# The Regular Charge-Monopole Theory at a Glance

Contents
• A brief explanation showing how the well established principles of electrodynamics [1] are used for the construction of the Regular Charge-Monopole Theory (RCMT) [2,3].
• Important features of RCMT.
• A brief description of the relevance of RCMT to strong interactions.
• A comparison between QCD and RCMT demonstrating the net superiority of RCMT.

The Magnetic Monopole Formulation
Ordinary classical electrodynamics is a theory of electromagnetic fields and electrically charged matter. Maxwell equations are the fields’ equation of motion [1]

The Lorentz law of force is the equation of motion of charged particles

Standard notation is used [1] and the speed of light c = 1. The subscript (e) denotes that these equations hold for a system of charges without monopoles.
Duality transformations are defined as follows:

where g denotes the monopole strength. Applying the duality transformations (3) to the equations of motion of ordinary electrodynamics, (1) and (2), one obtains

This theory holds for monopole systems containing no electric charge, as denoted by the subscript (m).
The next problem is to construct a unified charge-monopole theory that is consistent with two sub-theories: for systems of charges without monopoles, it must agree with the ordinary Maxwellian electrodynamics; for systems of monopoles without charges it must agree with the dual theory described above.

For this purpose, RCMT is derived in [2] and, alternatively, in [3] using well known principles of Relativistic Electrodynamics [1], and demonstrates a regular Lagrangian for the particles, a regular Lagrangian density for the fields, the corresponding equations of motion and the fields’ energy-momentum tensor. The analysis relies on a decomposition of the fields into bound and radiation components [4].

Appealing Theoretical Properties of the RCMT
The main results of RCMT are:
• charges do not interact with bound fields of monopoles; monopoles do not interact with bound fields of charges; radiation fields of the systems are identical and charges as well as monopoles interact with them.
• The elementary monopole unit is a free parameter.
• The electric polar dipole differs from the electric axial dipole. Analogous relations hold for the corresponding magnetic dipoles.

These properties have several appealing aspects, three of which are pointed out below.

A. It is consistent with the following theoretical issue. Consider the Lorentz scalar B2−E2 [1]. This quantity is negative for bound fields of a charge; it is positive for bound fields of a monopole; it vanishes for radiation fields emitted from one source. Therefore, only radiation fields of charge-monopole systems are theoretically indistinguishable.

B. The theory is based on a regular Lagrangian density. Therefore, the standard route towards quantum mechanics can be taken.

C. It settles the conceptual problem concerning the asymmetry between electricity and magnetism. Thus, like electric charges, magnetic monopoles do exist and actual differences between electricity and magnetism stem from the size of the elementary unit of charge and monopole.

The description of the hadronic structure is based on the assumption that each quark carries a monopole unit. Thus, mesons are bound quark-antiquark systems. On the other hand, baryons must have a core that carries 3 monopole units. The core attracts the 3 valence quarks, each of which carries one negative monopole unit and the entire baryon resembles a neutral atom. This baryonic structure provides straightforward explanations for phenomena of strongly interacting systems. This section mentions a few examples illustrating this issue.

