Lorenz gauge condition explained

In electromagnetism, the Lorenz gauge condition or Lorenz gauge (after Ludvig Lorenz) is a partial gauge fixing of the electromagnetic vector potential by requiring

\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 ^{\mu} \mapsto A ^{\mu} + \partial ^{\mu} f, where \partial ^{\mu} is the four-gradient and f is any harmonic scalar function: that is, a scalar function obeying \partial ^{\mu} _{\mu\partial} 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$$\backslash partial\_\backslash mu\; A^\backslash mu\; \backslash equiv\; A^\backslash mu\_\; =\; 0,$$where

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$$\backslash nabla\backslash cdot\; +\; \backslash frac\backslash frac\; =\; 0,$$where

A is the magnetic vector potential and \varphi is the electric potential;^{[1] [2]} see also gauge fixing.

In Gaussian units the condition is^{[3] [4]} $$\backslash nabla\backslash cdot\; +\; \backslash frac\backslash frac\; =\; 0.$$

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:$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\; -\backslash frac\; =\; -\; \backslash frac$$

Therefore,$$\backslash nabla\; \backslash times\; \backslash left(\backslash mathbf\; +\; \backslash frac\backslash right)\; =\; 0.$$

Since the curl is zero, that means there is a scalar function

\varphi such that $$-\backslash nabla\backslash varphi\; =\; \backslash mathbf\; +\; \backslash frac.$$

This gives a well known equation for the electric field:$$\backslash mathbf\; =\; -\backslash nabla\; \backslash varphi\; -\; \backslash frac.$$

This result can be plugged into the Ampère–Maxwell equation,

This leaves$$\backslash nabla\backslash left(\backslash nabla\; \backslash cdot\; \backslash mathbf\; +\; \backslash frac\backslash frac\backslash right)\; =\; \backslash mu\_0\backslash mathbf\; -\; \backslash frac\; \backslash frac\; +\; \backslash nabla^2\backslash mathbf.$$

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$$\backslash Box\backslash mathbf\; =\; \backslash left[\backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; -\; \backslash nabla^2\; \backslash right]\backslash mathbf\; =\backslash mu\_0\backslash mathbf.$$

A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield$$\backslash Box\backslash varphi\; =\; \backslash left[\backslash frac\{1\}\{c^2\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; t^2\}\; -\; \backslash nabla^2\; \backslash right]\; \backslash varphi\; =\; \backslash frac\backslash rho\; .$$

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

Here$$c\; =\; \backslash frac$$ is the vacuum velocity of light, and

\Box is the d'Alembertian operator with the metric signature. These equations are not only valid under vacuum conditions, but also in polarized media,^[5] if \rho and \vec{J} are source density and circulation density, respectively, of the electromagnetic induction fields \vec{E} and \vec{B} calculated as usual from \varphi and \vec{A} by the equations

The explicit solutions for

\varphi and A – unique, if all quantities vanish sufficiently fast at infinity – are known as retarded potentials.

History

When originally published in 1867, Lorenz's work was not received well by James Clerk Maxwell. Maxwell had eliminated the Coulomb electrostatic force from his derivation of the electromagnetic wave equation since he was working in what would nowadays be termed the Coulomb gauge. The Lorenz gauge hence contradicted Maxwell's original derivation of the EM wave equation by introducing a retardation effect to the Coulomb force and bringing it inside the EM wave equation alongside the time varying electric field, which was introduced in Lorenz's paper "On the identity of the vibrations of light with electrical currents". Lorenz's work was the first use of symmetry to simplify Maxwell's equations after Maxwell himself published his 1865 paper. In 1888, retarded potentials came into general use after Heinrich Rudolf Hertz's experiments on electromagnetic waves. In 1895, a further boost to the theory of retarded potentials came after J. J. Thomson's interpretation of data for electrons (after which investigation into electrical phenomena changed from time-dependent electric charge and electric current distributions over to moving point charges).

See also

• Gauge fixing

External links and further reading

General

• Web site: Weisstein . E. W. . Lorenz Gauge . Wolfram Research.

Further reading

• Lorenz . L. . 1867 . On the Identity of the Vibrations of Light with Electrical Currents . . Series 4 . 34 . 230 . 287–301.
• van Bladel . J. . 1991 . Lorenz or Lorentz? . . 33 . 2 . 69 . 10.1109/MAP.1991.5672647. 21922455 .
  • See also Bladel . J. . 1991 . Lorenz or Lorentz? [Addendum] . . 33 . 4 . 56 . 1991IAPM...33...56B . 10.1109/MAP.1991.5672657.
• Book: Becker, R. . 1982 . Electromagnetic Fields and Interactions . Chapter 3 . Dover Publications.
• Book: O'Rahilly, A. . 1938 . Electromagnetics . Chapter 6 . Longmans, Green and Co.

History

• Nevels . R. . Shin . Chang-Seok . 2001 . Lorenz, Lorentz, and the gauge . . 43 . 3 . 70–71. 2001IAPM...43...70N . 10.1109/74.934904 . .
• Book: Whittaker, E. T.. E. T. Whittaker . 1989 . . 1–2 . . 268.

Notes and References

1. Book: Jackson, John David. Classical Electrodynamics. 3rd. 240. John Wiley & Sons. 978-0-471-30932-1. 1999 . Classical Electrodynamics (book).
2. Book: Keller, Ole. Quantum Theory of Near-Field Electrodynamics. 2012-02-02. Springer Science & Business Media. 9783642174100 . 19. en. 2011qtnf.book.....K.
3. Book: Gbur, Gregory J. . Greg Gbur . Mathematical Methods for Optical Physics and Engineering . 2011. 59. Cambridge University Press. 978-0-521-51610-5. 2011mmop.book.....G.
4. Book: Heitler, Walter . The Quantum Theory of Radiation. Walter Heitler . 1954. Courier Corporation. 9780486645582. 3. en .
5. For example, see hep-th/0310036. Riemann-Silberstein representation of the complete Maxwell equations set. Cheremisin. M. V.. Okun. L. B.. 2003.