SPIN(8) GAUGE FIELD THEORY Frank D. (Tony) Smith, Jr. P. O. Box 1032 Cartersville, Georgia 30120 U.S.A. (article in Clifford Algebras and Their Applications in Mathematical Physics, Chisholm and Common, eds., Reidel 1986) ABSTRACT. The 16-dimensional spinor representation of Spin(8)reduces to two irreducible 8-dimensional half-spinor representations that can correspond to the 8 fundamental fermion lepton and quark first generation particles and to their 8 antiparticles in a gauge field theory whose gauge group is Spin(8) and whose base manifold is S4. Numerical values for force strength constants and ratios of particle masses to the electron mass are given. No predictions of transitions or decay rates have yet been made. 1. INTRODUCTION. Spin(8) gauge field theory, as defined herein, begins with the Yang Mills principal fibre bundle having gauge group Spin(8) and base manifold s8 described by Grossman, Kephart, and Stasheff (1984). The base manifold s8 is reduced to S4 by a geometric Higgs symmetry breaking mechanism described in Smith (1986). The reduced base manifold S4 corresponds to space-time. The quaternionic structure of S4 naturally corresponds to a (3,3,4,3) lattice structure of space time that has a natural quaternionic structure. The gauge group Spin(8) decomposes at the Weyl group level into Spin(5), SU(3), Spin(4), and the maximal torus U(1)4. The Spin(5) component corresponds to a gauge theory of de Sitter gravitation with a cosmological term as described by MacDowell and Mansouri (1977). The SU(3) component corresponds to a gauge theory of the color force. The Spin(4) = SU(2)xSU(2) component corresponds to a gauge theory of the weak force with a geometric form of spontaneous symmetry breaking from Spin(4) to SU(2). Details are given in Smith (1986). The U(1)4 component corresponds to the four components of the photon in the path integral formulation of quantum electrodynamics. It may be useful to consider the decomposition of Spin(8) in terms of the parameterization of Spin(8) given by Chisholm and Farwell (1984). For spinor matter fields, there is an associated bundle to the principal bundle that is related to the spinor representation of Spin(8) One of the two mirror image 8-dimensional half-spinor spaces, denoted by Q8+, corresponds to the first-generation fermion particles. The other, denoted by Q8_, corresponds to the first-generation anti particles. Q8+ can be parameterized by triples of U and D spinors. Equivalently, it can also be parameterized by the octonions, for which a basis can be taken to be (1, Ol, 02, 03, 04, 05, 06, 07), the real unit 1 and the 7 octonion imaginary units. The octonion parameterization is particularly useful in considering Q8+ as the compact manifold RPlxS7, where RPl corresponds to the real axis and S7 to the imaginary octonions. Compact manifolds are useful in Spin(8) gauge field theory calculations because they have finite volume and ratios of volumes can be well-defined. In Spin(8) gauge field theory, the first-generation fermion particles correspond to triples of spinors or octonions as follows: Fermion: Triple of Spinors: Octonion: electron UxUxU O7 green up quark DxUxU O3 blue up quark UxDxU O2 red up quark UxUxD O1 green down quark UxDxD O6 blue down quark DxUxD O5 red down quark DxDxU O4 neutrino DxDxD 1 The mirror image 8-dimensional half-spinor space corresponds similarly to the first-generation fermion antiparticles. Second and third generation fermions correspond to second-order and third-order direct products of the irreducible half-spinor representations. 2. HISTORICAL BACKGROUND. Armand Wyler (1971) wrote a paper in which he purported to calculate the fine structure constant to be a = 1/137.03608 and the proton electron mass ratio to be mp/me = 6pi5 = 1836.118 from the volumes of homogeneous symmetric spaces. Although the numerical values were close to experimental data, the physical reasons he gave for using the particular volumes he chose were not clear. Freeman Dyson invited Wyler to the Institute for Advanced Study in Princeton for a year to see if Wyler could develop good physical reasons. However, Wyler was primarily a mathematician and did not produce a convincing physical basis for his numerical calculations. With no clear physical basis, Wyler's results were dismissed by many physicists, such as those writing letters in the November 1971 issue of Physics Today, as mere unproductive numerology. As far as I know, no further work was done on Wyler's results. It seemed to me that even though Wyler didn't come up with a good physical basis, there might be one. To look for one, I started by