From Generalized Dirac Equations to a Candidate for Dark Energy

Abstract

We consider extensions of the Dirac equation with mass terms m1 + i γ 5 m2 and i m1 + γ 5 m2. The corresponding Hamiltonians are Hermitian and pseudo-Hermitian ("γ 5 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 ǫ 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").

Weighing the neutrino

“…(A2) have an convenient structure in helicity basis [55]. Furthermore, they fulfill projector sum rules, discussed in Refs.…”

“…[22] and recently subject to a few more investigations reported in Refs. [52][53][54][55]. It is based on the Lagrangian…”

“…Quite recently [52,53,55], it has been shown that the solutions of Eq. (A2) have an convenient structure in helicity basis [55].…”

“…Furthermore, they fulfill projector sum rules, discussed in Refs. [52,55], permitting the evaluation of the tachyonic Dirac propagator as the inverse of the kinetic operator (Dirac Hamiltonian), which is crucial in formulating the corresponding field theory. In view of the structure of the Lagrangian (A1), the role of the Dirac adjoint in the tachyonic sector is assumed by the pseudoscalar adjoint ψ γ 5 , and the generalized Dirac propagator is calculated as the time-ordered product S T (x, y) = 0 T ψ(x) ψ(y)γ 5 0 .…”

A new approach on the stability analysis in ELKO cosmology

“…DE is something even more mysterious, since their behavior is gravitationally repulsive [13][14][15][16]. From a theoretical point of view it is very common to interpret these entities as being some kind of field (scalar fields [17][18][19][20][21][22][23][24][25][26][27] or spinor fields [28][29][30], for example). Other models also consider the possibility of interaction between DM and DE [31][32][33][34][35][36][37][38][39][40][41][42][43][44], which could alleviate the coincidence problem for instance.…”