Covariant Canonical Gauge Theory of Classical Gravitation for Scalar, Vector, and Spin-1/2 Particle Fields

Abstract: The framework of the Covariant Canonical Gauge theory of Gravity (CCGG) is described in detail. CCGG emerges naturally in the Palatini formulation, where the vierbein and the spin connection are independent fields. Neither torsion nor non-metricity are excluded. The manifestly covariant gauge process is based on canonical transformations in the De Donder-Weyl Hamiltonian formalism, starting from a small number of basic postulates. Thereby, the original system of matter fields in flat spacetime, represented by non-degenerate Hamiltonian densities, is amended by spacetime fields. The coupling of matter and spacetime fields leaves the action integral of the combined system invariant under active local Lorentz transformations and passive diffeomorphisms, aka Principle of General Relativity. We consider the Klein-Gordon, Maxwell-Proca, and Dirac fields and derive the corresponding equations of motion. Albeit the coupling of the given matter fields to the gauge fields are unambiguously determined by CCGG, the dynamics of the free gauge fields must be postulated based on physical reasoning. Our choice allows to derive Poisson-like equations of motion also for curvature and torsion. The latter is proven to be totally anti-symmetric. The affine connection is a function of the spin connection and vierbein fields. Requesting the spin connection to be anti-symmetric gives naturally metric compatibility. The canonical equations combine to an extension of the Einstein-Hilbert action with a quadratic Riemann-Cartan concomitant that endows spacetime with inertia. Moreover, a non-degenerate, quadratic version of the free Dirac Lagrangian is deployed. When coupled to gravity, the Dirac equation is endowed with an emergent mass parameter, a curvature-dependent mass correction, and novel interactions between particle spin and spacetime torsion.