Donald C. Salisbury

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

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. @S0556-2821~97!06202-4#

Issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity

Diffeomorphism-induced symmetry transformations and time evolution are distinct operations in generally covariant theories formulated in phase space. Time is not frozen. Diffeomorphism invariants are consequently not necessarily constants of the motion. Time-dependent invariants arise through the choice of an intrinsic time, or equivalently through the imposition of time-dependent gauge fixation conditions. One example of such a time-dependent gauge fixing is the Komar-Bergmann use of Weyl curvature scalars in general relativity. An analogous gauge fixing is also imposed for the relativistic free particle and the resulting complete set time-dependent invariants for this exactly solvable model are displayed. In contrast with the free particle case, we show that gauge invariants that are simultaneously constants of motion cannot exist in general relativity. They vary with intrinsic time.

Gauge transformation in Einstein-Yang-Mills theories

We discuss the relation between space–time 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 space–time metric plus associated gauge transformations. Our results are generalizations of earlier results by ourselves and by Salisbury and Sundermeyer.

Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity

We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar’s complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac’s method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms.

ROSENFELD, BERGMANN, DIRAC AND THE INVENTION OF CONSTRAINED HAMILTONIAN DYNAMICS

In a paper appearing in Annalen der Physik in 1930 Leon Rosenfeld invented the first procedure for producing Hamiltonian constraints. He displayed and correctly distinguished the vanishing Hamiltonian generator of time evolution, and the vanishing generator of gauge transformations for general relativity with Dirac electron and electrodynamic field sources. Though he did not do so, had he chosen one of his tetrad fields to be normal to his spacetime foliation, he would have anticipated by almost thirty years the general relativisitic Hamiltonian first published by Paul Dirac.

Reduced phase space: quotienting procedure for gauge theories

We present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration-velocity space. Local expressions for a basis of this kernel are obtained using phase-space procedures; the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented. We show that this reduction procedure is equivalent to the standard Dirac method as long as the Dirac conjecture holds: that the Dirac Hamiltonian, containing the primary first-class constraints, with their Lagrange multipliers, can be enlarged to an extended Dirac Hamiltonian which includes all first-class constraints without any change of the dynamics. The quotienting procedure is always equivalent to the extended Dirac theory, even when it differs from the standard Dirac theory. The differences occur when there are ineffective constraints, and in these situations we conclude that the standard Dirac method is preferable - at least for classical theories. An example is given to illustrate these features, as well as the possibility of having phase-space formulations with an odd number of physical degrees of freedom.

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

Intrinsic time-dependent invariants are constructed for classical, flat, homogeneous, anisotropic cosmology with a massless scalar material source. Invariance under the time reparameterization-induced canonical symmetry group is displayed explicitly.

Peter Bergmann and the Invention of Constrained Hamiltonian Dynamics

It has always been the practice of those of us associated with the Syracuse “school” to identify the algorithm for constructing a canonical phase space description of singular Lagrangian systems as the Dirac–Bergmann procedure. I learned the procedure as a student of Peter Bergmann, and I should point out that he never employed that terminology. Yet it was clear from the published record at the time (the 1970s) that his contribution was essential. Constrained Hamiltonian dynamics constitutes the route to canonical quantization of all local gauge theories, including not only conventional general relativity, but also grand unified theories of elementary particle interactions, superstrings, and branes. Given its importance and my suspicion that Bergmann has never received adequate recognition from the wider community for his role in the development of the technique, I have long intended to explore this history in depth. This paper is merely a tentative first step, in which I will focus principally on the work of Peter Bergmann and his collaborators in the late 1940s and early 1950s, indicating where appropriate the relation of this work to later developments. I begin with a brief survey of the prehistory of work on singular Lagrangians, followed by some comments on the life of Peter Bergmann. These are included in part to commemorate Peter in this first meeting on the History of General Relativity since his death in October 2002. Then I will address what I perceive to be the principal innovations of his early Syracuse career. Josh Goldberg has already covered some of this ground in his 2005 report (Goldberg 2005), but I hope to contribute some new perspectives. I shall conclude with a partial list of historical issues that remain to be explored.

Gauge symmetries in Ashtekar's formulation of general relativity

Abstract It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekars formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry generators serve as Hamiltonians on members of equivalence classes of solutions of Einsteins equations and are thus intimately related to the so-called “problem of time” in an eventual quantum theory of gravity.

A GENERALIZED SCHRÖDINGER EQUATION FOR LOOP QUANTUM COSMOLOGY

A temporally discrete Schroedinger time evolution equation is proposed for isotropic quantum cosmology coupled to a massless scalar source. The approach employs dynamically determined intrinsic time and produces the correct semiclassical limit.