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On the relativistic dynamics of polarized systems. II
Physica
41 (1969)
368-378
@ North-Holland
ON THE
RELATIVISTIC
POLARIZED SIMPLIFICATION
Publishing
DYNAMICS
SYSTEMS.
OF M0LLER’S
Instituut-Lorentz,
OF
II
EQUATIONS
AND THE ENERGY-MOMENTUM J. VLIEGER
Co., Amsterdam
OF MOTION
TENSOR
and S. EMID Leiden,
Nederland
Received 4 July 1968
synopsis The classical equations of motion for electric and magnetic dipole atoms (or molecules) in an external electromagnetic field of force, derived in a previous paper on the basis of Mraller’s theory of the relativistic dynamics of systems with an internal angular momentum, are simplified by showing that certain terms, which contain an unphysical trembling motion (‘
3 1. Irttrodzlction. In a preceding paperr) on the relativistic dynamics of polarized systems, in the following denoted by I, we have derived the equations of motion of electric and magnetic dipole atoms in an external electromagnetic field of force. The derivation was based on Moller’s theorys) of the dynamics of relativistic systems with an internal angular momentum in an arbitrary (non-gravitational) field of force. In this theory the equations of motion and of the change of the intrinsic angular momentum are found, using the definition * (I.1 1) for the centre of gravity, or rather, using M@ller’s words, the pseudo-centres of gravity of the system. These equations were then applied to the special model of electric and magnetic dipole atoms (or molecules) and used in order to derive the relativistic energy-momentum tensor for a system, consisting of N of these atoms. It was found that this four-tensor is exactly of the same form as the one derived by de Groot and * Formulae of paper I will be denoted here by (I.l), (1.2), etc. 368
ON THE
RELALIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
369
Suttorpsp a), although these authors have used a slightly different definition for the reference point within the atom from the one used here. The motion of the pseudo-centres of gravity, following from Merller’s equationss), is still rather unphysical as it contains, superposed on what may be interpreted as the real translational motion of the system, a very small (order of the Compton-wavelength) trembling motion (frequency higher than which has no physical interpretation 1Osr), the so-called “Zitterbewegung”, and is purely a consequence of the definition (I. 11) of these reference points. It will be shown in 3 2 that from the above equations of motion, which are third order differential equations, one can derive, by using an iterative procedure proposed by Plahtes), equations of the usual second order type, which possess solutions which also satisfy the original equations, but which do not give rise to a trembling motion and can therefore be considered as the correct equations, describing the motion of the system as a whole in an external field of force. It appears that for atoms or molecules this iteration method is rapidly convergent, and gives already in lowest order negligible results. The equations of motion (and of course also the angular momentum equation) are therefore simplified, and this is also the case with the atomic energy-momentum tensor derived with the help of these equations. In $ 3 we find that the field part of this tensor is exactly the same as obtained in I, but that its material part is slightly different. It is shown that the total energy-momentum tensor can be obtained in a symmetrical form, using the same procedure as in I, $4. The theory developed in the present paper is based on a classical model of the atoms or molecules. Radiative effects are neglected, as this has also been done in I. We hope to come back to this latter point in a future paper. 5 2. Simplification
of Mtiller’s equations of motiolz in the case of electric and of the equations of motion (1.41) of electric and magnetic dipole atoms in a given external electromagnetic field of force, starting from Moller’s equations (I. 15) for the motion of arbitrary relativistic systems with an internal angular momentum. The latter equations were derived by Mprllers), using the condition (cf. (I. 11))
magnetic dipole atoms. In I we have given a derivation
for the reference point within the system. It follows from the definition (1.8) of Q@ (internal angular momentum four-tensor with respect to the reference point) and the fact that Ub is the derivative of the time-space coordinate four-vector X0 of this point with respect to the eigentime T, that this condition has the form of a first-order differential equation in X, and does therefore not define one single reference point, but an infinity of these points, which are called pseudo-centres of gravity by Moller. This is the reason, that the differential equations (1.15) for the motion of these
J. VLIEGER
370
AND S. EMID
pseudo-centres of gravity are not of the second order in X, as usual, but of the third order. Consequently the energy-momentum four-tensor derived in I with the help of these equations, contains a material part, which does possess not only velocity-dependent terms, but also terms depending on the accelerations of the atoms (cf. (1.67)). Th’is result has also been obtained by de Groot and Suttorp 394), which authors have used, however, a different definition for the reference points within the atoms (molecules) as the one used by us. The problem of solving the equations (I. 1.5) and in particular the physical significance of the solutions obtained, have been a matter of long discussions in the past, (see refs. 6 and 7). One obtains a trembling motion of a very high frequency (> lOsi), the so-called “Zitterbewegung”, superposed on a motion of a much smoother character, (in the free case a straight line). From Moller’s point of views), however, this trembling motion is only a consequence of the definition (1) of the reference points, and has therefore no physical meaning *. Moreover he finds a very small amplitude of the trembling motion, namely of the order of the Compton wave length, which is for an atom about lo-13 cm, much smaller than its dimensions. It is therefore evident, that one must try to find new differential equations, which describe the above mentioned smooth motion, but not the unphysical trembling motion. A straightforward way to achieve this is the following iteration procedure, which is formally analogous to the method of Plahte5) for the elimination of the “Zitterbewegung” of the classical spinning electron. Consider the equation of motion (I. 15A) and the internal angular momentum equation (I. 15B) :
&3 + f
,‘-J~Q&J0, -
&
(,lDQw 0, = A “&@“C,
PB)
where M* is the rest mass of the system, defined by (I. 12), Fa the four-vector of the external force acting upon it (eq. (1.5)), da@ the four-tensor of the moment of this force with respect to the reference point (eq. I.lO)), and
with B&pthe elements of the unit four-tensor. (The dots mean differentiation with respect to the eigentime 7.) Since it is the second term at the left-hand * One should also note that a motion of the extreme high frequency of the trembling motion, has never been observed in physics.
