Jonathan M Robbins

Indistinguishability for quantum particles: spin, statistics and the geometric phase

The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwingers scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.

Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction

Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2; 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinear-waves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the three-wave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.

Chaotic classical and half-classical adiabatic reactions: geometric magnetism and deterministic friction

We study the dynamics of a heavy (slow) classical system coupled, through its position, to a classical or quantal light (fast) system, and derive the first-order velocity-dependent corrections to the lowest adiabatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-order forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force–force correlation function of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the ‘magnetic field’ is the classical limit of the 2-form generating the quantum geometric phase. The symmetric part is deterministic friction, dissipating slow energy into the fast chaos; previously found by Wilkinson, this involves the same correlation function as governs the fluctuations and drift of the adiabatic invariant. In the ‘half-classical’ case where the fast system is quantal with a discrete spectrum of adiabatic states, the only first-order slow force is geometric magnetism; there is no friction. This discordance between classical and quantal fast motion is explained in terms of the clash between the semiclassical and adiabatic limits. A generalization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, the symplectic form in the lowest-order hamiltonian dynamics is modified.

Maslov indices in the Gutzwiller trace formula

The Gutzwiller trace formula is a semi-classical approximation for the density of states of a bound quantum system, expressed in terms of a sum over periodic orbits of the corresponding classical system. The authors establish the existence of a topological invariant associated with classical periodic orbits, namely the winding number of their invariant manifolds, and show that this winding number is precisely the index which appears in the trace formula. The proof is valid in any number of dimensions. Explicit formulae for the index are given.

Quantum boundary conditions for torus maps

The quantum states of a dynamical system whose phase space is the two-torus are periodic up to phase factors under translations by the fundamental periods of the torus in the position and momentum representations. These phases, and , are conserved quantities of the quantum evolution. We show that for a large and important class of quantum maps, and are restricted to being the coordinates of the fixed points of the automorphism induced on the fundamental group of the torus by the underlying classical dynamics. As a consequence, if the classical map commutes with lattice translations in it can be quantized for any choice of the phases, but otherwise it can be quantized for only a finite set. This result is a special case of a more general condition on the phases, which is also derived. The cat maps, perturbed cat maps, and the kicked Harper map are discussed as specific examples.

Topology and bistability in liquid crystal devices.

We study nematic liquid crystal configurations in a prototype bistable device -- the post aligned bistable nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector field n , in a model version of the PABN cell. First, we identify four distinct topologies in this geometry. We explicitly construct trial configurations with these topologies which are used as initial conditions for a numerical solver, based on the finite-element method. The morphologies and energetics of the corresponding numerical solutions qualitatively agree with experimental observations and suggest a topological mechanism for bistability in the PABN cell geometry.

Classical Geometric Forces of Reaction: An Exactly Solvable Model

We illustrate the effects of the classical ‘magnetic’ and ‘electric’ geometric forces that enter into the adiabatic description of the slow motion of a heavy system coupled to a light one, beyond the Born–Oppenheimer approximation of simple averaging. When the fast system is a spin S and the slow system is a massive particle whose spatial position R is coupled to S with energy (fast hamiltonian) S · R, the magnetic force is that of a monopole of strength I (= adiabatic invariant S · R/R) centred at R = 0, and the electric force is inverse-cube repulsion with strength S2 – I2. Confining the slow particle to the surface of a sphere eliminates the Born–Oppenheimer and electric forces, and generates motion with precession and nutation exactly equivalent to that of a heavy symmetrical top. In the adiabatic limit the nutation is small and the averaged precession is precisely reproduced by the magnetic force. Alternatively, choosing the exactly conserved total angular momentum to vanish eliminates the Born–Oppenheimer and magnetic forces, and generates as exact orbits a one-parameter family of curly ‘antelope horns’ coiling in from infinity, reversing hand, and receding to infinity. In the adiabatic limit the repulsion of the ‘guiding centre’ of these coils is exactly reproduced by the electric force. A by-product of the ‘antelope horn’ analysis is a determination of the shape of a curve with a given curvature k and torsion T in terms of the evolution of a quantum 2-spinor driven by a planar ‘magnetic field’ with components k and T.

Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals

We investigate prototypical profiles of point defects in two-dimensional liquid crystals within the framework of Landau–de Gennes theory. Using boundary conditions characteristic of defects of index k/2, we find a critical point of the Landau–de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime,

Elastic energy of liquid crystals in convex polyhedra

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