• This baryonic structure, combined with RCMT’s main results mentioned above, automatically account for the following experimental evidence: electrons (pure charges) do not participate in strong interactions [5] whereas real photons do [6].
• The duality relations between electrodynamics of charges and electrodynamics of monopoles yield similarity of experimental features. In particlar, the electromagnetic Coulomb law shows that the field outside a uniformly charged spherical shell is independent of the radius of the shell. This property yields the rapid decrease of the van der Waals force with the increase of distance. Another aspect of the Coulomb law is atomic screening effect. Illustrations of these principles are given in the following three items, where electromagnetic systems of molecules are compared to nuclear systems.
• The graph describing the distance dependence of the van-der-Waals potential between neutral molecules takes the same form as the graph of the potential of the strong force between nucleons [7,8].
• The uniform density of molecules in a liquid corresponds to the uniform density of nucleons in nuclei (with the exception of some very few light nuclei).
• The 1st EMC effect shows that the self volume of nucleonic valence quarks increases together with the number of nucleons in a nucleus. This effect corresponds to a similar effect in atomic electrons. Both effects stem from electromagnetic screening.
• It is known for about a decade that in an energetic proton-proton collision, both general and elastic cross sections rise with the collision energy [9]. The rise of the elastic cross section shows that the proton contains a solid component that can take the heavy collision and keep the proton’s integrity. This proton property is completely different from the well known electron-proton deep inelastic scattering, where the cross section decreases with increasing energy and the elastic cross section decreases even faster [10]. The dramatic differences of the proton-proton and electron-proton cross section graphs prove that the solid proton’s component is electrically neutral. The fact that this proton property shows up only in very high energy indicates that, beside the valence quarks, the proton has closed shells of quarks. By analogy with the Franck-Hertz experiment, these closed shells of quarks become active only for high enough energy.
• The classical reason for formulating QCD is the existence of the Delta++ baryon which has a total spin J=3/2, even parity and isospin I=3/2. Text books claim that each of the three quarks is in an s-state and that three spin-1/2 fermions cannot be in such a state without having another degree of freedom [11]. Another textbook states that without QCD, the situation is a fiasco (see [5], p. 5). This is simply not true. Take for example the very simple case of the ground state of the He atom, which has two electrons in a total spin J=0 and even parity. Calculations prove that the He atom ground state is written as a linear combination of configurations and that some of which are built from electrons’ single particle wave functions whose spatial angular momentum l>0 [12]. The following arguments show that this property is much stronger in the case of the Delta baryons. Thus, in comparison to a two particle total spin J=0 of the He atom ground state, a three particle total spin J=3/2 allows many more configurations based on different single particle states. Moreover, in hadrons, the relative strength of spin-dependent interactions is much stronger than in an electronic system. An example showing how the Delta++ baryon state can be built from several configurations of three Dirac particles can be found here [13].

A Comparison between QCD and RCMT

 Experimental finding QCD RCMT Quarks’ momentum is about 45% of the nucleon’s momentum Hard-pressed explanation (Gluons take the rest of the momentum) Explained easily (the baryonic core must carry momentum) Hard real photon scattering on nucleons Wrong explanation (VMD contradicts Special Relativity) Easily explained (real photons interact with monopoles) Similarity between the form of the van-der-Waals and the nuclear forces Unexplained Explained easily Nucleon density is uniform in atomic nuclei Unexplained Explained easily 1st EMC effect Unexplained Explained easily Larger volume of anti-quark in a nucleon Unexplained Explained easily Rise in the p-p cross section graph Unexplained Explained (baryons must have inner closed shells of quarks) Pentaquarks have not been found Unexplained Explained easily SQM has not been found Unexplained Explained easily Glueballs have not been found Unexplained Explained easily The neutron’s null electric dipole moment Explained Explained (the polar electric dipole moment vanishes) The nuclear tensor force and its sign Untreated in textbooks Explained easily

References:
[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Elsevier,
Amsterdam, 2005).
[2] E. Comay, Nuovo Cimento 80B, 159 (1984).
[3] E. Comay, Nuovo Cimento 110B, 1347 (1995).
[4] E. Comay, Am. J. Phys. 65, 862 (1997).
[5] F. Halzen and A. D. Martin, Quarks and Leptons, An Introductory Course in
Modern Particle Physics (John Wiley, New York,1984) p. 2.
[6] T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin, Rev. Mod. Phys.
50 261 (1978). (see p. 269).
[7] H. Haken and H. C. Wolf, Molecular Physics and Elements of Quantum Chemistry (Springer, Berlin, 1995). p. 15
[8] S. S. M. Wong, Introductory Nuclear Physics (Wiley, New York, 1998). P. 97
[9] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010). See p. 12 of pdg.lbl.gov/2010/reviews/rpp2010-rev-cross-section-plots.pdf
[10] D. H. Perkins, Introduction to High Energy Physics (Addison-Wesley, Menlo Park, CA, 1987).
[11] F. E. Close, Introduction to Quarks and Partons (Academic Press, London, 1979). (See pp. 9, 338).
[12] A. W. Weiss, Phys. Rev. 122, 1826 (1961).
[13] www.tau.ac.il/~elicomay/d1232_Hilbert.html