We study application of the Clifford algebra and the Grassmann algebra to image recognitions in (... more We study application of the Clifford algebra and the Grassmann algebra to image recognitions in (3 + 1)D using quaternions. We construct a quaternion based wave function model with fermions and bosons of equal degrees of freedom, similar to Cartan's supersymmetric model. The Clifford algebra A 3,1 is compared with A 2,1 and the model applied to the (2 + 1)D nondestructive testing is extended. The fixed point lattice actions are calculated for 7 paths in (3 + 1)D space with lengths less than or equal to 8 lattice units. Comparison with the quaternion time approach of Ariel, quaternion Fourier transform of Hitzer and the tensor renormalization group approach to classical lattice models are also discussed.
simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analyzed. We assu... more simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analyzed. We assume materials are filled with Weyl spinors located on 256 × 256 2D lattice points, which are expressed by quaternions H. The topology of solitonic phonon propagation is defined by modifying fixed point actions of 4D Quantum Chromo Dynamics action to (2+1)D action, replacing Dirac fermions by Weyl fermions, and changing the electric charge current flow to the energy flow. We consider type loops whose path are on a 2D plane, and type loops which contain two parallel links that connect two 2D plane on different time slices. The length of loops are restricted to be less than or equal to 8 lattice units. At the moment spatial lattice unit and time lattice unit are same. They can be chosen arbitrarily when one compares hysteresis effects with experimental data Using the quaternion expression of Porteous, we calculate the plaquette part of loop actions and the link part of loop actions. Link actions of type loops cancel with each other, but those of type loops depends on whether the spin rotation is clockwise or that is counterclockwise. In the present work we consider average of clockwise rotating and counterclockwise rotating loop contributions.
Proceedings of The 39th International Symposium on Lattice Field Theory — PoS(LATTICE2022)
simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analysed. We assu... more simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analysed. We assume materials are filled with Weyl spinors located on 256 × 256 2D lattice points, which are expressed by quaternions H. The topology of solitonic phonon propagation is defined by modifying fixed point actions of 4D QCD action to (2+1)D action, replacing Dirac fermions by Weyl fermions, and changing the electric charge current flow to the energy flow. We consider A type loops whose path are on a 2D plane, and B type loops which contain two parallel links that connect two 2D plane on different time slices. The length of loops are restricted to be less than or equal to 8 lattice units. At the moment spacial lattice unit and time lattice unit are same. They can be chosen arbitrarily when one compares hysteresis effects with experimental data Using the quaternion expression of Porteous, we calculate the plaquette part of loop actions and the link part of loop actions. Link actions of A type loops cancel with each other, but those of B type loops depends on whetehr the spin rotation is clockwise or that is counterclockwise. In the present work we consider average of clockwise rotating and counterclockwise rotating loop contributions.
The differential cross section for ..pi../sup -/ photoproduction on carbon 12 at 319 MeV/c to 569... more The differential cross section for ..pi../sup -/ photoproduction on carbon 12 at 319 MeV/c to 569 MeV/c is calculated. (SDF)
Proceedings of the 14th International Symposium on Nuclei in the Cosmos (NIC2016), 2017
E. Cartan's supersymmetry in which 4 dimensional spinors couple to 4 dimensional vector is extend... more E. Cartan's supersymmetry in which 4 dimensional spinors couple to 4 dimensional vector is extended to spinors in 5 dimensional Anti de Sitter (AdS) spacetime q = q 1 σ x +q 2 σ y +q 3 σ z +q 4 I±(iβ)e couple to 5 dimensional vector allows a new interpretation of finite temperature electromagnetic phenomena. It explains qualitative difference of pp → Higgs(125GeV) → γγ experiments and pp → S (750GeV) → γγ experiments studied by ATLAS collaboration and CMS collaboration. In this spacetime, cyclic paths of colour freedom of quarks on S 3 sphere appears, and relation of the coupling constants 1/α 1 (∞) ≃ 1/α 2 (−∞) ≃ 6/α 3 (∞) is obtained.
Bulletin of the American Physical Society, Sep 20, 2005
dependence on the polarization due to asymmetry of the lattice, it is observed that c is consiste... more dependence on the polarization due to asymmetry of the lattice, it is observed that c is consistent with 1. Presence of infrared fixed point of α 0 ∼ 2−2.5 irrespective of the fermion actions in the continuum and in the chiral limit is suggested. In comparison with pQCD results in M OM scheme up to the 4-loop level, the MILC data of α s in the region 1 ∼ 3GeV exhibit presence of dimension 2 condensates and dimension 4 condensates with sign opposite to the dimension 2 condensates. We observed that Zwanzigers horizon condition is satisfied within errors. The dimension 2 condensates is interpreted as the square norm of the gauge field which is related to the Zwanziger's horizon condition and the dimension 4 condensates is interpreted as the quark condensates.
