General relativity as an extended canonical gauge theory

Abstract

It is widely accepted that the fundamental geometrical law of nature should follow from an action principle. The particular subset of transformations of a system's dynamical variables that maintain the form of the action principle comprises the group of canonical transformations. In the context of canonical field theory, the adjective "extended" signifies that not only the fields but also the space-time geometry is subject to transformation. Thus, in order to be physical, the transition to another, possibly noninertial frame of reference must necessarily constitute an extended canonical transformation that defines the general mapping of the connection coefficients, hence the quantities that determine the space-time curvature and torsion of the respective reference frame. The canonical transformation formalism defines simultaneously the transformation rules for the conjugates of the connection coefficients and for the Hamiltonian. As will be shown, this yields unambiguously a particular Hamiltonian that is form-invariant under the canonical transformation of the connection coefficients and thus satisfies the general principle of relativity. This Hamiltonian turns out to be a quadratic function of the curvature tensor. Its Legendre-transformed counterpart then establishes a unique Lagrangian description of the dynamics of space-time that is not postulated but derived from basic principles, namely the action principle and the general principle of relativity. Moreover, the resulting theory satisfies the principle of scale invariance and is renormalizable.