Electrodynamics' spin
- From: khrapko_ri@xxxxxxxxxxx
- Date: 23 Dec 2005 09:06:46 -0800
We solve the angular momentum problems of the electrodynamics in this
paper. We show that a desire of using a spin density proved to be
correct for evaluating of the electrodynamics spin because the moment
of momentum is an orbital angular momentum density and does not
encompass the spin. However, the expression E x A is not a true
expression for the spin density. This expression yields correct results
in simplest cases randomly. Its improvement is hopeless. Instead we
present a true spin density of electromagnetic waves, \Upsilon^{ijk},
and demonstrate a way for its deducing. Our result can be expressed by
a formula for a sum of the orbital and spin angular momentums density:
j = r x (E x B) + \Upsilon^{ijk}
There is a scandalous situation in the modern electrodynamics. On the
one hand,
s = E x A
is considered as a spin volume density of an electromagnetic field. I
present a series of quotations for a confirmation of my statement [1 -
5].
Please note that a corollary of this definition of the spin density is
a circularly polarized plane wave, has the spin density,
s = U/\omega,
is that the quantum theory prescribes (U is the energy density).
At the same time, however, on the other hand, all physicists insist
that a circularly polarized plane wave does not carry spin. It is a
matter of common opinion that
j = r x (E x B)
encompasses both the spin and orbital angular momentum density of
electromagnetic field. Therefore a circularly polarized plane wave
carry neither orbital angular momentum nor spin in direct contradiction
to the quantum theory. I present a series of quotations for a
confirmation of my statement [6 - 8]
Naturally, this contradiction shows strong evidence for a defect of the
classical field theory, and this defect causes many conflicts,
vagueness, and paradoxes concerning electrodynamics angular momentum.
This was recognized long ago, and I present here a series of quotations
for a confirmation of my statement.
1) Zambrini, and Barnett [9]: "Experimental observations appear to be
in conflict with theoretical considerations".
2) Nieminen et al. [4]: "If the above expression j = r x (E x B) was in
fact the correct angular momentum density, then the angular momentum of
a circularly polarized plane would be zero. Since the correct classical
angular momentum density must agree with the classical limit of the
quantum angular momentum density, this must be incorrect."
3) Allen and Padgett [10]: "A circularly polarised plane wave has a
linear momentum density only in the z-direction. When this is crossed
with r to give the angular momentum density, there is no contribution
in the z-direction. Thus, such a beam has no angular momentum to
transfer to a waveplate. Yet, Beth was able to make such a transfer - a
paradox."
4) Khrapko [11]: "The classical experiment of Beth [12, 13] was carried
out almost 70 years ago. However, this experiment raises questions. The
Poynting vector was everywhere equal to zero in the experiment. How is
it therefore that the plate experienced a torque and rotated?"
5) Khrapko [14]: "Suppose that a quasiplane wave is absorbed by a round
flat target which is divided concentrically into outer and inner parts.
According to previous reasoning, the inner part of the target will not
perceive a torque. Nevertheless R. Feynman [15] clearly showed how a
circularly polarized plane wave transfers a torque to an absorbing
medium. What is true? And if R. Feynman is right, how can one express
the torque in terms of ponderomotive forces?"
Unfortunately, authors [1 - 5] do not explain that the expression E x
A is a component of the canonical spin tensor. The point is that
physicists obtain the canonical energy-momentum tensor and the
canonical spin tensor using a canonical Lagrangian, by the Lagrange
formalism. However, the canonical tensors are not electrodynamics
tensors. They obviously contradict experiments. And so, physicists are
forced to modify the canonical tensors. Following [16, 17], physicists
accomplish a Belinfante-Rosenfeld procedure. They add specific terms to
the canonical tensors and arrive to the standard energy-momentum tensor
and the standard spin tensor, which is zero.
Unfortunately, the standard energy-momentum tensor obviously
contradicts experiments as well. It is not symmetric and has a wrong
divergence. But the main defect of the standard electrodynamics is the
absence of spin. In contrast to the canonical pair, the standard pair
is defective. Standard energy-momentum tensor is not accompanied by a
spin tensor.
The absence of spin in the standard electrodynamics implies an absurd
corollary: a circularly polarized plane wave has no angular momentum at
all [1, 2, 6 - 8, 18, 19] because E x B is parallel to the direction
of propagation and
j = r x (E x B) = 0.
But this corollary is in direct contradiction to quantum theory [20].
In accordance with this absurdity, the author [8] uses mystical
concepts of "angular momentum in an actual form" and "angular momentum
in an potential form".
j = 0 in an important case of the Beth experiment [12, 13] because E x
B =0. In the Beth experiment a beam of circularly polarized light
exerted a torque on a doubly refracting plate, which changes the state
of polarization of the light beam. But, it is evident that the Poynting
vector equals to zero in the experiment because the passed beam is
added with the reflected one [11]. Therefore the result of the Beth
experiment cannot be understood in the frame of the standard
electrodynamics without spin. So, we must add a concept of spin to the
standard electrodynamics.
