Einstein claimed that one cannot de？ne global time, and proposed de？ning local time additionally. Such approach was adopted also by E. Cartan, in which fermions are described by spinors with 16 bases and interact wi...Einstein claimed that one cannot de？ne global time, and proposed de？ning 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 xi （i = 1, …, 4） and xi （i = 1, …,4）. In Cartan’s theory, spinors and vectors transform by super symmetric transformations G23, G12, G13, G123 and G132 and bases of fermion spinors consist of ξ0,ξi （i = 1, ···, 4）, ξ1234, ξ234, ξ134, ξ124, ξ123 and ξi,j （i /= j ∈{1,2,3,4}）. Except G23, the transformations mix spinors and vectors, and operations of G23 on spinors contain G23 ξ4 = ξ0 and G23 ξ123 = ξ1234, and operations of G23 on vectors contain G23x4 = ？x4＇ and G23 x4＇= ？x4. Therefore, there are 14 independent spinor bases and 7 independent vector bases, which corresponds to the number of bases of the G2 symmetry. From the bases of non-commutative geometry, Connes took two ？bers from a point of S3 basis, and on top of ？bers allowed two times propagate following von Neumann algebra, but evolution of the system was assumed to be de？ned by one-parameter group of transformation.Steenrod stated that the S7 symmetry can be regarded as S3 symmetry covered over S4 symmetry, which allows decomposition of S7 × R8 → （S3 × R4） × （S3 × R4）. We assume there is a space-time representation by an algebra C（V） of smooth function and matrix algebra Mnand transformations A are expressed as A = C（V ） Mn. 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） × SOn^2-1 in Minkowski space. We choose n = 3 and construct SO8 on R8 ？ R4,4. We apply this model to understanding experimentally observed CP violation in pp→ tt or bb and in pp → （H →bb） ＋ ll ＋jets and Time Reversal Based Nonlinear Elastic Wave Spectroscopy （TR-NEWS） method.展开更多
Einstein claimed that one cannot de？ne global time, and in order to formulate physical dynamics, it is useful to adopt ？ber bundle structure. We de？ne topological space E which consists of base space X and ？bers ...Einstein claimed that one cannot de？ne global time, and in order to formulate physical dynamics, it is useful to adopt ？ber bundle structure. We de？ne topological space E which consists of base space X and ？bers F = Π-1（X）, where Π is a projection of an event on the base space. Relations between initial data and ？nal data are de？ned by group G and a Fiber bundle is de？ned as as set （E, Π, F, G, X）.Tangent bundle TX of real linear space X is de？ned by the projection πTX = TX → X; （x,a） → a for any a ∈ X and a sphere Sn any non negative integer n may be thought to be a smooth submanifold of Rn＋1 and TSn is identi？ed as {（x,a） ∈Rn＋1 ×Sn ： x·a = 0} Connes proposed that when one adopts non-commutative geometry, one can put two ？bers at each point of X and on top of the two ？bers de？ne the initial input event and the ？nal detection event. When one considers dynamics of leptons de？ned by Dirac equation, group G is given by quaternions H, and the base space X is usually taken to be S3. E. Cartan studied dynamics of spinors which are described by octonions or Cayley numbers which is an ordered product of two quaternions. The asymptotic form Y of this system is S7. Cayley numbers of S7 are de？ned as a 3-sphere bundle over S4 with group S3. Therefore in T X there are two manifolds S3 × R4 and S3＇ × R4 and the direction of propagation of time on S3 and S＇3 are not necessarily same. We apply this formulation to experimentally observed violation of time reversal symmetry in pp→ tt process and for understanding the result of time reversal based nonlinear elastic wave spectroscopy （TR-NEWS） in memoducers.展开更多
Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conju...Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conjugation in the affine group A(V).展开更多
文摘Einstein claimed that one cannot de？ne global time, and proposed de？ning 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 xi （i = 1, …, 4） and xi （i = 1, …,4）. In Cartan’s theory, spinors and vectors transform by super symmetric transformations G23, G12, G13, G123 and G132 and bases of fermion spinors consist of ξ0,ξi （i = 1, ···, 4）, ξ1234, ξ234, ξ134, ξ124, ξ123 and ξi,j （i /= j ∈{1,2,3,4}）. Except G23, the transformations mix spinors and vectors, and operations of G23 on spinors contain G23 ξ4 = ξ0 and G23 ξ123 = ξ1234, and operations of G23 on vectors contain G23x4 = ？x4＇ and G23 x4＇= ？x4. Therefore, there are 14 independent spinor bases and 7 independent vector bases, which corresponds to the number of bases of the G2 symmetry. From the bases of non-commutative geometry, Connes took two ？bers from a point of S3 basis, and on top of ？bers allowed two times propagate following von Neumann algebra, but evolution of the system was assumed to be de？ned by one-parameter group of transformation.Steenrod stated that the S7 symmetry can be regarded as S3 symmetry covered over S4 symmetry, which allows decomposition of S7 × R8 → （S3 × R4） × （S3 × R4）. We assume there is a space-time representation by an algebra C（V） of smooth function and matrix algebra Mnand transformations A are expressed as A = C（V ） Mn. 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） × SOn^2-1 in Minkowski space. We choose n = 3 and construct SO8 on R8 ？ R4,4. We apply this model to understanding experimentally observed CP violation in pp→ tt or bb and in pp → （H →bb） ＋ ll ＋jets and Time Reversal Based Nonlinear Elastic Wave Spectroscopy （TR-NEWS） method.
文摘Einstein claimed that one cannot de？ne global time, and in order to formulate physical dynamics, it is useful to adopt ？ber bundle structure. We de？ne topological space E which consists of base space X and ？bers F = Π-1（X）, where Π is a projection of an event on the base space. Relations between initial data and ？nal data are de？ned by group G and a Fiber bundle is de？ned as as set （E, Π, F, G, X）.Tangent bundle TX of real linear space X is de？ned by the projection πTX = TX → X; （x,a） → a for any a ∈ X and a sphere Sn any non negative integer n may be thought to be a smooth submanifold of Rn＋1 and TSn is identi？ed as {（x,a） ∈Rn＋1 ×Sn ： x·a = 0} Connes proposed that when one adopts non-commutative geometry, one can put two ？bers at each point of X and on top of the two ？bers de？ne the initial input event and the ？nal detection event. When one considers dynamics of leptons de？ned by Dirac equation, group G is given by quaternions H, and the base space X is usually taken to be S3. E. Cartan studied dynamics of spinors which are described by octonions or Cayley numbers which is an ordered product of two quaternions. The asymptotic form Y of this system is S7. Cayley numbers of S7 are de？ned as a 3-sphere bundle over S4 with group S3. Therefore in T X there are two manifolds S3 × R4 and S3＇ × R4 and the direction of propagation of time on S3 and S＇3 are not necessarily same. We apply this formulation to experimentally observed violation of time reversal symmetry in pp→ tt process and for understanding the result of time reversal based nonlinear elastic wave spectroscopy （TR-NEWS） in memoducers.
文摘Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conjugation in the affine group A(V).
