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Lorenz gauge : ウィキペディア英語版
Lorenz gauge condition
In electromagnetism, the Lorenz gauge or Lorenz gauge condition is a partial gauge fixing of the electromagnetic vector potential. The condition is that \partial_\mu A^\mu=0. This does not completely fix the gauge: one can still make a gauge transformation A^\mu\to A^\mu+\partial^\mu f where f is a harmonic scalar function (that is, a scalar function satisfying \partial^\mu\partial_\mu f=0, the equation of a massless scalar field).
The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2,1/2) representation of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.
The Lorenz condition is named after Ludvig Lorenz. It is a Lorentz invariant condition, and is frequently called the "Lorentz condition" because of confusion with Hendrik Lorentz, after whom Lorentz covariance is named.
==Description==
In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials.〔 and (【引用サイトリンク】url=http://www.physics.princeton.edu/~mcdonald/examples/jefimenko.pdf ).〕 The condition is
:\partial_A^\mu \equiv A^\mu} + \frac\frac=0.
where \vec is the magnetic vector potential and \,\varphi is the electric potential; see also Gauge fixing.
In Gaussian units the condition is:
:\nabla\cdot\frac=0.
A quick justification of the Lorenz gauge can be found using Maxwell's equations and the relation between the magnetic vector potential and the magnetic field:
:\nabla\times\vec=-\frac=\nabla\times-\frac
Therefore,
:\nabla\times(\vec+\frac)=0
Since the curl is zero, that means there is a scalar function \,\varphi such that -\nabla\,\varphi=\vec+\frac. This gives the well known equation for the electric field, \vec=-\nabla\,\varphi-\frac. This result can be plugged into another one of Maxwell's equations,
: )}=}=\frac}=\nabla\frac-\frac\frac}
This leaves,
:\nabla(\nabla\cdot\vec+\frac\frac)=\mu_0\vec-\frac\frac+\nabla^2\vec
To have Lorentz invariance, the time derivatives and spatial derivatives must be treated equally (i.e. of the same order). Therefore it is convenient to choose the Lorenz gauge condition, which gives the result
:\Box \vec=\left() \varphi = \frac\rho\,.
These are simpler and more symmetric forms of the inhomogeneous Maxwell's equations. Note that the Coulomb gauge also fixes the problem of Lorentz invariance, but leaves a coupling term with first-order derivatives.
Here c=\frac and \vec B=\nabla\times \vec A\,. The explicit solutions for \,\varphi and \vec A – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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