The graded product of real spectral triples

Abstract

Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.

15 pages, no figures

References in corpus (8)

• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Noncommutative Geometry and the standard model with neutrino mixing
• Why the Standard Model
• Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model
• Particle Physics from Almost Commutative Spacetimes
• Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry
• The Standard Model as an extension of the noncommutative algebra of forms
• A new algebraic structure in the standard model of particle physics

Cited by in corpus (7)

• Lorentz signature and twisted spectral triples
• The standard model, the Pati-Salam model, and "Jordan geometry"
• Spectral Noncommutative Geometry, Standard Model and all that
• The geometry of physical observables
• A new algebraic structure in the standard model of particle physics
• Twisted reality and the second-order condition
• Particle models from special Jordan backgrounds and spectral triples