mass term, The dispersion relation is 𝐸= 𝑝 +𝑚1.Thecorresponding Dirac Hamiltonian reads 𝜇 5 (i𝛾 𝜕𝜇 − i𝑚1 −𝛾 𝑚2)𝜓(𝑥) =0. (9) (1) 𝐻 = 𝛼⋅⃗ 𝑝+𝛽𝑚⃗ 1. (3) √ ⃗2 2 2 The dispersion relation is 𝐸= 𝑝 −𝑚1 −𝑚2.Thecorre- Extensions of the Dirac equation with pseudoscalar mass sponding Hamiltonian reads terms that contain the fifth current have been introduced in the literature. In [4], it is shown that for a mass term 𝐻󸀠 = 𝛼⋅⃗ 𝑝+⃗ 𝛽𝑚 +𝛽𝛾5𝑚 5 i 1 2 (10) of the form 𝑀=𝑚1 + i𝛾 𝑚2, the fermion propagator 5 󸀠 5 󸀠+ 5 may obtain nontrivial gradient corrections already at the and is 𝛾 Hermitian, 𝐻 =𝛾𝐻 𝛾 .For𝑚2 =0,(9)hasbeen first order in derivative expansion, for a position-dependent discussed in [24, 25]. mass. In that case, the fermion self-energy may contribute Itisourgoalheretopresentthefundamentaleigenspinors to a conceivable explanation for CP-violation during elec- corresponding to the plane-wave solutions of (2), (4), (7), and troweak baryogenesis, as pointed out in [4]. We thus study (9) in a unified and systematic manner. Furthermore, we dis- the following generalized form of the tardyonic (subluminal) cuss the second-quantized versions of the fermionic theories Dirac equation: described by the generalized Dirac equations. Anticipating

𝜇 5 part of the results, we may point out that the massless Dirac (i𝛾 𝜕𝜇 −𝑚1 − i𝛾 𝑚2)𝜓(𝑥) =0. (4) equation “interpolates” between the tardyonic equations (2) and (4) and the tachyonic equations (7)and(9). For zero √ ⃗2 2 2 mass, helicity and chirality are equal. Helicity and chirality The dispersion relation is 𝐸= 𝑝 +𝑚1 +𝑚2.TheHermi- “depart” from each other in very specific directions, when the tian tardyonic Hamiltonian operator reads as tardyonic and tachyonic mass terms are “switched on,” as we (𝑡) 5 shall discuss in the following. 𝐻 = 𝛼⋅⃗ 𝑝+𝛽𝑚⃗ 1 + i𝛽𝛾 𝑚2. (5)

We may indicate a further motivation for our study; namely, 1.2. Tachyonic Dirac Equation and Neutrinos: Possible Con- the unitarity of the 𝑆 matrix implies the existence of useful nections. The tachyonic generalized Dirac equations7 ( )and 2𝑛 relations [5]fortheevenpowers(𝑚2) obtained upon (9) describe the motion of superluminal particles, which may expanding a one-loop amplitude, formulated with a mass either be important for astrophysical studies (neutrinos) or 5 term 𝑚1 + i𝛾 𝑚2,inpowersof𝑚2. This implies that a better for artificially generated environments such as honeycomb understanding of the tardyonic equation with two mass terms photonic lattices in which pertinent dispersion relations could be of much more general interest. become practically important [26]. The existence of superlu- It has not escaped our attention that the chiral transfor- minal particles would not falsify Einstein’s theory of special mation relativity [27], which according to common wisdom is based on the following postulates. (i) The principle of relativity 5 5 5 𝜇 exp (i𝛾 𝜃) = 𝜇 cos 𝜃+i𝛾 𝜇 sin 𝜃=𝑚1 + i𝛾 𝑚2 (6) states that the laws of physics are the same for all observers in uniform motion relative to one another. (ii) The speed of (1) (𝑡) connects the two Hamiltonians 𝐻 and 𝐻 for 𝑚1 =𝜇cos 𝜃 light in a vacuum is the same for all observers, regardless of and 𝑚2 =𝜇sin 𝜃, but it is computationally easier and more theirrelativemotionorofthemotionofthesourceofthe instructive to consider the real and imaginary parts of the light. Predictions of relativity theory regarding the relativity mass term separately. of simultaneity, time dilation, and length contraction would Within a systematic approach to generalized Dirac equa- not change if superluminal particles did exist. Furthermore, tions with pseudoscalar mass terms, we also consider tachy- as shown by Sudarshan et al. [28–31]andFeinberg[32, 33], 5 onic (superluminal) mass terms of the form 𝑀=𝛾𝑚 which the existence of tachyons, which are superluminal particles 2 ⃗2 𝐸=√𝑝⃗2 −𝑚2 fulfilling a Lorentz-invariant dispersion relation 𝐸 = 𝑝 − induce a superluminal dispersion relation .The 𝑚2 corresponding generalized Dirac equation has been named 𝜈, is fully compatible with special relativity and Lorentz the “tachyonic Dirac equation” and reads as follows [6–10]: invariance. According to special relativity, it is forbidden to accelerate a particle “through” the light barrier (because 𝐸= 𝜇 5 𝑚/√1−𝑣2 →∞ 𝑣→1 (i𝛾 𝜕𝜇 −𝛾 𝑚) 𝜓 (𝑥) =0. (7) for ), but a genuinely superluminal particle remains superluminal upon Lorentz transformation. The corresponding Hamiltonian reads Significant problems are encountered when one attempts to quantize the tachyonic theories, but again, as shown in [9], 5 𝐻5 = 𝛼⋅⃗ 𝑝+𝛽𝛾⃗ 𝑚. (8) these problems may not be as serious as previously thought. ISRN High Energy Physics 3

In particular, the so-called reinterpretation of solutions prop- helicity are different; hence a 𝑉−𝐴coupling of the form 𝜇 5 agating into the past according to the Feynman prescription 𝛾 (1 − 𝛾 ) no longer vanishes for massive Dirac neutrinos [31] is a cornerstone of modern field theory. Furthermore, it if one uses the canonical eigenstates of the massive Dirac has been shown in [9] that tachyonic particles can be localized equation. One therefore has to invoke additional exotic and equal-time anticommutators of the spin-1/2 tachyonic mechanisms in order to ensure the “sterility” of the right- field involve an unfiltered Dirac-𝛿 (see equation (37) of [9]). handed Dirac neutrinos.) Wheeler also disliked (Professor Despite these arguments, we can say that, from the point M. Fink from the University of Austin (Texas) was engaged of view of fundamental symmetries, accepting a superluminal in discussions with Professor J. A. Wheeler, who found the neutrino would be equivalent to “an ugly duckling.” Adding notion of a nonvanishing neutrino mass so unappealing that to the difficulties, we notice that recent experimental claims he discouraged experimentalists from undertaking any effort regarding the conceivable observation of highly superluminal to measure the neutrino mass, based on arguments described neutrinos have turned out to be false. One may point out in Section 1 (M. Fink, private communication, 2012.)) the −5 that a relative deviation 𝑣=𝑐(1+𝛿)with 𝛿∼10 at notion of a Majorana neutrino, arguing that the charge 𝐸∼30GeV, as claimed by some recent experimental collabo- conjugation invariance condition imposed on the Majorana rations, would correspond to a negative neutrino mass square particle precludes the existence of plane-wave solutions to the 2 in the order of ∼ −(100 MeV) , if one assumes a Lorentz- Majorana equation and maximally violates lepton number. invariant dispersion relation [34]. Still, there is at present no Again, these arguments (We are grateful to Professor M. Fink conclusive answer regarding the conceivable superluminality for the clarification and confirmation.) are in full agreement of at least one neutrino flavor [35–39], and it is intriguing with those recently given in [52]. that all available direct measurements of the neutrino mass The original standard model thus called for manifestly square have resulted in negative expectation values, still com- massless neutrinos. The commonly accepted observation of patible with zero within experimental uncertainty, whereas neutrino oscillations precludes the possibility that all three published experimental best estimates for the neutrino speed generations of neutrino mass eigenstates are massless. Lepton [40–43]havebeensuperluminal,againstillcompatiblewith number conservation is based on the global gauge symmetry 𝜓→𝜓 ( Λ) the speed of light within experimental error. The recent exp i , applied simultaneously to all lepton fields. ICARUS result [43]isconsistentwiththistrend[44–50]; A Majorana neutrino would destroy lepton number as a thebestestimatefortheneutrinovelocityissuperluminal, global symmetry but solve the “autobahn paradox,”because a but the deviation from 𝑣𝜈 =𝑐is statistically insignificant. Majorana neutrino would be equal to its own antiparticle and The OPERA collaboration [51] has indicated a preliminary, thus, looking back, the right-handed neutrino state would +3.8 −6 consist of the same particle = antiparticle. revised result of (𝑣 − 𝑐)/𝑐 = (2.7 ± 3.1(stat)−2.8(sys.)) × 10 . Neither subluminal nor superluminal propagation velocities On the other hand, if we assume that the neutrino is are excluded based on the available experimental data. The described by the tachyonic Dirac equation, then the following “ugly duckling of a superluminal neutrino” is not beautiful; if statements are valid. we are to consider accepting it, then we should be able to hope (i)Statement1:wecanproperlyassignleptonnumber that the emergence of at least one “added insight” should be and use plane-wave eigenstates for incoming and theresultofthisoperation. outgoing particles, while allowing for nonvanishing Before we discuss the possible emergence of these ben- mass terms and thus mass square differences among efits, let us include some historic remarks. According to the neutrino mass (not flavor) eigenstates. reliable sources (Professor M. Fink from the University of (ii) Statement 2: there is a natural resolution for the Austin (Texas) was engaged in discussions with Professor J. A. “autobahn paradox” because a left-handed spacelike Wheeler, who found the notion of a nonvanishing neutrino neutrino always remains spacelike upon Lorentz mass so unappealing that he discouraged experimentalists transformation and cannot be overtaken. from undertaking any effort to measure the neutrino mass, based on arguments described in Section 1 (M. Fink, private (iii) Statement 3: the right-handed particle and left- communication, 2012.)), Professor J. A. Wheeler, in his later handed antiparticle states are suppressed due to neg- years at the University of Austin (Texas), used to argue that ative Fock-space norm. the neutrino has to be massless, necessarily, and that in his (iv) Statement 4: at least qualitatively, tachyonic neutrinos opinion, it could only be a massless Weyl particle with definite could yield an explanation for a repulsive force on helicity (and chirality). Recently presented arguments [52] intergalactic distance scales as they are repulsed, regarding the possibility of overtaking a subluminal, left- like all tachyons, by gravitational interactions (dark handed neutrino, looking back and seeing a right-handed energy). neutrino, were supposedly already used by Wheeler in order Pauli [53] postulated the existence of neutrinos, on the basis to dispel the conceivable existence of a neutrino mass term. of the conservation of angular momentum and energy, and If we assume that right-handed neutrinos are much more also introduced pseudo-Hermitian operators [13]. Here, we massive than their left-handed counterparts, then this leads describe conceivable connections of neutrino physics and to a paradox unless one assumes that all light, right-handed pseudo-Hermitian operators. Final clarification can only neutrinos are sterile. (The problem with a right-handed sterile come from experiment. When in 1956, Reines and Cowan massive neutrino is that for massive neutrinos, chirality and [54] discovered the electron neutrino, two and a half years 4 ISRN High Energy Physics

̂⃗ ⃗ ⃗ before Pauli’s death, Pauli replied [55]bytelegram“Thanks These fulfill the fundamental relations (𝜎⋅⃗ 𝑘)𝑎𝜎(𝑘) = 𝜎𝑎𝜎(𝑘), ̂ for message. Everything comes to him who knows how to ∑ 𝑎 (𝑘)⊗𝑎⃗ +(𝑘)⃗ = 1 ∑ 𝜎𝑎 (𝑘)⊗𝑎⃗ +(𝑘)⃗ = 𝜎⋅⃗ 𝑘⃗ wait. Pauli.”In defense of the tachyonic hypothesis, we would as well as 𝜎 𝜎 𝜎 2 and 𝜎 𝜎 𝜎 , ̂⃗ ⃗ ⃗ like to stress that a tachyonic Dirac neutrino would allow us to where 𝑘=𝑘/|𝑘| and the sum over 𝜎 is over the values ±1. retain lepton number conservation as a symmetry of nature. The sums over the fundamental bispinors 𝑢 and 𝑣 fulfill the Wewouldthusliketowriteupthesethoughtsinthecurrent following sum rules: paper, with attention to detail. We should point out that 󵄨 󵄨 ∑2 󵄨𝑘⃗󵄨 𝑢 (𝑘)⃗ ⊗ 𝑢 (𝑘)⃗ = 𝑘, our approach fully conserves Lorentz invariance, in contrast Sum rule I: 󵄨 󵄨 𝜎 𝜎 ¡ (15) to the extensions of the Standard Model based on Lorentz- 𝜎 violating terms which can otherwise lead to superluminal as well as propagation (see Tables 11 and 13 of [56]). 󵄨 󵄨 ℎ=𝑐=𝜖 =1 ∑2 󵄨𝑘⃗󵄨 (−𝜎) 𝑢 (𝑘)⃗ ⊗ 𝑢 (𝑘)⃗ 𝛾5 = 𝑘, Units with 0 are used throughout the Sum rule II: 󵄨 󵄨 𝜎 𝜎 ¡ (16) paper. The organization is as follows: in Section 2,wediscuss 𝜎 massless and tardyonic theories. Tachyonic extensions of the SumruleIcanbeobtainedbyaquickexplicitcalculation,and Dirac equation are discussed in Section 3. Connections of sum rule II holds because in the massless limit, helicity equals the fundamental tensor sums over the eigenspinors with ± chirality (positive sign for positive energy, negative sign for the derivation of the time-ordered propagator are analyzed 𝑢 (𝑘)⃗ = in Section 4. A candidate for dark energy is presented in negative energy). We denote the Dirac adjoint by 𝜎 𝑢+(𝑘)𝛾⃗ 0 Section 5. Conclusions are reserved for Section 6. 𝜎 . One can easily check by an explicit calculation that ⃗ 5 5 0 ⃗ + ⃗ ⃗ 5 𝑢𝜎(𝑘)𝛾 =(𝛾𝛾 𝑢𝜎(𝑘)) =(−𝜎)𝑢𝜎(𝑘) and 𝑣𝜎(𝑘)𝛾 = 5 0 ⃗ + ⃗ 2. Generalized Dirac Equations: Massless and (𝛾 𝛾 𝑣𝜎(𝑘)) =(−𝜎)𝑣𝜎(𝑘). We can thus introduce a factor 2 Tardyonic Theories (−𝜎) =1under the summation over spins in (15)and replace one of the factors (−𝜎) by a multiplication of the Dirac 2.1. Massless Dirac Theory. The massless Dirac equation and adjointspinorfromtherightbythefifthcurrent.TheLorentz- the massless Dirac Hamiltonian read as invariant normalization of the massless solutions vanishes; 𝜇 ⃗ 𝑢 (𝑘)𝑢⃗ (𝑘)⃗ = 𝑣 (𝑘)𝑣⃗ (𝑘)⃗ = 0 i𝛾 𝜕𝜇𝜓 (𝑥) =0, 𝐻0 = 𝛼⋅⃗ 𝑝. (11) that is, 𝜎 𝜎 𝜎 𝜎 . Eigenstates of the massless Hamiltonian 𝐻0 = 𝛼⋅⃗ 𝑝⃗ have to 5 𝛾5 We note that 𝐻0 is both Hermitian as well as 𝛾 Hermitian; be eigenstates of the chirality operator because the chirality 5 + 5 5 ⃗ [𝛾 ,𝐻0]= that is, 𝐻0 =𝛾𝐻0 𝛾 . The dispersion relation is 𝐸=|𝑘|.With commutes with the Hamiltonian, in the sense that 𝜇 ⃗ 0.Furthermore, 𝑘 =(𝐸,𝑘), we seek positive-energy and negative-energy solutions of the form 󵄨 󵄨 Σ⋅⃗ 𝑝⃗ 𝐻 = 𝛼⋅⃗ 𝑝=⃗ 󵄨𝑝⃗󵄨 𝛾5 ( ), ⃗ ⃗ 0 󵄨 󵄨 󵄨 ⃗󵄨 (17) 𝜓 (𝑥) =𝑢𝜎 (𝑘) exp (−i𝑘⋅𝑥) ,𝜙(𝑥) =𝑣𝜎 (𝑘) exp (i𝑘⋅𝑥) , 󵄨𝑝󵄨

