Noncommutative Geometry Framework and The Feynman's Proof of Maxwell Equations

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 $m[x_j,\dot{x_k}]=i\hslash {\delta }_{jk}+im{\theta }_{jk}f.$ 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 $di{v}_{\theta }B={\eta }_{\theta }$ and $\frac{\partial B_s}{\partial t}+{\epsilon }_{kjs}\frac{\partial E_j}{\partial x_k}=A_1\frac{d^{2}f}{d{t}^2}+A_2\frac{df}{dt}+A_3,$ 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 $di{v}_{\theta }B={\eta }_{\theta }$ is not vanishing in general.

LaTeX file, 16 pages

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• Quantum Field Theory on Noncommutative Spaces
• Feynman's derivation of Maxwell equations and extra dimensions
• Covariantly Quantized Spinning Particle and its Possible Connection to Non-Commutative Space-Time
• Supersymmetric Moyal-Lax Representations
• Effective Field Theory for Noncommutative Spacetime: A Toy Model
• Modelling a Noncommutative Two-Brane
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Cited by in corpus (5)

• Electrodynamics on $\kappa$-Minkowski space-time
• Geodesic equation in $k$-Minkowski spacetime
• Feynman problem in the noncommutative case
• Equations of motion, Noncommutativity and Quantization
• Maxwell's equations and Lorentz force in doubly special relativity