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From Generalized Dirac Equations to a Candidate for Dark Energy

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From Generalized Dirac Equations to a Candidate for Dark Energy 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 1 + i 2 and i1 +2. The corresponding Hamiltonians 5 are Hermitian and pseudo-Hermitian ( 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 (i 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.” 1. Introduction choose the space-time metric as = diag(1,−1,−1,−1). The denotes the partial derivative / with respect to the 1.1. Generalized Dirac Equations: Mass Terms and Dispersion space-time coordinate =(,)⃗ .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 Dirac equations (i − )(), =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==0 =1andusetheDiracmatricesinthe 2 relations of the form =√⃗ +2, whereas tachyonic mass standard representation terms lead to superluminal dispersion relations of the form 2 0 12×2 0 0 ⃗ =√⃗ −2,where is the energy and ⃗ is the momentum. ==( ), =(⃗ ), 0−12×2 −0⃗ The generalized, matrix-valued mass term enters the Dirac ( − )() =0 4×4 (1) equation in the form i .The are 5 0 12×2 =( ), matrices that fulfill the relations { , }=2 ,wherewe 12×2 0 2 ISRN High Energy Physics 0 0 + 0 and define =⃗ ⃗. For the ordinary Dirac theory, one has The relation 5 =5 (−)⃗ has been given in [9, 10]. =1 (one should say more precisely =114×4)witha However, it is much more instructive to observe that 5 is 5 5 + 5 5 real mass 1, Hermitian; that is, 5 =5 . The concept of Hermiticity is known in lattice theory [11, 12] and is otherwise (i −1)() =0. (2) called pseudo-Hermiticity [13–23]. An obvious generalization of the tachyonic case contains 5 √ ⃗2 2 an imaginary mass and a mass term, The dispersion relation is = +1.Thecorresponding Dirac Hamiltonian reads 5 (i − i1 − 2)() =0. (9) (1) = ⋅⃗ +⃗ 1. (3) √ ⃗2 2 2 The dispersion relation is = −1 −2.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 = ⋅⃗ +⃗ +5 5 i 1 2 (10) of the form =1 + i 2, the fermion propagator 5 5 + 5 may obtain nontrivial gradient corrections already at the and is Hermitian, = .For2 =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 5 part of the results, we may point out that the massless Dirac (i −1 − i 2)() =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 = +1 +2.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 () 5 shall discuss in the following. = ⋅⃗ +⃗ 1 + i 2. (5) We may indicate a further motivation for our study; namely, 1.2. Tachyonic Dirac Equation and Neutrinos: Possible Con- the unitarity of the matrix implies the existence of useful nections. The tachyonic generalized Dirac equations7 ( )and 2 relations [5]fortheevenpowers(2) 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 1 + i 2,inpowersof2. 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 exp (i ) = cos +i sin =1 + i 2 (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) () connects the two Hamiltonians and for 1 =cos light in a vacuum is the same for all observers, regardless of and 2 =sin , 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 = which the existence of tachyons, which are superluminal particles 2 ⃗2 =√⃗2 −2 fulfilling a Lorentz-invariant dispersion relation = − induce a superluminal dispersion relation .The 2 corresponding generalized Dirac equation has been named , 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 = 5 /√1−2 →∞ →1 (i − ) () =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 5 = ⋅⃗ +⃗ . (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 −coupling of the form 5 agating into the past according to the Feynman prescription (1 − ) 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- (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.
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