The quark pair-creation model or the [sup 3]P[sub 0] model is applied to low-energy [bar p]N anni... more The quark pair-creation model or the [sup 3]P[sub 0] model is applied to low-energy [bar p]N annihilation into mesons and compared with experimental data. In the decay into three mesons, overlaps of resonance bands are important in producing structures in the cos[theta] distribution. The experimentally observed resonances AX(1565), [zeta](1480), and X(1110) could be produced from the overlap of two nonexotic resonance bands. 28 refs., 10 figs., 3 tabs.
Cartan's supersymmetry can be applied not only to electromagnetic interaction but also to weak in... more Cartan's supersymmetry can be applied not only to electromagnetic interaction but also to weak interaction of leptons and quarks. We studied the decay of the possible higgs partner h 0 (11GeV) to ϒ(bb)γ(ℓl), and extended the model to study the decay branching ratios of Higgs boson H 0 (125GeV) to WW , ZZ and γγ. * Speaker.
Effects of the quark field on the ghost propagator of the lattice Landau gauge are investigated b... more Effects of the quark field on the ghost propagator of the lattice Landau gauge are investigated by using the unquenched SU(3) configurations produced by the MILC collaboration and compared with quenched gauge configurations of SU(2) first copy and the parallel tempering (PT) gauge fixed samples and quenched SU(3) 56 4 configurations. We measure the color symmetric and the color antisymmetric ghost propagator and the Binder cumulant of the l 1 norm and the l 2 norm of color antisymmetric ghost propagators and investigate deviation from those of Gaussian distributions. In the first copy samples of quenched SU(2) we observe a large fluctuation in the Binder cumulant at the lowest momentum point. This fluctuation is reduced in the PT gauge fixed samples. The color anti-symmetric ghost propagator of quenched SU(3) configurations depends on the lattice size and is small as compared to the symmetric one in the large lattice of 56 4. The Binder cumulants of the quenched SU(2) and the unquenched SU(3) are almost consistent with 3-d and 8-d Gaussian distribution, respectively. In most cases they are independent of the momentum, but in the case of MILC f configuration of βimp = 7.11, the Binder cumulant has an anomalous momentum dependence. This anomaly is correlated with an anomaly of mass function M (q) of the corresponding quark wave function of bare mass m0 = 27.2MeV. A comparison of the SU(3) unquenched configurations and quenched configurations indicates that the dynamical quarks have the effect of making color antisymmetric ghost propagator closer to the Gaussian distribution and the Kugo-Ojima color confinement parameter c closer to 1.
We made a chaotic memristive circuit proposed by Chua and Muthuswamy-Chua, and analyzed the behav... more We made a chaotic memristive circuit proposed by Chua and Muthuswamy-Chua, and analyzed the behavior of the voltage of the capacitor, electric current in the inductor and the voltage of the memristor by adding an external sinusoidal oscillation of a type γω cos ωt toẏ(t) ≃i L (t), when theẋ(t) ≃v C (t) is expressed by y(t)/C, and studied Devil's staircase route to chaos.
The results of Coulomb gauge and Landau gauge lattice QCD simulation do not agree completely with... more The results of Coulomb gauge and Landau gauge lattice QCD simulation do not agree completely with continuum theory. There are indications that the ghost propagator in the infrared region has strong fluctuation whose modulus is compatible with that of the color diagonal ghost propagator. After presenting lattice simulation of configurations produced with Kogut-Susskind fermion (MILC collaboration) and those with domain wall fermion (RBC/UKQCD collaboration), I investigate in triple gluon vertex and the ghost-gluon-ghost vertex how the square of the color antisymmetric ghost contributes. Then the effect of the vertex correction to the gluon propagator and the ghost propagator is investigated. Recent Dyson-Schwinger equation analysis suggests the ghost dressing function G(0) = finite and no infrared enhancement or α G = 0. But the ghost propagator renormalized by the loop containing a product of color antisymmetric ghost is expected to behave as < cc > r = − G(q 2) q 2 with G(q 2) ∝ q −2α G with α G = 0.5, if the fixed point scenario is valid. I interpret the α G = 0 solution should contain a vertex correction. The infrared exponent of our lattice Landau gauge gluon propagator of the RBC/UKQCD is α D = −0.5 and that of MILC is about-0.7. A possible interpretation of the origin of the fluctuation is given.
