Quantum diffusions and the noncommutative torus
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.
Quantum diffusions on the algebra of smooth elements of the noncommutative torus
canonical outer derivations generating the natural action of the torus group on the algebra
cohomological obstruction to the existence of quantum diffusion
scientific article; zbMATH DE number 4050305
scientific article