JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 40, NUMBER 10, (pp. 4911−4943)#
OCTOBER 1999
Uniqueness theorems for classical four-vector fields
in Euclidean and Minkowski spaces†
Dale A. Woodsidea)
Department of Physics, Macquarie University-Sydney, New South Wales 2109, Australia
(Received 8 December 1998; accepted for publication 14 July 1999)
Euclidean and Minkowski four-space uniqueness theorems are derived which yield a new perspective of classical four-vector fields. The Euclidean four-space uniqueness theorem is based on
a Euclidean four-vector identity which is analogous to an identity used in Helmholtz’s theorem on
the uniqueness of three-vector fields. A Minkowski space identity and uniqueness theorem can be
formulated from first principles and the space components of this identity turn out to reduce to
the three-vector Helmholtz’s identity in a static Newtonian limit. A further result is a uniqueness
theorem for scalar fields based on an identity which is proved to be a static Newtonian limit of the
zeroth or scalar component of the Minkowski space extension of the Helmholtz identity. Lastly, the
three-vector Helmholtz identity and uniqueness theorem and their four-space extensions to Minkowc 1999 American Institute of Physics.
ski space are generalized to mass damped fields. [S0022-2488(99)00810-5]
†
This version of the article was typeset by the author using LATEX 2e and REVTEX v3.1.
Note, page numbering differs slightly from the original journal article. [See heading at top of page.]
#
I. INTRODUCTION
In Sec. II a review of Helmholtz’s theorem on the uniqueness of three-vector fields is first presented. Helmholtz’s
theorem is concerned with irrotational fields, which have zero curl everywhere in space, and solenoidal fields, which
have zero divergence everywhere in space. Now, the divergence and curl of a vector field over all of a Euclidean threespace uniquely determine this vector field. Based on these properties a theorem, that will be called the Helmholtz
theorem, states that the most general continuous three-vector field defined everywhere in a Euclidean three-space,
that along with its first derivatives vanishes sufficiently rapidly at infinity, may be uniquely represented as a sum of
an irrotational and a solenoidal part, up to a possible additive vector constant. The theorem can be extended to
finite volumes as well. In proving Helmholtz’s theorem, a vector identity is used that is commonly referred to as the
Helmholtz identity, which in turn is derived from a delta function property of the Laplacian operator.
A comparison of the three-space Laplacian operator with the Minkowski space d’Alembertian operator then suggests
that a three-divergence and a three-curl naturally generalize into a four-divergence and a four-curl when a fourth
dimension is added. A review of the Sommerfeld four-space integral solution of the four-vector potential d’Alembertian
wave equation, translated into modern notation, is then presented.
In Secs. III and IV of this article a number of theorems are put forward. Of these theorems, Theorems I, II, IV,
V, and VII-XII appear to be new results for three-vector and four-vector fields in a flat space. The new four-vector
results of this paper follow from a different starting point than another series of attempts to extend Helmholtz’s
theorems to Minkowski four-space1 in that the present paper adopts a definition of the four-curl as given by the
Maxwell field tensor F µν , equation (14b), following the definition in Møller,2 while these other attempts choose the
dual of the Maxwell field tensor as their four-curl. This later definition appears to be oriented toward the analysis of
the uniqueness of electromagnetic fields in the presence of hypothetical magnetic monopoles. The advantage of the
approach of the present paper, on the other hand, is that it leads to a more natural generalization of Helmholtz’s
three-space theorems to four-space. In particular, it yields a four-space Helmholtz identity whose space components
reduce to the three-space Helmholtz identity in a static Newtonian limit.
In Sec. III a comparison of Sommerfeld’s four-space integral solution with a Green’s function approach yields a
four-space Euclidean Laplacian delta function identity. On observing that the original three-space Helmholtz identity
is commonly derived from an analogous three-space Laplacian operator delta function identity, a parallel approach
is taken and the four-vector Euclidean Laplacian delta function identity is used to derive a Euclidean four-space
generalization of the Helmholtz identity. As with Sommerfeld’s Euclidean four-space integral, the Euclidean fourspace delta function identity can be analytically continued to Minkowski space by choosing an appropriate integration
contour which takes timelike causality into account. Rather than working with this analytic continuation technique
to derive a Minkowski space result, it turns out to be more convenient to derive the Minkowski four-space extension
4911
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4912
of the Helmholtz identity from first principles using a Minkowski space retarded Green’s function. These Euclidean
and Minkowski four-space extensions of the three-space Helmholtz identity are explicitly stated in Theorems I and II,
respectively, and form the basis for several uniqueness theorems for four-vector fields.
Next, it is shown that this Minkowski four-space identity reduces to the three-space Helmholtz identity in a static
Newtonian limit. The zeroth or scalar component of this Minkowski four-space extension of the Helmholtz identity is
then shown to reduce to a three-vector integral identity for scalar fields in a static Newtonian limit. This three-vector
integral identity for scalar fields, explicitly stated in Theorem III, is also proved by direct application of the three-space
Laplacian operator delta function identity.
Uniqueness theorems in Euclidean and Minkowski four-spaces, Theorems IV and V, are then proved using their
respective four-vector Helmholtz identities. It is found that the specification of the four-curl and four-divergence of
the four-vector field throughout the four-volume V4 , as well as the four-tangential and four-normal projections of the
four-vector field everywhere on the bounding three-surface Σ, are sufficient to obtain a unique four-vector field. A
further result based on Theorems IV and V, and stated later in Theorem X in Sec. IV, is that a four-vector field is
uniquely specified by the sum of a four-irrotational and a four-solenoidal part. This latter theorem corresponds to a
four-space generalization of Helmholtz’s uniqueness theorem.
Also in Sec. III, a uniqueness theorem for scalar fields, Theorem VI, is proved using the scalar field integral identity
of Theorem III. In this three-vector field case it is found that the gradient of the scalar field throughout the volume
V, as well as the magnitude of the scalar field on the bounding surface S, are sufficient to obtain a unique scalar field
in a Euclidean three-space.
Finally, in Sec. IV there is a discussion on whether or not the Helmholtz identity and its relativistic extensions to
Euclidean and Minkowski spaces can be generalized to fields with mass. First, by adding an exponential damping
factor for the mass to the Euclidean three-space Laplacian delta function identity, an exponentially damped threevector Helmholtz identity is stated in Theorem VII. A uniqueness theorem, Theorem VIII, is then proved using this
three-vector identity. Next, in the Euclidean four-vector case it is shown that adding an exponential damping factor for
the mass leads to a Euclidean four-space Laplacian delta function identity with an additional cross term which makes
the development of an analogous four-vector identity problematic. On the other hand, in the Minkowski four-space
case, the existence of a massive scalar Green’s function over timelike separations allows one to obtain an exponentially
damped Minkowski space four-vector identity which is explicitly stated in Theorem IX. But the four-space extension
of Helmholtz’s uniqueness theorem does not appear to carry over to the exponentially damped case. However, an
exponentially damped version of uniqueness Theorem V is stated in Theorem XII which relies on a theorem on the
vanishing of a four-vector field, i.e., Theorem XI.
Although it is tempting to interpret the exponentially damped results as generalizations of the earlier results to
massive four-vector fields, no limitation was imposed on their derivations other than that the four-vector fields are
assumed to be sufficiently smooth. The exponentially damped results do, however, appear to be oriented towards
application to massive vector fields or alternately to fields undergoing spatial diffusion in three or four dimensions,
respectively.
II. HISTORICAL AND MATHEMATICAL BACKGROUND
A. Historical survey of uniqueness theorems in three dimensions
A natural starting point in the development of a uniqueness theorem for classical four-vector fields is an examination
of the various statements of uniqueness theorems for ordinary three-vector fields in a flat Euclidean three-space.
Historically, the first statement of a uniqueness theorem for three-vector fields is found in Stokes’ article on diffraction
in 1849.3 The next statement is found in Helmholtz’s article on the hydrodynamics of vortex motion in 1858.4 Modern
texts often attach Helmholtz’s name to a uniqueness theorem for three-vector fields, but Sommerfeld5 makes a point
of acknowledging Stokes’ contribution as well. Also, modern statements of uniqueness theorems for three-vector fields
adopt vector notation, which is absent from their 1850’s counterparts, and in addition differ in their emphasis and
presentation.6–9 A short review of these modern uniqueness statements is therefore warranted.
Firstly, it is important to note that the defining properties of irrotational and solenoidal fields are the essential
underpinnings of the uniqueness theorems for three-vector fields. Namely, an irrotational field has zero curl everywhere
in space, i.e.,
0 = ∇×A,
while a solenoidal field has zero divergence everywhere in space, i.e.,
(1a)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4913
0 = ∇ · A.
(1b)
Next, a preliminary uniqueness theorem can be stated as follows:
Theorem U: The divergence and curl of a three-vector field over a volume V in a Euclidean three-space, along
with its normal components on a closed surface S bounding the volume V, uniquely determines the three-vector field
over the volume V and on the surface S.
In other words, one must specify
∇×F = j(x, y, z),
(2a)
∇ · F = ρ(x, y, z),
(2b)
over the volume V, and the normal component Fn (x, y, z) on the surface S, where for example in an electromagnetic
context, j is a “circulation current density” and ρ is a “source charge density”. A proof of this theorem is given in
Arfken.8
A uniqueness theorem that can be attributed to Helmholtz can now be stated as follows:
Theorem H1 (Helmholtz’s theorem on the uniqueness of three-vector fields over all of a Euclidean
three-space): A general continuous three-vector field defined everywhere in a Euclidean three-space, that along with
its first derivatives vanishes sufficiently rapidly at infinity, may be uniquely represented as a sum of an irrotational
and a solenoidal part, up to a possible additive constant vector.
A proof of this theorem is given in Sommerfeld.5
A modern alternate form of this theorem can be stated by restricting the domain of definition of the three-vector
field to a finite volume as follows:
Theorem H2 (Alternate form of Theorem H1 for a finite volume in a Euclidean three-space): A
general continuous three-vector field that is defined everywhere in a finite volume V of a Euclidean three-space and
whose tangential and normal components on the bounding closed surface S are given may be uniquely represented as
a sum of an irrotational and a solenoidal part.
In order to prove either Theorem H1 or Theorem H2, it turns out that it is sufficient to prove only that the
three-vector F can be written as
F(x, y, z) = −∇Φ(x, y, z) + ∇×A(x, y, z),
(3)
∇×∇Φ≡0,
(4)
since by the three-vector identity
one has by (1a) that −∇Φ is irrotational, and since by the three-vector identity
∇ · ∇×A≡0,
(5)
one has by (1b) that ∇×A is solenoidal. In fact, Eq. (3) is used by King as a brief statement of his version of
Helmholtz’s theorem.9 In another notable case, Eq. (3) is referred to in the very extensive treatment by Plonsey and
Collin as “the mathematical statement of the second part of Helmholtz’s theorem.”7
The proof of (3) is based on the assumption that there exists a solution for the three-vector potential A of the
vector Poisson equation in Cartesian coordinates, which in the electromagnetic case reads
∇2 A(x, y, z) = −µ0 J(x, y, z),
(6)
where µ0 = 1/0 c2 is the free space permeability. Now, in Cartesian coordinates, the three components of A are
each separately a solution of a scalar Poisson equation. The scalar Poisson equation can then be solved in terms of
a two-point scalar Green’s function G(r, r0 ) which connects its unit delta function source located at the source point
r0 = (x 0 , y 0 , z 0 ) to a measurement at the field point r = (x, y, z), i.e.,
∇2 G(r, r0 ) = −δ 3 (r − r0 ) = −δ(x − x0 )(y − y 0 )(z − z 0 ).
(7)
The well known identity over the Euclidean three-space R3 , namely
∇2
1
≡ −δ 3 (r − r0 )
4πr
∀ r, r0 ∈ R3 ,
(8)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4914
where r ≡ |r − r0 |, yields by comparison with (7) for the case of an infinite spatial domain, the Green’s function of
the Laplacian operator as G(r, r0 ) = 1/4πr. The inhomogeneous solution of (6) for the vector potential then follows
from the integral
Z
Z
µ0 J(r0 ) 3 0
A(r) =
µ0 J(r0 )G(r, r0 )d3 r0 =
(9)
0 d r .
V0
V 0 4π |r − r |
Although (9) is an inhomogeneous solution to the vector Poisson equation (6) in an infinite spatial domain, it is the
delta function property of the vector identity (8) which is of importance in the proof of Theorems H1 or H2, namely
Z
Z
−1
0 0 0 3
0 3 0
0 0 0
2
F(x, y, z) =
F(x , y , z )δ (r − r )d r =
F(x , y , z )∇
d3 r0 .
(10)
4πr
V0
V0
Then, since the Laplacian operator acts only on the field coordinates, it can be brought outside of the integration.
