Videnskabshistorisk Selskab

Donald Salisbury,
Physics Department, Austin College, Sherman, Texas
Max-Planck-Institut für Wissenschaftsgeschichte, Berlin

Leon Rosenfeld invented indeterminate gauge dynamics in 1930

In 1929 and 1930 Heisenberg and Pauli published groundbreaking papers presenting schemes for quantizing the electromagnetic field in interaction with a second-quantized Dirac electron matter field. At the same time Fermi independently proposed his own distinct approach. The principle problem that needed to be addressed was how do deal with the identical vanishing of the momentum conjugate to the temporal component of the electromagnetic potential. Heisenberg and Pauli invented two procedures, each of which destroyed the manifest symmetry under gauge transformations of the potential and matter field. Pauli in particular expressed doubts about the validity of the method. Fermi's construction destroyed manifest Lorentz symmetry.

In 1929 Leon Rosenfeld was working with Pauli in Zurich. With Pauli's encouragement and assistance he was able to show that no unphysical interaction with longitudinal components of the electromagnetic field resulted from the first Heisenberg-Pauli modified Lagrangian. But neither Pauli or Rosenfeld were satisfied with the seemingly ad hoc suggestions for dealing with the vanishing momentum. Again at Pauli's urging and with an apparently essential (though unidentified) idea from Pauli, Rosenfeld undertook the first formal investigation of the nature of local symmetries under the transition from Lagrangian to Hamiltonian dynamics. The resulting analysis was published in Annalen der Physik in 1930. Rosenfeld was able to show that both the Heisenberg-Pauli, and Fermi methods were mathematically justified. In the process he invented a significant portion of the machinary of constrained Hamiltonian dynamics, now the basis of all canonical treatments of local gauge theories, including Yang-Mills, gravity, and superstrings. Furthermore, Rosenfeld treated as an example Dirac matter fields and electromagnetism in interaction with general relativity. It was primarily in this domain in 1949 that Dirac and Bergmann began to produce the constrained Hamiltonian algorithm that now bears their name.

One naturally wonders why Rosenfeld's contribution has not received the attention that it merits. It was surely of crucial interest to Pauli, and demonstrably also to Dirac. In fact, after having satisfied himself of the validity of the Fermi procedure, Rosenfeld published in 1932 a demonstration that Dirac's relativistic electron theory was equivalent to that of Heisenberg and Pauli. Dirac was indeed in correspondence with Rosenfeld before Dirac, Fock, and Podolsky published their "improved" demonstration of this equivalence. Dirac had written Rosenfeld asking him detailed questions about the appearance of arbitrary functions in Rosenfeld's constraint analysis.

It appears that Rosenfeld's contribution to quantum field theory simply came too early. It suffered the same fate in the coming two decades as did quantum field theory itself. Coming to terms with the apparently insurmountable infinities that plagued quantum electrodynamics, Rosenfeld himself declared in his 1932 overview that there was no longer any reason to think that second quantization could lead to an acceptable theory of interaction of material particles. This may also explain why Rosenfeld's contribution was initially unknown to Peter Bergmann in 1949 when he set himself the task of inventing a procedure for tracking the indeterminate time evolution of the gravitational field.

tirsdag, den 6. maj 2008, kl. 17.00

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