CPTM Symmetry for the Dirac Equation and Its Extended Version Based on the Vector Representation of the Lorentz Group

Abstract

We revisit the CPT theorem for the Dirac equation and its extended version based on the vector representation of the Lorentz group. Then it is proposed that CPTM may apply to this fundamental equation for a massive fermion a s a singlet or a doublet with isospin. The symbol M stands here for reversing the sign of the mass in the Dirac equation, which can be accomplished by operation on it with the so-called gamma-five matrix that plays an essential role for the chirality in the Standard Model. We define the CPTM symmetry for the standard and extended Dirac equation and discuss its physical implications and some possible consequences for general relativity.

Spinor fields, CPTM symmetry and smallness of cosmological constant in framework of extended manifold

“…Correspondingly considering the mutual vacuum 15 state for the creation and annihilation operators in both manifolds we have:…”

“…The motivation of the introduction of the negative mass in the different cosmological models is very clear. In any scenario, see [5,6] for example, the presence of some kind of the repulsive gravitation forces or an additional gravitational field in our Universe helps with an explanation of the existence of dark energy in the models, see also [7][8][9][10][11][12][13][14][15] and references therein. In the present construction we, nevertheless, consider a different realization of the appearance of the negative mass.…”

Spinor fields, CPTM symmetry and smallness of cosmological constant in framework of extended manifold

“…The motivation of the introduction of the negative mass in the different cosmological models is very clear. In any scenario, see [5,6] for example, the presence of some kind of the repulsive gravitation forces or an additional gravitational field in our Universe helps with an explanation of the existence of dark energy in the models, see also [7][8][9][10][11][12][13][14][15] and references therein. In the present construction we, nevertheless, consider a different appearance of the negative mass, the way of two metrics introduced is more similarly to the bimetric models perhaps, see [7,8] for some examples.…”

“…Performing Fourier transform we obtain that this contribution is zeroInt 2d = m d 4 p i (2π) 4 s(p i ) d 4 p f s(p f )δ 4 (p i + p f ) d 4 p (2π) 4 γ µ γ ρ p ρ + m p 2 − m 2 + ıε γ ρ1 (p − p f ) ρ1 + m (p − p f ) 2 − m 2 + ıε p f µ = = m 2 d 4 p i (2π) 4 s(p i ) d 4 p f s(p f )δ 4 (p i + p f ) µ γ ρ p ρ + γ ρ1 (p − p f ) ρ1 (p 2 − m 2 + ıε ) ((p − p f ) 2 − m 2 + ıε) p f µ = 0 . (C 15).…”