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P.B. 1029. BLINDERN, OSLO 3, NORWAY Theoretic Papers 1983 Nr 2

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P.B. 1029. BLINDERN, OSLO 3, NORWAY Theoretic Papers 1983 Nr 2 INIS-mf—9691 THEORETIC PAPERS THE BLINDERN THEORETIC RESEARCH TEAM P.B. 1029. BLINDERN, OSLO 3, NORWAY Theoretic Papers 1983 Nr 2 THE MASSES OF ELEMENTARY PARTICLES. APPLICATIONS OP SEMI-CLASSICAL ENERGY LEVELS IN MAGNETIC MONOPOL SYSTEMS by Nils Aall Barricelli Institute of Matbenatics University of Oslo, Blindern, Norway THE MASSES OF ELEMENTARY PARTICLES. APPLICATIONS OF SEMI-CLASSICAL ENERGY LEVELS IN MAGNETIC MONOPOL SYSTEMS. by Nils Aall Barricelli Department of Mathematics, University of Oslo, Blindern, Norway Abstract The semi-classical method of calculating the energy levels in a system of two magnetic monopoles, presented in a preceding paper (Barricelli, 1982) is applied in the interpretation of the masses and angular momenta of various elementary particles. The interpretation involves new proposals concerning the organization of magnetic monopoles (configurations) in various elementary particles. It is worth noticing that no other theory known today has been capable of calculating the masses of so many elementary particles by using so few free parameters (meaning parameters neither expressed in terms of universal constants nor calculated by the theory) and with a precision comparable to that obtained in the model presented here. - 1 - 1. Introduction In a preceding paper (Barricelli, 1982) the problem of calculating the energy levels by semi-classical (Bohr and Sommerfeld) methods in a system of two magnetic monopoles was examined. It was found that if semi-classical theory is applied rigorously the magnetic monopole can not be considered a point charge, and must be assigned a finite (classical) radius rQ related to its magnetostatic mass m and its magnetic charge g by the formula (The ro~radius defined this way is one half of what is commonly called "classical radius". We will call it "semi-classical radius" because it presents some advantages in semi-classical theory). When the magnetic monopoles are ascribed this kind of a finite radius, the energy levels in a system of two magnetic monopoles can easily be calculated by semi-classical theory in spite of their large magnetic charges which according to Dirac are expressed by integer numbers in terms of an elementary charge g (Dirac monopole) about —p— times larger than the charge of the electron e: (2) q _ 137.036 c _ \/137.036fic where c=Speed og light, f^-j^' h being Plank's constant. If we decided to ignore the finite size of the magnetic monopoles and tried to calculate the energy levels assuming a coulombian field all the way to the lowest distances, we would find that no less than 17 energy levels would give for example circular orbits with a radius lower than rQ. The potential and bindings energies at these low levels would have negative values far greater than the magnetostatic energies (or masses) of the two magnetic monopoles orbiting each other. The system would have negativ energy if we use non relativistic formulas, and imaginary or complex energies if we use relativistic formulas. The result would be completly absurd. - 2 - This is the reason of earlier failures to calculate such energy levels by semi-classical theory. If we want to obtain meaningful results, as shown in the preceding paper (Barricelli, 1982) we have to take into account that inside the semi-classical radius rQ the magnetic field does not grow to infinity when the distance r between the two charges approaches 0, but on the contrary the force becomes smaller as a growing portion of the magnetic charge remains out- side and does not contribute to the field. The attraction force goes to 0 and the potential energy for two equal magnetic charges of opposite sign and equal mass (or magnetostatic energy) m goes to -2mc2 when r-*0. This makes it impossible to find an energy level with negative total energy if the potential field is consistent with semi-classical theory because the minimum poten- tial energy -2mc2 is barely sufficient to make up for the sum 2mc2 of the magnetostatic energies of the two magnetic charges. The absurdities which have frustrated earlier attempts to calculate energy levels by semi-classical theory are due to a mistaken application of the theory. In wave mechanics it is customary to consider both electrons and magnetic monopoles as point charges. As long as that approach is used there is perhaps no much possibility of finding meaningful energy levels inside the semi-classical radius rQ. However, if the same potential fields which give meaningful energy levels by a semi-classical approach were used in a wave mechanical(for example a Dirac equation) approach, we