Un algorithme générateur de structures quantiques. (A productive algorithm for quantical structures).

*(French)*Zbl 0608.58028
Élie Cartan et les mathématiques d’aujourd’hui, The mathematical heritage of Elie Cartan, Semin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 341-399 (1985).

[For the entire collection see Zbl 0573.00010.]

The geometrical quantization is studied by using diffeological spaces. Such a space X is characterized not by local charts but by ”plates” which are maps from the numerical space \(R^ p\) to X. Some theories such as homotopy, fibering, etc. can be introduced in analogy with the manifold geometry. The general diffeological structures are defined and their differential forms on diffeological spaces are studied. The exterior derivative, the pull-backs, etc., are introduced in a ”classical” way. For every diffeological group G one associates the invariant p-forms (by left translation) and the exterior derivative is defined as a cohomology on their direct sum. The Maurer-Cartan forms are classified and it is shown that the non-singular forms (for example the nilpotent forms) are proportional with an ”integral” form. The ”quantum” forms, i.e. nilpotent integral forms with some cohomological obstruction, are considered as a particular class of differential forms of a diffeological space. It is shown that they have quantizations with the property: to every quantization a quantum space can be associated which is a principal fiber-bundle over the circle with an \(\ell\)-form invariant under the action of the structural group. It is also mentioned that some results about the unitary representations of a diffeological group will be given in a forthcoming paper. These representations are fundamental for the quantum description of the matter.

The geometrical quantization is studied by using diffeological spaces. Such a space X is characterized not by local charts but by ”plates” which are maps from the numerical space \(R^ p\) to X. Some theories such as homotopy, fibering, etc. can be introduced in analogy with the manifold geometry. The general diffeological structures are defined and their differential forms on diffeological spaces are studied. The exterior derivative, the pull-backs, etc., are introduced in a ”classical” way. For every diffeological group G one associates the invariant p-forms (by left translation) and the exterior derivative is defined as a cohomology on their direct sum. The Maurer-Cartan forms are classified and it is shown that the non-singular forms (for example the nilpotent forms) are proportional with an ”integral” form. The ”quantum” forms, i.e. nilpotent integral forms with some cohomological obstruction, are considered as a particular class of differential forms of a diffeological space. It is shown that they have quantizations with the property: to every quantization a quantum space can be associated which is a principal fiber-bundle over the circle with an \(\ell\)-form invariant under the action of the structural group. It is also mentioned that some results about the unitary representations of a diffeological group will be given in a forthcoming paper. These representations are fundamental for the quantum description of the matter.

Reviewer: G.Zet