Quantum connection, charges and virtual particles

Geometrically, quantum mechanics is defined by a complex line bundle L_hbar over the classical particle phase space T^*{R}^3cong{R}⁶ with coordinates x^a and momenta p_a, a,...=1,2,3. This quantum bundle L_hbar is endowed with a connection A_hbar, and its sections are standard wave functions ψ obeying the Schrödinger equation. The components of covariant derivatives nabla_Ahbar^{} in L_hbar are equivalent to operators {hat x}^a and {hat p}_a. The bundle L_hbar=: L_C^+ is associated with symmetry group U(1)_hbar and describes particles with quantum charge q=1 which is eigenvalue of the generator of the group U(1)_hbar. The complex conjugate bundle L^-_C:={overline{L_C^+}} describes antiparticles with quantum charge q=-1. We will lift the bundles L_C^pm and connection A_hbar on them to the relativistic phase space T^*{R}^3,1 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 {R}^3,1. This equation does not contain interaction with the quantum connection A_hbar on bundles L^pm_C→ T^*{R}^3,1 because A_hbar has non-vanishing components only along p_a-directions in T^*{R}^3,1. To enable the interaction of elementary fermions Ψ with quantum connection A_hbar on L_C^pm, we will extend the Dirac equation to the phase space while maintaining the condition that Ψ depends only on t and x^a. 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.