In previous work [7], we developed quantum physics on the Moyal plane with time-space noncommutativity, basing ourselves on the work of Doplicher et al. [1]. Here we extend it to certain noncommutative versions of the cylinder, R 3 and R × S 3 . In all these models, only discrete time translations are possible, a result known before in the first two cases [2]-[6]. One striking consequence of quantised time translations is that even though a time independent Hamiltonian is an observable, in scattering processes, it is conserved only modulo 2π θ , where θ is the noncommutative parameter. (In contrast, on a one-dimensional periodic lattice of lattice spacing a and length L = N a, only momentum mod 2π L is observable (and can be conserved).) Suggestions for further study of this effect are made. Scattering theory is formulated and an approach to quantum field theory is outlined.
4-dimensional spin-foam model with quantum Lorentz group J. Math. Phys. 52, 072501 (2011) Attributing sense to some integrals in Regge calculus J. Math. Phys. 52, 022502 (2011) When do measures on the space of connections support the triad operators of loop quantum gravity? J. Math. Phys. 52, 012503 (2011) Some results concerning the representation theory of the algebra underlying loop quantum gravity J. Math. Phys. 52, 012502 (2011) Additional information on J. Math. Phys. We study the free-fall of a quantum particle in the context of noncommutative quantum mechanics ͑NCQM͒. Assuming noncommutativity of the canonical type between the coordinates of a two-dimensional configuration space, we consider a neutral particle trapped in a gravitational well and exactly solve the energy eigenvalue problem. By resorting to experimental data from the GRANIT experiment, in which the first energy levels of freely falling quantum ultracold neutrons were determined, we impose an upper-bound on the noncommutativity parameter. We also investigate the time of flight of a quantum particle moving in a uniform gravitational field in NCQM. This is related to the weak equivalence principle. As we consider stationary, energy eigenstates, i.e., delocalized states, the time of flight must be measured by a quantum clock, suitably coupled to the particle. By considering the clock as a small perturbation, we solve the ͑stationary͒ scattering problem associated and show that the time of flight is equal to the classical result, when the measurement is made far from the turning point. This result is interpreted as an extension of the equivalence principle to the realm of NCQM.
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