From symmetries of partial differential equations towards secondary (``quantized'') calculus
A pertinacious criticism of foundations of quantum field theory is performed in rather emotional terms to propose the solution: a general and a somewhat mystical secondary or quantized differential calculus, the only possible language by which various ``renormalizations'', ``anomalities'', ``fantastic creations'', etc., could be deleted. In brief terms, this tool which should resolve all troubles is but the formal theory of infinitely prolonged systems of partial differential equations, so called diffieties. The paper provides a very informal introduction into this realm by using rather elementary and figurative arguments: a vague differential caluclus on infinite-dimensional manifolds equipped with a filtered contact structure with symmetries (the familiar Bäcklund transformations) and infinitesimal symmetries (the ``secondary quantized vector fields''), multiple variational integrals with de Rham homologies organized in spectral sequences, and wave concepts interrelated with characteristics and singularities of solutions. (It should be noted that many contemporary achievements are passed over in full silence.).
scientific article; zbMATH DE number 663780
scientific article