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Derivation of Maxwell's equations
Physica 31 125-140
de Groot, S. R. Vlieger, J. 1965
DERIVATION THE
OF MAXWELL’S ATOMIC FIELD
EQUATIONS
EQUATIONS
by S. R. DE GROOT Instituut voor theoretische fysica, Amsterdam, Nederland
and J. VLIEGER Instituut-Lorentz,
Leiden, Nederland
synopsis From the electromagnetic equations for fields, produced by point particles (electrons and nuclei) the field equations, taking into account the existence of stable atoms (or molecules, or ions) are derived in covariant form. These “atomic field equations” involve, besides the field tensor (e, b) an atomic polarization tensor (p, m), or alternatively the field tensor (d, h) = (e + p, b - m). The atomic polarization tensor is given (up to second order in the internal atomic variables) as a function of atomic positions and velocities; proper electric dipole, quadrupole and magnetic dipole moments; and time derivatives of these quantities. (The “proper atomic multipole moments” are defined in Lorentz frames in which the atoms are momentarily at rest).
5 1. Introductim Maxwell’s equations in material media will be derived in two steps in this article and the following. First, from the equations for electromagnetic fields produced by point particles (electrons and nuclei), valid at what we shall call the “sub-atomic level”, the equations, which take into account the existence of stable atoms (molecules, or ions) v&illbe derived. The derivation of these “atomic field eq%atiom” will be described in this article. A following paper will be concerned with the second step: the statistical averaging process, which must be performed to obtain the Maxwell equations, valid at the “macroscopic level”, from the atomic field equations. In this article we start with the sub-atomic field equations of “electron theory”, as given originally by Lorentz, for the electromagnetic field produced by point charges ($2). If these point charges are grouped into stable atoms, one can expand the source terms in these equations into powers of the internal coordinates (5 3). These series expansions contain certain combinations of the internal variables. In $3 5,6,7 these combinations This research was supported in part by the Office of Aerospace Research, United States Air Force, through its European Office.
-
125 -
126
S. R. DE GROOT
AND
J. VLIEGER
will be expressed in terms of the proper atomic multipole moments, which are defined in $4 as the moments in momentary rest frames of inertia for the atoms. We finally discuss the resulting atomic field equations. The latter contain a polarization tensor which is expressed in terms of the atomic multipole moments, the atomic positions and velocities, and time derivatives of these quantities
($3 8, 9).
$ 2. The mb-atomic field eqtiations. Lorentz’ original “electron theory” equationsr) for the fields e and b, produced by point particles j with charges ej, positions &(t) and velocities l&(t) can be written in the form V-b
= 0,
(1)
dab + V A e = 0, V-e -doe
= 2 ej6(Rj i
(2)
R),
+ V A b = x ej(&/c) ~(RJ 3
(3) R).
They are valid in a time-space Lorentz’frame (ct, R) (c = velocity The time and space differentiations are written as
(4 of light).
do E d/dct, V = dldx, dldy, d/d.z.
(5)
These derivatives are total derivatives in the sense that the fields depend not only explicitly on t and R, but also implicitly, because they are functions of R&Q, &(t,) and higher time derivatives, where tj = t - ~3 is the retarded time with 7, = T~(R, t) following from TJ = IRj(t - TJ) - RI/c. In covariant notation we write with
d, = (do, V), da = (da, V) and the anti-symmetric
= 0, 1, 2, 3)
(6)
field tensor
f@(% p = 0, 1, 293) Then the homogeneous
da = -da(a
with
equations
f01, f02, fO3 = e, f23, f31, fl2 = b.
(7)
(1) and (2) read
daf#fSY + dflfya + dyf”@ = 0,
(8)
with C& = 1 2 3 in (1) and 023, 031 and 012 in (2). The inhomogeneous equations (3) and (4) b ecome the cases CY= 0 and c( = 1, 2 and 3 of daf@ = C e&/c) I
8(Rf -
R)
(a = 0, 1, 2, 3),
(9)
where Ri = ct and Rt, Rf, Ri = R,. The right-hand side of (9) is a fourvector: in fact it transforms as R;, because the three-dimensional deltafunction transforms as a time. The equations were written for a collection of arbitrary charged point
DERIVATION
OF MAXWELL’S
127
EQUATIONS
particles. We shall now choose (spinless) electrons and nuclei as the particles and refer to (l)-(4) or (8), (9) as the subatomic field equations. In the next sections we consider how we may transform these equations, if the electrons and nuclei are grouped into stable atoms, molecules or ions. We now suppose
5 3. The atomic series expansion.
that the electrons
and nuclei are grouped into stable “atoms”, indexed by k. By atoms we shall understand any stable group, such as a molecule, ion or atom and also single free electrons*). We number the constituents of these groups by i. So, replacing j by ki, we have for (3) and (4)
v-6? =
c
eki
6(&t
-
R),
(10)
k,i -doe
+ v A b = x ‘Z&k&)
6(&i
-
R).
(11)
k,i
For later use, it will be convenient e =
c, ek,
b
=
to split the fields according I: bk
(Or f”o = 2
to
j$‘).
(12)
We can then write separate equations for all values of k: ‘17’ek = --dock
+
z ekZ d(Rkt i
v A bk =
c
ekt(fik&)
R),
(13)
6(Rkt
-
R).
i
Let us now choose a point Rk describing the whole, and int reduce internal coordinates rkZ and . . Rkz = Rk + rki, Rkt = Rk +
(14
position of atom k as a velocities ik2 by . rkt.
(15)
We now consider the case in which the solutions (retarded fields) of the field equations can be written as converging series expansions in rkt/l& - R ], in accordance with the assumption that stable atomic structures exist. We can therefore develop the delta-function into powers of rk{ around Rk - R. We shall call this procedure the atomic series expansion. (We follow in this section essentially the formalism of Mazur and Nijboers), but without averaging). For (13) we thus obtain V’t?k = x eki(l -
i
rk6.V + &rkg
:vv - ...) d(Rk- R),
(16)
*) Formally one can choose as the “group” anything between (and including) the “elementary” particles (here the electrons and nuclei) and the complete system. Physically one wishes to choose stable groups as the sub-units of the system. Atoms, molecules but it may be useful under certain circumstances to consider group, e.g., vibrating formula (15)).
antennas
or plasmas
observed
and ions are most obvious choices, the complete system as one single
from a large
distance
(see the text
following
128
S. R.
DE GROOT
AND
J. VLIEGER
where V operates on R. This can be written as V.C?k= pk - V’pk,
(17)
with the notations pk = 2 ek#6(Rk - R) = ek d(Rk i
R)
(18)
for the charge density of atom k(ek = & e& is the atomic charge), and pk
=
(:
ek#ki
-
4 x i
ektrktrki’v
+
. ..)
d(Rk
-
R).
(19)
The CpUtitieS & ektrkt and +xczek&lrkf, appearing in this equation, have the form of atomic electric dipole and quadrupole moments, and in the non-relativistic approximation can be interpreted as such. However, in a relativistic theory such as ours this is not possible, since these quantities are defined in the (ct, R)-frame, whereas the proper atomic moments must be defined in a proper atomic Lorentz frame. We deal with this problem in the next section. We now write (14) with the same Taylor expansion and use also (15) : -doQ
+ v A bk
=
x ekl{(@k/C) i
-
-(&k/C) +
C-l(
v’(rk# iki
-
-
+rk(rki’v
ik(rki’V
+
+ . ..)}
. ..)
d(Rk
+
R).
-
(20)
With the current density of atom k, defined as ik = r, e&k i
6(Rk - R) = t?k& d(Rk - R)
(21)
and with (19)) one can write for (20) -do&Q + v A bk = h/c +
C-l
(&k/C) v.pk x
ek((
ik(
-
+ &rk$‘V
+
. ..)
