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) * supported by Studienstiftung des Deutschen Volkes, BASF-fellow, a

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We study the commutation relations, uncertainty relations and spectra of position and momentum operators within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of minimal nonzero uncertainties in the positions and momenta. The usual quantum mechanical behaviour is recovered as a limiting case for not too small and not too large distances and momenta.DAMPT/93-65 and hep-th/9311147 * supported by Studienstiftung des deutschen Volkes, BASF-fellow

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 rotation and translation invariance can be preserved. An example geometry is studied in detail.

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 about the Planck scale. Here, we propose a framework for field theories on curved backgrounds with a plausible type of ultraviolet cutoff. We find an explicit mechanism by which during cosmic expansion new (comoving) modes are generated continuously. Our results allow the numerical calculation of a prediction for the CMB perturbation spectrum.Several problems of standard big bang cosmology, such as the horizon and the flatness problems, can be explained under the assumption that the very early universe underwent a period of extremely rapid inflation, driven by the potential of some assumed inflaton field. In particular, the inflationary scenario is also able to explain the observed perturbations in the cosmic microwave background (CMB), namely as originating ultimately from quantum fluctuations of the inflaton field. Indeed, the inflationarily predicted gaussianity and near scale invariance of the perturbations' spectrum closely matches the current experimental evidence, see e.g. [1]. However, it has also been pointed out that in typical inflationary models, such as simple models of chaotic inflation, space-time inflated to the extent that modes which are now of cosmological size originated as modes with wavelengths that were at least several orders of magnitude smaller than the Planck length. Until recently, reason to believe that the inflationary prediction of the CMB spectrum might be insensitive to structure at the Planck scale was provided by the analogy with black hole radiation, which suffers from a similar transplanckian problem: any asymptotic Hawking photon with a medium range frequency should have had a far transplanckian proper frequency 1

Abstract. Physical theories are developed to describe phenomena in particular regimes, and generally are valid only within a limited range of scales. For example, general relativity provides an effective description of the Universe at large length scales, and has been tested from the cosmic scale down to distances as small as 10 meters [1,2]. In contrast, quantum theory provides an effective description of physics at small length scales. Direct tests of quantum theory have been performed at the smallest probeable scales at the Large Hadron Collider, ∼10 −20 meters, up to that of hundreds of kilometers [3]. Yet, such tests fall short of the scales required to investigate potentially significant physics that arises at the intersection of quantum and relativistic regimes. We propose to push direct tests of quantum theory to larger and larger length scales, approaching that of the radius of curvature of spacetime, where we begin to probe the interaction between gravity and quantum phenomena. In particular, we review a wide variety of potential tests of fundamental physics that are conceivable with artificial satellites in Earth orbit and elsewhere in the solar system, and attempt to sketch the magnitudes of potentially observable effects. The tests have the potential to determine the applicability of quantum theory at larger length scales, eliminate various alternative physical theories, and place bounds on phenomenological models motivated by ideas about spacetime microstructure from quantum gravity. From a more pragmatic perspective, as quantum communication technologies such as quantum key distribution advance into Space towards large distances, some of the fundamental physical effects discussed here may need to be taken into account to make such schemes viable. arXiv:1206.4949v2 [quant-ph] 5 Oct 2012Fundamental quantum optics experiments conceivable with satellites 2

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 uncertainties in positions do have the power to regularise. These studies are motivated with the ansatz that nonzero minimal uncertainties in positions and in momenta arise from gravity. Algebraic techniques are used that have been developed in the field of quantum groups. * supported by Studienstiftung des Deutschen Volkes, BASF-fellow 1

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 canonical commutation relations. In order to study the full implications on the concept of locality it is crucial to find the physical states of then maximal localisation. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta.