From symmetries of partial differential equations towards secondary (``quantized'') calculus
type: Item
instance of: scholarly article
title: From symmetries of partial differential equations towards secondary (``quantized'') calculus
zbMATH Open document ID: 0815.58028
DOI: 10.1016/0393-0440(94)90005-1
published in: Journal of Geometry and Physics
publication date: 1995-06-14T00:00:00Z
review text: 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.).
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 58A20
Mathematics Subject Classification ID: 81P05
Mathematics Subject Classification ID: 58C99
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 58J47
Mathematics Subject Classification ID: 81S99
Mathematics Subject Classification ID: 81T99
Mathematics Subject Classification ID: 58H10
zbMATH DE Number: 663780
zbMATH Keywords: diffiety
zbMATH Keywords: jet space
zbMATH Keywords: secondary calculus
zbMATH Keywords: quantum field theory
zbMATH Keywords: symmetry
author: A. M. Vinogradov
MaRDI profile type: Publication
full work available at URL: 0393-0440(94)90005-1
OpenAlex ID: W2008522166
Wikidata QID: Q115363093
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