Cohomology and

*quanta*. Quantum Observability

To any concept of "traditional" Differential Mathematics - like vector fields, differential forms, etc. - it can be associated in a natural way a cohomological theory called C-Spectral Sequence. Elements from the first term of such a spectral sequence are the analogues in Secondary Calculus of the original objects. In other words, secondary vector fields, secondary differential forms, etc. are elements from the first term of the C-Spectral Sequence of a diffiety. Therefore the solution of the secondarization problem for a concept of Primary Calculus amounts to the construction of the corresponding C-Spectral Sequence after which one must face the technical problem of describing its first term.

Due to what has been said

**all the subjects of Secondary Calculus over a Diffiety can be expressed in terms of the corresponding C-Spectral Sequences**. Thus the associated with it theory of Spectral Sequences and the whole Homological Algebra become an indispensable tool for the modern geometrical theory of the non-linear PDEs.

*A priori*, all of this would have been

**impossible to be foreseen**.

Since the objects of Secondary Calculus are cohomology classes of various differential complexes on diffieties, the identification of the Secondary Calculus with the "Quantum" Calculus means that

**the mathematical formalism which describes the quantum phenomena posses an essentially homological nature**and that the characteristic quantum features - like the non-commutativity - are just aspects of such a nature. Moreover this means that all the quantum quantities have a cohomological nature and so has the quantum observability mechanism. It can be said that this mechanism is analogous to the "observation" mechanism for smooth manifold performed by means of the characteristic classes.

Both theories of Homology and Quantum Mechanics - the two most important new theories of Mathematics and Physics from the last century - were born nearly in the same time about the thirties. It is natural that in that epoch - and also later - it was not possible to suspect the existence of any link between them. The most adapted tool for describing quantum phenomena which could be found in the Mathematics of those years was the theory of self-adjoint operators in Hilbert spaces.

**Such a choice was passively accepted**by the physicians' community due to the

**seductiveness**of the emerging Functional Analysis and the

**authority**of John von Neumann the author of the idea. Yet the theoretical scheme built on Hilbert spaces is not localizable and is also very poor. Once Dirac happened to notice that the physically essential interactions in Fields Theory are so violent that they throw the state vector away from the Hilbert space in the shortest possible time.

Failures were also many attempts during the forties to find a analogous of the Schrodinger Equation in Fields Theory by replacing differential derivatives by the functional ones. Actually these were the first attempts to write secondary differential equations in Quantum Fields Theory. But the

**brutal replacement of the partial derivative by the functional ones**could not take into account the deep cohomological nature of the quantities and operators of Secondary Calculus. And this was

**the main reason for its failure**. Only now it has become more clear that the discovery of the cohomological methods is nothing but the discovery of the "quantum" description in Mathematics.