Uncertainty relation in quantum mechanics with quantum group symmetry

Kempf, Achim

doi:10.1063/1.530798

Cited by 397 publications

(513 citation statements)

References 13 publications

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“…For example, the ultraviolet regularity of loop graphs in such field theories has been shown. See [28,29,30]. In work by Brout et al, [11], it has been shown that this type of short-distance cut-off without energy-momentum cut-off, when built into quantum theory could resolve the transplanckian energies paradox of black hole radiation -without invoking superoscillations.…”

Section: Discussionmentioning

confidence: 99%

Black holes, bandwidths and Beethoven

Kempf

2000

Journal of Mathematical Physics

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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. It has been claimed that even the recording of an entire Beethoven symphony can occur as part of a signal with 1Hz bandwidth. Bandlimited functions also occur as ultraviolet regularized fields. Their superoscillations have been suggested, for example, to resolve the transplanckian frequencies problem of black hole radiation.Here, we give an exact proof for generic superoscillations. Namely, we show that for every fixed bandwidth there exist functions which pass through any finite number of arbitrarily prespecified points. Further, we show that, in spite of the presence of superoscillations, the behavior of bandlimited functions can be characterized reliably, namely through an uncertainty relation: The standard deviation ∆T of samples φ(t n ) taken at the Nyquist rate obeys: ∆T ≥ 1/4ω max . This uncertainty relation generalizes to variable bandwidths. We identify the bandwidth as the in general spatially variable finite local density of the degrees of freedom of ultraviolet regularized fields.

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Section: Discussionmentioning

confidence: 99%

Black holes, bandwidths and Beethoven

Kempf

2000

Journal of Mathematical Physics

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“…The derivation of this GUP by quantum mechanical tools in the one dimensional space and its generalization to arbitrary dimensional case can be found in a series of papers by Kempf and co-workers [24,[57][58][59]. It is shown in these papers how the quantum mechanics with GUP can be formulated.…”

Section: Particles Creation With Minimal Lengthmentioning

confidence: 99%

Influence of a minimal length on the creation of scalar particles

Haouat

Nouicer

2014

Phys. Rev. D

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In this paper we have studied the problem of scalar particles pair creation by an electric field in the presence of a minimal length. Two sets of exact solutions for the Klein Gordon equation are given in momentum space. Then the canonical method based on Bogoliubov transformation connecting the "in" with the "out" states is applied to calculate the probability to create a pair of particles and the mean number of created particles. The number of created particles per unit of time per unit of length, which is related directly to the experimental measurements, is calculated. It is shown that, with an electric field less than the critical value, the minimal length minimizes the particle creation. It is shown, also, that the limit of zero minimal length reproduces the known results corresponding to the ordinary quantum fields.

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“…[12,13,14,15]. In fact, any theory with this type of ultraviolet cutoff can be written, equivalently, as a continuum theory and as a lattice theory, see [16].…”

Section: Introductionmentioning

confidence: 99%

Minimum length cutoff in inflation and uniqueness of the action

2005

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According to most inflationary models, fluctuations that are of cosmological size today started out much smaller than any plausible cutoff length such as the string or Planck lengths. It has been shown that this could open an experimental window for testing models of the short-scale structure of space-time. The observability of effects hinges crucially, however, on the initial conditions imposed on the new comoving modes which are continually being created at the cutoff length scale. Here, we address this question while modelling spacetime as obeying the string and quantum gravity inspired minimum length uncertainty principle. We find that the usual strategy for determining the initial conditions faces an unexpected difficulty because it involves reformulating the action and discarding a boundary term: we find that actions that normally differ merely by a boundary term can differ significantly when the minimum length is introduced. This is possible because the introduction of a minimum length comes with an ordering ambiguity much like the ordering ambiguity that arises with the introduction ofh in the process of quantization.

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Uncertainty relation in quantum mechanics with quantum group symmetry

Cited by 397 publications

References 13 publications

Black holes, bandwidths and Beethoven

Black holes, bandwidths and Beethoven

Influence of a minimal length on the creation of scalar particles

Minimum length cutoff in inflation and uniqueness of the action

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