On a mathematical hypothesis of quantum space-time

A space-time in field theory, except noncommutative field theory, is traditionally described as a finite-dimensional smooth manifold, locally homeomorphic to an Euclidean topological space E. The following fact enables us to think that a space-time might be a wider space of Schwartz distributions on E.

Let E be an Euclidean topological space. Let D(E) be a space of smooth complex functions F of compact support on E. The space of continuous forms on D(E)  is the space D'(E) of Schwartz distributions on E, which includes the subspace T(E) of Dirac’s delta-functions dl_x such that, for any function F on E, we have dl_x(F)=F(x).

A key point is that there exists a homeomorphism x->dl_x of E onto the subset T(E) of delta-functions of D'(E). Moreover, the injection E-> T(E)-> D'(E) is smooth. Therefore, we can identify E with a topological subspace E=T(E) of the space of Schwartz distributions. Herewith, any smooth function F of compact support on E= T(E) is extended to a continuous form

F’(dl_x+w)=F(x) + F’(w)

on the space of Schwartz distributions D'(E). One can think of this extension F’ as being a quantum deformation of F.

In quantum models, one therefore should replace integration of functions over E with that over D'(E).

Reference:
G.Sardanashvily, On the mathematical origin of quantum space-time, arXiv: 0709.3475