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46
UDC 530.1; 539.1
On the semi-classical origin of the electron anomalous magnetic moment
A. V. Kozlov, E. A. Nemchenko
Department of Theoretical Physics, Tomsk State University, Tomsk, 634050, Russia.
E-mail: tf.kozlov@gmail.com, katusha77@sibmail.com
For modem theoretical physics there is tendency to review methods of describing quantum-mechanical systems in terms of relationship with classical analogues. We attempted to calculate the anomalous magnetic moment from a semiclassical view (with consideration for Zitterbewegung) on the basis of electromagnetic fields generated by the electron in the immediate vicinity of the charge point. Derived value of anomalous magnetic moment is in full agreement with the Schwinger’s calculations.
Keywords: radiation, anomalous magnetic moment, Zitterbewegung, semi-classical theory.
1 Introduction In our work to obtain the theoretical value of the
In 1941 W. Pauli fl] showed that it’s possible to add a relative invariant an additional to Bohr magneton mo = eH/2m0c magnetic moment part Ma describing interaction with external electromagnetic field. At first it seemed to be purely abstract theoretical construct. In 1947 G. Breit [2], analyzing experimental data for measuring the hyperfine structure of the hydrogen atom spectrum, suggested that the magnetic moment of the electron has the intrinsic anomalous part:
M = Mo + Ma.
The first theoretical calculations of the anomalous magnetic moment of the electron were provided by J. Schwinger [3] in 1948. These calculations were excellent application of the developing quantum electrodynamics. The result obtained by Schwinger:
, a Ma = Mo(l + ),
where a = e2/Hc - fine structure constant, which was in good agreement with experiment [4]. Further calculations and measurements, which took into account higher approximations of the fine structure constant
[5], depending on the presence of a strong external fields and the number of electron energy level [6] resulted only with a refinement of the obtained expressions for the Schwinger’s anomalous magnetic moment (see review works [7,8] and others).
The discovery of the anomalous magnetic moment was a milestone in the construction of modern quantum field theory. Attempts of better visibility of the semi-classical presentation goes back to the Welton and Koba works [9,10] (see also [8]). Of course, the Welton-Koba method still represents only a qualitative interest for the interpretation of the anomalous magnetic moment. And it can not claim to be serious quantitative results.
anomalous magnetic moment used exact methods of purely relativistic classical electrodynamics [11]. Only at the last stage of calculations the electron’s self-field is determined by semi-classically introducing Compton wavelength. This allows to define more precisely the effective electromagnetic mass of an electron, and with it the anomalous magnetic moment. But the classical goal is not important in itself. The quantum mechanical picture when expressed in terms of invariance principles will show the relationship between the classical variables. This is our main motivation for the classical analysis of spinning particles: to finally obtain a thorough quantum scheme [12].
2 Electomagnetic self-action of an electron in classical electrodynamics
In the classical field theory interaction of a charged particle with an external and its own electromagnetic field can be formulated using a differential conservation law of momentum density [13]:
DvVr + DvPXt + DvV^v + DvVMv + Dv= 0, (1) here vr = Mo J v^(r)vv(t)S(RP - rp(T))dT
- momentum density tensor of particle with effective
Mo
-p^v =_____— (HWH v + - flMv H aHa/3)
1 ext = 4nc (H Hp +4 3 Hat3H )
- momentum density tensor of external field with stress H^t. The last three members in the eq. 1 take into account the field produced by the charged particle. It is composed of the convective field HT ~ J? and the radiation field H ~ J Therefore, the momentum density
tensor of the self-field of the charge consists of three parts
where
-p_
4n (r pvp)4 2
(ô-
r»vv ~+ rvv»
r pvp
(rpvp)2 '
corresponds to the convective field.
_ 1 (c2 (rPWP)2 — WPWP )~^~v „
4nc (r pvp)2 (r pvp)4 (r pvp)2 '
describes radiation field.
~ 1 e2c2 ,r»wv — rv w» r^vv + rv vM
= -—,(--------—----------„ rnw-
4n (rpvp)3 rpvp
them DM = dM dr- obtain the equations of
motion of a charged particle the eq. 1 should be integrated throughout the four-dimensional space around the world line of the segment of length cdT. In the first two members integration is carried out using a four-J-function and a second pair of Maxwell equations in the second term. The result of these procedure is the usual Lorentz force. The remaining three members are integrated with the theorem of Gauss-Ostrogradsky in the volume enclosed between the two light cones and the time-like hyper band (see fig. ). It is called Bha-bha tube with elements of the hypersurface:
da
e2((1 +—~wpep)eu +—r-wpepvv )dQcdr, c2 c3
(rpvp)2 p 2c2r» rv r pwp 1
----------------- ) rs,/ -
(rpcp)3 r3
where dQ - element of solid angle in the rest frame of
ev
r v -
v 1 v v v n
ev e = 1, e = -c-----------------v , Vv e = 0.
r pvp c
- corresponds to part of mutual interference. In V^if field are defined in the retarded time and therefore for
Invariant quantity e = rpep plays the role of a spherical radius.
2
c
1
4
As a result of integration over the four-dimensional volume, taking into account the delay of the radiation
we get
e 2e2 3c3
Mow^(T ) = -H?xtvv (t) — 3^ (wa.w°v» +~^w^)T .
