en>Dcrjsr: many minor corrections of typos, word order, etc

2013-09-14T16:35:26Z

many minor corrections of typos, word order, etc

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{{electromagnetism}}

In [[mathematical physics]], in particular [[electromagnetism]], the '''Riemann–Silberstein vector''', named after [[Bernhard Riemann]] and [[Ludwik Silberstein]], (or sometimes ambiguously called the "electromagnetic field") is a [[complex number|complex]] [[vector space|vector]] that combines the [[electric field]] '''E''' and the [[magnetic field]] '''B'''.<ref>
{{Cite journal
|last=Silberstein |first=Ludwik
|year=1907
|title=Elektromagnetische Grundgleichungen in bivectorieller Behandlung
|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_em_equations_in_bivector_fomr.pdf
|journal=[[Annalen der Physik]]
|volume=327 |issue= 3|pages=579–586
|bibcode=1907AnP...327..579S
|doi=10.1002/andp.19073270313
}}</ref><ref>
{{Cite journal
|last=Silberstein |first=Ludwik
|year=1907
|title=Nachtrag zur Abhandlung über 'Elektromagnetische Grundgleichungen in bivectorieller Behandlung'
|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_addendum.pdf
|journal=[[Annalen der Physik]]
|volume=329 |pages=783–784
|bibcode=1907AnP...329..783S
|doi=10.1002/andp.19073291409
|issue=14
}}</ref>

==Definition==
Given an electric field '''E''' and a magnetic field '''B''' defined on a common [[region (mathematical analysis)|region]] of [[spacetime]], the Riemann–Silberstein vector is
: <math>\mathbf{F} = \mathbf{E} + ic \mathbf{B}</math>
where ''c'' is the [[speed of light]], with some authors preferring to multiply the right hand side by an overall constant <math>\sqrt{\epsilon_0 / 2}</math> where ε<sub>0</sub> 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]]. The value of '''F''' at an event was a [[bivector (complex)|bivector]].<ref>
{{cite journal
|last=Aste |first=Andreas
|year=2012
|title=Complex representation theory of the electromagnetic field
|journal=[[Journal of Geometry and Symmetry in Physics]]
|volume=28 |page=47-58
|doi=10.7546/jgsp-28-2012-47-58
|arxiv = 1211.1218 }}</ref>

==Application==

{{main|Maxwell's equations}}
Bernhard Riemann used <math>\mathfrak{E} + i\ \mathfrak{M}</math> to illustrate consolidation of Maxwell’s equations. According to lectures published by [[Heinrich Martin Weber]] in 1901, the real and imaginary components of the equation
:<math>c \operatorname{curl}(\mathfrak{E} + i\ \mathfrak{M}) = i\ \frac {\partial (\mathfrak{E} + i\ \mathfrak{M})} {\partial t} </math>
are an interpretation of Maxwell’s equations without charges or currents. Riemann’s lectures are available on-line from [[University of Michigan]] Historical Math Collection in Weber’s book, §138, S. 348, under the title ''Zusammenziehung der Maxwell’schen Gleichungen'' (Consolidation of Maxwell’s Equations).

The Riemann–Silberstein vector is used as a point of reference in the [[mathematical descriptions of the electromagnetic field#Geometric algebra (GA) formulation|geometric algebra formulation of electromagnetism]]. Maxwell's ''four'' equations in [[vector calculus]] reduce to ''one'' equation in the [[algebra of physical space]]:

:<math> \left(\frac{1}{c}\dfrac{\partial }{\partial t} + \boldsymbol{\nabla}\right)\mathbf{F} = \frac{1}{\epsilon_0}\left( \rho - \frac{1}{c}\mathbf{J} \right).</math>

Expressions for the [[classification of electromagnetic fields#Invariants|fundamental invariants]] and the [[energy density#Energy density of electric and magnetic fields|energy density]] and [[momentum]] density also take on simple forms:

:<math> \mathbf{F}^2 = \mathbf{E}^2 - c^2\mathbf{B}^2 + 2 i c\mathbf{E} \cdot \mathbf{B}</math>
:<math> \frac{\epsilon_0}{2}\mathbf{F}^{\dagger}\mathbf{F} = \frac{\epsilon_0}{2}\left( \mathbf{E}^2 + c^2\mathbf{B}^2 \right) + \frac{1}{c} \mathbf{S},</math>

where '''S''' is the [[Poynting vector]].

The Riemann–Silberstein vector is used for an [[Matrix Representation of Maxwell Equations|exact matrix representations of Maxwell's equations in an inhohogeneous medium with sources]].<ref>
{{cite journal
|last=Bialynicki-Birula |first=Iwo
|year=1996
|title=Photon wave function
|journal=[[Progress in Optics]]
|volume=36 |issue= |pages=245–294
|arxiv=quant-ph/0508202
|bibcode=
|doi=10.1016/S0079-6638(08)70316-0
|series=
|isbn=978-0-444-82530-8
}}</ref><ref>
{{cite journal
|last1=Khan |first1=Sameen Ahmed
|year=2005
|title=An Exact Matrix Representation of Maxwell's Equations
|journal=[[Physica Scripta]]
|volume=71 |issue=5 |pages=440
|arxiv=physics/0205083
|bibcode=2005PhyS...71..440K
|doi=10.1238/Physica.Regular.071a00440
}}</ref>

==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 equation|Schrödinger]] [[wave function]]s of non-relativistic wave mechanics. Yet defense is mounted on the basis of practicality: describing quantum states of excitation of a free field, electromagnetic fields acting on a medium, vacuum excitation of virtual positron-electron pairs, and to present the photon among quantum particles that do have wave functions.

==References==
{{reflist}}

==Further reading==
* [[Heinrich Martin Weber]] (1901) ''Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen'', S. 348, Braunscheweig: Friedrich Vieweg und Sohn, available from [http://quod.lib.umich.edu/u/umhistmath University of Michigan Historical Math Collection].

{{DEFAULTSORT:Riemann-Silberstein vector}}
[[Category:Electromagnetism]]
[[Category:Geometric algebra]]

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en>Dcrjsr: many minor corrections of typos, word order, etc