  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.