trying to study generalizations of complex manifolds to quaternionic manifolds that have the structure of space-time. Joseph Wolf (1965) wrote a paper in which he classified the 4-dimensional Riemannian symmetric spaces with quaternionic structure. There are just 4 equivalence classes, with the following representatives: T4 = U(1)4 S2xS2 = SU(2)/U(l) x SU(2)/U(l) CP2 = SU(3)/S(U(2)xU(l)) S4 = Spin(5)/Spin(4) Although Wolf's paper was pure mathematics with no attempt at physical application, it seemed to me that the occurrence of the gauge group of electromagnetism U(l), the gauge group of the weak force SU(2), the gauge group of the color force SU(3), and the gauge group of de Sitter gravitation Spin(5) might be physically significant. Another indication of the possible physical significance of quaternionic structure was the paper of David Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser (1963) in which they used quaternionic Structure to construct a geometric spontaneous symmetry breaking mechanism producing two charged massive vector bosons and one massless neutral photon. I then started trying to construct a gauge field theory with a 4-dimensional base manifold having quaternionic structure and a gauge group that would include U(1)4, SU(2)xSU(2) = Spin(4), SU(3), and Spin (5). Such a gauge group should have dimension at least 4+6+8+10= 28 If U(1)4 is taken to be part of a maximal torus, the rank of the gauge group should be at least 4. As Spin(8) has rank 4 and dimension 28, it is a natural candidate. However, it does not even include U(l)xSU(2)xSU(3) as a subgroup. To have U(1)4, Spin(4), SU(3), and Spin(5) included in it, Spin(8) must be decomposed at the Weyl group level rather than decomposed into subgroups. Then, as described in Smith (1986), the 28-dimensional adjoint representation of Spin(8), after reduction to 24 dimensions by the geometric Higgs mechanism, corresponds to the gauge bosons; the 8-dimensional vector representation of Spin(8), after reduction to 4 dimensions by the geometric Higgs mechanism, corresponds to space time; and the 16-dimensional reducible spinor representation of Spin(8) corresponds to the 8 first-generation fermion particles and their 8 antiparticles, each corresponding to a mirror image irreducible 8-dimensional half-spinor representation of Spin (8). 3. CALCULATION OF FORCE STRENGTH CONSTANTS. The relative strengths of the four forces of gravitation, the color force, the weak force, and electromagnetism should be determined in part by thei-r geometric relationships to the base manifold S4 and each half-spinor manifold Q8+ of Spin(8) gauge field theory. The other part of their relative strengths should be determined by considering them to be proportional to the ratio of the square of the electron mass to the square of the characteristic mass, if any, associated with the forces. The electron mass me is the only mass term that is not calculable in Spin(8) guage field theory, in which it is a fundamental quantity like the speed of light and Planck's constant. Only gravitation, involving the Planck mass, and the weak force, involving the sum of the squares of the weak vector boson masses, have mass terms. To calculate the geometric part of the relative force strengths, proceed in three steps. 3.1. Base Manifold Component Of Geometric Part. Each of the four forces has a natural global action on a part of the base manifold S4. Gravitation has gauge group Spin(5), which has a natural global action on Spin(5)/Spin(4) = S4. Therefore the base manifold component of the geometric part of the strength of gravitation is S4, and the volume V(S4) = 8pi2/3. The color force has gauge group SU(3), which has a natural global action on SU(3)/S(U(2)xU(l)) = CP2. CP2 is the same as S4 except for structure at infinity. Therefore the base manifold component of the geometric part of the strength of the color force is S4, and the volume V(S4) = 8pi2/3. The weak force has gauge group Spin(4) = SU(3)xSU(2), but that is reducible by spontaneous symmetry breaking to SU(2), which has a natural global action on SU(2)/U(l) = S2. The base manifold component of the geometric part of the strength of one half of the weak force is S2, which is contained in S4. V(S2) = 4pi. Electromagnetism has gauge group U(1)4, the maximal torus of Spin(8), but that is reducible to U(l) by considering each of the four U(l)'s in U(l)4 to be one space-time component of the photon in the Feynman path-integral formulation of quantum electrodynamics. U(l) has a natural global action on U(l) = Sl. S4 contains Sl. The base manifold component of the geometric part of the strength