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Classical elementary measurement electrodynamics
ANNALS
OF PHYSICS:
51, 561-575 (1969)
Classical Elementary
Measurement
DARRYL
Electrodynamics
J. LEITER
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02167
We consider a classical electrodynamic theory, in which the operational paradigm of Elementary Measurement implies that particles and fields are interdependent degrees of freedom in an action principle associated with the “elementary measurement”, and are not elementary in themselves. In this model Maxwell’s equations, with an empirically chosen Green function, are identities associated with each charged particie current in the elementary interaction. Making the action stationary with respect to the interacting degrees of freedom gives the equations of motion of the measurement. These are shown to be equivalent to those of the “Action-at-a-Distance” electrodynamics of Wheeler and Feynman, with the distinct advantage that no “complete absorber” assumption is ever needed. The difficulties associated with infinite self-energies and mass renormalization are absent from this theory and a consistent set of energy-momentum and angularmomentum conservation laws follows from the associated energy-momentum tensor. The mechanism of “measurement” radiation is discussed and is shown to predict similar results as that of Maxwell-Lorentz electrodynamics.
I. INTRODUCTION
Physical events always involve an “observer” and an “object” to be observed. Since the very act of trying to observe an “elementary” charged particle or electromagnetic field involves other charged particles and fields, the existence of an indivisible nonzero unit of electric charge makes this “absolute” observation impossible. Consequently, one is lead to the paradigm of Elementary Measurement, which postulates that, in a physical event, it is the mutual measurementinteraction between the “observer” and the “observed” which is elementary, In the context of a relativistic theory of Classical Elementary Measurement Electrodynamics, charged particles (associated with JhK’(x); K = l,..., N) and electromagnetic fields (associated with ALK’(x); K = l,..., N) are not elementary in themselves, but are merely interdependent degrees of freedom in an elementary measurement. This means that the scalar elementary measurement field J:K)Au(J); K # ] = l,..., N is more fundamental than either JLK’(x) or A?)(x). As a consequence, in this model, Maxwell’s equations (with the proper Green function) are interpreted to be merely a set of covariant identities which give a prescription for converting particle currents J,!K)(x) into their associated electromagnetic fields 561
562
LEITER
Ap(x); K = l,..., N. Since all electromagnetic fields will be required to be connected to particle currents through their associated Maxwell identities, then no “free” uncoupled electromagnetic fields will exist in the theory. Hence, the phenomena of radiation will occur as the by-product of the propagation of mutual electromagnetic interactions between charged particles, and will not be “elementary” in itself. It should be noted that on the basis of the paradigm which underlies this model, “self-measurement” fields JiK)AU(K); K = I,..., N are excluded, a priori, as being unphysical. In the following sections, we shall construct the action for the relativistic “elementary measurement” of N classical charged “point” particles and their associated electromagnetic fields. Making the variation of the action stationary with respect to arbitrary variations in the interacting degrees of freedom gives the equations of motion of the measurement. With the proper choice of green function for the Maxwell tautologies, the equations of motion of the measurement have the same form as that of the “Action-at-a-Distance” theory of Wheeler and Feynman (I). However, it has a distinct advantage in that no “complete absorber” assumption is ever needed or used, and the electromagnetic fields are not eliminated from the theory a priori, as is done in the Wheeler-Feynman theory. The difficulties associated with infinite self-energies and mass renormalization are absent from Elementary Measurement Electrodynamics, and a consistent set of energymomentum and angular-momentum conservation laws follows from an associated conserved energy-momentum tensor. The extension of this theory (2), to include wave mechanics, would imply that the “photon” is not an elementary particle. II. THE ELEMENTARY
MEASUREMENT
OF N CHARGED
POINT
PARTICLES
To begin the development of the theory, before the variation, we define the interacting degrees of freedom which go into the action principle of the “elementary measurement”. We assume that each particle degree of freedom, which is not to be considered as “elementary,” is described by a dynamic variable associated with the kth particle trajectory $?(t). We shall represent the totality of these by the single iv-column
/y”= &) i ; i XL) 1
p = 0, 1,2, 3.
If 87~~) is the kth particle’s proper time differential, 4-velocity N-column is
(1)
then, using equation (l), the (2)
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563
ELECTRODYNAMICS
where the N @ N proper time matrix dr is
dr = dt
(0
= dt(y-l), (3)
and (ii) The N-column
current density is
(4) where the current associated with the kth particle trajectory is
dW -d& dt ‘% - 8dt))
J&,(4 = 7
K = l,..., N
(5)
and q(K) is the invariant charge of the Kth particle. The N @ N mass matrix associated with the masses of the charged particle degree of freedom is
I
and the N-column
electromagnetic
(6)
field dynamic variable is
(7) In A we choose ALK) to be the “Lorentz” definid before the variation as
6) (ii)
Af’(x)
vector potential
degree of freedom,
3 ay()(x) - au / di4 D(x - x’) a;a”(q(K)
q D(x - x’) = 84(x- x’).
