Quantum mechanics on manifolds and topological effects

A unique classification of the topological effects associated to quantum mechanics on manifolds is obtained on the basis of the invariance under diffeomorphisms and the realization of the Lie-Rinehart relations between the generators of the diffeomorphism group and the algebra of infinitely differentiable functions on the manifold. This leads to a unique ("Lie-Rinehart") C* algebra as observable algebra; its regular representations are shown to be locally Schroedinger and in one to one correspondence with the unitary representations of the fundamental group of the manifold. Therefore, in the absence of spin degrees of freedom and external fields, the first homotopy group of the manifold appears as the only source of topological effects.Comment: A few comments have been added to the Introduction, together with related references; a few words have been changed in the Abstract and a Note added to the Titl

Refined Algebraic Quantization in the oscillator representation of SL(2,R)

We investigate Refined Algebraic Quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T^*R^4 and gauge group SL(2,R). The reduced phase space M is connected and contains four mutually disconnected `regular' sectors with topology R x S^1, but these sectors are connected to each other through an exceptional set where M is not a manifold and where M has non-Hausdorff topology. The RAQ physical Hilbert space H_{phys} decomposes as H_{phys} = (direct sum of) H_i, where the four subspaces H_i naturally correspond to the four regular sectors of M. The RAQ observable algebra A_{obs}, represented on H_{phys}, contains natural subalgebras represented on each H_i. The group averaging takes place in the oscillator representation of SL(2,R) on L^2(R^{2,2}), and ensuring convergence requires a subtle choice for the test state space: the classical analogue of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space H_{phys} and a finitely-generated observable subalgebra of A_{obs} is recovered through both Ashtekar's Algebraic Quantization and Isham's group theoretic quantization.Comment: 30 pages, REVTeX v3.1 with amsfonts. (v4: Published version.

Developing a Framework for Child Welfare Supervision

The roles of the child welfare supervisor in guiding practice and in retaining child welfare workers are well established in the literature. In this article, we discuss a framework for child welfare supervision that was developed and implemented in the state of Iowa with support from the Children’s Bureau through a five-year grant to improve recruitment and retention in public child welfare. The framework supports family centered practice through a parallel process of supervision reflecting these guiding principles: strength-based, competency-based, culturally competent, reflective, individualized to workers’ learning styles and stages of development, and aimed at enhancing worker skill, autonomy, teamwork, and commitment to the organization. We present key elements of the framework, an overview of implementation, and evaluation results regarding knowledge gain, use of skills, and rates of worker retention

Complete chaotic synchronization in mutually coupled time-delay systems

Complete chaotic synchronization of end lasers has been observed in a line of mutually coupled, time-delayed system of three lasers, with no direct communication between the end lasers. The present paper uses ideas from generalized synchronization to explain the complete synchronization in the presence of long coupling delays, applied to a model of mutually coupled semiconductor lasers in a line. These ideas significantly simplify the analysis by casting the stability in terms of the local dynamics of each laser. The variational equations near the synchronization manifold are analyzed, and used to derive the synchronization condition that is a function of the parameters. The results explain and predict the dependence of synchronization on various parameters, such as time-delays, strength of coupling and dissipation. The ideas can be applied to understand complete synchronization in other chaotic systems with coupling delays and no direct communication between synchronized sub-systems.Comment: 22 pages, 6 figure

Theory for all-optical responses in topological materials: The velocity gauge picture

High-order harmonic generation (HHG), which has been widely studied in atomic gas, has recently been expanded to solids to study the highly nonlinear electronic response in condensed matter and produce coherent high-frequency radiation. Recently, attention has turned to topological materials and the use of HHG to characterize topological bands and invariants. However, the theoretical interpretation of the nonlinear electronic response in topological materials presents many challenges. In particular, the Bloch wavefunction phase of topological materials has undefined points in the Brillouin zone. This leads to singularities in the calculation of the interband and intraband transition dipole matrix elements of the semiconductor Bloch equations (SBEs). Here, we use the laser-electromagnetic velocity gauge p⋅A(t) to numerically integrate the SBEs and treat the singularity in the production of the electrical currents and HHG spectra with better numerical efficiency and more straightforward implementation. We used a prototype of Chern insulators (CIs), the Haldane model, to demonstrate our approach. The validity of the velocity gauge approach is demonstrated in the following way: for topologically trivial materials such as MoS2, qualitative agreement is achieved with the results of the length gauge approach and the time-dependent density functional theory. For the application of the velocity gauge approach to topological materials, Chern insulator is taken, using the two-band Haldane model. We found a good qualitative agreement between the velocity gauge and the length gauge approach in view of (i) the selection rules, (ii) the linear cutoff law scaling, and (iii) anomalous circular dichroism. We conclude that the velocity-gauge approach for HHG provides a theoretical tool to investigate topological materials

Group averaging in the (p,q) oscillator representation of SL(2,R)

We investigate refined algebraic quantisation with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.Comment: 34 pages. LaTeX with amsfonts, amsmath, amssymb. (References added; minor typos corrected.