see no reason why meaning- ful energy levels could not be obtained also by the wave mechan- ical approach. The argument that a potential with a minimum value -2mc2 can not give negative energies in a system of two magnetic monopoles with masses equal to m is valid for wave mechanics as well as for semi-classical mechanics. A machine program is being developed for the purpose of solving by wave mechanical methods the same problems which have been solved by semi-classical methods. That can give us the possibility to verify whether the wave mechanical methods will give results substantially different from those obtained by semi-classical methods. The purpose of this paper is to show how the possibility of calculating the energy levels in a system of magnetic monopoles can be helpful in the development of a magnetic quark theory for - 3 - the interpretation of the properties of elementary particles. Particularly the masses of elementary particles are a character- istic which has been difficult to calculate earlier in the absence of a method for identifying meaningful energy levels. In order to carry out this program we will, however, have to introduce a set of new assumptions both concerning the nature of the magnetic potential fields inside the semi-classical radius and concerning the way in which quarks and other magnetic monopoles (gluons) are supposedly put together in the various elementary particles. One may agree or disagree on the particular assumptions we are going to introduce and perhaps others will be able to do a better job. But the investigation we are going to present seem to give evidence that the calculation of energy levels made possible by a rigorous application of semi-classical theory is the way to obtain information concerning the possibility of interpreting the masses and other properties of elementary particles by a magnetic monopole theory. While in the earlier paper (Barricelli, 1982) no new assumption was introduced; only a rectification of a common error and a self- consistent application of semi-classical theory was involved, in the present paper we will have to introduce a series of new assumptions in order to achieve an interpretation of elementary particles properties. 2. The quest for a potential distribution The first thing to do in order to identify a suitable dis- tribution was to write machine programs capable of calculating energy levels and other orbital characteristics of two magnetic monopoles for any given asymptotic coulombian potential distribu- tion fulfilling the requirements imposed by semi-classical theory. Two programs, a relativistic and a non relativistic one, whose characteristics are described in the preceding paper (Barricelli, 1982, table 1A,1B and 1C) , solve this problem for circular orbits. Of course the relativistic one is the one which is normally used. The other one has the purpose of verifying the validity of the relativistic one for energy levels involving low velocities compared with the velocity of light. Table 1A Masses of binary systems of magnetic monopoles with respective rest masses M »M9n and the respective magnetic charges S-iig» (s being the Dirao monopole, and — the nonopolar unit of mass). 0 Energy level n-1 Energy level n«2 4M 9 4M 9 «20 «o Q «o "o 0 "o S "2g «10 «1 2 -g -3g -g "2g -3g «o e 0. 08307 1. 07933 4.O8467 0. 19111 1. 187 23 4. 20033 2 1. 07933 0. 1. 18723 0. 1. 12998 4MO « 05671 I.O5279 13884 9M0 3g 4. 08467 1. 05279 0.04416 4. 20033 1. 12998 0. IO952 Table IB Maximum reciprocal distance (rQ»l) reached by the two monopoles in their linear oscillations. Energy level n«1 Energy level n-2 M 4M 9Mo M 9M «20 o O 0 4«0 O "2g -3g «1 g2 -g -g -2g -3g M g 0. 36882 0.29541 0.26802 0. 48906 0.38309 0.34473 0 4Mo 2g 0. 29541 0.22168 0.19500 0. 3830^ 0.284a> 0.2489 i 9M 0. 26802 O.195OO 34473 tr. 24893 o 3g 0.16717 Or. 21268 Table 1C Maximum velocities v./c , vg/c of the two monopoles (c 'beiRg the speed Of light). Energy level n«l Energy level n=2 4Mo 9 M 4M «20 «0 «o o Q 9Mo 4 0 -3g «10 *1 g2 -a« -g -2g -3g M g 0.27955 0. 33946 0. 36895 O.4O042 0.49080 0. 52941 0 T 0.27955 0. O8966 0. 64407 O.4O842 0.13946 0. 06917 4M. 2g v-i/° 0.08986 0. 11848 .0. 13418 0.13943 0.18392 0. 20833 0.33946 0. 11848 0. O6OO7 O.4908O 0.18392 0. 09425 9M0 3g O.O44O7 0. 06 007 0. 06991 0.06917 O.O9425 0. IO98I 0.36895 0. 13418 0. 06991 0.52941 0.20833 0. IO98I * - 4 - It was immediately clear that in order to interpret the low spins of many elementary particles, circular orbits could not be used. Orbits with low angular momentum, often with angular momentum equal to 0, were needed. The next two programs, a relativistic and a non relativistic one, solve the same problem for linear oscillation orbits, which are orbits of angular momentum equal to 0.
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