6(&
-
R).
a
(22)
This equation can be written in a simpler form by applying a vector identity to the second term and to half of the fourth term at the right-hand side. Taking Q independent of R, one has a(V.b) = (a.V) b - V A (b A a).
(23)
Equation (22) then becomes: -d&Q + v
A
bk = ikjc + dapk + v
A mk,
(24)
where the following quantity has been introduced mk =
(T&~
x
i
ek(rkd
A ik(
+
e..)
d(&
-
R) + pk A &c/C.
(25)
DERIVATION
OF MAXWELL’S
EQUATIONS
129
Here again an atOn& qUaIItity, &-l z:t C?k&$ A &, appears. It has the form of a magnetic moment, but it is defined in the (ct, R)-frame, not in the proper atomic frame. The last term of (25) gives the effect of pk moving with the atomic Velocity tik. We thus find the forms (17) and (24) for the inhomogeneous field equations (13) and (14), with the charge-current density (18) and (21) and the quantities pk and mk defined by the series expansions (19) and (25). Let us introduce the covariant notation for the (anti-symmetric) field tensor /$(a,/3 = 0, 1,2,3)
with
f;‘, fi2, fok3 = ek,fE3,f;‘, fi2 = bk,
(26)
the charge-current density four-vector j$(oz= 0, 1) 2, 3)
with
j; = ,QC,j;, j;, j,” = ik
(27)
and let us define the quantities my = - ?‘?Ze(cc,j? = 0, 1, 2, 3) with P?Z;‘,?‘PZ;‘,m;’ =pk, ??Z13, P?Z;‘,FBi2= mk. (28) Then the atomic field equations (17) and (24) are the cases a = 0 and = 1, 2, 3 of
dsf$ = i;lc + dm;zB.
a =
(29)
In the following we shall limit ourselves in the expressions (19) and (25) only to terms up to the second order in the internal variables rkt and &. It can then be shown that the quantities rniP (28) are the components of an (anti-symmetric) tensor, which will be called the polarization tensor. With (12) and the similar espressions*) **). (30) (31)
we find from the inhomogeneous equations (17) and (24) the equations V.e = p -doe
V-p,
+ V A b = i/c + dop + V A m,
(32) (33)
or, in covariant notation, dFf”fl = ja/c + dBrn@.
(34)
*) It may be noted that while the charges eb are invariants, the atomic charge-current density (PC,I) is a four-vector. If, therefore, in a particular case, one has a charge density p = 0 in one Lorentz frame, then in a different Lorentz frame the charge density will in general be different from zero s). **) Note that from (18), (21) and (30) follows the law of conservation of charge: dp/dt + + V.f = 0 or,in covariant notation d& = 0.
130
S. R. DE
GROOT
AND
J. VLIEGER
With the help of the definitions d = e + p, h = b the following
alternative
(35)
forms of (32) and (33) are obtained:
-ded or, in covariant
m (or MB = f@ - m@),
V.d = p,
(36)
+ V A h = i/c,
(37)
notation, dsh@ = j”/c.
(38)
The “atomic field equations” (32) and (33) contain the polarization tensor (p, n-z), given by (31) and the series expansions (19) and (25). These expansions contain characteristic combinations of the internal variables. In the following sections we shall express these combinations and therefore the polarization tensor, in terms of the proper atomic multipole moments. $ 4. The atomic mdtipole moments. We must first define the proper atomic multipole moments. The combinations of internal variables Y~Zand & which occur in the polarization tensor (see (19) and (25)) are defined with respect to our reference frame (ct, R). A quantity like x;a ek(%, for instance, has the form of an atomic dipole moment, but rki is defined in the reference frame (ct, R). The proper atomic multipole moments, however, are defined with the help of internal variables in a Lorentz frame in which the atom is at rest. This “atomic frame” has a (constant) velocity u = &(t) with respect to the reference frame, where @k(t) is the velocity of atom h at time t on the latter frame. Since the atom suffers in general an acceleration, it is only at rest in the atomic frame at one particular time t; in other words: the atomic frame is a momentary rest frame of inertia for the atom. The procedure ($9 5, 6, 7) will therefore be the following. We express internal quantities like rk$ and ika, defined in the reference frame (ct, R\, in terms of quantities rig and &, defined on the momentary atomic frame, by means of a Lorentz transformation, involving a constant velocity U, at a certain fixed time t. We repeat this at all times t, which means that in the results obtained in the procedure described above, we have to replace v by &. This leads to expressions depending on rit and iii through the atomic multipole moments, which are defined as: pk
the electric dipole moment,
=
c
i
ek&t
(39)
DERIVATION
OF MAXWELL’S
EQUATIONS
131
the electric quadrupole moment,
the magnetic dipole moment, etc. Furthermore we wish to consider these proper atomic quantities as functions of rkf and iki in the original reference frame, because the atomic field equations, which involve the polarization tensor, contain quantities all measured in the original reference frame. So we have to consider also the Lorentz transformation, which is simply the reverse of the transformation described above, and which gives rit and ijG(in terms of rk2 and ixc. Again consideration of a succession of such Lorentz transformations at all times t means replacement of v by tit. If proper atomic quantities are considered in the reference frame (ct, R), we shall denote them with a bar, e.g. x:( e$& and 8 ECektrk& § 5. The Lore&z transformation from the reference frame to the atomic frame. We wish to express rkt and ik( as measured in the reference frame (ct, R) in terms of the quantities rig and & as measured in the atomic frame. For this we need the Lorentz transformations with v 3 /?c the velocity of the atomic frame (with dashed quantities) with respect to the reference frame:
(42) (43) Rk@) = G(Gc) + (7 t = yti
1) NC(G) *P/VP + rtipc,
+ yl?gtg
*p/c.
(44) (45)
From the difference of (43) and (45) one finds tic - Gc= -(P/C) *(Gc(&)
-
Rh(G)).
(46)
From the difference of (42) and (44) it follows with (15) and (46) rt#)
= Q(P) .{N&)
- K(G)},
(47)
1) /V/P,
(4.8)
where the three-tensor Q is defined as Q(P) = U + (41 - P2 -
with U the unit three-tensor. We want to express all quantities at the right-hand side of (47) at the same time, which we take to be the time ti. This can be achieved by first
132
S. R. DE GROOT
developing $a -
the right-hand
ti = -(P/c)
AND
J. VLIEGER
side of (46) in a Taylor series
+$c&)
+ &&)(Gz
-
G) + 9iii&i)(Gc2
where terms up to second order have been included.
-
G)2),
(49)
Then solving tit -
tic
up to second order and inserting the result into (47), where one has developed &(tjJ in a Taylor series around ti (see appendix I), one obtains, with (15) and fik~ = & + i;k~, up to second order
.Ir&
I.
-
p.rid&
+ 8 (1
1 + a;*/+
(p -riilc) 2 +
.. ,
1
Rk.
&p/c)2
(50)
This is the result of a Lorentz transformation at a fixed time t. We now consider such transformations at all times. This means that we replace p by Pk s fit/C. Furthermore, in the atomic system the atom is at rest, so that li;(t;) vanishes. In this way (50) becomes rkt(t)
=
ak’{rj(
-
Pk.&%&
+
8(Pk’rb/c)2~>p
where the bar indicates that the proper atomic quantities to be considered in the original reference frame. Furthermore the abbreviation ak
An alternative
From
=
n(pk)
=
u +
(41
-
/f$ -
(51)
rit, etc., have we have used
1) PkPk/t$
(52)
form of (51) is
this connection
we obtain,
with the multipole
From (53) one can also obtain the other internal in pk. ( 19), up to second order :
moments
quantity
(39)-(41)
which appears
(55)
We also need the internal
quantity,
occurring
in mk (25).
