Peter Bergmann and the Invention of Constrained Hamiltonian Dynamics

Abstract

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.