[Paper Review] Uncertainty of Velocity in kappa-Minkowski Spacetime

Ken-Ichi Tezuka|ArXiv.org|Feb 17, 2003

Noncommutative and Quantum Gravity Theories28 references3 citations

TL;DR

This paper investigates velocity uncertainty in $κ$-Minkowski spacetime, a non-commutative geometry where the Poincaré group is deformed into a quantum group with parameter $κ$. By analyzing the spacetime non-commutativity through coordinate uncertainties, it derives a velocity uncertainty that depends on the particle's propagation length $L$, yielding a minimum uncertainty $\delta V^2_{\text{min}} \approx -\frac{3c^2}{|\kappa L|}$, independent of particle energy but sensitive to $L$ and $\kappa$. This provides a novel mechanism for energy-dependent light speed without relying on momentum-space velocity definitions.

ABSTRACT

A velocity of a point particle in the kappa-Minkowski spacetime is investigated. Characteristic points of the spacetime are that the Poincare group becomes a quantum group with kappa, which is a mass dimension parameter, and is a kind of non-commutative geometry. We consider a particle in a coordinate space instead of it in a momentum space which is discussed in many articles. We see that the particle's velocity has an uncertainty which depends on a length of particle's propagation.

Motivation & Objective

• To investigate the velocity of a free particle in $κ$-Minkowski spacetime, a non-commutative spacetime where the Poincaré algebra is deformed into a quantum group.
• To address the lack of rigorous justification for using $v = \partial E / \partial p$ in non-commutative spacetimes, which leads to inconsistent velocity predictions depending on the algebraic basis.
• To derive the velocity uncertainty directly from spacetime non-commutativity, avoiding reliance on momentum-space definitions.
• To determine whether the speed of light can deviate from $c$ due to quantum spacetime structure, even when the action is $κ$-independent.

Proposed method

• Formalism is based on the $κ$-Poincaré algebra in the bicrossproduct basis, with non-commutative spacetime coordinates satisfying $[x^0, x^i] = i\kappa^{-1} x^i$.
• The particle's action is shown to be independent of $κ$, implying no direct $κ$-deformation of dynamics.
• Spacetime uncertainty is derived from the commutator $[x^0, x^i]$, leading to a Heisenberg-type uncertainty relation $\langle(\Delta X^i)^2\rangle\langle(\Delta T)^2\rangle \geq \frac{1}{4c^2\kappa^2}|\langle X^i\rangle|^2$.
• Velocity uncertainty $\delta V^2$ is computed perturbatively up to second order in $\delta X^i$ and $\delta T$, assuming minimal deviation under the uncertainty constraint.
• The minimum velocity uncertainty is found by canceling first-order terms, yielding $\delta V^2_{\text{min}} \approx -\frac{3v^3}{|\kappa L|c}$, which simplifies to $\delta V^2_{\text{min}} \approx -\frac{3c^2}{|\kappa L|}$ for massless particles.
• The constant $c$ is interpreted as the expectation value of photon velocity, with deviations arising from non-commutative geometry.

Experimental results

Research questions

• RQ1 Can particle velocity in $κ$-Minkowski spacetime be consistently defined without relying on $v = \partial E / \partial p$?
• RQ2 How does spacetime non-commutativity induce uncertainty in the velocity of a free particle?
• RQ3 What is the dependence of velocity uncertainty on the particle's propagation length $L$ and the $κ$-scale?
• RQ4 Does the speed of light deviate from $c$ in $κ$-Minkowski spacetime even when the action is $κ$-independent?
• RQ5 What observational bounds can be placed on $κ$ from velocity uncertainty in light propagation?

Key findings

• The velocity of a free particle in $κ$-Minkowski spacetime exhibits intrinsic uncertainty due to non-commutative spacetime coordinates.
• The minimum velocity uncertainty for a massless particle is $\delta V^2_{\text{min}} \approx -\frac{3c^2}{|\kappa L|}$, which depends on the propagation length $L$ and the $κ$-scale but not on the photon's energy.
• The constant $c$ is interpreted as the expectation value of the photon's velocity, with deviations arising from quantum spacetime structure.
• For the observed value of $c$ to be consistent, the bound $|L\kappa| > 10^{20}$ must hold, implying $|\kappa| > 10^{17}$ m for typical laboratory-scale light paths.
• The velocity uncertainty is negligible in the non-relativistic limit ($v/c \to 0$), consistent with the fact that only the Lorentz boost sector is $κ$-deformed.
• The result provides an alternative mechanism for Lorentz violation in quantum gravity, distinct from momentum-space dispersion relations, based purely on spacetime non-commutativity.