Canonical Transformation Path to Gauge Theories of Gravity

TL;DR

The paper develops a canonical-transformation framework to derive a gauge-theoretic description of gravity starting from a globally Lorentz-invariant Hamiltonian for matter fields, promoting to local spacetime invariance via affine connections that act as gauge fields.A second, internal stage treats the gauge fields as dynamical by introducing their conjugates, yielding a closed set of eight canonical field equations in a generally covariant geometrodynamics with possible torsion and non-metricity, and a Riemann–Cartan curvature emerges naturally from the formalism.A free-gravity Hamiltonian is required to procure the dynamics of the gauge fields; a concrete example leads to an Einstein-type equation with additional quadratic curvature terms, recovering GR in vacuum metrics while allowing deviations in matter-filled spacetimes.The approach makes explicit how spin-0 and spin-1 fields source spacetime geometry and shows that the curvature and connection enter through a gauge-theoretic construction, offering a unifying, principle-based route to geometrodynamics without prescribing a Lagrangian for gravity a priori.

Abstract

In this paper, the generic part of the gauge theory of gravity is derived, based merely on the action principle and on the general principle of relativity. We apply the canonical transformation framework to formulate geometrodynamics as a gauge theory. The starting point of our paper is constituted by the general De Donder-Weyl Hamiltonian of a system of scalar and vector fields, which is supposed to be form-invariant under (global) Lorentz transformations. Following the reasoning of gauge theories, the corresponding locally form-invariant system is worked out by means of canonical transformations. The canonical transformation approach ensures by construction that the form of the action functional is maintained. We thus encounter amended Hamiltonian systems which are form-invariant under arbitrary spacetime transformations. This amended system complies with the general principle of relativity and describes both, the dynamics of the given physical system's fields and their coupling to those quantities which describe the dynamics of the spacetime geometry. In this way, it is unambiguously determined how spin-0 and spin-1 fields couple to the dynamics of spacetime. A term that describes the dynamics of the free gauge fields must finally be added to the amended Hamiltonian, as common to all gauge theories, to allow for a dynamic spacetime geometry. The choice of this "dynamics Hamiltonian" is outside of the scope of gauge theory as presented in this paper. It accounts for the remaining indefiniteness of any gauge theory of gravity and must be chosen "by hand" on the basis of physical reasoning. The final Hamiltonian of the gauge theory of gravity is shown to be at least quadratic in the conjugate momenta of the gauge fields -- this is beyond the Einstein-Hilbert theory of General Relativity.