From Lorenz to Coulomb and other explicit gauge transformations
Abstract
The main purposes of this paper are (i) to illustrate explicitly by a number of examples the gauge functions χ(x,t) whose spatial and temporal derivatives transform one set of electromagnetic potentials into another equivalent set; and (ii) to show that, whatever propagation or nonpropagation characteristics are exhibited by the potentials in a particular gauge, the electric and magnetic fields are always the same and display the experimentally verified properties of causality and propagation at the speed of light. The example of the transformation from the Lorenz gauge (retarded solutions for both scalar and vector potential) to the Coulomb gauge (instantaneous, actionatadistance, scalar potential) is treated in detail. A transparent expression is obtained for the vector potential in the Coulomb gauge, with a finite nonlocality in time replacing the expected spatial nonlocality of the transverse current. A class of gauges (vgauge) is described in which the scalar potential propagates at an arbitrary speed ν relative to the speed of light. The Lorenz and Coulomb gauges are special cases of the vgauge. The last examples of gauges and explicit gauge transformation functions are the Hamiltonian or temporal gauge, the nonrelativistic Poincaré or multipolar gauge, and the relativistic FockSchwinger gauge.
 Publication:

American Journal of Physics
 Pub Date:
 September 2002
 DOI:
 10.1119/1.1491265
 arXiv:
 arXiv:physics/0204034
 Bibcode:
 2002AmJPh..70..917J
 Keywords:

 01.50.i;
 41.20.q;
 Educational aids;
 Applied classical electromagnetism;
 Physics  Classical Physics;
 High Energy Physics  Phenomenology;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 pdf file, 30 pages, vgauge references added, submitted for publication