At this point, one makes a decomposition of the Laplacian operator using identity (20), the significance of which is
discussed in Sec. II B of this article. The identity ∇(1/r) = −∇0 (1/r) and a fair amount of vector analysis then yields
the identity6,7
Z
I
∇0 · F(x0 , y 0 , z 0 ) 0
F(x0 , y 0 , z 0 ) · n0 0
F(x, y, z) = − ∇
dV −
dS
4πr
4πr
V0
S0
Z
I
∇0 ×F(x0 , y 0 , z 0 ) 0
F(x0 , y 0 , z 0 )×n0 0
+ ∇×
dV +
dS ,
(11a)
4πr
4πr
V0
S0
where n0 is the unit surface normal pointing out of the volume V 0 bounded by the closed surface S 0 . The proof of
the volume integral terms of (11a) can be traced back to Stokes.3 Proof of the surface integral terms of (11a) can be
found in Field Theory of Guided Waves by Collin.6 Equation (11a) is of the desired form (3) and can therefore be
considered as completing the proof of Theorem H2, provided of course that the integrals are well defined. In order
for the integrals to be well defined, one must make the additional assumption that the field F must vanish sufficiently
rapidly at infinity, i.e., at least as fast as 1/r2 in order to avoid logarithmic divergences. To prove H1, one takes the
surface S as going to infinity to include all of Euclidean three-space. The surface integral terms vanish as r → ∞
under the same assumption that the field F falls off at least as fast as 1/r2 . Equation (11a), but without the surface
integral terms, is sometimes referred to as the “Helmholtz identity”, (cf. Ref. 10). However, the more general result,
the full Eq. (11a), will be referred to as the Helmholtz identity in this article.
It should be noted in passing that application of identity (11a) to a field F falling off only as fast as 1/r, (e.g. a
potential), would presumably require a cut-off procedure in the integrals. However, the vector derivatives which stand
in front of the integrals act only on the field point coordinates so that, for a twice continuously differentiable vector
field F, one can move these derivatives inside of the integrals over the source point coordinates to give
Z
0
I
∇ · F(x0 , y 0 , z 0 )
F(x0 , y 0 , z 0 ) · n0
0
0
F(x, y, z) = −
∇
dV
−
∇
dS
4πr(r, r0 )
4πr(r, r0 )
0
S0
ZV
0
I
∇ ×F(x0 , y 0 , z 0 )
F(x0 , y 0 , z 0 )×n0
0
0
+
∇×
dV
+
∇×
dS
,
(11b)
4πr(r, r0 )
4πr(r, r0 )
V0
S0
as an alternate form of the Helmholtz identity. Now, since the integrands of identity (11b) involve an extra vector
derivative over the field point coordinates r, the convergence properties of the integrands are improved and the
integrals are now well defined for vector potentials F falling off only as fast as 1/r.
Note, Eqs. (11) should be thought of as identities for representing a general (static) three-vector field rather than
as a general solution to a partial differential equation. Indeed, Eqs. (11) follow from the vector identity (8) over
an infinite spatial domain and its subsequent use in the delta function property (10). Consequently, Eqs. (11) are
vector identities that apply to all of the Euclidean three-space, and so must hold for a finite sub-volume of it as well.
The Green’s function for an inhomogeneous vector Poisson equation in an infinite spatial domain is only mentioned
in passing, and certainly no use is made of the solutions of the associated source free homogeneous vector Laplace
equation. This does not necessarily reduce the utility or applicability of (11a), for example. In fact, if one makes the
replacements (2), for suitable volume source densities, and then makes the replacements
σ ≡ −F · n,
(12a)
K ≡ F×n,
(12b)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4915
with for example σ taken as a surface “charge” density and K taken as a surface “current” density, one can deduce
from a general point of view basically all of the (static) macroscopic integral equations for an electromagnetic field in
material media expressed in terms of its sources. This program, i.e., using (11a), is carried to its logical completion
for the electromagnetic case in the thorough treatment of Plonsey and Collin.7
From the point of view of the present article, it turns out that identity (11a), as well as theorems H1 or H2, are
actually static, (i.e., nontime varying), cases of a more general identity and theorem in a pseudo-Euclidean 3+1 space,
(hereafter taken to be Minkowski space). Indeed, in the nontime varying case, identity (11a) follows as the space
components of a four-vector identity in a static Newtonian limit, and in addition, a three-space identity arises from
the fourth or scalar component of the four-vector identity. In the general time varying case in Minkowski space, the
new identity and uniqueness theorem imply that the three-space notions of uniquely specifying a general vector field
by its divergence and curl in a Euclidean volume V, along with its normal and tangential components on the closed
bounding surface S, must be generalized to specifying its four-divergence and its four-curl in a four-volume V4 of
Minkowski space, along with its four-normal and four-tangential components on the closed bounding three-surface Σ.
B. Lagrangian formulation of four-vector fields in the flat space-time of special relativity
Before proceeding, a few preliminary definitions and assumptions are made. First, the nonzero components of the flat
space Minkowski metric tensor ηµν are taken as: −η00 = η11 = η22 = η33 = 1. So, the ordinary four-vector derivatives
are taken as: ∂µ = ((1/c)∂/∂t, ∇) and ∂ µ = (−(1/c)∂/∂t, ∇). Similarly, the position four-vector xν = (ct, x, y, z), and
so xν = (−ct, x, y, z). [The adoption of this (− + ++) signature metric in this article will aid in the comparison of
the results of this article with historical results expressed using complex Minkowski space notation, here taken as
xν = (ict, x, y, z).]
Next, it is assumed that the most general form of Lagrangian density for a four-vector field, which is no more than
quadratic in its variables and their derivatives, is given by the so-called Stueckelberg Lagrangian density,11,12 (in SI
units where 0 is the free space permittivity and c is the speed of light),
L=−
0 c2
λ0 c2
0 c2 µ2
2
Fµν F µν + jµ Aµ −
(∂µ Aµ ) −
(Aµ Aµ ) ,
4
2
2
(13)
where j µ = (ρc, j) is the usual four-vector current, where the positive real constant λ is a Lagrange multiplier for the
Lorentz constraint term, and where µ = 2π/λC = 2πmc/h is the Compton wave number for photons of mass m. A
choice of λ = 0 and µ = 0 yields what many physicists believe to be the electromagnetic theory, with its massless
photons, i.e., when an appropriate constraint is externally imposed. However, the choice of λ = 0 has the distinct
disadvantage of implying a vanishing momentum canonically conjugate to the zeroth component of the four-vector
potential Aν = (φ/c, A). The incorporation of the Lorentz constraint term, with its ∂φ/∂t functionality, eliminates
this deficiency, and yields an added bonus in terms of the ease of renormalization of the theory. A particularly simple
choice of λ = 1, (and µ = 0), then yields a Lagrangian density which is equivalent, (i.e., differs by no more than a fourdivergence), to the so-called Fermi Lagrangian density.13 The Fermi Lagrangian density is the most straightforward
take off point for field quantization in terms of harmonic oscillators which correspond to massless photons, (cf. Ref.
13). The Stueckelberg Lagrangian density (13) also has the advantage of explicitly including the four-divergence and
four-curl of Aµ , which are in turn sufficient for the unique specification of a four-vector field as is shown in Secs. II
F and III B. Therefore, the point of view is taken that the choice λ = 1 in (13) is the most natural choice for the
development of a uniqueness theorem for four-vector fields.
The Maxwell field tensor Fµν in terms of Aν is taken as following from a Bianchi identity, namely
∂λ Fµν + ∂µ Fνλ + ∂ν Fλµ = 0
⇒
Fµν = ∂µ Aν − ∂ν Aµ .
The covariant form of the Euler-Lagrange equations of motion
∂L
∂L
∂µ
−
= 0,
∂ (∂µ Aν )
∂Aν
then yields for the Stueckelberg Lagrangian density (13), the covariant equation of motion
∂µ ∂ µ − µ2 Aν − (1 − λ)∂ ν (∂σ Aσ ) = −j ν /0 c2 .
In the so-called Feynman gauge, one takes λ = 1 so that (16) reduces to the following:
(14a)
(14b)
(15)
(16)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4916
2Aν − µ2 Aν = −j ν /0 c2 ,
(17)
where 2 ≡ ∂µ ∂ µ is the d’Alembertian operator.
One can now rewrite the d’Alembertian operator acting on Aν by adding and subtracting a term, i.e.,
2Aν = ∂µ ∂ µ Aν − ∂µ ∂ ν Aµ + ∂µ ∂ ν Aµ .
(18)
The first two terms combine naturally using the field tensor F µν , while the ordinary four-vector derivatives in the last
term commute in flat space-time to yield the identity
2Aν ≡ ∂µ ∂ µ Aν = ∂µ F µν + ∂ ν (∂µ Aµ ) .
(19)
Equation (19) is a special relativistic generalization to 3+1 space-time of the well known three-space identity
∇2 A ≡ −∇×(∇×A) + ∇(∇ · A).
(20)
Specifically, the curl of A is generalized into the four-curl of Aν , (i.e., F µν ), the divergence of A is generalized into
the four-divergence of Aµ , (i.e., ∂µ Aµ ), and the Laplacian is generalized into the d’Alembertian.
This special relativistically invariant decomposition, using the identity (19), is the defining property which leads to a
uniqueness theorem for four-vector fields in a flat space-time. Indeed, in the same way that the delta function property
(10), with its Laplacian operator, leads to the Helmholtz identity (11a), so too does a more general Minkowski space
delta function property, with a d’Alembertian operator, lead to a new special relativistically invariant identity.
C. Integration of the four-vector potential wave equation
Assuming that the four-vector potential wave equation (17), as following from the Stueckelberg Lagrangian (13) with
λ = 1, is the most natural starting point for a four-dimensional analysis, one now proceeds in a manner analogous to
the three-dimensional case outlined in Sec. II A. Initially, however, µ is taken as zero yielding the massless four-vector
potential wave equation as
1 ∂2
j µ (xν )
µ
ν
2
2A (x ) ≡ ∇ − 2 2 Aµ (xν ) = −
.
(21)
c ∂t
0 c2
In Sec. IV B the mass term is restored and an attempt at a more general result is made. Now, the first step in the
analysis leading to a uniqueness theorem for classical four-vector fields is to integrate the four-vector potential wave
equation (21).
The integration of the massless four-vector potential wave equation (21) was first done in a special relativistically
invariant way by Arnold Sommerfeld in an important article in 1910.14 The method was later translated into English
and appeared in Volume III: Electrodynamics of his well known Lectures on Theoretical Physics.15 Additional revealing
descriptions of the method were given by Møller2 and Stratton16 in their classic texts.
To formulate the integral, Sommerfeld replaces Newton’s 1/r potential in three dimensions by an analogous fourdimensional scalar potential, i.e.,
U=
1
,
R2
where R is the distance in a Euclidean four-space between the source point x0ν and field point xν ,
p
R ≡ |xν − x0ν | = (xν − x0ν ) (xν − x0ν ),
(22)
(23)
and where the Euclidean metric tensor is just ηµν = I, the identity matrix. As pointed out by Møller,2 Sommerfeld
derives the integral in Euclidean space, and then in order to get physically reasonable results, he analytically continues
the four-current density by deforming the time integration from the real axis onto the imaginary axis in such a way as
to take timelike causality into account as is appropriate for the complex Minkowski space used in his treatment. On the
other hand, modern notation using covariant and contravariant indices is used in this partial review of Sommerfeld’s
method. Therefore, after the analytic continuation from Euclidean four-space to complex Minkowski space is done,
the results are adjusted to take into account the replacement of the Euclidean metric tensor by the Minkowski metric
tensor. The contour integration itself is described in Refs. 15 and 2.
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4917
Just as the three-space Laplacian of Newton’s 1/r potential is zero everywhere except at the source point r0 , [see
(8)], the Euclidean version of the d’Alembertian, (i.e., the four-space Laplacian ∆νν ), of the scalar potential U is
likewise zero everywhere except at the four-space source point x0ν . To demonstrate this one first calculates
∂
1
2 ∂R
2 (xν − x0ν )
=
−
=
−
,
(24)
∂xν R2
R3 ∂xν
R4
where use is made of the four-vector derivative of R as follows:
1 ∂ ((xν − x0ν ) (xν − x0ν ))
(xν − x0ν )
∂R
=
=
.
∂xν
2R
∂xν
R
It is obvious from the calculation (24) that one also has the useful identity:
∂
1
∂
1
=− ν
.
0ν
2
∂x
R
∂x
R2
The Euclidean four-space Laplacian of U now follows from a four-vector derivative contraction of (24) as
1
∂ ∂
1
∂
2 (xν − x0ν )
−2 · 4 2 · 4 (xν − x0ν ) ∂R
ν
∆ν
≡
=
−
=
+
.
R2
∂xν ∂xν R2
∂xν
R4
R4
R5
∂xν
Substitution of a contravariant version of (25) into (27) yields finally
1
ν
∆ν
=0
∀ xν 6= x0ν ,
R2
(25)
(26)
(27)
(28)
that is except at the singular point at R = 0.
Next, using a Green’s theorem integral approach, Sommerfeld obtains the Euclidean four-space integral: (cf. Refs.
