Toward a quantum theory of gravity: Syracuse 1949-1962

TL;DR

This paper reviews the historical development of canonical quantum gravity from 1949 to 1962, highlighting key ideas, challenges, and the evolving understanding of spacetime symmetry in the Hamiltonian formulation of Einstein's theory.

Contribution

It provides a detailed historical analysis of the early canonical quantization efforts and the conceptual debates surrounding spacetime invariance in quantum gravity.

Findings

Early efforts focused on constructing classical invariants.

The Hamiltonian approach faced challenges in preserving full spacetime symmetry.

The ADM and Wheeler-DeWitt programs abandoned full four-dimensional covariance.

Abstract

Peter Bergmann and his students embarked in 1949 on a mainly canonical quantization program whose aim was to take into account the underlying four-dimensional diffeomorphism symmetry in the transition from a Lagrangian to a Hamiltonian formulation of Einstein's theory. They argued that even though one seemed to destroy the full covariance through the focus on a temporal foliation of spacetime, this loss was illusory. Early on they convinced themselves that only the construction of classical invariants could adequately reflect the fully relativistic absence of physical meaning of spacetime coordinates. Efforts were made by Bergmann students Newman and Janis to construct these classical invariants. Then in the late 1950's Bergmann and Komar proposed a comprehensive program in which classical invariants could be constructed using the spacetime geometry itself to fix intrinsic spacetime landmarks. At roughly the same time Dirac formulated a new criterion for identifying initial phase space variables, one of whose consequences was that Bergmann himself abandoned the gravitational lapse and shift as canonical variables. Furthermore, Bergmann in 1962 interpreted the Dirac formalism as altering the very nature of diffeomorphism symmetry. One class of infinitesimal diffeomorphism was to be understood as depending on the perpendicular to the given temporal foliation. Thus even within the Bergmann school program the preservation of the full four-dimensional symmetry in the Hamiltonian program became problematic. Indeed, the ADM and associated Wheeler-DeWitt program that gained and has retained prominence since this time abandoned the full symmetry. There do remain dissenters - raising the question whether the field of quantum gravity has witnessed a Renaissance in the ensuing decades - or might the full four-dimensional symmetry yet be reborn?