aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Comment on “Defining the electromagnetic potentials”
Hendrik van Hees ∗ Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany (Dated: June 23, 2020)In this comment it is argued that the argument for a unique determination of the electro-magnetic potentials in classical electrodynamics in is flawed. To the contrary the “gaugefreedom” of the electromagnetic potentials has proven as one of the most important proper-ties in the development of modern physics, where local gauge invariance with its extensionto non-Abelian gauge groups is a key feature in the formulation of the Standard Model ofelementary particles in terms of a relativistic quantum field theory. I. INTRODUCTION In the author claims that contrary to the standard treatment of the electromagnetic potentialsin all textbooks like, e.g., on classical Maxwell theory the potentials are to be chosen as thoseof the Lorenz gauge. As shall be argued in the following, this is not only mathematically wrongbut also misleading from a physical (as well as didactical) point of view since the gauge invari-ance of electromagnetism is the paradigmatic example for a local gauge symmetry demonstratinga general important concept for the formulation of the Standard Model of elementary particlephysics, describing all hitherto observed elementary particles and their interactions in terms ofa (renormalizable) relativistic quantum field theory. In this sense the claim of any fundamentala-priori preference for any specific gauge is also highly misleading from a pedagogical point of view.Ironically the choice of the Lorenz gauge itself is not even a complete gauge fixing to begin with.From the theoretical-physics point of view it is quite commonly accepted for a long time that thefundamental laws governing the realm of classical electrodynamics are the “microscopic” Maxwellequations in differential form (for the historical context see, e.g., the remark in the introductorychapter in ), ∇ × E + 1 c ∂ t B = 0 , (1) ∇ · B = 0 , (2) ∇ × B − c ∂ t E = 1 c j (3) ∇ · E = ρ, (4)where the Heaviside-Lorentz system of units has been used, which is more convenient for theoreticalpurposes than the SI units used in .This paper is organized as follows: II. HELMHOLTZ’S THEOREM
First it is important to note that Helmholtz’s theorem is applicable to time-dependent as well asto time-independent vector fields and states in a quite general form that if a vector field V andits first derivatives, which are themselves differentiable, vanish at infinity, it can be decomposed as V = V + V such that ∇ × V = 0 and ∇ · V = 0. With given source, ~ ∇ · V = ~ ∇ · V = J , andcurl ~ ∇ × V = ∇ × F = C , the decomposition is unique up to additive constants for the vectorfields V and V . In the following we tacitly assume the conditions on the fields needed for thefollowing manipulations being justified.Further there are theorems that any curl-free vector field can be written (at least in any simplyconnected region of space) as the gradient of a scalar potential, i.e., V = − ∇ Φ, where Φ is uniqueup to a constant and any source-free vector field can be written as the curl of a vector potential V = ∇ × A , and of course A is unique only up to an arbitrary gradient field, and this freedomcan be used to impose one constraint condition (“gauge condition”) on A .Defining ∇ · V = J and ∇ × V = C we have ∇ · V = ∇ · V = − ∆Φ = J, ∇ × V = ∇ × V = ∇ × ( ∇ × A ) = ∇ ( ∇ · A ) − ∆ A . (5)Since we know from electrostatics how to solve the Poisson equation with the Green’s function of theLaplace operator (here for “free space”, i.e., without boundary conditions for Cauchy or Neumannproblems as needed in electrostatics at presence of conductors or dielectrics), it is convenient toimpose the additional constraint ∇ · A = 0 (“Coulomb gauge condition”), such thatΦ( x ) = Z R d x ′ J ( x ′ )4 π | x − x ′ | , A ( x ) = Z R d x ′ C ( x ′ )4 π | x − x ′ | . (6)These formulae can be proven using Green’s theorem. Then one has V = − ∇ Φ + V (0)1 , V = ∇ × A + V (0)2 , (7)where V (0)1 = const and V (0)2 = const. Of course, if it is known that V and V vanish, e.g., if J and C have compact support, at infinity these constants are both determined to vanish given thepotentials (6).As we shall see, however, Helmholtz’s decomposition theorem is not of prime importance tointroduce the electromagnetic potentials. For this it is sufficient that a curl-free vector field canbe written as the gradient of a scalar potential and that a source-free field can be written as thecurl of a vector potential. For a given curl-free vector field its scalar potential is defined up to anadditive constant, and for a given source-free vector field its vector potential is only determinedup to a gradient of an arbitrary scalar field. In fact, as we shall see, for the solution of Maxwell’sequation with given sources ρ and j the Helmholtz decomposition theorem is of not too muchpractical use. One rather needs a Green’s function of the D’Alembert operator (cid:3) = ∆ − /c ∂ t ,of which in classical electrodynamics usually the retarded propagator is the relevant one (forreasons of causality). III. THE ELECTROMAGNETIC POTENTIALS
