Geometric quantization and foliation reduction

A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact Kähler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-Kähler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds.

In this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the Kähler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or “admissible”, values of momentum.

We first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems.

We then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces.

Finally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context.

A significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.

Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

Discrete Modeling of Granular Media: A NURBS-based Approach

This dissertation is concerned with the development of a new discrete element method (DEM) based on Non-Uniform Rational Basis Splines (NURBS). With NURBS, the new DEM is able to capture sphericity and angularity, the two particle morphological measures used in characterizing real grain geometries. By taking advantage of the parametric nature of NURBS, the Lipschitzian dividing rectangle (DIRECT) global optimization procedure is employed as a solution procedure to the closest-point projection problem, which enables the contact treatment of non-convex particles. A contact dynamics (CD) approach to the NURBS-based discrete method is also formulated. By combining particle shape flexibility, properties of implicit time-integration, and non-penetrating constraints, we target applications in which the classical DEM either performs poorly or simply fails, i.e., in granular systems composed of rigid or highly stiff angular particles and subjected to quasistatic or dynamic flow conditions. The CD implementation is made simple by adopting a variational framework, which enables the resulting discrete problem to be readily solved using off-the-shelf mathematical programming solvers. The capabilities of the NURBS-based DEM are demonstrated through 2D numerical examples that highlight the effects of particle morphology on the macroscopic response of granular assemblies under quasistatic and dynamic flow conditions, and a 3D characterization of material response in the shear band of a real triaxial specimen.

Nonclassical excitation and quantum interference in a three level atom

Non-classical properties and quantum interference (QI) in two-photon excitation of a three level atom (|1〉), |2〉, |3〉) in a ladder configuration, illuminated by multiple fields in non-classical (squeezed) and/or classical (coherent) states, is studied. Fundamentally new effects associated with quantum correlations in the squeezed fields and QI due to multiple excitation pathways have been observed. Theoretical studies and extrapolations of these findings have revealed possible applications which are far beyond any current capabilities, including ultrafast nonlinear mixing, ultrafast homodyne detection and frequency metrology. The atom used throughout the experiments was Cesium, which was magneto-optically trapped in a vapor cell to produce a Doppler-free sample. For the first part of the work the |1〉 → |2〉 → |3〉 transition (corresponding to the 6S_1/2F = 4 → 6P_3/2F' = 5 → 6D_5/2F" = 6 transition) was excited by using the quantum-correlated signal (Ɛ_s) and idler (Ɛ_i) output fields of a subthreshold non-degenerate optical parametric oscillator, which was tuned so that the signal and idler fields were resonant with the |1〉 → |2〉 and |2〉 → |3〉 transitions, respectively. In contrast to excitation with classical fields for which the excitation rate as a function of intensity has always an exponent greater than or equal to two, excitation with squeezed-fields has been theoretically predicted to have an exponent that approaches unity for small enough intensities. This was verified experimentally by probing the exponent down to a slope of 1.3, demonstrating for the first time a purely non-classical effect associated with the interaction of squeezed fields and atoms. In the second part excitation of the two-photon transition by three phase coherent fields Ɛ₁ , Ɛ₂ and Ɛ₀, resonant with the dipole |1〉 → |2〉 and |2〉 → |3〉 and quadrupole |1〉 → |3〉 transitions, respectively, is studied. QI in the excited state population is observed due to two alternative excitation pathways. This is equivalent to nonlinear mixing of the three excitation fields by the atom. Realizing that in the experiment the three fields are spaced in frequency over a range of 25 THz, and extending this scheme to other energy triplets and atoms, leads to the discovery that ranges up to 100's of THz can be bridged in a single mixing step. Motivated by these results, a master equation model has been developed for the system and its properties have been extensively studied.

The astrophysics of strongly interacting systems

This thesis presents investigations in four areas of theoretical astrophysics: the production of sterile neutrino dark matter in the early Universe, the evolution of small-scale baryon perturbations during the epoch of cosmological recombination, the effect of primordial magnetic fields on the redshifted 21-cm emission from the pre-reionization era, and the nonlinear stability of tidally deformed neutron stars.