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
side of eq. (2A), which causes the microscopically
SYSTEMS.
II
371
small trembling motion, one
can obviously write eq. (2A) in lowest order by neglecting this term, and then solve 0, as a function of X, and U, from the approximate second order differential equation, which is left. If this result is now substituted for 00 and I?,, into eqs. (2A) and (2B), one obtains in first approximation the influence of the external field on the motion through the “Zitter-term” c-s(d/dT) (L?“G&), and the terms c-stY~~%~~,, and c_sUflsZ~‘~~, causing the so-called Thomas-precession of the internal angular momentum of the system. In this way one obtains for (2A) a second- and for (2B) a first-order differential equation, and one can easily see, that this set of equations describes, within the approximation of their derivation, the motion of the system (with intrinsic angular momentum) in the external field of force, however, z~ithout the unphysical trembling motion. With the above mentioned iteration method of Plahtes), one could obtain also higher approximations of this set of differential equations, but we shall show below that for atoms or molecules in an electromagnetic field the first approximation is already negligibly small. The equations of Moller for the motion of electric and magnetic dipole atoms, derived in I, can therefore be simplified. In order to prove the above statements, we remind that for these dipole atoms (cf. eqs. (1.36) and (1.37)) :
F” = f fWJ,g - ;
(i3,f"fl) /_J~Y-
f
&
(f"$q'U,,),
and: &S =
-!C (jWJpydAd8 -
f8V,q,dAda),
(5)
where e is the total charge of the atom (or molecule), f@ the tensor of the external electromagnetic field ((fol,f02,f03)= e and (f23,f31,f12)= b, with e the electric and b the magnetic field strength) and ,u@ a tensor given by* :
where e(t) are the charges of the constituent particles (electrons and nuclei) of the atom (or molecule), x& the relative time-space coordinates of these particles with respect to the centre of gravity of the atom and kg, the derivatives of these quantities with respect to the eigentime T of this reference point. Substituting eqs. (4) and (5) into (2), we obtain, after
* This formula is directly found by comparing eq. (17) of ref. 8 with eq. (20) of our paper I. We have only changed the notations in (6).
J. VLIEGER
372
multiplication c
S. EMID
with c :
d”,(MV”)
--
AND
+ f
$
(Qm7~) = efdlwjj -- (aypq pjp -
&{(fay~ye - fSypya) ‘%}+f -$-(U"UBfSY,hdUd),
,i
(cf. (I.41)). Omitting the second term on the left-hand find in lowest order approximation: c&
(M*Ua) N efaWo
--
& {(fay~Y5 - fSYpY") u,}+ f $
,i
-
(7A)
side of eq. (7A), we
(i3,f@) p$J -
(~*u~f5y~ydud).
(8)
Let us now first disregard the terms with ,ua@ at the right-hand side of this equation. (We shall see below that they give rise in the iterated equations to terms of third and fourth order in the internal variables X& and %g,, which are neglected in the present theory.) One can then easily solve i?a from the remaining equation, obtaining :
(9) Substituting
this result into (7A) and (7B), we get
c$(M’U.)+-&g +
(
-
@Yf”B)
+ 7
1
Pay
z
d
-
;
-g
Da@fflYUy N efWJ~ -
WYPY@
-
fflYPY9
%I
+
(‘04
(UaUSfsyhdUd, e
CtiaS+ ___
M*c2
N A~,A$(fyy~
UC&!~yfydUd
-
-
&
Ufl@yfydUd
=
fc”py”),
where we have used the relation
(10B) (1). If we introduce
the tensor:
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
373
eqs. (10A) and (IOB) can be written as
c$
(M*Ua) 1: efaWfl - (&f”fl).j@ - $
+ i c2
d dr
cfias = +
L
$
(fbypyWfl) +
(124
(ui3fBfsyjiydd,Ja),
(fay,n!dys
-
foY/dya)
Udfdy(kyflUa
-
+
&
(fayUB
-
fsyua)
,uydU,,
+
fiyaUB),
(12B)
c2
using also the relation (1). Now the four-tensor LW is given by (1.8), and taking for the energymomentum tensor T@(x), appearing in this formula, in first approximation only its material part :
~(4)(&t,(w) -
x}dw,
(13)
--m
where m(g) are the rest masses of the constituent particles of the atom (molecule), X& their time-space coordinates and ~(1) their eigentimes, one derives in a straightforward way, making also use of eq. (1) and the relation x&Ua = 0 (cf. (1.23)), that
J-J)“@ = z:z W)(&X(i)‘Y - &x&,)
LP,Lvy,
(14B)
also the results obtained by de Groot and Suttorpsp4)). One easily verifies, with the help of the antisymmetry of the field tensor f@ and the property d,Yd,S = d,fl (cf. (1.40)), that the last terms at the right-hand sides of eqs. (12A) and (12B) respectively can be rewritten in the following forms : (cf.
