Quantum Evolution of Hopf Algebra Hamiltonians

TL;DR

The paper analyzes whether Hopf-algebra deformations of time-evolution generators can generate Lindblad-type decoherence for a qubit. Using $q$-deformed $U_q(\mathfrak{su}(2))$ and general leading-order deformations, it shows that most deformed adjoint-action prescriptions fail to preserve positivity, and that a unique linear combination of left and right adjoint actions reproduces the standard von Neumann evolution, sometimes with a deformed Hamiltonian. In the $\kappa$-deformed spacetime setting, a Lindblad-type form is not physically viable; the evolution remains unitary with a deformed effective Hamiltonian. Overall, the work constrains Planck-scale decoherence scenarios from Hopf-algebra deformations, indicating that physically acceptable dynamics are of von Neumann type rather than genuine Lindblad evolution within the examined models.

Abstract

In recent years, growing attention has been devoted to the possibility that theories with deformed symmetries, associated with certain models of non-commutative spacetime, may encode a fundamental form of decoherence. This effect should be described by a Lindblad-like evolution governed by the non-trivial Hopf algebra structure of the time-evolution generators. In this work we provide a detailed analysis of such possibility for similar Hopf algebra deformations of the Hamiltonian of a qubit. Starting from a critical examination of the very definition of time evolution through the generalized adjoint action, we explore whether a coherent and physically viable framework can be established. In particular, our analysis shows that a more general combination of adjoint actions always guarantees a von Neumann dynamics and, also in the case of deformed spacetime symmetries considered in the literature, a physically viable Lindblad evolution cannot be established.