2023/10/10 by Alexander D. Popov, Popov, Alexander D.
Computer Science · Physics and Astronomy · Social Sciences · #Computational Physics and Python Applications #FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph) #High Energy Physics - Theory (hep-th) #International Science and Diplomacy #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.2310.06507
openalex publication_date 2023/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Geometrically, quantum mechanics is defined by a complex line bundle L_ℏ over the classical particle phase space T^*R3≅R6 with coordinates xa and momenta pa, a,...=1,2,3. This quantum bundle L_ℏ is endowed with a connection A_ℏ, and its sections are standard wave functions ψ obeying the Schrödinger equation. The components of covariant derivatives ∇A_ℏ in L_ℏ are equivalent to operators xa and pa. The bundle L_ℏ=: LC+ is associated with symmetry group U(1)_ℏ and describes particles with quantum charge q=1 which is eigenvalue of the generator of the group U(1)_ℏ. The complex conjugate bundle L-C:=LC+ describes antiparticles with quantum charge q=-1. We will lift the bundles LC^± and connection A_ℏ on them to the relativistic phase space T^*R3,1 and couple them to the Dirac spinor bundle describing both particles and antiparticles. Free relativistic quarks and leptons are described by the Dirac equation on Minkowski space R3,1. This equation does not contain interaction with the quantum connection A_ℏ on bundles L^±C→ T^*R3,1 because A_ℏ has non-vanishing components only along pa-directions in T^*R3,1. To enable the interaction of elementary fermions Ψ with quantum connection A_ℏ on LC^±, we will extend the Dirac equation to the phase space while maintaining the condition that Ψ depends only on t and xa. The extended equation has an infinite number of oscillator-type solutions with discrete energy values as well as wave packets of coherent states. We argue that all these normalized solutions describe virtual particles and antiparticles living outside the mass shell hyperboloid. The transition to free particles is possible through squeezed coherent states.