(154
and : f
Udfdy(pySUa
-
- y&2$
Fy”uo)
Ua)
=
f
udfdy
,u.$A~~A~~-
,u&A~~da~ - +Slya (
)
Ufl . 1
(15B)
J. VLIEGER
374
AND
S. EMID
Futhermore it can be shown (see appendix), that for atoms and molec&es the following inequality holds for each o( and /3:
(16) We may therefore neglect the terms containing sZ@ at the right-hand sides of eqs. (15A) and (15B) with respect to the terms with ~“0 which means that we may replace the tensor p ~0 in the expressions at the left-hand sides of these equations in very good approximation by ,u@. Eqs. (12A) and( 12B) then finally become :
d
1 c2
>,-
{(fayh'b
fay,h'a)u,>
-
cd?i”@N AaeAfls(f”y,$
-
+
f
& (UdUDfBy,k'dUd),
f++,“).
(17A) (17B)
These are the simplified Moller-equations in the case of dipole atoms (or molecules), describing their relativistic motion with a very high degree of accuracy, however, without the physically irrelevant trembling motion. About the above derivation we want to make two remarks. First of all we note that the contribution of the intra-atomic field to the tensor Q@ has been neglected in our calculations. De Groot and SuttorpJ) have discussed this contribution in the so-called “Darwin-approximation”, but it appears to be much smaller than that of T&, calculated above. The (strong) inequality (16) will therefore certainly also hold for the exact angular momentum tensor O@. As a second point there is the fact that we have taken into account only the effect of the Lorentz-force term of eq. (8) in the iteration procedure and neglected the effect of the terms, containing ,u@. Especially when the total atomic charge e = 0, these are the only terms left at the right-hand side of (8). If we now consider e.g. the term -(&f”o) ,L@ in (8), this will give rise to:
(184 in the iterated
equation
of motion
(cf. (lOA)), and to:
(189 in the angular momentum
equation
(cf. (10B)).
The characteristic
quantity
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
375
appearing in (18) is
(19) a third and fourth order quantity in the internal atomic variables (just as a magnetic quadrupole or an electric octupole). Consistent with the dipole approximation, made in the present paper, the terms (18) have therefore to be neglected in the iterated equations of motion. Along the same lines one proves, that also the other ,u@b-terms in (8) give rise to negligible effects in these equations. Since the left-hand side of eq. (17A) does not contain the intrinsic angular momentum tensor SZ@, as this was the case with the original Moller-equations (2A) and (7A), or the iterated equation (1OA), the equations (17A) and (17B) appear at first sight to be uncoupled. However, as SZ@ is related to (cf. (14)) :
which is a part of the tensor ,ua@, see eq. (6), appearing at the right-hand side of the equation of motion (17A), we come to the conclusion, that in reality there is no question of uncoupling of eqs. (17A) and (17B). On the contrary, one should also write down a differential equation for the first term at the right-hand side of eq. (6), i.e. for: P$j = Cr %)($$Ufl
-
+W,
(21)
in order to solve (17A). The reason, that we do not derive such equation here is, that we shall always consider the right-hand side of (17A) (and (17B)) as given, as this was also the case with the original Moller-equations (2A) and (2B). Since the trembling motion originated from the special choice (1) of the reference points, and had to be eliminated afterwards as an unphysical motion, the question arises whether other conditions than (1) could be used, which lead directly to “physical” second order equations of motion without “Zitter-terms”? This is indeed possible: Dixona) uses e.g. the following condition
for the reference point :
Q@P(j = 0,
(24
where PO is the energy-momentum four-vector of the system. In contrast with (1)) eq. (22) defines one single reference point : the centre of energy of the system in the Lorentz frame, for which the space-part of PO (i.e. the momentum) is zero. It can be shown that the condition (22) leads immediately to the second-order equations of motion (10) or (17), but the calculations are not very much shorter since we cannot use any longer Moller’s results, which were based on eq. (1). Although eq. (22) would have been a slightly
J. VLIEGEK
376
AND
S. EMID
better starting point for the derivation of the equations of motion than (l), we have taken the latter condition, since we could then prove in paper I, that one obtains formally the same results as de Groot and Suttorp394) for the atomic energy-momentum tensor of a polarized system, whereas we know that both conditions lead to the same final forms (17A) and (17B) for the equation of motion and the intrinsic angular momentum equation of the dipole atoms (or molecules). $3. The atomic energy-momentum tensor. In a completely analogous way as in I, 3 4, one can derive the energy-momentum four-tensor for a system of N dipole atoms or molecules, starting now from the equations (17). It is evident, that the field part t$ of this tensor will be the same as derived in I, since the Moller-equations (7) and the simplified type (17) do not differ for what concerns their field parts. We therefore obtain again the expression (1.59) for t$. However, since the “material parts” of eqs. (7A) and (17A) (i.e. the left-hand sides) are different, this will also be the case with the material parts of the energy-momentum tensors, derived with the help of these equations. It is easily seen, that in the material part, found with the simplified Moller-equations (17A), the “acceleration term” will be lacking, so that we simply have, instead of (1.60) : t;::
=
Ck
P;k;$&kp
(23)
where p&i is the rest mass density of atom K (k = 1, 2, . . . . N), defined by (1.53), and u;yk)the four-velocity of this atom. In a completely similar way as in paper I, one can now again symmetrize the total energy-momentum tensor, by adding the divergence-free tensor: 4
Ck
&{f’&~;kq”
to its material tT:j =
xk
+
$k,@)’
-
u[k,o;kg” >
(24
part (see (1.65)), so that this now becomes [P;‘UU k)
(k&k,
+
!&~“;yk)u~~~
+
ufk,a;kS’
-
a[k,‘;kgp
>I*
(25)
with a;$ the atomic angular momentum density tensor, defined by (1.54). This symmetry can be proved again by applying the atomic intrinsic angular momentum balance, which follows from eq. (17B). (Note that this balance equation will have a different form from (1.62)). We know that the symmetrical total energy-momentum tensor t@ = t& + t$ has the advantage of implying both the conservation law of total energy-momentum (cf. (1.66)) and of total angular momentum of the system of polarized atoms (cf. refs. 3 and 4). Th e material energy-momentum tensor (25) is slightly different from the expression found in paper I (eq. (1.67)), which had been derived earlier by de Groot and Suttorp39 4). As we have seen, this is due to the elimination of the unphysical trembling motion of the (pseudo)centres of gravity of the atoms.