(8)
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LEITER
Then arbitrary SaLK)with &LK) # 0 will imply arbitrary SALK)for which @‘AI(~) P = 0. Hence, from equations (7) and (8) @A, = 0
(9
is valid for arbitrary variations SA, . Then the associated electromagnetic N-columns are (9 (ii>
tention
pv G av,p _ @Av, j%v z &uvuBFaB,
(10)
auF#,= 0
(11)
and
holds as an identity for all Au. The electric and magnetic field N-columns Et =
(0 (ii)
Fii =
pi
,
are related to FuP as
(i, j, K, = 1,2,3)
,tiZBZ
Then the action principle associated with the “elementary measurement” charges and their related electromagnetic fields is
w
of N-point
where the N @ N matrix Q is
Q=
(14)
Q is the electromagnetic coupling matrix which excludes u priori all direct “self” measurement fields JLK)AwfK); K = i,..., N from the action, equation (13). This is as required by the basic paradigm of the theory that only “mutual” measurement
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565
ELECTRODYNAMICS
fields #“)AJ‘(‘); K f J = 1, -N contribute to the description of an electroIr magnetic physical event. Then we set SZ = 0 with respect to arbitrary variations in the dynamical variables X, ; A, ; X,‘; A,+ to get the equations of motion of the measurement which are avii’Fu, = SZJ, ,
6) (ii)
@Au = 0, dP” = Q/c W2FVU(traj.), dt
(iii)
(15)
@F,, = 0,
(iv) Where we have eliminated (3) and
the proper time in equation (15-iii) by use of equation
is the charge matrix and velocity matrix of the theory, respectively. Since Sz-l exists as Q-1 = Q - Z(N - 2) . (N >, 2) (17) N-l ’ where, N 3 2 is the number of charged particles in the “elementary then equations (15-i) and (15-ii) become
q A,
interaction,”
= J,, .
(18)
Now equation (18) is to be interpreted as an identity. This physical requirement must be mathematically specified. To do this, we define the N @ N matrix Green function D(x - x’) through c]D(x - x’) = 84(x - x’)
(19)
which has the solution (i) where (ii)
D(x - x’) = (I) D+(x - x’) + AD-(x - x’),
D*(x _ x,) = [‘(t - t’) s(t’ - t + R/c) f o(t’ - t, s(t’ - t - R/c)] 87rR
(iii)
RE
(g-~‘1,
f (20)
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LEITER
where A is a constant N @ N matrix to be determined empirically. Once h is chosen from empirical considerations, then it follows that the associated “A” Maxwell identity (which implies that if J, = 0 then A, = 0 and vice-versa) is A,(x) = s ~x’~[(Z) D+(x - x’) + ho-(x - x’)] J,(x’). Upon insertion of equation (21) into equation (&ii),
00
F& = (a “A&) - PAi;,),
(iii)
A;
=
(21)
we-have
(22)
I dx’” D&(x - x’) J”(x’).
Equation (22) can be written in the form
Now to determine A we note that it is empirically well known that a charged particle, on being accelerated, sends out electromagnetic energy and itself loses energy. This loss is empirically interpreted as caused by a force of “radiative reaction” given in magnitude and direction by d=P(r.r.) = ~ J$(24)
f
in the nonrelativistic
(
w
1
limit, and by f7r.r.j
=
(&)
r$+
-
&
($
T)]
(25)
in the relativistic case. Equation (23) has the proper form of the radiation reaction force on each charged particle in the N-particle interaction, if in (23) we empirically choose (GA - In) = Z
or
h = Sa-yQ + I).
(26)
Then equation (23) becomes
=
fket.)
+
f7r.r.)
-
(27)
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ELECTRODYNAMICS
Equations (27) are very similar to the equations of motion of the Wheeler-Feynman “Action-at-a-Distance” electrodynamics. However within the context of Elementary Measurement Electrodynamics, no “complete absorber” condition is ever invoked, which is a distinct advantage. This occurs because the empirically chosen value of h in (26) implies that, in equation (22) the Kth charged particle sees the effective field $eff.) 11”
=
f
FE;+,
+
$t-,).
J#K
However the identity (29) implies that this effective field is
where the Wheeler-Feynman “complete Absorber” condition EN= FtJ) = 0 has not been used, since the above cancellation of the C,“==,(F$! ,) reri ii e$&ons (28), (29), (30), occurs independent of the value of C,“=, (F$t)). To gain some physical insight into how the empirical “radiation reaction” effect is generated by the choice of x = Q-l@ + I) we re-examine equation (27). It can be written as
(9 (ii) Hence the kth particle sees the sum of “retarded” Lorentz force fields J # K and the “radiation reaction” force as an interference between the sum of the “time-symmetric” Lorentz force fields f ';+,and the “total coupled radiation field” of the interaction f’;-, . The physics of fT-:-, can be seen from examining the associated electromagnetic field A?-‘_,. Since we may write D-(x - x’) as D-(x
- x’) = l dk4(--2+k,))
8(kyky) e-ik~(s”--r’w’,
(32)
then equations (32) and (22-iii) give 6) (ii)
AL-‘(x)
= s dkQ e-j’@
6(k&“)
JU(k) = 1 dx’4 eik@‘” (-24
e(k,) J,(k), J,(x’).
(33)
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LEITER
We see that Ah?(x) is an N-column wave packet obeying the homogeneous wave whose Fourier co-efficient Ju(k) is directIy connected to the current N-column J,(x). Hence the term “coupled” radiation field. However, in r;;l_, the “total” current (In + I) J,, appears; hence Fr-, is the electromagnetic force associated with the “total coupled radiation field” of the interaction. For the special case of a single particle, N = 1, then in equation (13) 9 = 0, which implies, upon setting 61= 0 the equations of motion (34) Hence, in Elementary Measurement Electrodynamics, a single charged particle by itself cannot interact mutually with itself and does not represent a physical measurement. The case of a particle in an “external” field corresponds to the N 2 2 case where a single particle xyl, is artificially isolated from the others and the approximation is made that the effect of the “aggregate” on the single particle (the “external” field) is negligibly affected by the reaction of the single particle back on the “aggregate”. This approximation may be represented by the, “elementary measurement”, action principle
whereJ&t. ) is the “fixed” and prescribed external current and
then we set 61 = 0 with respect to xc, and Au, Au+ where PAA, = 0 as before. This gives the equations of motion
(ii)
av&, = 0
(iii)
dxy P *4* df%, -q(1) dt -z-= ( c 1 F(ext.)v I
where if P1(Q A“(x)
(37)
+ I) = A, then = 1 dx’*[ID+(x
- x’) + D-l(Q
+ r) D-(x - x’)] J”(x’)
(38)
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569
ELECTRODYNAMICS
which, with equation (36) and the fact that (39) gives for the AyeXt., component A&t.(x)
of A, in equation (38)
= j dxt4 D+(x - x’) J&.(x’)
+ j- dx14 D-(x
- x’)(J;,(x’)
+ J&.(x’))
(40) which, when inserted into equation (37-iii), gives for the equation of motion of
which again shows how the “time symmetric” Lorentz force of the “external” current interferes with the “total coupled radiation field” of the system to produce the “retarded” Lorentz force of the external current and the “radiation reaction” force on the particle. It should be noted that the structure of this theory has the advantage of giving a physical interpretation of the phenomena of radiation reaction, within the context of a variational principle, without recourse to any cosmological assumptions about “complete absorption.” At the same time, the fact that the electromagnetic fields are not eliminated apriori allows a consistent set of electromagnetic conservation laws to be deduced. It should be understood that a consistent interpretation of equations (27) or (41) as physically consistent equations of motion without “run-away” solutions requires the imposition of the asymptotic boundary conditions that dxpldt G 0 which leads to a set of well-known integrodifferential equations discussed elsewhere (4). III.