15, 2, and 16 for details)
ν
φ (x ) = −
Z
V40
∂ 0µ ∂µ0 φ (x0ν ) 4 0
d x,
4π 2 R2 (xν , x0ν )
(29)
where the surface area 2π 2 R2 of a three-sphere of radius R is incorporated in the denominator of (29). As pointed out
by Møller2 Eq. (29) holds for any regular (i.e., analytic), function φ, so if φ = A0 c is taken as satisfying the zeroth
component of (21), i.e., of ∂ 0µ ∂µ0 φ(x0 ) = −ρ(x0 )/0 , one obtains
φ (xν ) =
Z
V40
ρ (x0ν )
d4 x0 .
(xν , x0ν )
4π 2 0 R2
(30a)
Equation (30a) allows one to calculate φ at every point in V40 when the source charge density ρ is known over all
00
of V40 . However, in
real life physical problems ρ is given only for purely imaginary x values corresponding to
Im x0 > Im x0 0 i.e., the zeroth component of (21) is a Minkowski space relation and the integral (30a) can no
longer be limited to Euclidean space. This timelike causality assumption requires that a signal from the source point
reaches the field point only after traveling the distance R at a the finite speed c. Therefore, to take into account the
timelike causal data required for the charge density ρ, one can then analytically continue in x00 = ct0 in the integrand
of (30a). The analytically continued version of (30a) is therefore
Z I
ρ (x0ν )
ν
0
φ (x ) =
c dt d3 x0 ,
(30b)
2
2
ν
0ν
V0
t0 4π 0 R (x , x )
where a suitable integration contour in the complex t0 plane has to be chosen to satisfy timelike causality of the data.
That is, such a contour integration is taken as being compatible with timelike causality. Performance of the integral
(30b) under the specified timelike causal data yields, in terms of a “retarded” charge density, a Minkowski space
result. In order to obtain results in modern metric notation, the complex Minkowski space result is then mapped
to a Minkowski metric space by replacement of the Euclidean metric tensor by the Minkowski metric tensor, (thus
absorbing any factors of the imaginary unit i), which here amounts to retention of raised and lowered index notation.
An analytic continuation in the complex t0 plane of a time integral of the type in (30b) is detailed in Refs. 15 and 2.
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4918
Again, as pointed out by Møller,2 since Eq. (29) holds for any regular function of φ it will hold for the A1 , A2 , and
A spatial Cartesian components of Aµ in the massless four-vector potential wave equation (21). The combined result
for the four-vector potential Aµ at the space-time field point xν , is therefore given by a four-dimensional integration
over the space-time source coordinates x0ν of the timelike causal four-vector current density j µ , i.e.,
Z
Z
j µ (x0ν )
µ0 j µ (x0ν )
4 0
Aµ (xν ) =
d
x
=
d4 x0 ,
(31)
2
2
2
ν
0ν
2 |xν − x0ν |2
0
0
4π
c
R
(x
,
x
)
0
4π
V4
V4
3
where the integral (31) is to be interpreted as a Minkowski space integral in the same sense as in (30b). Equation
(31) is the integral result first obtained by Sommerfeld,14 retraced here in modern notation, as desired.
III. DERIVATION OF UNIQUENESS THEOREMS
A. Green’s function approach and a delta function identity
Rather than pursuing further the inhomogeneous four-vector wave equation (21), it turns out to be more convenient
for the purpose of this article to focus instead on the inhomogeneous scalar wave equation for the scalar potential φ,
ρ (xν )
1 ∂2
2φ (xν ) = ∇2 − 2 2 φ (xν ) = −
.
(32)
c ∂t
0
The wave equation (32) can then be solved in terms of a two-point scalar Green’s function G (xν , x0ν ) which connects
its unit delta function source located at the space-time source point x0ν to a measurement at the space-time field
point xν in Minkowski space, i.e.,
1 ∂2
ν
0ν
2
2G (x , x ) = ∇ − 2 2 G (xν , x0ν ) = −δ (4) (xν − x0ν ) ,
(33)
c ∂t
where δ (4) (xν − x0ν ) = δ x0 − x00 δ 3 (r − r0 ). If the Green’s function G (xν , x0ν ) is known over all space-time, the
inhomogeneous solution of (32) then follows from the Minkowski space integral
Z
ρ (x0ν )
φ (xν ) =
G (xν , x0ν ) d4 x0 .
(34)
0
0
V4
Interestingly, Eq. (34) is of the same general form as the Euclidean space integral (30a). Therefore, if one starts with
(30b), i.e., the analytic continuation of (30a) to Minkowski space, it should be possible to extract a Green’s function
G (xν , x0ν ) through simple comparison of (34) and (30b), which would integrate to the same result under mutually
compatible causality conditions. As additional evidence notice that for xν 6= x0ν , Eq. (33) reduces to the following
Minkowski space result:
2G (xν , x0ν ) = 0
∀ xν 6= x0ν .
(35)
The Euclidean four-space result (28) has the same general form as Eq. (35). And, the result (28) holds in Minkowski
space as well, (since e.g. ∂ ν xν = 4 in either space). This implies, (by the uniqueness of the solutions of the Cauchy
problem for constant coefficient wave equations), that the Green’s function G (xν , x0ν ) can be taken as the scalar
field U = 1/R2 times a numerical constant. It is reasonable to assume therefore that the numerical constant can be
obtained from a comparison of (30b) and (34) and is just 1/4π 2 . The desired Green’s function G (xν , x0ν ) is therefore
given by the ansatz
G (xν , x0ν ) =
2
1
,
4π 2 R2
(36)
where R2 = |xν − x0ν | . It should be emphasized that (36) is an appropriate Green’s function for Minkowski space only
in the context of an analytically continued time integral as in (30b), and therefore appropriate causality conditions
must subsequently be applied to obtain physically reasonable results. Equation (36) would of course be suitable in a
Euclidean four-space without any assumptions involving analytic continuation.
The result (36) differs from the familiar retarded Green’s function as derived for example in Cushing17 or Jackson,18
which for the metric signature (− + ++) and sign of the source term in (32) would be
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4919
Gret (r, r0 ; t, t0 ) =
1 δ (|r − r0 | − c (t − t0 ))
4π
|r − r0 |
∀ t > t0 .
(37)
This is because the retarded Green’s function (37) is derived via a spectral decomposition of the delta function, while
also taking into account homogeneous boundary conditions on a closed spatial surface, as well as timelike causality
with the initial conditions
G (r, r0 ; t, t0 ) = 0,
∂G (r, r0 ; t, t0 ) /∂t = 0
∀ t > t0 ,
(38)
and with the Green’s function symmetry relation
G (r, r0 ; t, t0 ) = G (r0 , r; −t0 , −t) .
(39)
On the other hand, in obtaining the result (36), no such spectral decomposition is made and the calculation of the
time integral, with an appropriate contour taking causality into account, is performed only later in this article. In
contradistinction to the Green’s function (37), (36) might therefore be referred to as an “acausal” Minkowski space
Green’s function, or more simply as a Euclidean four-space Green’s function.
Substitution of (36) into a Euclidean space version of (33) then yields a delta function identity over all of the
Euclidean four-space R4 as follows:
1
∆αα
= −δ (4) (xν − x0ν )
∀ xν , x0ν ∈ R4 .
(40)
4π 2 R2
Although (40) follows from ostensibly Euclidean space calculations it can be applied to a space-time integral when
an analytic continuation of the time integral is performed under appropriate causality conditions in order to obtain a
Minkowski space result.
Equation (34) with the Green’s function (36) is, under suitable analytic continuation of the time integral, the same
result as (30b). [Indeed, the numerical constant 1/4π 2 in (36) was obtained by a short cut comparison between the
two.] And since the A1 , A2 , A3 spatial Cartesian components of (21) separately satisfy a scalar wave equation like
(32), one can then write the inhomogeneous solution of (21) for the four-vector potential Aµ as
Z
Z I
j µ (x0ν )
j µ (x0ν )
ν
0ν
4 0
0
G
(x
,
x
)
d
x
=
c
dt
d3 x0 ,
(41)
Aµ (xν ) =
2
2 c2 R2 (xν , x0ν )
0
c
4π
0
0
0
0
V4
V
t
which is the same result as (31). Compare (36), (40), and (41) with the three-dimensional case (8) and (9). It is
shown in Sommerfeld15,14 in terms of retarded potentials that an expression like (9), as well as a scalar potential
integral, follows from (31) via a contour integration over the time coordinate.
Although the result (31) is a historically important result, it is the delta function property of identity (40) which
is of importance in the proof of a vector identity in Minkowski space analogous to the Helmholtz identity, namely
Z
Z I
−1
0
Aµ (xν ) =
Aµ (x0ν ) δ (4) (xν − x0ν ) d4 x0 =
Aµ (x0ν ) ∆αα
c
dt
d3 x0 .
(42)
4π 2 R2
V40
V0
t0
Equation (42) and a Euclidean four-space version of (42) are used by the author in the next section to derive four-vector
identities analogous to the three-vector Helmholtz identity (11a).
B. Euclidean and Minkowski four-space analogues of the Helmholtz identity
Equation (42) is now used by the author to derive four-vector identities analogous to, but more general than, the
three-vector Helmholtz identity (11a). To a certain extent, the derivation parallels the three-vector derivation of (11a)
as detailed in Refs. 6 and 7.
Initially, a theorem containing a four-vector identity is stated and proved for Euclidean space. Next, a corollary
to the Euclidean theorem is stated for four-vectors in Minkowski space based on the contour integral used in (42).
Subsequently, a second theorem containing a four-vector identity is stated which is based on a direct relativistically
invariant integration over Minkowski space involving a Minkowski space Green’s function of the form (37).
It is assumed in what follows, unless otherwise stated, that the term “sufficiently smooth” refers to functions which
are scalar fields or components of vector fields that are C 2 V 4 , i.e., twice continuously differentiable functions on
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4920
the closure of V4 which in turn is comprised of the union of V4 and its bounding three-surface Σ (i.e., V 4 = V4 ∪ Σ).
The boundary three-surface Σ is itself assumed to be sufficiently smooth.
Theorem I: The following identity holds for sufficiently smooth four-vector fields Aµ (xσ ) in the Euclidean fourspace R4 :
"Z
I
Aν (x0σ ) n0ν
∂ν0 Aν (x0σ )
4 0
0
µ
σ
µ
d
x
−
dΣ
A (x ) = − ∂
2 2
σ
0σ
2 2
σ
0σ
V40 4π R (x , x )
Σ0 4π R (x , x )
"Z
#
I
∂ 0α Aµ (x0σ ) − ∂ 0µ Aα (x0σ ) 4 0
Aα (x0σ ) n0µ − Aµ (x0σ ) n0α 0
− ∂α
d x +
dΣ ,
(43)
4π 2 R2 (xσ , x0σ )
4π 2 R2 (xσ , x0σ )
V40
Σ0
2
where R2 (xσ , x0σ ) = |xσ − x0σ | , and where n0µ is the four-vector outward unit normal of the three-surface Σ0 which
encloses the four-volume V40 .
Proof: The proof is based on a Euclidean space version of the four-vector delta function property (42), namely
Z
Z
−1
µ
ν
µ
0ν
(4)
ν
0ν
4 0
µ
0ν
α
A (x ) =
A (x ) δ (x − x ) d x =
A (x ) ∆α
d4 x0 .
(44)
4π 2 R2
V40
V40
The important thing to realize about (44) is that the Euclidean four-space Laplacian operator acts only on the fourspace field coordinates xν and so for sufficiently smooth four-vector fields Aµ (x0ν ) it can be brought outside of the
integration over the four-space source coordinates x0ν as follows:
Z
Z
−1
Aµ (x0ν ) 4 0
µ
ν
µ
0ν
α
4 0
α
A (x ) =
A (x ) ∆α
d
x
=
−∆
d x.
(45)
α
2 2
4π 2 R2
V40
V40 4π R
At this point, one makes a decomposition of the Euclidean Laplacian operator ∆α
α , analogous to that discussed in
(18), by adding and subtracting a term in (45) yielding
Z
Z
Z
Aµ (x0 ) 4 0
Aα (x0 ) 4 0
Aν (x0 ) 4 0
µ
α
µ
µ
A (x) = −∂α ∂
d
x
+
∂
∂
d
x
−
∂
∂
d x,
(46)
α
ν
2 2
2 2
2 2
V40 4π R
V40 4π R
V40 4π R
where the four-vector superscripts in the functional dependencies are again suppressed. Now, in Euclidean space, (and
also in flat Minkowski space), one can commute the derivatives in the third term of (46). And since the four-vector
derivatives act only on the field coordinates they can be passed inside the integrations over the source coordinates.
Also identity (26) and its contravariant derivative counterpart
1
1
µ
0µ
∂
= −∂
,
(47)
R2
R2
allow one to change the unprimed field point derivatives of 1/R2 in (46) into primed source point derivatives. The
net result is that one can rewrite (46) as
"Z
#
"Z
#
Z
Aµ (x0 ) 0α 1
Aα (x0 ) 0µ 1
Aν (x0 ) 0
1
µ
4 0
4 0
µ
4 0
A (x) = ∂α
∂
d x −
∂
d x +∂
∂
d x .
(48)
4π 2
R2
4π 2
R2
4π 2 ν R2
V40
V40
V40
Setting the bracketed part of the first term of (48) equal to Aαµ , i.e.,
"Z
#
Z
Aµ (x0 ) 0α 1
Aα (x0 ) 0µ 1
αµ
4 0
4 0
A ≡
∂
d x −
∂
d x ,
4π 2
R2
4π 2
R2
V40
V40
and the bracketed part of the second term of (48) equal to A , i.e.,
"Z
#
Aν (x0 ) 0
1
4 0
A≡
∂
d x ,
4π 2 ν R2
V40
allows one to write (48) as
(49)
(50)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4921
Aµ = ∂α Aαµ + ∂ µ A,
(51)
which is reminiscent of the decomposition (19).
Working on (50) first using the four-vector derivative product rule while using a Euclidean four-space version of
Gauss’ divergence theorem on the appropriate term of the resulting equation then yields
I
Z
Aν (x0 ) n0ν 0
∂ν0 Aν (x0 ) 4 0
A=
dΣ
−
d x,
(52)
2 2
4π 2 R2
Σ0
V40 4π R
where n0ν is the four-vector outward unit normal of the three-surface Σ0 which encloses the four-volume V40 . Equation
(52) is in its final form.
Working on (49) next using the four-vector derivative product rule on both terms yields after some rearrangement
the following:
µ 0 α 0 Z
Z
Z
A (x )
A (x )
∂ 0α Aµ (x0 ) − ∂ 0µ Aα (x0 ) 4 0
4 0
0µ
4 0
Aαµ =
∂ 0α
d
x
−
∂
d
x
−
d x.
(53)
2
2
2
2
4π R
4π R
4π 2 R2
V40
V40
V40
The third term of (53) is in its final form. It remains then to show that the four-vector derivative ∂α [see (51)], of the
first two terms of (53) combine to yield the Euclidean four-space identity
α 0 µ 0 Z I
A (x )
A (x )
Aα (x0 ) n0µ − Aµ (x0 ) n0α 0
0α
0µ
4 0
∂α
∂
−
∂
d
x
=
−∂
dΣ .
(54)
α
4π 2 R2
4π 2 R2
4π 2 R2
V40
Σ0
To prove (54), one can define a four-vector aµ with constant magnitude and constant but arbitrary direction in
Euclidean four-space which is chosen once and then held fixed. Next consider the two point function
Aα (x0σ )
Kµ (xσ , x0σ ) ≡ aµ ∂α
.
(55)
4π 2 R2 (xσ , x0σ )
Using a Euclidean four-space version of Gauss’ divergence theorem on (55), namely
Z
I
(∂ 0µ Kµ ) d4 x0 =
Kµ n0µ dΣ0 ,
V40
(56)
Σ0
where in (56) and in what follows n0µ is the four-vector outward unit normal of the three-surface Σ0 which encloses
the four-volume V40 , one obtains
Z I Aα (x0σ )
Aα (x0σ )
4 0
aµ
∂ 0µ ∂α
d
x
=
a
∂
n0µ dΣ0 ,
(57)
µ
α
4π 2 R2 (xσ , x0σ )
4π 2 R2 (xσ , x0σ )
V40
Σ0
where the four-vector aµ factors out of the integral since it is a constant. Now, since |aµ | =
6 0 and aµ has an arbitrary
fixed direction, then its four-contractions in (57) cannot everywhere vanish and so (57) reduces to the identity
Z I Aα (x0σ )
Aα (x0σ )
4 0
∂α
∂ 0µ
d
x
=
∂
n0µ dΣ0 ,
(58)
α
2 R2 (xσ , x0σ )
2 R2 (xσ , x0σ )
0
4π
4π
0
V4
Σ
where the field point derivatives ∂α have been moved out of the source point integrations and where the field and
source point derivatives on the left-hand side (lhs) of (58) have been commuted. Equation (58) shows that the second
term on the lhs of (54) is equal to the first term on the right-hand side (rhs) of (54). The remaining two terms in (54)
follow in a similar fashion. One can again define a four-vector aµ with constant magnitude and constant but arbitrary
direction in the Euclidean four-space which is chosen once and then held fixed. Next consider the two point function
Aµ (x0σ )
σ
0σ
Iα (x , x ) ≡ aµ ∂α
.
(59)
4π 2 R2 (xσ , x0σ )
Using (59) in the Euclidean four-space Gauss’ divergence theorem (56) one obtains
Z I Aµ (x0σ )
Aµ (x0σ )
4 0
aµ
∂ 0α ∂α
d
x
=
a
∂
n0α dΣ0 ,
µ
α
2 R2 (xσ , x0σ )
2 R2 (xσ , x0σ )
0
4π
4π
0
V4
Σ
(60)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4922
where the four-vector aµ factors out of the integral since it is a constant. Now, since |aµ | =
6 0 and aµ has an arbitrary
fixed direction, then its four-contractions in (60) cannot everywhere vanish and so (60) reduces to the identity
Z I Aµ (x0σ )
Aµ (x0σ )
0α
4 0
∂α
∂
d x = ∂α
n0α dΣ0 ,
(61)
4π 2 R2 (xσ , x0σ )
4π 2 R2 (xσ , x0σ )
V40
Σ0
where the field point derivatives ∂α have been moved out of the source point integrations and where the field and
source point derivatives on the lhs of (60) have been commuted. Equation (61) shows that the first term on the lhs
of (54) is equal to the second term on the rhs of (54), which when combined with (58) completes the proof of the
Euclidean four-space identity (54).
Combining the results (51), (52), (53), and (54) completes the proof of the Euclidean four-space identity (43) and
Theorem I.
♣
Note the similarity in structure of identity (43) and the Helmholtz identity (11a). In particular, the factors of
1/4πr in (11a) appropriate for spherically symmetric functions in R3 are replaced in (43) by factors of 1/4π 2 R2 which
are appropriate for hyperspherically symmetric functions in R4 . It appears then that identity (43) is a Euclidean
four-space generalization of the (Euclidean three-space) Helmholtz identity (11a). Therefore, just as (11a) can be
used to prove the Helmholtz uniqueness theorems H2 and H1 of (static) three-vector fields, identity (43) is used later
in this article to prove a uniqueness theorem for four-vector fields in the Euclidean four-space R4 .
A corollary is now stated which extends the Euclidean four-space Theorem I to Minkowski space.
Corollary I to Theorem I: When, in the complex t0 plane, a time integration contour is taken which is compatible with
timelike causality, and for sufficiently smooth four-vector fields Aµ (xσ ), the following identity holds in the Minkowski
space R3+1 :
Z I 0 ν 0σ
ν 0σ Z I
∂ν A (x )
A (x )
µ
σ
µ
0
3 0
0
0
3 0
A (x ) = − ∂
c dt d x −
∂ν
c dt d x
4π 2 R2
4π 2 R2
0
0
V0
t0
ZV It 0α µ 0σ
∂ A (x ) − ∂ 0µ Aα (x0σ )
c dt0 d3 x0
− ∂α
4π 2 R2 (xσ , x0σ )
V0
t0
µ 0σ α 0σ Z I
Z I
A (x )
A (x )
0α
0
3 0
0µ
0
3 0
−
∂
c dt d x +
∂
c dt d x ,
(62)
4π 2 R2
4π 2 R2
V0
t0
V0
t0
2
where R2 (xσ , x0σ ) = |xσ − x0σ | in R4 , and where V 0 is a spatial volume in R3 .
Proof: The proof is based on the four-vector delta function property (42) and in all important respects parallels
the proof of the Euclidean four-space identity (43) of Theorem I. The details of the basic approach are not repeated
here. However, in order to express the time integration in terms of a contour integral over the complex t0 plane, it is
convenient to forgo the transformations from four-volume to three-surface integrals as in (52) and (54). The result is
identity (62). An integration contour compatible with timelike causality is specified in order to interpret the identity
as a Minkowski space integral. These contours are discussed in general in Refs. 15 and 2. The contour integration
is performed using the residue theorem and assumes that the integrand, aside from the poles in the 1/R2 factor, is
analytic and without singularities over the domain enclosed by or on the chosen contour. The actual details of the
choice of contour, as well as performing the integration itself are not necessary for the proof of Corollary I to Theorem
I. It is only necessary for the sake of a physical interpretation that a contour compatible with timelike causality is
chosen.
♣
The similarity in structure of identity (62) and the Helmholtz identity (11a) is now only apparent through the fourvolume integrals in (62) containing the four-divergence and four-curl of Aν . The normal and tangential three-surface
integrals in (43) no longer appear in (62). That (62) can be interpreted as a Minkowski space integral instead of an
integral in R4 is a result of the analytic continuation of the time integral. Identity (62) could be interpreted as a
Minkowski space generalization of the (Euclidean three-space) Helmholtz identity (11a).
However, identity (62) is not of the same general form as identity (11a) because it lacks surface integral terms.
Consequently, identity (62) is not a convenient starting point for proving a four-vector uniqueness theorem analogous
to the three-space Helmholtz uniqueness theorem H2. Therefore, a second theorem is now developed in R3+1 , through
a direct integration in Minkowski space, which states an identity that is formally analogous to identity (43) in R4 .
It is central to the development of this second theorem that identities for the four-vector derivatives of the Green’s
functions satisfying (33) in R3+1 analogous to (26) and (47) in R4 be obtained. A reciprocal four-space integral
representation of the Green’s functions satisfying (33) appears to be the easiest way to obtain the desired derivative
properties. Therefore, a spectral decomposition of the delta function δ (4) (xν − x0ν ) appearing in (33) is made via the
Fourier integral
J. Math. Phys., Vol. 40, No. 10, October 1999
1
δ (4) (xν − x0ν ) = q
4
(2π)
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4923
Z


Z
−ikµ x0µ
µ
0µ
µ
e
q
 eikµ x d4 k = 1
eikµ (x −x ) d4 k.
4
4
(2π)
(2π)
In the usual way, Green’s functions satisfying (33) then follow from
!
Z
Z ikµ (xµ −x0µ )
1
1
e
ikµ (xµ −x0µ ) 4
ν
0ν
−1
d4 k.
G (x , x ) = −2
e
d
k
=
4
4
k2
(2π)
(2π)
(63)
(64)
The desired covariant and contravariant derivative properties in R3+1 follow immediately from the result (64), for any
appropriate contour, as
∂µ G (xν , x0ν ) = −∂µ0 G (xν , x0ν ) ,
µ
0ν
ν
0µ
ν
0ν
∂ G (x , x ) = −∂ G (x , x ) ,
(65a)
(65b)
which are of the same general form as the properties (26) and (47) in R4 , as might be expected.
An example of a Green’s function which satisfies (64) is the retarded Green’s function (37). However, as a manifestly
covariant identity is desired for this theorem, the retarded Green’s function, and optionally an advanced Green’s
function, must be able to be restated in relativistically covariant form. This is readily done using the delta function
identity18
2
2
2
δ (xν − x0ν ) = δ |r − r0 | − c2 (t − t0 )
= δ ((|r − r0 | − c (t − t0 )) (|r − r0 | + c (t − t0 )))
δ ((|r − r0 | − c (t − t0 )) + δ (|r − r0 | + c (t − t0 )))
=
.
2 |r − r0 |
(66)
Using (66), the retarded Green’s function (37), along with an advanced Green’s function Gadv , can be stated in
relativistically covariant form as follows:18
1
2
θ x0 − x00 δ (x − x0 ) ,
2π
1
2
0
θ x00 − x0 δ (x − x0 ) ,
Gadv (x, x ) =
2π
Gret (x, x0 ) =
(67a)
(67b)
where the theta function is defined as follows:
0
00
θ x −x
=
1
0
for x0 > x00
.
for x00 > x0
(68)
A theorem can now be stated for four-vector fields in Minkowski space.
Theorem II: Given a covariant scalar two-point Green’s function G (xν , x0ν ) which is a solution of (33) and which
satisfies the derivative properties (65a,b), the following identity holds for sufficiently smooth four-vector fields Aµ (xσ )
in the Minkowski space R3+1 :
"Z
#
I
Aµ (x) = −
−
V40
"Z
V40
∂ µ ((∂ν0 Aν (x0 )) G (x, x0 )) d4 x0 −
Σ0
∂ µ ((Aν (x0 ) n0ν ) G (x, x0 )) dΣ0
I
∂α ((∂ 0α Aµ (x0 ) − ∂ 0µ Aα (x0 )) G (x, x0 )) d4 x0 +
Σ0
∂α ((Aα (x0 ) n0µ − Aµ (x0 ) n0α ) G (x, x0 )) dΣ0 , (69)
where n0µ is the four-vector outward unit normal of the three-surface Σ0 which encloses the four-volume V40 , and where
the three-surface Σ0 is defined covariantly with respect to a general Lorentz transformation.
