Quantum diffusions and the noncommutative torus
type: Item
instance of: scholarly article
title: Quantum diffusions and the noncommutative torus
zbMATH Open document ID: 0644.46047
DOI: 10.1007/BF00416571
author: B. George
published in: Letters in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: Quantum diffusions on the algebra of smooth elements of the noncommutative torus [\textit{A. Connes}, Noncommutative differential geometry, Publ. Math., Inst. Hautes Etud. Sci. 62, 41-144 (1985; Zbl 0592.46056)] are studied. The diffusion equation is \[ dj_ t(x)=j_ t(\alpha (x))dA^++j_ t(\alpha^+(x))dA+j_ t(\tau (x))dt, \] where \(\alpha\) is a derivation of the algebra, \(\tau\) is a map satisfying \(\tau (xy)=\tau (x)y+x\tau (y)+\alpha^+(x)\alpha (y)\). There are two canonical outer derivations generating the natural action of the torus group on the algebra. When \(\alpha\) is a nontrivial complex linear combination of the canonical outer derivations, no such map \(\tau\) can be found. It is shown that this cohomological obstruction to the existence of quantum diffusion is overcome by extending to a larger algebra, on which the required diffusion exists. An explicit form for this diffusion is given, which is likely to prove of interest in interpreting the spectrum of the generalization of the Landau oscillator Hamiltonian which lies in the noncommutative torus algebra and is thus of significance in solid-state physics.
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 4050305
zbMATH Keywords: Quantum diffusions on the algebra of smooth elements of the noncommutative torus
zbMATH Keywords: canonical outer derivations generating the natural action of the torus group on the algebra
zbMATH Keywords: cohomological obstruction to the existence of quantum diffusion
zbMATH Keywords: extending to a larger algebra
zbMATH Keywords: Landau oscillator Hamiltonian
zbMATH Keywords: solid-state physics
MaRDI profile type: Publication
cites work: Quantum mechanical Wiener processes
cites work: Existence of quantum diffusions
cites work: Q3793211
cites work: Q3034372
cites work: Quantum Ito's formula and stochastic evolutions
full work available at URL: https://doi.org/10.1007/bf00416571
OpenAlex ID: W2057665266
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reviewed by: Alexander S. Holevo
description: scientific article; zbMATH DE number 4050305