ELEMENTARY
ELECTROMAGNETIC CONSERVATION
MEASUREMENT LAWS
Equations (15-i) and (15~iv) imply the following differential conservation laws a L Tuv =
where the electromagnetic
J !4+QFuv
measurement energy-momentum
(42)
tensor is (43)
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LEITER
Equations (42) and (15-iii) imply the consistent integral conservation laws for the total Cmomentum of the “elementary measurement” as g-[--f1
and the corresponding f
J’S’“’ + j/x3
To,,] = -f,dS”Tul
total 4-angular momentum
[ g1 (PY-2)
-
P!=‘x?‘)
Z-Y-- dS1(xuTvl Is
+ //x3h$-,v
- x&J]
x,Tuz)
(45)
where in equations (44) and (45), V is the 3-space volume containing the entire N-particle interaction, and S is a closed surface surrounding V. It is to be noted that no “infinite” self-interactions or self-energies appear in equations (44) and (45) or in the equations of motion (15-iii). This occurs because of the presence of the coupling matrix Q, which was inserted, before the variation, on the basis of the paradigm that only “mutual” measurement fields go into the action principle of the “elementary measurement.” (The concept of “self-interaction” is unphysical and was excluded a priori from Elementary Measurement Electrodynamics.) This also implies that the particle masses occuring in the “elementary measurement” are not “renormalized” in the equations of motion. Hence they are the “physical” phenomenological mass parameters which are determined empirically. IV.
ELEMENTARY
MEASUREMENT
RADIATION
In Elementary Measurement Electrodynamics, measurement radiation is not an elementary process associated with a single current, but involves both the “emitter” and the receiver. To illustrate this, we consider the simple example of a “receiver” made up of two localized currents (5) JF’ and JF’, constrained in a volume I’,, , and moving with a quasi-periodic motion for a characteristic time T. The “emitter” is a single current Jf’ which is considered to be localized in a volume V, very far away from V,, , then only the “wave zone” fields of A:’ will affect the “receiver” significantly. If Jf’ is time dependent due to acceleration of particle one, then the wave zone fields which pass through the surface S,, (surrounding Vzs) at a large distance r from VI , are E(l) N &x (ii)
p)
N
AaA
& (-
_
rj k
x
a t, 1
r
a&P
c at, )
(46)
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ELECTRODYNAMICS
where fl = _r/r and t, = t - r/c is the retarded time. As the wave zone fields (46) become important in V,, the subsequent absorption measurement radiation energy is related to the induced changes in the state of the receiver, as given by the equations of motion of the particles which make up the receiver. In a quasi-linear, nonrelativistic approximation, where the wave zone fields and the mutual retarded fields dominate the radiation reaction fields, we have for the induced acceleration of the receiver charges C(K)
N
4(K) y--
(1) (B
+
(K # J = 2, 3).
El')t re. ,I- ,
(47)
Then the power response of the receiver is from equation (44), applied to V,, and S,, _ dE(2,3)
= - - $
dt
=C
[PO(~) + p,(3) + j- yggdx3W)
- E(3) + B(2) - 8(3)1]
dS . [E(2) x B(3) + E(3) x B(2)]. s 18-
Then using the Lienard-Weichert potential currents .I:’ and Jf’, equation (48) gives -
dE(2,3) dt
=
(48)
solutions associated with &function
2q(2) q(3) ; C(2) - $3) 477~ N 1 I* cs
(49)
The “induced power response” is defined as the difference between the power response of the receiver under the influence of the wave zone fields, and that of the receiver before it is influenced by the wave zone fields. Then (47) and (49) imply that the “induced power response” is -A ($ E(2,3))
N_ 2qi;)$3)
(;) [@“’
+ lp)
* (E(l) + $2)) - p
* p].
(50)
If the receiver subtends a solid angle A,$ra oriented at an angle 0 with respect to the z-axis, where A,, is the cross-sectional area of V,, , then the time average of the induced power response per unit solid angle averaged over the characteristic time T is (6) -<-Wdt
E(2, 3))> c= W43N2/4~mc32 &ha
A33
cr2(p)2>
(51)
where we have assumed q(1) = q(2) = e; m(l) = m(2) = m. If we define the “active area ratio” of the receiver, S, as the ratio of the “effective” classical electron
572
LEITER
cross-sectional areas in the receiver, to the total cross-sectional area A,, for our receiver 6 3 (16rr/3)(e2/4rmc2)2
then (52)
A23
the (51) and (52) imply that the time average of the induced power response of the receiver, per unit solid angle, per active area ratio, is p 5 (--d(wta29 (A23/r2)
3))) N
(r2(p')2>
(53)
. (6)
which by (46) is
Then, in this approximation, this definition of radiation, (54), is equivalent to that of the “Maxwell-Lorentz” distribution of radiation from Jf’. Hence, measurement radiation involves an “emitter” and a physical “receiver” of finite size and dimensions. The time average of the induced power response of the receiver, per unit solid angle, per active area ratio, represents quantitatively the physical act of observing the radiation from the emitter at large distances.
