aa r X i v : . [ phy s i c s . g e n - ph ] D ec Magnetic monopole and the nature of the static magnetic field
Xiuqing Huang , ∗ Department of Physics and National Laboratory of Solid State Microstructure, Nanjing University, Nanjing 210093, China Department of Telecommunications Engineering ICE,PLA University of Science and Technology, Nanjing 210016, China (Dated: October 31, 2018)We investigate the factuality of the hypothetical magnetic monopole and the nature of the staticmagnetic field. It is shown from many aspects that the concept of the massive magnetic monopolesclearly is physically untrue. We argue that the static magnetic field of a bar magnet, in fact, is thestatic electric field of the periodically quasi-one-dimensional electric-dipole superlattice, which canbe well established in some transition metals with the localized d -electron. This research may shedlight on the perfect unification of magnetic and electrical phenomena. PACS numbers: 14.80.Hv, 03.50.De, 75.70.Cn
I. INTRODUCTION
The general concept of symmetry plays an importantrole in physics and other fields of science. It is well knownthat the behavior of electric and magnetic fields can becompletely described by Maxwell’s equations. For time-varying fields, the differential form of these four impor-tant equations in cgs (short for centimeter, gram, second)is given by ∇ · E = 4 πρ e , ∇ · B = 0 , ∇ × E = − c ∂ B ∂t , ∇ × B = 1 c ∂ E ∂t + 4 πc J e , (1)where E is the electric field, B is the magnetic field, ρ e is the electric charge density, J e is the electric currentdensity and c is the speed of light in a vacuum.Without electromagnetic sources ( ρ e = 0; J e = 0),we can see clearly that the set of Eq. (1) will remaininvariant under the following duality transformations E → B ; B → − E . (2)This implies that electric and magnetic fields are sym-metrical and equivalent in this special case. Obviously,the electric-magnetic duality symmetry is no longer truewhen ρ e = 0 (or J e = 0). However, Dirac believed thatthe electromagnetic laws should have the “dual nature”under any circumstances, in other words, the electric andmagnetic fields may have a general intrinsic symmetryand the Maxwell’s equations of Eq. (1) are incomplete.In 1931 [1], Dirac claimed that the mathematical intro-duction of magnetic monopole (a basic unit of magneticcharge) into the Maxwell’s equations would lead to a sym- ∗ Electronic address: [email protected] g B V electric fieldmagnetic field magnetic field electron (e) (a) E (c) B electric field (b)(d) E V monopole (g)e
FIG. 1: Electric ( E ) and magnetic ( B ) field lines generated byelectron (or monopole) and by their motion with velocity V .(a) The electric field of a static electron with electric charge e ,(b) the magnetic field of a moving electron. (c) The magneticfield of a static magnetic monopole with magnetic charge g ,(d) the electric field of a moving monopole. metrical form of the Maxwell-Dirac equations ∇ · E = 4 πρ e , ∇ · B = 4 πρ m , ∇ × E = − c ∂ B ∂t − πc J m , ∇ × B = 1 c ∂ E ∂t + 4 πc J e . (3)where ρ m is the magnetic charge density and J m isthe magnetic current density. The above four equationswould also be invariant under the following transforma-tions E → B ; B → − E ,ρ e → ρ m ; ρ m → − ρ e , J e → J m ; J m → − J e . (4)Dirac’s monopole theory [1] results into the follow-ing relation between an electric charge ( e ) and magneticcharge ( g ) eg = hc π n = ¯ hc n, ( n = 1 , , , · · · ) (5)where h is the Plank’s constant, ¯ h = h/ π and c is thespeed of light.It should be pointed out that the magnetic monopoleis merely a hypothetical particle whose existence is pos-tulated based on the duality symmetry. The equivalenceof the electric charge (electron) and the magnetic charge(monopole) is explicitly shown in Fig. 1. Interestingly,Dirac linked the magnetic monopoles with the quanti-zation of electric charge by Eq. (5). Such appealingproposal exhilarated a number of theoretical and experi-mental investigations since then. The numerous attemptsof experimental search for these magnetic monopoles ataccelerators and in cosmic rays have been done. Andvarious techniques of detection in the experiments tosearch for magnetic monopole have been developed, forinstance, the magnetometer SQUID. Unfortunately, upto now, no positive