It contains
DERIVATION
OF MAXWELL’S
133
EQUATIONS
~6 A ikt. The quantity & follows from differentiation of (50), where we can neglect the last two terms, since we need ~2 A ik( only up to second order. With & = 0 and dtldti, = y,
(56)
which follows then from (45), and replacing p by /3k, we then obtain ik( = y;lsk’{ii(
-
(57)
Z(&Z/C)},
where we have written yk
=
(1 - /9$-t.
(58)
The internal variables have now been expressed as shown in equations (54), (55) and (57). In the first expression an as yet unencountered “atomic quantity” appears in the third term. It can, however, be eliminated, as we shall show in 5 7. Furthermore, (54) and (57) contain the acceleration ii?. We shall prefer to express it in terms of the acceleration fik in the reference frame (ct, R). This will be done in the next section, before we further develop the internal variables. 5 6. The acceleration in the atomic and in the reference frames. To express the acceleration & in the atomic reference system in terms of the acceleration fik: in the reference frame (ct, R) one needs the Lorentz transformation
JGW = &c(t) + (y -- 1) &c(t) -p/3/P
tic= yt -
y&(t)
-
-p/c,
y,%t,
(59) (60)
which is the inverse of the transformation (44), (45). By differentiating twice with respect to time one finds iii in terms of iit, /& = kk/C and /3 = V/C (see appendix II). One then replaces in this expression /3 by /?k to obtain the relation (61)
valid for all times. When this is inserted into (54) and (57), we obtain (see also appendix II): 2
ekfrkt
=
fik’pk
+
Pk A vk -
i
+kt =
,‘;‘nk’i$
-
Yk(Pk’x$)
fik/C.
(63)
The last result and (53) permit us to calculate (up to second order) the
134 following
S. R. DE GROOT
quantity,
AND
J. VLIEGER
which occurs in nzk (25) (appendix
+
Ykfik
II):
A (ak*qk.Pk/C2).
(64)
We have now found the expressions quantities
occurring
(62), (55) and (64) for the internal in pk (19) and mk (25).
5 7. Internal variables expressed in terms of the atomic mzlltipole moments. The expression (62) for the internal variable & e&& contains the atomic VdOCity hk, the at0II-k XCekratiOn fik, the atOmiC m.dtipOk IrIOmentS pk, vk and qk, and also the atomic quantity J$ ekc(rj&t + ii&). We wish to express now the latter quantity in terms of the others. A simple way to achieve this is first to express rig and ii8 in terms of rkg and +kgby means of the inverse transformation of (50). We need it only to first order if we wish to calculate the atomic quantity mentioned above up to second order. We obtain in this way (see appendix III) for the last two terms of & ek&t (62) :
(65) Now with (55) we can express the right-hand side of this formula in terms of the quadrupole
moment
-Y$k’do(~k’qk’~k)
-
qk. It becomes then
Y;(Pk’iik)
Pk’~k.qk’~k/C2 =
=
-YkPk’dO(Yk~k’qk’~k),
where the last result follows because time differentiation identity Pk.&
=
C2yi3
(66)
of (58) yields the
dayk.
(67)
With the result (66) for the last two terms of (62) we have 2 i
ektrki
=
ak’pk
f
Pk A vk
-
YkPk’dO(YkQk’qk’ak)r
(68)
whereby we have expressed the internal quantity 26 ek@kf as a function of the atomic multipole moments pk, qk and vg, the atomic velocity @k = pkc and a time derivative of these quantities. The other internal quantities (55) and (64), which occur in pk (19) and mk (25), were already expressed in this way.
DERIVATION
Ej8. The polarization
OF MAXWELL’S
135
EQUATTONS
tensor in terms of the atomic multipole moments. The
atomic polarization tensor m@(p, m) is, according to (31), the sum over k of the tensors my(pk, mk), which are given in terms of internal quantities by the expressions (19) and (25). These internal quantities have now been expressed in terms of the atomic multipole moments pk, ok and vk, the atomic velocity fik = pkc, and time derivatives of these quantities according to the formulae (68), (55) and (64). We thus obtain immediately p=~{~k’(~kk
‘jk’nk’v)
+ Pk A vk -
do(~k’qk’~kJ’k)
‘Pkyk}
a(Rk
-
R),
(69)
and after some elementary vector manipulations (see appendix IV) : m = x {Qk’vk k
-
Pk A (pk
+
qk’fik.V) do(Pk
+
* qk’akyk)
‘Pk)‘k)
d(Rk
-
R)*
(70)
In this way the polarization tensor at the time-space point (ct, R) is expressed up to second order in terms of the proper atomic multipole moments pk, qk and vk, the atomic positions Rk and velocities & = pkc, and time derivatives of these quantities. (Recall that the quantities & (52) and yk (58) are functions of the atomic velocities). To discuss the physical meaning of the polarization tensor, it may be useful to write down first the non-relativistic approximation (obtained by neglecting terms with c-s). Then the last two formulae become p = 2 (pk m =
x (vk k
4k.v)
p k A (pk
-
a(Rk qk’v)}
-
R), d(Rk
(71) -
R).
(74
(Note that the magnetic moment vk contains a factor c-r, so that its contribution to p is of order c-z). These formulae follow also directly from (19) and (25) with the non-relativistic forms of (68), (55) and (64):
which were used by Mazur and Nijboers). The formulae (71) and (72) for the non-relativistic polarization tensor show the electric multipoles occurring in the “polarization vector’ ’ p, whereas in the “magnetization vector” m we find the effect of moving charges; the motion of particles inside the atoms gives rise to the atomic magnetic moment vk, while the motions of the electrically polarized atoms as a whole (terms with /?k) give the second contribution. Let us now turn back to the discussion of the relativistic expressions (69) and (70) for the polarization tensor. We then see that, as compared to the
136
S. R. DE GROOT
AND
1. VLIEGER
non-relativistic approximation, correction factors S2k involving the atomic velocity appear. Furthermore we find now in p a contribution from magnetic dipoles vk in motion (Pk A vk), analogous to the contribution to rrz from electric multipoles in motion. In the third place new terms with time derivatives, containing the electric quadrupole moments, appear both in p and m. We note that the expressions for p and m were obtained in $ 3 as series expansions in the internal coordinates of the atoms up to second order. The expressions were covariant up to that order. The expressions (69) and (70), however, are series expansions in which the terms are not classified according to their order in the internal coordinates, but according to the different proper multipole moments, which are characteristic atomic quantities, independent of the special reference frame used. If we limit ourselves therefore to one or more terms in the series (69) and (70), the expressions are strictly covariant. We may, for instance, choose to limit ourselves to dipole terms only. Then (69) and (70) get the symmetric form: p = T
(ak’pk
m = 2
(ak’vk
+
-
Pk
Pk
A vk)
d(Rk
-
R),
(74)
A ,uk)
d(Rk
-
R).
(75)
k
An instructive alternative form of these expressions is obtained, if we split the dipole moments into their components perpendicular and parallel to the atomic motion Pk: pk = p;
+ ,&
vk = v;
+ v;.