Detector
QJl) “Source”
t-t-l,A-= 4 i4 1
Y
t Incoming wave zone source field assumed collimated
As another example of the measurement radiation case of Thompson scattering (7). This is a physical quasi-plane wave-zone fields of a localized “source” localized “target” current JF’ and causes it to emit
wave zone source field
process, we will discuss the situation where the incident current JF’ impinges on a wave-zone fields of its own,
CLASSICAL
573
ELECTRODYNAMlCS
which are then observed by a “detector” (J, ‘3), JF’) according to the induced “detector” power response. If one defines the differential scattering crossection as the ratio of the time average of the induced power response, per unit solid angle, of the detector with the target present; to the time average of the induced power response, per unit cross-sectional area, of the detector, when the target is absent, then we find the usual Thompson scattering result arises. To see this in more detail, if the incoming wave zone field is &( 1); B(1) from the “source”, this causes the “target” to be accelerated as
This acceleration produces a radiation field which propagates away from the “target” to the “detector” a distance r away. If the “detector” is in the wave zone of the radiation fields E(2) N (e/h)
* (e/w@(l))) sin @ r
(9 (ii)
B(2)
= (r/r> x
(56)
ECW
generated by the acelleration of the target, then these fields produce an induced detector power response in the same manner as described in equations (47) through (50). This implies, taking the time average as before, (57)
(where the detector is assumed to subtend a solid angle A23/r2 = 52(2,3) at an angle 0 with respect to the z-axis taken along the propagation vector of the incoming “source” fields) The “background” induced detector power response is obtained by removing the target and observing the induced detector response when it is placed on the z-axis. In this case the detector is influenced by the source fields only (assuming the source fields to be colimated) and <--d(d/dt
a39 4)))
BACKGR.
=
(2e2/4?r)(e/m
(E(1)))2
(8)
(5)’
(58)
Then the differential cross-section is du dI-2
-=
<--d(W
E(3,4))> 52(3,4>
<-4W I
N ((e2/47rmc2)2 sin2 0). 595/S/3-13
E(3,
4)))BACKGRL
A34
(60)
574
LEITER V. CONCLUSIONS
We have shown how the paradigm of Elementary Measurement, when applied to classical electrodynamics of point charges, which are not elementary in themselves, leads to a consistent theory, free of the paradoxes associated with self-energy. Retardation and radiation reaction are successfully accounted for within the context of a Lagrangian theory, without the use of “complete absorber” assumptions. Moreover, no mass renormalization is required, since only the empirical masses occur in the theory. The paradigm of Elementary Measurement applies to a generalization of this theory to include wave mechanics, but leads to a nonlinear wave-mechanical theory for which the classical theory is a correspondence limit. Again, no selfenergy occurs, but the question of mass and charge renormalization is more complicated, due to the nonlocal character of the wavefunctions. Such a theory is presently being developed and will represent a completely operational theory of wave-mechanical measurement in which both the “object” and the measuring “apparatus” are always represented. However, if something corresponding to a “photon” existed in such a theory, it would not be an elementary particle, since the radiation process involves the “emitter” and the “receiver” (8).
ACKNOWLEDGMENTS
The author wishes to thank Dr. S. Schwebel and Dr. R. Becker for many stimulating discussions and criticisms. Appreciation is also expressed to Sharon Leiter for her help in editing and typing the final manuscript. RECEIVED: June 21, 1968
REFERENCES
AND FOOTNOTES
AND R. P. FEYNMAN, Rev. Mod. Phys. 17, 157 (1945); 21,425 (1949). 2. D. LEITER, Elementary Measurement Electrodynamics II. In preparation. 3. It is to be understood that the terms in the action which give the propagation of “free” particles are not to be interpreted as “self-interactions” since they are not directly involved in the physical definition of the electromagnetic measurements via J$)AP(‘) (n # j) but instead define the nature of the kinematics of the particle degrees of freedom. 4. F. ROHRLICH, “Classical Charged Particles.” Addison-Wesley, Reading, Mass., 1965. 5. Conceptually, there is no reason why a single particle can’t play the role of a “receiver”, as long as there is another particle in the system to act as the “emitter”; since the reception of the measurement radiation energy is related to the induced changes in the state of the motion of the “receiver” described by the associated equations of motion. However the advantage of using two or more particles to define the receiver, lies in the mathematical convenience 1. J. A. WHEELER
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ELECTRODYNAMICS
575
of being able to use the conservation laws and the equations of motion together in a quasilinear, nonrelativistic approximation. If a single particle receiver is used, then the conservation laws are not so easily utilized, and an exact solution of the equations of motion is required to extract the desired “receiver” information. 6. This occurs because the time average of the dot product of (EC’) . _E!t’,.> j = 2,3 vanishes because of its quasi-periodicity during the time interval T ofinterest. Hence even though the mutual retarded fields, in the receiver, cannot be neglected with respect to the incoming wave zone fields in the equations of motion of the receiver (47) they will not contribute to the “induced” power response, (51) of the receiver when averaged over T. 7. See Fig. 1. 8. See Ref. 2. Another approach to this problem, using the same paradigm, but differing in detail and interpretation occurs in the paper by Sachs and Schwebel, Nuovo Cimento Supp. 2, XXI, 197, (1961).
Addendum The technique of splitting a covariant propagator into two noncovariant parts has been extensively examined by A. Ramakrishnan: J. Math. Anal. Appl. 17, 68 (1967) and 18, 175 (1967); J. Math. Phys. Sci. 1, 57 (1967).
