Three-vector and scalar field identities and uniqueness theorems in ...

and so by the definitions (53) the fields E, B, and C areuniquely specified and theorem U2 is proved. Note that ifEq. (54a) is substituted into identity (44), and Eq. (54b)is substituted into identity (49), we obtain∫A(x) = µ 0 j(x ′ )G(x, x ′ )d 4 x ′ , (60a)φ(x)c= 1 cV 4′∫V ′4ρ(x ′ )ɛ 0G(x, x ′ )d 4 x ′ , (60b)which are inhomogeneous solutions of the wave equations(59) in terms of a suitable Green’s function. It was theGreen’s function solution of Eq. (17b) which led via thedelta function identity (23a) to the spatial components(37) and time component (45) of the A µ identity itself. 6Equations (54a) and (54b) reduce to Maxwell’s sourceequations if we set the four-divergence C = ∂ µ A µ = 0,that is, with the Lorentz gauge condition (more descriptivelycalled a relativistic transverse gauge condition). 6,7It is easily proved that once this relativistic transversegauge condition is assumed, the resulting Maxwell’sequations decouple to give the wave equations (59). Themost general Lagrangian density for a (massless) fourvectorfield A µ that is no more than quadratic in its variablesand their derivatives, 6 and which embodies theoremU2 via the covariant formalism (55)−(57), appears to beL = − ɛ 0c 24 F µνF µν − λɛ 0c 22(∂ µ A µ ) 2 + j µ A µ , (61)where j µ = (ρc, j) and λ is a Lagrange multiplier for theLorentz constraint term. The covariant Lagrange equationof motion which follows from Eq. (61) is−j ν /ɛ 0 c 2 = ∂ µ F µν + λ∂ ν (∂ µ A µ )(62a)= ∂ µ ∂ µ A ν − (1 − λ)∂ ν (∂ µ A µ ). (62b)If we make the physical assumption that all derivatives∂ µ have equal weight, we may set λ = 1, which reducesEq. (62b) to□A ν = ∂ µ ∂ µ A ν = −j ν /ɛ 0 c 2 = −µ 0 j ν , (63)which is the wave equations (59) in covariant notation.There is another covariant gauge that I have dubbedthe relativistic longitudinal gauge, 6,7 where the four-curlF µν = 0, which implies via Eq. (55) that E = 0 andB = 0 as well. With this relativistic longitudinal gaugecondition the Lagrange equation of motion (62b) reduces(with λ = 1) to−j ν /ɛ 0 c 2 = ∂ ν (∂ µ A µ ) = ∂ µ (∂ ν A µ ) = ∂ µ (∂ µ A ν ), (64)where ∂ µ A ν = ∂ ν A µ in this gauge via Eq. (56), which isalso the wave equations (59) in covariant notation. Similarly,in three-vector notation the field equations (54)and definitions (53) of theorem U2 for the relativisticlongitudinal gauge condition reduce to−∇(∇ · A) − 1 ∂(∇φ)c 2 = −∇C = µ 0 j,∂t(65a)∇ · ∂A∂t + 1 ∂ 2 φc 2 ∂t 2 = ∂C∂t = ρ .ɛ 0(65b)The relativistic longitudinal gauge condition F µν = 0implies thatE = −∇φ − ∂A∂t = 0,B = ∇ × A = 0,(66a)(66b)so that, with the identity (8), it follows that Eqs. (65a)and (65b) reduce to the wave equations (59).VI.CONCLUDING REMARKSIt can be argued that theorem U2 uniquely specifiesthe most general three-vector and scalar field (component)equations in Minkowski space for an associated(massless) classical four-vector field A µ that is assumedto satisfy an inhomogeneous hyperbolic wave equation.That is, the field definitions and equations of theoremU2 follow directly from the use of a Green’s functionsolution technique using an operator delta function fourvectoridentity. Thus, it should not be surprising thatboth the relativistic transverse and relativistic longitudinalcovariant gauge conditions, when used in conjunctionwith theorem U2, lead to the same wave equations. Ihave previously shown that the relativistic transverse andrelativistic longitudinal covariant gauge conditions leadrespectively to only two classes of classical four-vectorfields which are potentially physical. 7 By the latter it ismeant that a suitable fully quadratic Lagrangian density(of the type used here but more general in that it has afield mass term and can have complex valued charges) isbounded from below when one or the other of these twocovariant gauge conditions is applied. 7ACKNOWLEDGMENTSThe author thanks Andrew Stewart of the ResearchSchool of Physical Sciences and Engineering at The AustralianNational University for prompting the author toinvestigate the uniqueness of time-varying three-vectorfields in relation to his work in Ref. 17, and for subsequentreview of this paper. The author also thanks J. V.Corbett of the Department of Mathematics of MacquarieUniversity−Sydney for detailed review of this paper. Theauthor is indebted to James Cresser of the Departmentof Physics of Macquarie University−Sydney for supportof the author’s research position in the department.9