(76)
Then, with the explicit form (52) for !&, one obtains instead of (74) and (75) : k
(77)
From these formulae it is seen how the components (76) of the dipole moments enter into the formalism. In particular it shows how one obtains the non-relativistic approximation by neglecting terms in c-s (the magnetic moment vk contains a factor c-r). In an ordinary gaseous or condensed system Pk is of the order of the sound velocity divided by the velocity of light and therefore very small (1 O-s). The internal atomic velocity is about 1O-2 times the velocity of light, such that the last term of (77) is of more importance than the relativistic correction in the second term. The ultra-relativistic form of (77) and (78), obtained by putting Pk = c/c, is also interesting. Then only the perpendicular components of the dipole moments remain. For ordinary atoms this limit is only of academic interest,
DERIVATION
since the atoms destroy
OF MAXWELL’S
137
EQUATIONS
each other. However,
for a system of elementary
particles, suffering elastic collisions only, the ultra-relativistic form may be useful. A similar splittjng into perpendicular and parallel components can be performed,
if the quadrupole
moment is taken into account.
$9. D isczcssion. Let us state briefly the results obtained. We have derived the atomic field equations. The homogeneous atomic equations which contain only the fields e and b are simply identical to the field equations (1) and (2) (or (8), in covariant notation) of electron theory. The non-homogeneous field equations (32) and (33) (or (34), in covariant notation) contain, besides the field tensor (e, II), also the polarization tensor (p, m). The set of atomic field equations is completed by the explicit, covariant expressions (69) and (70) of the polarization tensor in terms of the proper atomic electric dipole and quadrupole moments pk and qk, and the magnetic moments vk, and furthermore the atomic positions Rk and velocities Rk, and the derivatives of these quantities. As early as 19 19, D all e nb ac ha) had stressed the desirability of expressing the polarization tensor explicitly in terms of the proper atomic quantities in a covariant way. Neither Dallenbach himself, nor later authors seem to have fulfilled this program. We shall have to postpone the discussion of their work to a later articles), since in most papers the theory starts from the outset with averaging procedures (to obtain the macroscopic Maxwell equations) which we have not yet considered at this stage.
APPENDIX
I
On the Lorentz transformation of the internal atomic quantities (9 5). Solving for the “internal time difference” from (49) one finds, up to second order in rit:
tlkt -
p* fiir
p *rialc
tic= -
1
+
p*
(P*G/c) 2
-2c
&iicr,C
(A 1)
(1 + p - &i/c)3
With the splittjng fi;, this becomes, tk -
tk = -
= &
+ iit,
itic+ St,
Rl* =
again up to second order,
+ 1 -I- p*sic/c
(1 + P*lit/c)2 (P *Gz)2
-- pii; 2~3
With
a Taylor
, (A 2)
expansion
of Rk(tk)
around
(1 +
pa hi/c)3
tjE, up to second
*
(A 3)
order, the
S.
138 internal coordinate r&t)
R. DE GROOT
AND
J. VLIEGER
~kt of (47) becomes:
= Q(p) +jd&)
+ I&&;)(&
-
G) + &&&)(Gr
-
Mz}>
(A 4)
which with (A 2) reads up to second order rk&) = Q(P) .(ri&)
+
With (A 3) for the time difference, and retaining terms up to second order only, this becomes (50)) where all quantities on the right-hand side are taken at the same time ti. APPENDIX
II
On the Lore&z transformation of the acceleratiofi (3 6). From the differentials of the Lorentz transformation (59), (60) one finds first for dRi/dth:
(A 6) the addition theorem of velocities with y = (1 -
jP)-*. A second differentia-
tion yields iii = [y(1 -
Pdk/C){iik
+
(r -
+{lik
+
1)pp'fik/p2}
(y -
+
1)pp"hk/p2 7F3(
With p = Pk = R k/ c and Yk = (1 text. Formulae
(62) and (63) follow
-
#}YP*fik/Cl . p *Rk/C)-3.
1-
(AT)
&)--* this becomes formula (61) of the
from
(54) and (57) with (61), and the
relations Qk*{U
+
ak’(Pkh
where the definition Formula
of &
(Yk vk)
1) bkPk/b’%) =
Pkh
=
u,
vk>
(A
8)
(A
9)
(52) has been used.
(64) follows from the first order approximation rkg = &‘rjz,
(which is sufficient to calculate of the identity
of (53) (A 10)
(64) up to second order) with (63), because
(where (52) has been used) and the definitions (40) and (41) of the electric quadrupole and the magnetic dipole moments.
DERIVATION
OF MAXWELL’S
APPENDIX
139
EQUATIONS
III
On atomic internal variables in terms of interlzal variables in the reference system ($7). The inverse transformation of (50) reads up to first .order
BkPlC 1-
’ rk&
Rlc’~/C
With (56) we have from (A 12) also
+
u+
1
IikPlC 1 fik.p,Jrki
_
1
.
(A 13)
In these formulae we must now replace p by Pk = I&/c. For the expressions which occur in the last two terms of (62) we find the following by straightforward calculations : ii&‘% pk.% ak’xl pk’ikz
=
=
=
YkikC
?@k’ikt
= rk&
(A 14)
Ykpk’rkh
(A 15)
+ f
(A 16)
Yz*k(pk’rki)/&
(A 17)
Yt(pk' fik)(Pk.fkt)/C.
With these expressions the third terms of (62) becomes -& c
ekt{$Pk’dO(rkirk#)
+
C-2r~(Pk’rkd)(Pk.rkziik
+
(A 18)
Pk' fikrk&
i
whereas the fourth term becomes 4c
ektC-2y#k’rk#)2
(A 19)
fik.
i
The last two expressions together yield (65). APPENDIX
Cdczdations
IV
m. From (25) one obtains with (64), (69)
of the magnetization
and (31): fflk = &2{U + +
ykc-2fik
-
Pk
*
(Yk
-
1) PkPk/@;}+k
A (nk’qk.Pk) (pk
ft vk)
+
?‘kPk
+ Pk
A (pk
-
qk’ak’v)
A do(y&k*qk*~k)
‘pk]
d(Rk
-
R),
(A
20)
where we have also used the relation (a is vector) : Pk h (&‘a)
= Pk A a.
(A21)
140
DERIVATION
Equation
OF MAXWELL’S
(A 20) can be transformed Pk
*
(Pk
* vk)
=
EQUATIONS
with the help of the identity (Pk’vk)
Pk
-
@vk.
(A
22)
If We use yk = (1 - j$j-’ and the definition (52) of &, then by collecting the terms in vk of equation (A 20) we arrive at the simple result: f& .vk. Thus we have, instead of (A 20), mk
=
{ak’vk
-
pk
A (pk
-
qk%‘V)
+
ykc-‘fk
+
YkPk
-t-
A (ak’qk’pk)
+
A do(yk~k’qk’~k)‘Pk}
d(Rk
-
R).
(A 23)
This can also be written as mk
=
{ak’vk
Since &‘Pk: result (70).
-
pk
A (pk
-
qk’ak’\J)
+
ykc-‘iik
-
?‘k(do
Pk)
+
do(Pk
A ykak.qk’fik)
= yk’pk,
+
A (ak’qk’pk) *
-
(yknk.qk’nk’Pk)
two terms
+
‘YkPk} Card
The authors wish to thank Professor cussions.
d(Rk
-
R).
and one obtains
G. E. Uhlenbeck
(A 24) the desired
for useful dis-
Received l-9-64
REFERENCES
1) Lorentz, H. A., Proc. roy. Acad., Amsterdam (1902) 254. 2) Mazur, P. and Nijboer, B. R. A., Physica 19 (1953) 971. 3) Pauli, W., Enc. math. Wiss. V 2, fast. 4; RelativitBtstheorie, Leipzig (1921); Theory of relativity, London (1958). W., Ann. Physik 58 (1919) 523. 4) DHllenbach, J., Physica, to be published; for results see Phys. Letters 10 5) de Groot, S. R. and Vlieger, (1964) 71, 294.
de Groot, S. R. Vlieger, J. 1965
DERIVATION THE
OF MAXWELL’S ATOMIC FIELD
EQUATIONS
EQUATIONS
by S. R. DE GROOT Instituut voor theoretische fysica, Amsterdam, Nederland
and J. VLIEGER Instituut-Lorentz,
Leiden, Nederland
synopsis From the electromagnetic equations for fields, produced by point particles (electrons and nuclei) the field equations, taking into account the existence of stable atoms (or molecules, or ions) are derived in covariant form. These “atomic field equations” involve, besides the field tensor (e, b) an atomic polarization tensor (p, m), or alternatively the field tensor (d, h) = (e + p, b - m). The atomic polarization tensor is given (up to second order in the internal atomic variables) as a function of atomic positions and velocities; proper electric dipole, quadrupole and magnetic dipole moments; and time derivatives of these quantities. (The “proper atomic multipole moments” are defined in Lorentz frames in which the atoms are momentarily at rest).
