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Applied Mathematics

Uncertainty Principle and Bifurcations in the SU(2) Nonlinear Semiquantum Dynamics

R. O. Hansen, Claudia M. Sarris, Angelo Plastino

Published Jan 1, 2018• 0 citations
Publisher

Abstract

In this paper, a nonlinear semiquantum Hamiltonian associated to the special unitary group SU(2) Lie algebra is studied so as to analyze its dynamics. The treatment here applied allows for a reduction in: 1) the system’s dimension, as well as 2) the number of system’s parameters (to only three). We can now discern clear patterns in: 1) the complete characterization of the system’s fixed points and 2) their stability. It is shown that the parameter associated to the uncertainty principle, which constitutes a very strong constraint, is the key one in determining the presence of fixed points and bifurcation curves in the parameter’s space.

DOI
10.4236/am.2018.91001

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In this paper, a nonlinear semiquantum Hamiltonian associated to the special unitary group SU(2) Lie algebra is studied so as to analyze its dynamics. The treatment here applied allows for a reduction in: 1) the system’s dimension, as well as 2) the number of system’s parameters (to only three). We can now discern clear patterns in: 1) the complete characterization of the system’s fixed points and 2) their stability. It is shown that the parameter associated to the uncertainty principle, which constitutes a very strong constraint, is the key one in determining the presence of fixed points and bifurcation curves in the parameter’s space.
We describe a general method to study the nonlinear semiquantum systems: as the dynamics of the system develops in a semiquantum space V spanned by the mean values of a Complete Set of Noncommuting observables (CSNCO) and the conjugate classical variables, X, P, it is possible to define a positive definite metric for the quantum manifold (QM). Through the components of the second rank covariant metric tensor, a close connection between this metric and the uncertainty principle can be established, meanwhile, the classical manifold (CM) of the system, is preserved as a symplectic one. We present an example of an entangled nonlinear semiquantum system where different dynamics take place.
How does the classical phase-space structure for a composite system relate to the entanglement characteristics of the corresponding quantum system? We demonstrate how the entanglement in nonlinear bipartite systems can be associated with a fixed-point bifurcation in the classical dynamics. Using the example of coupled giant spins we show that when a fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state---the ground state---achieves its maximum amount of entanglement near the critical point. We conjecture that this will be a generic feature of systems whose classical limit exhibits such a bifurcation.
We describe how, departing from the Shannon entropy, it is possible to deal with semiquantum time-independent nonlinear Hamiltonians. The interplay between the quantal and classical degrees of freedom can be easily seen, and the set of differential equations that govern the temporal evolution of the quantal mean values and the classical variables is obtained. We find invariants of motion and, particularly, we describe under which conditions, the uncertainty principle remains as an invariant of motion too. Through the analysis of these invariants, it is possible to follow the transition of the system from quantum to classical regime and to conclude that the uncertainty principle behaves as an indicator telling us whether the system is in regular or irregular regime. A simple example is shown.
We analyze the nonlinear dynamics of a quartic semiclassical system able to describe the interaction of matter with a field. We do it in both dissipative and conservative scenarios. In particular, we study the classical limit of both frameworks and compare the associated features. In the two environments, we heavily use a system's invariant, related to the Uncertainty Principle, that helps to determine how the dynamics tends to the pertinent classical limit. We exhibit the convergence to the classical limit and also verify that the Uncertainty Principle is complied with during the entire process, even in the presence of dissipation.
We describe the motion of an SU(1,1) pseudospin vector in the frame of the mean-field approximation induced by the variational principle on linear-plus-quadratic Hamiltonians. The dynamics of the SU(1,1) states of the Perelomov type obeys a nonlinear Bloch or torquelike equation, and each orbit can be interpreted as the intersection of two quadrics, one representing the energy surface and the other the group manifold, both in the space of the averaged algebra generators or semiclassical pseudospin. The fixed points of the flow can be also determined by resorting to strictly geometric considerations. The evolution of the phase diagram in parameter space is investigated as well for selected Hamiltonians. The bifurcation sets are constructed and the nonthermodynamic phase transitions can be clearly identified for the systems under consideration.
The analytical determination of critical scaling in three-dimensional strongly coupled field theories is inherently restricted by the Borel non-summability of perturbative expansions. In this paper, we establish a symplectic isomorphism that maps the infrared dynamics of scalar quantum field theories onto the finite-strain bifurcation mechanics of hyperelastic continua. By reformulating the renormalization group flow as an effective spatial Hamiltonian evolution within a symplectic phase space, we identify the critical fixed point with the spectral degeneration of a continuous Riccati-Lyapunov dynamical system. The algebraic integrity of this geometric correspondence is rigorously verified against exactly solvable limits, recovering both the Gaussian free-field scaling and the localized Jackiw-Rebbi zero-mode of the continuum Su-Schrieffer-Heeger model. Applying this framework to the 3D Ising universality class, we construct a self-consistent algebraic scaling Ansatz for the anomalous dimension ηηη. This derivation dispenses with arbitrary perturbative truncations, demonstrating instead that interaction-induced non-perturbative screening emerges as a strict consequence of the nonlinear self-consistency demanded by the continuous Algebraic Riccati Equation. The resulting rational scaling law functions as a geometric Padé approximant that asymptotically satisfies the unitarity bound limits, providing a mathematically transparent, first-principles macroscopic impedance analogue for the dynamic suppression of strong-coupling divergences.