(12) 5 where Σ=𝛾⃗ 𝛼⃗ is the vector of 4×4spin matrices, and 𝜎=± ⃗ where denotes a quantum number which is equal to the the helicity operator is identified as Σ⋅𝑝/|⃗ 𝑝|⃗ .Let𝜆1 be the helicity for positive-energy states and equal to the negative of eigenvalue of chirality and let 𝜆2 be the eigenvalue of the 𝑘=𝛾𝜇𝑘 the helicity for negative-energy states. With ¡ 𝜇,wehave helicity operator. Then, the eigenvalue of the Hamiltonian 𝑘𝑢¡ ±(𝑘) = 𝑘𝑣¡ ±(𝑘) =. 0 In the massless limit, the solutions to is 𝐸0 =|𝑝|𝜆⃗ 1𝜆2.Since𝜆1 =±1and 𝜆2 =±1,we the Dirac equation are given as (see Chapter 2 of [57]) easily recover the known fact that helicity equals chirality for ⃗ ⃗ positive energy, whereas the relation is reversed for negative- 1 𝑎+ (𝑘) 1 𝑎− (𝑘) 𝑢 (𝑘)⃗ = ( ), 𝑢 (𝑘)⃗ = ( ), energy states (see also Chapter 2.4 of [57]).Weareawareof + √ ⃗ − √ ⃗ the fact that the considerations reported in the current section 2 𝑎+ (𝑘) 2 −𝑎− (𝑘) partly refer to the literature, but we give them in some detail (13a) because they are essential for the following considerations. ⃗ ⃗ ⃗ ⃗ 𝑣+ (𝑘) =−𝑢+ (𝑘) ,𝑣− (𝑘) =−𝑢− (𝑘) . (13b) 2.2. Massive Dirac Theory. We start from the ordinary Dirac 𝜇 The well-known helicity spinors are recalled as equation given in (2), which reads (i𝛾 𝜕𝜇 −𝑚1)𝜓(𝑥).In =0 𝜃 the helicity basis, the fundamental spinor solutions read as ( ) cos 2 (1) ⃗ ⃗ 𝜓 (𝑥) =𝑈 (𝑘) exp (−i𝑘⋅𝑥) , 𝑎+ (𝑘) = ( ), ± 𝜃 𝜑 (18) sin ( ) ei (1) ⃗ 2 𝜙 (𝑥) =𝑉± (𝑘) exp (i𝑘⋅𝑥) . (14) 𝜃 − 𝜑 − sin ( ) e i The algebraic relations that have to be fulfilled by the bispinor 2 𝑈(1)(𝑘)⃗ 𝑉(1)(𝑘)⃗ 𝑎 (𝑘)⃗ = ( ). amplitudes ± and ± read as follows: − 𝜃 cos ( ) (1) ⃗ (1) ⃗ 2 (𝑘−𝑚¡ 1)𝑈± (𝑘) = 0,𝑘+𝑚 (¡ 1)𝑉± (𝑘) = 0. (19) ISRN High Energy Physics 5

2 (1) √ ⃗ 2 One can change the normalization according to The dispersion relation is 𝐸 = 𝑘 +𝑚1. In the helicity basis, the solutions of (19)withatardyonic𝑚1 mass term are 1/2 (𝑘−𝑚¡ )(𝑘+𝑚¡ )=𝑘2 − 𝐸(1) easily written down, using the identity 1 1 U(1) (𝑘)⃗ = ( ) 𝑈(1) (𝑘)⃗ , 𝑚2 =0 𝜎 𝜎 1 . With an appropriate normalization factor and after 𝑚1 some algebraic simplification, the positive-energy solutions (23) (1) 1/2 read as follows: (1) ⃗ 𝐸 (1) ⃗ V𝜎 (𝑘) = ( ) 𝑉𝜎 (𝑘) . 𝑚1 (𝑘+𝑚 )𝑢 (𝑘)⃗ (1) ⃗ ¡ 1 + 𝑈+ (𝑘) = 󵄨 󵄨 2 The Lorentz-invariant normalization is equal to one for the √(𝐸(1) − 󵄨𝑘⃗󵄨) +𝑚2 󵄨 󵄨 1 fundamental positive-energy bispinors and equal to minus one for the fundamental negative-energy bispinors, (1) (20a) 𝐸 +𝑚1 √ 𝑎 (𝑘)⃗ (1) 2𝐸(1) + U (𝑘)⃗ U(1) (𝑘)⃗ = 1, =( ), 𝜎 𝜎 (24) 𝐸(1) −𝑚 (1) √ 1 𝑎 (𝑘)⃗ V (𝑘)⃗ V(1) (𝑘)⃗ = −1. 2𝐸(1) + 𝜎 𝜎

(𝑘+𝑚 )𝑢 (𝑘)⃗ A little algebra is sufficient to reproduce the following known (1) ⃗ ¡ 1 − 𝑈− (𝑘) = sums over bispinors: 󵄨 󵄨 2 √(𝐸(1) − 󵄨𝑘⃗󵄨) +𝑚2 󵄨 󵄨 1 (1) 𝑘+𝑚¡ ∑U(1) (𝑘)⃗ ⊗ U (𝑘)⃗ = 1 , 𝜎 𝜎 2𝑚 (1) (20b) 𝜎 1 √ 𝐸 +𝑚1 ⃗ 𝑎− (𝑘) (25) 2𝐸(1) (1) 𝑘−𝑚¡ =( ). ∑V(1) (𝑘)⃗ ⊗ V (𝑘)⃗ = 1 . 𝜎 𝜎 2𝑚 𝐸(1) −𝑚 𝜎 1 −√ 1 𝑎 (𝑘)⃗ 2𝐸(1) − In accordance with general wisdom of the tardyonic case, these do not involve a helicity-dependent prefactor. The sum The negative-energy eigenstates of the tardyonic equations in rule (25) is of type I (see (15)). thehelicitybasisaregivenas 2.3. Two Tardyonic Mass Terms. Inspired by the discussion in (𝑚 − 𝑘) 𝑣 (𝑘)⃗ (1) ⃗ 1 ¡ + Section 1, we consider an equation with two tardyonic mass 𝑉+ (𝑘) = 󵄨 󵄨 2 terms, which has already been indicated in (2) and reads √(𝐸(1) − 󵄨𝑘⃗󵄨) +𝑚2 𝜇 5 (𝑡) 󵄨 󵄨 1 (i𝛾 𝜕𝜇 −𝑚1 − i𝛾 𝑚2)𝜓 (𝑥) =. 0 For the corresponding bispinors in the fundamental plane-wave solutions, this 𝐸(1) −𝑚 (21a) implies that −√ 1 𝑎 (𝑘)⃗ 2𝐸(1) + =( ), 5 (𝑡) ⃗ (𝑘−𝑚¡ 1 − i𝛾 𝑚2)𝑈± (𝑘) = 0, (26a) 𝐸(1) +𝑚 −√ 1 𝑎 (𝑘)⃗ (1) + 5 (𝑡) ⃗ 2𝐸 (−𝑘−𝑚¡ 1 − i𝛾 𝑚2)𝑉± (𝑘) = 0. (26b) ⃗ (𝑚1 − 𝑘)¡ − 𝑣 (𝑘) 2 𝑉(1) (𝑘)⃗ = (𝑡) √ ⃗ 2 2 − The dispersion relation is 𝐸 = 𝑘 +𝑚1 +𝑚2.The 󵄨 󵄨 2 √(𝐸(1) − 󵄨𝑘⃗󵄨) +𝑚2 fundamental positive-energy bispinors read as follows: 󵄨 󵄨 1 (𝑘+𝑚 − 𝛾5𝑚 )𝑢 (𝑘)⃗ (1) (21b) (𝑡) ¡ 1 i 2 + 𝐸 −𝑚1 ⃗ −√ 𝑎 (𝑘)⃗ 𝑈+ (𝑘) = (1) − 󵄨 󵄨 2 2𝐸 √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 =( ). 󵄨 󵄨 1 2 (1) 𝐸 +𝑚1 √ 𝑎 (𝑘)⃗ (𝑡) 󵄨 󵄨 (1) − 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ 2𝐸 𝑚1 − i𝑚2 +𝐸 − 󵄨𝑘󵄨 + (27a) 󵄨 󵄨 2 √2 √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ( 󵄨 󵄨 1 2 ) These solutions are consistent with those given in3 [ ]andin = ( 󵄨 󵄨 ) , 𝜎=± ( (𝑡) 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ ) Chapter 23 of [58], and the normalizations are ( ) 𝑚1 − i𝑚2 −𝐸 + 󵄨𝑘󵄨 + 󵄨 󵄨 2 √2 (1)+ ⃗ (1) ⃗ (1)+ ⃗ (1) ⃗ √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 𝑈𝜎 (𝑘)𝜎 𝑈 (𝑘) =𝜎 𝑉 (𝑘)𝜎 𝑉 (𝑘) = 1. (22) ( 󵄨 󵄨 1 2 ) 6 ISRN High Energy Physics

(𝑘+𝑚 − 𝛾5𝑚 )𝑢 (𝑘)⃗ (𝑡) ⃗ ¡ 1 i 2 − In the normalization 𝑈− (𝑘) = 󵄨 󵄨 2 √ (𝑡) 󵄨 ⃗󵄨 2 2 (𝐸 − 󵄨𝑘󵄨) +𝑚1 +𝑚2 󵄨 󵄨 1/2 𝐸(𝑡) 󵄨 󵄨 U(𝑡) (𝑘)⃗ = ( ) 𝑈(𝑡) (𝑘)⃗ , 𝑚 + 𝑚 +𝐸(𝑡) − 󵄨𝑘⃗󵄨 𝑎 (𝑘)⃗ 𝜎 𝜎 1 i 2 󵄨 󵄨 − 𝑚1 (27b) 󵄨 󵄨 2 √2 (30) √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 (𝑡) 1/2 ( 󵄨 󵄨 1 2 ) (𝑡) ⃗ 𝐸 (𝑡) ⃗ = ( 󵄨 󵄨 ) . V𝜎 (𝑘) = ( ) 𝑉𝜎 (𝑘) , ( (𝑡) 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ ) 𝑚 −𝑚1 − i𝑚2 +𝐸 − 󵄨𝑘󵄨 − 1 󵄨 󵄨 2 √2 √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ( 󵄨 󵄨 1 2 ) the positive-energy solutions acquire a “positive Lorentz- invariant norm,” whereas the negative-energy solutions have The negative-energy eigenstates of the equation with two “negative Lorentz-invariant norm,” tardyonic mass terms are given as

5 (−𝑘− 𝛾 𝑚 +𝑚 )𝑣 (𝑘)⃗ (𝑡) (𝑡) (𝑡) (𝑡) (𝑡) ¡ i 2 1 + ⃗ ⃗ ⃗ ⃗ ⃗ U𝜎 (𝑘) U𝜎 (𝑘) = 1, V𝜎 (𝑘) V𝜎 (𝑘) = −1. (31) 𝑉+ (𝑘) = 󵄨 󵄨 2 √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 󵄨 󵄨 1 2 󵄨 󵄨 After some algebra, one can derive the following sums over −𝑚 + 𝑚 +𝐸(𝑡) − 󵄨𝑘⃗󵄨 𝑎 (𝑘)⃗ 1 i 2 󵄨 󵄨 + bispinors: (28a) 󵄨 󵄨 2 √2 (√(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ) ( 󵄨 󵄨 1 2 ) = 󵄨 󵄨 , 5 ( −𝑚 + 𝑚 −𝐸(𝑡) + 󵄨𝑘⃗󵄨 𝑎 (𝑘)⃗ ) (𝑡) 𝑘+𝑚¡ − 𝛾 𝑚 1 i 2 󵄨 󵄨 + ∑𝑈(𝑡) (𝑘)⃗ ⊗ 𝑈 (𝑘)⃗ = 1 i 2 , 𝜎 𝜎 2𝑚 󵄨 󵄨 2 √2 𝜎 1 √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 (32) ( 󵄨 󵄨 1 2 ) 5 (𝑡) 𝑘−𝑚¡ + 𝛾 𝑚 ∑𝑉(𝑡) (𝑘)⃗ ⊗ 𝑉 (𝑘)⃗ = 1 i 2 . 𝜎 𝜎 2𝑚 for negative helicity (positive chirality in the massless limit) 𝜎 1 and

