Reviews in Mathematical Physics

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although in the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular, in the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d′_L, which coincides exactly with the spectral distance d_D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator — together with their translations — d′_L and d_D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d′_L and d_D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

We study two versions of quasicrystal model, both subcases of Jacobi matrices. For the off-diagonal model, we show an upper bound of dynamical exponent and the norm of the transfer matrix. We apply this result to the off-diagonal Fibonacci Hamiltonian and obtain a sub-ballistic bound for coupling large enough. In the diagonal case, we improve previous lower bounds on the fractal box-counting dimension of the spectrum.

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession. We provide an example of a state together with observables such that ordinary spatial correlations are local, while the spatiotemporal correlations reveal nonlocality. Moreover, we find an extended version of Tsirelson's problem which, for each nontrivial Bell scenario, is equivalent to the QWEP conjecture. This extended version can be conveniently formulated in terms of steering the system of a third party. Finally, a comprehensive mathematical appendix offers background material on complete positivity, tensor products of C*-algebras, group C*-algebras, and some simple reformulations of the QWEP conjecture.