Difficulties in Evaluating Lyapunov Exponents for Lie Governed Dynamics

We consider here an environment in which the fact that a semiquantum Hamiltonian obeys SU(2) symmetries poses serious difficulties if one wants to compute Lyapunov exponents. 1. Introduction An extreme complexity of phase space's trajectories that are very sensitive to small changes in the initial conditions is the signature of classical chaos, accompanied by (i) an ostensibly random allotment of phase points on a Poincare's surface of section and (ii) an exponentially rapid separation of two initially close trajectories [1]. Instead, it is clear that the state vector of a closed quantum system cannot exhibit chaotic motion in Hilbert space. The interaction between a quantum system and a classical one may instead lead to authentic chaotic motion of the quantum component, a phenomenon known as semiquantum chaos [2, 3]. Remark that the vocable semiquantum is reserved to systems for which neither the quantum part nor the classical part would be chaotic by themselves. If chaos ensues, this happens because of the classical-quantum coupling. Semiquantumness implies that one part is treated classically and the other one in quantal fashion [3]. We consider here an environment in which the semiquantum Hamiltonian obeys symmetries and consider the difficulties that arise if one wants to compute Lyapunov exponents [4]. The paper is organized as follows: Section 2 deals with some background materials [5], while Section 3 explicates Hamiltonian details. 2. Background Consider a system that possesses both quantum and classic degrees of freedom, with a coupling amongst them, that we call semiquantum [3, 6–10]. The associated Hamiltonian is of the general form [11] where , , and are the quantum, classical, and interaction parts, respectively. There exist many situations in which a semiquantum description has been attempted [6, 9, 12]. Porter [6] made an exhaustive compilation of physical systems for which this kind of description is relevant. One may highlight vibrating quantum billiards as a useful abstraction of the ensuing semiquantum dynamics [13]. Indeed, many semiquantum Hamiltonians are found in the literature [3, 14–23]. In this work, via the Maximum Entropy Principle (MEP) vantage point [5], we show that serious difficulties arise, in the case of these systems, if one wants to compute Lyapunov exponents, because the quantum degrees of freedom of the system must abide by the generalized uncertainty principle (GUP) [24] which in the case of an underlying Lie algebra becomes a dynamic invariant. In turn, the other dynamic invariants (the energy) involve both