(−𝑘− 𝛾5𝑚 +𝑚 )𝑣 (𝑘)⃗ These are easily identified as the positive- and negative- (𝑡) ⃗ ¡ i 2 1 − 𝑉− (𝑘) = energy projectors. The sum rules do not involve helicity- 󵄨 󵄨 2 dependent prefactor and are of type I (see (15)). √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 󵄨 󵄨 1 2 The solutions (27a), (27b), (28a), and (28b)approachthe 󵄨 󵄨 massless solutions if one replaces 𝑚2 →0firstandthenlet −𝑚 − 𝑚 +𝐸(𝑡) − 󵄨𝑘⃗󵄨 𝑎 (𝑘)⃗ 1 i 2 󵄨 󵄨 − 𝑚1 →0.Theyarethususefulforsystemswherethe𝑚1 mass (28b) 𝑚 𝑚 ≫𝑚 󵄨 󵄨 2 √2 is greater than 2.For 2 1,onewouldliketocalculate √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ( 󵄨 󵄨 1 2 ) solutions that approach the massless case for the sequence = ( 󵄨 󵄨 ) , 𝑚 →0 𝑚 →0 ( (𝑡) 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ ) 1 ,then 2 .Thesereadasfollows: 𝑚1 + i𝑚2 +𝐸 − 󵄨𝑘󵄨 − 󵄨 󵄨 2 √2 √(𝐸(𝑡) − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ( 󵄨 󵄨 1 2 ) 󸀠(𝑡) ⃗ (𝑡) ⃗ 󸀠(𝑡) ⃗ (𝑡) ⃗ 𝑈𝜎 (𝑘) = i𝜎𝑈𝜎 (𝑘) ,𝑉𝜎 (𝑘) = i𝜎𝑉𝜎 (𝑘) . (33) for positive helicity (negative chirality in the massless limit). 𝐸(𝑡) →|𝑘|⃗ 𝑚 →0 In the massless limit (first ,then 2 ,and In comparison to the solutions (27a), (27b), (28a), and (28b), 𝑚 →0 then 1 ), we again reproduce the massless solutions they acquire a nontrivial phase factor. (𝑡) ⃗ ⃗ (𝑡) ⃗ ⃗ (𝑡) ⃗ ⃗ 𝑈+ (𝑘) →𝑢 +(𝑘), 𝑈− (𝑘) →𝑢 −(𝑘), 𝑉+ (𝑘) →𝑣 +(𝑘) and (𝑡) ⃗ ⃗ 𝑉− (𝑘) →𝑣 −(𝑘). Of course, in the limit 𝑚1 →0,onealso has to expand the normalization denominators in powers of 3. Generalized Dirac Equations: 𝑚1.For𝑚2 =0the solutions (27a), (27b), (28a), and (28b) Tachyonic Mass Terms reduce to the solutions of the ordinary Dirac equation in (20a), (20b), (21a), and (21b) and can be expanded in 𝑚2 to 3.1. Tachyonic Dirac Equation. The tachyonic Dirac equation ( 𝛾𝜇𝜕 −𝛾5𝑚)𝜓(𝑥) =0 yield corrections to the ordinary Dirac equation for small 𝑚2; is given in (7) and reads i 𝜇 .Thefunda- that is, 𝑚1 ≫𝑚2. The states are normalized with respect to mental bispinors entering the equations fulfill the equations the condition

(𝑡)+ ⃗ (𝑡) ⃗ (𝑡)+ ⃗ (𝑡) ⃗ 5 ⃗ 5 ⃗ 𝑈𝜎 (𝑘)𝜎 𝑈 (𝑘) =𝜎 𝑉 (𝑘)𝜎 𝑉 (𝑘) = 1. (29) (𝑘−𝛾¡ 𝑚)± 𝑈 (𝑘) = 0, (𝑘+𝛾¡ 𝑚)± 𝑉 (𝑘) = 0. (34) ISRN High Energy Physics 7

5 5 2 2 Using (𝑘−𝛾¡ 𝑚)(𝑘−𝛾¡ 𝑚) = 𝑘 +𝑚 andsomealgebra,the The sum rule fulfilled by the fundamental plane-wave spinors prefactors in the fundamental bispinors (for positive energy) is of type II (see (16)). For the positive-energy spinors, we have take a very simple form, 󵄨 󵄨 󵄨𝑘⃗󵄨 +𝑚 󵄨 󵄨 5 √ 𝑎 (𝑘)⃗ 5 𝑘−𝛾¡ 𝑚 5 󵄨 ⃗󵄨 + ∑ (−𝜎) U (𝑘⃗) ⊗ U (𝑘⃗) 𝛾 = , (𝛾 𝑚−𝑘)¡ 𝑢 (𝑘)⃗ 2 󵄨𝑘󵄨 𝜎 𝜎 (40a) ⃗ + ( 󵄨 󵄨 ) 𝜎 2𝑚 𝑈+ (𝑘) = = , 󵄨 󵄨 2 󵄨 ⃗󵄨 √(𝐸 − 󵄨𝑘⃗󵄨) +𝑚2 󵄨𝑘󵄨 −𝑚 󵄨 󵄨 √ 󵄨 󵄨 𝑎 (𝑘)⃗ 󵄨 ⃗󵄨 + ( 2 󵄨𝑘󵄨 ) 󵄨 󵄨 where for the negative-energy spinors, the sum rule reads (35a) 󵄨 󵄨 󵄨𝑘⃗󵄨 −𝑚 󵄨 󵄨 ⃗ 5 √ 󵄨 󵄨 𝑎− (𝑘) 5 𝑘+𝛾¡ 𝑚 5 ⃗ 󵄨 ⃗󵄨 ∑ (−𝜎) V (𝑘⃗) ⊗ V (𝑘⃗) 𝛾 = . (𝑘−𝛾¡ 𝑚)− 𝑢 (𝑘) 2 󵄨𝑘󵄨 𝜎 𝜎 (40b) ⃗ ( 󵄨 󵄨 ) 𝜎 2𝑚 𝑈− (𝑘) = = . 󵄨 󵄨 2 󵄨 󵄨 󵄨 ⃗󵄨 2 󵄨𝑘⃗󵄨 +𝑚 √(𝐸 − 󵄨𝑘󵄨) +𝑚 󵄨 󵄨 ⃗ 󵄨 󵄨 −√ 󵄨 󵄨 𝑎− (𝑘) 󵄨 ⃗󵄨 ( 2 󵄨𝑘󵄨 ) The expressions on the right-hand sides are the positive- and (35b) negative-energy projectors. For negative energy, the solutions read as follows: 󵄨 󵄨 󵄨𝑘⃗󵄨 −𝑚 3.2. Two Tachyonic Mass Terms. We study the equation 󵄨 󵄨 ⃗ 𝜇 5 −√ 󵄨 󵄨 𝑎+ (𝑘) (i𝛾 𝜕𝜇 − i𝑚1 −𝛾𝑚2)𝜓(𝑥),asgivenin( =0 9). The funda- 5 ⃗ 2 󵄨𝑘⃗󵄨 (𝛾 𝑚+𝑘)¡ + 𝑣 (𝑘) 󵄨 󵄨 󸀠 ⃗ 󸀠 ⃗ ⃗ ( ) mental spinors, which we denote 𝑈±(𝑘) and 𝑉±(𝑘),fulfillthe 𝑉+ (𝑘) = = , 󵄨 󵄨 2 󵄨 󵄨 󵄨 ⃗󵄨 2 󵄨𝑘⃗󵄨 +𝑚 following equations: √(𝐸 − 󵄨𝑘󵄨) +𝑚 󵄨 󵄨 ⃗ 󵄨 󵄨 −√ 󵄨 󵄨 𝑎+ (𝑘) 󵄨 ⃗󵄨 ( 2 󵄨𝑘󵄨 ) (36a) 5 󸀠 ⃗ 󵄨 󵄨 (𝑘−¡ i𝑚1 −𝛾 𝑚2) 𝑈 (𝑘) =0, (41a) 󵄨𝑘⃗󵄨 +𝑚 ± 󵄨 󵄨 ⃗ −√ 𝑎 (𝑘) 5 󸀠 5 󵄨 ⃗󵄨 + ⃗ (−𝑘−𝛾 𝑚) 𝑣 (𝑘)⃗ 2 󵄨𝑘󵄨 (𝑘+¡ i𝑚1 +𝛾 𝑚2)𝑉± (𝑘) = 0. (41b) ⃗ ¡ − ( 󵄨 󵄨 ) 𝑉− (𝑘) = = . 󵄨 󵄨 2 󵄨 󵄨 󵄨 ⃗󵄨 2 󵄨𝑘⃗󵄨 −𝑚 √(𝐸 − 󵄨𝑘󵄨) +𝑚 󵄨 󵄨 ⃗ 󵄨 󵄨 √ 󵄨 󵄨 𝑎+ (𝑘) 󵄨 ⃗󵄨 ( 2 󵄨𝑘󵄨 ) The positive-energy solutions are obtained using the identity (36b) 5 5 2 2 2 (𝑘−¡ i𝑚1 −𝛾𝑚2)(𝑘+¡ i𝑚1 −𝛾𝑚2)=𝑘 +𝑚1 +𝑚2.With 𝜎=± 2 The normalization condition is ( ) 󸀠 √ ⃗ 2 2 𝐸 = 𝑘 −𝑚1 −𝑚2, they read as follows: + ⃗ ⃗ + ⃗ ⃗ 𝑈𝜎 (𝑘)𝜎 𝑈 (𝑘) =𝜎 𝑉 (𝑘)𝜎 𝑉 (𝑘) = 1. (37)

One can change to Lorentz-invariant normalization by a (|𝑘|/𝑚)⃗ 1/2 󵄨 󵄨 multiplication with , 󸀠 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ i𝑚1 +𝑚2 −𝐸 + 󵄨𝑘󵄨 + 󵄨 ⃗󵄨 1/2 󵄨𝑘󵄨 󵄨 󵄨 2 √2 U (𝑘)⃗ = (󵄨 󵄨) 𝑈 (𝑘)⃗ , (√(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ) 𝜎 𝜎 󸀠 ⃗ 󵄨 󵄨 1 2 𝑚 𝑈 (𝑘) = ( 󵄨 󵄨 ) , (42a) + ( 𝑚 +𝑚 +𝐸󸀠 − 󵄨𝑘⃗󵄨 𝑎 (𝑘)⃗ ) (38) i 1 2 󵄨 󵄨 + 󵄨 󵄨 1/2 󵄨𝑘⃗󵄨 󵄨 󵄨 2 √2 󵄨 󵄨 √(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 V (𝑘)⃗ = ( ) 𝑉 (𝑘)⃗ . 󵄨 󵄨 1 2 𝜎 𝑚 𝜎 ( ) 󸀠 󵄨 ⃗󵄨 ⃗ i𝑚1 +𝑚2 +𝐸 − 󵄨𝑘󵄨 𝑎− (𝑘) The “calligraphic” spinors fulfill the following helicity-depen- 󵄨 󵄨 dent normalizations: 󵄨 󵄨 2 √2 (√(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ) 󸀠 ⃗ ( 󵄨 󵄨 1 2 ) U (𝑘)⃗ U (𝑘)⃗ = 𝜎, V (𝑘)⃗ V (𝑘)⃗ = −𝜎, 𝑈− (𝑘) = 󵄨 󵄨 . (42b) 𝜎 𝜎 𝜎 𝜎 (39) ( 󸀠 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ ) −i𝑚1 −𝑚2 +𝐸 − 󵄨𝑘󵄨 − where we observe that 𝜎 is a good quantum number because 󵄨 󵄨 2 √2 √(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 the helicity operator commutes with the Hamiltonian (8). ( 󵄨 󵄨 1 2 ) 8 ISRN High Energy Physics

