Asymptotic generalized extended uncertainty principle
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Regular Article - Theoretical Physics
Asymptotic generalized extended uncertainty principle Mariusz P. D¸abrowski1,2,3,a , Fabian Wagner1,b 1
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland National Centre for Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland 3 Copernicus Center for Interdisciplinary Studies, Szczepa´ nska 1/5, 31-011 Kraków, Poland
2
Received: 9 June 2020 / Accepted: 15 July 2020 © The Author(s) 2020
Abstract We present a formalism which allows for the perturbative derivation of the Extended Uncertainty Principle (EUP) for arbitrary spatial curvature models and observers. Entering the realm of small position uncertainties, we derive a general asymptotic EUP. The leading 2nd order curvature induced correction is proportional to the Ricci scalar, while the 4th order correction features the 0th order Cartan invariant 2 (a scalar quadratic in curvature tensors) and the curved space Laplacian of the Ricci scalar all of which are evaluated at the expectation value of the position operator i.e. the expected position when performing a measurement. This result is first verified for previously derived homogeneous space models and then applied to other non-trivial curvature related effects such as inhomogeneities, rotation and an anisotropic stress fluid leading to black hole “hair”. Our main achievement combines the method we introduce with the Generalized Uncertainty Principle (GUP) by virtue of deformed commutators to formulate a generic form of what we call the Asymptotic Generalized Extended Uncertainty Principle (AGEUP).
1 Introduction The standard uncertainty principle of quantum mechanics in its fundamental form does not take into account effects which are expected to arise from an underlying theory of quantum gravity. Taking inspiration from string theory [1,2], the Heisenberg uncertainty principle was generalized by the inclusion of the gravitational photon-electron interaction (acceleration) leading to the Generalized Uncertainty Principle (GUP) [3–6]. As usually assumed, the GUP takes into account the gravitational uncertainty of position related to the minimum fundamental length scale in physics. However, it may not be the a e-mail:
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only gravitationally induced change. In fact, the curvature of space-time does exert an influence over quantum mechanical uncertainty relations. This is the regime of the Extended Uncertainty Principle (EUP) [7–10] which takes into account the uncertainty related to the background space-time. Both components can be formulated in terms of the standard deviations of position x and momentum p ˆ 2 σx2 = xˆ 2 − x
σ p2 = pˆ 2 − p ˆ 2
and combined to yield the most general Generalised Extended Uncertainty Principle (GEUP) [11,12] α0 l 2p 2 β0 2 h¯ σx σ p ≥ (1) 1 + 2 σ p + 2 σx , 2 rc h¯ where l p denotes the Planck length, h¯ the Planck constant, rc some curvature scale related to the back
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