Proof: In contradistinction to identity (43) of Theorem I, the unprimed four-vector derivatives are now included
in the integrands of identity (69) of Theorem II. Consequently, the convergence properties of these integrands are
improved in comparison to those in (43) and so the integrals in (69) are well defined for four-vector fields Aµ falling
off only as fast as 1/r. For sufficiently smooth four-vector fields the unprimed derivatives can still be factored out of
the integrals over the primed coordinates if required.
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4924
The proof is based on a Minkowski space version of the four-vector delta function property (42), but now written in
terms of a covariant scalar two-point Green’s function G (xν , x0ν ) which is assumed to be a solution of (33), as follows:
Z
Z
µ
ν
µ
0ν
(4)
ν
0ν
4 0
A (x ) =
A (x ) δ (x − x ) d x =
Aµ (x0ν ) 2 (−G (xν , x0ν )) d4 x0 .
(70)
V40
V40
At this point, one makes a decomposition of the d’Alembertian operator as in (18), by adding and subtracting a term
in (70), yielding
Z
Z
Z
µ
α
µ
0
0
4 0
µ
α
0
0
4 0
A (x) = −
∂α ∂ (A (x ) G (x, x )) d x +
∂α ∂ (A (x ) G (x, x )) d x −
∂ν ∂ µ (Aν (x0 ) G (x, x0 )) d4 x0 ,
V40
V40
V40
(71)
where the four-vector superscripts in the functional dependencies are again suppressed. The rest of the proof proceeds
in a manner parallel to the proof of the Euclidean four-space identity (43) of Theorem I and most of the details will be
condensed. In flat Minkowski space one can commute the derivatives in the third term of (71). Then certain unprimed
derivatives can be commuted with primed coordinate dependent fields. Then using identities (65a) and (65b) one can
change the unprimed field point derivatives of G (x, x0 ) in (71) into primed source point derivatives. The net effect of
these changes allows one to rewrite (71) as
"Z
#
"Z
#
Z
Aµ (x) = ∂α
V40
Aµ (x0 ) ∂ 0α G (x, x0 ) d4 x0 −
Aα (x0 ) ∂ 0µ G (x, x0 ) d4 x0 + ∂ µ
V40
V40
Aν (x0 ) ∂ν0 G (x, x0 ) d4 x0 ,
(72)
where for sufficiently smooth four-vector fields, certain unprimed derivatives have been factored out of the primed
coordinate integrals of (72) for later reference. Setting the first bracketed term of (72) equal to Aαµ , i.e.,
#
"Z
Z
Aαµ ≡
V40
Aµ (x0 ) ∂ 0α G (x, x0 ) d4 x0 −
Aα (x0 ) ∂ 0µ G (x, x0 ) d4 x0 ,
and the bracketed part of the second term of (72) equal to A , i.e.,
"Z
A≡
0
ν
A (x
V40
(73)
V40
) ∂ν0 G (x, x0 ) d4 x0
#
,
(74)
allows one to write (72) as
Aµ = ∂α Aαµ + ∂ µ A,
(75)
which is reminiscent of the decomposition (19). This decomposition of a four-vector field into the sum of a fourirrotational and a four-solenoidal part will be used in Sec. IV B in connection with Theorem X.
Next, working on the second bracketed term of (72) first (but without the unprimed derivatives factored out),
using the four-vector derivative product rule, while at the same time using a Minkowski four-space version of Gauss’
divergence theorem as appropriate for a second rank tensor integrand, (cf. p. 130 of Ref. 2), namely
Z
I
I
0 µν 4 0
µν
0
∂ν Z d x =
Z dσν =
Z µν n0ν dΣ0 ,
(76)
V40
Σ0
Σ0
on the appropriate term of the resulting equation (i.e., after commuting ∂ µ and ∂ν0 ), then yields
Z
I
Z
∂ µ (Aν (x0 ) ∂ν0 G (x, x0 )) d4 x0 =
∂ µ (Aν (x0 ) n0ν G (x, x0 )) d4 x0 −
∂ µ ((∂ν0 Aν (x0 )) G (x, x0 )) d4 x0 ,
V40
Σ0
(77)
V40
where n0ν is the four-vector outward unit normal of the three-surface Σ0 which encloses the four-volume V40 . Equation
(77) is in its final form.
Working on the first bracketed term of (72) next (but again without the unprimed derivatives factored out), using
the four-vector derivative product rule on both terms yields after some rearrangement the following:
J. Math. Phys., Vol. 40, No. 10, October 1999
Z
V40
∂α (Aµ (x0 ) ∂ 0α G (x, x0 )) d4 x0 −
−
Z
V40
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4925
Z
∂α (Aα (x0 ) ∂ 0µ G (x, x0 )) d4 x0 =
V40
Z
∂α ∂ 0α (Aµ (x0 ) G (x, x0 )) d4 x0
V40
∂α ∂ 0µ (Aα (x0 ) G (x, x0 )) d4 x0 −
Z
V40
∂α ((∂ 0α Aµ (x0 ) − ∂ 0µ Aα (x0 )) G (x, x0 )) d4 x0 .
(78)
The third term on the rhs of (78) is in its final form. It can be shown in an entirely analogous manner as for
the Euclidean four-space identity (54), (using a four-vector aµ with constant magnitude and constant but arbitrary
direction in Minkowski space), that the first two terms on the rhs of (78) combine to yield the Minkowski space
identity
Z
Z
0α
µ
0
0
4 0
∂α ∂ (A (x ) G (x, x )) d x −
∂α ∂ 0µ (Aα (x0 ) G (x, x0 )) d4 x0
V40
V40
=−
I
Σ0
∂α ((Aα (x0 ) n0µ − Aµ (x0 ) n0α ) G (x, x0 )) dΣ0 .
(79)
Combining the results (77), (78), and (79) completes the proof of the Minkowski space identity (69) and Theorem
II.
♣
In passing, consider a representative case where the Green’s function is chosen to be the covariant retarded Green’s
function (67a). A convenient covariant three-surface Σ0 , (bounding a Minkowski four-volume V40 ), could then be taken
as the union of a finite section of the forward light cone Σ0C , (a lightlike hypersurface), and its end cap comprised for
example by an intersecting three-spherical conic section Σ0cap (a spacelike hypersurface). The outward surface normal
n0µ to the forward light cone is a spacelike unit four-vector, while the outward surface normal to the three-spherical
end cap of a finite section of the forward light cone is a timelike unit four-vector. But since the forward light cone is
the only surface where the retarded Green’s function is different than zero, only the surface integrals over the finite
section of the forward light cone Σ0C can be nonzero!
Finally, note the similarity in structure of the Minkowski space identity (69) and the Euclidean four-space identity
(43). Aside from the incorporation of the unprimed derivatives in the integrals of (69) (which was done to improve
the convergence of the integrals), the only difference is that the Euclidean four-space Green’s function 1/4π 2 R2 is
replaced by an appropriate Minkowski space Green’s function G(x, x0 ).
It will be shown in Sec. III D that identity (69) is a Minkowski space generalization of the (Euclidean three-space)
Helmholtz identity (11a). Therefore, just as (11a) can be used to prove the Helmholtz uniqueness theorems H2 and
H1 of (static) three-vector fields, identity (69) will be used in Secs. III E and IV B to prove uniqueness theorems for
four-vector fields in R3+1 .
C. Scalar field identity in Euclidean three-space
The Euclidean three-space identity (8) is next substituted into a delta function property as follows:
Z
Z
−1
0
3
0
3 0
0
2
φ(r) =
φ (r ) δ (r − r ) d r =
φ (r ) ∇
d3 r0 .
4πr
V0
V0
(80)
Identity (80) is now used to state a theorem.
Theorem III: The following identity holds for twice continuously differentiable (static) scalar fields φ in the
Euclidean three-space R3 :
Z
I
−∇0 φ (r0 ) 0
φ (r0 ) n0 0
φ(r) = ∇ ·
dV +
dS ,
(81)
4πr
4πr
V0
S0
where the three-dimensional distance r ≡ |r − r0 |, and where n0 is the three-vector outward unit normal of the twosurface S 0 which encloses the three-volume V 0 .
Proof: Starting with (80), for a twice continuously differentiable field φ, one can factor part of the Laplacian operator
acting on the unprimed field point coordinates out of the integration over the primed source point coordinates to yield
Z
−φ (r0 )
1 3 0
φ(r) = ∇ ·
∇
d x .
(82)
4π
r
V0
Using the vector identity
J. Math. Phys., Vol. 40, No. 10, October 1999
∇
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4926
1
|r − r0 |
= −∇0
1
|r − r0 |
,
(83)
one can rewrite (82) as
φ(r) = ∇ ·
Z
V0
φ (r0 ) 0 1 3 0
∇
d x .
4π
r
(84)
The vector identity
∇ (φψ) = φ∇ψ + ψ∇φ,
(85)
with ψ ≡ 1/|r − r0 | becomes
∇
φ
|r − r0 |
= φ∇
φ (r0 )
r
1
|r − r0 |
+
∇φ
,
|r − r0 |
(86)
allowing one to rewrite (84) as
φ(r) = ∇ ·
Z
1
4π
V0
∇
0
∇0 φ (r0 )
−
r
3 0
d x .
(87)
Now, using the vector identity7
Z
V
∇Φ dV =
I
Φ n dS,
(88)
S
which follows from Gauss’ divergence theorem in a flat space-time, on the first term of (87) yields immediately
identity (81), proving Theorem III. Identity (81) is of a form which is reminiscent of the Helmholtz identity (11a),
but is applicable to static scalar fields.
♣
D. Derivation of the Helmholtz identity and a scalar field identity from a Minkowski space Helmholtz identity
The author will now show that identity (69) of Theorem II is a Minkowski space generalization of the (Euclidean
three-space) Helmholtz identity (11a) in a static Newtonian limit. In addition, the fourth or scalar field component
of (69) will be shown to yield identity (81) in a similar static Newtonian limit.
To prove that the Helmholtz identity (11a) follows from identity (69) in a static Newtonian limit, it is convenient
to start with an identity which follows from (69), i.e.,
"Z
#
ν 0
Z
0 ν
0
∂
(x
)
A
A
(x
)
ν
Aµ (x) = − ∂ µ
δ (r − c (t − t0 )) d4 x0 −
δ (r − c (t − t0 )) d4 x0
∂ν0
4πr
4πr
V40
V40
"Z
Z
∂ 0α Aµ (x0 )
∂ 0µ Aα (x0 )
− ∂α
δ (r − c (t − t0 )) d4 x0 −
δ (r − c (t − t0 )) d4 x0
4πr
4πr
V40
V40
#
µ 0
α 0
Z
Z
A
(x
)
A
(x
)
−
∂ 0α
δ (r − c (t − t0 )) d4 x0 +
∂ 0µ
δ (r − c (t − t0 )) d4 x0 ,
(89)
4πr
4πr
V40
V40
where r ≡ |r − r0 |. Identity (89) follows from (69) as a result of using the retarded Green’s function (37), Gauss’
divergence theorem on the surface integral in the first bracketed term of (69), and identity (79) on the surface integral
in the second bracketed term of (69), while at the same time factoring the unprimed derivatives out of all the integrals.
One next takes the following as an intermediate approximation of the space components of identity (89):
"Z
#
k 0
Z
A (x )
∂k0 Ak (x0 )
j
j
0
4 0
0
0
4 0
A (x) = − ∂
δ (r − c (t − t )) d x −
∂k
δ (r − c (t − t )) d x
4πr
4πr
V40
V40
"Z
Z
∂ 0i Aj (x0 )
∂ 0j Ai (x0 )
0
4 0
− ∂i
δ (r − c (t − t )) d x −
δ (r − c (t − t0 )) d4 x0
4πr
4πr
V40
V40
#
j 0
i 0
Z
Z
A (x )
A (x )
0i
0
4 0
0j
0
4 0
−
∂
δ (r − c (t − t )) d x +
∂
δ (r − c (t − t )) d x .
(90)
4πr
4πr
V40
V40
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4927
Note, all terms involving partial time derivatives, e.g., ∂0 = (1/c)∂/∂t, are omitted in (90) since a detailed dimensional
analysis in the speed of propagation c shows that these terms vanish in a Newtonian limit where c → ∞. The author
will delay the application of the Newtonian limit for the terms retained in (90), however, until later in this derivation.
As usual, Roman indices i,j,k, etc., are used here and in what follows to denote three-vectors. One next performs the
t0 time integrations in (90) yielding
Z
Z
∇0 · A (r0 , t − r/c) 0
A (r0 , t − r/c)
A (r, t) = − ∇
dV −
∇0 ·
dV 0
4πr
4πr
V0
V0
Z
Z
0
0
∇ ×A (r , t − r/c) 0
A (r0 , t − r/c)
+ ∇×
dV −
∇0 ×
dV 0 .
