Gauge transformation in Einstein–Yang–Mills theories

Abstract

We discuss the relation between spacetime diffeomorphisms and gauge transformations in theories of the Yang-Mills type coupled with Einstein's General Relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Legendre map. Although pure Yang-Mills gauge transformations are projectable by themselves, diffeomorphisms are not. Instead the projectable symmetry group arises from infinitesimal diffeomorphism-inducing transformations which must depend on the lapse function and shift vector of the spacetime metric plus associated gauge transformations. Our results are generalizations of earlier results by ourselves and by Salisbury and Sundermeyer. 04.20.Fy, 04.40.-b, 11.10.Ef, 11.15.-q gr-qc/9912086

Reparameterization invariants for anisotropic Bianchi I cosmology with a massless scalar source

“…They were, however, the first to observe that an enlarged diffeomorphism symmetry group existed, and that it possessed a compulsory dependence on the lapse and shift [15]. The resulting classical diffeomorphism-induced canonical transformation group has recently been studied, also in models in which additional gauge symmetries are present [3,4,5,6]. Recognizing the existence of this symmetry we are able to complement the previous use of intrinsic coordinates with a demonstration that phase space variables constructed in intrinsic coordinates are indeed invariant under this group.…”

“…We are thus able to demonstrate in detail that our phase space invariants do not change when the coordinate time is altered under the action of the canonical diffeomorphism-induced symmetry group presented by J. Pons, D.C. Salisbury and L. C. Shepley. [3,4,5,6]. We construct invariants by choosing the evolving value of the scalar field as an intrinsic time, i.e., by establishing correlations between the scalar field and the spacetime metric components, including the lapse component.…”

Léon Rosenfeld’s general theory of constrained Hamiltonian dynamics

“…The expressions (82), (83), (85) have this generic structure, although the first class property need not hold for the coefficients of the ξ A andξ A in the Rosenfeld-Noether generators above. Indeed, we expect that one needs to work with modified transformations in order to respect the Legendre projectability of the transformations [36]. These questions will be addressed in a future publication.…”

Canonical Noether symmetries and commutativity properties for gauge systems

“…This method has been implicitly used in a series of papers [24][25][26] that analyze the relationship between the Lagrangian and Hamiltonian descriptions of the gauge group structure for generally covariant theories. …”