The negative-energy solutions for the tachyonic equation For definiteness, we consider the solution of the tachyonic with two mass terms are given as Dirac equation (Section 3.1). The generalization to other 󵄨 󵄨 generalized Dirac equations is straightforward. We start from 󸀠 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ i𝑚1 −𝑚2 −𝐸 + 󵄨𝑘󵄨 + the field operator9 [ ] √ 󵄨 󵄨 2 2 3 (√(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ) d 𝑘 𝑚 󸀠 󵄨 󵄨 1 2 𝜓 (𝑥) = ∫ 𝑉 (𝑘)⃗ = ( 󵄨 󵄨 ) , 3 + ( 󸀠 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ ) (43a) (2𝜋) 𝐸 i𝑚1 −𝑚2 −𝐸 − 󵄨𝑘󵄨 + − 𝑘⋅𝑥 + 𝑘⋅𝑥 󵄨 󵄨 2 √2 × ∑ {𝑏 (𝑘) U (𝑘)⃗ i +𝑑 (𝑘) V (𝑘)⃗ i }, √(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 𝜎 𝜎 e 𝜎 𝜎 e ( 󵄨 󵄨 1 2 ) 𝜎=± 󵄨 󵄨 󸀠 󵄨 ⃗󵄨 ⃗ √ 2 −i𝑚1 −𝑚2 +𝐸 − 󵄨𝑘󵄨 𝑎− (𝑘) ⃗ ⃗ 2 󵄨 󵄨 𝑘=(𝐸,𝑘), 𝐸=𝐸𝑘⃗ = 𝑘 −𝑚 − i𝜖, 󵄨 󵄨 2 √2 (46) (√(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 ) 󸀠 ⃗ ( 󵄨 󵄨 1 2 ) 𝑉− (𝑘) = ( 󸀠 󵄨 󵄨 ) . (43b) + 󵄨 ⃗󵄨 𝑎 (𝑘)⃗ where 𝑏𝜎 annihilates particles and 𝑑 creates antiparticles. i𝑚1 +𝑚2 +𝐸 − 󵄨𝑘󵄨 − 𝜎 The following anticommutators vanish: 󵄨 󵄨 2 √2 √(𝐸󸀠 − 󵄨𝑘⃗󵄨) +𝑚2 +𝑚2 󵄨 󵄨 1 2 󸀠 + + 󸀠 ( ) {𝑏𝜎 (𝑘) ,𝑏𝜌 (𝑘 )}={𝑏𝜎 (𝑘) ,𝑏𝜌 (𝑘 )}=0, (47a) 󸀠+ 󸀠 󸀠+ 󸀠 ⃗ ⃗ ⃗ ⃗ 󸀠 + + 󸀠 The normalization condition is 𝑈𝜎 (𝑘)𝑈𝜎(𝑘) =𝜎 𝑉 (𝑘)𝑉𝜎(𝑘) = {𝑑𝜎 (𝑘) ,𝑑𝜌 (𝑘 )}={𝑑𝜎 (𝑘) ,𝑑𝜌 (𝑘 )}=0. (47b) 1. We use a definition of the “calligraphic” spinors analogous to (38), We assume the following general form for the nonvanishing 󵄨 󵄨 1/2 anticommutators, 󵄨𝑘⃗󵄨 󸀠 󵄨 󵄨 󸀠 U (𝑘)⃗ = ( ) 𝑈 (𝑘)⃗ , 𝐸 󸀠 𝜎 𝑚 𝜎 {𝑏 (𝑘) ,𝑏+ (𝑘󸀠)} = 𝑓 (𝜎, 𝑘)⃗ (2𝜋)3 𝛿3 (𝑘−⃗ 𝑘⃗ )𝛿 , 2 𝜎 𝜌 𝑚 𝜎𝜌 (44) 󵄨 󵄨 1/2 (48a) 󵄨𝑘⃗󵄨 V󸀠 (𝑘)⃗ = ( 󵄨 󵄨 ) 𝑉󸀠 (𝑘)⃗ . 𝐸 󸀠 𝜎 𝑚 𝜎 {𝑑 (𝑘) ,𝑑+ (𝑘󸀠)}=𝑔(𝜎,𝑘)⃗ (2𝜋)3 𝛿3 (𝑘−⃗ 𝑘⃗ )𝛿 , 2 𝜎 𝜌 𝑚 𝜎𝜌 (48b) In analogy to (40a)and(40b), a sum rule of type II (see (16)) is fulfilled by the fundamental plane-wave spinors, with arbitrary 𝑓 and 𝑔 functions of the quantum numbers 5 𝜎 𝑘⃗ 𝑓 𝑔 󸀠 𝑘+¡ 𝑚 −𝛾 𝑚 and . One might argue that since and must be ∑ (−𝜎) U󸀠 (𝑘)⃗ ⊗ U (𝑘)⃗ 𝛾5 = i 1 2 , 𝜎 𝜎 2𝑚 dimensionless, they can depend only on the dimensionless 𝜎 2 𝜎 𝑘/𝑚⃗ (45) arguments and , but that is a detail of the discussion 5 󸀠 𝑘−¡ 𝑚 +𝛾 𝑚 which we do not pursue any further. Our only assumption ∑ (−𝜎) V󸀠 (𝑘)⃗ ⊗ V (𝑘)⃗ 𝛾5 = i 1 2 . 𝜎 𝜎 2𝑚 concerns the fact that the field anticommutators should be 𝜎 2 diagonal in the helicity and wave vector quantum numbers, 𝛿 We thus obtain that the desired projectors onto positive- and leading to the corresponding Kronecker and Dirac- ’s. Γ negative-energy solutions for the Dirac equation with two We assume that the spin-matrix either constitutes a 𝜇 tachyonic mass terms (9). The generalized equation (i𝛾 𝜕𝜇 − Lorentz scalar or a pseudoscalar quantity, which is a scalar 5 under the proper orthochronous Lorentz group. The time- i𝑚1 −𝛾𝑚2)𝜓(𝑥) =0 is fully compatible with the Clifford- ordered product of field operators reads as algebra-based approach recently described in [59]. 󵄨 󵄨 ⟨0 󵄨𝑇𝜓 (𝑥) 𝜓(𝑦)Γ󵄨 0⟩ 4. Theorems for Generalized Dirac Fields 3𝑘 𝑚 = ∫ d 4.1. Spinor Sums and Time-Ordered Propagator. Our central (2𝜋)3 𝐸 postulate regarding the quantized fermionic theory is that the time-ordered vacuum expectation value of the field operators ×{Θ(𝑥0 −𝑦0) −i𝑘⋅(𝑥−𝑦) should yield the time-ordered (Feynman) propagator, which, e in the momentum representation, is equal to the inverse of the (49) 𝛾0 ⃗ ⃗ ⃗ Hamiltonian (upon multiplication with ). This postulate × ∑𝑓(𝜎,𝑘) U𝜎 (𝑘) ⊗ U𝜎 (𝑘) Γ implies that under rather general assumptions regarding the 𝜎=± mathematical form of the elementary field anticommutators, 0 0 i𝑘⋅(𝑥−𝑦) sum rules have to be fulfilled by the tensor sums over the −Θ(𝑦 −𝑥 ) e fundamental spinor solutions. It is perhaps not surprising that these sum rules are precisely of the form investigated in ⃗ ⃗ ⃗ ×∑𝑔(𝜎,𝑘) V𝜎 (𝑘) ⊗ V𝜎 (𝑘) Γ} . Sections 2 and 3 of this paper. 𝜎=± ISRN High Energy Physics 9

This equation contains the same coefficient functions 𝑓 and postulatesgivenin(50),butitistheonlystructurallysimple 𝑔 that enter into (48a)and(48b). In order to proceed with choice that we have found. the derivation of the propagator, we must postulate that the For the egregiously simple choice (54), let us study the following sum rules hold: transition to the massless limit (16) in some further detail. Indeed, in the limit 𝑚→0, the denominator of the spin 𝑘−𝛾¡ 5𝑚 ∑𝑓(𝜎,𝑘)⃗ U (𝑘)⃗ ⊗ U (𝑘)⃗ Γ = , sums in (40a)and(40b) vanishes, and a finite limit is obtained 𝜎 𝜎 2𝑚 𝜎 2𝑚 after multiplication with , (50) 𝑘+𝛾5𝑚 ∑2𝑚 (−𝜎) U (𝑘)⃗ ⊗ U (𝑘)⃗ 𝛾5 = 𝑘, ⃗ ⃗ ⃗ ¡ lim 𝜎 𝜎 ¡ (55a) ∑𝑔(𝜎,𝑘) V𝜎 (𝑘) ⊗ V𝜎 (𝑘) Γ = . 𝑚→0 𝜎 𝜎 2𝑚 ∑2𝑚 (−𝜎) V (𝑘)⃗ ⊗ V (𝑘)⃗ 𝛾5 = 𝑘. lim 𝜎 𝜎 ¡ (55b) The sum rule (50) is crucial for the further steps in the 𝑚→0 𝜎 derivation of the time-ordered propagator. Introducing a suitable complex integral representation for the step function, In order to compare the normalizations of the fundamental one obtains from (49), using (50), after a few steps which we spinors in the massless limit, we calculate the following do not discuss in further detail (see also equations (3.169) and quantities: (3.170) of [57]) 𝐸 󵄨 󵄨 U (𝑘)⃗ 𝛾5𝛾0U (𝑘)⃗ = −𝜎 , ⟨0 󵄨𝑇𝜓 (𝑥) 𝜓(𝑦)Γ󵄨 0⟩ 𝜎 𝜎 𝑚 󵄨 󵄨 (56) 3 𝑢 (𝑘)⃗ 𝛾5𝛾0𝑢 (𝑘)⃗ = −𝜎, d 𝑘 𝑚 d𝑘0 − 𝑘 ⋅(𝑥0−𝑦0)+ 𝑘⋅(⃗ 𝑥−⃗ 𝑦)⃗ 𝜎 𝜎 = i ∫ ∫ 𝑒 i 0 i (2𝜋)3 𝐸 2𝜋 ⃗ 5 0 ⃗ 𝐸 V𝜎 (𝑘) 𝛾 𝛾 V𝜎 (𝑘) = −𝜎 , 𝛾0𝑘 − 𝛾⋅⃗ 𝑘−𝛾⃗ 5𝑚 𝑚 × 0 (57) 2𝑚 (𝑘 −𝐸+ 𝜖) ⃗ 5 0 ⃗ 0 i (51) 𝑣𝜎 (𝑘) 𝛾 𝛾 𝑣𝜎 (𝑘) = −𝜎, 3 d 𝑘 𝑚 d𝑘0 − 𝑘 ⋅(𝑥0−𝑦0)+ 𝑘⋅(⃗ 𝑥−⃗ 𝑦)⃗ 𝑢 𝑣 + i ∫ ∫ e i 0 i . where the fundamental spinors 𝜎 and 𝜎 of the massless (2𝜋)3 𝐸 2𝜋 equation have been given in Section 2.1.Observingthat𝐸= ⃗ ⃗ |𝑘| in the massless limit, the identifications √𝑚U𝜎(𝑘) → −𝛾0𝑘 + 𝛾⋅⃗ 𝑘+𝛾⃗ 5𝑚 × 0 . √ ⃗ ⃗ ⃗ √ ⃗ ⃗ |𝑘|𝑢𝜎(𝑘) and √𝑚V𝜎(𝑘) → |𝑘|𝑣𝜎(𝑘), as implied by (56) 2𝑚 0(𝑘 +𝐸−i𝜖) and (57), show that the identities (55a)and(55b)precisely 3 reduce to the sum rule (16) in the massless limit. The convention is that in any integrals ∫ d 𝑘,thecomponent 2 √ ⃗ 2 𝑘0 is set equal to 𝐸= 𝑘 −𝑚 in the integrand when it 4.2. Generalized Field Anticommutators for Tardyonic and occurs in scalar products of the form 𝑘⋅(𝑥−𝑦),andsoforth, Tachyonic Fields. For definiteness, we have considered the 4 but if the integral is over the full d 𝑘, then the integration case of the tachyonic Dirac field in the above derivation. The interval is the full 𝑘0 ∈(−∞,∞).Withtheconventionwhich decisive observation is that the tachyonic choice, is adopted in many quantum field theoretical textbooks, ⃗ ⃗ 5 including [57, 60], we finally obtain the result tachyonic choice: 𝑓(𝜎,𝑘) = 𝑔 (𝜎, 𝑘) = −𝜎, Γ =𝛾 , (58) 4𝑘 𝑘−𝛾5𝑚 󵄨 󵄨 d −i𝑘⋅(𝑥−𝑦) ¡ ⟨0 󵄨𝑇𝜓 (𝑥) 𝜓(𝑦) Γ󵄨 0⟩ = i ∫ e . (2𝜋)4 𝑘2 +𝑚2 + i𝜖 is consistent with both massive tachyonic fields discussed in (52) Sections 3.1 and 3.2, whereas the tardyonic choice, ⃗ ⃗ The tachyonic propagator 𝑆𝑇 is identified, under the integral tardyonic choice: 𝑓(𝜎,𝑘) = 𝑔 (𝜎, 𝑘) = 1, Γ = 14×4, sign, as (59) 5 1 𝑘−𝛾¡ 𝑚 yields the time-ordered propagator for both massive tardy- 𝑆𝑇 (𝑘) = = . (53) 𝑘−𝛾¡ 5 (𝑚+i𝜖) 𝑘2 +𝑚2 + i𝜖 onic fields discussed in Sections 2.2 and 2.3.Thenonvanish- ing anticommutators for tardyons take the simple form (cf. The sum rules (40a)and(40b) imply that the derivation is (48a)and(48b)), validforthechoice tardyonic anticommutators: 𝑓(𝜎,𝑘)⃗ = 𝑔 (𝜎, 𝑘)⃗ = −𝜎, Γ =𝛾5, (54) 𝐸 󸀠 {𝑏 (𝑘) ,𝑏+ (𝑘󸀠)} = (2𝜋)3 𝛿3 (𝑘−⃗ 𝑘⃗ )𝛿 , 𝜎 𝜌 𝑚 𝜎𝜌 (60) in which case the relations given in (50)arefulfilled.Note that this observation does not imply that the choice (54) 𝐸 󸀠 {𝑑 (𝑘) ,𝑑+ (𝑘󸀠)} = (2𝜋)3 𝛿3 (𝑘−⃗ 𝑘⃗ )𝛿 . necessarily is the only one for which we are able to fulfill the 𝜎 𝜌 𝑚 𝜎𝜌 10 ISRN High Energy Physics

Again, compared with (48a)and(48b), the nonvanishing 4.3. Tachyonic Gordon Identities. It is useful to illustrate anticommutators for tachyons take the simple form, the derivation outlined above by exploring its connection to tachyonic Gordon identities. For definiteness, we again tachyonic anticommutators: concentrate on the tachyonic Dirac equation discussed in 𝐸 󸀠 Section 3.1. The matrix element of the vector current finds {𝑏 (𝑘) ,𝑏+ (𝑘󸀠)} = (−𝜎)(2𝜋)3 𝛿3 (𝑘−⃗ 𝑘⃗ )𝛿 , the following Gordon decomposition for positive-energy 𝜎 𝜌 𝑚 𝜎𝜌 (61) spinors: 𝐸 󸀠 + 󸀠 3 3 ⃗ ⃗ 󸀠 󸀠 {𝑑𝜎 (𝑘) ,𝑑𝜌 (𝑘 )} = (−𝜎)(2𝜋) 𝛿 (𝑘−𝑘 )𝛿𝜎𝜌 . ⃗ 𝜇 ⃗ 𝑚 U± (𝑘 )𝛾 U± (𝑘 )