The electromagnetic potentials are introduced using the homogeneous Maxwell equations (1)and (2). Though they have profound physical meaning, from a mathematical point of view theyare merely constraint conditions on the electric and magnetic fields, but nevertheless necessary tomake the solutions of the inhomogeneous Maxwell equations (3) and (4) unique, which describethe charge and current densities as the sources of the electromagnetic field and thus provide thedynamical equations of motion.From our discussion in the previous Sect. it is clear that we should start to use (2) to introducea vector potential for the magnetic field, B ( t, x ) = ∇ × A ( t, x ) . (8)As is clear from this argument, A is not an observable field but just a way to identically fulfill (2).For a given physical field configuration ( E , B ) the vector potential A is only determined up to thegradient of an arbitrary scalar field, χ , i.e., any alternative vector potential A ′ = A − ∇ χ (9)is physically equivalent to any other vector potential.Using now (8) in (1) leads to ∇ × E + 1 c ∂ t ∇ × A = ∇ × (cid:18) E + 1 c ∂ t A (cid:19) = 0 . (10)Now again referring to the mathematical theorems summarized in the previous Sect. one can writethe vector field in the parentheses as a gradient of a scalar potential Φ, E + 1 c ∂ t A = − ∇ Φ ⇒ E = − c ∂ t A − ∇ Φ . (11)As the scalar and vector potentials are not a observable fields and their non-uniqueness is physicallyunimportant. What, however, is of utmost importance is that the “gauge freedom” (9) for thevector potential can be compensated by a redefinition of the scalar potential such that also theelectric field E remains unchanged, as the latter is an observable physical field. Indeed using A ′ instead of A in (11) and also a new scalar potential Φ ′ we find E = − c ∂ t A − ∇ Φ ! = − c ∂ t A ′ − ∇ Φ ′ = − c ∂ t A − ∇ (cid:18) Φ ′ − c ∂ t χ (cid:19) . (12)Thus we can compensate for the redefinition of A by settingΦ ′ = Φ + 1 c ∂ t χ. (13)Together with (9) this defines the full arbitrariness in the choice of the vector and scalar potentialsfor the electromagnetic field ( E , B ). Since only the latter is an observable physical field, thisarbitrariness is no conceptual problem but can be used to the advantage for simplifying the taskto solve the Maxwell equations, because one has still the freedom to impose one constraint on thepotentials (Φ , A ) to simplify the equations for a specific physical situation without changing thephysical content of the solutions for ( E , B ). In other words: Two sets of em. potentials (Φ , A )and (Φ ′ , A ′ ), connected by a gauge transformation (9) and (13) are physically equivalent. This isknown as the gauge invariance of classical electrodynamics.Using (8) and (11) in the inhomogeneous Maxwell equations (3) and (4) yields − (cid:3) A + ∇ (cid:18) ∇ · A + 1 c ∂ t Φ (cid:19) = 1 c j , (14) − ∆Φ − c ∂ t ∇ · A = ρ. (15)Here, the d’Alembert operator is used with the sign convention as in , i.e., (cid:3) = ∆ − /c ∂ t . It isclear that these two equations alone do not resolve the ambiguity in the choice of the potentialssince these equations are of course still gauge invariant, because they are formulated originally interms of the Maxwell equations (3) and (4) involving only the gauge invariant fields ( E , B ). Thus(14) and (15) do not provide any constraint for the choice of gauge, i.e., we can still impose oneconstraint on the potentials to facilitate the solution of the equations (14) and (15).A glance at (14) immediately shows that a promising choice for a gauge constraint is the Lorenzgauge condition, ∇ · A L + 1 c ∂ t Φ L = 0 . (16)The index L indicates the Lorenz-gauge potentials. Then − (cid:3) A L = 1 c j , (17)i.e., in the Lorenz gauge the equations for the components of the vector potential decouple fromeach other as well as from the scalar potential. Using the