The coupling constants and the form factor of NN annihilation into one meson are calculated in th... more The coupling constants and the form factor of NN annihilation into one meson are calculated in the 'P,, model. This allows one to estimate the one-meson annihilation potential for the NN system and to give information on the effect of pair creation in the nucleus. The form factor of relative S-wave NN coupling to an s-wave meson has a zero at f-value around 5 GeV' but the relative D-wave NN coupling to a meson decreases slower than that of S-wave and the sign does not change. We study in the same framework the decay of a meson into two or three mesons. The decay widths of w, p, E, B and A, are calculated. The decays of w, A, and virtual m into three rr are also studied. We include the final-state interaction by multiplying the amplitude of 71~ scattering on the three-meson production amplitudes. The model is tested in the Dalitz-plot distribution of three pions in the decay of A, meson production from 7. The selection rules that are obtained from the annihilation model A2, in which a p is treated as a I@, and from the annihilation model A3, in which a p is the two r interacting in the final state are compared, and the experimental data of NN annihilation into mesons and meson decay into mesons are discussed.
There is a wrong correspondence between the K and K' parameters of the ECQF Hamil-K tonian (3) an... more There is a wrong correspondence between the K and K' parameters of the ECQF Hamil-K tonian (3) and the values given in Fig. 5. In this figure, the values-~-and-K' should be read instead of K and K', respectively.
The experimental data for charged pion electroproduction on '2C and '60 measured at Mainz and Sen... more The experimental data for charged pion electroproduction on '2C and '60 measured at Mainz and Sendai are analyzed in a modified equivalent photon approximation. For the Sendai experiment we use distorted waves for the pion, while for the Mainz experiment we adopt a closure sum with a finite Fermi-gas model. We can roughly reproduce both sets of experimental data. l. Introduction Recently, double differential cross sections for charged pion electroproduction on 12C and 160 near threshold were measured at Mainz 1) and Sendai 2). In the case of
Journal of mathematics and system science, Jan 28, 2018
Einstein claimed that one cannot define global time, and proposed defining local time additionall... more Einstein claimed that one cannot define global time, and proposed defining local time additionally. Such approach was adopted also by E. Cartan, in which fermions are described by spinors with 16 bases and interact with vectors with 8 bases, that consists of a couple of 4 dimensional vectors x i (i = 1, •••, 4) and x i (i = 1, •••, 4). In Cartan's theory, spinors and vectors transform by super symmetric transformations G 23 , G 12 , G 13 , G 123 and G 132 and bases of fermion spinors consist of ξ 0 ,ξ i (i = 1, •••, 4), ξ 1234 , ξ 234 , ξ 134 , ξ 124 , ξ 123 and ξ i,j (i /= j ∈{1,2,3,4}). Except G 23 , the transformations mix spinors and vectors, and operations of G 23 on spinors contain G 23 ξ 4 = ξ 0 and G 23 ξ 123 = ξ 1234 , and operations of G 23 on vectors contain G 23 x 4 = −x 4' and G 23 x 4' = −x 4. Therefore, there are 14 independent spinor bases and 7 independent vector bases, which corresponds to the number of bases of the G 2 symmetry. From the bases of non-commutative geometry, Connes took two fibers from a point of S 3 basis, and on top of fibers allowed two times propagate following von Neumann algebra, but evolution of the system was assumed to be defined by one-parameter group of transformation. Steenrod stated that the S 7 symmetry can be regarded as S 3 symmetry covered over S 4 symmetry, which allows decomposition of S 7 × R 8 → (S 3 × R 4) × (S 3 × R 4). We assume there is a space-time representation by an algebra C(V) of smooth function and matrix algebra M n and transformations A are expressed as A = C(V) M n. In order to make total momentum space to remain 4 dimensional, the group of A becomes SO(3 + n 2 −1,1) SO(3,1) × SO n 2 −1 in Minkowski space. We choose n = 3 and construct SO 8 on R 8 R 4,4. We apply this model to understanding experimentally observed CP violation in p → t or b and in pp → (H →b) + ℓℓ +jets and Time Reversal Based Nonlinear Elastic Wave Spectroscopy (TR-NEWS) method.
