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

## No comments:

## Post a Comment