evidence for its existence has beenfound. In theoretical physics, ’t Hooft [2] pointed outthat a unified gauge theory in which electromagnetismis embedded in a semisimple gauge group would pre-dict the existence of the magnetic monopole as a solitonwith spontaneous symmetry breaking. Wu and Yang [3]first described magnetic monopoles in terms of a prin-cipal of fiber bundle. Seiberg and Witten [4] developedthe famous magnetic monopole equations. The standardSU(5) model predicts that the magnetic monopoles areextremely heavy with a mass at least 10 GeV/ c (themass about 10 protons), moreover, whose mass is esti-mated to be even higher (up to the Planck mass of 10 GeV/ c ) by the Kaluza-Klein model.What we are most concerned about is why no mag-netic monopoles have been detected after it had beenhypothesized for 77 years. The experimental status ofmonopoles had led Dirac to doubt his theory: “I am in-clined now to believe that monopoles do not exist” [5].In fact, several errors of the Dirac monopole theory havebeen pointed out a long time ago [6, 7, 8]. In this paper,we provide a solid argument that the hypothetical mag-netic monopoles aren’t naturally real or the concept ofmagnetic monopole is only a well-known particle (elec-tron) of different representation. SN - e - g (b) Magnetic monopoles (Dirac's hypothesis)
N SN NN NS SSS + g (c) Positive proton (or nucleus ) and negative electron (Our viewpoint) + e Magnetic Bar ?? (a) FIG. 2: What is the most essential (smallest) component ofa magnet? (a) As a basic knowledge in electromagnetism, nomatter how many times a bar magnet is cut in half, there isalways a north and a south pole. (b) Dirac put forward theidea of the magnetic monopoles: the isolated N -pole (+ g )and the isolated S -pole ( − g ). (c) Our viewpoint is that thesmallest magnet is composed of a single proton (or nucleus)and a single electron. II. MAGNET: ELECTRIC CHARGES ORMAGNETIC CHARGES?
It is well accepted that if a bar magnet is cut in halfrepeatedly, then each half of the magnet becomes a sepa-rate magnet with its own north and south poles, as shownin Fig. 2 (a). Since the discovery of the peculiar featureof the magnetic materials, many people are curious aboutwhat it would look like if there was the smallest magnet,moreover, can the smallest magnet be further isolated?According to Dirac’s opinion, similar to electric charges,there would have net magnetic charges (a magnet withonly one pole) in the universe, as shown in Fig. 2 (b).Although the assumption of the existence of the mag-netic monopoles sounds interesting, there are two fatalproblems with this idea.First, if the hypothetical magnetic particle is true inthe natural world, apparently, there should be plenty ofthe Dirac’s magnetic monopoles inside the permanentmagnet materials. Hence, there is much more possibilityto detect the magnetic monopoles in the magnet mate-rials than in the accelerators and cosmic rays. In ouropinion, no evidence for the monopole’s existence in thesources (magnet materials) for the magnetic monopolesmay indicate that monopoles do not exist at all.Second, assuming there occurs a magnetic to non-magnetic transition in a material, how and where arethe magnetic monopoles going? If there are some mag-netic monopoles escaping from the material during thetransition, as a result, the mass of the material shouldbe greatly reduced due to the theoretical predication ofthe massive magnetic monopoles. Of course, if one con-siders that all the magnetic monopoles still remain in thematerial after the transition, then he has to explain whatare the differences between the monopole’s states beforeand after the transition and why these differences can notbe experimentally detected.From the viewpoint of the objectiveness and rational-ity of physics, when a permanent magnet material is cutin the way of Fig. 2, there is no doubt that ultimatelywe will obtain one positively charged proton and onenegatively charged electron, rather than the hypotheticmagnetic monopoles, as shown in Fig. 2 (c). Now thequestion turn out to be “Can the real particles of protonand electron be used to interpret the extremely commonnatural phenomenon described in Fig. 2 (a)?” In thefollowing sections, we will try to answer this importantquestion in a very intuitive way.
