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IL NUOVO CIMENTO

NOTE BREVI

VOL. 109 A, N. 11 Novembre 1996

A remark on Dirac's magnetic monopoles

A. LOINGER Dipartimento di Fisicc~ Universit(t di Milano - Via Celoria 16, 20133 Milano, Italy

(ricevuto il 26 Luglio 1996; approvato il 5 Agosto 1996)

Summary. -- The only reasonable magnetic ,,monopoles- are the poles of very long and very thin magnetic needles.

PACS 14.80.Hv - Magnetic monopoles.

Many papers have been written on Dirac's magnetic monopoles, but the fundamental ones remain those of Dirac [1] and Fierz [2]. I point out here that, for a valid reason, the only realistic magnetic ,,monopoles- are the poles of very thin and very long rectilinear permanent magnets. The proof is simple, and it is sufficient to consider the classical theory.

As is well known, the system M of differential equations describing an assembly of n Dirac's magnetic monopoles interacting through their own electromagnetic field is mathematically identical with the system C of the Maxwell-Lorentz equations plus the equations of dynamics describing an assembly of n interacting electric charges. This means that each one of the two systems, M and C, can be transformed into the other by performing suitable interchanges of the involved physical magnitudes. Therefore, an assembly of interacting charges, e.g., can be equivalently described either by system C or by system M. Thus monopoles and charges are physically undistinguishable when they are considered apart, the inhabitants of a world constituted solely of monopoles and those of a world constituted solely of charges would experience identical phenomena--accordingly, the two worlds are identical. But it is commonly asserted that the difference between a charge and a monopole becomes manifest through their motions, when they are coupled.

Now, this statement, which is a trivial consequence of the basic equations of monopole theory, would be really convincing from the physical point of view if we were allowed to regard the magnetic charges of the monopoles as generated by some microstructural ,,mechanism,. But, in accordance with Dirac's conception, the monopoles are true elementary particles, devoid of any microstructure. Therefore the desired difference between charge and monopole can be brought into existence only par dgcret, with the peculiar postulate that endows the monopole with the magnetic charge of the (non-elementary) pole of an infinitely long and infinitely thin ideal magnetic needle. N.B.: auxiliary entities such as Dirac's strings,

1609

1610 A. LOINGER

singularity lines or -obstructions,, etc., which are useful in relativistic quantum formulations of the theory, do not represent material objects.

In conclusion, one can say that the Maxwell-Lorentz-Dirac field equations, which intend to describe the e.m. fields produced by an assembly of electric charges and magnetic monopoles, describe essentially the e.m. fields produced by an assembly of charges and poles of infinitely long and infinitely thin ideal magnetic needles. Of course, the motions of the members of an assembly only composed of n interacting poles of this kind are approximately equal to the motions of the members of an assembly only composed of n interacting charges, but if one of the above poles interacts with a charge, the microst ructure of the pole becomes responsible for the difference existing between the field of the charge and the field of the pole.

A useful discussion with my friend P. Bocchieri is ~-atefully acknowledged.

APPENDIX

In an unconnected space-time manifold (i.e. a differential manifold on which neither affinity nor metric has been impressed) Maxwell equations can be written as follows [3]:

(A .1 ) 3~'" - or'; ~'~'" - 0, ~x" ~x ~

where ~'~ and ~"" are contravariant antisymmetric tensor densities representing the fields H, D and E, B, respectively, and _~ is a contravariant vector density representing the electric current j and the electric charge Q.

The formulae

(A .2 ) ~ '~ - tt'; 3f~'~' - 0 ~x ~ ~x"

give the fields generated by a monopole vector density t ~' = (k, ~). The so-called material equations, which give the relations between E and D, and

between B and H, require the introduction of the Riemannian relativistic metric ds2= g~,,,dx~'dx v. In vacuum we have

1 ~pw~fl (A.3) ~"" = 2( - g)l/, ig,Tgz~ ~' ,

dx F' dx F' (A.4) ~ = ( -g ) I /200- - ; t ~' = - (g)1/2 ~o - - ,

ds ds

where e uv"~ is the contravariant antisymmetric tensor density for which e ~ 1.

A REMARK ON I)IRAC~S MAGNETIC MONOPOLES 1611

Now, the structural identity of (A.1) and (A.2) becomes quite evident:

~f .... le,,,,,/~ ~ [ f,/~ ] (A.I') 8x" - ~'; --2 8x ~ (_g)l/'2 0",

1 .... o 3 [ ~,~ (A.2') 8~'" - t'" -~ ' ' ' - - 8x" ' 2 ~x" [ ( _ g)l/2

=0.

Equations (A.2') are a mathematical duplicate of Maxwell equations (A.I'). Further, as is well known, if we schematize the permanent magnets as bodies having a field-independent magnetization plus a possible induced magnetization, proportional to the field, we can derive from Maxwell equations for the material media a system of equations, which describes the fields produced by an assembly of very long and very thin needles. This system is substantially identical with system (A.2').

Finally, if we imagine to concentrate into the disposable pole the whole mass of an infinitely long and infinitely thin ideal magnetic needle, we get an object which simulates almost perfectly a Dirac's monopole. This last is a sort of superfetation product, generated by the desire to have an exact symmetry between the Maxwell equations concerning the fields H, D and those concerning the fields E, B. Its fictive nature is absolutely clear from the standpoint of the electromagnetic theory of distant actions (A. D. Fokker and others): here the fields are ignored, all e.m. interactions are interactions at a distance of electric charges [4].

REFERENCES

[1] DIRAC P. A. M., Proc. R. Soc. London, Set. A, 133 (1931) 60; Phys. Rev., 74 (1948) 817. [2] FIERZ M., Heir. Phys. Acta, 17 (1944) 27, and also BANDERET P. P., Helv. Phys. Acta, 19

(1946) 503. See further: BOCCHIERI P. and LOINGER i., Nuovo Cimento A, 59 (1980) 121; LOINGER A., Riv. Nuovo Cimento, 10 (1987) 1.

13] SCtIRODINGER E., Space-Time Stmtcture (The University Press, Cambridge) 1960, p. 24 and following.

[4] ..... one musn't believe in anything too strongly; one must always be prepared that various beliefs one has had for a long time may be overthrown.,, (From the Lectures on Quantum Field Theory by P. A. M. DIRAC (Academic Press Inc., New York, N.Y.) 1966, p. 8.)