The last term on the right side in the radiation friction force is a divergent expression, since e = c(t* — t) ^ 0 then, to be excluded from the definition of force, let taking into account the delay of the radiation
w»(t *) = w»(t ) — kw »(t )At,
where k - constant factor, which can be determined from the condition of the space-like radiation friction
force. Putting k = 4/3, we obtain
e2 e 2e2 -
(M0+2ec2 )W^(T ) = ~C HexctVv (t ) + 3r^5 (wM-~ci wv wv V^).
Thus on the right side we get the Lorentz force and the standard expression of the radiative friction force, on the left - the definition of four-dimensional force vector with a massive multiplier that allows for self-action of the electromagnetic field. Interesting that the delay parameter of the radiation
4 A 4e 2e2
— At = — = -r- = T0
3 3c m0c3
coincides with the quantum of time that proposed by
P. Caldirola [14], if under m0 is understood electromagnetic mass of an electron:
mo =
2ec2 '
Mo = mo(l -
2emoc2
).
eh
2M0c
2em0 c2
).
the electron. Clearly, this is the minimum distance at which the electromagnetic field, leaving the electron leaves it "naked". Compton wavelength of an electron is known to be an indicator of the distance at which quantum effects become important, considering
To obtain the standard expression for the Lorentz force, taking into account the radiative friction force should take
Mo + mo = mo.
3 Semi-classical self-action and the anomalous magnetic moment
Thus, we have a well-known equation
— ~ mow» = -H vv + F * c
which is satisfied if in the original equation the effective mass equals to
2 Ac
we get
nh
mo c
a
Mo = mo(l - —), 2n
In fact, that it is mass of "naked" particle in the absence of the self-interaction with the electromagnetic field. It is obvious that the magnetic moment of an electron with an anomalous addition is determined by this mass:
The idea, that the anomalous magnetic moment of the electron is directly related to the effective mass of the particle, found its confirmation in quantum electrodynamics [15]. This idea has been actively developed in classical electrodynamics by MacGregor [16], but his calculations have essentially approximate nature.
Until now we have been using only the methods of relativistic electrodynamics. Our further calculations will be associated with the refinement of the concept of radius of the spherical electromagnetic field around
then according to the definition
a \
m = Mo(i + >.
It is Schwinger’s result for the anomalous magnetic moment. It’s very interesting that the value of e can be associated with the Schrodinger’s Zitterbewegung, considering
hi^zb = 2moc2.
e
2nc nh
e = clzb =-------= ------.
^Zb moc
4 Conclusion
Feature of our work is that the derivating of the anomalous magnetic moment of the electron, we started from a purely classical theory - exact relativistic radiation. Only at the last stage of calculations we attracted semiclassical postulates to describe the interaction of an electron with a quantized electromagnetic field.
Our result confirms the correctness of our assumptions and the validity of the principle of correspondence of classical and quantum field theory. It opens up new possibilities in the study of the effects of phenomena such as electromagnetic mass of an electron, Zitterbewegung and quantization of time and etc.
2
e
2
e
2
e
References
[1] Pauli W. 1941 Rev. Mod. Phys. 13 203
[2] Breit G. 1974 Phys. Rev. 72 984
[3] Schwinger J. 1948 Phys. Rev. 73 416
[4] Kusch P., Foley H. M. 1948 Phys. Rev. 74 250
[5] Aldins J., Brodsky S. J., Dufner A. J., Kinoshita T. 1970 Phys. Rev. ID 2378
[6] Ternov I. M., Bagrov V. G., Bordovitsyn V. A., Dorofeev 0. F. 1968 Sov. Phys. JETF 55 2273 (in Russian)
[7] Vonsovskii S. V. 1974 Magnetism (Wiley, New York, in two volumes, 1974).
[8] Ternov I. M. 1977 Introduction to spin physics of relativistic particles (MSU Press)
[9] Welton T. A. 1948 Phys. Rev. 74 1157
[10] Koba Z. 1949 Progr. Theor. Phys. 4 98
[11] Bordovitsyn V. A. 2002 In: Radiation of Relativistic Particles (Moscow, Nauka, p. 21-151)
[12] Rivas M. 2000 Kinematical Theory of Spinning Particles (Bilbao, Fundamental Theories of Physics)
[13] Bordovitsyn V. A., Pozdeeva T.O. 2006 Russian Physics J. 1 72 (in Russian)
[14] Caldirola P. 1956 Suppl. Nuovo Cimento 3 343
[15] Ritus V. I. 1986 Proc. Levedev Phys. Inst. 168 52
[16] MacGregor M. H. 1989 Found. Phys. Lett. 2 577
Received 01.10.2012
А. В. Козлов, E. А. Немченко
О ПОЛУКЛАССИЧЕСКОМ ПРОИСХОЖДЕНИИ АНОМАЛЬНОГО МАГНИТНОГО
МОМЕНТА ЭЛЕКТРОНА
В данной работе предпринята попытка рассчитать аномальный магнитный момент электрона с полуклассической точки зрения с учётом явления Zitterbewegung и на основании электромагнитных полей, создаваемых электроном в непосредственной близости от заряда. Полученное значение аномального магнитного момента находится в полном согласии с первоначальными расчетами Швингера.
Ключевые слова: конвективное электромагнитное поле, аномальный магнитный момент, Zitterbewegung, полуклассическая теория.
Козлов А. В., студент.
Томский государственный университет.
Пр. Ленина, 36, Томск, Россия, 634050.
E-mail: tf.kozlov@gmail.com
Немченко Е. А., аспирант.
Томский государственный университет.
Пр. Ленина, 36, Томск, Россия, 634050.
E-mail: katusha77@sibmail.com