5 1. Introductim Maxwell’s equations in material media will be derived in two steps in this article and the following. First, from the equations for electromagnetic fields produced by point particles (electrons and nuclei), valid at what we shall call the “sub-atomic level”, the equations, which take into account the existence of stable atoms (molecules, or ions) v&illbe derived. The derivation of these “atomic field eq%atiom” will be described in this article. A following paper will be concerned with the second step: the statistical averaging process, which must be performed to obtain the Maxwell equations, valid at the “macroscopic level”, from the atomic field equations. In this article we start with the sub-atomic field equations of “electron theory”, as given originally by Lorentz, for the electromagnetic field produced by point charges ($2). If these point charges are grouped into stable atoms, one can expand the source terms in these equations into powers of the internal coordinates (5 3). These series expansions contain certain combinations of the internal variables. In $3 5,6,7 these combinations This research was supported in part by the Office of Aerospace Research, United States Air Force, through its European Office.
-
125 -
126
S. R. DE GROOT
AND
J. VLIEGER
will be expressed in terms of the proper atomic multipole moments, which are defined in $4 as the moments in momentary rest frames of inertia for the atoms. We finally discuss the resulting atomic field equations. The latter contain a polarization tensor which is expressed in terms of the atomic multipole moments, the atomic positions and velocities, and time derivatives of these quantities
($3 8, 9).
$ 2. The mb-atomic field eqtiations. Lorentz’ original “electron theory” equationsr) for the fields e and b, produced by point particles j with charges ej, positions &(t) and velocities l&(t) can be written in the form V-b
= 0,
(1)
dab + V A e = 0, V-e -doe
= 2 ej6(Rj i
(2)
R),
+ V A b = x ej(&/c) ~(RJ 3
(3) R).
They are valid in a time-space Lorentz’frame (ct, R) (c = velocity The time and space differentiations are written as
(4 of light).
do E d/dct, V = dldx, dldy, d/d.z.
(5)
These derivatives are total derivatives in the sense that the fields depend not only explicitly on t and R, but also implicitly, because they are functions of R&Q, &(t,) and higher time derivatives, where tj = t - ~3 is the retarded time with 7, = T~(R, t) following from TJ = IRj(t - TJ) - RI/c. In covariant notation we write with
d, = (do, V), da = (da, V) and the anti-symmetric
= 0, 1, 2, 3)
(6)
field tensor
f@(% p = 0, 1, 293) Then the homogeneous
da = -da(a
with
equations
f01, f02, fO3 = e, f23, f31, fl2 = b.
(7)
(1) and (2) read
daf#fSY + dflfya + dyf”@ = 0,
(8)
with C& = 1 2 3 in (1) and 023, 031 and 012 in (2). The inhomogeneous equations (3) and (4) b ecome the cases CY= 0 and c( = 1, 2 and 3 of daf@ = C e&/c) I
8(Rf -
R)
(a = 0, 1, 2, 3),
(9)
where Ri = ct and Rt, Rf, Ri = R,. The right-hand side of (9) is a fourvector: in fact it transforms as R;, because the three-dimensional deltafunction transforms as a time. The equations were written for a collection of arbitrary charged point
DERIVATION
OF MAXWELL’S
127
EQUATIONS
particles. We shall now choose (spinless) electrons and nuclei as the particles and refer to (l)-(4) or (8), (9) as the subatomic field equations. In the next sections we consider how we may transform these equations, if the electrons and nuclei are grouped into stable atoms, molecules or ions. We now suppose
5 3. The atomic series expansion.
that the electrons
and nuclei are grouped into stable “atoms”, indexed by k. By atoms we shall understand any stable group, such as a molecule, ion or atom and also single free electrons*). We number the constituents of these groups by i. So, replacing j by ki, we have for (3) and (4)
v-6? =
c
eki
6(&t
-
R),
(10)
k,i -doe
+ v A b = x ‘Z&k&)
6(&i
-
R).
(11)
k,i
For later use, it will be convenient e =
c, ek,
b
=
to split the fields according I: bk
(Or f”o = 2
to
j$‘).
(12)
We can then write separate equations for all values of k: ‘17’ek = --dock
+
z ekZ d(Rkt i
v A bk =
c
ekt(fik&)
R),
(13)
6(Rkt
-
R).
i
Let us now choose a point Rk describing the whole, and int reduce internal coordinates rkZ and . . Rkz = Rk + rki, Rkt = Rk +
(14
position of atom k as a velocities ik2 by . rkt.
(15)
We now consider the case in which the solutions (retarded fields) of the field equations can be written as converging series expansions in rkt/l& - R ], in accordance with the assumption that stable atomic structures exist. We can therefore develop the delta-function into powers of rk{ around Rk - R. We shall call this procedure the atomic series expansion. (We follow in this section essentially the formalism of Mazur and Nijboers), but without averaging). For (13) we thus obtain V’t?k = x eki(l -
i
rk6.V + &rkg
:vv - ...) d(Rk- R),
(16)
*) Formally one can choose as the “group” anything between (and including) the “elementary” particles (here the electrons and nuclei) and the complete system. Physically one wishes to choose stable groups as the sub-units of the system. Atoms, molecules but it may be useful under certain circumstances to consider group, e.g., vibrating formula (15)).
antennas
or plasmas
observed
and ions are most obvious choices, the complete system as one single
from a large
distance
(see the text
following
128
S. R.
DE GROOT
AND
J. VLIEGER
where V operates on R. This can be written as V.C?k= pk - V’pk,
(17)
with the notations pk = 2 ek#6(Rk - R) = ek d(Rk i
R)
(18)
for the charge density of atom k(ek = & e& is the atomic charge), and pk
=
(:
ek#ki
-
4 x i
ektrktrki’v
+
. ..)
d(Rk
-
R).
(19)
The CpUtitieS & ektrkt and +xczek&lrkf, appearing in this equation, have the form of atomic electric dipole and quadrupole moments, and in the non-relativistic approximation can be interpreted as such. However, in a relativistic theory such as ours this is not possible, since these quantities are defined in the (ct, R)-frame, whereas the proper atomic moments must be defined in a proper atomic Lorentz frame. We deal with this problem in the next section. We now write (14) with the same Taylor expansion and use also (15) : -doQ
+ v A bk
=
x ekl{(@k/C) i
-
-(&k/C) +
C-l(
v’(rk# iki
-
-
+rk(rki’v
ik(rki’V
+
+ . ..)}
. ..)
d(Rk
+
R).
-
(20)
With the current density of atom k, defined as ik = r, e&k i
6(Rk - R) = t?k& d(Rk - R)
(21)
and with (19)) one can write for (20) -do&Q + v A bk = h/c +
C-l
(&k/C) v.pk x
ek((
ik(
-
+ &rk$‘V
+
. ..)
