Noncommutative integration calculus

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Abstract
A noncommutative integration calculus arising in the mathematical description of Schwinger terms of fermion–Yang–Mills systems is discussed. The differential complexes of forms u0[ε,u1]...[ε,un] with ε a grading operator on a Hilbert space ℋ and ui bounded operators on ℋ which naturally contains the compactly supported de Rham forms on Rd (i.e., ε is the sign of the free Dirac operator on Rd and ℋ is a L2-space on Rd) are considered. An elementary proof is presented in which the integral of d-forms ∫RdtrN(X0dX1...dXd) for Xi∈C0∞(Rd;glN) is equal, up to a constant, to the conditional Hilbert space trace of ΓX0[ε,X1]...[ε,Xd] where Γ=1 for d odd and Γ=γd+1 (‘γ5-matrix’) a spin matrix anticommuting with ε for d even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes’ noncommutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace.
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