We study application of the Clifford algebra and the Grassmann algebra to image recognitions in (... more We study application of the Clifford algebra and the Grassmann algebra to image recognitions in (3 + 1)D using quaternions. We construct a quaternion based wave function model with fermions and bosons of equal degrees of freedom, similar to Cartan's supersymmetric model. The Clifford algebra A 3,1 is compared with A 2,1 and the model applied to the (2 + 1)D nondestructive testing is extended. The fixed point lattice actions are calculated for 7 paths in (3 + 1)D space with lengths less than or equal to 8 lattice units. Comparison with the quaternion time approach of Ariel, quaternion Fourier transform of Hitzer and the tensor renormalization group approach to classical lattice models are also discussed.
simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analyzed. We assu... more simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analyzed. We assume materials are filled with Weyl spinors located on 256 × 256 2D lattice points, which are expressed by quaternions H. The topology of solitonic phonon propagation is defined by modifying fixed point actions of 4D Quantum Chromo Dynamics action to (2+1)D action, replacing Dirac fermions by Weyl fermions, and changing the electric charge current flow to the energy flow. We consider type loops whose path are on a 2D plane, and type loops which contain two parallel links that connect two 2D plane on different time slices. The length of loops are restricted to be less than or equal to 8 lattice units. At the moment spatial lattice unit and time lattice unit are same. They can be chosen arbitrarily when one compares hysteresis effects with experimental data Using the quaternion expression of Porteous, we calculate the plaquette part of loop actions and the link part of loop actions. Link actions of type loops cancel with each other, but those of type loops depends on whether the spin rotation is clockwise or that is counterclockwise. In the present work we consider average of clockwise rotating and counterclockwise rotating loop contributions.
Proceedings of The 39th International Symposium on Lattice Field Theory — PoS(LATTICE2022)
simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analysed. We assu... more simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analysed. We assume materials are filled with Weyl spinors located on 256 × 256 2D lattice points, which are expressed by quaternions H. The topology of solitonic phonon propagation is defined by modifying fixed point actions of 4D QCD action to (2+1)D action, replacing Dirac fermions by Weyl fermions, and changing the electric charge current flow to the energy flow. We consider A type loops whose path are on a 2D plane, and B type loops which contain two parallel links that connect two 2D plane on different time slices. The length of loops are restricted to be less than or equal to 8 lattice units. At the moment spacial lattice unit and time lattice unit are same. They can be chosen arbitrarily when one compares hysteresis effects with experimental data Using the quaternion expression of Porteous, we calculate the plaquette part of loop actions and the link part of loop actions. Link actions of A type loops cancel with each other, but those of B type loops depends on whetehr the spin rotation is clockwise or that is counterclockwise. In the present work we consider average of clockwise rotating and counterclockwise rotating loop contributions.
The differential cross section for ..pi../sup -/ photoproduction on carbon 12 at 319 MeV/c to 569... more The differential cross section for ..pi../sup -/ photoproduction on carbon 12 at 319 MeV/c to 569 MeV/c is calculated. (SDF)
Proceedings of the 14th International Symposium on Nuclei in the Cosmos (NIC2016), 2017
E. Cartan's supersymmetry in which 4 dimensional spinors couple to 4 dimensional vector is extend... more E. Cartan's supersymmetry in which 4 dimensional spinors couple to 4 dimensional vector is extended to spinors in 5 dimensional Anti de Sitter (AdS) spacetime q = q 1 σ x +q 2 σ y +q 3 σ z +q 4 I±(iβ)e couple to 5 dimensional vector allows a new interpretation of finite temperature electromagnetic phenomena. It explains qualitative difference of pp → Higgs(125GeV) → γγ experiments and pp → S (750GeV) → γγ experiments studied by ATLAS collaboration and CMS collaboration. In this spacetime, cyclic paths of colour freedom of quarks on S 3 sphere appears, and relation of the coupling constants 1/α 1 (∞) ≃ 1/α 2 (−∞) ≃ 6/α 3 (∞) is obtained.
Bulletin of the American Physical Society, Sep 20, 2005
dependence on the polarization due to asymmetry of the lattice, it is observed that c is consiste... more dependence on the polarization due to asymmetry of the lattice, it is observed that c is consistent with 1. Presence of infrared fixed point of α 0 ∼ 2−2.5 irrespective of the fermion actions in the continuum and in the chiral limit is suggested. In comparison with pQCD results in M OM scheme up to the 4-loop level, the MILC data of α s in the region 1 ∼ 3GeV exhibit presence of dimension 2 condensates and dimension 4 condensates with sign opposite to the dimension 2 condensates. We observed that Zwanzigers horizon condition is satisfied within errors. The dimension 2 condensates is interpreted as the square norm of the gauge field which is related to the Zwanziger's horizon condition and the dimension 4 condensates is interpreted as the quark condensates.
