On Nonlocality, Lattices and Internal Symmetries

Abstract

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.

6 pages, LaTeX

References in corpus (2)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Gravitational Quantum Limit for Length Measurements

Cited by in corpus (13)

• Matrix D-brane Dynamics, Logarithmic Operators and Quantization of Noncommutative Spacetime
• Generalized uncertainty principles and localization in discrete space
• Spacetime could be simultaneously continuous and discrete in the same way that information can
• A Covariant Information-Density Cutoff in Curved Space-Time
• Mimimal Length Uncertainty Principle and the Transplanckian Problem of Black Hole Physics
• Quantum gravity, information theory and the CMB
• Unsharp Degrees of Freedom and the Generating of Symmetries
• On the only three Short Distance Structures which can be described by Linear Operators
• Optimal uncertainty relations in a modified Heisenberg algebra
• Nonlocality and gravity-induced CP violation
• A framework for nonrelativistic isotropic models based on generalized uncertainty principles
• Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional Quantum Mechanics
• Particle in a cavity in one-dimensional bandlimited quantum mechanics