ON THE RELATIVISTIC 3 4. Concluding remark.
DYNAMICS
OF POLARIZED
The theory has been developed
SYSTEMS.
II
377
in the previous
and present papers only for the case of electric and magnetic dipole atoms or molecules. The electric quadrupole moments of the atoms, which are of the same (second) order in the internal atomic variables as the magnetic dipole moments, have always been neglected. As Rosenfeldra) has pointed out, this is in general certainly not allowed. We shall therefore investigate in the next paper (III), what are the changes in the theory, if atomic molecular electric quadrupole moments are also taken into account.
or
14B) in the following
Proof of the inequality form:
(A.11 where m,i and M(c) are the rest masses of an electron and the ith nucleus respectively, whereas the symbol x’ denotes summation over the electrons and z” over the nuclei. Multiplying this expression with e/M* and using the fact that* le/lcf*) %ll G (%l/qlr) Ieel I and leM(ti,/M*I 5 e(a), with eel the (negative) elementary charge, mpr the rest mass of the proton and e(z) the (positive) charge of the ith nucleus, we obtain the following inequality for each 01and ,9:
+
Cl le(&$$,
- ~;i,&) A@&I.
(A4
We now compare the right-hand side of this inequality with the absolute value of the corresponding element of the tensor ,&AasAQ which can be written as ,&Aa,A~~ = $e,l z;1 (x&2& + & IX:5e(&&&
-
3i;&,)
~&v&) A%AQ
Aa,Aflc + (A.4
where we have used the fact that p$,AaBAflc = 0, which follows from eqs. (21) and (1.40). In the momentary atomic rest frame the tensor ,uECA~,A@C possesses only nonzero space-space components (just as the tensor Q@), and one can prove, using formula (56) of ref. 8, that these are the components of the magnetic dipole moment of the atom, as long as one neglects moments of higher than the second order in the internal variables. Since this atomic * The validity of these inequalities is easily checked by means of elementary calculations.
378
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
(molecular) magnetic dipole moment is the sum of an electronic and a nuclear moment, and since the nuclear contribution is according to Van Vleckii) completely negligible with respect to the electronic contribution, we can neglect the second sum of (A.3) against the first, so that we get : 5534 1e,l Cl (x;&,
IpQ@E&/
-
i$v&)
da&~\.
We then see, that the first sum at the right-hand
(A.4
side of (A.2) is much smaller
than the second member of (A.4), because the factor (mei/mpr) is of the order 10-s. But the second sum of (A.2), which is again a nuclear contribution, is also negligibly small, so that we may finally conclude, that:
This proves the validity of the inequality ( 16). Remark. The reason, that we have rewritten the last terms of eqs. (12A) and (12B) in the forms as given by the right-hand sides of eqs. (15A) and (15B) is, that we cannot apply the above arguments of Van Vleck to the tensor ,u@. Though ,L$“) gives no contribution to those terms, the remaining part ,L$., has nonzero space-time components in the momentary atomic rest-frame which depends on the atomic (molecular) electric quadrupole moment. For this moment, however, it is generally no longer true that the nuclear contributions are negligible with respect to the electronic contributions. This is of course irrelevant in the present paper, where we have always neglected electric quadrupole moments in the dipole approximation but the above given proof of the possibility of replacing p@ by ,u@ in the interated Moller-equations is also valid if these moments are no longer neglected, as will be done in the next paper (III).
REFERENCES
1) 4 3) 4) 5)
Vlieger, Moller,
J., Physica C., Ann.
de Groot,
37 (1967)
Inst.
165.
H. Poincare
11 (1949-50)
284.
de Groot, S. R. and Suttorp, L. G., Physica 37 (1967) Plahte, E., Nuovo Cimento Suppl. 4 (1966) 291.
297.
Mathisson,
8)
de Groot,
M., Acta
Weyssenhoff, Weyssenhoff,
Polon.
J. and Raabe, J., Acta phys.
Dixon, W. G., Nuovo Rosenfeld, L., Theory
11)
Van Vleck,
163, 218.
L. G., Physica
31 (1965)
Cimento 38 (1965) 1616. of electrons, North-Holland
J. H., The theory
(Oxford,
6 (1937)
A., Acta phys. Polon. 9 (1947) Polon. 9 (1947) 26, 34.
S. R. and Suttorp,
9) 10)
Press
phys.
L. G., Physica
251.
37 (1967)
6) 7)
S. R. and Suttorp,
1932) 259, 277.
of electric
1713.