It was explained [11, 23, 24] that the Belinfante-Rosenfeld's
modification [16, 17] of the canonical pair does not lead to true
energy- momentum and spin tensors. We must change this standard
procedure. We must use another addends. The standard addends lead to
the defective standard pair. But, using our addends we get, instead of
standard pair, the Maxwell energy-momentum tensor and a tensor, which
was proved to be a doubled electric part of the spin tensor,
2A^{[i}\partial^kA^{j]}.
This result was submitted to "JETP Letters" on May 12, 1998. But, this
result was not final one. The true spin tensor must depend
symmetrically on the magnetic vector potential and on the electric
vector potential \Pi^i. So the spin tensor of electromagnetic waves has
the form
\Upsilon^ijk= A^{[i}\partial^kA^{j]}+Pi^{[i}\partial^k\Pi^{j]}.
and the total angular momentum density has the form
j= r x (E x B) + \Upsilon^{ijk}
instead of
j = r x (E x B)
A new spin tensor was presented and applied in a series of works [25]
and also at web sites http://www.sciprint.org,
http://www.mai.ru/projects/mai_works/.
This paper conveys new physics. We briefly review existing works
concerning electrodynamics spin and indicate that existing theory is
insufficient to solve spin problems because spin tensor of modern
electrodynamics is zero. Then we show how a change of the
Belinfante-Rosenfeld procedure resolves the difficulty by introducing a
true electrodynamics spin tensor. Our spin tensor, in particular,
doubles a predicted angular momentum of a circularly polarized light
beam without an azimuth phase structure and explains the Beth
experiment.
Unfortunately, materials of this paper were rejected more than 350
times by scientific journals. For example (I show an approximate number
of the rejections in parentheses): JETP Lett. (8), JETP (13), TMP (10),
UFN (9), RPJ (70), AJP (14), EJP (4), EPL (5), PRA (3), PRD (4), PRE
(2), APP (5), FP (6), PLA (9), OC (2), JPA (4), JPB (1), JMP (4), JOPA
(3), JMO (2), CJP (1), OL (1), NJP (2), MREJ (3), arXiv (70). In
particular, PRA rejected a paper "Beth's experiment modification"
submitted on Sun, 16 Nov 2003 06:36:00.
I am deeply grateful to Professor Robert H. Romer for publishing my
question [14] (was submitted on Oct. 7, 1999) and to Professor Timo
Nieminen for valuable discussions (Newsgroups: sci.physics.electromag).
Unfortunately, Jan Tobochnik, the present-day Editor of AJP, rejected
my papers more than 20 times.
[1] J. D. Jackson, Classical Electrodynamics (Wiley, 1999)
[2] H. C. Ohanian, Amer. J. Phys. 54, 500-5 (1986)
[3] M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg & H.
Rubinsztein-Dunlop, Nature 394, 348-350 (1998)
[4] A. Nieminen et al. arXiv:physics/0408080.
[5] J. H. Crichton and P. L. Marston, Electronic Journal of
Differential Equations Conf. 04, 37 (2000).
[6] W. Heitler, The Quantum Theory of Radiation (Oxford: Clarendon,
1954)
[7] J. W. Simmonds, M. J. Guttmann, States, Waves and Photons
(Addison-Wesley, Reading, MA, 1970)
[8] A. M. Stewart, Eur. J. Phys., 26, 635 (2005)
[9] R. Zambrini, S. M. Barnett, J. Mod. Optics 52 1045 (2005)
[10] L. Allen, M. J. Padgett, Opt. Commun. 184, 67 (2000).
[11] R. I. Khrapko, Measurement Techniques 46 No. 4, 317 (2003)
[12] R. A. Beth, Phys. Rev., 48, 471 (1935)
[13] R. A. Beth, Phys. Rev., 50, 115 (1936)
[14] R. I. Khrapko, Amer. J. Phys., 69, 405 (2001)
[15] R. P. Feynman et al., The Feynman Lectures on Physics
(Addison-Wesley, London, 1965) V. 3, p. 17-10.
[16] F. J. Belinfante, Physica 6, 887 (1939).
[17] L. Rosenfeld, Memoires de l'Academie Royale des Sciences de
Belgiques 8 No 6 (1940).
[18] D. E. Soper, Classical Field Theory (N.Y.: John Wiley, 1976).
[19] F. Rohrlich, Classical Charged Particles (Addison-Wesley, Mass.
1965)
[20] L. H. Ryder, Quantum Field Theory (Cambridge, 1985)
[21] J Humblet, Physica, 10, 585 (1943)
[22] J. Crichton et al., Gen. Relat. Grav., 22, 61 (1990)
[23] R. I. Khrapko, Gravitation & Cosmology, 10, 91 (2004)
[24] R. I. Khrapko, physics/0102084, physics/0105031
[25] R. I. Khrapko mp_arc@xxxxxxxxxxxxxxxxxx 03-307, 03-311, 03-315
This paper is published at www.sciprint.org
and is submitted to PRA on Dec. 22, 2005
R. Khrapko
.
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