摘要
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.
