Quantum walks, deformed relativity and Hopf algebra symmetries

Area of Science:

• Quantum physics
• Theoretical physics
• Mathematical physics

Background:

• The Weyl quantum walk, based on D'Ariano & Perinotti (2014), exhibits nonlinear Lorentz symmetry.
• Hopf algebras offer a framework for describing algebraic structures in quantum mechanics.

Purpose of the Study:

• To introduce Hopf algebras for position and momentum within the Weyl quantum walk framework.
• To explore the application of specific Hopf algebra models, namely the Poincaré and κ-Poincaré algebras.

Main Methods:

• Derivation of the Weyl quantum walk from established principles.
• Application of Hopf algebra theory to quantum mechanical concepts of position and momentum.
• Analysis of the Poincaré and κ-Poincaré algebras in the context of the quantum walk.

Main Results:

• Demonstration that the Weyl quantum walk allows for the introduction of Hopf algebras for position and momentum.
• Successful integration of the Poincaré and κ-Poincaré algebras into the quantum walk model.
• Highlighting the nonlinear Lorentz symmetry inherent in the dynamics.

Conclusions:

• The Weyl quantum walk provides a novel pathway to incorporate Hopf algebraic structures in quantum mechanics.
• The study successfully models particle properties using specific Hopf algebras, advancing theoretical frameworks.
• This research bridges quantum walk dynamics with advanced algebraic structures, opening new avenues for theoretical exploration.