OF PHYSICS:
51, 561-575 (1969)
Classical Elementary
Measurement
DARRYL
Electrodynamics
J. LEITER
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02167
We consider a classical electrodynamic theory, in which the operational paradigm of Elementary Measurement implies that particles and fields are interdependent degrees of freedom in an action principle associated with the “elementary measurement”, and are not elementary in themselves. In this model Maxwell’s equations, with an empirically chosen Green function, are identities associated with each charged particie current in the elementary interaction. Making the action stationary with respect to the interacting degrees of freedom gives the equations of motion of the measurement. These are shown to be equivalent to those of the “Action-at-a-Distance” electrodynamics of Wheeler and Feynman, with the distinct advantage that no “complete absorber” assumption is ever needed. The difficulties associated with infinite self-energies and mass renormalization are absent from this theory and a consistent set of energy-momentum and angularmomentum conservation laws follows from the associated energy-momentum tensor. The mechanism of “measurement” radiation is discussed and is shown to predict similar results as that of Maxwell-Lorentz electrodynamics.
I. INTRODUCTION
Physical events always involve an “observer” and an “object” to be observed. Since the very act of trying to observe an “elementary” charged particle or electromagnetic field involves other charged particles and fields, the existence of an indivisible nonzero unit of electric charge makes this “absolute” observation impossible. Consequently, one is lead to the paradigm of Elementary Measurement, which postulates that, in a physical event, it is the mutual measurementinteraction between the “observer” and the “observed” which is elementary, In the context of a relativistic theory of Classical Elementary Measurement Electrodynamics, charged particles (associated with JhK’(x); K = l,..., N) and electromagnetic fields (associated with ALK’(x); K = l,..., N) are not elementary in themselves, but are merely interdependent degrees of freedom in an elementary measurement. This means that the scalar elementary measurement field J:K)Au(J); K # ] = l,..., N is more fundamental than either JLK’(x) or A?)(x). As a consequence, in this model, Maxwell’s equations (with the proper Green function) are interpreted to be merely a set of covariant identities which give a prescription for converting particle currents J,!K)(x) into their associated electromagnetic fields 561
562
LEITER
Ap(x); K = l,..., N. Since all electromagnetic fields will be required to be connected to particle currents through their associated Maxwell identities, then no “free” uncoupled electromagnetic fields will exist in the theory. Hence, the phenomena of radiation will occur as the by-product of the propagation of mutual electromagnetic interactions between charged particles, and will not be “elementary” in itself. It should be noted that on the basis of the paradigm which underlies this model, “self-measurement” fields JiK)AU(K); K = I,..., N are excluded, a priori, as being unphysical. In the following sections, we shall construct the action for the relativistic “elementary measurement” of N classical charged “point” particles and their associated electromagnetic fields. Making the variation of the action stationary with respect to arbitrary variations in the interacting degrees of freedom gives the equations of motion of the measurement. With the proper choice of green function for the Maxwell tautologies, the equations of motion of the measurement have the same form as that of the “Action-at-a-Distance” theory of Wheeler and Feynman (I). However, it has a distinct advantage in that no “complete absorber” assumption is ever needed or used, and the electromagnetic fields are not eliminated from the theory a priori, as is done in the Wheeler-Feynman theory. The difficulties associated with infinite self-energies and mass renormalization are absent from Elementary Measurement Electrodynamics, and a consistent set of energymomentum and angular-momentum conservation laws follows from an associated conserved energy-momentum tensor. The extension of this theory (2), to include wave mechanics, would imply that the “photon” is not an elementary particle. II. THE ELEMENTARY
MEASUREMENT
OF N CHARGED
POINT
PARTICLES
To begin the development of the theory, before the variation, we define the interacting degrees of freedom which go into the action principle of the “elementary measurement”. We assume that each particle degree of freedom, which is not to be considered as “elementary,” is described by a dynamic variable associated with the kth particle trajectory $?(t). We shall represent the totality of these by the single iv-column
/y”= &) i ; i XL) 1
p = 0, 1,2, 3.
If 87~~) is the kth particle’s proper time differential, 4-velocity N-column is
(1)
then, using equation (l), the (2)
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ELECTRODYNAMICS
where the N @ N proper time matrix dr is
dr = dt
(0
= dt(y-l), (3)
and (ii) The N-column
current density is
(4) where the current associated with the kth particle trajectory is
dW -d& dt ‘% - 8dt))
J&,(4 = 7
K = l,..., N
(5)
and q(K) is the invariant charge of the Kth particle. The N @ N mass matrix associated with the masses of the charged particle degree of freedom is
I
and the N-column
electromagnetic
(6)
field dynamic variable is
(7) In A we choose ALK) to be the “Lorentz” definid before the variation as
6) (ii)
Af’(x)
vector potential
degree of freedom,
3 ay()(x) - au / di4 D(x - x’) a;a”(q(K)
q D(x - x’) = 84(x- x’).
(8)
564
LEITER
Then arbitrary SaLK)with &LK) # 0 will imply arbitrary SALK)for which @‘AI(~) P = 0. Hence, from equations (7) and (8) @A, = 0
(9
is valid for arbitrary variations SA, . Then the associated electromagnetic N-columns are (9 (ii>
tention
pv G av,p _ @Av, j%v z &uvuBFaB,
(10)
auF#,= 0
(11)
and
holds as an identity for all Au. The electric and magnetic field N-columns Et =
(0 (ii)
Fii =
pi
,
are related to FuP as
(i, j, K, = 1,2,3)
,tiZBZ
Then the action principle associated with the “elementary measurement” charges and their related electromagnetic fields is
w
of N-point
where the N @ N matrix Q is
Q=
(14)
Q is the electromagnetic coupling matrix which excludes u priori all direct “self” measurement fields JLK)AwfK); K = i,..., N from the action, equation (13). This is as required by the basic paradigm of the theory that only “mutual” measurement
CLASSICAL
565
ELECTRODYNAMICS
fields #“)AJ‘(‘); K f J = 1, -N contribute to the description of an electroIr magnetic physical event. Then we set SZ = 0 with respect to arbitrary variations in the dynamical variables X, ; A, ; X,‘; A,+ to get the equations of motion of the measurement which are avii’Fu, = SZJ, ,
6) (ii)
@Au = 0, dP” = Q/c W2FVU(traj.), dt
(iii)
(15)
@F,, = 0,
(iv) Where we have eliminated (3) and
the proper time in equation (15-iii) by use of equation
is the charge matrix and velocity matrix of the theory, respectively. Since Sz-l exists as Q-1 = Q - Z(N - 2) . (N >, 2) (17) N-l ’ where, N 3 2 is the number of charged particles in the “elementary then equations (15-i) and (15-ii) become
q A,
interaction,”
= J,, .
