Maximal localization in the presence of minimal uncertainties in positions and in momenta

@article{Hinrichsen1996MaximalLI,
  title={Maximal localization in the presence of minimal uncertainties in positions and in momenta},
  author={Haye Hinrichsen and Achim Kempf},
  journal={Journal of Mathematical Physics},
  year={1996},
  volume={37},
  pages={2121-2137}
}
Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small ‘‘noncommutative geometric’’ corrections to the… 

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References

SHOWING 1-10 OF 46 REFERENCES
Quantum groups and quantum field theory with nonzero minimal uncertainties in positions and momenta
Various regularisation techniques are presently emerging from the field of noncommutative geometry. We focus on the possibility of introducing nonzero minimal uncertainties in positions and momenta
Uncertainty relation in quantum mechanics with quantum group symmetry
The commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann)
Hilbert space representation of the minimal length uncertainty relation.
TLDR
The quantum mechanical structure which underlies the generalized uncertainty relation which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements is studied.
Small-scale structure of spacetime as the origin of the gravitational constant
We suggest a means of incorporating the Planck length as a fundamental constant determined by the structure of spacetime. In this scheme the spacetime symmetry group is taken as the de Sitter group
Quantum groups, gravity, and the generalized uncertainty principle.
  • Maggiore
  • Mathematics, Physics
    Physical review. D, Particles and fields
  • 1994
TLDR
The result indicates that in the $\ensuremath{\kappa}$-deformed Poincar\'e algebra a minimal observable length emerges naturally.
Quantum Gravity and Minimum Length
The existence of a fundamental scale, a lower bound to any output of a position measurement, seems to be a model-independent feature of quantum gravity. In fact, different approaches to this theory
Quantum group symmetric Bargmann–Fock space: Integral kernels, Green functions, driving forces
Raising and lowering operators that transform under the SUq(n)‐quantum group get deformed commutation relations. They are represented as adjoint operators on a Hilbert space of noncommutative
...
...