Lorenz gauge condition

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 $${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }\equiv A^{\mu }{}_{,\mu }=0,}$$ where ${\displaystyle A^{\mu }}$ 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.

In ordinary vector notation and SI units, the condition is $${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}=0,}$$ where ${\displaystyle \mathbf{A}}$ is the magnetic vector potential and ${\displaystyle \phi }$ is the electric potential; see also Gauge fixing.

In Gaussian units the condition is $${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}=0.}$$

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: $${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}=-\frac{\mathrm{\partial }(\mathrm{\nabla }\times \mathbf{A})}{\mathrm{\partial }t}}$$

Therefore, $${\displaystyle \mathrm{\nabla }\times \left(\mathbf{E}+\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}\right)=0.}$$

Since the curl is zero, that means there is a scalar function ${\displaystyle \phi }$ such that $${\displaystyle -\mathrm{\nabla }\phi =\mathbf{E}+\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}.}$$

This gives a well known equation for the electric field: $${\displaystyle \mathbf{E}=-\mathrm{\nabla }\phi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}.}$$

This result can be substituted in the Ampère–Maxwell equation, $${\displaystyle \begin{array}{cc}\mathrm{\nabla }\times \mathbf{B}& ={\mu }_0\mathbf{J}+\frac{1}{c^2}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}\\ \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathbf{A}\right)& ={\mu }_0\mathbf{J}+\frac{1}{c^2}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(-\mathrm{\nabla }\phi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}\right)\\ \Rightarrow \mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{A}\right)-{\mathrm{\nabla }}^2\mathbf{A}& ={\mu }_0\mathbf{J}-\frac{1}{c^2}\frac{\mathrm{\partial }(\mathrm{\nabla }\phi )}{\mathrm{\partial }t}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}.\end{array}}$$

This leaves $${\displaystyle \mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}\right)={\mu }_0\mathbf{J}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}+{\mathrm{\nabla }}^2\mathbf{A}.}$$

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 makes the left hand side zero and gives the result $${\displaystyle ◻\mathbf{A}=\left[\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\right]\mathbf{A}={\mu }_0\mathbf{J}.}$$

A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield $${\displaystyle ◻\phi =\left[\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2\right]\phi =\frac{1}{{\epsilon }_0}\rho .}$$

These are simpler and more symmetric forms of the inhomogeneous Maxwell's equations.

Here $${\displaystyle c=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}}$$ is the vacuum velocity of light, and ${\displaystyle ◻}$ is the d'Alembertian operator with the (+ − − −) metric signature. These equations are not only valid under vacuum conditions, but also in polarized media, if ${\displaystyle \rho }$ and ${\displaystyle \mathbf{J}}$ are source density and circulation density, respectively, of the electromagnetic induction fields ${\displaystyle \mathbf{E}}$ and ${\displaystyle \mathbf{B}}$ calculated as usual from ${\displaystyle \phi }$ and ${\displaystyle \mathbf{A}}$ by the equations $${\displaystyle \begin{array}{cc}\mathbf{E}& =-\mathrm{\nabla }\phi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}\\ \mathbf{B}& =\mathrm{\nabla }\times \mathbf{A}\end{array}}$$

The explicit solutions for ${\displaystyle \phi }$ and ${\displaystyle \mathbf{A}}$ – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.