[Paper Review] Decoherence limit of quantum systems obeying generalized uncertainty principle: new paradigm for Tsallis thermostatistics

Petr Jizba, Gaetano Lambiase+2 more|arXiv (Cornell University)|Jan 19, 2022

Noncommutative and Quantum Gravity Theories72 references41 citations

TL;DR

This paper establishes a direct link between the generalized uncertainty principle (GUP) and Tsallis non-extensive thermostatistics by showing that GUP-induced coherent states in momentum representation exactly match Tsallis probability amplitudes, with the non-extensivity parameter q monotonically increasing with the GUP deformation parameter β. For β < 0 (q < 1), the GUP is shown to be fully equivalent to Beckner–Babenko inequality-based uncertainty relations derived from Tsallis entropy-power, and the maximal entropy principle reveals that the quasi-classical limit of GUP quantum theory naturally aligns with non-extensive thermostatistics, offering a new paradigm for quantum gravity and analog gravity models.

ABSTRACT

The generalized uncertainty principle (GUP) is a phenomenological model whose purpose is to account for a minimal length scale (e.g., Planck scale or characteristic inverse-mass scale in effective quantum description) in quantum systems. In this Letter, we study possible observational effects of GUP systems in their decoherence domain. We first derive coherent states associated to GUP and unveil that in the momentum representation they coincide with Tsallis' probability amplitudes, whose non-extensivity parameter $q$ monotonically increases with the GUP deformation parameter $\beta$. Secondly, for $\beta < 0$ (i.e., $q < 1$), we show that, due to Bekner-Babenko inequality, the GUP is fully equivalent to information-theoretic uncertainty relations based on Tsallis-entropy-power. Finally, we invoke the Maximal Entropy principle known from estimation theory to reveal connection between the quasi-classical (decoherence) limit of GUP-related quantum theory and non-extensive thermostatistics of Tsallis. This might provide an exciting paradigm in a range of fields from quantum theory to analog gravity. For instance, in some quantum gravity theories, such as conformal gravity, aforementioned quasi-classical regime has relevant observational consequences. We discuss some of the implications.

Motivation & Objective

• To explore observational consequences of the generalized uncertainty principle (GUP) in the decoherence regime.
• To derive coherent states (CSs) for GUP in both position and momentum representations, focusing on their connection to Tsallis statistics.
• To establish equivalence between GUP uncertainty relations and Tsallis entropy-power-based uncertainty relations (EPUR) for β < 0.
• To demonstrate that the quasi-classical (decoherence) limit of GUP quantum theory is naturally described by non-extensive thermostatistics of Tsallis.
• To propose a new theoretical framework linking GUP, coherent states, and Tsallis thermostatistics for applications in quantum gravity and cosmology.

Proposed method

• Derive GUP coherent states using deformed commutation relations [ˆx, ˆp] = iℏ(1 + β ˆp²/m²p), focusing on momentum representation.
• Solve the Schrödinger-type minimum-uncertainty condition (ˆp − iγˆx)|ψ⟩ = 0 in momentum space to obtain wavefunctions ψ(p) proportional to [1 + βp²/m²p]⁻(m²p/(2βγℏ) + 1/2) for β > 0 and [1 − |β|p²/m²p]⁺^(m²p/(2|β|γℏ) − 1/2) for β < 0.
• Identify that for β > 0, the momentum-space wavefunction matches the Tsallis probability amplitude with non-extensivity parameter q = 1 + 2βγℏ/m²p.
• For β < 0, reformulate the GUP uncertainty relation using Tsallis entropy-power-based uncertainty relations (EPUR), showing saturation by GUP coherent states.
• Apply the maximum entropy principle (MEP) to connect the quasi-classical limit of GUP to Tsallis non-extensive thermostatistics.
• Use Beckner–Babenko inequality to prove that for β < 0 (q < 1), the GUP is fully equivalent to Tsallis-entropy-power uncertainty relations.

Experimental results

Research questions

• RQ1 How do coherent states of GUP systems relate to Tsallis probability amplitudes in momentum representation?
• RQ2 Can GUP uncertainty relations be reformulated as Tsallis entropy-power-based uncertainty relations (EPUR) for β < 0?
• RQ3 Is there a fundamental equivalence between GUP and information-theoretic uncertainty relations based on Tsallis entropy for β < 0?
• RQ4 How does the maximal entropy principle connect the quasi-classical limit of GUP to non-extensive thermostatistics?
• RQ5 What are the cosmological and gravitational implications of this GUP–Tsallis unification in the context of entropic gravity and analog gravity models?

Key findings

• GUP coherent states in momentum representation are mathematically identical to Tsallis probability amplitudes, with the non-extensivity parameter q increasing monotonically with the GUP deformation parameter β.
• For β < 0 (q < 1), the GUP is fully equivalent to Beckner–Babenko inequality-based uncertainty relations derived from Tsallis entropy-power, due to the saturation of these relations by GUP coherent states.
• The quasi-classical limit of GUP quantum theory is naturally described by non-extensive thermostatistics of Tsallis, as confirmed by the maximum entropy principle applied to GUP coherent states.
• The derived uncertainty relation in terms of Tsallis entropy-power (EPUR) is saturated by GUP coherent states and takes the form M_T^{q/2}(|ψ|²) M_T^{q'/2}(|ψ̃|²) ≥ ℏ²/4, with a universal function f(q) that ensures the bound is independent of unknown parameters.
• The non-extensivity parameter q is bounded below by 1 (from Beckner–Babenko inequality) and increases with β, with q < 2 for β < 0, consistent with the physical requirement of non-extensivity.
• In the ultra-relativistic limit, the momentum variance (∆p)²_ψ ≈ 12(kBT)² is derived using the equipartition theorem, suggesting that thermal and quantum fluctuations are of the same order in semi-classical regimes, supporting the consistency of the GUP–Tsallis framework in cosmological settings.