Moyal Planes are Spectral Triples

Abstract

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R 2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes-Lott functional action, are given for these noncommutative hyperplanes.

Renormalization of the Orientable Non-commutative Gross–Neveu Model

“…The algebra A Θ of "functions on R 2 Θ " may be defined as S(R 2 ) (it may also be extended to an algebra of tempered distributions, see [6][7][8]30] for rigorous descriptions) endowed with the associative non-commutative Moyal product: The skew-symmetric matrix Θ is…”

“…These last two types of decreasing functions are equivalent to the branch delta functions. The method we use to get the power counting now depends on the topology of the considered graph 7 . We only consider graphs with at least two external legs.…”

Index Theory and Adiabatic Limit in QFT

“…Therefore no superposition exist between (elements of spaces of) in-equivalent irreducible representations of A; thus no "superposition" of two different spacetime "points" can exist; as the "points" of A correspond to equivalence classes of irreducible representations of A. 13 Parameters numbering the irreducible representations are in a one-to-one correspondence with the spectrum of a commutative subalgebra A cl . Assume for a moment (only for heuristic aims) that A cl is a subalgebra of A and therefore it is equal to the center of A.…”

“…12 I mean the well known FAPP-type methods of H.Żurek and his school. 13 Therefore the "parameters" numbering irreducible representations of the spacetime algebra cannot superpose. In passing: also the classical manifold (in the sense: commutative) can be described by a noncommutative algebra Morita equivalent to the commutative algebra of smooth functions on the manifold, compare e.g.…”

“…We rejected those propositions whose construction is based on foliations into Cauchy hyper-surfaces, which seem to be less general. The non-compact riemannian case (non-unital A) is worked out in [13]. Actually first steps has been prepared only in this non-compact direction, but no fundamental difficulties are expected here.…”