Heisenberg and Noncommutative Algebra

CLASSICAL MODEL: the algebra of observable physical quantities can ben directly read from the group G of emitted frequencies.

Since G is a commutative group, the convolution algebra is commutative.

In reality one is not dealing with a group of frequencies, but rather, due to the Ritz-Ridberg combination principle, with a groupoid:

D={(i,j) : i,j \in I}

having the composition rule

(i,j)*(j,k)=(i,k)

The convolution algebra of the groupoid D is none other than the algebra od matrices, since the convolution product may be written

(a,b)_i,k = \sum_i,j a(i,j)b(j,k)

which is identical to the product rule for matrices.

=> REPLACE THE COMMUTATIVE CONVOLUTION ALGEBRA OF THE GROUP G BY THE NON COMMUTATIVE CONVOLUTION ALGEBRA OF THE GROUPOID D.

source: Alain Connes, Non Commutative Geometry