In the first part of the thesis, we study the asymmetry-driven resonant production of 7 keV-scale sterile neutrino dark matter in the primordial Universe at temperatures T >~ 100 MeV. We report final DM phase space densities that are robust to uncertainties in the nature of the quark-hadron transition. We give transfer functions for cosmological density fluctuations that are useful for N-body simulations. We also provide a public code for the production calculation.

In the second part of the thesis, we study the instability of small-scale baryon pressure sound waves during cosmological recombination. We show that for relevant wavenumbers, inhomogenous recombination is driven by the transport of ionizing continuum and Lyman-alpha photons. We find a maximum growth factor less than ≈ 1.2 in 10⁷ random realizations of initial conditions. The low growth factors are due to the relatively short duration of the recombination epoch.

In the third part of the thesis, we propose a method of measuring weak magnetic fields, of order 10^-19 G (or 10^-21 G if scaled to the present day), with large coherence lengths in the inter galactic medium prior to and during the epoch of cosmic reionization. The method utilizes the Larmor precession of spin-polarized neutral hydrogen in the triplet state of the hyperfine transition. We perform detailed calculations of the microphysics behind this effect, and take into account all the processes that affect the hyperfine transition, including radiative decays, collisions, and optical pumping by Lyman-alpha photons.

In the final part of the thesis, we study the non-linear effects of tidal deformations of neutron stars (NS) in a compact binary. We compute the largest three- and four-mode couplings among the tidal mode and high-order p- and g-modes of similar radial wavenumber. We demonstrate the near-exact cancellation of their effects, and resolve the question of the stability of the tidally deformed NS to leading order. This result is significant for the extraction of binary parameters from gravitational wave observations.

Magnetic alignment of high-aspect ratio microwires into vertical arrays

Fundamental studies of magnetic alignment of highly anisotropic mesostructures can enable the clean-room-free fabrication of flexible, array-based solar and electronic devices, in which preferential orientation of nano- or microwire-type objects is desired. In this study, ensembles of 100 micron long Si microwires with ferromagnetic Ni and Co coatings are oriented vertically in the presence of magnetic fields. The degree of vertical alignment and threshold field strength depend on geometric factors, such as microwire length and ferromagnetic coating thickness, as well as interfacial interactions, which are modulated by varying solvent and substrate surface chemistry. Microwire ensembles with vertical alignment over 97% within 10 degrees of normal, as measured by X-ray diffraction, are achieved over square cm scale areas and set into flexible polymer films. A force balance model has been developed as a predictive tool for magnetic alignment, incorporating magnetic torque and empirically derived surface adhesion parameters. As supported by these calculations, microwires are shown to detach from the surface and align vertically in the presence of magnetic fields on the order of 100 gauss. Microwires aligned in this manner are set into a polydimethylsiloxane film where they retain their vertical alignment after the field has been removed and can subsequently be used as a flexible solar absorber layer. Finally, these microwires arrays can be protected for use in electrochemical cells by the conformal deposition of a graphene layer.

An integrated design approach to power systems : from power flows to electricity markets

Power system is at the brink of change. Engineering needs, economic forces and environmental factors are the main drivers of this change. The vision is to build a smart electrical grid and a smarter market mechanism around it to fulfill mandates on clean energy. Looking at engineering and economic issues in isolation is no longer an option today; it needs an integrated design approach. In this thesis, I shall revisit some of the classical questions on the engineering operation of power systems that deals with the nonconvexity of power flow equations. Then I shall explore some issues of the interaction of these power flow equations on the electricity markets to address the fundamental issue of market power in a deregulated market environment. Finally, motivated by the emergence of new storage technologies, I present an interesting result on the investment decision problem of placing storage over a power network. The goal of this study is to demonstrate that modern optimization and game theory can provide unique insights into this complex system. Some of the ideas carry over to applications beyond power systems.

