Canonical Transformation Path to Gauge Theories of Gravity

Abstract

In this paper, the generic part of the gauge theory of gravity is derived, based 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.

arXiv comment20 pages

References in corpus (4)

• Covariant Hamiltonian Field Theory
• General relativity as an extended canonical gauge theory
• General U(N) gauge transformations in the realm of covariant Hamiltonian field theory
• Identity for scalar-valued functions of tensors and its applications to energy-momentum tensors in classical field theories and gravity

Cited by in corpus (24)

• Inflation from fermions with curvature-dependent mass
• The dark side of the torsion: Dark Energy from propagating torsion
• Quadratic curvature theories formulated as Covariant Canonical Gauge theories of Gravity
• Torsional dark energy in quadratic gauge gravity
• Dark energy and inflation invoked in CCGG by locally contorted space-time
• Torsion driving cosmic expansion
• A correspondence between $1^{st}$ and $2^{nd}$ order formalism by a metricity constraint
• Conservation of energy-momentum of matter as the basis for the gauge theory of gravitation
• Covariant canonical gauge theory of gravitation for fermions
• Covariant Canonical Gauge Gravitation and Cosmology
• Canonical transformation path to gauge theories of gravity II --- Spacetime coupling of spin-0 and spin-1 particle fields
• Low-Redshift Constraints on Covariant Canonical Gauge Theory of Gravity
• Covariant Momentum Map Thermodynamics for Parametrized Field Theories
• Energy transfer from space-time into matter and a bouncing inflation from Covariant Canonical Gauge theory of Gravity
• Generic Theory of Geometrodynamics from Noether's theorem for the Diff(M) symmetry group
• On the cosmological constant in the deformed Einstein-Cartan gauge gravity in De Donder-Weyl Hamiltonian formulation
• Predictions of the pseudo-complex theory of Gravity for EHT observations- II. Theory and predictions
• Hamiltonian consistency of the gravitational constraint algebra under deformations
• Identity for scalar-valued functions of tensors and its applications to energy-momentum tensors in classical field theories and gravity
• Gauge theory of Gravity based on the correspondence between the $1^{st}$ and the $2^{nd}$ order formalisms
• Rigorous derivation of dark energy and inflation as geometry effects in Covariant Canonical Gauge Gravity
• On CCGG, the De Donder-Weyl Hamiltonian formulation of canonical gauge gravity
• Vanishing torsion coupling of the Maxwell field in canonical gauge theory of gravity
• Cosmology in the De Donder-Weyl Formulation of Einstein-Cartan Gravity