Publ.
and magnetic
7;
Co. (Amsterdam,
susceptibilities,
1951).
Clarendon
41 (1969)
368-378
@ North-Holland
ON THE
RELATIVISTIC
POLARIZED SIMPLIFICATION
Publishing
DYNAMICS
SYSTEMS.
OF M0LLER’S
Instituut-Lorentz,
OF
II
EQUATIONS
AND THE ENERGY-MOMENTUM J. VLIEGER
Co., Amsterdam
OF MOTION
TENSOR
and S. EMID Leiden,
Nederland
Received 4 July 1968
synopsis The classical equations of motion for electric and magnetic dipole atoms (or molecules) in an external electromagnetic field of force, derived in a previous paper on the basis of Mraller’s theory of the relativistic dynamics of systems with an internal angular momentum, are simplified by showing that certain terms, which contain an unphysical trembling motion (‘
3 1. Irttrodzlction. In a preceding paperr) on the relativistic dynamics of polarized systems, in the following denoted by I, we have derived the equations of motion of electric and magnetic dipole atoms in an external electromagnetic field of force. The derivation was based on Moller’s theorys) of the dynamics of relativistic systems with an internal angular momentum in an arbitrary (non-gravitational) field of force. In this theory the equations of motion and of the change of the intrinsic angular momentum are found, using the definition * (I.1 1) for the centre of gravity, or rather, using M@ller’s words, the pseudo-centres of gravity of the system. These equations were then applied to the special model of electric and magnetic dipole atoms (or molecules) and used in order to derive the relativistic energy-momentum tensor for a system, consisting of N of these atoms. It was found that this four-tensor is exactly of the same form as the one derived by de Groot and * Formulae of paper I will be denoted here by (I.l), (1.2), etc. 368
ON THE
RELALIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
369
Suttorpsp a), although these authors have used a slightly different definition for the reference point within the atom from the one used here. The motion of the pseudo-centres of gravity, following from Merller’s equationss), is still rather unphysical as it contains, superposed on what may be interpreted as the real translational motion of the system, a very small (order of the Compton-wavelength) trembling motion (frequency higher than which has no physical interpretation 1Osr), the so-called “Zitterbewegung”, and is purely a consequence of the definition (I. 11) of these reference points. It will be shown in 3 2 that from the above equations of motion, which are third order differential equations, one can derive, by using an iterative procedure proposed by Plahtes), equations of the usual second order type, which possess solutions which also satisfy the original equations, but which do not give rise to a trembling motion and can therefore be considered as the correct equations, describing the motion of the system as a whole in an external field of force. It appears that for atoms or molecules this iteration method is rapidly convergent, and gives already in lowest order negligible results. The equations of motion (and of course also the angular momentum equation) are therefore simplified, and this is also the case with the atomic energy-momentum tensor derived with the help of these equations. In $ 3 we find that the field part of this tensor is exactly the same as obtained in I, but that its material part is slightly different. It is shown that the total energy-momentum tensor can be obtained in a symmetrical form, using the same procedure as in I, $4. The theory developed in the present paper is based on a classical model of the atoms or molecules. Radiative effects are neglected, as this has also been done in I. We hope to come back to this latter point in a future paper. 5 2. Simplification
of Mtiller’s equations of motiolz in the case of electric and of the equations of motion (1.41) of electric and magnetic dipole atoms in a given external electromagnetic field of force, starting from Moller’s equations (I. 15) for the motion of arbitrary relativistic systems with an internal angular momentum. The latter equations were derived by Mprllers), using the condition (cf. (I. 11))
magnetic dipole atoms. In I we have given a derivation
for the reference point within the system. It follows from the definition (1.8) of Q@ (internal angular momentum four-tensor with respect to the reference point) and the fact that Ub is the derivative of the time-space coordinate four-vector X0 of this point with respect to the eigentime T, that this condition has the form of a first-order differential equation in X, and does therefore not define one single reference point, but an infinity of these points, which are called pseudo-centres of gravity by Moller. This is the reason, that the differential equations (1.15) for the motion of these
J. VLIEGER
370
AND S. EMID
pseudo-centres of gravity are not of the second order in X, as usual, but of the third order. Consequently the energy-momentum four-tensor derived in I with the help of these equations, contains a material part, which does possess not only velocity-dependent terms, but also terms depending on the accelerations of the atoms (cf. (1.67)). Th’is result has also been obtained by de Groot and Suttorp 394), which authors have used, however, a different definition for the reference points within the atoms (molecules) as the one used by us. The problem of solving the equations (I. 1.5) and in particular the physical significance of the solutions obtained, have been a matter of long discussions in the past, (see refs. 6 and 7). One obtains a trembling motion of a very high frequency (> lOsi), the so-called “Zitterbewegung”, superposed on a motion of a much smoother character, (in the free case a straight line). From Moller’s point of views), however, this trembling motion is only a consequence of the definition (1) of the reference points, and has therefore no physical meaning *. Moreover he finds a very small amplitude of the trembling motion, namely of the order of the Compton wave length, which is for an atom about lo-13 cm, much smaller than its dimensions. It is therefore evident, that one must try to find new differential equations, which describe the above mentioned smooth motion, but not the unphysical trembling motion. A straightforward way to achieve this is the following iteration procedure, which is formally analogous to the method of Plahte5) for the elimination of the “Zitterbewegung” of the classical spinning electron. Consider the equation of motion (I. 15A) and the internal angular momentum equation (I. 15B) :
&3 + f
,‘-J~Q&J0, -
&
(,lDQw 0, = A “&@“C,
PB)
where M* is the rest mass of the system, defined by (I. 12), Fa the four-vector of the external force acting upon it (eq. (1.5)), da@ the four-tensor of the moment of this force with respect to the reference point (eq. I.lO)), and
with B&pthe elements of the unit four-tensor. (The dots mean differentiation with respect to the eigentime 7.) Since it is the second term at the left-hand * One should also note that a motion of the extreme high frequency of the trembling motion, has never been observed in physics.
