We revisit the Extended Uncertainty Principle (EUP) from an operational viewpoint, replacing wavefunction-based widths with apparatus-defined position constraints such as a finite slit of width Δx or a geodesic ball of radius R. Using Hermitian momentum operators consistent with the EUP algebra, we prove a sharp lower bound on the product of momentum spread and preparation size in one dimension and show that it reduces smoothly to the standard quantum limit as the deformation vanishes. We then extend the construction to dimensions two and three on spaces of constant curvature and obtain the corresponding bound for spherical confinement, clarifying its geometric meaning via an isometry to S2 and S3. The framework links curvature-scale effects to operational momentum floors and suggests concrete tests in diffraction, cold-atom, and optomechanical settings.
We revisit the Extended Uncertainty Principle (EUP) from an operational viewpoint, replacing wavefunction-based widths with apparatus-defined position constraints such as a finite slit of width Δx or a geodesic ball of radius R. Using Hermitian momentum operators consistent with the EUP algebra, we prove a sharp lower bound on the product of momentum spread and preparation size in one dimension and show that it reduces smoothly to the standard quantum limit as the deformation vanishes. We then extend the construction to dimensions two and three on spaces of constant curvature and obtain the corresponding bound for spherical confinement, clarifying its geometric meaning via an isometry to S2 and S3. The framework links curvature-scale effects to operational momentum floors and suggests concrete tests in diffraction, cold-atom, and optomechanical settings.
In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a region σ̃g>0 in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by setting σg≃ℏG/c3 , the Planck length, and σ̃g≃ℏΛ/3 , where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.
A momentum-dependent translation operator, inspired by the position-dependent translation operator (PDTO) framework, is introduced to describe a particle in a minimum-length scenario. We show that a chosen momentum-dependent translation profile G ( p ) induces a deformed commutation relation between position and momentum and the corresponding Generalized Uncertainty Principle (GUP), providing a structural construction rather than postulating the commutator independently. The construction is the momentum-space dual of the position-space operator that yields the Extended Uncertainty Principle, so that EUP and GUP emerge as the two faces of a single translation principle. We then adapt the split-operator method, using the eigenfunctions of the momentum operator modulated by a Gaussian in the representation of Brau and Pedram, to describe a particle under three potentials: free, linear, and quadratic. In all cases the quantum mean values track the classical dynamics obtained from the modified Poisson brackets, with a leading correction proportional to β ⟨ p ⟩(Δ p ) 2 that we identify analytically. Propagating a classical ensemble with the same initial spread shows this correction to be statistical rather than quantum: it is reproduced to machine precision for free and linear forces, and the genuinely quantum residue in the oscillator amounts to 5 × 10 − 5 of the amplitude. We also bound the error of the first-order truncation by comparison with the exact tangent Hamiltonian. This places the deformed quantum–classical correspondence on a quantitative footing, complementing earlier work on coherent states and on packet dispersion in a minimum-length scenario.
This work establishes a link between deformed angular momentum operators and modifications of the quantum uncertainty principle. Starting from an algebra in which the standard angular momentum operators are deformed by two parameters, [Formula: see text] and [Formula: see text], we construct the corresponding momentum operators via [Formula: see text] where the radial operator [Formula: see text] is determined by the requirement that the standard momentum operator is recovered in the limit [Formula: see text], [Formula: see text]. The resulting commutation relations take the form [Formula: see text] where [Formula: see text] is a position-dependent deformation matrix. This leads to an uncertainty relation [Formula: see text] which is state-dependent, anisotropic and position-dependent. We use the term Generalized Uncertainty Principle (GUP) in its broad sense referring to any modification of the Heisenberg relation motivated by quantum-gravity ideas. However, the model does not yield a fixed minimal length of the kind familiar from conventional GUP models (e.g. [Formula: see text]). In the isotropic, limit the trace of the deformation matrix simplifies the bound to [Formula: see text] which can be considered as a rescaling of the uncertainty constant. Our work thus provides a geometric origin for modified uncertainty relations from symmetry breaking in the rotation group, and offers a framework in which quantum-gravity effects manifest themselves through state- and direction-dependent measurement limits, without necessarily introducing a universal minimal length.
Abstract We study and generalize the momentum operator satisfying the extended uncertainty principle relation (EUP). This generalized extended momentum operator (GEMO) consists of an arbitrary auxiliary function of position operator, μ x , in such a combination that not only GEMO satisfies the EUP relation but also it is Hermitian. Next, we apply the GEMO to construct the generalized one-dimensional Schrödinger equation. Upon using the so called point canonical transformation (PCT), we transform the generalized Schrödinger equation from x -space to z -space where in terms of the transformed coordinate, z , it is of the standard form of the Schrödinger equation. In continuation, we study two illustrative examples and solve the corresponding equations analytically to find the energy spectrum.
A translation operator acting in a space with a diagonal metric is introduced to describe the motion of a particle in a quantum system. We show that the momentum operator and, as a consequence, the uncertainty relation now depend on the metric. It is also shown that, for any metric expanded up to second order, this formalism naturally leads to an extended uncertainty principle (EUP) with a minimum momentum dispersion. The Ehrenfest theorem is modified to include an additional term related to a tidal force arriving from the space curvature introduced by the metric. For one-dimensional systems, we show how to map a harmonic potential to an effective potential in Euclidean space using different metrics.
The Heisenberg Uncertainty Principle is traditionally formulated within the mathematical idealization of perfectly flat Euclidean space. However, physical reality dictates thatspace is inherently curved by gravitational fields, cosmological dynamics, and perpetualquantum vacuum fluctuations. This paper proposes a novel theoretical framework that generalizes three fundamental uncertainty principles—Position-Momentum, Energy-Time, andAngle-Angular Momentum—by directly incorporating spatial curvature on a cosmologicalscale. We introduce a continuous spatial modulating function, f(x), governed by the netspatial curvature R(x), representing the dynamic interplay between attractive and repulsive geometric forces. To rigorously preserve the quantum mechanical axiom that physicalobservables correspond to Hermitian operators, we employ Weyl symmetrization to construct modified differential operators for momentum and angular momentum. Through exactmathematical derivations utilizing wave packet phase-space dynamics, a modified Ehrenfesttheorem, and the resolution of topological boundary discontinuities via mapping to a Universal Covering Space with strict Dirichlet boundary conditions, we establish absolute lowerbounds for these uncertainties. Ultimately, we demonstrate that the minimum bound ofquantum uncertainty is not a rigid universal constant, but is strictly positive and dynamically dictated by local geometric curvature, providing a robust bridge between quantummechanics and spacetime geometry.Keywords: Generalized Uncertainty Principle; Spatial Curvature; Weyl Symmetrization; Hermitian Operators; Modified Ehrenfest Theorem; Universal Covering Space; Cosmological Scale.PACSNumbers: 03.65.Ta(Foundationsof quantummechanics); 04.60.-m (Quantum gravity);03.65.Ca (Formalism).MSC2020 Codes: 81P05 (General and philosophical foundations of quantum mechanics);81S05 (Canonical quantization, commutation relations and statistics)