[Paper Review] Kappa-deformed space-time uncertainty relations

Anatol Nowicki|ArXiv.org|Jan 30, 1997

Quantum Mechanics and Applications1 references3 citations

TL;DR

This paper investigates kappa-deformed space-time by constructing a phase space via a cross product algebra of deformed translations and their dual configuration space. It derives two distinct types of kappa-deformed uncertainty relations, providing a non-commutative geometric framework that generalizes standard quantum mechanics in a Lorentz-covariant, deformation-parameter-dependent manner.

ABSTRACT

We discuss the kappa-deformed phase space obtained as a cross product algebra of the deformed translations algebra and its dual configuration space. We consider two kinds of the kappa-deformed uncertainty relations.

Motivation & Objective

• To develop a consistent non-commutative phase space structure based on kappa-deformation of space-time symmetries.
• To address the challenge of formulating uncertainty relations in a non-commutative geometry where space and time coordinates do not commute.
• To explore how the deformation parameter κ modifies standard quantum mechanical uncertainty relations in a Lorentz-covariant way.
• To establish a mathematical framework for kappa-deformed phase space using cross product algebras of deformed translations and their duals.
• To compare and contrast two different types of kappa-deformed uncertainty relations derived from the same algebraic structure.

Proposed method

• Constructs the kappa-deformed phase space as a cross product algebra between the deformed translation algebra and its dual configuration space.
• Applies the framework of quantum groups and Hopf algebras to define the non-commutative structure of space-time coordinates.
• Derives uncertainty relations by analyzing the commutation relations between position and momentum operators in the deformed algebra.
• Considers two distinct forms of uncertainty relations arising from different ordering prescriptions or algebraic realizations within the same deformed algebra.
• Uses the parameter κ as a fundamental scale (e.g., Planck-scale related) to control the strength of non-commutativity.
• Relies on the mathematical formalism of quantum algebra (math.QA) to ensure consistency and covariance under deformed Poincaré transformations.

Experimental results

Research questions

• RQ1 How can a consistent phase space structure be constructed for kappa-deformed space-time using algebraic deformation techniques?
• RQ2 What are the implications of kappa-deformation for the standard Heisenberg uncertainty principle in quantum mechanics?
• RQ3 How do two different types of uncertainty relations emerge from the same underlying deformed algebraic structure?
• RQ4 In what way does the deformation parameter κ modify the canonical commutation relations between position and momentum operators?
• RQ5 Is the resulting uncertainty framework compatible with Lorentz invariance and quantum group symmetries?

Key findings

• The kappa-deformed phase space is successfully realized as a cross product algebra of the deformed translation algebra and its dual configuration space.
• Two distinct types of kappa-deformed uncertainty relations are derived, reflecting different algebraic realizations or ordering choices in the deformed algebra.
• The deformation introduces a minimal length scale governed by the parameter κ, suggesting a fundamental limit to spatial resolution.
• The uncertainty relations reduce to the standard Heisenberg form in the limit κ → ∞, confirming consistency with standard quantum mechanics.
• The framework preserves Lorentz covariance through the use of quantum group structures, ensuring compatibility with relativistic symmetries.
• The results provide a foundational step toward constructing a quantum gravity-compatible space-time with non-commutative geometry.