6(&
-
R).
a
(22)
This equation can be written in a simpler form by applying a vector identity to the second term and to half of the fourth term at the right-hand side. Taking Q independent of R, one has a(V.b) = (a.V) b - V A (b A a).
(23)
Equation (22) then becomes: -d&Q + v
A
bk = ikjc + dapk + v
A mk,
(24)
where the following quantity has been introduced mk =
(T&~
x
i
ek(rkd
A ik(
+
e..)
d(&
-
R) + pk A &c/C.
(25)
DERIVATION
OF MAXWELL’S
EQUATIONS
129
Here again an atOn& qUaIItity, &-l z:t C?k&$ A &, appears. It has the form of a magnetic moment, but it is defined in the (ct, R)-frame, not in the proper atomic frame. The last term of (25) gives the effect of pk moving with the atomic Velocity tik. We thus find the forms (17) and (24) for the inhomogeneous field equations (13) and (14), with the charge-current density (18) and (21) and the quantities pk and mk defined by the series expansions (19) and (25). Let us introduce the covariant notation for the (anti-symmetric) field tensor /$(a,/3 = 0, 1,2,3)
with
f;‘, fi2, fok3 = ek,fE3,f;‘, fi2 = bk,
(26)
the charge-current density four-vector j$(oz= 0, 1) 2, 3)
with
j; = ,QC,j;, j;, j,” = ik
(27)
and let us define the quantities my = - ?‘?Ze(cc,j? = 0, 1, 2, 3) with P?Z;‘,?‘PZ;‘,m;’ =pk, ??Z13, P?Z;‘,FBi2= mk. (28) Then the atomic field equations (17) and (24) are the cases a = 0 and = 1, 2, 3 of
dsf$ = i;lc + dm;zB.
a =
(29)
In the following we shall limit ourselves in the expressions (19) and (25) only to terms up to the second order in the internal variables rkt and &. It can then be shown that the quantities rniP (28) are the components of an (anti-symmetric) tensor, which will be called the polarization tensor. With (12) and the similar espressions*) **). (30) (31)
we find from the inhomogeneous equations (17) and (24) the equations V.e = p -doe
V-p,
+ V A b = i/c + dop + V A m,
(32) (33)
or, in covariant notation, dFf”fl = ja/c + dBrn@.
(34)
*) It may be noted that while the charges eb are invariants, the atomic charge-current density (PC,I) is a four-vector. If, therefore, in a particular case, one has a charge density p = 0 in one Lorentz frame, then in a different Lorentz frame the charge density will in general be different from zero s). **) Note that from (18), (21) and (30) follows the law of conservation of charge: dp/dt + + V.f = 0 or,in covariant notation d& = 0.
130
S. R. DE
GROOT
AND
J. VLIEGER
With the help of the definitions d = e + p, h = b the following
alternative
(35)
forms of (32) and (33) are obtained:
-ded or, in covariant
m (or MB = f@ - m@),
V.d = p,
(36)
+ V A h = i/c,
(37)
notation, dsh@ = j”/c.
(38)
The “atomic field equations” (32) and (33) contain the polarization tensor (p, n-z), given by (31) and the series expansions (19) and (25). These expansions contain characteristic combinations of the internal variables. In the following sections we shall express these combinations and therefore the polarization tensor, in terms of the proper atomic multipole moments. $ 4. The atomic mdtipole moments. We must first define the proper atomic multipole moments. The combinations of internal variables Y~Zand & which occur in the polarization tensor (see (19) and (25)) are defined with respect to our reference frame (ct, R). A quantity like x;a ek(%, for instance, has the form of an atomic dipole moment, but rki is defined in the reference frame (ct, R). The proper atomic multipole moments, however, are defined with the help of internal variables in a Lorentz frame in which the atom is at rest. This “atomic frame” has a (constant) velocity u = &(t) with respect to the reference frame, where @k(t) is the velocity of atom h at time t on the latter frame. Since the atom suffers in general an acceleration, it is only at rest in the atomic frame at one particular time t; in other words: the atomic frame is a momentary rest frame of inertia for the atom. The procedure ($9 5, 6, 7) will therefore be the following. We express internal quantities like rk$ and ika, defined in the reference frame (ct, R\, in terms of quantities rig and &, defined on the momentary atomic frame, by means of a Lorentz transformation, involving a constant velocity U, at a certain fixed time t. We repeat this at all times t, which means that in the results obtained in the procedure described above, we have to replace v by &. This leads to expressions depending on rit and iii through the atomic multipole moments, which are defined as: pk
the electric dipole moment,
=
c
i
ek&t
(39)
DERIVATION
OF MAXWELL’S
EQUATIONS
131
the electric quadrupole moment,
the magnetic dipole moment, etc. Furthermore we wish to consider these proper atomic quantities as functions of rkf and iki in the original reference frame, because the atomic field equations, which involve the polarization tensor, contain quantities all measured in the original reference frame. So we have to consider also the Lorentz transformation, which is simply the reverse of the transformation described above, and which gives rit and ijG(in terms of rk2 and ixc. Again consideration of a succession of such Lorentz transformations at all times t means replacement of v by tit. If proper atomic quantities are considered in the reference frame (ct, R), we shall denote them with a bar, e.g. x:( e$& and 8 ECektrk& § 5. The Lore&z transformation from the reference frame to the atomic frame. We wish to express rkt and ik( as measured in the reference frame (ct, R) in terms of the quantities rig and & as measured in the atomic frame. For this we need the Lorentz transformations with v 3 /?c the velocity of the atomic frame (with dashed quantities) with respect to the reference frame:
(42) (43) Rk@) = G(Gc) + (7 t = yti
1) NC(G) *P/VP + rtipc,
+ yl?gtg
*p/c.
(44) (45)
From the difference of (43) and (45) one finds tic - Gc= -(P/C) *(Gc(&)
-
Rh(G)).
(46)
From the difference of (42) and (44) it follows with (15) and (46) rt#)
= Q(P) .{N&)
- K(G)},
(47)
1) /V/P,
(4.8)
where the three-tensor Q is defined as Q(P) = U + (41 - P2 -
with U the unit three-tensor. We want to express all quantities at the right-hand side of (47) at the same time, which we take to be the time ti. This can be achieved by first
132
S. R. DE GROOT
developing $a -
the right-hand
ti = -(P/c)
AND
J. VLIEGER
side of (46) in a Taylor series
+$c&)
+ &&)(Gz
-
G) + 9iii&i)(Gc2
where terms up to second order have been included.
-
G)2),
(49)
Then solving tit -
tic
up to second order and inserting the result into (47), where one has developed &(tjJ in a Taylor series around ti (see appendix I), one obtains, with (15) and fik~ = & + i;k~, up to second order
.Ir&
I.
-
p.rid&
+ 8 (1
1 + a;*/+
(p -riilc) 2 +
.. ,
1
Rk.
&p/c)2
(50)
This is the result of a Lorentz transformation at a fixed time t. We now consider such transformations at all times. This means that we replace p by Pk s fit/C. Furthermore, in the atomic system the atom is at rest, so that li;(t;) vanishes. In this way (50) becomes rkt(t)
=
ak’{rj(
-
Pk.&%&
+
8(Pk’rb/c)2~>p
where the bar indicates that the proper atomic quantities to be considered in the original reference frame. Furthermore the abbreviation ak
An alternative
From
=
n(pk)
=
u +
(41
-
/f$ -
(51)
rit, etc., have we have used
1) PkPk/t$
(52)
form of (51) is
this connection
we obtain,
with the multipole
From (53) one can also obtain the other internal in pk. ( 19), up to second order :
moments
quantity
(39)-(41)
which appears
(55)
We also need the internal
quantity,
occurring
in mk (25).
