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Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations in four-dimensional Minkowski space The, Dennis

Abstract

We derive new nonlocal symmetries and corresponding new nonlocal conservation laws for the free-space Maxwell's equations (ME) in 3+1 space-time dimensions. These results arise from a detailed point symmetry analysis of several potential systems associated with ME: (1) LPS: the standard Lagrangian potential system, (2) A-LPS: LPS augmented by the Lorentz gauge, (3) PS: a natural non-Lagrangian potential system with dual vector potentials, and (4) A-PS: PS augmented by dual Lorentz gauges. The well-known (local) space-time symmetries and (local) energy-momentum conservation laws are shown to be recovered from the point symmetries of LPS and A-LPS. The point symmetry structure of A-PS is much richer: inversion and vector rotation / boost symmetries arise which yield nonlocal symmetries of ME. Through an embedding of the set of local PS and A-PS symmetries into the set of (local and nonlocal) ME adjoin-symmetries, an explicit formula is given for constructing ME conservation laws from PS and A-PS symmetries. The resulting ME conservation laws are completely classified with respect to their locality or nonlocality. This classification relies on an important cohomology result for ME for which a (partial) proof is given using tensorial methods. The work in this thesis extends the previously known classes of symmetries and conservation laws admitted by ME, and emphasizes the utility of potential systems and gauge constraints in their construction.

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