(91)
4πr
4πr
V0
V0
It should be noted in passing that (91) is only an intermediate result where time derivative terms have already been
neglected. Taking a Newtonian limit where c → ∞ reduces (91) further to
Z
Z
A (r0 , t)
∇0 · A (r0 , t) 0
0
0
dV −
∇ ·
dV
A (r, t) = − ∇
4πr
4πr
V0
V0
Z
Z
∇0 ×A (r0 , t) 0
A (r0 , t)
0
0
+ ∇×
dV −
∇×
dV .
(92)
4πr
4πr
V0
V0
Now, applying Gauss’ divergence theorem to the second term of the first bracketed term of (92), while using the
three-vector identity6,7
Z
I
A (r0 ) 0
A (r0 ) ×n0 0
−
∇0 ×
dV =
dS
(93)
4πr
4πr
V0
S0
on the second term of the second bracketed term of (92), and taking t = 0 for convenience, yields identity (11) as
desired (since the t dependence is the same on both sides of the resulting equation and can be ignored in what amounts
to a static field assumption). The Helmholtz identity (11a) therefore follows from the space components of (69) in a
static Newtonian limit.
Next, to prove that identity (81) of Theorem III follows directly from the Minkowski space Helmholtz identity (69)
in a static Newtonian limit, it is convenient to again start with identity (89) which follows from (69). One next takes
the following as an intermediate approximation of the zeroth component of identity (89):
"Z
#
0 0
Z
0i 0
0
∂
A
(x
)
A
(x
)
A0 (x) = −∂i
δ (r − c (t − t0 )) d4 x0 −
δ (r − c (t − t0 )) d4 x0 .
(94)
∂ 0i
4πr
4πr
V40
V40
Note, all terms involving partial time derivatives, e.g., ∂0 = (1/c)∂/∂t, are omitted in (94) since a detailed dimensional
analysis in the speed of propagation c shows that these terms vanish in a Newtonian limit where c → ∞. The author
will delay the application of the Newtonian limit for the terms retained in (94), however, until later in this derivation.
One next performs the t0 time integrations in (94), while using three-vector notation and setting φ = cA0 , yielding
Z
Z
−∇0 φ (r0 , t − r/c) 0
φ (r0 , t − r/c)
0
0
dV +
∇ ·
dV .
(95)
φ (r, t) = ∇ ·
4πr
4πr
V0
V0
It should be noted in passing that (95) is only an intermediate result where time derivative terms have already been
neglected. Taking a Newtonian limit where c → ∞ reduces (95) further to
Z
Z
−∇0 φ (r0 , t) 0
φ (r0 , t)
φ (r, t) = ∇ ·
dV +
∇0 ·
dV 0 .
(96)
4πr
4πr
V0
V0
Now, applying the three-vector Gauss’ divergence theorem to the second term on the rhs of (96) and taking t = 0
for convenience, yields identity (81) as desired, (since the t dependence is the same on both sides of the resulting
equation and can be ignored in what amounts to a static field assumption). The scalar field identity (81) of Theorem
III therefore follows from the zeroth or time component of (69) in a static Newtonian limit.
So, in a static Newtonian limit the space components of the Minkowski space identity (69) reduce to the Helmholtz
identity (11a) and the zeroth component of (69) reduces to the scalar field identity (81). It seems reasonable to
conclude therefore that identity (69) is a Minkowski space generalization of the (three-space) Helmholtz identity
(11a).
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4928
E. Uniqueness theorems for four-vector fields in Euclidean and Minkowski four-spaces
In this section, two uniqueness theorems will be proved. First, a uniqueness theorem for four-vector fields in
Euclidean space will be proved using identity (43) of Theorem I, i.e., using the Euclidean four-space generalization
of the Helmholtz identity. Then, a uniqueness theorem for four-vector fields in Minkowski space will be proved using
identity (69) of Theorem II, i.e., using the Minkowski four-space generalization of the Helmholtz identity.
A uniqueness theorem for four-vector fields in Euclidean four-space is now stated.
Theorem IV: A sufficiently smooth four-vector field Aν in the Euclidean four-space R4 is uniquely specified by giving
its four-divergence and its four-curl within a four-space region V4 , as well as its normal and tangential components
on the bounding three-surface Σ. That is, one must specify the following:
∂ν Aν (xσ ) ≡ s,
A (x ) nν ≡ Anorm (xσ ) ,
∂ α Aµ (xσ ) − ∂ µ Aα (xσ ) ≡ cαµ ,
σ
Aα (xσ ) nµ − Aµ (xσ ) nα ≡ Aαµ
tang (x ) ,
ν
σ
(97a)
(97b)
(97c)
(97d)
where nν is the four-vector outward unit normal of the three-surface Σ which encloses the four-volume V4 .
Proof: In order to demonstrate the uniqueness of the four-vector field Aν , one first postulates the existence of a
second four-vector B ν , which also satisfies equations (97). That is, one only replaces Aν on the lhs of Eq. (97) by B ν
while the rhs of Eq. (97) remain unchanged. The four-vector field Aν is unique if one can show that
W ν ≡ Aν − B ν = 0.
(98)
Now, taking the four-divergence of W ν and using (97a) yields
∂ν W ν = ∂ν Aν − ∂ν B ν = s − s = 0,
(99)
everywhere in the four-volume V4 . Also, calculating the magnitude of the normal component of W ν along the surface
normal nν and using (97b) yields
Wnorm ≡ W ν nν = Aν nν − B ν nν = Anorm − Anorm = 0,
(100)
everywhere on the bounding three-surface Σ. Next, taking the four-curl of W ν and using (97c) yields
∂ α W µ − ∂ µ W α = (∂ α Aµ − ∂ µ Aα ) − (∂ α B µ − ∂ µ B α ) = cαµ − cαµ = 0,
(101)
everywhere in the four-volume V4 . Also, calculating the rank two tangential components with respect to nν of W ν
and using (97d) yields
αµ
W α nµ − W µ nα = (Aα nµ − Aµ nα ) − (B α nµ − B µ nα ) = Aαµ
tang − Atang = 0,
(102)
everywhere on the bounding three-surface Σ. Finally, substituting the results (99)-(102) for the four-vector field W ν
into identity (43) of Theorem I, one obtains the result W ν = 0 which implies [via (98)], that Aν = B ν everywhere in
the four-space region V40 and on its bounding three-surface Σ0 . This proves that the four-vector field Aν is uniquely
determined by Eq. (97) thus proving Theorem IV on the uniqueness of four-vector fields in Euclidean four-space. ♣
A uniqueness theorem for four-vector fields in Minkowski space is now stated.
Theorem V: A sufficiently smooth four-vector field Aµ (xσ ) in the Minkowski space R3+1 which satisfies identity
(69), (i.e., Theorem II), is uniquely specified by giving its four-divergence and its four-curl within a space-time region
V4 , as well as its normal and tangential components on the bounding three-surface Σ. That is, one must specify the
following:
∂ν Aν (xσ ) ≡ s,
∂ α Aµ (xσ ) − ∂ µ Aα (xσ ) ≡ cαµ ,
(103a)
(103b)
throughout the space-time region V4 , as well as
Aν (xσ ) nν ≡ Anorm (xσ ) ,
σ
Aα (xσ ) nµ − Aµ (xσ ) nα ≡ Aαµ
tang (x ) ,
(103c)
(103d)
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4929
everywhere on the bounding three-surface Σ, where nν is the four-vector outward unit normal of the three-surface Σ
which encloses the space-time four-volume V4 .
Proof: The proof is based on the Minkowski space Theorem II. The proof proceeds in a parallel manner to the proof
of the Euclidean four-space Theorem IV. One postulates the existence of a second four-vector B ν which also satisfies
Eq. (103). The four-vector field Aν is unique if one can show, as in Theorem IV, that
W ν ≡ Aν − B ν = 0.
(104)
Equations (103) then lead, as before with (97), to results analogous to (99)-(102), which when substituted into identity
(69) shows that W ν = 0 which implies [via the definition of W ν in (104)] that Aν = B ν everywhere in the Minkowski
space region V40 and on its bounding three-surface Σ0 . This proves that the four-vector field Aν is uniquely determined
by Eq. (103) specified over Minkowski space, thus proving Theorem V on the uniqueness of four-vector fields in Minkowski space.
♣
It has already been demonstrated that identity (69) of Theorem II reduces to the Helmholtz identity (11a) when an
appropriate (static) Newtonian limit is taken. And it is known that the Helmholtz identity (11a) can be used to prove
the Helmholtz uniqueness theorem for a finite volume of R3 , i.e., Theorem H2. Theorem IV and Theorem V are later
used in Sec. IV B to prove Theorem X, which extends Theorem H2 to four-vector fields in Euclidean and Minkowski
spaces. [If one makes the usual assumption that the four-vector fields Aµ vanish sufficiently rapidly at infinity, then the
surface integral terms in (43) or (69) vanish and so a four-space generalization of the Helmholtz uniqueness Theorem
H1 over the entire four-volume of Euclidean or Minkowski space follows from Theorem X as well.] Also, Theorem’s IV
and V can be interpreted as extensions of Theorem U in Sec. II A to four-vector fields in Euclidean and Minkowski
spaces, respectively. The Helmholtz identity (11a) and the three-vector uniqueness theorems U, H1, and H2 of Sec.
II A can therefore be generalized readily to four-vector fields in Euclidean and Minkowski spaces.
F. Uniqueness theorem for scalar fields in Euclidean three-space
In this section, a uniqueness theorem for scalar fields in Euclidean three-space will be proved using identity (81) of
Theorem III. The theorem will now be stated.
Theorem VI: A twice continuously differentiable (static) scalar field φ(r) in the Euclidean three-space R3 is uniquely
specified by giving its gradient everywhere within a spatial volume V, as well as its value on the bounding surface S.
That is, one must specify the following:
−∇φ (r) ≡ E (r)
(105a)
φ (r) |S ≡ φS (r)
(105b)
throughout the volume V, as well as
on the bounding surface S.
Proof: In order to demonstrate the uniqueness of the scalar field φ (r), one first postulates the existence of a second
scalar field ξ (r), which also satisfies Eq. (105). That is, one replaces φ (r) on the lhs of Eq. (105) by ξ (r), while the
rhs of Eq. (105) remain unchanged. The scalar field φ (r) is unique if one can show that
Φ (r) ≡ φ (r) − ξ (r) = 0.
(106)
Now, taking minus one times the gradient of Φ (r) and using (105a) yields
−∇Φ (r) = −∇φ (r) + ∇ξ (r) = E (r) − E (r) = 0
(107)
for all r in the spatial volume V. Next, evaluating Φ on the boundary surface S and using (105b) yields
Φ (r) |S = φ (r) |S − ξ (r) |S = φS (r) − φS (r) = 0,
(108)
for all r on the surface S. Finally, using the results (107) and (108) for the scalar field Φ (r) in identity (81) of Theorem
III, one obtains the result Φ (r) = 0 which implies [via the definition of Φ (r) in (106)] that φ (r) = ξ (r) everywhere in
the three-space region V 0 and on its bounding surface S 0 . This proves that the scalar field φ (r) is uniquely determined
by Eq. (105) thus proving Theorem VI on the uniqueness of scalar fields in Euclidean three-space.
♣
Parenthetically, one should take note of the definition of the vector field E in (105a), which for example in electromagnetism could be interpreted as the static electric field. From this point of view, the uniqueness of a static electric
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4930
scalar potential φ (r) requires the specification of the static electric field in a volume V, as well as the value of the
scalar potential evaluated on the bounding surface S.
It should also be noted that Theorem VI bears little resemblance to a typical statement of a uniqueness theorem
for a scalar field in Euclidean three-space (cf. pp. 38-45 in Jackson − Ref. 18). This is because Theorem VI is
based on identity (81) of Theorem III, which is based on identity (80) alone. Equation (80) in turn follows from a
solution of the inhomogeneous scalar wave equation, i.e., from an inhomogeneous Green’s function approach. The
homogeneous solutions which would have lead to Dirichlet or Neumann boundary conditions have not been included.
The importance of Theorem VI appears to be that it emphasizes the role of the electric field, i.e., the negative gradient
of φ, in obtaining a unique scalar potential.
Essentially the same remarks as above would apply to Theorems IV and V, but in a four-vector potential context.
The importance of the four-vector result appears to be that it emphasizes the role of the four-divergence and the
four-curl of Aµ in obtaining a unique four-vector potential.
IV. EXTENSION TO FIELDS WITH MASS
A. Helmholtz identity and uniqueness theorem for fields with mass
In order to extend the Helmholtz identity (11a) to vector fields with mass it is convenient to start by adding a µ2
mass term to the scalar Poisson equation for the static scalar potential φ, as follows:
ρ (x, y, z)
∇2 − µ2 φ (x, y, z) = −
.
0
(109)
The massive scalar Poisson equation (109) can be solved in terms of a two-point scalar Green’s function G (r, r0 ) which
connects its unit delta function source located at the source point r0 = (x0 , y 0 , z 0 ) to a measurement at the field point
r = (x, y, z), i.e.,
∇2 − µ2 G (r, r0 ) = −δ 3 (r − r0 ).
