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Non-commutative differential calculus and lattice gauge theory. (English) Zbl 0789.58010
The authors consider consistent deformations of the classical differential calculus on algebras of functions such that differentials and functions satisfy non-trivial commutation relations. Via this deformations a continuum theory is replaced by a corresponding lattice theory. The studying motivation of non-commutative geometry is to remove ultraviolet divergencies in quantum field theories by replacing coordinates by non-commuting operators. The present work shows that such a regularization can be achieved by keeping coordinates commutative, and deforming the differential calculus in such a way that coordinates (or functions) no longer commute with differentials.

MSC:
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
81T05 Axiomatic quantum field theory; operator algebras
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