Extended canonical field theory of matter and space‐time

Abstract

AbstractAny physical theory that follows from an action principle should be invariant in its form under mappings of the reference frame in order to comply with the general principle of relativity. The required form‐invariance of the action principle implies that the mapping must constitute a particular extended canonical transformation. In the realm of the covariant Hamiltonian formulation of field theory, the term “extended” implies that not only the fields but also the space‐time geometry is subject to transformation. A canonical transformation maintains the general form of the action principle by simultaneously defining the appropriate transformation rules for the fields, the conjugate momentum fields, and the transformation rule for the Hamiltonian. Provided that the given system of fields exhibits a particular global symmetry, the associated extended canonical transformation determines an amended Hamiltonian that is form‐invariant under the corresponding local symmetry. This will be worked out for a Hamiltonian system of scalar and vector fields that is presupposed to be form‐invariant under space‐time transformations x µ ∼→ X µ with δX µ/δxv = const., hence under global space‐time transformations such as the Poincar´e transformation. The corresponding amended system that is form‐invariant under local space‐time transformations δX µ/δxv ∼ const. then describes the coupling of the fields to the space‐time geometry and thus yields the dynamics of space‐time that is associated with the given physical system. Non‐zero spin matter determines thereby the space‐time curvature via a well‐defined source term in a covariant Poisson‐type equation for the Riemann tensor. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)