of one fourth of electromagnetism is Sl, and V(Sl) = 2pi. 3.2. Half-Spinor Component of Geometric Part. Each of the four forces has a natural local action on a part of the half-spinor manifold Q8+ = RPlxS7. Gravitation has gauge group Spin(5), so that it has a natural local action on an irreducible symmetric bounded domain of type IV5, D5+ = Spin(7)/Spin(5)xU(l). D5+ has Silov boundary Q5+ = S4xRPl. Q8+ ContainS Q5+. The half-spinor component of the geometric part of the strength of gravitation is Q5+ = S4xRPl, and V(Q5+) = 8pi3/3. The color force has gauge group SU(3), so that it has a natural local action on an irreducible symmetric bounded domain of type I1,3, D1,3+ = SU(4)/S(U(3)xU(l)) = B6. Dl,3+ has Silov boundary Q1,3+ = S5. Q8+ contains Q1,3+. The half-spinor component of the geometric part of the strength of the color force is Ql,3+ = S5, and V(Q1,3)+ = 4pi3. Each of the SU(2) gauge groups in the Spin(4) of the weak force has a natural local action on an irreducible symmetric bounded domain of type IV3, D3+ = Spin(5)/SU(2)xU(1) D3+ has Silov boundary Q3+ = S2xRPl, which is contained in Q8+. The half-spinor component of the geometric part of the strength of one half of the weak force is Q3+ = S2xRPl, and V(Q3+) = 4pi2. Each of the U(l) gauge groups in the U(1)4 of electromagnetism has a natural local action on U(l) = Sl. Q8+ contains Sl. The half spinor component of the geometric part of the strength of one fourth of electromagnetism is S , and V(S ) = 2pi. 3.3. Dimensional Adjustment Factor of Geometric Part. Note that in some cases the dimension of the base manifold component differs from the dimension of the half-spinor component. In those cases, Spin(8) gauge field theory requires use of a dimensional adjustment factor. The dimensional adjustment factor is intuitively the hardest part of Spin(8) gauge field theory to understand. A similar factor was used by Wyler (1971) in calculating his value of the electromagnetic fine structure constant, and the difficulty in finding a physical interpretation for it was a major factor in criticisms of Wyler's work (Gilmore (1972)). Gravitation has a 5-dimensignal half-spinor component Q5+ =S4xRPl, which is the Silov boundary of D5+. The base manifold component S4 is 4-dimensional. The gravitational dimensional adjustment factor is the fourth root of the volume of D5+, V(D5+)(1/4) = (pi5/2715)(1/4). The color force has a 5-dimensional half-spinor component Ql 3+ = S5, which is the Silov boundary of D1,3+. The base manifold component S4 is 4-dimensional. The color force dimensional adjustment factor is the fourth root of the volume of Dl 3+, V(Dl 3+)(/4) = (pi3/6)(1/4). One half of the weak force has a 3-dimensional half-spinor component Q3+ = S2xRPl, which is the Silov boundary of D3+. The base manifold component S2 is 2-dimensional. The dimensional adjustment factor for one half of the weak force is the square root of the volume of D3+, V(D3+)(1/2) = (pi3/24)(1/2). The electromagnetism half-spinor component and base manifold component are both Sl for one fourth of the electromagnetic force, so no dimensional adjustment factor is needed. 3.4. Final Force Strength Calculation. The geometric part of the force strengths is then: for gravitation, VG = V(S4)V(Q5+)/V(D5+)(1/4) = 3444.0924; for the color force, VC = V(S4)V(Ql,3+)/V(Dl,3+)(1/4) = 2164.978; for one half of the weak force, VW = V(S2)V(Q3 )/V(D3+)(1/2) = 436.46599; and for one fourth of electromagnetism, VE = V(Sl) = 6.2831853. Therefore, mass factors for the weak force and gravitation aside, the relative strengths of the forces are: gravitation, VG/VG = l; the color force, VC/VG = 0.6286062; the weak force, 2VW/VG = 0.2534577; and electromagnetism, 4VE/VG = 1/137.03608. When the mass factors are taken into account, Spin(8) gauge field theory gives the following values for force constants: fine structure constant for electromagnetism = 1/137.608; weak Fermi constant times protQn mass squared = 1.03 x 10(-5); color force constant (at about 10-13 cm.) = 0.6286; and gravitational constant times proton mass squared = 3.4-8.8 x 10(-39). The corresponding experimental values are, respectively: 1/137.03604; 1.02 x 10(-5); about 1; and 5.9 x 10(-39). 