The quark pair-creation model or the [sup 3]P[sub 0] model is applied to low-energy [bar p]N anni... more The quark pair-creation model or the [sup 3]P[sub 0] model is applied to low-energy [bar p]N annihilation into mesons and compared with experimental data. In the decay into three mesons, overlaps of resonance bands are important in producing structures in the cos[theta] distribution. The experimentally observed resonances AX(1565), [zeta](1480), and X(1110) could be produced from the overlap of two nonexotic resonance bands. 28 refs., 10 figs., 3 tabs.
Cartan's supersymmetry can be applied not only to electromagnetic interaction but also to weak in... more Cartan's supersymmetry can be applied not only to electromagnetic interaction but also to weak interaction of leptons and quarks. We studied the decay of the possible higgs partner h 0 (11GeV) to ϒ(bb)γ(ℓl), and extended the model to study the decay branching ratios of Higgs boson H 0 (125GeV) to WW , ZZ and γγ. * Speaker.
Effects of the quark field on the ghost propagator of the lattice Landau gauge are investigated b... more Effects of the quark field on the ghost propagator of the lattice Landau gauge are investigated by using the unquenched SU(3) configurations produced by the MILC collaboration and compared with quenched gauge configurations of SU(2) first copy and the parallel tempering (PT) gauge fixed samples and quenched SU(3) 56 4 configurations. We measure the color symmetric and the color antisymmetric ghost propagator and the Binder cumulant of the l 1 norm and the l 2 norm of color antisymmetric ghost propagators and investigate deviation from those of Gaussian distributions. In the first copy samples of quenched SU(2) we observe a large fluctuation in the Binder cumulant at the lowest momentum point. This fluctuation is reduced in the PT gauge fixed samples. The color anti-symmetric ghost propagator of quenched SU(3) configurations depends on the lattice size and is small as compared to the symmetric one in the large lattice of 56 4. The Binder cumulants of the quenched SU(2) and the unquenched SU(3) are almost consistent with 3-d and 8-d Gaussian distribution, respectively. In most cases they are independent of the momentum, but in the case of MILC f configuration of βimp = 7.11, the Binder cumulant has an anomalous momentum dependence. This anomaly is correlated with an anomaly of mass function M (q) of the corresponding quark wave function of bare mass m0 = 27.2MeV. A comparison of the SU(3) unquenched configurations and quenched configurations indicates that the dynamical quarks have the effect of making color antisymmetric ghost propagator closer to the Gaussian distribution and the Kugo-Ojima color confinement parameter c closer to 1.
We made a chaotic memristive circuit proposed by Chua and Muthuswamy-Chua, and analyzed the behav... more We made a chaotic memristive circuit proposed by Chua and Muthuswamy-Chua, and analyzed the behavior of the voltage of the capacitor, electric current in the inductor and the voltage of the memristor by adding an external sinusoidal oscillation of a type γω cos ωt toẏ(t) ≃i L (t), when theẋ(t) ≃v C (t) is expressed by y(t)/C, and studied Devil's staircase route to chaos.
The results of Coulomb gauge and Landau gauge lattice QCD simulation do not agree completely with... more The results of Coulomb gauge and Landau gauge lattice QCD simulation do not agree completely with continuum theory. There are indications that the ghost propagator in the infrared region has strong fluctuation whose modulus is compatible with that of the color diagonal ghost propagator. After presenting lattice simulation of configurations produced with Kogut-Susskind fermion (MILC collaboration) and those with domain wall fermion (RBC/UKQCD collaboration), I investigate in triple gluon vertex and the ghost-gluon-ghost vertex how the square of the color antisymmetric ghost contributes. Then the effect of the vertex correction to the gluon propagator and the ghost propagator is investigated. Recent Dyson-Schwinger equation analysis suggests the ghost dressing function G(0) = finite and no infrared enhancement or α G = 0. But the ghost propagator renormalized by the loop containing a product of color antisymmetric ghost is expected to behave as < cc > r = − G(q 2) q 2 with G(q 2) ∝ q −2α G with α G = 0.5, if the fixed point scenario is valid. I interpret the α G = 0 solution should contain a vertex correction. The infrared exponent of our lattice Landau gauge gluon propagator of the RBC/UKQCD is α D = −0.5 and that of MILC is about-0.7. A possible interpretation of the origin of the fluctuation is given.
