Gauge Theory of the Maxwell Equations

Gerhard W. Bruhn, Darmstadt University of Technology

 

Summary

In paraphysical contributions of the internet a misunderstood gauge theory of the Maxwell equations is used to produce "strange" effects, e.g. Tesla's longitudinal electromagnetic waves [2]. The reasons of that wrong calculations are deviations from a correct gauge theory, which is represented here in detail.

1. The physical basis

The Maxwell equations are experiment based relations between the electromagnetic observables E,B, J and ρ. We assume constant material coefficients ε and μ here. Let c² := 1/εμ.

(1.1)                                 curl E + Bt = 0                 (Faraday's Law),

(1.2)                           curl B1/Et = μJ                 (Amper's Law),

(1.3)                                         div E = ρ/ε                 (Gauss' Law 1),

(1.4)                                         div B = 0                   (Gauss' Law 2).

There is a hidden dependency between the densities ρ and J: By evaluating the combination div(1.2) + /∂t(1.3) we obtain

(1.5)                                 ρt + div J = 0                   (Conservation of charge),

i.e. the change of the density ρ is just caused by the sources of the current density J.


2. Potential representations and gauge transformations

Equation (1.3) means that the field B can be derived from a vector potential A:

(2.1)                                                 B = curl A.

By inserting (2.1) into (1.1) we obtain

(2.2)                                                 curl (E + At) = 0.

Therefore E + At is representable as some gradient

(2.3)                                                 E + At = − grad Φ ,

hence

(2.4)                                                 E = − (grad Φ + At) .

Thus, both E and B can be represented by means of some potentials A and Φ; and for each choice of A and Φ the Maxwell equations (1.1) and (1.4) are fulfilled. The identity

(2.4')                 E = − (grad Φ + At) = − [grad(Φ − Ψt) + (A + grad Ψ)t]

shows that the substitution

(2.5)                 Φ → Φ' := Φ − Ψt ,                 AA' := A + grad Ψ

generates equivalent potentials Φ', A' for the representation of the observables E, B for arbitrary functions Ψ. Also the observables ρ and J, given by the eqns. (1.2) and (1.3), remain unchanged under the transformation (2.5).

In other words:

The transformation (2.5) leaves the observables E, B, J and ρ invariant for arbitrary functions Ψ.

The substitutions (2.5) are called a gauge transformation with the generating function Ψ.

Let now the densities ρ and J be given. Then we have to show the existence of potentials A and Φ, which fulfil the Maxwell equations (1.2) and (1.3). Inserting the representations (2.1) and (2.4) into (1.2) yields

                                       curl curl A + 1/(Att + grad Φt) = μJ

or due to curl curl = grad div − Δ

(2.6)                         1/Att − ΔA + grad (div A + 1/Φt) = μJ

Analogously by insertion into (2.3) we obtain

                                                − (ΔΦ + div At) = ρ/ε ,

or

(2.7)                                 1/Φtt − ΔΦ − (div A + 1/Φt)t = ρ/ε .

As the reader himself should check the application of the gauge transformation (2.5) transforms (2.6) and (2.7) to

(2.6')                                 1/A'tt − ΔA' + grad (div A' + 1/Φ't) = μJ

and

(2.7')                                 1/Φ'tt − ΔΦ' − (div A' + 1/Φ't)t = ρ/ε ,

i.e. both equations are gauge invariant.

Remark At the present state of our theory we cannot guarantee the existence of solutions A,Φ of the coupled differential equations (2.6'),(2.7') for given data ρ and J. However, using the freedom of gauge transformations we shall show below that the equations can be transformed in a decoupled version, which is solvable. This implies that the equations (2.6'),(2.7') are solvable too.


3. The Coulomb gauge

The problem in resolving the eqns. (2.6) and (2.7) is their coupling. One way of decoupling them is introducing the additional condition

(3.1)                                                  div A = 0 ;

i.e. we have to remove the term div A in (2.7) by applying an appropriate gauge transformation. Since (2.5) transforms

(3.2)                                 div A → div A + ΔΨ

we merely have to choose the generating function Ψ as a solution of

(3.3)                                 ΔΨ = − div A.

This gauge transformation transforms (2.7) into

                                1/Φ'tt − ΔΦ' − 1/Φ'tt = ρ/ε ,

or

(2.7')                                 ΔΦ' = − ρ/ε ,

which is the Poisson equation with the well-known integral solution

(3.2)                                 Φ'(x,t) = 1/4πε ρ(y,t)/|xy| dy .

