Article

# Dirac monopole with Feynman brackets

Massachusetts Institute of Technology, Cambridge, Massachusetts, United States

Physics Letters A (Impact Factor: 1.68). 04/2000; 254(3-4). DOI: 10.1016/S0375-9601(99)00016-X Source: arXiv

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**ABSTRACT:**In 1992, Dyson published Feynman's proof of the homogeneous Maxwell equations assuming only the Newton's law of motion and the commutation relations between position and velocity for a nonrelativistic particle. Recently Tanimura gave a generalization of this proof in a relativistic context. Using the Hodge duality we extend his approach in order to derive the two groups of Maxwell equations with a magnetic monopole in flat and curved spaces. (C) 1999 American Institute of Physics. [S0022-2488(99)00408-9].Journal of Mathematical Physics 07/1999; 40(8):3732-3737. DOI:10.1063/1.532923 · 1.24 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The dynamic of a classical system can be expressed by means of Poisson brackets. In this paper we generalize the relation between the usual non covariant Hamiltonian and the Poisson brackets to a covariant Hamiltonian and new brackets in the frame of the Minkowski space. These brackets can be related to those used by Feynman in his derivation of Maxwell's equations. The case of curved space is also considered with the introduction of Christoffel symbols, covariant derivatives, and curvature tensors.International Journal of Theoretical Physics 05/2000; 39(4). DOI:10.1023/A:1003654525047 · 1.18 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The main focus of the present work is to study the Feynman's proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman's approach. This allows us to go beyond the standard case and discover non-trivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption $m[x_{j},\dot{x_{k}}]=i\hbar \delta_{jk}+im\theta_{jk}f.$ The results extracted from the second formulation are more significant since they are associated to a non trivial $\theta $-extension of the Bianchi-set of Maxwell equations. We find $div_{\theta}B=\eta_{\theta}$ and $\frac{\partial B_{s}}{\partial t}+\epsilon_{kjs}\frac{\partial E_{j}}{\partial x_{k}}=A_{1}\frac{d^{2}f}{dt^{2}}+A_{2}\frac{df}{dt}+A_{3},$ where $\eta_{\theta}$, $A_{1}$, $A_{2}$ and $A_{3}$ are local functions depending on the NC $\theta $-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charges sources that we can associate to the magnetic monopole since $div_{\theta}B=\eta_{\theta}$ is not vanishing in general. Comment: LaTeX file, 16 pagesJournal of Mathematical Physics 08/2003; 44(12). DOI:10.1063/1.1625891 · 1.24 Impact Factor