(18)
Now equation (18) is to be interpreted as an identity. This physical requirement must be mathematically specified. To do this, we define the N @ N matrix Green function D(x - x’) through c]D(x - x’) = 84(x - x’)
(19)
which has the solution (i) where (ii)
D(x - x’) = (I) D+(x - x’) + AD-(x - x’),
D*(x _ x,) = [‘(t - t’) s(t’ - t + R/c) f o(t’ - t, s(t’ - t - R/c)] 87rR
(iii)
RE
(g-~‘1,
f (20)
566
LEITER
where A is a constant N @ N matrix to be determined empirically. Once h is chosen from empirical considerations, then it follows that the associated “A” Maxwell identity (which implies that if J, = 0 then A, = 0 and vice-versa) is A,(x) = s ~x’~[(Z) D+(x - x’) + ho-(x - x’)] J,(x’). Upon insertion of equation (21) into equation (&ii),
00
F& = (a “A&) - PAi;,),
(iii)
A;
=
(21)
we-have
(22)
I dx’” D&(x - x’) J”(x’).
Equation (22) can be written in the form
Now to determine A we note that it is empirically well known that a charged particle, on being accelerated, sends out electromagnetic energy and itself loses energy. This loss is empirically interpreted as caused by a force of “radiative reaction” given in magnitude and direction by d=P(r.r.) = ~ J$(24)
f
in the nonrelativistic
(
w
1
limit, and by f7r.r.j
=
(&)
r$+
-
&
($
T)]
(25)
in the relativistic case. Equation (23) has the proper form of the radiation reaction force on each charged particle in the N-particle interaction, if in (23) we empirically choose (GA - In) = Z
or
h = Sa-yQ + I).
(26)
Then equation (23) becomes
=
fket.)
+
f7r.r.)
-
(27)
CLASSICAL
567
ELECTRODYNAMICS
Equations (27) are very similar to the equations of motion of the Wheeler-Feynman “Action-at-a-Distance” electrodynamics. However within the context of Elementary Measurement Electrodynamics, no “complete absorber” condition is ever invoked, which is a distinct advantage. This occurs because the empirically chosen value of h in (26) implies that, in equation (22) the Kth charged particle sees the effective field $eff.) 11”
=
f
FE;+,
+
$t-,).
J#K
However the identity (29) implies that this effective field is
where the Wheeler-Feynman “complete Absorber” condition EN= FtJ) = 0 has not been used, since the above cancellation of the C,“==,(F$! ,) reri ii e$&ons (28), (29), (30), occurs independent of the value of C,“=, (F$t)). To gain some physical insight into how the empirical “radiation reaction” effect is generated by the choice of x = Q-l@ + I) we re-examine equation (27). It can be written as
(9 (ii) Hence the kth particle sees the sum of “retarded” Lorentz force fields J # K and the “radiation reaction” force as an interference between the sum of the “time-symmetric” Lorentz force fields f ';+,and the “total coupled radiation field” of the interaction f’;-, . The physics of fT-:-, can be seen from examining the associated electromagnetic field A?-‘_,. Since we may write D-(x - x’) as D-(x
- x’) = l dk4(--2+k,))
8(kyky) e-ik~(s”--r’w’,
(32)
then equations (32) and (22-iii) give 6) (ii)
AL-‘(x)
= s dkQ e-j’@
6(k&“)
JU(k) = 1 dx’4 eik@‘” (-24
e(k,) J,(k), J,(x’).
(33)
568
LEITER
We see that Ah?(x) is an N-column wave packet obeying the homogeneous wave whose Fourier co-efficient Ju(k) is directIy connected to the current N-column J,(x). Hence the term “coupled” radiation field. However, in r;;l_, the “total” current (In + I) J,, appears; hence Fr-, is the electromagnetic force associated with the “total coupled radiation field” of the interaction. For the special case of a single particle, N = 1, then in equation (13) 9 = 0, which implies, upon setting 61= 0 the equations of motion (34) Hence, in Elementary Measurement Electrodynamics, a single charged particle by itself cannot interact mutually with itself and does not represent a physical measurement. The case of a particle in an “external” field corresponds to the N 2 2 case where a single particle xyl, is artificially isolated from the others and the approximation is made that the effect of the “aggregate” on the single particle (the “external” field) is negligibly affected by the reaction of the single particle back on the “aggregate”. This approximation may be represented by the, “elementary measurement”, action principle
whereJ&t. ) is the “fixed” and prescribed external current and
then we set 61 = 0 with respect to xc, and Au, Au+ where PAA, = 0 as before. This gives the equations of motion
(ii)
av&, = 0
(iii)
dxy P *4* df%, -q(1) dt -z-= ( c 1 F(ext.)v I
where if P1(Q A“(x)
(37)
+ I) = A, then = 1 dx’*[ID+(x
- x’) + D-l(Q
+ r) D-(x - x’)] J”(x’)
(38)
CLASSICAL
569
ELECTRODYNAMICS
which, with equation (36) and the fact that (39) gives for the AyeXt., component A&t.(x)
of A, in equation (38)
= j dxt4 D+(x - x’) J&.(x’)
+ j- dx14 D-(x
- x’)(J;,(x’)
+ J&.(x’))
(40) which, when inserted into equation (37-iii), gives for the equation of motion of
which again shows how the “time symmetric” Lorentz force of the “external” current interferes with the “total coupled radiation field” of the system to produce the “retarded” Lorentz force of the external current and the “radiation reaction” force on the particle. It should be noted that the structure of this theory has the advantage of giving a physical interpretation of the phenomena of radiation reaction, within the context of a variational principle, without recourse to any cosmological assumptions about “complete absorption.” At the same time, the fact that the electromagnetic fields are not eliminated apriori allows a consistent set of electromagnetic conservation laws to be deduced. It should be understood that a consistent interpretation of equations (27) or (41) as physically consistent equations of motion without “run-away” solutions requires the imposition of the asymptotic boundary conditions that dxpldt G 0 which leads to a set of well-known integrodifferential equations discussed elsewhere (4). III.
