Asymptotic generalized extended uncertainty principle

Eur. Phys. J. C (2020) 80:676 https://doi.org/10.1140/epjc/s10052-020-8250-x Regular Article - Theoretical Physics Asymptotic generalized extended uncertainty principle Mariusz P. Da̧browski1,2,3,a , Fabian Wagner1,b 1 Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland 2 National Centre for Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland 3 Copernicus Center for Interdisciplinary Studies, Szczepańska 1/5, 31-011 Kraków, Poland 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: (corresponding author) b e-mail: 0123456789().: V,-vol 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 σx2 = x̂ 2  − x̂2 σ p2 =  p̂ 2  −  p̂2 and combined to yield the most general Generalised Extended Uncertainty Principle (GEUP) [11,12]   α0 l 2p 2 β0 2 h̄ σx σ p ≥ (1) 1 + 2 σ p + 2 σx , 2 rc h̄ where l p denotes the Planck length, h̄ the Planck constant, rc some curvature scale related to the background space-time and α0 and β0 the GUP- and EUP-parameters, respectively. The GEUP as given by (1) has its heuristic Newtonian analogue [12] which represents the acceleration of an electron induced by both the gravitational interaction of a photon of energy E and the cosmological Hubble horizon r H = c/H = (/3)1/2 of de Sitter space  2 2  ¨r = − G(E/c ) + c r r , (2) r2 3 r with the cosmological constant , the photon-electron distance r, the Hubble parameter H, the gravitational constant G and the speed of light c (the latter two will be set equal to 1 throughout the remainder of this paper). A derivation of the EUP based on the notion of geodesic balls was performed in Refs. [13,14]. It reflects the influence of spatial curvature on quantum-mechanics on 3-dimensional spacelike hypersurfaces of space-time. The method was applied to homogeneous and isotropic geometries of constant curvature K and the corresponding EUP was calculated. In our recent paper [15] we applied this method to calculate the EUP for Rindler and Friedmann horizons and 123 676 Page 2 of 14 Eur. Phys. J. C obtained corrections to the Hawking temperature and Bekenstein entropy of black holes. In said derivation emphasis was put on the local observer in an accelerated frame or at the center of symmetry seeing the effects related to such a specific choice of frame. Yet, a coordinate-independent (covariant) relation for the EUP was the same as derived earlier in [14]. This was further commented on in Ref. [16]. The purpose of this paper is to derive a general expression for the EUP in the case of small position uncertainties (to be specified below) and thus consider non-trivial (e.g. nonhomogeneous) effects of curvature on the EUP. The paper is organized as follows. Section 2 outlines a method of deriving the EUP as it was applied to homogeneous (constant curvature) spaces in Ref. [14]. In Sect. 3 we present our perturbative approach to apply it further to small position uncertainties in Sect. 4. Moreover, in Sect. 5 several types of perturbations are discussed to provide exemplary applications of the asymptotic EUP to various geometries of non-trivial curvature. Finally, Sect. 7 is intended to summarize our results. σp =  (2020) 80:676    p̂ 2  = h̄ − ψ| |ψ ≥ h̄ λ1 (5) with the 1st eigenvalue of the eigenvalue problem (3). Multiplying by ρ, the uncertainty relation is obtained. In Ref. [14] it was found that the uncertainty relation for Riemannian 3-manifolds of constant curvature K reads  K (6) σ p ρ ≥ π h̄ 1 − 2 ρ 2 . π Note that the uncertainty relation derived this way is not of the same kind as the one described by (1) because it features the characteristic length of confinement ρ, a generalisation of Heisenberg’s slit width. Thus, ρ should rather be interpreted as uncertainty and does not represent the standard deviation of position. 3 Weak curvature EUP formalism Consider a generic weak curvature effect on spacelike 3Dhypersurfaces, i.e. for a spatial metric splitting 2 Background geometry determined EUP (1) ds 2 = gi(0) j + gi j + The reasoning behind Refs. [13,14] restricts our perspective to spacelike hypersurfaces of the underlying space-time (it is impossible to even define a mathematically sound uncertainty relation in standard quantum mechanics otherwise). Thus, as other effects which combine quantum mechanics and gravity, this method is observer dependent. Consider a free wave function ψ living on said Riemannian manifold but confined to a geodesic ball (Bρ ) of radius ρ. Due to their invariance under diffeomorphisms, geodesic balls are the natural generalization of the Heisenberg slit to curved manifolds where ρ corresponds to the slit width thus providing a measure of position uncertainty. Confinement to this domain is ensured by imposing Dirichlet boundary conditions on the wave function. As the Laplace–Beltrami-operator (repre (...truncated)