Quantum Gravity 347.0: Italy Relates Symmetries of a Quantum System to Geometry, Algebra, and Analysis (Latter includes Probability!)

Subject:	Quantum Gravity 347.0: Italy Relates Symmetries of a Quantum System to Geometry, Algebra, and Analysis (Latter includes Probability!)
Date:	Fri, 4 Dec 2009 22:41:58 -0800 (PST)

>From Osher Doctorow

G. De Nittis and G. Ponati of SISSA and U. di Roma "La Sapienza"
respectively (SISSA is Sezione Internazionale Superiore di Studi
Avanzati, Trieste Italy) in "The geoemtry emerging from the symmetries
of a quantum system," arXiv: 0911.5270 v2 [math-ph] 29 Nov 2009, 34
pages, find incredible relationships between/among symmetries of a
quantum system, geometry, analysis, and algebra. Analysis, via
Lebesgue Measure and its variants including finite Lebesgue and
Lebesgue-like measures, includes Probability, although the latter is
usually studied as a separate branch of mathematics because it has so
many applications and also because it is the main mathematical tools
of Statistics.

In geometry, the Hilbert Bundle is the concept related to the Analysis
Continuous Field of Hilbert Spaces, while in algebra these are related
to the Hilbert Module. Some or all of these can be looked up online,
for example Wikipedia articles on them.

The paper does not describe "all" physics explicitly, but a good part
of physics with specifically mentioned Quantum Hall Effect,
Macroscopic Polarization, Quantum Pumps, Symmetries induced by finite
grops, periodic systems (periodic crystal electron dynamics), Mathieu-
like Hamiltonians for 1-dimensional torus, etc.

There are some surprises, including the remarkable role played by Z^d,
the Cartesian product of d (integer > 1) spaces of integers Z and the
Bloch-Floquet transform and its generalization which involves a
negative exponential. Z^d automatically brings in the importance of
Combinatorics, which is a "partner" of Probability, and Number Theory
(ditto).

Osher Doctorow