L\'eon Rosenfeld's general theory of constrained Hamiltonian dynamics

TL;DR

This paper reviews Léon Rosenfeld's 1930 work on phase-space constraints, highlighting his pioneering methods in Hamiltonian dynamics, symmetries, and conservation laws, with historical context and recent insights.

Contribution

It clarifies Rosenfeld's original contributions to constrained Hamiltonian dynamics and uncovers overlooked aspects still relevant today.

Findings

Rosenfeld derived symmetry generators without explicit time evolution reference.

He applied his theory to Einstein-Maxwell-Dirac systems.

Many of Rosenfeld's discoveries predate later re-discoveries by decades.

Abstract

This commentary reflects on the 1930 discoveries of L\'eon Rosenfeld in the domain of phase-space constraints. We start with a short biography of Rosenfeld and his motivation for this article in the context of ideas pursued by W. Pauli, F. Klein, E. Noether. We then comment on Rosenfeld's General Theory dealing with symmetries and constraints, symmetry generators, conservation laws and the construction of a Hamiltonian in the case of phase-space constraints. It is remarkable that he was able to derive expressions for all phase space symmetry generators without making explicit reference to the generator of time evolution. In his Applications, Rosenfeld treated the general relativistic example of Einstein-Maxwell-Dirac theory. We show, that although Rosenfeld refrained from fully applying his general findings to this example, he could have obtained the Hamiltonian. Many of Rosenfeld's discoveries were re-developed or re-discovered by others two decades later, yet as we show there remain additional firsts that are still not recognized in the community.