Gauge Fixing and Observables in General Relativity

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest.Comment: Contribution to the Proceedings of Spacetime and Fundamental Interactions: Quantum Aspects, May, 2003, honoring the 65'th birthday of A. P. Balachandra

Leon Rosenfeld and the challenge of the vanishing momentum in quantum electrodynamics

Leon Rosenfeld published in 1930 the first systematic Hamiltonian approach to Lagrangian models that possess a local gauge symmetry. The application of this formalism to theories with local internal symmetries, such as electromagnetism in interaction with charged matter fields, is valid and complete, and predates by two decades the work by Dirac and Bergmann. Although he provided a group-theoretical justification for gauge fixing procedures that had just been implemented in the first expositions of quantum electrodynamics by Heisenberg and Pauli, and also by Fermi, his contribution went largely unnoticed. This lack of impact seems to be related to a generalized disenchantment with second quantization in the 1930's and 1940's.Comment: 32 pages, submitted to Proceedings of the HQ2 Conference on the History of Quantum Physics, Utrecht, 14-17 July, 200