III. MAGNETIC FIELD OR ELECTRIC FIELD?
According to the traditional physics, the magnetic fieldlines of a bar magnet form closed lines. The field directionis taken to be outward from the North pole ( N ) and into the South pole ( S ) of the magnet, as shown in Fig. 3.The magnetic field lines, which can be traced out withthe use of the compasses (see also Fig. 3), are clearlymore concentrated around the two poles of the magnet.Basically, the space with a denser magnetic field linesindicates a stronger magnetic field in that region.If there really exist the monopoles with Dirac magneticcharges + g and − g , then the magnetic field lines associ-ated with a magnetic dipole can be readily obtained, asshown in Fig. 4(a). As a comparison between the mag- N S
The magnetic field lines of a bar magnet (N S)
FIG. 3: The static magnetic field of a bar magnet. The corre-sponding magnetic field lines (the white curves) can be tracedout with the use of the compasses. electric fieldmagnetic field - g (b)(a) Electric dipoleMagnetic monopoles + g - e + e FIG. 4: A comparison of the static magnetic field and thestatic electric field. (a) The theoretical magnetic field lines(the white curves) of a pair of hypothetical monopoles (+ g and − g ), (b) the static electric field lines (the green curves)of the simplest electric dipole consists of one proton (+ e ) andone electron ( − e ). netic field of the artificial magnetic dipole and the electricfield of the real electric dipole, in Fig. 4(b) we plot thewell-known electric field lines for the electric dipole. Sim-ilar to the case of the magnetic dipole of Fig. 4(a), theelectric field lines produced by positive charge + e willend in the negative charge − e . It is not difficult to findthat the two figures are identical. In fact, there is noeffective experimental means which can be used to dis-tinguish between the magnetic field of the so-called mag-netic dipole and the electric field of the electric dipole. Inour opinion, the physical definition of the static magneticfield is essentially an electric-dipole field. Namely, thewidely accepted physical concept of the static magneticfield most likely do not exist in practice, it is thereforeunnecessary to discuss the possible existence of the mag-netic monopoles (the sources of the static magnetic field)in the nature. IV. THE NATURE OF THE STATICMAGNETIC FIELD
In order to make our argument of the nature of thestatic magnetic field sounded, we try to design a bar“magnet” and some compass needles with the positiveand negative charges. As shown in Fig. 5, the “mag-netic” bar and the compasses have a superlattice struc-ture comprising some pairs of layers of positive and neg-ative electric charges. With an appropriate “magnetic”bar (structure, size and shape), the exactly same mag- - q + q - Q The electric field lines of a charge-superlattice bar +Q FIG. 5: The static magnetic field of a bar magnet of Fig. 3can be perfectly generated by a electric bar of the periodicallymodulated quasi-one-dimensional charge superlattice.
S N nucleus (or proton) electron electric dipole SN CompassCompassMagnetic Bar
Cut ( + Q) ( - Q) S N
Cut
NN SS
FIG. 6: The superlattice of Fig. 5 with the alternate posi-tive and negative charges periodic structure can be expectedin some transition metals where the nucleus and the corre-sponding localized d -electron form a electric dipole. All themagnetic properties can be well explained by this picture, asshown in the figure. netic field lines of Fig. 3 can be generated by this peri-odic electric charge bar. At the same time these electriccompasses (see Fig. 5) can play the same role as themagnetic compasses in Fig. 3. Now, the key question has been whether such a periodically modulated chargestructure can exist in real magnetic systems.To the best of our knowledge, why some elements havethe so-called intrinsic magnetic property (IMP) is stillan unsolved problem in condensed matter physics. In ac-cordance with the picture of Fig. 5, it now seems moreclear that, to exhibit the IMP, a quasi-one-dimensionalperiodic structure of the positive and negative chargesmust be naturally formed in the elements (or materials).In some transition metals with the IMP, it is reasonableto assume that each atom contains one nucleus carry-ing one net positive basic charge and one localized d -electron carrying one negative basic charge that form asmallest electric dipole. As shown in Fig. 6, the nuclearand electrons can organize into a electric-dipole crystalwith the alternate positive and negative charges periodicstructure. With the help of this figure, all the so-calledmagnetic properties occurring in nature can be well ex-plained. For example, when a bar of the electric-dipole iscut arbitrarily across the axis direction, each piece alwayshas its own positive charge end (or N -pole) and negativecharge end (or the S -pole), as indicated in Fig. 6. V. CONCLUSION
In this paper, on one hand, we have studied the possi-bility of the existence of the Dirac’s magnetic monopoles,on the other hand, we have attempted to uncover thephysical nature of the static magnetic field generatedby a bar magnet. It was shown clearly that the con-cept of the massive magnetic monopoles is physically un-true. The hypothetical particle is likely to be the well-known electron. This result indicates that any attemptsto search for the magnetic monopole in the universe willbe proved to be in vain. We have found that the tra-ditional static magnetic field of a bar magnet, in fact,is the static electric field of the periodically quasi-one-dimensional electric-dipole superlattice. It seems thatwe had misdefined the static electric field of the electric-dipole lattice as the magnetic field of the magnet (or themagnetic monopoles). Interestingly, this new concept ofperiodic structure of the positive and negative chargesmay proved to be true in some transition metals withthe intrinsic magnetic property. Further related researchis being conducted. [1] P. A. M. Dirac, Proc. Roy. Soc. A , 60 (1931).[2] G. ’t Hooft, Nucl. Phys. B , 276 (1974).[3] T. T. Wu and C. N. Yang, Phys. Rev. D , 384 (1975).[4] N. Seiberg and E. Witten, Nucl. Phys. B , 484 (1994).[5] P. A. M. Dirac, A letter to A. Salam, published in Monopoles in Quantum Field Theory . (World Scientific, Singapore, 1982).[6] D. Zwanziger, Phys. Rev. B , 647 (1965).[7] S. Weinberg, Phys. Rev. B , 988 (1965).[8] C. R. Hagen, Phys. Rev. B140