Feynman’s proof of the Maxwell equations and the Lorentz force equations

Idea

Starting with given Poisson brackets, Feynman posed the problem to determine the most general second-order dynamical system compatible with a prescribed set of fundamental Poisson brackets. In other words, the method employs consistency constraints and no action principle (or, alternatively, Hamiltonian approach) is assumed. The problem is related with the inverse problem in calculus of variations.

History

Feynman’s alternative derivation of the Maxwell equations and the Lorentz force equations (shown in 1948 to Freeman Dyson), has been conceived as a witty joke. Dyson has published a treatment of the “joke” in 1990, and since that time it has been of much interest and various generalizations have been thought of to encompass nonabelian gauge theories, gravity and so on.

References

F.J. Dyson, Feynman’s proof of the Maxwell equations, Amer. J. Phys. 58, 209-211 (1990), MR90m:78005.

J.F. Carinena, L.A. Ibort, G. Marmo, A. Stern, The Feynman problem and the inverse problem for Poisson dynamics, Physics Reports 263, n.3, 1995 , pp. 153-212(60), doi

Shogo Tanimura, Relativistic generalization and extension to the nonabelian gauge theory of Feynman’s proof of the Maxwell equations, Ann. Physics 220 (1992), no. 2, 229–247, doi; Relativistic generalization of Feynman’s proof of the Maxwell equations, Proc. Int. Symp. on Quantum Physics and the Universe (Tokyo, 1992). Vistas Astronom. 37 (1993), no. 1-4, 329–332, doi.

M. C. Land, N. Shnerb, L. P. Horwitz, On Feynman’s approach to the foundations of gauge theory, J. Math. Phys. 36 (1995), no. 7, 3263–3288, MR96e:81110, doi, pdf

S. A. Hojman, L. C. Shepley, No Lagrangian? No quantization!, J. Math. Phys. 32 (1991), no. 1, 142–146, MR92a:81079, doi

A Bérard, H Mohrbach, J Lages, P Gosselin, Y Grandati, H Boumrar, F Ménas, From Feynman proof of Maxwell equations to noncommutative quantum mechanics, J. Phys.: Conf. Ser. 70 012004, arxiv/0706.2751, doi; J. Lages, A. Bérard, H. Mohrbach, Y. Grandati, P. Gosselin, Noncommutative quantum nechanics viewed from Feynman formalism, arxiv/0709.0816

Last revised on August 11, 2010 at 16:07:53.
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