Finite quantum physics and noncommutative geometry

Remarks on Quantum Physics and Noncommutative Geometry

The quantum-event / prime ideal in a category/ noncommutative-point alternative to classical-event / commutative prime ideal/ point is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered as mathematical tools appropriate to model quantum mechanics. The space-time framework is to be reconstructed from the spectrum of the path category of a quiver....

Noncommutative Geometry and Physics

The two existing theories which successfully encode our knowledge of space-time are : • General Relativity • The Standard Model General relativity describes space-time as far as large scales are concerned (cf. Figure 1 and [2] for many more suggestive thoughts and pictures) and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincaré, Einstein, ...

Classification of Differentials on Quantum Doubles and Finite Noncommutative Geometry

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the ‘quantum groups approach’. We apply the author’s previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum codouble C(G)◮<CG = D∗(G) of a finite group G. We give D∗(S3) as an example. We ...

Quantum Groups and Noncommutative Geometry

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birt...

Noncommutative Geometry and Quantum Groups