Therefore Φ' is now a known function for the other equation, which can be rewritten as

(2.6')                                          1/A'tt − ΔA' = μ J* ,

where J* := J − ε grad Φt is a known function.

Again a solution of (2.6') in integral form is known:

(3.3)                                 A(x,t) = μ/ J*(y,t |xy|/c)/|xy| dy .

However, something is left: One has to show that the solution (3.3) fulfils the condition (3.1), i.e. div A = 0. The reader will find the (rather technical) proof in [1; p.607].


4. The Lorenz gauge (falsely attributed to H.A. Lorentz)

The second method of decoupling the eqns. (2.6) and (2.7) is more symmetrical: We remove the coupling term by introducing the so-called Lorenz convention:

(4.1)                                 div A + 1/Φt = 0.

This goal can be attained by an appropriate gauge transformation: Under the gauge transformation (2.5) we have

(4.2)                                 div A + 1/ Φt → div A + 1/Φt + (ΔΨ − 1/Ψtt) .

Thus, if we choose the generating function Ψ" to be a solution of the inhomogenous wave equation:

(4.3)                                 1/Ψ"tt − ΔΨ" = div A + 1/ Φt ,

we obtain equivalent potentials A" and Φ" that fulfil

(2.6")                                 1/ A"tt − ΔA" = μJ

and

(2.7")                                 1/ Φ"tt − ΔΦ" = ρ/ε .

As is well known both equations have solutions in integral form

(4.4)                                 A"(x,t) = μ/ J(y,t |xy|/c)/|xy| dy ,

(4.5)                                 Φ"(x,t) = 1/4πε ρ(y,t |xy|/c)/|xy| dy .

Again the question arises whether the solutions (4.4),(4.5) fulfil the Lorenz convention (4.1). The reader is kindly requested to have a look at [1;p.597f.] where a proof is given.


5. The equivalence of the Coulomb gauge and the Lorenz gauge

On first view the reader might doubt whether Coulomb gauge and Lorenz gauge would yield the same observables E and B for identical data J,ρ. However, it is easy to show that both solutions A',Φ' and A",Φ" are transformable into each other by a gauge transformation.

Let (E,B)(A,Φ) denote the observables E,B derived from the potentials A,Φ by eqns. (2.1) and (2.4) respectively. Then due to the gauge equivalence between A,Φ and A',Φ' (cf. Sect. 3) we have

(5.1)                                 (E,B)(A',Φ') = (E,B)(A,Φ)

and due to the gauge equivalence between A,Φ and A",Φ" (cf. Sect. 4)

(5.2)                                 (E,B)(A",Φ") = (E,B)(A,Φ).

Therefore we obtain

(5.3)                                 (E,B)(A',Φ') = (E,B)(A",Φ").

This result is a special case of the general result on gauge invariance of the observables that we had attained already in Sect. 2.

Remark The possibility of representing the observables of the Maxwell theory by different (gauge-equivalent) potentials means that within the Maxwell theory the electrodynamic potentials A and Φ possess no physical reality. The same holds for the vector potential A in context of the Aharonov-Bohm effect where only that part of information contained in A that is also contained in the observable B = curl A is of physical relevance (cf. [4]).



References

[1]            Ernst Schmutzler: Grundlagen der Theoretischen Physik Vol. I
                BI Wissenschaftsverlag 1989, ISBN 3-411-03145-X

[2]            Koen van Vlaenderen: Generalised Classical Electrodynamics
                for the prediction of scalar field effects
                                http://home.wanadoo.nl/raccoon/

[3]            K.J. van Vlaenderen and A. Waser: Electrodynamics with the scalar field
        http://www.info.global-scaling-verein.de/Documents/ElectrodynamicsWithTheScalarField03.pdf

[4]            Gerhard W. Bruhn: Zur Rolle des magnetischen Vektorpotentials
                beim Aharonov-Bohm-Effekt
                        http://www.mathematik.tu-darmstadt.de/~bruhn/Aharonov.htm



Remark on the Lorenz-Convention falsely attributed to H.A. Lorentz


The Lorenz condition stems from the Danish mathematician and physicist Ludvig Valentin Lorenz (18. Januar 1829 - 9. Juni 1891):

Lorenz, L. "On the Identity of the Vibrations of Light with Electrical Currents." Philos. Mag. 34, 287-301, 1867.

Further References

Wikipedia (in Danish)
Wikipedia (in German)
van Bladel, J. "Lorenz or Lorentz?" IEEE Antennas Prop. Mag. 33, 69, 1991.
Whittaker, E. A History of the Theories of Aether and Electricity, Vols. 1-2. New York: Dover, p. 268, 1989.