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
side of eq. (2A), which causes the microscopically
SYSTEMS.
II
371
small trembling motion, one
can obviously write eq. (2A) in lowest order by neglecting this term, and then solve 0, as a function of X, and U, from the approximate second order differential equation, which is left. If this result is now substituted for 00 and I?,, into eqs. (2A) and (2B), one obtains in first approximation the influence of the external field on the motion through the “Zitter-term” c-s(d/dT) (L?“G&), and the terms c-stY~~%~~,, and c_sUflsZ~‘~~, causing the so-called Thomas-precession of the internal angular momentum of the system. In this way one obtains for (2A) a second- and for (2B) a first-order differential equation, and one can easily see, that this set of equations describes, within the approximation of their derivation, the motion of the system (with intrinsic angular momentum) in the external field of force, however, z~ithout the unphysical trembling motion. With the above mentioned iteration method of Plahtes), one could obtain also higher approximations of this set of differential equations, but we shall show below that for atoms or molecules in an electromagnetic field the first approximation is already negligibly small. The equations of Moller for the motion of electric and magnetic dipole atoms, derived in I, can therefore be simplified. In order to prove the above statements, we remind that for these dipole atoms (cf. eqs. (1.36) and (1.37)) :
F” = f fWJ,g - ;
(i3,f"fl) /_J~Y-
f
&
(f"$q'U,,),
and: &S =
-!C (jWJpydAd8 -
f8V,q,dAda),
(5)
where e is the total charge of the atom (or molecule), f@ the tensor of the external electromagnetic field ((fol,f02,f03)= e and (f23,f31,f12)= b, with e the electric and b the magnetic field strength) and ,u@ a tensor given by* :
where e(t) are the charges of the constituent particles (electrons and nuclei) of the atom (or molecule), x& the relative time-space coordinates of these particles with respect to the centre of gravity of the atom and kg, the derivatives of these quantities with respect to the eigentime T of this reference point. Substituting eqs. (4) and (5) into (2), we obtain, after
* This formula is directly found by comparing eq. (17) of ref. 8 with eq. (20) of our paper I. We have only changed the notations in (6).
J. VLIEGER
372
multiplication c
S. EMID
with c :
d”,(MV”)
--
AND
+ f
$
(Qm7~) = efdlwjj -- (aypq pjp -
&{(fay~ye - fSypya) ‘%}+f -$-(U"UBfSY,hdUd),
,i
(cf. (I.41)). Omitting the second term on the left-hand find in lowest order approximation: c&
(M*Ua) N efaWo
--
& {(fay~Y5 - fSYpY") u,}+ f $
,i
-
(7A)
side of eq. (7A), we
(i3,f@) p$J -
(~*u~f5y~ydud).
(8)
Let us now first disregard the terms with ,ua@ at the right-hand side of this equation. (We shall see below that they give rise in the iterated equations to terms of third and fourth order in the internal variables X& and %g,, which are neglected in the present theory.) One can then easily solve i?a from the remaining equation, obtaining :
(9) Substituting
this result into (7A) and (7B), we get
c$(M’U.)+-&g +
(
-
@Yf”B)
+ 7
1
Pay
z
d
-
;
-g
Da@fflYUy N efWJ~ -
WYPY@
-
fflYPY9
%I
+
(‘04
(UaUSfsyhdUd, e
CtiaS+ ___
M*c2
N A~,A$(fyy~
UC&!~yfydUd
-
-
&
Ufl@yfydUd
=
fc”py”),
where we have used the relation
(10B) (1). If we introduce
the tensor:
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
373
eqs. (10A) and (IOB) can be written as
c$
(M*Ua) 1: efaWfl - (&f”fl).j@ - $
+ i c2
d dr
cfias = +
L
$
(fbypyWfl) +
(124
(ui3fBfsyjiydd,Ja),
(fay,n!dys
-
foY/dya)
Udfdy(kyflUa
-
+
&
(fayUB
-
fsyua)
,uydU,,
+
fiyaUB),
(12B)
c2
using also the relation (1). Now the four-tensor LW is given by (1.8), and taking for the energymomentum tensor T@(x), appearing in this formula, in first approximation only its material part :
~(4)(&t,(w) -
x}dw,
(13)
--m
where m(g) are the rest masses of the constituent particles of the atom (molecule), X& their time-space coordinates and ~(1) their eigentimes, one derives in a straightforward way, making also use of eq. (1) and the relation x&Ua = 0 (cf. (1.23)), that
J-J)“@ = z:z W)(&X(i)‘Y - &x&,)
LP,Lvy,
(14B)
also the results obtained by de Groot and Suttorpsp4)). One easily verifies, with the help of the antisymmetry of the field tensor f@ and the property d,Yd,S = d,fl (cf. (1.40)), that the last terms at the right-hand sides of eqs. (12A) and (12B) respectively can be rewritten in the following forms : (cf.
