Léon Rosenfeld's pioneering steps toward a quantum theory of gravity

Abstract

In an article published in 1930 Léon Rosenfeld invented a general Hamiltonian formalism that purported to realize general coordinate, local Lorentz, and U(1) symmetries as canonical phase space transformations. He applied the formalism to a q-number version of tetrad gravity in interaction with both the electromagnetic field and a spinorial Dirac electron matter field. His procedure predated by almost two decades the algorithms of Dirac and Bergmann, and with regard to internal (non-spacetime) symmetries is fully equivalent to them. Dirac was in fact already in 1932 familiar with Rosenfelds work, although as far as I can tell he never acknowledged in print his perhaps unconscious debt to Rosenfeld. I will review the general formalism, comparing and contrasting with the work of Dirac, Bergmann and his associates. Although Rosenfeld formulated a correct prescription for constructing the vanishing Hamiltonian generator of time evolution, he evidently did not succeed in carrying out the construction. Nor did he have the correct phase space generators of diffeomorphism-induced symmetry variations. He did not take into account that some of the Lagrangian symmetries are not projectable under the Legendre transformation to phase space.