Lorentz-gauge constraint (16) in (15)leads to − (cid:3) Φ L = ρ. (18)In this gauge we have four simple inhomogeneous wave equations for each of the Cartesian com-ponents of A L as well as Φ L .Of course, the inhomogeneous wave equation with a given source is also not uniquely solvablebut one has to impose initial as well as boundary conditions to make its solution unique, becauseits solutions are only determined up to a solution of the homogeneous wave equation, and thiscan be constrained by imposing initial conditions as well as boundary conditions. For the herediscussed case of the microscopic Maxwell equations the boundary conditions are usually imposedat spacial infinity implied by the physical situation. E.g., one usually has charges and currentsonly in a compact spatial region and thus looks for solutions of the wave equations (16) and (17)describing waves radiating outwards from these sources. Indeed, as correctly stated in also froma causality argument it is justified to choose the retarded solution for the potentials,Φ L ( t, r ) = Z R d r ′ ρ ( t − | r − r ′ | /c, r ′ )4 π | r − r ′ | , A L ( t, r ) = Z R d r ′ j ( t − | r − r ′ | /c, r ′ )4 πc | r − r ′ | . (19)The initial condition can then be satisfied by adding an appropriate solution of the homogeneouswave equations, (cid:3) Φ L = 0 and (cid:3) A L = 0. This of course implies that also the physical fields aregiven by retarded solutions and thus fulfill the demand of causal solutions that the observableelectromagnetic field are “caused” by the presence of the charge and current densities as sourcesdepends at time t only on the configuration of these sources at the earlier times t ret = t − | r − r ′ | /c .It is clear, though, that imposing this “causality constraint” on the potentials is not a priorinecessary, since only the observable em. field ( E , B ) needs to be “causally connected” functionalsof the sources ( ρ, j ). Indeed, since the fields are given by derivatives of the potentials, the “causalchoice” (19), using the retarded Green’s function for the (cid:3) -operator, as the solution of the equations(14) and (15) implies that also ( E , B ) are retarded solutions and thus fulfill the causality condition.It is also important to note that (18) provide only a solution to Maxwell’s equations if theLorenz condition (15) indeed is fulfilled. A simple calculation shows that this is indeed the case, ifthe continuity equation, ∂ t ρ + ∇ · j = 0 , (20)i.e., the local form of charge conservation is fulfilled. This is anyway a necessary integrabilitycondition for the Maxwell equations, and thus independent of the introduction of the potentialsand the choice of their gauge.One should also note that the Lorenz-gauge constraint (15) does not uniquely determine thepotentials, since a change of gauge by an arbitrary scalar field χ according to (9) and (13) constrains χ only to obey the homogeneous wave equation, (cid:3) χ = 0 , (21)which has non-zero solutions (even such vanishing at spatial infinity like, e.g., the spherical wave χ ( t, x ) = χ sin[ k ( ct − | r | ) / | r | ]). This arbitrariness of course is again irrelevant for the just deter-mined causal physical fields ( E , B ), because these do not depend on the gauge and are given asretarded integrals over the sources ( ρ, j ).From this line of arguments it is already clear that the potentials do not need to be necessarilyretarded solutions to fulfill the causal connection between sources and physical fields, and thus anyother gauge, which may not allow for entirely retarded solutions is as justified as the Lorenz gauge.This is of course, even in an extreme sense, illustrated by the other most commonly usedgauge fixing, the Coulomb gauge . It is motivated by starting from (15) and observing that withimposing the constraint, ∇ · A C = 0 , (22)one decouples A C from the equation for the scalar field, which now obeys a Poisson equation as inelectrostatics (but of course in general with a time-dependent charge density), − ∆Φ C = ρ (23)with the solution Φ C ( t, r ) = Z R d r ′ ρ ( t, r ′ )4 π | r − r ′ | . (24)This is of course still in some sense a “causal solution”, because the integrand only depends on thepresent time t but not on times > t , but it obviously seems to violate “Einstein causality” in thesense of relativity, according to which causal effects should “propagate” (at most) with the speedof light. Here the scalar potential at time t is determined by the present charge configuration atthis time t but gets “instantaneous contributions” from points