(154
and : f
Udfdy(pySUa
-
- y&2$
Fy”uo)
Ua)
=
f
udfdy
,u.$A~~A~~-
,u&A~~da~ - +Slya (
)
Ufl . 1
(15B)
J. VLIEGER
374
AND
S. EMID
Futhermore it can be shown (see appendix), that for atoms and molec&es the following inequality holds for each o( and /3:
(16) We may therefore neglect the terms containing sZ@ at the right-hand sides of eqs. (15A) and (15B) with respect to the terms with ~“0 which means that we may replace the tensor p ~0 in the expressions at the left-hand sides of these equations in very good approximation by ,u@. Eqs. (12A) and( 12B) then finally become :
d
1 c2
>,-
{(fayh'b
fay,h'a)u,>
-
cd?i”@N AaeAfls(f”y,$
-
+
f
& (UdUDfBy,k'dUd),
f++,“).
(17A) (17B)
These are the simplified Moller-equations in the case of dipole atoms (or molecules), describing their relativistic motion with a very high degree of accuracy, however, without the physically irrelevant trembling motion. About the above derivation we want to make two remarks. First of all we note that the contribution of the intra-atomic field to the tensor Q@ has been neglected in our calculations. De Groot and SuttorpJ) have discussed this contribution in the so-called “Darwin-approximation”, but it appears to be much smaller than that of T&, calculated above. The (strong) inequality (16) will therefore certainly also hold for the exact angular momentum tensor O@. As a second point there is the fact that we have taken into account only the effect of the Lorentz-force term of eq. (8) in the iteration procedure and neglected the effect of the terms, containing ,u@. Especially when the total atomic charge e = 0, these are the only terms left at the right-hand side of (8). If we now consider e.g. the term -(&f”o) ,L@ in (8), this will give rise to:
(184 in the iterated
equation
of motion
(cf. (lOA)), and to:
(189 in the angular momentum
equation
(cf. (10B)).
The characteristic
quantity
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
375
appearing in (18) is
(19) a third and fourth order quantity in the internal atomic variables (just as a magnetic quadrupole or an electric octupole). Consistent with the dipole approximation, made in the present paper, the terms (18) have therefore to be neglected in the iterated equations of motion. Along the same lines one proves, that also the other ,u@b-terms in (8) give rise to negligible effects in these equations. Since the left-hand side of eq. (17A) does not contain the intrinsic angular momentum tensor SZ@, as this was the case with the original Moller-equations (2A) and (7A), or the iterated equation (1OA), the equations (17A) and (17B) appear at first sight to be uncoupled. However, as SZ@ is related to (cf. (14)) :
which is a part of the tensor ,ua@, see eq. (6), appearing at the right-hand side of the equation of motion (17A), we come to the conclusion, that in reality there is no question of uncoupling of eqs. (17A) and (17B). On the contrary, one should also write down a differential equation for the first term at the right-hand side of eq. (6), i.e. for: P$j = Cr %)($$Ufl
-
+W,
(21)
in order to solve (17A). The reason, that we do not derive such equation here is, that we shall always consider the right-hand side of (17A) (and (17B)) as given, as this was also the case with the original Moller-equations (2A) and (2B). Since the trembling motion originated from the special choice (1) of the reference points, and had to be eliminated afterwards as an unphysical motion, the question arises whether other conditions than (1) could be used, which lead directly to “physical” second order equations of motion without “Zitter-terms”? This is indeed possible: Dixona) uses e.g. the following condition
for the reference point :
Q@P(j = 0,
(24
where PO is the energy-momentum four-vector of the system. In contrast with (1)) eq. (22) defines one single reference point : the centre of energy of the system in the Lorentz frame, for which the space-part of PO (i.e. the momentum) is zero. It can be shown that the condition (22) leads immediately to the second-order equations of motion (10) or (17), but the calculations are not very much shorter since we cannot use any longer Moller’s results, which were based on eq. (1). Although eq. (22) would have been a slightly
J. VLIEGEK
376
AND
S. EMID
better starting point for the derivation of the equations of motion than (l), we have taken the latter condition, since we could then prove in paper I, that one obtains formally the same results as de Groot and Suttorp394) for the atomic energy-momentum tensor of a polarized system, whereas we know that both conditions lead to the same final forms (17A) and (17B) for the equation of motion and the intrinsic angular momentum equation of the dipole atoms (or molecules). $3. The atomic energy-momentum tensor. In a completely analogous way as in I, 3 4, one can derive the energy-momentum four-tensor for a system of N dipole atoms or molecules, starting now from the equations (17). It is evident, that the field part t$ of this tensor will be the same as derived in I, since the Moller-equations (7) and the simplified type (17) do not differ for what concerns their field parts. We therefore obtain again the expression (1.59) for t$. However, since the “material parts” of eqs. (7A) and (17A) (i.e. the left-hand sides) are different, this will also be the case with the material parts of the energy-momentum tensors, derived with the help of these equations. It is easily seen, that in the material part, found with the simplified Moller-equations (17A), the “acceleration term” will be lacking, so that we simply have, instead of (1.60) : t;::
=
Ck
P;k;$&kp
(23)
where p&i is the rest mass density of atom K (k = 1, 2, . . . . N), defined by (1.53), and u;yk)the four-velocity of this atom. In a completely similar way as in paper I, one can now again symmetrize the total energy-momentum tensor, by adding the divergence-free tensor: 4
Ck
&{f’&~;kq”
to its material tT:j =
xk
+
$k,@)’
-
u[k,o;kg” >
(24
part (see (1.65)), so that this now becomes [P;‘UU k)
(k&k,
+
!&~“;yk)u~~~
+
ufk,a;kS’
-
a[k,‘;kgp
>I*
(25)
with a;$ the atomic angular momentum density tensor, defined by (1.54). This symmetry can be proved again by applying the atomic intrinsic angular momentum balance, which follows from eq. (17B). (Note that this balance equation will have a different form from (1.62)). We know that the symmetrical total energy-momentum tensor t@ = t& + t$ has the advantage of implying both the conservation law of total energy-momentum (cf. (1.66)) and of total angular momentum of the system of polarized atoms (cf. refs. 3 and 4). Th e material energy-momentum tensor (25) is slightly different from the expression found in paper I (eq. (1.67)), which had been derived earlier by de Groot and Suttorp39 4). As we have seen, this is due to the elimination of the unphysical trembling motion of the (pseudo)centres of gravity of the atoms.
