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Mathematical Physics, math-ph,High Energy Physics - Theory, hep-th,Mathematics, Mathematical Physics, math.MP
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.
