The category of differential equations and its significance for physics. (English) Zbl 0558.35060

Vinogradov, A. M.

Differential geometry and its applications, Proc. Conf., Nové Město na Moravě/Czech. 1983, Pt. 2, 289-301 (1984).

[For the entire collection see Zbl 0546.00021.]
In this lecture we consider the following very general but, as we hope, not very trivial problem: What mathematics is necessary for Quantum Field Theory?
Our answer may be shortly formulated as follows: (1) This sort of mathematics is to be discovered and developed. (2) The classical differential calculus is the mathematics for the classical physics (if the word calculus is interpreted widely enough to include the calculus of partial differential operators, partial differential equations, etc.). The quantum physics requires a differential calculus of a next (secondary) level. (3) The category of differential equations is the tool to build the secondary differential calculus.

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35A30	Geometric theory, characteristics, transformations in context of PDEs
81P05	General and philosophical questions in quantum theory

Keywords:

Quantum Field Theory; quantum physics; category of differential equations; secondary differential calculus

Citations:

Zbl 0546.00021