ON THE RELATIVISTIC 3 4. Concluding remark.
DYNAMICS
OF POLARIZED
The theory has been developed
SYSTEMS.
II
377
in the previous
and present papers only for the case of electric and magnetic dipole atoms or molecules. The electric quadrupole moments of the atoms, which are of the same (second) order in the internal atomic variables as the magnetic dipole moments, have always been neglected. As Rosenfeldra) has pointed out, this is in general certainly not allowed. We shall therefore investigate in the next paper (III), what are the changes in the theory, if atomic molecular electric quadrupole moments are also taken into account.
or
14B) in the following
Proof of the inequality form:
(A.11 where m,i and M(c) are the rest masses of an electron and the ith nucleus respectively, whereas the symbol x’ denotes summation over the electrons and z” over the nuclei. Multiplying this expression with e/M* and using the fact that* le/lcf*) %ll G (%l/qlr) Ieel I and leM(ti,/M*I 5 e(a), with eel the (negative) elementary charge, mpr the rest mass of the proton and e(z) the (positive) charge of the ith nucleus, we obtain the following inequality for each 01and ,9:
+
Cl le(&$$,
- ~;i,&) A@&I.
(A4
We now compare the right-hand side of this inequality with the absolute value of the corresponding element of the tensor ,&AasAQ which can be written as ,&Aa,A~~ = $e,l z;1 (x&2& + & IX:5e(&&&
-
3i;&,)
~&v&) A%AQ
Aa,Aflc + (A.4
where we have used the fact that p$,AaBAflc = 0, which follows from eqs. (21) and (1.40). In the momentary atomic rest frame the tensor ,uECA~,A@C possesses only nonzero space-space components (just as the tensor Q@), and one can prove, using formula (56) of ref. 8, that these are the components of the magnetic dipole moment of the atom, as long as one neglects moments of higher than the second order in the internal variables. Since this atomic * The validity of these inequalities is easily checked by means of elementary calculations.
378
ON THE
RELATIVISTIC
DYNAMICS
OF POLARIZED
SYSTEMS.
II
(molecular) magnetic dipole moment is the sum of an electronic and a nuclear moment, and since the nuclear contribution is according to Van Vleckii) completely negligible with respect to the electronic contribution, we can neglect the second sum of (A.3) against the first, so that we get : 5534 1e,l Cl (x;&,
IpQ@E&/
-
i$v&)
da&~\.
We then see, that the first sum at the right-hand
(A.4
side of (A.2) is much smaller
than the second member of (A.4), because the factor (mei/mpr) is of the order 10-s. But the second sum of (A.2), which is again a nuclear contribution, is also negligibly small, so that we may finally conclude, that:
This proves the validity of the inequality ( 16). Remark. The reason, that we have rewritten the last terms of eqs. (12A) and (12B) in the forms as given by the right-hand sides of eqs. (15A) and (15B) is, that we cannot apply the above arguments of Van Vleck to the tensor ,u@. Though ,L$“) gives no contribution to those terms, the remaining part ,L$., has nonzero space-time components in the momentary atomic rest-frame which depends on the atomic (molecular) electric quadrupole moment. For this moment, however, it is generally no longer true that the nuclear contributions are negligible with respect to the electronic contributions. This is of course irrelevant in the present paper, where we have always neglected electric quadrupole moments in the dipole approximation but the above given proof of the possibility of replacing p@ by ,u@ in the interated Moller-equations is also valid if these moments are no longer neglected, as will be done in the next paper (III).
REFERENCES
1) 4 3) 4) 5)
Vlieger, Moller,
J., Physica C., Ann.
de Groot,
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de Groot, S. R. and Suttorp, L. G., Physica 37 (1967) Plahte, E., Nuovo Cimento Suppl. 4 (1966) 291.
297.
Mathisson,
8)
de Groot,
M., Acta
Weyssenhoff, Weyssenhoff,
Polon.
J. and Raabe, J., Acta phys.
Dixon, W. G., Nuovo Rosenfeld, L., Theory
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Van Vleck,
163, 218.
L. G., Physica
31 (1965)
Cimento 38 (1965) 1616. of electrons, North-Holland
J. H., The theory
(Oxford,
6 (1937)
A., Acta phys. Polon. 9 (1947) Polon. 9 (1947) 26, 34.
S. R. and Suttorp,
9) 10)
Press
phys.
L. G., Physica
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S. R. and Suttorp,
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1713.
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