(110)
The well-known identity, (cf. Ref. 19), over the Euclidean three-space R3 , namely
∇2 − µ2
e−µr
≡ −δ 3 (r − r0 )
4πr
∀ r, r0 ∈ R3 ,
(111)
0
where r ≡ |r −
r |, yields by comparison with (110) for the case of an infinite spatial domain, the Green’s function of
2
2
the ∇ − µ operator as
0
e−µr
e−µ|r−r |
G (r, r ) =
=
.
4πr
4π |r − r0 |
0
(112)
One recognizes in (111) and (112) an exponential damping factor depending in this example on the Compton wavenumber µ = 2π/λC = mc/~ of interaction bosons of mass m. The inhomogeneous solution of (109) for the scalar potential
then follows from the integral
φ (r) =
Z
V0
ρ (r0 )
G (r, r0 ) d3 r0 =
0
Z
V0
0
ρ (r0 ) e−µ|r−r | 3 0
d r.
4π0 |r − r0 |
(113)
Although (113) is an inhomogeneous solution to the massive scalar Poisson equation (109) in an infinite spatial
domain, it is the delta function property of identity (111) which is of importance in the proof of the new identity,
namely
!
0
Z
Z
−e−µ|r−r |
0
3
0
3 0
0
2
2
F (r) =
F (r ) δ (r − r ) d r =
F (r ) ∇ − µ
d3 r0 .
(114)
4πr
0
0
V
V
In addition, the identity ∇(1/r) = −∇0 (1/r) used in the derivation of (11a) must be replaced by the identity
∇G (r, r0 ) = −∇0 G (r, r0 ) ,
(115)
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Dale A. Woodside : Uniqueness theorems for classical four- vector... 4931
which follows from the Green’s function (112). A new identity now follows from (114) and (115) which allows one to
state the following theorem.
Theorem VII: The following identity holds for a continuous (static) three-vector field F(r) in a Euclidean threespace R3 :
Z
I
∇0 · F(r0 ) −µr 0
F(r0 )· n0 −µr 0
F(r) = − ∇
e
dV −
e
dS
4πr
4πr
V0
S0
Z
I
Z
∇0 ×F(r0 ) −µr 0
F(r0 )×n0 −µr 0
F (r0 ) −µr 0
+ ∇×
e
dV +
e
dS + µ2
e
dV ,
(116)
4πr
4πr
V0
S0
V 0 4πr
where r ≡ |r − r0 | and where n0 is the unit surface normal pointing out of the volume V 0 bounded by the closed surface
S0.
Proof: Starting with (114), since the Laplacian operator acts only on the field coordinates, it can be brought
outside of the integration. At this point, one makes a decomposition of the Laplacian operator using identity (20)
as before in the derivation of (11a).6,7 Then, since identity (115) retains the same functional form as the identity
∇(1/r) = −∇0 (1/r) with respect to the overall derivation, the derivation retains the same form as the derivation of
(11), with the minor exception that an extra µ2 term is carried along unchanged. Equation (116) therefore follows
readily as a new identity for static three-vector fields thus proving Theorem VII.
♣
In contrast to the Helmholtz identity (11a), the integrands of identity (116) contain an additional exponential mass
damping factor which improves their convergence. The integrals are therefore well defined even for fields F falling off
only as fast as 1/r (e.g., potentials).
Note, Eq. (116) should be thought of as an identity for representing a general (static) three-vector field rather than
as a general solution to a partial differential equation. Indeed, (116) follows from the identity (111) over an infinite
spatial domain and its subsequent use in the delta function property (114). Consequently, (116) is a vector identity
that applies to all of the Euclidean three-space, and so must hold for a finite volume of it as well. The Green’s function
for an inhomogeneous massive scalar Poisson equation in an infinite spatial domain is only mentioned in passing, and
certainly no use is made of the solutions of the associated source free homogeneous scalar equation.
It is interesting at this point to inquire whether or not identity (116) of Theorem VII can be used to obtain an
alternate version of Theorem H2 of Sec. II A. However, the first thing to note about identity (116) is that F is no
longer simply of the form (3) i.e., involving only the gradient of a scalar field (an irrotational part), and the curl of a
vector field (a solenoidal part), but is now of the form
Z
F (r0 ) −µr 0
2
F (r) = −∇Φ (r) + ∇×A (r) + µ
e
dV ,
(117)
V 0 4πr
where the first and second bracketed terms on the rhs of (116) are set equal to Φ (r) and A (r) respectively, and where
the last term on the rhs of (117) is an extra nonzero term which is neither irrotational nor solenoidal. Consequently,
an alternate version of Theorem H2 does not appear to follow from identity (116).
On the other hand, one can use identity (116) to prove the following uniqueness theorem for three-vector fields.
Theorem VIII: A twice continuously differentiable (static) three-vector field F (r) in the Euclidean three-space R3 ,
which satisfies identity (116), is uniquely specified by giving its divergence and curl within the volume V, its normal
and tangential components on the bounding surface S, and the value of the real constant µ. That is, one must specify
the constant µ and the following:
∇ · F = ρ,
(118a)
∇×F = j,
(118b)
throughout the volume V, along with the normal and tangential components
F · n = −σ,
(118c)
F×n = K,
(118d)
respectively, on the surface S bounding the volume V, where n is the outward unit normal vector of the surface S which
encloses the volume V.
Proof: In order to demonstrate the uniqueness of massive vector fields which satisfy identity (116) and Eqs. (118a)(118d), one first postulates the existence of a second vector G which also satisfies identity (116) and Eqs. (118a)-(118d).
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4932
That is, one replaces F on the lhs of Eqs. (118a)-(118d) by G while the rhs of these equations remain unchanged.
The vector field F is unique if one can show that
W ≡ F − G = 0.
(119)
Now, taking the divergence of W and using (118a) yields
∇ · W = ∇ · F − ∇ · G = ρ − ρ = 0,
(120)
everywhere in the volume V. Also, calculating the magnitude of the normal component of W along the surface normal
n and using (118c) yields
Wnorm ≡ W · n = F · n − G · n = −σ − (−σ) = 0,
(121)
everywhere on the bounding surface S. Next, taking the curl of W and using (118b) yields
∇×W = ∇×F − ∇×G = j − j = 0,
(122)
everywhere in the volume V . Also, calculating the tangential components with respect to n of W and using (118d)
yields
Wtang ≡ W×n = F×n − G×n = K − K = 0,
(123)
everywhere on the bounding surface S. Finally, substituting the results (120)-(123) for the vector field W into identity
(116) yields
Z
W (r0 ) −µr 0
W (r) = µ2
e
dV .
(124)
V 0 4πr
Then, if one applies the operator ∇2 − µ2 , which acts only on the unprimed coordinates, to (124), while using (111)
and (114), one obtains
Z
Z
e−µr
∇2 − µ2 W (r) = µ2
W (r0 ) ∇2 − µ2
dV 0 = −µ2
W (r0 ) δ (r − r0 ) dV 0 = −µ2 W (r) .
(125)
4πr
V0
V0
The µ2 terms in (125) cancel and so W satisfies the vector Laplace equation
∇2 W (r) = 0.
(126)
Equation (126) also follows from a direct expansion of the Laplacian operator using (20) as follows:
∇2 W = ∇ (∇ · W) − ∇× (∇×W) = 0,
(127)
where the first term in (127) vanishes by (120) and the second term in (127) vanishes by (122). A further restriction
on W is that its normal component vanishes on the surface S by (121), which combined with the fact that W is
irrotational by (122) and is therefore expressible as W = −∇φ, implies via Green’s theorem that W = 0 throughout
the volume V as well.8 So, only the trivial solution W = 0 of (124) satisfies the boundary conditions and therefore
[via (119)], the vector field F = G everywhere in the volume V and on its bounding surface S. This proves that a
static vector field F in a Euclidean three-space satisfying identity (116) is uniquely determined by specifying the real
scalar constant µ and the relations (118a)-(118d), thus proving Theorem VIII.
♣
It is tempting to interpret Theorem VIII as a uniqueness theorem for static massive vector fields, i.e., those which
satisfy the inhomogeneous massive vector Poisson equation
∇2 − µ2 F (r) = −j (r) .
(128)
However, the Green’s function which was used to derive identity (116) followed from an inhomogeneous massive scalar
Poisson equation which is a much simpler problem. That is, equations of the form (128) would in general use, for
example, a solution technique which involves a two-point dyadic Green’s function, i.e., a Green’s function which is
not a three-vector.20 Nevertheless, a close inspection of the proofs of identity (116) and Theorem VIII reveals that no
limitations are imposed on the vector field F other than that its components must be twice continuously differentiable.
The vector field F could therefore be either a massless or a massive vector field. Naturally, identity (116) appears to be
oriented towards application to massive vector fields due to its incorporation of a mass damping factor. Alternately,
identity (116) could be used in situations involving spatial diffusion, where the parameter µ would be interpreted as
a diffusion parameter.
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Dale A. Woodside : Uniqueness theorems for classical four- vector... 4933
B. Extending the four-space Helmholtz identities and uniqueness theorems to four-vector fields with mass
In attempting to extend the four-space Helmholtz identities to four-vector fields with mass, one can start by adding
a µ2 mass term to (32), the inhomogeneous scalar wave equation for the scalar potential φ, as follows:
2φ (xν ) − µ2 φ (xν ) = −ρ (xν ) /0 ,
(129)
where 2 ≡ ∂µ ∂ µ is the d’Alembertian operator, and where the (− + ++) metric signature is again convenient. One
can then attempt a solution of the massive inhomogeneous scalar wave equation (129) in terms of a two-point massive
scalar Green’s function G (xν , x0ν ) as in (33), but now with a µ2 mass term as follows:
1 ∂2
2
ν
0ν
2
2
2 − µ G (x , x ) = ∇ − 2 2 − µ G (xν , x0ν ) = −δ (4) (xν − x0ν ) ,
(130)
c ∂t
where δ (4) (xν − x0ν ) = δ x0 − x00 δ 3 (r − r0 ). For the previous scalar field case with the mass factor µ = 0, comparison of the Euclidean space analytically continued integral (30b) with the Minkowski space integral (34) yielded
the Euclidean four-space Green’s function (36). One could therefore try to follow a course paralleling the three-space
analysis in Sec. IV A by simply adding a mass damping factor to the Euclidean four-space Green’s function (36)
thereby obtaining a trial Green’s function
Gtrial (xν , x0ν ) ≡
ν
0ν
e−µR
e−µ|x −x |
=
2,
4π 2 R2
4π 2 |xν − x0ν |
(131)
where R is the distance in a Euclidean four-space between the source point x0ν and field point xν as defined previously
in (23). However, it turns out that Gtrial satisfies a different identity, namely
µ
2 − µ2 −
Gtrial (xν , x0ν ) = −δ (4) (xν − x0ν ) ,
(132)
R
where the third term on the lhs of (132) is an additional cross term. To demonstrate this one uses the result ∂ ν xν = 4
to calculate
−µR µ
µ2
e
2
= −4π 2 δ (4) (xν − x0ν ) e−µR + (2 − 4 + 3) 3 e−µR + 2 e−µR ,
(133)
2
R
R
R
which proves (132) since the exponential in the first term on the rhs of (133) drops out in a delta function distribution
integral context. Contrast (133) with the Euclidean three-space identity (110) rewritten in index notation, and using
the result ∂ j xj = 3, as
−µr −µr
e
µ −µr µ2 −µr
∂ ∂
3
j
0j
=
−4πδ
x
−
x
e
+
(1
−
3
+
2)
e
+
e
,
(134)
∂xj ∂xj
r
r2
r
where the additional cross term, the middle term on the rhs of (134), clearly drops out.
However, a solution for the massive scalar two-point Green’s function has been obtained by DeWitt21 using a
2
Fourier “momentum” space method for the case of timelike separations (xν − x0ν ) < 0 as follows:
(2)
GM (xν , x0ν ) = −
(2)
where H1
µ2 H1 (iµR)
,
8π
iµR
is the Hankel function of the second kind of order 1 with an integral representation21
Z
1
1
1
1
(2)
H1 (z) =
exp z 1 −
du,
iπ C u2
2
u
(135)
(136)
over the contour C defined in Ref. 21. Since z = iµR = iµ |xν − x0ν | in (136), the four-vector derivative of the massive
scalar Green’s function (135) has the property
∂
∂
GM (xν , x0ν ) = − ν GM (xν , x0ν ) ,
∂x0ν
∂x
(137)
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Dale A. Woodside : Uniqueness theorems for classical four- vector... 4934
which is of the same functional form as properties (65a,b). Consequently, (137) can be used in the same way as in
the derivation of identity (69). An alternate form of Theorem II can therefore be stated as follows:
Theorem IX: Given that a massive scalar two-point Green’s function GM (xν , x0ν ) exists for timelike separations
2
ν
(x − x0ν ) < 0, whose dependencies on the coordinates xν and x0ν occur only through the variable R = |xν − x0ν |,
the following identity holds for sufficiently smooth four-vector fields Aµ (xσ ) in the Minkowski space R3+1 for timelike
2
separations (xν − x0ν ) < 0:
"Z
#
I
Aµ (x) = −
−
V40
"Z
V40
µ
∂ µ ((∂ν0 Aν (x0 )) GM (x, x0 )) d4 x0 −
Σ0
∂ µ ((Aν (x0 ) n0ν ) GM (x, x0 )) dΣ0
∂α ((∂ 0α Aµ (x0 ) − ∂ 0µ Aα (x0 )) GM (x, x0 )) d4 x0 +
0
0α
0
− A (x ) n ) GM (x, x )) dΣ
0
#
2
+µ
Z
I
∂α ((Aα (x0 ) n0µ
Σ0
Aµ (x0 ) GM (x, x0 ) d4 x0 ,
(138)
V40
where n0µ is the four-vector outward unit normal of the three-surface Σ0 which encloses the four-volume V40 , and where
the three-surface Σ0 is defined covariantly with respect to a general Lorentz transformation.