r ′ which are arbitrarily far fromthe observational point r . As we shall see, that is not a problem at all since with the appropriatechoice of solutions one finally ends up with the same retarded physical fields ( E , B ) as with theretarded potentials from the Lorenz-gauge potentials.This can be seen by using (23) and the Coulomb-gauge condition (21) in (14), which gets − (cid:3) A C = 1 c j ⊥ (25)with j ⊥ ( t, x ) = j ( t, x ) − ∂ t ∇ Φ C ( t, x ) = j ( t, x ) − ∇ Z R d r ′ ∂ t ρ ( t, r ′ )4 π | r − r ′ | . (26)To see that (25) is consistent with the Coulomb-gauge condition (22), i.e., with ∇ · j ⊥ we againneed the continuity equation (20) to rewrite (26) to j ⊥ ( t, x ) = j ( t, x ) − ∇ Z R d r ′ − ∇ ′ · j ( t, r ′ )4 π | r − r ′ | . (27)Now taking the divergence of this equation indeed gives ∇ · j ⊥ ( t, x ) = ∇ · j ( t, x ) − ∆ Z R d r ′ − ∇ ′ · j ( t, r ′ )4 π | r − r ′ | = 0 . (28)To get retarded solutions for the fields, it seems to be appropriate to solve (25) with the retardedpropagator, i.e., A C ( t, r ) = Z R d r ′ j ⊥ ( t ret , r ′ )4 πc | r − r ′ | . (29)In the following we like to show that indeed the Coulomb-gauge potentials can be written as agauge transformation of the retarded Lorenz-gauge potentials, which of course implies that thephysical fields are the same retarded fields as derived using the Lorenz-gauge potentials. This thenclearly demonstrates that imposing the causality condition on the physical fields does not imply aunique definition of the “right potentials” to be the retarded Lorenz-gauge potentials.For this proof it is convenient to introduce the vector field, j k ( t, x ) = j ( t, x ) − j ⊥ ( t, x ) = ∇ ∂ t Φ C ( t, x ) = ∇ ∂ t Z R d r ′ ρ ( t, r ′ )4 π | r − r ′ | = ∇ ∂ t Φ C ( t, r ) (30)where we have used (26). Then we can write (29) in the form A C ( t, r ) = A L ( t, r ) − Z R d r ′ Z R d t ′ δ ( t ′ − t + | r − r ′ | /c )4 πc | r − r ′ | ∇ ′ ∂ t ′ Φ C ( t ′ , r ′ ) . (31)Integration by parts yields A C ( t, r ) = A L ( t, r ) − Z R d r ′ Z R d t ′ Φ C ( t ′ , r ′ ) ∇ ′ ∂ t ′ δ ( t ′ − t + | r − r ′ | /c )4 πc | r − r ′ | = A L ( t, r ) − c ∂ t ∇ Z R d r ′ Z R d t ′ Φ C ( t ′ , r ′ ) δ ( t ′ − t + | r − r ′ | /c )4 π | r − r ′ | | {z } Ψ CL ( t, r ) . (32)With this first we have A C ( t, r ) = A L ( t, r ) − ∇ χ CL (33)with the scalar field defining the gauge transformation from the Lorenz- to the Coulomb-gaugepotentials, χ CL = 1 c ∂ t Ψ CL ( t, r ) . (34)All we have to show to complete our proof of the gauge equivalence of the Coulomb-gauge and theLorenz-gauge potentials is that with this definition we also fulfillΦ C ( t, r ) = Φ L ( t, x ) + 1 c ∂ t χ CL . (35)Now 1 c ∂ t χ CL = 1 c ∂ t Ψ CL = (∆ − (cid:3) )Ψ CL . (36)The first term is immediately calculted from the definition of Ψ Cl in (32) since the defining integralis just the retarded solution of the inhomogeneous wave equation (cid:3) Ψ CL = − Φ C . (37)Further we have, again using the definition of Ψ CL in (32), integrating by parts, using (23) and(19) ∆Ψ CL ( t, r ) = Z R d r ′ Z R d t ′ Φ C ( t ′ , r ′ )∆ δ ( t ′ − t + | r − r ′ | /c )4 π | r − r ′ | = Z R d r ′ Z R d t ′ Φ C ( t ′ , r ′ )∆ ′ δ ( t ′ − t + | r − r ′ | /c )4 π | r − r ′ | = Z R d r ′ Z R d t ′ δ ( t ′ − t + | r − r ′ | /c )4 π | r − r ′ | ∆ ′ Φ C ( t ′ , r ′ ) (23) = − Z R d r ′ Z R d t ′ δ ( t ′ − t + | r − r ′ | /c )4 π | r − r ′ | ρ ( t ′ , r ′ ) (19) = − Φ L ( t, r ) . (38)Using (37) and (38) in (36) indeed leads to (35), i.e., indeed the Coulomb-gauge potentials, with thechoice of a retarded solution (29) of (25), are just a gauge transforamtion of the retarded Lorenz-gauge potentials and thus the resulting electromagnetic fields are the same retareded solutions asderived from the Lorenz-gauge potentials, again underlining the fact that two sets of em. potentialsconnected by a gauge transformation with an arbitrary gauge field χ describe the same physicalsituation.The Lorenz-gauge potentials are in some respects more convenient to use since (a) they admitpurely retarded solutions which are usually what is needed in the physical applications and thusthese potentials admit a manifestly “causal connection” with the sources and (b) the Lorenz-gaugecondition is manifestly covariant under Lorentz transformations since it reads ∂ µ A µ = 0 in four-vector notation (where ( x µ ) = ( ct, x ), and ∂ µ = ∂/∂x µ are contra- and covariant four-vectorcomponents in Minkowski space).Nevertheless in some respects the Coulomb gauge has also some advantages. Among them isthat it fixes the gauge more stringently