Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories

Abstract

We study spacetime diffeomorphisms in the Hamiltonian and Lagrangian formalisms of generally covariant systems. We show that the gauge group for such a system is characterized by having generators which are projectable under the Legendre map. The gauge group is found to be much larger than the original group of spacetime diffeomorphisms, since its generators must depend on the lapse function and shift vector of the spacetime metric in a given coordinate patch. Our results are generalizations of earlier results by Salisbury and Sundermeyer. They arise in a natural way from using the requirement of equivalence between Lagrangian and Hamiltonian formulations of the system, and they are new in that the symmetries are realized on the full set of phase space variables. The generators are displayed explicitly and are applied to the relativistic string and to general relativity.

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.…”

“…The dependence on the lapse is required in order to produce variations of the metric and of the scalar field that are projectable under the Legendre map from configuration-velocity space (the tangent bundle) to phase space (the cotangent bundle) [3]. The resulting phase space variations are canonical transformations, generated in the present model by…”

“…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.…”

Three denials of time in the interpretation of canonical gravity

“…From the persecutive of the derivation of canonical from covariant general relativity the basis of this discrepancy between the symmetry transformations realised in the two formalism is well understood -it can be explained in terms of the spacelike nature of the otherwise arbitrary embedding (see Isham and Kuchař (1985)) or (relatedly) in the context of noncomplete projectability between the symmetry transformations defined in the relevant tangent bundle and cotangent bundle structures (see Pons, Salisbury, and Shepley (1997)). …”

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. …”