With these universal choices, the theory of the tardyonic and 1 󸀠 = U (𝑘⃗ )𝛾5 [(𝑘󸀠𝜇 −𝑘𝜇)+ 𝜎𝜇𝜈 (𝑘󸀠 +𝑘 )] U (𝑘)⃗ . tachyonic spin-1/2 fields can be unified. The time-ordered 2𝑚 ± i 𝜈 𝜈 ± propagator is given as (66) 󵄨 󵄨 ⟨0 󵄨𝑇𝜓 (𝑥) 𝜓(𝑦)Γ󵄨 0⟩ = i𝑆(𝑥−𝑦), For negative-energy solutions, the identity reads as 4𝑘 (62) 󸀠 d −i𝑘⋅(𝑥−𝑦) ⃗ 𝜇 ⃗ 𝑆(𝑥−𝑦)=∫ e 𝑆 (𝑘) . V± (𝑘 )𝛾 V± (𝑘) (2𝜋)4 1 󸀠 =− V (𝑘⃗ )𝛾5 [(𝑘󸀠𝜇 −𝑘𝜇)+ 𝜎𝜇𝜈 (𝑘󸀠 +𝑘 )] V (𝑘)⃗ . We use the sum rules for the tensor sums over fundamental 2𝑚 ± i 𝜈 𝜈 ± spinors given in (25) (for the tardyonic Dirac field), (32) (67) (for the tardyonic Dirac field with two mass terms),40a ( ) 󸀠 and (40b)(forthetachyonicDiracfield),and(45)(forthe For 𝑘 =𝑘,onehas Dirac field of imaginary mass). Going through the exact same ⃗ 𝜇 ⃗ i ⃗ 5 𝜇𝜈 ⃗ derivation as outlined above in between (49)and(53), we U (𝑘) 𝛾 U (𝑘) = U (𝑘) 𝛾 𝜎 𝑘 U (𝑘) , (68a) obtain the following results for the time-ordered propagators ± ± 𝑚 ± 𝜈 ± of tardyonic and tachyonic fields. For the tardyonic Dirac field V (𝑘)⃗ 𝛾𝜇V (𝑘)⃗ = − i V (𝑘)⃗ 𝛾5𝜎𝜇𝜈𝑘 V (𝑘)⃗ . (Section 2.2), one has ± ± 𝑚 ± 𝜈 ± (68b) 1 𝑘+𝑚¡ 𝑆(1) (𝑘) = = 1 . The matrix element of the axial current reads 2 2 (63) 𝑘−𝑚¡ 1 + i𝜖 𝑘 −𝑚1 + i𝜖 ⃗󸀠 5 𝜇 ⃗ U± (𝑘 )𝛾 𝛾 U± (𝑘) For the tardyonic field with two mass terms (Section 2.3), the Feynman propagator is easily found as 1 ⃗󸀠 󸀠𝜇 𝜇 𝜇𝜈 󸀠 ⃗ =− U± (𝑘 )[(𝑘 +𝑘 )+i𝜎 (𝑘𝜈 −𝑘𝜈)] U± (𝑘) , 1 2𝑚 𝑆(𝑡) (𝑘) = (69a) 5 𝑘−𝑚¡ 1 + i𝜖−i𝛾 (𝑚2 −𝑖𝜂) 5 (64) whereas for negative-energy solutions 𝑘+𝑚¡ 1 − i𝛾 𝑚2 = , 󸀠 2 2 2 ⃗ 5 𝜇 ⃗ 𝑘¡ −𝑚1 −𝑚2 + i𝜖 V± (𝑘 )𝛾 𝛾 V± (𝑘) where 𝜖 and 𝜂 are infinitesimal imaginary parts. Both tardy- 1 ⃗󸀠 󸀠𝜇 𝜇 𝜇𝜈 󸀠 ⃗ = V± (𝑘 )[(𝑘 +𝑘 )+i𝜎 (𝑘 −𝑘𝜈)] V± (𝑘) . onic mass terms acquire an infinitesimal negative imaginary 2𝑚 𝜈 part,andtheprefactor𝑚/𝐸 from (46)inthefieldoperator (69b) 𝑚 /𝐸(𝑡) needstobereplacedby 1 , where the tardyonic energy 󸀠 2 𝑘 =𝑘 (𝑡) √ ⃗ 2 2 For , this simplifies to is 𝐸 = 𝑘 +𝑚1 +𝑚2. For the tachyonic Dirac field 1 (Section 3.1 in [9]), one has the result given in (53). Finally, U (𝑘⃗) 𝛾5𝛾𝜇U (𝑘⃗) =− U (𝑘⃗) 𝑘𝜇U (𝑘⃗) , fortheDiracfieldwithtwotachyonicmassterms,wehave ± ± 𝑚 ± ± (70a)

5 1 𝑘+¡ 𝑚 −𝛾 𝑚 ⃗ 5 𝜇 ⃗ ⃗ 𝜇 ⃗ 𝑆󸀠 (𝑘) = i 1 2 . V± (𝑘) 𝛾 𝛾 V± (𝑘) = V± (𝑘) 𝑘 V± (𝑘) . (70b) 2 2 2 (65) 𝑚 𝑘 +𝑚1 +𝑚2 + i𝜖 The results (69a), (69b), (70a), and (70b) for the tachyonic In the latter case, the prefactor 𝑚/𝐸 from (46)inthefield 󸀠 axial vector currenthaveasimilarstructureastheGordon operator needs to be replaced by 𝑚1/𝐸 ,wherethetachyonic 2 decomposition for the tardyonic vector current obtained with 󸀠 √ ⃗ 2 2 (2.54) energy is 𝐸 = 𝑘 −𝑚1 −𝑚2. For both tachyonic fields the ordinary Dirac equation (see equation of [57]). The discussed here, the mass acquires an infinitesimal positive role of the Dirac adjoint for the tardyonic case is taken over by ⃗ 5 imaginary part, as manifest in the results given in (53)and the “chiral adjoint” U±(𝑘)𝛾 for the tachyonic particle. Here, (65). the designation “chiral adjoint” is inspired by the fact that ISRN High Energy Physics 11

⃗ 5 ⃗ U±(𝑘)𝛾 U±(𝑘) transforms as a pseudoscalar under Lorentz For clarification, the corresponding Gupta-Bleuler con- transformations. dition should be indicated explicitly. In full analogy to the The structure of (68a), (68b), (70a), and (70b)issome- instructive discussion of the Gupta-Bleuler mechanism for what peculiar with regard to parity. In (68a), an apparent thephotonfield,asgiveninfullclarityinChapter(9b)of vector current on the left-hand side appears to transform into Schweber’s textbook [61], we select the positive- and negative- an axial current on the right-hand side, whereas in (70a), an frequency component of the (𝜎=1)-component of the apparent axial vector on the left-hand side of the equation neutrino field operator, becomes what appears to be a vector on the right-hand side. 3𝑘 𝑚 The reason lies in the more complicated behavior of the 𝜓(+) (𝑥) =∫ d 𝑏 (𝑘) U (𝑘⃗) −i𝑘⋅𝑥, 𝜎=1 3 𝜎=1 𝜎=1 e (73a) tachyonic Dirac equation under parity as investigated in [10]. (2𝜋) 𝐸 Namely, the tachyonic Dirac equation (7)containsaterm 3𝑘 𝑚 which transforms as a scalar under parity, (−) d + ⃗ i𝑘⋅𝑥 𝜓 (𝑥) = ∫ 𝑑 (𝑘) V𝜎=1 (𝑘) e , (73b) 𝜎=1 (2𝜋)3 𝐸 𝜎=1 P 𝛾𝜈𝜕 󳨀→𝛾 0 ( 𝛾0𝜕 + 𝛾𝑖 (−𝜕 )) 𝛾0 = 𝛾𝜈𝜕 , (71a) i 𝜇 i 0 i 𝑖 i 𝜇 and postulate that it annihilates any physical Fock state |Ψ⟩ of the tachyonic field, as well as a term which transforms as a pseudoscalar, ⟨Ψ| 𝜓(−) (𝑥) =𝜓(+) (𝑥) |Ψ⟩ =0. P 𝜎=1 𝜎=1 (74) 𝛾5𝑚 󳨀→𝛾 0 (𝛾5𝑚) 𝛾0 =−𝛾5𝑚. (71b) These relations automatically imply that the Gupta-Bleuler The mass term in the tachyonic Dirac equation is pseu- condition also is realized in terms of the expectation value doscalar and changes sign under parity. Indeed, in [10], the ⟨Ψ| 𝜓 (𝑥) |Ψ⟩ = ⟨Ψ| 𝜓(−) (𝑥) +𝜓(+) (𝑥) |Ψ⟩ =0, tachyonic Dirac equation has been shown to be separately 𝜎=1 𝜎=1 𝜎=1 CP invariant, and T invariant, but not P invariant, due to (75) thechangeinthemassterm. but the condition (74) is stronger. We recall that the Gupta- 𝜇 In order to put this observation into perspective, we Bleuler condition on the photon field reads ⟨Ψ𝛾|𝜕 𝐴𝜇|Ψ𝛾⟩= recall that the entries of the electromagnetic field strength 0,where|Ψ𝛾⟩ isaFockstateofthephotonfield.As tensor are composed of axial vector components (magnetic stressed in Schweber’s book [61]onpage246,thecondition 𝐵⃗ 𝐸⃗ 𝜇 field), as well as vector components (electric field). ⟨Ψ𝛾|𝜕𝜇𝐴 |Ψ𝛾⟩=0is not sufficient for the suppression of The transformation properties of the electromagnetic field the longitudinal and scalar photons, but one must postulate strength tensor under the proper orthochronous Lorentz 𝜇 (+) (+) that 𝜕 𝐴𝜇 |Ψ𝛾⟩=0,where𝐴𝜇 is the positive-frequency group are nevertheless well defined. component of the photon field operator. Because the tachy- The transformation71b ( ) can be interpreted as a trans- onic fermion, unlike the photon, is not equal to its own formation 𝑚→−𝑚under parity. Thus, if we interpret the 𝑚 antiparticle,weneedtwoconditions,givenin(74). mass as a pseudoscalar quantity, then the right-hand sides A crucial question now concerns the possibility of revers- of (68a)and(70a) transform as a vector and an axial vector, ing the helicity-dependence, that is, the question of whether respectively. It is the parity noninvariance of the mass term or not a different choice for the helicity-dependent factors in the tachyonic Dirac equation which leads to the somewhat in (48a)and(48b) exists that would imply negative norm peculiar structure of (68a)and(70a). for left-handed particles and right-handed antiparticles. A related question is whether other tachyonic Dirac Hamilto- ⃗ ⃗ 4.4. Helicity-Dependence and Gupta-Bleuler Condition. The nians exist for which the choice 𝑓(𝜎, 𝑘) = 𝑔(𝜎, 𝑘) = +𝜎 anticommutator relations for tardyons given in (60) imply instead of (−𝜎) would fulfill our general postulate, namely, the that both left-handed and right-handed helicity states, for sum rule (50). For reasons outlined in the following, we can both particles and antiparticles, have positive norm. How- ascertainthatthisisnotthecase;tachyonicspin-1/2particles ever, the anticommutator relations for tachyons given in should always be left handed. (61) imply that right-handed particle as well as left-handed The arguments supporting this conclusion are as follows. antiparticle states acquire negative norm. This is shown in First, if we assume that the tachyonic field fulfills a sum equations (31) and (32) of [9]. Indeed, for one-particle states |1 ⟩=𝑏+(𝑘)|0⟩ rule of type II which for massless fields is given in16 ( ), 𝑘,𝜎 𝜎 , then it is impossible to replace (−𝜎) by (+𝜎) in the sum 󵄨 󵄨 󵄨 ⟨1 󵄨1 ⟩= ⟨0󵄨𝑏 (𝑘) 𝑏+ (𝑘)󵄨 0⟩ rule because of the necessity to preserve a smooth massless 𝑘,𝜎 󵄨 𝑘,𝜎 󵄨 𝜎 𝜎 󵄨 limit. The considerations in the text following (16) imply 󵄨 + 󵄨 thatifoneweretoreplace(−𝜎) by (+𝜎) in the massless =⟨0󵄨{𝑏𝜎 (𝑘) ,𝑏𝜎 (𝑘)}󵄨 0⟩ (72) case,thenonewouldviolatethesumrule(16). The second 𝐸 argumentisobtainedbyexplicitcalculation.Wehavechecked = (−𝜎) 𝑉 , 𝑚 that if one replaces 𝑚→−𝑚in the tachyonic and imaginary-mass Dirac equations (7)and(9), then the sum 3 3 where 𝑉=(2𝜋)𝛿 (0)⃗ is the normalization volume in rules fulfilled by the corresponding fundamental spinors still coordinate space. The Fock-space norm ⟨1𝑘,𝜎|1𝑘,𝜎⟩ is negative contain the characteristic factor (−𝜎). For the imaginary- for 𝜎=1. mass Dirac equation, this result is obtained in [25]. Intuitively, 12 ISRN High Energy Physics we can understand this result as follows: the mass 𝑚 in the 5. Physical Interpretation: From Tachyonic denominators of the right-hand sides of (40a), (40b), and (45) Neutrinos to Cosmology is obtained as the modulus √𝑚2 =|𝑚|and does not change if we replace 𝑚→−𝑚in the superluminal Dirac equation. 5.1. Arguments for and against Tachyonic Neutrinos. In the The mass in the numerator of the right-hand sides of (40a), absence of conclusive experimental evidence, the hypothesis (40b), and (45) changes sign, but this is consistent with the of tachyonic neutrinos has been controversially discussed in obvious change in the functional form of the positive-energy the literature. Three main arguments62 [ ]havebeenbrought and negative-energy projectors as we change the sign of the forward against tachyonic neutrinos. (1) They would require mass term. Again, this consideration supports the conclusion us to give up the notion of a Lorentz-invariant vacuum state, that we cannot invert the helicity-dependence by choosing a andeventhevacuumwouldbecomeunstableinthepresence different Hamiltonian; the factor (−𝜎) persists. of tachyonic fields. (2) Given the tachyonic dispersion rela- The third argument comes from the tachyonic Gordon 2 𝐸=√𝑘⃗ −𝑚2 |𝑘|⃗ < 𝑚 identities discussed in Section 4.3. We use the Gordon iden- tion ,theroleofstateswith needs tity (70a)and(70b) and the normalization (39)tocalculate to be clarified. (3) The physical (probability!?) interpretation 0 the bispinor trace (with a 𝛾 multiplied from the right) of the of the conserved Noether current of the free tachyonic Dirac left-hand side of (40a), equationhasbeencalledintoquestion[62], and it has been argued that no consistent interpretation can be given ⃗ ⃗ 5 0 because certain zero components of the conserved current tr (∑ (−𝜎) U𝜎 (𝑘) ⊗ U𝜎 (𝑘) 𝛾 𝛾 ) 𝜎 were conjectured to vanish for all tachyonic momentum eigenstates [62]. = ∑ (−𝜎) U (𝑘)⃗ 𝛾5𝛾𝜇=0U (𝑘)⃗ 𝜎 𝜎 (76a) A possible answer for question (1) has been proposed in 𝜎 [9]. Summarizing the argument, it has been concluded in 𝑘0 𝐸 [9] that one can solve the problem in two ways. (i) One can = ∑ (−𝜎) (− ) U (𝑘)⃗ U (𝑘)⃗ =2 . 𝑚 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝜎 𝜎 𝑚 Lorentz transform the vacuum state and Lorentz transform 𝜎 =𝜎 all fundamental creation and annihilation operators of the The bispinor trace of the right-hand side of (40a)is fermionfield(someofthesewillchangefromannihilators to creators upon transformation, due to the spacelike nature 5 0 𝑘−𝛾¡ 𝑚 0 𝑘¡ 𝐸 𝐸 of the tachyons). (ii) One keeps a Lorentz-invariant vacuum tr (𝛾 )=tr (𝛾 )=4 =2 , (76b) 𝜇 2𝑚 2𝑚 2𝑚 𝑚 state,andonlytransformthespace-timearguments 𝑘 and 𝜇 𝑥 of the field operators, keeping all creation operators which shows the consistency of the bispinor sum (40a)with as creators and annihilation operators as annihilators. The the Gordon decomposition (70a)and(70b). If we were to (−𝜎) (+𝜎) amplitudes, cross sections, and so forth, obtained using replace by , the two sides of the relation (76a)would approach (ii), then depend on scalar product of four vectors differ by a minus sign. The bispinor trace of the left-hand side which are equal to the result obtained by first calculating the of (40b)is process in the original Lorentz frame and then performing ⃗ ⃗ 5 0 the Lorentz transformation into the moving frame. tr (∑ (−𝜎) V𝜎 (𝑘) ⊗ V𝜎 (𝑘) 𝛾 𝛾 ) 𝜎 The conjecture regarding an expansion of a “false” vac- uum in the presence of tachyons can be traced to the = ∑ (−𝜎) V (𝑘)⃗ 𝛾5𝛾𝜇=0V (𝑘)⃗ fact that most of the tachyonic theories discussed so far 𝜎 𝜎 (77a) 𝜎 in the literature are scalar [28–33]. Indeed, scalar tachyons 𝑘0 𝐸 have a problem with instability, because of the structure of = ∑ (−𝜎) ( ) V (𝑘)⃗ V (𝑘)⃗ =2 . themassterminrelationtothefieldHamiltonians,which ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝜎 𝜎 2 2 𝜎 𝑚 𝑚 =−𝜎 changes sign 𝑚 →−𝑚 in a tachyonic theory, suggesting that the field energy can be lowered by creating tachyons. We have used the tachyonic Gordon decomposition for The problem does not occur in tachyonic spin-1/2 theories negative-energy states as given in (70a)and(70b), which because a linear, not quadratic, mass term enters the field differs from the positive-energy Gordon decomposition by a Lagrangian and Hamiltonian. Provided one reinterprets the minus sign, but an additional minus sign is obtained from spin-1/2 antiparticle solutions in the usual way (negative the Lorentz-invariant normalization of the negative-energy energy for propagation into the past becomes positive energy fundamental bispinors. From the right-hand side of (40b), we for propagation into the future), it then becomes immediately have clear that the vacuum energy cannot be lowered upon spin- 5 0 𝑘+𝛾¡ 𝑚 0 𝑘¡ 𝐸 1/2 tachyon antitachyon pair production. tr (𝛾 ) = tr (𝛾 ) =2 , (77b) 2𝑚 2𝑚 𝑚 A solution to problem (2) has also been proposed in [9]. 2 √ ⃗ 2 ⃗ which again is fully consistent, but only because we have Namely, the energies with 𝐸=± 𝑘 −𝑚 − i𝜖 (for |𝑘| <) 𝑚 a factor (−𝜎) in the sum rule (40a)and(40b), which find a natural interpretation in terms of complex resonance combines with the factor (−𝜎) from the Lorentz-invariant and antiresonance energies, which describe unstable states ⃗ ⃗ normalization of the V𝜎(𝑘)V𝜎(𝑘),togive2𝐸/𝑚 as a final which decay in time. Particle resonances are damped for result in (77a)and(77b). propagation into the future, and antiparticle antiresonances ISRN High Energy Physics 13

