Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models

Abstract

Modifications in quantum mechanical phase space lead to the changes in the Heisenberg uncertainty principle, which can result in the Generalized Uncertainty Principle (GUP) or the Extended Uncertainty Principle (EUP), introducing quantum gravitational effects at small and large distances, respectively. A combination of GUP and EUP, the Generalized Extended Uncertainty Principle (GEUP or EGUP), further generalizes these modifications by incorporating noncommutativity in both coordinates and momenta. This paper examines the impact of GEUP on the analogue of the Liouville theorem in statistical physics and density of states within the classical limit of non-relativistic quantum mechanics framework. We find a weighted phase space volume element, invariant under the infinitesimal time evolution, in the cases of Snyder-de Sitter and Yang models, presenting how GEUP alters the density of states, potentially affecting physical (thermodynamical) properties. Special cases, obtained in certain limits from the above models are discussed. New higher order types of GEUP and EUP are also proposed.

18 pages, matching published version

References in corpus (43)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• Universality of Quantum Gravity Corrections
• Spectral Dimension of the Universe
• Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations
• The Effect of the Minimal Length Uncertainty Relation on the Density of States and the Cosmological Constant Problem
• Quantum-corrected black hole thermodynamics to all orders in the Planck length
• The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length
• GUP parameter from quantum corrections to the Newtonian potential
• (Anti-)de Sitter Black Hole Thermodynamics and the Generalized Uncertainty Principle
• Natural extension of the Generalised Uncertainty Principle
• Probing deformed commutators with macroscopic harmonic oscillators
• Doubly Special Relativity and de Sitter space
• Minimal Length Uncertainty Relation and gravitational quantum well
• Extended uncertainty principle and the geometry of (anti)-de Sitter space
• A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum II: Applications
• Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator
• Generalized Uncertainty Principle, Classical Mechanics, and General Relativity
• Classical and quantum mechanics of the nonrelativistic Snyder model
• Triply Special Relativity
• Minimal Length in Quantum Gravity, Equivalence Principle and Holographic Entropy Bound
• Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry
• Linear Form of 3-scale Relativity Algebra and the Relevance of Stability
• Extended GUP formulation and the role of momentum cut-off
• Linear and Quadratic GUP, Liouville Theorem, Cosmological Constant, and Brick Wall Entropy
• Bounds on GUP parameters from GW150914 and GW190521
• Modified commutators are not sufficient to determine a quantum gravity minimal length scale
• Effects of the Generalized Uncertainty Principle on Compact Stars
• The minimal length: a cut-off in disguise?
• Constraining the Generalized Uncertainty Principle with the light twisted by rotating black holes and M87*
• A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
• Effect of minimal length uncertainty on the mass-radius relation of white dwarfs
• Generalizations of Snyder model to curved spaces
• Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case
• Unveiling Phase Space Modifications: A Clash of Modified Gravity and the Generalized Uncertainty Principle
• Constraining Snyder and GUP models with low-mass stars
• Reinterpreting deformed Heisenberg algebras
• Compact stars in quantum spacetime
• Extended Uncertainty Principle: A Deeper Insight into the Hubble Tension?
• From Snyder space-times to doubly $\kappa$-dependent Yang quantum phase spaces and their generalizations
• Radial Oscillations and Dynamical Instability Analysis for Linear-Quadratic GUP-modified White Dwarfs
• Finite Temperature Considerations in the Structure of Quadratic GUP-modified White Dwarfs
• Generalized Yang Poisson Models on Canonical Phase Space

Cited by in corpus (6)

• Dark Energy within the Generalized Uncertainty Principle in Light of DESI DR2
• Towards new relativistic doubly $\kappa$-deformed D=4 quantum phase spaces
• Symplectic realization of generalized Snyder-Poisson algebra
• Combining the Generalized and Extended Uncertainty Principles
• Einstein crystals in Snyder and Snyder-de Sitter noncommutative backgrounds
• The Extended Uncertainty Principle from a Projector-Valued Measurement Perspective