ELEMENTARY
ELECTROMAGNETIC CONSERVATION
MEASUREMENT LAWS
Equations (15-i) and (15~iv) imply the following differential conservation laws a L Tuv =
where the electromagnetic
J !4+QFuv
measurement energy-momentum
(42)
tensor is (43)
570
LEITER
Equations (42) and (15-iii) imply the consistent integral conservation laws for the total Cmomentum of the “elementary measurement” as g-[--f1
and the corresponding f
J’S’“’ + j/x3
To,,] = -f,dS”Tul
total 4-angular momentum
[ g1 (PY-2)
-
P!=‘x?‘)
Z-Y-- dS1(xuTvl Is
+ //x3h$-,v
- x&J]
x,Tuz)
(45)
where in equations (44) and (45), V is the 3-space volume containing the entire N-particle interaction, and S is a closed surface surrounding V. It is to be noted that no “infinite” self-interactions or self-energies appear in equations (44) and (45) or in the equations of motion (15-iii). This occurs because of the presence of the coupling matrix Q, which was inserted, before the variation, on the basis of the paradigm that only “mutual” measurement fields go into the action principle of the “elementary measurement.” (The concept of “self-interaction” is unphysical and was excluded a priori from Elementary Measurement Electrodynamics.) This also implies that the particle masses occuring in the “elementary measurement” are not “renormalized” in the equations of motion. Hence they are the “physical” phenomenological mass parameters which are determined empirically. IV.
ELEMENTARY
MEASUREMENT
RADIATION
In Elementary Measurement Electrodynamics, measurement radiation is not an elementary process associated with a single current, but involves both the “emitter” and the receiver. To illustrate this, we consider the simple example of a “receiver” made up of two localized currents (5) JF’ and JF’, constrained in a volume I’,, , and moving with a quasi-periodic motion for a characteristic time T. The “emitter” is a single current Jf’ which is considered to be localized in a volume V, very far away from V,, , then only the “wave zone” fields of A:’ will affect the “receiver” significantly. If Jf’ is time dependent due to acceleration of particle one, then the wave zone fields which pass through the surface S,, (surrounding Vzs) at a large distance r from VI , are E(l) N &x (ii)
p)
N
AaA
& (-
_
rj k
x
a t, 1
r
a&P
c at, )
(46)
CLASSICAL
571
ELECTRODYNAMICS
where fl = _r/r and t, = t - r/c is the retarded time. As the wave zone fields (46) become important in V,, the subsequent absorption measurement radiation energy is related to the induced changes in the state of the receiver, as given by the equations of motion of the particles which make up the receiver. In a quasi-linear, nonrelativistic approximation, where the wave zone fields and the mutual retarded fields dominate the radiation reaction fields, we have for the induced acceleration of the receiver charges C(K)
N
4(K) y--
(1) (B
+
(K # J = 2, 3).
El')t re. ,I- ,
(47)
Then the power response of the receiver is from equation (44), applied to V,, and S,, _ dE(2,3)
= - - $
dt
=C
[PO(~) + p,(3) + j- yggdx3W)
- E(3) + B(2) - 8(3)1]
dS . [E(2) x B(3) + E(3) x B(2)]. s 18-
Then using the Lienard-Weichert potential currents .I:’ and Jf’, equation (48) gives -
dE(2,3) dt
=
(48)
solutions associated with &function
2q(2) q(3) ; C(2) - $3) 477~ N 1 I* cs
(49)
The “induced power response” is defined as the difference between the power response of the receiver under the influence of the wave zone fields, and that of the receiver before it is influenced by the wave zone fields. Then (47) and (49) imply that the “induced power response” is -A ($ E(2,3))
N_ 2qi;)$3)
(;) [@“’
+ lp)
* (E(l) + $2)) - p
* p].
(50)
If the receiver subtends a solid angle A,$ra oriented at an angle 0 with respect to the z-axis, where A,, is the cross-sectional area of V,, , then the time average of the induced power response per unit solid angle averaged over the characteristic time T is (6) -<-Wdt
E(2, 3))> c= W43N2/4~mc32 &ha
A33
cr2(p)2>
(51)
where we have assumed q(1) = q(2) = e; m(l) = m(2) = m. If we define the “active area ratio” of the receiver, S, as the ratio of the “effective” classical electron
572
LEITER
cross-sectional areas in the receiver, to the total cross-sectional area A,, for our receiver 6 3 (16rr/3)(e2/4rmc2)2
then (52)
A23
the (51) and (52) imply that the time average of the induced power response of the receiver, per unit solid angle, per active area ratio, is p 5 (--d(wta29 (A23/r2)
3))) N
(r2(p')2>
(53)
. (6)
which by (46) is
Then, in this approximation, this definition of radiation, (54), is equivalent to that of the “Maxwell-Lorentz” distribution of radiation from Jf’. Hence, measurement radiation involves an “emitter” and a physical “receiver” of finite size and dimensions. The time average of the induced power response of the receiver, per unit solid angle, per active area ratio, represents quantitatively the physical act of observing the radiation from the emitter at large distances.
