Non--commutative Integration Calculus

Abstract

We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0\ccr{\eps}{u_1}\cdots\ccr{\eps}{u_n}$ with $\eps$ a grading operator on a Hilbert space $\cH$ and $u_i$ bounded operators on $\cH$ which naturally contains the compactly supported de Rham forms on ${\mathbb{R}}^d$ (i.e.\ $\eps$ is the sign of the free Dirac operator on ${\mathbb{R}}^d$ and $\cH$ a $L^2$--space on ${\mathbb{R}}^d$). We present an elementary proof that the integral of $d$--forms $\int_{\R^d}\trac{X_0\dd X_1\cdots \dd X_d}$ for $X_i\in\Map(\R^d;\gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $ΓX_0\ccr{\eps}{X_1}\cdots\ccr{\eps}{X_d}$ where $\Gamma =1$ for $d$ odd and $\Gamma ={\gamma }_{d+1}$ (`${\gamma }_5$--matrix') a spin matrix anticommuting with $\eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes' non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace.

16 pages, latex

References in corpus (3)

• Fermion Current Algebras and Schwinger Terms in 3+1 Dimensions
• (3+1)-Dimensional Schwinger Terms and Non-commutative Geometry
• QCD(1+1) with massless quarks and gauge covariant Sugawara construction

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• Anomalies and Schwinger terms in NCG field theory models
• The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects
• Noncommutative momentum and torsional regularization
• Higher bracket structure of density operators in Weyl fermion systems and topological insulators