Abstract
This paper studies the relativistic angular momentum for the generalized electromagnetic field, described by r-vectors in (k, n) space-time dimensions, with exterior-algebraic methods. First, the angular-momentum tensor is derived from the invariance of the Lagrangian to space-time rotations (Lorentz transformations), avoiding the explicit need of the canonical tensor in Noether’s theorem. The derivation proves the conservation law of angular momentum for generic values of r, k, and n. Second, an integral expression for the flux of the tensor across a (k+n-1)-dimensional surface of constant ell -th space-time coordinate is provided in terms of the normal modes of the field; this analysis is a natural generalization of the standard analysis of electromagnetism, i. e. a three-dimensional space integral at constant time. Third, a brief discussion on the orbital angular momentum and the spin of the generalized electromagnetic field, including their expression in complex-valued circular polarizations, is provided for generic values of r, k, and n.
Highlights
The Maxwell equations can be derived by an application of the principle of stationary action [5, Ch. 19], [3, Sect. 8]
; the complex-valued paper, we provide an analogous formula for the angular-momentum flux and its split into center-of-motion, orbital angular momentum, and spin components, as described
In the presence of sources, the divergence ∂ × Mα can be seen as an angularmomentum density, and the volume integral of ∂ × Mα across an (k + n)-dimensional hypervolume Vk+n gives the transfer of relativistic angular momentum from the field to the sources in the volume
Summary
E0 to ek−1 (resp. ek to ek+n−1) and have metric −1 (resp. +1). The generalized Maxwell equations for arbitrary r , k, and n are the following pair of coupled differential equations:. The action is a quantity given by the integral over a (k + n)-dimensional space-time of a scalar Lagrangian density L(x). The Lagrangian density L is expressed in terms of the multivector dot (scalar) product [1, Sect. The Euler–Lagrange equations for the Lagrangian density L in (4) give the Maxwell equation (1) as vector derivatives of L with respect to the potential A and its exterior derivative ∂ A, namely [6, Sect. If we replace the potential A by a new field A = A + A + ∂ ∧ G, where Ais a constant (r − 1)-vector and G is an (r − 2)-vector gauge field, the homogenous Maxwell equation (2). For a given Maxwell field, there is some unavoidable (gauge) ambiguity on the value of the vector potential if r ≥ 2.
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