Negative-energy and tachyonic solutions in the Weinberg-Tucker-Hammer equation for spin 1

Abstract

We considered Weinberg-like equations in the article [1] in order to construct the Feynman-Dyson propagator for the spin-1 particles. This construction is based on the concept of the Weinberg field as a system of four field functions differing by parity and by dual transformations. We also analyzed the recent controversy in the definitions of the Feynman-Dyson propagator for the field operator containing the S=1/2 self/anti-self charge conjugate states in the papers by D. Ahluwalia et al and by W. Rodrigues Jr. et al. The solution to this mathematical controversy is obvious. I proposed the necessary doubling of the Fock Space (as in the Barut and Ziino works), thus extending the corresponding Clifford Algebra. Meanwhile, the N. Debergh et al article considered our old ideas of doubling the Dirac equation, and other forms of T- and PT-conjugation [5]. Both algebraic equation Det (\hat p - m) =0 and Det (\hat p + m) =0 for u- and v- 4-spinors have solutions with p_0= \pm E_p =\pm \sqrt{{\bf p}^2 +m^2}. The same is true for higher-spin equations (or they may even have more complicated dispersion relations). Meanwhile, every book considers the equality p_0=E_p for both u- and v- spinors of the (1/2,0)\oplus (0,1/2)) representation only, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of negative-energy solutions. It seems, that it is imposible to consider the relativistic quantum mechanics appropriately without negative energies, tachyons and appropriate forms of the discrete symmetries.