than the Lorenz-gauge condition. Indeed if we ask forspecial gauge transformations, A ′ = A C − ∇ χ, Φ ′ = Φ + 1 c ∂ t χ (39)such that the Coulomb-gauge condition still holds, this leads to ~ ∇ · A ′ = ∇ · A − ∆ χ = − ∆ χ ! = 0 . (40)This implies that, under the constraint that the new gauge potentials vanish at spatial infinityas the retarded solutions for localized sources (i.e., sources with compact spacial support) χ = 0,i.e., the Coulomb-gauge condition is more restrictive than the Lorenz-gauge condition. I.e., in thissense it provides a complete gauge fixing and thus is, e.g., most convenient to quantize theelectromagnetic field in the canonical operator formalism.It is of course clear that these retarded solutions can also be directly derived from the Maxwellequations (1-4) without first introducing the electromagnetic potentials, leading to the socalled Jefimenko equations , which are, of course equivalent, to the solutions provided by the retardedsolution of the Lorenz-gauge potentials.
IV. CONCLUSION
In this comment, we have clarified that the electromagnetic potentials are not uniquely de-termined by the (relativistic) causality constraints leading to a unique choice of the potentials asthe retarded solutions of the wave equations for the potentials in Lorenz gauge, as claimed in .We have illustrated that the ambiguity in the choice of the potentials are mathematical facts,but that ambuiguity, described as the gauge invariance of classical Maxwell theory, is irrelevantfor the observable electromagnetic fields since the different electromagnetic potentials related toeach other by a gauge transformation represent the same physics. The causality constraint, in-cluding the more stringent Einstein causality imposed by the relativistic spacetime structure, hasto be imposed only on the physically observable fields and are not a necessary condition for theunobservable electromagnetic potentials.This has been demonstrated by the two standard examples given in most standard textbooks,the Lorenz gauge, which leads to decoupled inhomogeneous wave equations for the scalar and thecomponents of the vector potential, and for them thus the causality condition can be fulfilled byusing the retarded propagator of the d’Alembert operator, leading to retarded potentials and thusalso to an retarded electromagnetic field. The retardation is given by the speed of light as to beexpected from a massless field as the electromagnetic field. Thus these retarded solutions obeyboth the causality and the more stringent Einstein causality constraints as it must be for a classicalrelativistic field theory.The other “extreme choice” with regard to retardation is the Coulomb gauge, which leads to aninstantaneous solution for the scalar potential and in turn to a wave equation for the vector poten-tial with a nonlocal distribution to the source. Here the retarded solution has to be chosen for thesolution. Now both the scalar and the vector potential contain non-retarded, instantaneous contri-butions. However, the vector potential in the Coulomb gauge consists of the sum of the retardedLorenz-gauge vector potential and a gradient field. This immediately implies that the magneticfield B calculated from the Coulomb-gauge potential is the same retarded field as calculated fromthe retarede Lorenz-gauge vector potential. Then we have demonstrated that the resulting gaugefield also connects the instantaneous Coulomb-gauge scalar potential with the retarded Lorenz-gauge scalar potential in the way described by the corresponding gauge transformation such thatalso the electric field turns out to be the same retarded field as calculated from the Lorenz-gaugescalar potential.Of course one can also solve the Maxwell equations without introducing the potentials, leadingto inhomogeneous wave equations, for which the unique physical choice is the retarded solutiondue to the usual causality arguments given above.It is interesting that one can choose a different class of gauge constraints, the socalled “velocitygauges” such that the em. potentials contain retarded contributions which, however, propagate notwith the speed of light but with arbitrary speeds, even larger than the speed of light. Of course,these fields are not observable but lead again to the same physical retarded solutions for the em.field as it must be . ∗ Electronic address: [email protected] A. Davis, Eur. J. Phys. , 045202 (2020), https://doi.org/10.1088/1361-6404/ab78a6 . A. Sommerfeld,
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