0 are damped for propagation into the past, as they should We conclude that J is positive for left-handed particle states be [9]. The occurrence of momentum eigenstates with real (𝜎=−1) and right-handed for antiparticle states (likewise, energies, and resonances with complex resonance energies, is 𝜎=−1; the helicity is equal to −𝜎 for antiparticles). For 𝐸=0, a well-known phenomenon all across physics (e.g., in atomic the “axial norm” of the momentum eigenstates given in (82a) physics, the ground-state energy of the helium atom is strictly and (82b)vanishes,butthisisaveryspecialcase.Namely, ⃗ 2 real, whereas autoionizing resonances in the three-particle for tachyons, 𝐸=0implies |𝑘| = 𝑚 =√ 𝑚𝑣/ 𝑣 −1 for system have a manifestly complex resonance energy). the momentum and thus corresponds to an “infinitely fast” Another “myth” which should be refuted concerns a tachyon (𝑣→∞), which remains “infinitely fast” upon conceivable “runaway reaction” where a moving tachyon Lorentz transformation. It would thus be wrong to conclude, releases an arbitrarily large amount of energy, as it loses as done in [62] in the text following (54) of [62], that all energy and accelerates, given its classical energy-velocity tachyonic states have zero axial norm, because this would relation 𝐸=𝑚/√𝑣2 −1. According to this relation, a correspond to a forbidden generalization of a result which tachyon indeed accelerates as its energy is lowered and holds asymptotically, for 𝑣→∞,toallfinitevaluesofthe becomes commensurate with the invariant mass square. (At tachyonic velocity 𝑣.Our(82a)and(82b) gives the explicit high energy, a tachyon approaches the light cone, though.) resultforanyfinitevalueoftheenergy𝐸. An infinitely fast tachyon takes the role of a tardyon at Thespatialintegralofthezerocomponentofthecon- rest [28]; the explicit eigenstates have been indicated in [9]. served current, However, energy conservation holds, and one cannot mix 3 0 3 5 𝜇 the notion of energy increase by acceleration, which only ∫ d 𝑟J (𝑥) = ∫ d 𝑟𝜓 (𝑥) 𝛾 𝛾 𝜓 (𝑥) holds for tardyonic particles, in order to “convert” a tachyon (83) losing energy by acceleration into a tardyon that gains energy 3 + 5 inthesameprocess.The“runawayreaction”isimpossible; =−∫ d 𝑟𝜓 (𝑥) 𝛾 𝜓 (𝑥) , only a finite amount of energy is released as the tachyon accelerates and the energy goes from 𝐸=𝑚/√𝑣2 −1 to is precisely equal (up to a sign) to the scalar product 3 + 5 zero. Energy conservation holds for tachyons, even if they ⟨𝜓1(𝑡),2 𝜓 (𝑡)⟩ ≡ ∫ d 𝑟𝜓1 (𝑡, 𝑟)𝛾⃗ 𝜓2(𝑡, 𝑟)⃗ which is conserved 5 accelerate, somewhat counterintuitively, when losing energy. under the time-evolution by the 𝛾 Hermitian (pseudo- An answer to question (3) has not yet been provided so far Hermitian) Hamiltonian 𝐻5.Thisscalarproductisnot intheliteraturetothebestofourknowledge.Here,weaimto positive definite, as already noticed in the work of Pauli [13], provide a possible physical interpretation for the conserved and precisely corresponds to the scalar product introduced current and scalar product and also show where certain by Pauli in equation (3) of [13], where in the notation 5 arguments presented originally by Hughes and Stephenson of [13]wehave𝜂=𝛾. Similar observations have been in their research paper [62]entitled“againsttachyonicneu- made in equations (34) and (52) of [62]. According to [63, trinos” become inconsistent. The Lagrangian density of the 64], one could otherwise define a so-called C operator, tachyonic Dirac particle reads, in first quantization6 [ ], which is not equal to the charge conjugation operator and 5 𝜇 5 “remedies” the problem of negative norm attained by some L (𝑥) = 𝜓 (𝑥) 𝛾 ( 𝛾 𝜕 −𝛾 𝑚) 𝜓 (𝑥) . 5 i 𝜇 (78) states under the 𝛾 norm, leading to a redefined, positive- This is equivalent, up to partial integration, to the symmetric definite scalar product. However, the negative norm finds a rather natural interpretation, and a redefinition of the scalar form [10] 3 + 5 product ⟨𝜓1(𝑡),2 𝜓 (𝑡)⟩ ≡ ∫ d 𝑟𝜓1 (𝑡, 𝑟)𝛾⃗ 𝜓2(𝑡, 𝑟)⃗ therefore is i 5 𝜇 5 𝜇 not required. L = (𝜓𝛾 𝛾 (𝜕𝜇𝜓) − (𝜕𝜇𝜓) 𝛾 𝛾 𝜓) −𝑚𝜓𝜓, (79) 2 For the ordinary Dirac equation, the conserved current 𝜇 𝜇 0 𝑥 = (𝑡, 𝑟)⃗ is 𝐽 (𝑥) = 𝜓𝑒(𝑥)𝛾 𝜓𝑒(𝑥).Itstimelikecomponentis𝐽 (𝑥) = where we suppress the space-time argument .The 𝜓+(𝑥)𝜓 (𝑥) 𝑒 nonsymmetric form (78) clearly exhibits the presence of the 𝑒 𝑒 ,wherethesubscript reminds us of the electron. 5 “chiral adjoint” 𝜓(𝑥)𝛾 in the Lagrangian. The conserved The latter can be interpreted as a positive-definite probability density which is conserved under the time evolution gener- current is (1) ated by the ordinary Hermitian Dirac Hamiltonian 𝐻 . 𝜇 5 𝜇 𝜇 𝜇 J (𝑥) = 𝜓 (𝑥) 𝛾 𝛾 𝜓 (𝑥) ,𝜕𝜇J (𝑥) =0. (80) The tachyonic Dirac current J is obtained from the 𝜇 5 ordinary Dirac 𝐽 by the replacement 𝜓𝑒(𝑥) → 𝜓(𝑥)𝛾 . The zero component We have seen that the scalar product for the tachyonic 0 5 0 J (𝑥) = 𝜓 (𝑥) 𝛾 𝛾 𝜓 (𝑥) (81) Dirac Hamiltonian is equal to an integral over the timelike component of the conserved Noether current of the Dirac assumes the following values in plane-wave eigenstates, equation and is not positive definite. In order to put this observation into perspective, it is instructive to recall that according to (56)and(57): 𝜇 2 for the Klein-Gordon equation (𝜕𝜇𝜕 +𝑚)𝜙(𝑥), =0 5 0 𝐸 𝜇 U (𝑘)⃗ 𝛾 𝛾 U (𝑘)⃗ = −𝜎 , (82a) the zero component of the conserved current 𝑗 (𝑥) = 𝜎 𝜎 𝑚 ∗ 𝜇 ∗ (i/2𝑚)(𝜙 (𝑥)𝜕𝜇𝜙(𝑥) − 𝜙(𝑥)𝜕 𝜙 (𝑥)) is not positive definite, 𝐸 either. Therefore, the zero component of the Klein-Gordon V (𝑘)⃗ 𝛾5𝛾0V (𝑘)⃗ = −𝜎 . 𝜎 𝜎 𝑚 (82b) current cannot be interpreted as a probability density but 14 ISRN High Energy Physics must be interpreted as a charge density, which is positive for principleforthe“advanced”partoftheFeynmanGreen particles and negative for antiparticles. function, which propagates antiparticle solutions into the This interpretation is not available for the zero component past. In order to avoid problems with regard to causality, the of the Noether current of the tachyonic Dirac equation, canonical quantum field theory of subluminal particles has to because the equation is not charge conjugation invariant and be supplemented by a reinterpretation principle, just like the is primarily proposed to describe neutrinos [10]. However, we tachyonic theory. The physical interpretation of the Noether can come closer to a physical interpretation of the timelike current is not affected by this consideration. component of the Noether current of the tachyonic equation Finally, let us point out that even without invoking rein- L if we compare the interaction Lagrangian QED of quantum terpretation, superluminal propagation can be compatible electrodynamics to the weak interaction L𝑊 of a neutrino with causality if we postulate that the tachyonic mass 𝑚 is so 0 and a “heavy photon,” that is, a 𝑍 boson, small that the superluminality is within the limits set forth by 𝐸=𝑚/√𝛽2 −1 ≈ L =−𝑒𝜓 𝛾𝜇𝜓 𝐴 the uncertainty relation. With an energy QED 𝑒 𝑒 𝜇 𝑚/√2𝛿 with 𝑣=1+𝛿and 𝛿≪1,wehaveΔ𝐸Δ𝑡 ≈ 5 √ 𝑒 𝜇 1−𝛾 (84) 𝑚/ 2𝛿Δ𝑡 ≤1 (in units with ℎ=1). Thus, for an infinitesimal ←→ L𝑊 =− 𝜓(𝛾 )𝜓𝑍𝜇, 𝑚≤ 2 sin 𝜃𝑊 cos 𝜃𝑊 2 tachyonic mass parameter which does not exceed √2𝛿/Δ𝑡, causality is preserved within the limits set by 𝐴 where 𝜃𝑊 is the Weinberg angle. The axial vector part L𝑊 of the uncertainty principle. Quantum limitations, the role of L𝑊 is unstable modes and quantum tunneling in superluminal 𝑒 propagation have been discussed in [65–68]. L𝐴 =− 𝜓𝛾5𝛾𝜇𝜓𝑍 , 𝑊 𝜇 (85) 2 sin (2𝜃𝑊) 5.2. Gupta-Bleuler Condition and Seesaw Mechanism. The where the ordering of the 𝛾 matrices is important. For commonly accepted mechanism for the suppression of right- the interaction with the timelike component of the vector handed neutrino and left-handed antineutrino states is the potential, we have the expressions seesaw mechanism [69]. After integrating the heavy degrees of freedom (the sterile neutrino), it contains a nonrenormal- L0 =−𝑒𝜓 𝛾0𝜓 𝐴 QED 𝑒 𝑒 0 izable dimension five operator and has a hierarchy problem: Namely, the neutrino masses are inversely proportional to the 𝐴,0 𝑒 5 0 (86) Λ ←→ L𝑊 =− 𝜓𝛾 𝛾 𝜓𝑍0. grandunification(GUT)scale and sensitively depend on 2 sin (2𝜃𝑊) fine-tuning of Λ. Let us consider equation (18) of [69]and 0 hypothetically consider a formulation that would result if the For quantum electrodynamics (QED), we interpret 𝜓𝑒𝛾 𝜓𝑒 = + physically observable neutrino were right handed. Then, we 𝜓𝑒 𝜓𝑒 as the probability density, and this suggests an inter- 5 0 could reformulate equation (18) of [69]as pretation of the expression 𝜓𝛾 𝛾 𝜓 as an “axial probability 1 density” or “axial interaction density” of the neutrino field eff ̃ ̃𝑇 𝑐 L =− ∑ [𝑅𝑟󸀠𝑅𝐻]𝑟 𝑌 󸀠𝑟 [𝐻 (𝑅𝑟𝑅) ]+h.c., 𝑍0 𝐼 Λ (87) with the timelike component of the boson. According 𝑟,𝑟󸀠 to (82a)and(82b), the “axial interaction density” is positive for the physically allowed states (left-handed particle and where right-handed antiparticle states) and negative for the physi- 𝜈 𝐻(+) cally forbidden states (right-handed particle and left-handed 𝑅 =( 𝑟𝑅), 𝐻=( ). 