Connes’ Distance Function

In classical geometry, the distance between two points p and q is given by the length of any shortest path from p to q. However, this definition is not valid in quantum mechanics, where the concept of path between two points is not well defined.  The idea introduced by Alain Connes in noncommutative geometry consists of defining a spectral distance d ( p , q ) from values taken by operator observables rather than from classical coordinates. In this way, the concept of geometrical point is not used, which allows the spectral distance to be applied to both classical and quantum spaces. Continue reading