4. PARTICLE MASSES AND KOBAYASHI-MASKAWA PARAMETERS. Values for particle masses and Kobayashi-Maskawa parameters can also be calculated in Spin(8) gauge field theory. Particle masses are calculated in terms of the electron mass, which is a fundamental constant in Spin(8) gauge field theory. The masses for quarks are constituent masses. The mass ratio of the down quark to the electron is related to the volume of the compact half-spinor space Q8+ = RPlxS7. The masses of the other leptons and quarks of all three generations come from straightforward consideration of symmetries of their representations as octonions or triples of spinors (for the first generation), as direct products of two octonions or two triples of spinors (for the second generation), and as direct products of three octonions or three triples of spinors (for the third generation). Spin(8) gauge field theory indicates that there should be three generations of weak bosons, with the second generation having masses around 300 to 400 Gev and the third generation having masses around 18 to 22 Tev, It is from the three generations of weak bosons that Spin(8) gauge field theory gives the Kobayashi-Maskawa parameters. Details of the calculations can be found in Smith (1985, 1986). The results are as follows: electron-neutrino mass = 0 (experimentally 0); down quark constituent mass = 312.8 Mev (experimentally about 350 Mev); up quark constituent mass = 312.8 Mev (experimentally about 350 Mev); muon mass = 104.8 Mev (experimentally 105.7 Mev); muon-neutrino mass = 0 (experimentally 0); strange quark constituent mass = 523 Mev (experimentally about 550 Mev); charm quark constituent mass = 1.99 Gev (experimentally about 1.7 Gev); tauon mass = 1.88 Gev (experimentally 1.78 Gev); tauon-neutrino mass = 0 (experimentally 0); charged W mass (first-generation) = 81 Gev (experimentally 81 Gev); neutral W mass (first-generation) = 99 Gev (experimentally 93 Gev); charged W mass (second-generation) = 329 Gev; neutral W mass (second-generation) = 403 Gev; charged W mass (third-generation) = 17.5 Tev; neutral W mass (third-generation) = 21.5 Tev; Planck mass = about 1-1.6 x 1019 Gev (experimentally 1.22 x 1019 Gev); Kobayashi-Maskawa-Chau-Keung sin(x)=0.239 (experimentally 0.23); Kobayashi-Maskawa-Chau-Keung sin(y)=0.0188; Kobayashi-Maskawa-Chau-Keung sin(z)=0.0046 (experimentally 0.005); beauty quark constituent mass = 5.63 Gev (experimentally about 5.2 Gev); truth quark constituent mass = 130 Gev; No experimental values for the second and third-generation W masses are given, because no experiments have been done at the energies needed to observe them directly. The Kobayashi-Maskawa-Chau-Keung parameter sin(y) is related to the truth quark constituent mass. As discussed in Smith (1986), current experimental results are consistent with a value of sin(y) = 0.019 if the truth quark mass is 130 Gev, but if the truth quark mass is 45 Gev, then sin(y) = 0.05. CERN has announced that the truth quark mass is about 45 Gev (Rubbia (1984)), but I think that the phenomena observed by CERN at 45 Gev are weak force phenomena that are poorly explained by the standard SU(2)xU(l) model. I further think that current CP-violation experimental results (Wojcicki (1985)) are consistent with a truth quark mass of about 130 Gev and the Kobayashi-Maskawa-Chau-Keung parameters calculated herein from Spin(8) gauge field theory, and that the CERN value of 45 Gev for the truth quark mass is not consistent with those experimental results. As of the summer of 1985, CERN has been unable to confirm its identification of the truth quark in the 45 Gev events, as the UAl experimenters have found a lot of events clustering about the charged first-generation W mass and the UA2 experimenters have not found anything convincing.(Miller (1985)) I think that the clustering of UAl events near the charged first generation W mass indicates that the events observed are nonstandard weak force phenomena. At this time, I have not yet calculated any transition amplitudes or decay rates. 5. ACKNOWLEDGEMENTS. I would like to thank David Hestenes, David Finkelstein and members of his Georgia Tech seminar, many participants at the 1985 Clifford Algebra workshop at the University of Kent at Canterbury, and the referee for their help and encouragement. 6. REFERENCES. Chisholm, J. S. R., and Farwell, R. S. (1984), Il Nuovo Cimento 82A, 145. Finkelstein, D., Jauch, J., Schiminovich, S., and Speiser, D. (1963), J. Math. Phys. 4, 788. Gilmore, R. (1972), Phys. Rev. Lett. 28, 462. Grossman, B., Kephart, T. W., and Stasheff, J. D. (1984) Commun. Math. Phys. 96, 431. Miller, D. (1985), Nature 317, 110. Rubbia, C. (1984), talk at A.P.S. D.P.F. annual meeting at Santa Fe Smith, F. (1985), Int. J. Theor. Phys. 24, 155. Smith, F. (1986), to be published in Int J. Theor. Phys. Wojcicki, S. (1985), 'Particle Physics' in The Santa Fe Meeting, by T. Goldman and M Nieto (World Scientific, Singapore). Wolf, J. (1965), J. Math. Mech. 14, 1033. Wyler, A. (1971), C. R. Acad. Sci. Paris A272, 186.

......