𝑟𝑅 𝑟 (0) (88) antiparticle states). This consideration is independent of the 𝑅 𝐻 suppression mechanism for the states of “wrong” helicity which, in second quantization, proceeds due to negative With the expectation value of the Higgs field, norm(seethediscussionfollowing(72)). ̃ 1 𝑣 In general, the physical interpretation of tachyonic theo- 𝐻0 = ( ), (89) √2 0 ries has been discussed by Feinberg [32, 33] and Saudarshan et al. [28–31]. The reinterpretation principle is a cornerstone instead of the left-handed Majorana neutrino masses, the of the theory. The only physically sensible quantities in right-handed ones would be small, a quantum theory are transition amplitudes. If the time ordering of superluminal events changes upon Lorentz trans- 𝑀 1 𝑅 𝑐 L =− ∑𝜈𝑟󸀠𝑅𝑀 󸀠 (𝜈𝑟𝑅) + h.c., formation, then one reinterprets the amplitude as connecting 2 𝑟𝑟 𝑟,𝑟󸀠 two space-time events whose coordinates are transformed (90) 2 according to the Lorentz transformation, so that the only 𝑅 𝑣 𝑀 = 𝑌 󸀠 . physically sensible quantity (transition amplitude) simply 𝑟𝑟󸀠 Λ 𝑟 𝑟 connects two events, one of which happens before the other [31]. This may seem counterintuitive at first, but it is perhaps The seesaw mechanism is not unique in suppressing a a little less counterintuitive if we take into account that definite helicity of the neutrino. It is unique provided one the accepted formulation of quantum field theory is based formulates it in terms of the left-handed fermion fields, but it on the Feynman propagator and on the reinterpretation could be formulated with inverted helicities if the helicity of ISRN High Energy Physics 15 the observed neutrinos at low energy were different. In the (“East-Coast” conventions were used in [72]), the result reads latter case, the right-handed neutrino mass would be small, as instead of the left-handed one. On the other hand, the seesaw 𝜕𝑏𝛽 mechanism has the distinct advantage that it is not necessary i 𝜈 𝛼 𝛼 𝜇𝜈 Γ𝑘 =− 𝑔𝜇𝛼 ( 𝑎𝛽 −Γ𝜈𝑘) 𝜎 , (92) to assume a superluminal character of the neutrino. 4 𝜕𝑥𝑘 The mechanism discussed here in the text following (72) 𝜇𝜈 𝜇 𝜈 𝛼 𝛽 is definite in making a prediction regarding the suppression of where 𝜎 =(i/2)[𝛾 , 𝛾 ] is the spin matrix. The 𝑎𝛽 and 𝑏𝜈 right-handed neutrino states, as explained by three indepen- coefficients transform the Dirac matrices to the local vierbein, dent arguments in Section 4.4.However,wehavetoassume 𝛼 𝛼 a superluminal neutrino. While an interacting superluminal 𝛾𝜌 =𝑏𝜌 𝛾𝛼,𝛾𝜌 =𝑎𝜌 𝛾𝛼, field theory is problematic, it is perhaps not as problematic as (93) 𝛾𝛼 =𝑎𝛼𝛾𝜌,𝛾𝛼 =𝑏𝛼𝛾𝜌. previously thought (see [70]andSection4of[9]). 𝜌 𝜌 Final clarification can only come from experiment. The ∇ =𝜕 −Γ seesaw mechanism is compatible with a Majorana neutrino. With the covariant derivative 𝜇 𝜇 𝜇, the gravitationally The tachyonic Dirac equation implies that the neutrino coupled Dirac equation reads as [72–74] cannot be equal to its antiparticle, because it does not allow ( 𝛾𝜇∇ −𝑚)𝜓(𝑥) =0. charge conjugation invariant solutions. It is only CP but not i 𝜇 (94) C invariant. Experimental evidence for neutrinoless double beta decay is disputed [71], and direct measurements of In the case of a tachyonic Dirac particle, it has to be the neutrino mass square currently exclude neither positive reformulated as follows: nor negative values [44–50]. The generally accepted seesaw 𝜇 5 (i𝛾 ∇𝜇 − 𝛾 (𝑥) 𝑚) 𝜓 (𝑥) =0. (95) mechanism implies that neutrino masses are generated by a nonrenormalizable interaction with a concomitant hierarchy 𝛾5(𝑥) problem, and the mechanism in itself could be reformulated The space-time coordinate-dependent matrix can be with opposite helicities. It is compatible with a Majorana defined as neutrino. By contrast, a tachyonic neutrino is “automatically” 𝜀 𝛾5 (𝑥) = i 𝛼𝛽𝛾𝛿 𝛾𝛼𝛾𝛽𝛾𝛾𝛾𝛿 left handed and not equal to its own antiparticle. It is 4! 𝛾5 √−𝑔 described by a Hermitian Hamiltonian and is plagued (96) with the conceptual difficulties associated with (ever so = i 𝜖 𝛾𝛼𝛾𝛽𝛾𝛾𝛾𝛿 = 𝛾0𝛾1𝛾2𝛾3, slightly)superluminalpropagation.Thismeansthatitis 4! 𝛼𝛽𝛾𝛿 i experimentally possible to test the models. 𝜇𝜈 where 𝑔=det 𝑔 is the determinant of the metric and 𝜀 = √−𝑔𝜀 𝜖 𝜀 5.3. Tachyonic Neutrinos as a Candidate for Dark Energy. The 𝛼𝛽𝛾𝛿 𝛼𝛽𝛾𝛿 is the local tensor, while 𝛼𝛽𝛾𝛿 is the totally antisymmetric Levi-Civita` tensor. The last identity96 in( )is formulationofagravitationalinteractionofaspin-1/2particle 𝜇𝜈 is nontrivial in the quantized formalism. Brill and Wheeler valid for a diagonal metric 𝑔 . [72] performed the pioneering steps in this direction. In order Let us first discuss94 ( ) very briefly. A projection onto to formulate the gravitational coupling of a Dirac particle, the upper and lower radial components 𝑓(𝑟) and 𝑔(𝑟) in 𝜇 one has to formulate generalized Dirac matrices 𝛾 ,which agravitationalfieldcanbefoundinequations(19)and fulfill anticommutation relations compatible with the local (20) of [74]. For vanishing electrostatic potential 𝑉→0, 𝜇𝜈 metric 𝑔 (𝑥) of curved space-time. Based on the Christoffel equation (20) of [74] is invariant under the replacement 𝜌 𝜌 symbols Γ =Γ (𝑥),oneformulatestheChristoffelaffine 𝑓(𝑟) ↔ 𝑔(𝑟) and 𝐸↔−𝐸.So,if𝐸 is an eigenvalue of the 𝜇𝜈 𝜇𝜈 −𝐸 connection matrices Γ𝜇 in spinor space and calculates the gravitationally coupled Dirac equation, so is .Invoking −𝐸 →𝐸 covariant derivative ∇𝜇 as follows [72–76]: reinterpretation and replacing for antiparticles, we find that the spectrum of the gravitationally coupled {𝛾𝜇 (𝑥) , 𝛾𝜇 (𝑥)}=2𝑔𝜇𝜈 (𝑥) , Dirac Hamiltonian is the same for particles and antiparticles. Therefore, the formalism makes the unique prediction that 1 𝜕𝑔 𝜕𝑔 𝜕𝑔 Γ𝜌 = 𝑔𝜌𝜎 ( 𝜈𝜎 + 𝜇𝜎 − 𝜇𝜈 ), tardyonic antiparticles, like tardyonic particles, are attracted 𝜇𝜈 2 𝜕𝑥𝜇 𝜕𝑥𝜈 𝜕𝑥𝜎 by a gravitational field. In passing, we note that the often cited motivation for the investigation of trapped antihydrogen (91) 𝜕𝛾𝜇 and its interaction with the gravitational field therefore is ∇ 𝛾 = −Γ𝜌 𝛾 + 𝛾 Γ −Γ𝛾 =0, 𝜈 𝜇 𝜕𝑥𝜈 𝜇𝜈 𝜌 𝜇 𝜈 𝜈 𝜇 faced with a unique theoretical prediction. Antiparticles are attracted by gravitation as much as particles are. 1 𝜕𝛾 Γ =− 𝛾𝜈 ( 𝜇 − 𝛾 Γ𝜎 ). It has been confirmed within the last two decades77 [ – 𝜇 4 𝜕𝑥𝜈 𝜎 𝜈𝜇 80] that the Universe expands more rapidly on large dis- tance scales than compatible with the attractive gravitional The above formula for Γ𝜇 is valid in the case of a diagonal attraction of the matter density in the Universe. Coupling to 𝜇𝜈 metric 𝑔 such as the Schwarzschild metric considered in a scalar field “quintessence” is usually invoked in order to [74]. For a general space-time metric, and with “West-Coast” explain the expansion of the expansion rate of the Universe 𝜇𝜈 conventions for the local vierbein 𝑔 = diag(1,−1,−1,−1) [81].Asthescalarquintessencefield“rollsdownitspotential,” 16 ISRN High Energy Physics it accelerates the expansion rate of the Universe. Because of a larger when the position-dependent mass 𝑚=𝑚(𝑥)gets self-attractingproperty(thequintessencefieldenergycanbe smaller. lowered by increasing the local density of the quintessence As neutrinos are ubiquitous within the cosmos, one field), quintessence has positive energy density but negative now needs to evaluate their conceivable contribution to the pressure. This property is necessary in order to constitute repulsive force on intergalactic distance scales. A detailed a candidate for dark energy [81]. In the standard model of discussion is beyond the scope of the current paper, but cosmology, dark energy accounts for about 73% of the total we can formulate some initial considerations and order-of- mass-energy of the Universe. The quintessence field thereby magnitude estimates. One can identify, a priori, two sources acts like a time-dependent cosmological constant. of neutrinos which may become important: (i) thermalized Wecanthusconcludethatmostoftheenergyinthe neutrinos which decouple in the early Universe [85–88] Universe actually is not gravitationally attractive; that is, that and (ii) nonthermalized, high-energy neutrinos which are gravity can repel. In the following, we shall present qualitative continuously generated through various cosmic processes, arguments which suggest that tachyonic neutrinos may play such as nuclear fusion in stars like the sun, supernovae, and a role in the expansion of the Universe and may contribute to high-energy cosmic background. “dark energy.” Conceivable connections of tachyonic physics It is generally assumed that today’s cosmic background and dark energy have been explored in the literature, employ- (CMB) radiation is accompanied by a neutrino background, ing either scalar fields82 [ ] or Lorentz-violating mechanisms which at some point (low energy) decoupled from the other [83]. In order to provide an alternative explanation for dark particles. The average energy of the background neutrinos energy, it is necessary to invoke a mechanism that leads is generally assumed to be of the same order-of-magnitude to a repulsive gravitational force on intergalactic distance as compared to the cosmic microwave background (CMB), scales in the Universe. It is not fully surprising that tachyonic and the currently accepted value for the temperature of 1/3 neutrinos may provide for such an alternative mechanism. the neutrino background is (4/11) = 0.714 times that Namely, both on the classical level [84], and on the level of the electromagnetic background radiation, that is, about of quantum theory (95), tachyonic particles are repulsed by 1.9 K. The Universe, according to this hypothesis, would be gravitational fields. filled with a sea of nonrelativistic background neutrinos. On the classical level, this is seen as follows. We start from This picture changes when we assume that the neutrino the familiar equation of motion of a particle with mass 𝑚 is tachyonic. In the earliest stages of the Universe, tachy- in curved space-time (see equation (86’) of [84]), which is a onic neutrinos are of course highly energetic, with 𝐸= geodesic, 2 √ ⃗ 2 ⃗ 𝑘 −𝑚 ≈|𝑘| ≫. 0 As they lose energy, they become 2𝑥𝜇 𝑥𝜌 𝑥𝜎 𝐸=𝑚/√𝑣2 −1 →0 d +Γ𝜇 d d =0. faster, because (in the classical 2 𝜌𝜎 (97) 2 𝑠 𝑠 √ ⃗ 2 ⃗ d 𝑠 d d theory). The energy 𝐸= 𝑘 −𝑚 vanishes as |𝑘| →𝑚 , 󸀠 |𝑘|⃗ < 𝑚 With a suitably redefined proper time𝑠 d , the zero geodesic within quantum theory. For , according to [9], the for a tachyon reads as (see equation (86’) of [84]) spectrum of the tachyonic Hamiltonian contains unstable resonances and antiresonances, with a purely imaginary 2 󸀠𝜇 󸀠𝜌 󸀠𝜎 resonance energy. Because the real part of the resonance d 𝑥 󸀠𝜇 d𝑥 d𝑥 +Γ =0. 