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Register a new userMaxwell electromagnetic theory from a viewpoint of differential forms
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In this paper, we discuss the Maxwell equations in terms of differential forms, both in the 3-dimensional space and in the 4-dimensional space-time manifold. Further, we view the classical electrodynamics as the curvature of a line bundle, and fit it into gauge theory.
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We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same $_2F_1$ hypergeometric function with different rational pullbacks. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus $_2F_1$ hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that emerged in a paper by Casale. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to $_3F_2$, hypergeometric functions, and show that one just reduces to the previous $_2F_1$ cases through a Clausen identity. In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or $_2F_1$ hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.
1) The differential equation considered in terms of exterior differential forms, as E.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet? 2) Why criteria of formal integrability of differential equations are never used in practice?
Invariance properties of classes in the variational sequence suggested to Krupka et al. the idea that there should exist a close correspondence between the notions of variationality of a differential form and invariance of its exterior derivative. It was shown by them that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form, so that the latter is locally the Euler--Lagrange form of a Lagrangian. We show that the corresponding local system of Euler--Lagrange forms is variationally equivalent to a global Euler--Lagrange form.
In the framework of J-bundles a vielbein formulation of unified Einstein-Maxwell theory is proposed. In the resulting scheme, field equations matching the gravitational and electromagnetic fields are derived by constraining a five-dimensional variational principle. No dynamical scalar field is involved.
The method, proposed in the given work, allows the application of well developed standard methods used in quantum mechanics for approximate solution of the systems of ordinary linear differential equations with periodical coefficients.
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