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Journal ArticleDOI

Hilbert space representation of the minimal length uncertainty relation.

15 Jul 1995-Physical Review D (American Physical Society)-Vol. 52, Iss: 2, pp 1108-1118
TL;DR: 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.
Abstract: 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 which underlies this uncertainty relation. DAMTP/94-105, hep-th/9412167, and Phys.Rev.D52:1108 (1995)
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Journal ArticleDOI
TL;DR: In this paper, it was shown that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of spacetimes whose short-distance structure is governed by an observer-independent length scale.
Abstract: I show that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of spacetimes whose short-distance structure is governed by an observer-independent length scale. The consistency of these postulates proves incorrect the expectation that modifications of the rules of kinematics involving the Planck length would necessarily require the introduction of a preferred class of inertial observers. In particular, it is possible for every inertial observer to agree on physical laws supporting deformed dispersion relations of the type E2-c2 p2-c4m2 + f(E, p, m; Lp) =0, at least for certain types of f.

1,204 citations

Journal ArticleDOI
06 Aug 2009-Nature
TL;DR: The observation of optomechanical normal mode splitting is reported, which provides unambiguous evidence for strong coupling of cavity photons to a mechanical resonator, which paves the way towards full quantum optical control of nano- and micromechanical devices.
Abstract: Achieving coherent quantum control over massive mechanical resonators is a current research goal. Nano- and micromechanical devices can be coupled to a variety of systems, for example to single electrons by electrostatic or magnetic coupling, and to photons by radiation pressure or optical dipole forces. So far, all such experiments have operated in a regime of weak coupling, in which reversible energy exchange between the mechanical device and its coupled partner is suppressed by fast decoherence of the individual systems to their local environments. Controlled quantum experiments are in principle not possible in such a regime, but instead require strong coupling. So far, this has been demonstrated only between microscopic quantum systems, such as atoms and photons (in the context of cavity quantum electrodynamics) or solid state qubits and photons. Strong coupling is an essential requirement for the preparation of mechanical quantum states, such as squeezed or entangled states, and also for using mechanical resonators in the context of quantum information processing, for example, as quantum transducers. Here we report the observation of optomechanical normal mode splitting, which provides unambiguous evidence for strong coupling of cavity photons to a mechanical resonator. This paves the way towards full quantum optical control of nano- and micromechanical devices.

848 citations

Journal ArticleDOI
TL;DR: This work reviews the current status of phenomenological programs inspired by quantum-spacetime research and stresses the significance of results establishing that certain data analyses provide sensitivity to effects introduced genuinely at the Planck scale.
Abstract: I review the current status of phenomenological programs inspired by quantum-spacetime research. I stress in particular the significance of results establishing that certain data analyses provide sensitivity to effects introduced genuinely at the Planck scale. My main focus is on phenomenological programs that affect the directions taken by studies of quantum-spacetime theories.

642 citations


Cites background or methods from "Hilbert space representation of the..."

  • ...[323, 322], which is centered on a modification of the Heisenberg principle...

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  • ...Especially this second point is rather significant since heuristic arguments of the type also used to motivate modified dispersion relations suggest [22, 122, 323, 415, 243, 408] that the Heisenberg commutator might have to be modified in the quantum-spacetime realm....

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Journal ArticleDOI
TL;DR: The question of whether the fundamental laws of nature limit the ability to probe arbitrarily short distances is reviewed, and what insights can be gained from thought experiments for probes of shortest distances are examined.
Abstract: We review the question of whether the fundamental laws of nature limit our ability to probe arbitrarily short distances. First, we examine what insights can be gained from thought experiments for probes of shortest distances, and summarize what can be learned from different approaches to a theory of quantum gravity. Then we discuss some models that have been developed to implement a minimal length scale in quantum mechanics and quantum field theory. These models have entered the literature as the generalized uncertainty principle or the modified dispersion relation, and have allowed the study of the effects of a minimal length scale in quantum mechanics, quantum electrodynamics, thermodynamics, black-hole physics and cosmology. Finally, we touch upon the question of ways to circumvent the manifestation of a minimal length scale in short-distance physics.

628 citations


Cites background from "Hilbert space representation of the..."

  • ...To that end, one changes the measure in momentum space to [175, 184]...

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  • ...It has been shown in [184] that the sharply-peaked wavefunctions with spread below the minimal position uncertainty carry an infinite energy, and thus do not represent a physically-meaningful basis....

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  • ...[175, 174, 184, 178] developed the mathematical basis of quantum mechanics that took into account a minimal length scale and ventured towards quantum field theory....

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  • ...In [184], the maximal localization states have been constructed in one spatial dimension for a 2nd-order expansion of the GUP....

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  • ...In this fashion, in the leading-order approximation, the harmonic oscillator in one dimension has been studied in [184, 167, 8, 125], the harmonic oscillator in arbitrary dimensions in [83, 177, 85, 93], the energy levels of the hydrogen atom in [167, 69, 296, 67, 258], the particle in a box in [7], Landau levels and the tunneling current in [8, 96, 97], the uniform gravitational potential in [244, 83], the inverse square potential in [65, 66], neutrino oscillations in [292], reflection and transmission coefficients of a potential step and potential barrier in [8, 97], the Klein paradox in [124], and corrections to the gyromagnetic moment of the muon in [143, 95]....

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Journal ArticleDOI
TL;DR: In this article, a Generalized Uncertainty Principle (GUP) consistent with string theory, Doubly Special Relativity and black hole physics was proposed, and it was shown that this can predict an upper bound on the quantum gravity parameter in the GUP, from current observations.

552 citations

References
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Book
01 Jan 1964

916 citations