2. Secondary Differential Calculus

In some sense diffieties are smooth manifolds with possible singularities and usually they have infinite dimension. This last fact does not allow to develop on these objects a reasonable Differential Calculus by using the traditional methods of Analysis and classical Differential Geometry. Yet such difficulties are easily overcome by using the algebraic approach to Differential Calculus—the aforementioned Primary Differential Calculus. Thus Primary Differential Calculus becomes the indispensable basis for building a consistent reasonable differential calculus on diffieties. That part of the Primary Calculus on diffieties which in some sense is respectful of the natural geometrical structures on diffiety is called Secondary Differential Calculus. Traditional theory of varieties—both smooth and algebraic—appears in the theory of diffieties as its 0-dimensional case. This means that the Secondary Calculus respects the mathematical correspondence principle. In other words, the traditional differential mathematics can be interpreted as the limit case when the diffiety-dimension goes to 0. [...]
The mathematical correspondence principle concretized in this way shows that any natural concept of the Differential Calculus of Classical Differential Geometry admits an analogue in Secondary Calculus. Therefore the general mathematical analogue  of the quantization problem becomes the secondarization problem—the "mathematical quantization". In other words, the latter is the problem of constructing the analogues in Secondary Calculus of all the concepts and natural operations of all the traditional "differential mathematics". The secondarization problem being more general and thus more transparent than the quatization problem in Physics admits an exact formalization which will be discussed later on. Presently the state of the art is such that in any concrete case the secondarization problem is an algorithmic problem, which does not mean that the realization of the corresponding algorithm is easy and immediate, but it is important to stress that this is not a problem of principle. Now the complete solution to the secondarization problem for the basis components of the classical fields theory is just a question of time.  Due to the huge dimensions of the finishing works, consistent human resources are needed for it, but it is already possible to clearly foresee that the quantum fields theory will become in a not so distant future the study of some secondary differential equations. Still now some touching point of Secondary Calculus with some branches of contemporary Theoretical Physics like BRST transform and anti-field formalism, can be found. These two chapters of Physics—sometimes referred to as Cohomological Physics—can be described in the language of Secondary Calculus in a natural and conceptually clear way, in spite of the fact that the latter has been developed in an absolutely independent way from any problem of Theoretical Physics.

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