Joule-Thomson coefficients of propane and n-butane

Experimental Joule-Thomson measurements were made on gaseous propane at temperatures from 100 to 280˚F and at pressures from 8 to 66 psia. Joule-Thomson measurements were also made on gaseous n-butane at temperatures from 100 to 280˚ and at pressures from 8 to 42 psia. For propane, the values of these measurements ranged from 0.07986˚F/psi at 280˚F and 8.01 psia to 0.19685˚F/psi at 100˚F and 66.15 psia. For n-butane, the values ranged from 0.11031˚F/psi at 280˚F and 9.36 psia to 0.30141˚F/psi at 100˚F and 41.02 psia. The experimental values have a maximum error of 1.5 percent.

For n-butane, the measurements of this study did not agree with previous Joule-Thomson measurements made in the Laboratory in 1935. The application of a thermal-transfer correction to the previous experimental measurements would cause the two sets of data to agree. Calculated values of the Joule-Thomson coefficient from other types of p-v-t data did agree with the present measurements for n-butane.

The apparatus used to measure the experimental Joule-Thomson coefficients had a radial-flow porous thimble and was operated at pressure changes between 2.3 and 8.6 psi. The major difference between this and other Joule-Thomson apparatus was its larger weight rates of flow (up to 6 pounds per hour) at atmospheric pressure. The flow rate was shown to have an appreciable effect on non-isenthalpic Joule-Thomson measurements.

Photographic materials on pages 79-81 are essential and will not reproduced clearly on Xerox copies. Photographic copies should be ordered.

I. The attenuation and dispersion of sound in a condensing medium. II. The flow of a gas-particle mixture downstream of a normal shock in a nozzle

I. The attenuation of sound due to particles suspended in a gas was first calculated by Sewell and later by Epstein in their classical works on the propagation of sound in a two-phase medium. In their work, and in more recent works which include calculations of sound dispersion, the calculations were made for systems in which there was no mass transfer between the two phases. In the present work, mass transfer between phases is included in the calculations.

The attenuation and dispersion of sound in a two-phase condensing medium are calculated as functions of frequency. The medium in which the sound propagates consists of a gaseous phase, a mixture of inert gas and condensable vapor, which contains condensable liquid droplets. The droplets, which interact with the gaseous phase through the interchange of momentum, energy, and mass (through evaporation and condensation), are treated from the continuum viewpoint. Limiting cases, for flow either frozen or in equilibrium with respect to the various exchange processes, help demonstrate the effects of mass transfer between phases. Included in the calculation is the effect of thermal relaxation within droplets. Pressure relaxation between the two phases is examined, but is not included as a contributing factor because it is of interest only at much higher frequencies than the other relaxation processes. The results for a system typical of sodium droplets in sodium vapor are compared to calculations in which there is no mass exchange between phases. It is found that the maximum attenuation is about 25 per cent greater and occurs at about one-half the frequency for the case which includes mass transfer, and that the dispersion at low frequencies is about 35 per cent greater. Results for different values of latent heat are compared.

II. In the flow of a gas-particle mixture through a nozzle, a normal shock may exist in the diverging section of the nozzle. In Marble’s calculation for a shock in a constant area duct, the shock was described as a usual gas-dynamic shock followed by a relaxation zone in which the gas and particles return to equilibrium. The thickness of this zone, which is the total shock thickness in the gas-particle mixture, is of the order of the relaxation distance for a particle in the gas. In a nozzle, the area may change significantly over this relaxation zone so that the solution for a constant area duct is no longer adequate to describe the flow. In the present work, an asymptotic solution, which accounts for the area change, is obtained for the flow of a gas-particle mixture downstream of the shock in a nozzle, under the assumption of small slip between the particles and gas. This amounts to the assumption that the shock thickness is small compared with the length of the nozzle. The shock solution, valid in the region near the shock, is matched to the well known small-slip solution, which is valid in the flow downstream of the shock, to obtain a composite solution valid for the entire flow region. The solution is applied to a conical nozzle. A discussion of methods of finding the location of a shock in a nozzle is included.