Technique of tensors with values in the Atiyah-Kähler algebra in the new representation of Dirac-Yang-Mills equations
type: Item
instance of: scholarly article
title: Technique of tensors with values in the Atiyah-Kähler algebra in the new representation of Dirac-Yang-Mills equations
zbMATH Open document ID: 1116.53018
DOI: 10.1134/S106192080603006X
published in: Russian Journal of Mathematical Physics
publication date: 2007-03-26T00:00:00Z
review text: In this paper the technique of so-called \(\Lambda\)-tensor calculus is presented, i.e., a tool which is used in the formulation of relativistic equations of field theory. This is a continuation of former work of the author. Using coordinate notation in a Spin 4-manifold \(X\), the author defines locally \(\Lambda\)-tensors and some natural operations related to them, that lead to the notion of \(\Lambda T\)-algebra. Indeed, the set \(\Lambda\) of 1-forms on \(X\) is endowed, besides the exterior multiplication, with a Clifford multiplication that naturally extends to tensors with values in \(\Lambda\). This is refered as the Atiyah-Kähler algebra \(\Lambda T\). On \(\Lambda T\) one can naturally define the differential, the codifferential and the Hodge star operator. For \(U\in \Lambda\), the first theorem gives the invariant subspaces of the operator \(F_U:\Lambda \rightarrow \Lambda\) defined by \[ F_U(V)=U^* V U. \] The second theorem gives for a pseudo-Riemannian space equipped with a compatible affine connection, the local expression of the affine curvature and some other tensors together with their transformation under a local Pin-transformation. Finally, using the \(\Lambda T\) language and the previous results, the author introduces briefly a Dirac-Yang-Mills equation in the context of 4-dimensional spinor manifold.
reviewed by: Julien Keller
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 83C60
zbMATH DE Number: 5136630
zbMATH Keywords: \(\Lambda\)-tensor calculus
zbMATH Keywords: relativistic
zbMATH Keywords: field theory
zbMATH Keywords: local Pin-transformation
zbMATH Keywords: Dirac-Yang-Mills equation
author: Nikolai Marchuk
MaRDI profile type: Publication
cites work: Dirac \(\gamma\)-equation, classical gauge fields and Clifford algebra
cites work: Dirac--Kähler equation
cites work: Q3994406
cites work: Q4344119
description: scientific article; zbMATH DE number 5136630