•  400
    We consider momentum operators on intrinsically curved manifolds. Given that the momentum operators are Killing vector fields whose integral curves are geodesics, it is shown that the corresponding manifold is either flat, or otherwise of compact type with positive constant sectional curvature and dimension equal to 1, 3 or 7. Explicit representations of momentum operators and the associated Casimir element will be discussed for the 3-sphere. It will be verified that the structure constants of t…Read more
  •  298
    We revise the extended uncertainty relations for the Rindler and Friedmann spacetimes recently discussed by Dabrowski and Wagner in [9]. We reveal these results to be coordinate dependent expressions of the invariant uncertainty relations recently derived for general 3-dimensional spaces of constant curvature in [10]. Moreover, we show that the non-zero minimum standard deviations of the momentum in [9] are just artifacts caused by an unfavorable choice of coordinate systems which can be removed…Read more
  •  203
    In this paper, we make a proposal for addressing the cosmological constant problem. Our approach will be based on a reinterpretation of two non-standard de Sitter solutions given by the Einstein vacuum equations with Λ>0. As a first result, we derive an uncertainty principle for both variants of the de Sitter space (Theorem). Subsequently, a decomposition of the cosmological constant in a pair of time-dependent pieces is introduced (Corollary). The time-dependence of the corresponding energy and…Read more
  •  41
    We consider the successive measurement of position and momentum of a single particle. Let P be the conditional probability to measure the momentum with precision dk, given a previously successful position measurement of precision dq. Several upper bounds of the probability P are derived. For arbitrary, but given precisions dq and dk, these bounds refer to the variation of the state vector of the particle. The first bound is given by the inequality P<=dkdq/h, where h is Planck's quantum of action…Read more
  •  54
    A Closer Look at the Uncertainty Relation of Position and Momentum
    with Ingo Hoffmann
    Foundations of Physics 39 (8): 958-963. 2009.
    We consider particles prepared by a single slit diffraction experiment. For those particles the standard deviation σ p of the momentum is discussed. We find out that σ p =∞ is not an exception but a rather typical case. A necessary and sufficient condition for σ p &lt;∞ is given. Finally, the inequality σ p Δx≥π ℏ is derived and it is shown that this bound cannot be improved
  •  20
    Uncertainty Principle on 3-Dimensional Manifolds of Constant Curvature
    Foundations of Physics 48 (6): 716-725. 2018.
    We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature K. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann–Lüders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and K. For hyperbolic spac…Read more