Proof: The proof is based on the four-vector delta function property (42) which in the present case uses (130) for
representing the four-space delta function as follows:
Z
Z
µ
µ
0
(4)
0
4 0
A (x) =
A (x ) δ (x − x ) d x = −
Aµ (x0 ) 2 − µ2 GM (x, x0 ) d4 x0 .
(139)
V40
V40
Then since the property (137) is of the same functional form as in (65), the proof of (138) parallels the proof of (69)
in all important respects, except that an extra µ2 mass term is carried along unchanged from (139), and so the details
will not be repeated.
♣
It is interesting at this point to inquire whether or not one can use identity (138) of Theorem IX to state a four-space
analog of the three-vector Theorem H2 of Sec. II A. For a sufficiently smooth four-vector field one can factor the
unprimed field point derivatives out of the integrals over the primed source point coordinates in (138). However, the
first thing to note about such a factored version of identity (138) is that Aµ is no longer simply of the form (75), i.e.,
involving only the four-gradient of a scalar field (a four-irrotational part), and the four-curl of a four-vector field (a
four-solenoidal part), but is now of the form
Z
µ
µ
αµ
2
A (x) = ∂ A(x) + ∂α A (x) + µ
Aµ (x0 ) GM (x, x0 ) d4 x0 ,
(140)
V40
where the first and second bracketed terms on the rhs of (138) are set equal to −A(x) and −Aαµ (x), respectively, and
where the last term on the rhs of (140) is an extra nonzero term which is neither four-irrotational nor four-solenoidal.
Therefore a four-space analog of the three-vector Theorem H2 of Sec. II A does not appear to follow from (138).
However, it is possible to state a theorem for four-vector fields based on identity (43) of Theorem I or on identity (69)
of Theorem II as follows:
Theorem X: A sufficiently smooth four-vector field Aµ (xσ ) that is defined everywhere in a finite volume V4 in a
Euclidean four-space R4 or in a Minkowski space R3+1 and whose tangential and normal components on the bounding
three-surface Σ are given may be uniquely represented as a sum of a four-irrotational and a four-solenoidal part.
Proof: It has already been shown that (44) leads to (51) in the Euclidean case, while in a similar fashion (70) leads
to (75) in the Minkowski case. Now, the second term of (51) or (75), is four-irrotational
∂ µ (∂ ν A) − ∂ ν (∂ µ A) = 0,
(141)
i.e., its four-curl is zero. Also, the first term of (51) or (75) is four-solenoidal
∂µ (∂α Aαµ ) = 0,
(142)
i.e., its four-divergence is zero, since it is a contraction of a symmetric factor ∂µ ∂α and an antisymmetric factor (53)
or (73). The decomposition defined by Eq. (51) or (75) is therefore a sum of a four-irrotational and a four-solenoidal
part and by identity (43) and (69) and by the arguments of the Euclidean uniqueness Theorem IV and the Minkowski
space uniqueness Theorem V the field Aµ is unique under this decomposition, thereby proving Theorem X.
♣
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Dale A. Woodside : Uniqueness theorems for classical four- vector... 4935
A theorem will now be stated that will be used later in this section.
Theorem XI: A sufficiently smooth four-vector field Aµ (xσ ) vanishes in a compact nonempty region V 4 = V4 ∪ Σ
of the Minkowski space R3+1 when the four-divergence and four-curl of Aµ (xσ ) vanish over the space-time region V4
and the four-normal and four-tangential components of Aµ (xσ ) vanish over the bounding three-surface Σ. That is,
one must specify the following:
∂ν Aν (xσ ) = 0,
∂ A (xσ ) − ∂ µ Aα (xσ ) = 0,
α
µ
(143a)
(143b)
throughout the space-time region V4 , as well as
Anorm (xσ ) ≡ Aν (xσ ) nν = 0,
σ
α
σ
µ
µ
σ
α
Aαµ
tang (x ) ≡ A (x ) n − A (x ) n = 0,
(143c)
(143d)
everywhere on the bounding three-surface Σ, where nν is the four-vector outward unit normal of the three-surface Σ
which encloses the space-time four-volume V4 .
σ
Proof: Since it is assumed that Aµ and nν are four-vectors, then Anorm (xσ ) and Aαµ
tang (x ) are covariant w.r.t.
general Lorentz transformations and identity (69) of Theorem II, which is defined under similar constraints, can be
used. Substitution of (143a)-(143d) into identity (69) then yields the result Aµ (xσ ) = 0 throughout V 4 , thus proving
Theorem XI.
♣
The next step is to state a four-vector uniqueness theorem analogous to the three-vector Theorem VIII as follows:
Theorem XII: A sufficiently smooth four-vector field Aµ (xσ ) in the Minkowski space R3+1 which satisfies identity
(138) (i.e., Theorem IX), is uniquely specified by giving its four-divergence and four-curl within the space-time region
V4 , its normal and tangential components on the bounding three-surface Σ, and the value of the real constant µ. That
is, one must specify the constant µ and the following:
∂ν Aν (xσ ) ≡ s,
∂ α Aµ (xσ ) − ∂ µ Aα (xσ ) ≡ cαµ ,
(144a)
(144b)
throughout the space-time region V4 , as well as
Aαµ
tang
Anorm (xσ ) ≡ Aν (xσ ) nν ,
(xσ ) ≡ Aα (xσ ) nµ − Aµ (xσ ) nα ,
(144c)
(144d)
everywhere on the bounding three-surface Σ, where nν is the four-vector outward unit normal of the three-surface Σ
which encloses the space-time four-volume V4 .
Proof: The proof proceeds in a parallel manner to the proof of Theorem V. One postulates the existence of a second
four-vector B ν which also satisfies Eqs. (144a)−(144d). The four-vector field Aν is unique if one can show, as in
Theorem V, that
W ν ≡ Aν − B ν = 0.
(145)
Equations (144a)-(144d) then lead, as before with (97a)-(97d), to results analogous to (99)-(102), which when substituted into identity (138) yields the result
Z
µ
2
W (x) = µ
W µ (x0 ) GM (x, x0 ) d4 x0 .
(146)
V40
Then, if one applies the operator 2 − µ2 , which acts only on the unprimed coordinates xν , to (146) while using
(139), one obtains
Z
µ
2
2
2 − µ W (x) = µ
W µ (x0 ) 2 − µ2 GM (x, x0 ) d4 x0
V40
2
= −µ
Z
V40
W µ (x0 ) δ (4) (x − x0 ) d4 x0 = −µ2 W µ (x).
(147)
The µ2 terms in (147) cancel and so W µ (x) satisfies the homogeneous wave equation
2W µ (x) = 0.
(148)
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Dale A. Woodside : Uniqueness theorems for classical four- vector... 4936
Equation (148) also follows from a direct expansion of the d’Alembertian operator using (19) as follows:
2W µ = ∂ν (∂ ν W µ − ∂ µ W ν ) + ∂ µ (∂ν W ν ) = 0,
(149)
where the first bracketed term of (149) vanishes because the four-curl vanishes by a result analogous to (101) and the
second bracketed term of (146) vanishes because the four-divergence vanishes by a result analogous to (99). It should
be clear from the result (149) that it is not necessary to solve the Cauchy problem for the wave equation (148) to
find W µ (x) since each bracketed term in (149) separately vanishes. In fact, each of the conditions (143a)-(143d) are
satisfied for W µ (x) throughout V 4 , and so by Theorem XI W µ (x) = 0 throughout V 4 . Thus, only the trivial solution
W µ (x) = 0 of (148) satisfies the boundary conditions and therefore, [via (145)], the four-vector field Aµ (x) = B µ (x)
everywhere in the four-volume V4 and on its bounding three-surface Σ. This proves that a four-vector field Aµ (x) in a
Minkowski 3+1 space-time satisfying identity (138) is uniquely specified by specifying the real scalar constant µ and
the relations (144a)-(144d), thus proving Theorem XII.
♣
It is tempting to interpret Theorem XII as a uniqueness theorem for massive four-vector fields, i.e., those which
satisfy the inhomogeneous massive four-vector wave equation
2 − µ2 F µ (x) = −j µ (x).
(150)
However, the Green’s function GM (x, x0 ), e.g., (135), which was used in identity (138) followed from an inhomogeneous
massive scalar wave equation which is a much simpler problem. That is, equations of the form (150) would in general
use a solution technique which involves a two-point second rank tensor Green’s function.22,23 Nevertheless, a close
inspection of the proofs of identity (138) and Theorem XII reveals that no limitations are imposed on the four-vector
field Aµ (x) other than that its components must be sufficiently smooth. The four-vector field Aµ (x) could therefore be
either a massless or a massive four-vector field. Naturally, identity (138) appears to be oriented towards application
to massive four-vector fields Aµ (x) due to its incorporation of a mass damping factor.
V. CONCLUSION
In conclusion, the three-space Helmholtz identity and its associated uniqueness theorems, which focus on the
curl and divergence of a vector field, provide insight into irrotational and solenoidal fields. The extension of the
Helmholtz identity and associated uniqueness theorems to Euclidean and Minkowski four-spaces presented in this
article demonstrates that the curl and divergence of a three-vector field generalize into the four-curl and four-divergence
of a four-vector field, and that irrotational and solenoidal three-vector fields naturally generalize into four-irrotational
and four-solenoidal four-vector fields, respectively.
Now, a four-solenoidal field is essentially a four-vector field in the Lorentz gauge, (sometimes referred to as a
relativistic transverse gauge), with zero four-divergence, as for example in the case of the electromagnetic field. The
author is currently investigating the associated concept of a four-irrotational four-vector field. This leads to the
development of what the author shall call a “relativistic longitudinal gauge” where the Maxwell field tensor itself is set
to zero, while the four-divergence of the four-vector field can in general be nonzero. Consider the case of a so-called
“pure gauge” field which arises in connection with the Meissner effect deep in a superconductor, where the magnetic
field is required to vanish, (cf. Ref. 24). Interestingly, pure gauge fields which are defined by the relation
Aµ ≡ ∂ µ Λ,
(151)
satisfy a “relativistic longitudinal gauge condition” as defined by the vanishing of its Maxwell field tensor:
F µν = ∂ µ Aν − ∂ ν Aµ = ∂ µ ∂ ν Λ − ∂ ν ∂ µ Λ = 0,
(152)
and further are four-irrotational since the four-curl of Aµ , i.e., F µν , is zero by (152). If they also have nonzero
four-divergence throughout a region V4 where (152) holds, i.e.,
∂µ Aµ (x) = ∂µ ∂ µ Λ(x) 6= 0
∀ x ∈ V4 ,
(153)
they would provide an example of a four-vector field in the relativistic longitudinal gauge. Indeed, if a pure gauge
field, defined for example over an unbounded space-time region, satisfied ∂µ Aµ = 0, then by Theorem XI one would
have the field Aµ vanishing everywhere! In a finite space-time region, on the other hand, one could still possibly have
∂µ Aµ = 0, F µν = 0, and Aµ 6= 0 all holding true if either (143c) and/or (143d) were nonzero on the three-surface Σ
bounding the four-volume V4 .
J. Math. Phys., Vol. 40, No. 10, October 1999
Dale A. Woodside : Uniqueness theorems for classical four- vector... 4937
ACKNOWLEDGMENTS
The style and presentation of this article has benefited from detailed discussions with the author’s thesis adviser
Associate Professor John Corbett. The author is deeply indebted to Professor Corbett for allowing him to freely
develop his own ideas while at the same time making him aware of their mathematical significance and limitations.
This work is a partial reporting of the results of the author’s Ph.D. thesis. The thesis was submitted on 28 August, 1998
to the Department of Physics at Macquarie University-Sydney, Australia, in partial fulfillment of the requirements for
the degree Doctor of Philosophy. Tuition exemption from the Higher Education Contribution Scheme (HECS) was
obtained via the Ph.D. by Research program.
a)
Electronic mail: [email protected]
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