Detector
QJl) “Source”
t-t-l,A-= 4 i4 1
Y
t Incoming wave zone source field assumed collimated
As another example of the measurement radiation case of Thompson scattering (7). This is a physical quasi-plane wave-zone fields of a localized “source” localized “target” current JF’ and causes it to emit
wave zone source field
process, we will discuss the situation where the incident current JF’ impinges on a wave-zone fields of its own,
CLASSICAL
573
ELECTRODYNAMlCS
which are then observed by a “detector” (J, ‘3), JF’) according to the induced “detector” power response. If one defines the differential scattering crossection as the ratio of the time average of the induced power response, per unit solid angle, of the detector with the target present; to the time average of the induced power response, per unit cross-sectional area, of the detector, when the target is absent, then we find the usual Thompson scattering result arises. To see this in more detail, if the incoming wave zone field is &( 1); B(1) from the “source”, this causes the “target” to be accelerated as
This acceleration produces a radiation field which propagates away from the “target” to the “detector” a distance r away. If the “detector” is in the wave zone of the radiation fields E(2) N (e/h)
* (e/w@(l))) sin @ r
(9 (ii)
B(2)
= (r/r> x
(56)
ECW
generated by the acelleration of the target, then these fields produce an induced detector power response in the same manner as described in equations (47) through (50). This implies, taking the time average as before, (57)
(where the detector is assumed to subtend a solid angle A23/r2 = 52(2,3) at an angle 0 with respect to the z-axis taken along the propagation vector of the incoming “source” fields) The “background” induced detector power response is obtained by removing the target and observing the induced detector response when it is placed on the z-axis. In this case the detector is influenced by the source fields only (assuming the source fields to be colimated) and <--d(d/dt
a39 4)))
BACKGR.
=
(2e2/4?r)(e/m
(E(1)))2
(8)
(5)’
(58)
Then the differential cross-section is du dI-2
-=
<--d(W
E(3,4))> 52(3,4>
<-4W I
N ((e2/47rmc2)2 sin2 0). 595/S/3-13
E(3,
4)))BACKGRL
A34
(60)
574
LEITER V. CONCLUSIONS
We have shown how the paradigm of Elementary Measurement, when applied to classical electrodynamics of point charges, which are not elementary in themselves, leads to a consistent theory, free of the paradoxes associated with self-energy. Retardation and radiation reaction are successfully accounted for within the context of a Lagrangian theory, without the use of “complete absorber” assumptions. Moreover, no mass renormalization is required, since only the empirical masses occur in the theory. The paradigm of Elementary Measurement applies to a generalization of this theory to include wave mechanics, but leads to a nonlinear wave-mechanical theory for which the classical theory is a correspondence limit. Again, no selfenergy occurs, but the question of mass and charge renormalization is more complicated, due to the nonlocal character of the wavefunctions. Such a theory is presently being developed and will represent a completely operational theory of wave-mechanical measurement in which both the “object” and the measuring “apparatus” are always represented. However, if something corresponding to a “photon” existed in such a theory, it would not be an elementary particle, since the radiation process involves the “emitter” and the “receiver” (8).
ACKNOWLEDGMENTS
The author wishes to thank Dr. S. Schwebel and Dr. R. Becker for many stimulating discussions and criticisms. Appreciation is also expressed to Sharon Leiter for her help in editing and typing the final manuscript. RECEIVED: June 21, 1968
REFERENCES
AND FOOTNOTES
AND R. P. FEYNMAN, Rev. Mod. Phys. 17, 157 (1945); 21,425 (1949). 2. D. LEITER, Elementary Measurement Electrodynamics II. In preparation. 3. It is to be understood that the terms in the action which give the propagation of “free” particles are not to be interpreted as “self-interactions” since they are not directly involved in the physical definition of the electromagnetic measurements via J$)AP(‘) (n # j) but instead define the nature of the kinematics of the particle degrees of freedom. 4. F. ROHRLICH, “Classical Charged Particles.” Addison-Wesley, Reading, Mass., 1965. 5. Conceptually, there is no reason why a single particle can’t play the role of a “receiver”, as long as there is another particle in the system to act as the “emitter”; since the reception of the measurement radiation energy is related to the induced changes in the state of the motion of the “receiver” described by the associated equations of motion. However the advantage of using two or more particles to define the receiver, lies in the mathematical convenience 1. J. A. WHEELER
CLASSICAL
ELECTRODYNAMICS
575
of being able to use the conservation laws and the equations of motion together in a quasilinear, nonrelativistic approximation. If a single particle receiver is used, then the conservation laws are not so easily utilized, and an exact solution of the equations of motion is required to extract the desired “receiver” information. 6. This occurs because the time average of the dot product of (EC’) . _E!t’,.> j = 2,3 vanishes because of its quasi-periodicity during the time interval T ofinterest. Hence even though the mutual retarded fields, in the receiver, cannot be neglected with respect to the incoming wave zone fields in the equations of motion of the receiver (47) they will not contribute to the “induced” power response, (51) of the receiver when averaged over T. 7. See Fig. 1. 8. See Ref. 2. Another approach to this problem, using the same paradigm, but differing in detail and interpretation occurs in the paper by Sachs and Schwebel, Nuovo Cimento Supp. 2, XXI, 197, (1961).
Addendum The technique of splitting a covariant propagator into two noncovariant parts has been extensively examined by A. Ramakrishnan: J. Math. Anal. Appl. 17, 68 (1967) and 18, 175 (1967); J. Math. Phys. Sci. 1, 57 (1967).