The coupling constants and the form factor of NN annihilation into one meson are calculated in th... more The coupling constants and the form factor of NN annihilation into one meson are calculated in the 'P,, model. This allows one to estimate the one-meson annihilation potential for the NN system and to give information on the effect of pair creation in the nucleus. The form factor of relative S-wave NN coupling to an s-wave meson has a zero at f-value around 5 GeV' but the relative D-wave NN coupling to a meson decreases slower than that of S-wave and the sign does not change. We study in the same framework the decay of a meson into two or three mesons. The decay widths of w, p, E, B and A, are calculated. The decays of w, A, and virtual m into three rr are also studied. We include the final-state interaction by multiplying the amplitude of 71~ scattering on the three-meson production amplitudes. The model is tested in the Dalitz-plot distribution of three pions in the decay of A, meson production from 7. The selection rules that are obtained from the annihilation model A2, in which a p is treated as a I@, and from the annihilation model A3, in which a p is the two r interacting in the final state are compared, and the experimental data of NN annihilation into mesons and meson decay into mesons are discussed.
There is a wrong correspondence between the K and K' parameters of the ECQF Hamil-K tonian (3) an... more There is a wrong correspondence between the K and K' parameters of the ECQF Hamil-K tonian (3) and the values given in Fig. 5. In this figure, the values-~-and-K' should be read instead of K and K', respectively.
The experimental data for charged pion electroproduction on '2C and '60 measured at Mainz and Sen... more The experimental data for charged pion electroproduction on '2C and '60 measured at Mainz and Sendai are analyzed in a modified equivalent photon approximation. For the Sendai experiment we use distorted waves for the pion, while for the Mainz experiment we adopt a closure sum with a finite Fermi-gas model. We can roughly reproduce both sets of experimental data. l. Introduction Recently, double differential cross sections for charged pion electroproduction on 12C and 160 near threshold were measured at Mainz 1) and Sendai 2). In the case of
Journal of mathematics and system science, Jan 28, 2018
Einstein claimed that one cannot define global time, and proposed defining local time additionall... more Einstein claimed that one cannot define global time, and proposed defining local time additionally. Such approach was adopted also by E. Cartan, in which fermions are described by spinors with 16 bases and interact with vectors with 8 bases, that consists of a couple of 4 dimensional vectors x i (i = 1, •••, 4) and x i (i = 1, •••, 4). In Cartan's theory, spinors and vectors transform by super symmetric transformations G 23 , G 12 , G 13 , G 123 and G 132 and bases of fermion spinors consist of ξ 0 ,ξ i (i = 1, •••, 4), ξ 1234 , ξ 234 , ξ 134 , ξ 124 , ξ 123 and ξ i,j (i /= j ∈{1,2,3,4}). Except G 23 , the transformations mix spinors and vectors, and operations of G 23 on spinors contain G 23 ξ 4 = ξ 0 and G 23 ξ 123 = ξ 1234 , and operations of G 23 on vectors contain G 23 x 4 = −x 4' and G 23 x 4' = −x 4. Therefore, there are 14 independent spinor bases and 7 independent vector bases, which corresponds to the number of bases of the G 2 symmetry. From the bases of non-commutative geometry, Connes took two fibers from a point of S 3 basis, and on top of fibers allowed two times propagate following von Neumann algebra, but evolution of the system was assumed to be defined by one-parameter group of transformation. Steenrod stated that the S 7 symmetry can be regarded as S 3 symmetry covered over S 4 symmetry, which allows decomposition of S 7 × R 8 → (S 3 × R 4) × (S 3 × R 4). We assume there is a space-time representation by an algebra C(V) of smooth function and matrix algebra M n and transformations A are expressed as A = C(V) M n. In order to make total momentum space to remain 4 dimensional, the group of A becomes SO(3 + n 2 −1,1) SO(3,1) × SO n 2 −1 in Minkowski space. We choose n = 3 and construct SO 8 on R 8 R 4,4. We apply this model to understanding experimentally observed CP violation in p → t or b and in pp → (H →b) + ℓℓ +jets and Time Reversal Based Nonlinear Elastic Wave Spectroscopy (TR-NEWS) method.
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Papers by Sadataka Furui