Minkowski Space and Quantum Mechanics

William George Unruh

Hrvoje J. Hrgovčić

László B. Szabados

Abstract Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.

Yu Xuegang

Clifford is the first scientist who gives the logic relation between matter and four-dimensional space.He introduced the hyperbolic imaginary unit,which the complex space corresponds with Minkowski space in relativity theory.Using Clifford algebra,we have abstracted the hyperbolic Hilbert space from Minkowski space,and applied to quantum mechanics.Thus a new philosophical interpretation has been obtained to the move of microscopic foreign body,so much as Schrdinger cat.