In electromagnetism, a branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers, and vector calculus.
[1] A single equation using 4 × 4 matrices is necessary and sufficient for any homogeneous medium.
[2] Maxwell's equations in the standard vector calculus formalism, in an inhomogeneous medium with sources, are:[3] The media is assumed to be linear, that is where scalar
× 10−7 H·m−1 One possible way to obtain the required matrix representation is to use the Riemann–Silberstein vector[4][5] given by If for a certain medium
satisfy Thus by using the Riemann–Silberstein vector, it is possible to reexpress the Maxwell's equations for a medium with constant
In order to obtain a single matrix equation instead of a pair, the following new functions are constructed using the components of the Riemann–Silberstein vector[6] The vectors for the sources are Then, where * denotes complex conjugation and the triplet, M = [Mx, My, Mz] is a vector whose component elements are abstract 4×4 matricies given by The component M-matrices may be formed using: where
Each of the four Maxwell's equations are obtained from the matrix representation.
This is done by taking the sums and differences of row-I with row-IV and row-II with row-III respectively.
Moreover, they satisfy the usual (quaternion-like) algebra of the Dirac matrices, including, The (Ψ±, M) are not unique.
[5] The Riemann–Silberstein vector is well known in classical electrodynamics and has certain interesting properties and uses.
[5] In deriving the above 4×4 matrix representation of the Maxwell's equations, the spatial and temporal derivatives of ε(r, t) and μ(r, t) in the first two of the Maxwell's equations have been ignored.
The exceptional ones are the ones that contain the three components of w(r, t), the logarithmic gradient of the resistance function.
The Maxwell's equations have been expressed in a matrix form for a medium with varying permittivity ε = ε(r, t) and permeability μ = μ(r, t), in presence of sources.
Moreover, the exact matrix representation has an algebraic structure very similar to the Dirac equation.
[2] Maxwell's equations can be derived from the Fermat's principle of geometrical optics by the process of "wavization"[clarification needed] analogous to the quantization of classical mechanics.
The matrix form of the Maxwell's equations is used as a candidate for the Photon Wavefunction.
[8] Historically, the geometrical optics is based on the Fermat's principle of least time.
Geometrical optics can be completely derived from the Maxwell's equations.
The derivation of the Helmholtz equation from the Maxwell's equations is an approximation as one neglects the spatial and temporal derivatives of the permittivity and permeability of the medium.
A new formalism of light beam optics has been developed, starting with the Maxwell's equations in a matrix form: a single entity containing all the four Maxwell's equations.
Such a prescription is sure to provide a deeper understanding of beam-optics and polarization in a unified manner.
[9] The beam-optical Hamiltonian derived from this matrix representation has an algebraic structure very similar to the Dirac equation, making it amenable to the Foldy-Wouthuysen technique.
[10] This approach is very similar to one developed for the quantum theory of charged-particle beam optics.