Untitled

Abstract

Einstein claimed that one cannot define global time, and in order to formulate physical dynamics, it is useful to adopt fiber bundle structure. We define topological space E which consists of base space X and fibers F = Π −1 (X), where Π is a projection of an event on the base space. Relations between initial data and final data are defined by group G and a Fiber bundle is defined as as set (E, Π, F, G, X).Tangent bundle TX of real linear space X is defined by the projection π TX = TX → X; (x,a) → a for any a ∈ X and a sphere S n any non negative integer n may be thought to be a smooth submanifold of R n+1 and TS n is identified asConnes proposed that when one adopts non-commutative geometry, one can put two fibers at each point of X and on top of the two fibers define the initial input event and the final detection event. When one considers dynamics of leptons defined by Dirac equation, group G is given by quaternions H, and the base space X is usually taken to be S 3 .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 S 7 . Cayley numbers of S 7 are defined as a 3-sphere bundle over S 4 with group S 3 . Therefore in T X there are two manifolds S 3 × R 4 and S 3 ' × R 4 and the direction of propagation of time on S 3 and S' 3 are not necessarily same. We apply this formulation to experimentally observed violation of time reversal symmetry in p  t process and for understanding the result of time reversal based nonlinear elastic wave spectroscopy (TR-NEWS) in memoducers.