We reply to a criticism made on the new equation of motion proposed by us for classical charged particles

Mo, Tse Chin

Papas, C. H.

Caltech Authors - Main

COMMENT ON A NEW EQUATION OF MOTION FOR. . . 2293
solves problems which have already been solved
for the Lorentz-Dirac equation. This weakens the
raison d' etre for their theory, and seems to sug-
gest that they were too hasty in suggesting that
their formalism replace the Lorentz-Dirac formal-
ism.
*Research supported by the National Research Council
of Canada under Grant No. A6286.
~T. C. Mo and C. H. Papas, Phys. Rev. D 4 3566 (1971).
2F. Rohrlich, Classical Charged Particles (Addison-
Wesley, Reading, Mass. , 1965).
3The conventional association is via Noether's theorem.
No extremum is required, but only a vanishing first-
variation and invariance under a. Lie group (the Poincare
group in this case). Also other derivations have been
given within the context of new field-theoretic points of
view [see Ref. 2, and also D. Leiter, Ann. Phys. (N.Y.)
51, 561 (1969); Lett. Nuovo Cimento 3, 347 (1970);
Proc. Phys. Soc. London (Gen. ) A3, 89 (1969)].
4See the section on the integral form of the Lorentz-
Dirac equation, and the associated boundary condition
which precludes runaway solutions, in Ref. 2.
~See J. D. Jackson, Classical Electrodynamics
(Wiley, New York, 1962) for a good discussion of this
point (e.g. , chap. 17). The preacceleration effects are
confined to times on the order of 10 sec. This is
clearly within the quantum domain, and hence the pre-
dictions of classical mechanics are breaking down long
before the effects can occur. For this reason they are
not classically observable.
P 8 Y SIC AL R K V IK%' D VOLUM E 6, NUMBER 8 15 OC TOBER 1972
Reply to Comment on the "New Equation of Motion for Classical Charged Particles"
Tse Chin Mo and C. H. Papas
Electrical Engineering DePartment, California Institute of Technology, Pasadena, California 91109
(Received 10 July 1972)
We reply to a criticism made on the new equation of motion proposed by us for classical
charged particles.
A new equation of motion for classical charged
particles' was proposed by us to account for radia-
tion reaction. ' Here we reply briefly to a criticism
of it made by Leiter' and invite attention to the fol-
lowing.
(1) That we did not find an action principle for
the new equation was explicitly pointed out in our
original paper. The existence or nonexistence of
such an action principle is still an open question.
(2) The validity or nonvalidity of an equation of
motion is not determined by the existence or non-
existence of associated action principle (at least
we believe so).
(3) The "validity" arguments for an "action prin-
ciple" of the Lorentz-Dirac equation4 are already
amPle in the literature. ' Acknowledging these,
Ref. 1 pointed out, just as a side remark, the con-
trary view that this "action principle" is not a
physically valid action principle at all. %e empha-
size that this has no essential bearing on the pro-
posed new equation.
(4) The energy-momentum conservation law re-
quired by and found for the new equation is "local"
and is "nonlocal. " It is "local'" in the sense of be-
ing a pointwise differential relation, which says
that something must happen in between if energy-
momentum decreases and increases at two places,
and in the sense of the t"' being retarded from the
sources. ' It is nonlocal in the sense that at each
space-time point x, T,"' is not entirely made of the
field E"'(x) at that point but has an additional piece
t"'(x) 'This was als. o pointed out in our original
paper (Ref. 1).
(5) The runaway and preacceleration phenomena
of the Lorentz-Dirac equation, which are well
known and usually considered "solved, '" are exact-
ly what dissatisfied and motivated us.
2294 TSE CHIN Mo AND C. H. PAPAS
~T. C. Mo and C. H. Papas, Phys. Rev. D 4, 3566 (1971).
2A further detail on the testability of the new equation
is to be published elsewhere.
3D. Leiter, preceding paper, Phys. Rev. D 6, 2292 (1972).
4P. A. M. Dirac, Proc. Roy. Soc. (London) A167, 148
(1938).
5See, e.g. , F. Rohrlich, Phys. Rev. Letters 12, 375
(1964).
See R. P. Feynman, The Eeynman I.ectures on Physics
(Addison-Wesley, Reading, Mass. , 1964), Sec. 27-1 of
Vol. 2 and Sec. 21-2 of Vol. 3.
7Ref. 1, Appendix.
See any text treating classical radiation for the pre-
acceleration including the integral-equation form of the
I orentz-Dirac equation, e.g. , J. D. Jackson, Classical
Electrodynamics /Wiley, New York, 1962), p. 597.
PHYSICAL REVIE W D VOLUME 6, NUMBER 8 15 OC TOBER 1972
Comment on the Breakdown of Dilatation Invariance*
Antonio Aurilia and Yasushi Takahashi
Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada
(Received 22 May 1972)
On the basis of our previous results concerning the breakdown of dilatation invariance we
find that two large classes of models in quantum field theory may lead to the same observable
quantities in spite of their different properties under scale transformation. Although the con-
ditions of equivalence have long been known we emphasize their relevance in the new context
of the dilatation transformation. The equivalence in question leads to the same physical mass-
es, observable coupling constants, and, possibly, to the same energy-momentum tensors
associated with the asymptotic fields in the two theories. It is suggested that the arbitrariness
in the choice of an interpolating field is a key point for the equivalence of the two theories. Our
analysis also underlines the difference between the notions of physical dimension and scale
dimension of a field operator.
It is commonly accepted by now that dilatation
invariance, in order to be a useful concept in par-
ticle physics, must be broken to a certain extent.
It is also well known that the symmetry may be
violated in any of the following ways:
(i) by explicitly introducing into the theory some
breaking parameters which, incidentally, happen
to be just the fundamental parameters of the theory,
namely, masses and (dimensional) coupling con-
stants (intrinsic breakdown);
(ii) by requiring that the vacuum be asymmetric
under the dilatation transformation (spontaneous
breakdown).
Qf course, one also has to take into account possi-
ble combinations of intrinsic and spontaneous
breakdown. In any case, however, in order to have
a meaningful theory, the dynamics must produce
massive observable particles.
We have remarked in previous papers' ' that
the above requirement poses a consistency problem
as far as spontaneous breakdown is concerned. In-
deed, one begins with a Lagrangian which is fully
invariant under the dilatation transformation and
eventually ends up with massive observable parti-
cles thereby apparently violating the original in-
variance of the theory.
In Ref. 1 we have introduced the notion of "di-
mensional invariance" which, being simply an ex-
pression of the dimensional consistency of the
theory, can never be violated. In Ref. 2 we have
shown that dilatation invariance, with only spon-
taneous breakdown, may consistently lead to a
physical mass and that the formal invariance of
the theory is recovered, among the massive as-
ymptotic fields, in the form of dimensional invari-
ance.
In this note we proceed one step further and in-
quire about the connection, if any, between intrin-
sic breakdown and spontaneous breakdown of scale
invariance. More precisely we wish to find out if
the two breaking techniques differ among them-
selves in some fundamental way or rather if they
may lead to the same physics with the same observ-
able particles. In the latter case we wigh to es-
tablish the conditions of equivalence.
Our findings are that, in the framework of ordin-
ary quantum field theory, there are well-known


Reply to Comment on the "New Equation of Motion for Classical Charged Particles"

https://authors.library.caltech.edu/80355/1/PhysRevD.6.2293.pdf

Reply to Comment on the "New Equation of Motion for Classical Charged Particles"

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main