It contains
DERIVATION
OF MAXWELL’S
133
EQUATIONS
~6 A ikt. The quantity & follows from differentiation of (50), where we can neglect the last two terms, since we need ~2 A ik( only up to second order. With & = 0 and dtldti, = y,
(56)
which follows then from (45), and replacing p by /3k, we then obtain ik( = y;lsk’{ii(
-
(57)
Z(&Z/C)},
where we have written yk
=
(1 - /9$-t.
(58)
The internal variables have now been expressed as shown in equations (54), (55) and (57). In the first expression an as yet unencountered “atomic quantity” appears in the third term. It can, however, be eliminated, as we shall show in 5 7. Furthermore, (54) and (57) contain the acceleration ii?. We shall prefer to express it in terms of the acceleration fik in the reference frame (ct, R). This will be done in the next section, before we further develop the internal variables. 5 6. The acceleration in the atomic and in the reference frames. To express the acceleration & in the atomic reference system in terms of the acceleration fik: in the reference frame (ct, R) one needs the Lorentz transformation
JGW = &c(t) + (y -- 1) &c(t) -p/3/P
tic= yt -
y&(t)
-
-p/c,
y,%t,
(59) (60)
which is the inverse of the transformation (44), (45). By differentiating twice with respect to time one finds iii in terms of iit, /& = kk/C and /3 = V/C (see appendix II). One then replaces in this expression /3 by /?k to obtain the relation (61)
valid for all times. When this is inserted into (54) and (57), we obtain (see also appendix II): 2
ekfrkt
=
fik’pk
+
Pk A vk -
i
+kt =
,‘;‘nk’i$
-
Yk(Pk’x$)
fik/C.
(63)
The last result and (53) permit us to calculate (up to second order) the
134 following
S. R. DE GROOT
quantity,
AND
J. VLIEGER
which occurs in nzk (25) (appendix
+
Ykfik
II):
A (ak*qk.Pk/C2).
(64)
We have now found the expressions quantities
occurring
(62), (55) and (64) for the internal in pk (19) and mk (25).
5 7. Internal variables expressed in terms of the atomic mzlltipole moments. The expression (62) for the internal variable & e&& contains the atomic VdOCity hk, the at0II-k XCekratiOn fik, the atOmiC m.dtipOk IrIOmentS pk, vk and qk, and also the atomic quantity J$ ekc(rj&t + ii&). We wish to express now the latter quantity in terms of the others. A simple way to achieve this is first to express rig and ii8 in terms of rkg and +kgby means of the inverse transformation of (50). We need it only to first order if we wish to calculate the atomic quantity mentioned above up to second order. We obtain in this way (see appendix III) for the last two terms of & ek&t (62) :
(65) Now with (55) we can express the right-hand side of this formula in terms of the quadrupole
moment
-Y$k’do(~k’qk’~k)
-
qk. It becomes then
Y;(Pk’iik)
Pk’~k.qk’~k/C2 =
=
-YkPk’dO(Yk~k’qk’~k),
where the last result follows because time differentiation identity Pk.&
=
C2yi3
(66)
of (58) yields the
dayk.
(67)
With the result (66) for the last two terms of (62) we have 2 i
ektrki
=
ak’pk
f
Pk A vk
-
YkPk’dO(YkQk’qk’ak)r
(68)
whereby we have expressed the internal quantity 26 ek@kf as a function of the atomic multipole moments pk, qk and vg, the atomic velocity @k = pkc and a time derivative of these quantities. The other internal quantities (55) and (64), which occur in pk (19) and mk (25), were already expressed in this way.
DERIVATION
Ej8. The polarization
OF MAXWELL’S
135
EQUATTONS
tensor in terms of the atomic multipole moments. The
atomic polarization tensor m@(p, m) is, according to (31), the sum over k of the tensors my(pk, mk), which are given in terms of internal quantities by the expressions (19) and (25). These internal quantities have now been expressed in terms of the atomic multipole moments pk, ok and vk, the atomic velocity fik = pkc, and time derivatives of these quantities according to the formulae (68), (55) and (64). We thus obtain immediately p=~{~k’(~kk
‘jk’nk’v)
+ Pk A vk -
do(~k’qk’~kJ’k)
‘Pkyk}
a(Rk
-
R),
(69)
and after some elementary vector manipulations (see appendix IV) : m = x {Qk’vk k
-
Pk A (pk
+
qk’fik.V) do(Pk
+
* qk’akyk)
‘Pk)‘k)
d(Rk
-
R)*
(70)
In this way the polarization tensor at the time-space point (ct, R) is expressed up to second order in terms of the proper atomic multipole moments pk, qk and vk, the atomic positions Rk and velocities & = pkc, and time derivatives of these quantities. (Recall that the quantities & (52) and yk (58) are functions of the atomic velocities). To discuss the physical meaning of the polarization tensor, it may be useful to write down first the non-relativistic approximation (obtained by neglecting terms with c-s). Then the last two formulae become p = 2 (pk m =
x (vk k
4k.v)
p k A (pk
-
a(Rk qk’v)}
-
R), d(Rk
(71) -
R).
(74
(Note that the magnetic moment vk contains a factor c-r, so that its contribution to p is of order c-z). These formulae follow also directly from (19) and (25) with the non-relativistic forms of (68), (55) and (64):
which were used by Mazur and Nijboers). The formulae (71) and (72) for the non-relativistic polarization tensor show the electric multipoles occurring in the “polarization vector’ ’ p, whereas in the “magnetization vector” m we find the effect of moving charges; the motion of particles inside the atoms gives rise to the atomic magnetic moment vk, while the motions of the electrically polarized atoms as a whole (terms with /?k) give the second contribution. Let us now turn back to the discussion of the relativistic expressions (69) and (70) for the polarization tensor. We then see that, as compared to the
136
S. R. DE GROOT
AND
1. VLIEGER
non-relativistic approximation, correction factors S2k involving the atomic velocity appear. Furthermore we find now in p a contribution from magnetic dipoles vk in motion (Pk A vk), analogous to the contribution to rrz from electric multipoles in motion. In the third place new terms with time derivatives, containing the electric quadrupole moments, appear both in p and m. We note that the expressions for p and m were obtained in $ 3 as series expansions in the internal coordinates of the atoms up to second order. The expressions were covariant up to that order. The expressions (69) and (70), however, are series expansions in which the terms are not classified according to their order in the internal coordinates, but according to the different proper multipole moments, which are characteristic atomic quantities, independent of the special reference frame used. If we limit ourselves therefore to one or more terms in the series (69) and (70), the expressions are strictly covariant. We may, for instance, choose to limit ourselves to dipole terms only. Then (69) and (70) get the symmetric form: p = T
(ak’pk
m = 2
(ak’vk
+
-
Pk
Pk
A vk)
d(Rk
-
R),
(74)
A ,uk)
d(Rk
-
R).
(75)
k
An instructive alternative form of these expressions is obtained, if we split the dipole moments into their components perpendicular and parallel to the atomic motion Pk: pk = p;
+ ,&
vk = v;
+ v;.
