About maximally localized states in quantum mechanics

We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimension higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variational principle.Comment: accepted for publication in PR

On Nonlocality, Lattices and Internal Symmetries

We study functional analytic aspects of two types of correction terms to the Heisenberg algebra. One type is known to induce a finite lower bound $\Delta x_0$ to the resolution of distances, a short distance cutoff which is motivated from string theory and quantum gravity. It implies the existence of families of self-adjoint extensions of the position operators with lattices of eigenvalues. These lattices, which form representations of certain unitary groups cannot be resolved on the given geometry. This leads us to conjecture that, within this framework, degrees of freedom that correspond to structure smaller than the resolvable (Planck) scale turn into internal degrees of freedom with these unitary groups as symmetries. The second type of correction terms is related to the previous essentially by "Wick rotation", and its basics are here considered for the first time. In particular, we investigate unitarily inequivalent representations.Comment: 6 pages, LaTe

On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta

We continue studies on quantum field theories on noncommutative geometric spaces, focusing on classes of noncommutative geometries which imply ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in positions and momenta. The case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered. The example of euclidean $\phi^4$-theory is studied in detail and in this example we can now show ultraviolet and infrared regularisation of all graphs.Comment: LaTex, 32 page

Nonpointlike Particles in Harmonic Oscillators

Quantum mechanics ordinarily describes particles as being pointlike, in the sense that the uncertainty $\Delta x$ can, in principle, be made arbitrarily small. It has been shown that suitable correction terms to the canonical commutation relations induce a finite lower bound to spatial localisation. Here, we perturbatively calculate the corrections to the energy levels of an in this sense nonpointlike particle in isotropic harmonic oscillators. Apart from a special case the degeneracy of the energy levels is removed.Comment: LaTeX, 9 pages, 1 figure included via epsf optio