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- Achim Kempf
- 1994

We introduce an algebraic theory of integration on quantum planes and other braided spaces. In the one dimensional case we obtain a novel picture of the Jackson q-integral as indefinite integration on the braided group of functions in one variable x. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. We show that the… (More)

The existence of a minimal observable length has long been suggested in quantum gravity as well as in string theory. In this context a generalized uncertainty relation has been derived which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements. Here we study in full detail the quantum mechanical structure… (More)

- Achim Kempf
- 2000

In most inflationary models, space-time inflated to the extent that modes of cosmological size originated as modes of wavelengths at least several orders of magnitude smaller than the Planck length. Recent studies confirmed that, therefore, inflationary predictions for the cosmic microwave background perturbations are generally sensitive to what is assumed… (More)

- Achim Kempf
- 1999

It is usually believed that a function φ(t) whose Fourier spectrum is bounded can vary at most as fast as its highest frequency component ω max. This is in fact not the case, as Aharonov, Berry and others drastically demonstrated with explicit counter examples, so-called superoscillations. The claim is that even the recording of an entire Beethoven symphony… (More)

- Achim Kempf
- 2008

The existence of a natural ultraviolet cutoff at the Planck scale is widely expected. In a previous Letter, it has been proposed to model this cutoff as an information density bound by utilizing suitably generalized methods from the mathematical theory of communication. Here, we prove the mathematical conjectures that were made in this Letter.

As a potentially accessible window to aspects of Planck scale physics it has been pointed out that the perturbation spectrum predicted by inflation may be sensitive to a natural ultraviolet cutoff. A fairly general classification of the possible short-distance cutoffs that one may encounter at the Planck scale has also recently been given. Indeed, various… (More)

Studies in string theory and quantum gravity suggest the existence of a finite lower limit ∆x 0 to the possible resolution of distances, at the latest on the scale of the Planck length of 10 −35 m. Within the framework of the euclidean path integral we explicitly show ultraviolet regularisation in field theory through this short distance structure. Both… (More)

- Achim Kempf
- 1994

A noncommutative geometric generalisation of the quantum field theoretical framework is developed by generalising the Heisenberg commutation relations. There appear nonzero minimal uncertainties in positions and in momenta. As the main result it is shown with the example of a quadratically ultraviolet divergent graph in φ 4 theory that nonzero minimal… (More)

- Achim Kempf
- 1996

Quantum mechanics ordinarily describes particles as being pointlike, in the sense that the uncertainty ∆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 per-turbatively calculate the corrections to the energy… (More)