(76)
Then, with the explicit form (52) for !&, one obtains instead of (74) and (75) : k
(77)
From these formulae it is seen how the components (76) of the dipole moments enter into the formalism. In particular it shows how one obtains the non-relativistic approximation by neglecting terms in c-s (the magnetic moment vk contains a factor c-r). In an ordinary gaseous or condensed system Pk is of the order of the sound velocity divided by the velocity of light and therefore very small (1 O-s). The internal atomic velocity is about 1O-2 times the velocity of light, such that the last term of (77) is of more importance than the relativistic correction in the second term. The ultra-relativistic form of (77) and (78), obtained by putting Pk = c/c, is also interesting. Then only the perpendicular components of the dipole moments remain. For ordinary atoms this limit is only of academic interest,
DERIVATION
since the atoms destroy
OF MAXWELL’S
137
EQUATIONS
each other. However,
for a system of elementary
particles, suffering elastic collisions only, the ultra-relativistic form may be useful. A similar splittjng into perpendicular and parallel components can be performed,
if the quadrupole
moment is taken into account.
$9. D isczcssion. Let us state briefly the results obtained. We have derived the atomic field equations. The homogeneous atomic equations which contain only the fields e and b are simply identical to the field equations (1) and (2) (or (8), in covariant notation) of electron theory. The non-homogeneous field equations (32) and (33) (or (34), in covariant notation) contain, besides the field tensor (e, II), also the polarization tensor (p, m). The set of atomic field equations is completed by the explicit, covariant expressions (69) and (70) of the polarization tensor in terms of the proper atomic electric dipole and quadrupole moments pk and qk, and the magnetic moments vk, and furthermore the atomic positions Rk and velocities Rk, and the derivatives of these quantities. As early as 19 19, D all e nb ac ha) had stressed the desirability of expressing the polarization tensor explicitly in terms of the proper atomic quantities in a covariant way. Neither Dallenbach himself, nor later authors seem to have fulfilled this program. We shall have to postpone the discussion of their work to a later articles), since in most papers the theory starts from the outset with averaging procedures (to obtain the macroscopic Maxwell equations) which we have not yet considered at this stage.
APPENDIX
I
On the Lorentz transformation of the internal atomic quantities (9 5). Solving for the “internal time difference” from (49) one finds, up to second order in rit:
tlkt -
p* fiir
p *rialc
tic= -
1
+
p*
(P*G/c) 2
-2c
&iicr,C
(A 1)
(1 + p - &i/c)3
With the splittjng fi;, this becomes, tk -
tk = -
= &
+ iit,
itic+ St,
Rl* =
again up to second order,
+ 1 -I- p*sic/c
(1 + P*lit/c)2 (P *Gz)2
-- pii; 2~3
With
a Taylor
, (A 2)
expansion
of Rk(tk)
around
(1 +
pa hi/c)3
tjE, up to second
*
(A 3)
order, the
S.
138 internal coordinate r&t)
R. DE GROOT
AND
J. VLIEGER
~kt of (47) becomes:
= Q(p) +jd&)
+ I&&;)(&
-
G) + &&&)(Gr
-
Mz}>
(A 4)
which with (A 2) reads up to second order rk&) = Q(P) .(ri&)
+
With (A 3) for the time difference, and retaining terms up to second order only, this becomes (50)) where all quantities on the right-hand side are taken at the same time ti. APPENDIX
II
On the Lore&z transformation of the acceleratiofi (3 6). From the differentials of the Lorentz transformation (59), (60) one finds first for dRi/dth:
(A 6) the addition theorem of velocities with y = (1 -
jP)-*. A second differentia-
tion yields iii = [y(1 -
Pdk/C){iik
+
(r -
+{lik
+
1)pp'fik/p2}
(y -
+
1)pp"hk/p2 7F3(
With p = Pk = R k/ c and Yk = (1 text. Formulae
(62) and (63) follow
-
#}YP*fik/Cl . p *Rk/C)-3.
1-
(AT)
&)--* this becomes formula (61) of the
from
(54) and (57) with (61), and the
relations Qk*{U
+
ak’(Pkh
where the definition Formula
of &
(Yk vk)
1) bkPk/b’%) =
Pkh
=
u,
vk>
(A
8)
(A
9)
(52) has been used.
(64) follows from the first order approximation rkg = &‘rjz,
(which is sufficient to calculate of the identity
of (53) (A 10)
(64) up to second order) with (63), because
(where (52) has been used) and the definitions (40) and (41) of the electric quadrupole and the magnetic dipole moments.
DERIVATION
OF MAXWELL’S
APPENDIX
139
EQUATIONS
III
On atomic internal variables in terms of interlzal variables in the reference system ($7). The inverse transformation of (50) reads up to first .order
BkPlC 1-
’ rk&
Rlc’~/C
With (56) we have from (A 12) also
+
u+
1
IikPlC 1 fik.p,Jrki
_
1
.
(A 13)
In these formulae we must now replace p by Pk = I&/c. For the expressions which occur in the last two terms of (62) we find the following by straightforward calculations : ii&‘% pk.% ak’xl pk’ikz
=
=
=
YkikC
?@k’ikt
= rk&
(A 14)
Ykpk’rkh
(A 15)
+ f
(A 16)
Yz*k(pk’rki)/&
(A 17)
Yt(pk' fik)(Pk.fkt)/C.
With these expressions the third terms of (62) becomes -& c
ekt{$Pk’dO(rkirk#)
+
C-2r~(Pk’rkd)(Pk.rkziik
+
(A 18)
Pk' fikrk&
i
whereas the fourth term becomes 4c
ektC-2y#k’rk#)2
(A 19)
fik.
i
The last two expressions together yield (65). APPENDIX
Cdczdations
IV
m. From (25) one obtains with (64), (69)
of the magnetization
and (31): fflk = &2{U + +
ykc-2fik
-
Pk
*
(Yk
-
1) PkPk/@;}+k
A (nk’qk.Pk) (pk
ft vk)
+
?‘kPk
+ Pk
A (pk
-
qk’ak’v)
A do(y&k*qk*~k)
‘pk]
d(Rk
-
R),
(A
20)
where we have also used the relation (a is vector) : Pk h (&‘a)
= Pk A a.
(A21)
140
DERIVATION
Equation
OF MAXWELL’S
(A 20) can be transformed Pk
*
(Pk
* vk)
=
EQUATIONS
with the help of the identity (Pk’vk)
Pk
-
@vk.
(A
22)
If We use yk = (1 - j$j-’ and the definition (52) of &, then by collecting the terms in vk of equation (A 20) we arrive at the simple result: f& .vk. Thus we have, instead of (A 20), mk
=
{ak’vk
-
pk
A (pk
-
qk%‘V)
+
ykc-‘fk
+
YkPk
-t-
A (ak’qk’pk)
+
A do(yk~k’qk’~k)‘Pk}
d(Rk
-
R).
(A 23)
This can also be written as mk
=
{ak’vk
Since &‘Pk: result (70).
-
pk
A (pk
-
qk’ak’\J)
+
ykc-‘iik
-
?‘k(do
Pk)
+
do(Pk
A ykak.qk’fik)
= yk’pk,
+
A (ak’qk’pk) *
-
(yknk.qk’nk’Pk)
two terms
+
‘YkPk} Card
The authors wish to thank Professor cussions.
d(Rk
-
R).
and one obtains
G. E. Uhlenbeck
(A 24) the desired
for useful dis-
Received l-9-64
REFERENCES
1) Lorentz, H. A., Proc. roy. Acad., Amsterdam (1902) 254. 2) Mazur, P. and Nijboer, B. R. A., Physica 19 (1953) 971. 3) Pauli, W., Enc. math. Wiss. V 2, fast. 4; RelativitBtstheorie, Leipzig (1921); Theory of relativity, London (1958). W., Ann. Physik 58 (1919) 523. 4) DHllenbach, J., Physica, to be published; for results see Phys. Letters 10 5) de Groot, S. R. and Vlieger, (1964) 71, 294.