Lorenz gauge condition
incomplete, Lorentz-invariant gauge condition setting the four-divergence of the electromagnetic four-potential to zero
In electromagnetism, the Lorenz gauge condition or Lorenz gauge (after Ludvig Lorenz) is a partial gauge fixing of the electromagnetic vector potential by requiring
∂
μ
A
μ
=
0
{\displaystyle \partial _{\mu }A^{\mu }=0}
. The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The Lorenz gauge condition does not completely determine the gauge: one can still make a gauge transformation
A
μ
↦
A
μ
+
∂
μ
f
{\displaystyle A^{\mu }\mapsto A^{\mu }+\partial ^{\mu }f}
, where
∂
μ
{\displaystyle \partial ^{\mu }}
is the four-gradient and
f
{\displaystyle f}
is any harmonic scalar function: that is, a scalar function obeying
∂
μ
∂
μ
f
=
0
{\displaystyle \partial _{\mu }\partial ^{\mu }f=0}
, the equation of a massless scalar field.
The Lorenz gauge condition is used to eliminate the redundant spin-0 component in Maxwell's equations when these are used to describe a massless spin-1 quantum field. It is also used for massive spin-1 fields where the concept of gauge transformations does not apply at all.
Description
In electromagnetism, the Lorenz condition is generally used in calculations of time-dependent electromagnetic fields through retarded potentials. The condition is
∂
μ
A
μ
≡
A
μ
,
μ
=
0
,
{\displaystyle \partial _{\mu }A^{\mu }\equiv A^{\mu }{}_{,\mu }=0,}
where
A
μ
{\displaystyle A^{\mu }}
is the four-potential, the comma denotes a partial differentiation and the repeated index indicates that the Einstein summation convention is being used. The condition has the advantage of being Lorentz invariant. It still leaves substantial gauge degrees of freedom.
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