Riemann–Silberstein vector
In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is a complex vector that combines the electric field E and the magnetic field B. History Heinrich Martin Weber published the fourth edition of "The partial differential equations of mathematical physics according to Riemann's lectures" in two volumes (1900 and 1901). However, Weber pointed out in the preface of the first volume (1900) that this fourth edition was completely rewritten based on his own lectures, not Riemann's, and that the reference to "Riemann's lectures" only remained in the title because the overall concept remained the same and that he continued the work in Riemann's spirit. In the second volume (1901, §138, p. 348), Weber demonstrated how to consolidate Maxwell’s equations using . The real and imaginary components of the equation are an interpretation of Maxwell’s equations without charges or currents. It was independently rediscovered and further developed by Ludwik Silberstein in 1907. Definition Given an electric field E and a magnetic field B defined on a common region of spacetime, the Riemann–Silberstein vector is where is the speed of light, with some authors preferring to multiply the right hand side by an overall constant , where is the permittivity of free space. It is analogous to the electromagnetic tensor F, a 2-vector used in the covariant formulation of classical electromagnetism. In Silberstein's formulation, i was defined as the imaginary unit, and F was defined as a complexified 3-dimensional vector field, called a bivector field. Application The Riemann–Silberstein vector is used as a point of reference in the geometric algebra formulation of electromagnetism. Maxwell's four equations in vector calculus reduce to one equation in the algebra of physical space: Expressions for the fundamental invariants and the energy density and momentum density also take on simple forms: where S is the Poynting vector. The Riemann–Silberstein vector is used for an exact matrix representations of Maxwell's equations in an inhomogeneous medium with sources. Photon wave function In 1996 contribution to quantum electrodynamics, Iwo Bialynicki-Birula used the Riemann–Silberstein vector as the basis for an approach to the photon, noting that it is a "complex vector-function of space coordinates r and time t that adequately describes the quantum state of a single photon". To put the Riemann–Silberstein vector in contemporary parlance, a transition is made: With the advent of spinor calculus that superseded the quaternionic calculus, the transformation properties of the Riemann-Silberstein vector have become even more transparent ... a symmetric second-rank spinor. Bialynicki-Birula acknowledges that the photon wave function is a controversial concept and that it cannot have all the properties of Schrödinger wave functions of non-relativistic wave mechanics. Yet defense is mounted on the basis of practicality: it is useful for describing quantum states of excitation of a free field, electromagnetic fields acting on a medium, vacuum excitation of virtual positron-electron pairs, and presenting the photon among quantum particles that do have wave functions.