2025/08/29 by Popov, Alexander D.
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2508.21590
We describe relativistic particles with spin as points moving in phase space X=T^* R1,3× C2L× C2R, where T^* R1,3=R1,3× R1,3 is the space of coordinates and momenta, and C2L and C2R are the spaces of representation of the Lorentz group of type (\frac12 , 0) and (0, \frac12). Passing from relativistic mechanics with a Lorentz-invariant Hamiltonian function H on the phase space X to quantum mechanics with a Hamiltonian operator H, we introduce two complex conjugate line bundles LC+ and LC- over X. Quantum particles are introduced as sections Ψ+ of the bundle LC+ holomorphic along the space C2L× C2R, and antiparticles are sections Ψ- of the bundle LC- antiholomorphic along the internal spin space C2L× C2R. The wave functions Ψ_± are characterized by conserved charges q_\sfv=± 1 associated with the structure group U(1)_\sfv of the bundles LC^±. Wave functions Ψ_± are governed by relativistic analogue of the Schrödinger equation. We show how fields with spin s=0 (Klein-Gordon), spin s=\frac12 (Dirac) and spin s=1 (Proca fields) arise from these equations in the zeroth, first, and second order expansions of the functions Ψ_± in the coordinates of the spin space C2L× C2R. The Klein-Gordon, Dirac and Proca equations for these fields follow from the Schrödinger equation on the extended phase space T^* R1,3× C2L× C2R. Using these results, we also introduce equations describing first quantized photons. We show that taking into account the charges q_\sfv=± 1 of the fields Ψ_± changes the definitions of the inner products and currents, which eliminates negative energies and negative probabilities from relativistic quantum mechanics.