2 2 𝜌𝜎 󸀠 󸀠 (98) √ 2 ⃗ d 𝑠󸀠 d𝑠 d𝑠 energy 𝐸=−i 𝑚 − 𝑘 vanishes, the unstable states form a “background” of fluctuating quantum states in the Universe. The force exerted on a tardyon reads, according to equation 2 √ 2 ⃗ (79a) of [84], Particle resonances (𝐸=−i 𝑚 − 𝑘 )andantiparticle √ 2 2 𝜇 𝜌 𝜎 𝐸=+𝑖 𝑚2 − 𝑘⃗ 𝜇 d 𝑥 𝜇 d𝑥 d𝑥 antiresonances ( ) are damped for propaga- 𝐹 =𝑚 =−𝑚Γ𝜌𝜎 . (99) tionintothefutureandpast,respectively(see[9]). The latter d𝑠2 d𝑠 d𝑠 case amounts to propagation into the future if one invokes However, there is a sign change for a tachyon (see equation reinterpretation for the antiparticle solutions [9]. (79b) of [84]), A very course-grained estimate regarding the role of the fluctuating resonance and antiresonance states can be given 2 󸀠𝜇 󸀠𝜌 󸀠𝜎 󸀠𝜇 d 𝑥 󸀠𝜇 d𝑥 d𝑥 as follows. We start from the grand canonical ensemble 𝐹 =−𝑚 =𝑚Γ𝜌𝜎 , (100) d𝑠󸀠2 d𝑠 d𝑠 𝑘 𝑇 ∞ 𝜇−𝐸(𝑘) Ω=−2 𝐵 𝑉 ∫ 𝑘𝑘2 (1 + ( )) , 2 d ln exp corresponding to the change in the energy-velocity relation 2𝜋 0 𝑘𝐵𝑇 √ 2 √ 2 𝐸=𝑚/1−𝑣 →𝐸=𝑚/ 𝑣 −1 for tardyon versus (101) tachyon, which leads to gravitational repulsion for tachyons. Another intuitive way to understand gravitational tachyonic where 𝑘𝐵 is the Boltzmann constant, 𝑉 is the normalization 2 𝜇 𝐸(𝑘) 𝐸=√𝑘⃗ +𝑚2 volume, is the chemical potential, and is the energy of a repulsion is to observe that the quantity gets 𝑘=|𝑘|⃗ 2 smaller when the position-dependent mass 𝑚=𝑚(𝑥)gets tachyon as a function of . The multiplicity factor takes 2 into account the particle and antiparticle resonances and is √ ⃗ 2 smaller,whereasthetachyonicenergy𝐸= 𝑘 −𝑚 gets supplemented here in comparison to the derivation presented ISRN High Energy Physics 17 in [89].Furthermore,wehaveassumedanisotropicenergy energy densities of various cosmologically relevant quanti- 3 2 dependence ∫ d 𝑘→4𝜋∫d𝑘𝑘 . The pressure is found as ties, 𝑎̇ 2 8𝜋𝐺 8𝜋𝐺 𝐻2 = ( ) = (𝜌 +𝜌 +𝜌) ≈ (𝜌 +𝜌 ) , Ω 𝑘 𝑇 ∞ 𝜇−𝐸(𝑘) |𝑎| 3 Λ 𝑀 𝑘 3 Λ 𝑀 𝑝=− = 𝐵 ∫ 𝑘𝑘2 (1 + ( )) . 2 d ln exp (102) 𝑉 𝜋 0 𝑘𝐵𝑇 (105a) 𝑎̈ 8𝜋𝐺 4𝜋𝐺 For 𝑇→0, the contribution of resonances and antireso- = 𝜌 − (𝜌 +3𝑝 +𝜌 +3𝑝 ) ⃗ Λ 𝑀 𝑀 𝑘 𝑘 nances with |𝑘| < 𝑚 to the pressure and energy density is |𝑎| 3 3 (105b) easily evaluated according to equations (27) and (28) of [89]. 4𝜋𝐺 For the pressure, we have ≈ (2𝜌 −𝜌 ). 3 Λ 𝑀 𝜌 𝜌 = 𝑚 𝐸 (𝑘) Here, is the matter density in the Universe and 𝑘 𝑝 = i ∫ 𝑘𝑘3 dIm −3𝑘/(8𝜋𝐺𝑎2) 0 2 d is the energy density associated with the cur- 3𝜋 0 𝑑𝑘 vatureoftheUniverse(𝑘=+1,0,−1), while 𝜌Λ is the (103) 𝑚 𝑘4 𝑚4 energy density corresponding to the cosmological constant. = i ∫ 𝑘 = i , 2 d The mass density is 𝜌𝑀 and 𝑝𝑀 ≈0.Weaimtosolve(105a) 3𝜋 0 √𝑚2 −𝑘2 16𝜋 and (105b)usingacomplexvariableansatz𝑎 = |𝑎|exp(𝑖𝜃) and 𝑡=𝑡exp(𝑖𝜑).Using(105a)and(105b), with a value 𝐻=71 −1 −1 whereas the energy density is km s Mpc for the Hubble constant, one easily reproduces the accepted value for the critical mass density of 𝜌 ≈ 9.34 × 10−33 −3 crit kg cm (in SI units) which corresponds 𝑚 −9 −3 i 2 𝜌 ≈8.39×10 𝜌 = ∫ 𝑘𝑘 𝐸 (𝑘) to a critical energy density of crit Jm (we 0 𝜋2 d Im 𝑐=1 𝜌 ≈ 0 have set ). In natural units, the result converts to crit (104) 9.99 × 10−45 4 𝑚 𝑚4 GeV . =− i ∫ 𝑘𝑘2√𝑚2 −𝑘2 =−i =−𝑝. We courageously set 𝑘=0in (105a)and(105b) 𝜋2 d 16𝜋 0 0 (flat Universe) and match the modulus of the (complex) accelerated expansion 𝑎/|𝑎|̈ against the currently accepted Ω =𝜌/𝜌 ≈ 0.27 Ω =𝜌 /𝜌 ≈ The equation of state fulfilled by the zero-temperature limit values of 𝑀 crit (see [94]) and Λ Λ crit 0.73. According to (105b), the matching is performed by of the tachyonic Dirac sea is 𝑤0 =𝑝0/𝜌0 =−1. This is the required equation of state for a “vacuum” energy density courageously assuming that the bulk of the cosmological that describes a nonvanishing cosmological constant [90]. If constant is given by the complex energy density (104)and equating the complex modulus as follows: 𝑝0 +𝜌0 =0, then there is no net energy gain upon pulling on a “piston” which contains the Universe (an illustrative 𝑚4 𝜌󸀠 ≈𝜌 =−i , discussioncanbefoundin[91]). Λ 0 16𝜋 Both results in (103)and(104) are imaginary. This raises (106) the pertinent question of how to incorporate the imaginary 󵄨 󸀠 󵄨 󵄨 󵄨2 ! 󵄨2𝜌 −𝜌 󵄨 = √4󵄨𝜌󸀠 󵄨 +𝜌2 =2𝜌 −𝜌 ≈ 1.19𝜌 . pressure and energy density of the neutrino field into the 󵄨 Λ 𝑀󵄨 󵄨 Λ󵄨 𝑀 Λ 𝑀 crit evolution equations of the Universe. In giving rise to unstable 𝜌󸀠 =−0.579𝜌 resonance states, the tachyonic Dirac field, if it exists, would The solution is Λ i crit,andsoourestimateforthe be different from any other known fundamental quantum neutrino mass reads as fields [57]. An exhaustive mathematical description would 󵄨 󵄨 𝑚4 (5.79 × 10−45 4)∼󵄨𝜌 󵄨 = 󳨐⇒ 𝑚 ∼ 0.0232 . require an extension of scattering theory to complex energy GeV 󵄨 0󵄨 16𝜋 eV and momentum exchanges, applied to low-energy collisions (107) with neutrinos, and describe the continuous decay and repopulation of the unstable tachyonic states in the Universe. If this treatment is valid, then the mass 𝑚 necessarily has to A detailed discussion is beyond the scope of the current be the mass of the “heaviest” neutrino mass eigenstate, that paper. However, in a first approximation, we can argue that is, the one with the largest modulus of the tachyonic mass an established technique for the mathematical treatment of 𝑚. Heavier eigenstates would lead to a larger cosmological complex resonances entails complex scaling (for reviews, see constant, because 𝜌0 is proportional to the fourth power of the [92, 93]), and we thus explore a complex scaling ansatz to the neutrino mass. In terms of a conversion to flavor eigenstates, solution of the evolution equations in the following. the heaviest mass eigenstates might well be the one closest to The imaginary results in(103)and(104) describe the the electron neutrino and electron antineutrino lepton flavor quantum fluctuations due to the unstable states. We thus con- eigenstates [95]. sider equations (8)–(11) of [90], which relate the accelerated We have interpreted the quantities 𝑝0 and 𝜌0 givenin(103) expansion rate 𝑎/|𝑎|̈ and the Hubble constant 𝐻=𝑎/|𝑎|̈ and (104) as contributions to a cosmological constant which (the dot denotes differentiation with respect to time) to the describes the evolution of the Universe in complex space and 18 ISRN High Energy Physics time directions 𝑎 = |𝑎|exp(𝑖𝜃) and 𝑡=𝑡exp(i𝜑).Weshould the factor (−𝜎) inthetensorsumsovertheeigenspinorsfor supplement the solutions for the complex rotation angles for the tachyonic equations. This fact is verified, on the basis ∘ ∘ space and time, which read as 𝜃 = 38.1 and 𝜑 = 70.6 , of tachyonic Gordon identities and related considerations, within our complex scaling approach. The evolution of the in Sections 4.3 and 4.4.Thepresenceofthefactor(−𝜎) energy density with the aging of the Universe is given by (see in the fundamental tachyonic field anticommutators in61 ( ) equation (12) of [90]) implies the suppression of right-handed particle and left- handed antiparticle states, due to negative norm, as shown −3(1+𝑤0) 𝜌0 =𝜌𝑖 (0) 𝑎 = const., 𝑤0 =−1. (108) in (72). Finally, in Section 5,weobservethatsincetachyonsare We therefore obtain a time-independent energy density 𝜌0, repelled by gravity, it might be worthwhile to investigate which is consistent with the fundamental character of the their conceivable role in the mechanism(s) responsible for fluctuating unstable resonances of the neutrino field. We the accelerated expansion of the Universe “dark energy.” reemphasize that the mass 𝑚 ∼ 0.0232 eV is tachyonic and Furthermore, the tachyonic resonances and antiresonance enters the gravitationally coupled tachyonic Dirac equation energies might play a role in the sum over states that enter (95). The given mass value is not currently excluded by any the thermodynamic potentials of a free tachyonic fermionic terrestrial experiment, and it is consistent with the observed gasinthelow-temperaturelimit.Ifweconsiderthestateswith time spread of the arrival times of neutrinos from the imaginary energy to be unstable, fluctuating states, then it is supernova SN1987A (see [96]). The hypothesis that very light intuitively obvious that they might contribute to a fluctuating elementary spin-1/2 particles could be tachyonic has been energy density and pressure on large distance scales in the pursued elsewhere [97]. Universe. As described in an approach in Section 5.3,an order-of-magnitudecalculationbasedonacomplexscaling 6. Conclusions transformation of the time evolution of the Universe leads to a tachyonic neutrino mass which is not excluded at present In the current investigation, we present the fundamental by any terrestrial experiment. We believe that it might be solutions of generalized Dirac equations in the helicity basis worthwhile to explore the physical consequences of the “ugly in a systematic and unified manner. Of particular importance duckling” (tachyonic neutrino) somewhat further. It allows us are the Dirac equation with two tardyonic mass terms 𝑚1 + to retain, among other things, lepton number conservation as 5 5 i𝛾 𝑚2 and two tachyonic mass terms 𝑖𝑚1 +𝛾𝑚2.Letus asymmetryofnature. summarize the main results. We have discussed the ordinary (tardyonic) Dirac equation in Section 2.2, a tardyonic Dirac Acknowledgments equation (with two mass terms) in Section 2.3,andtwo tachyonic Dirac equations in Sections 3.1 and 3.2.Wegive The authors acknowledge helpful conversations with C. the fundamental eigenspinors that enter the plane-wave Hirata, R. J. Hill, W. Rodejohann, P. J. Mohr, and I. Nandori.´ solutions of all of these equations (see (20a), (20b), (21a), (21b) This work was supported by the NSF (Grant PHY-1068547) (27a), (27b), (28a), (28b), (35a), (35b), (36b), (36a), (42a), and by the National Institute of Standards and Technology (42b), (43a)and(43b)). The eigenspinors are obtained using (Precision Measurement Grant). projector techniques as outlined in Chap. 2 of [57]. For the “normal” Dirac equation (see (20a), (20b), (21a), and (21b)), our results are consistent with [3]andChapter23of[58]. For References the generalized Dirac equations, the solutions have not yet [1] P. A. M. 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