Laguerre semigroup and Dunkl operators

We construct a two-parameter family of actions \omega_{k,a} of the Lie algebra sl(2,R) by differential-difference operators on R^N \setminus {0}. Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from the interpolation of the Weil representation of Mp(N,R) and the minimal unitary representation of O(N+1,2) keeping smaller symmetries. We prove that this action \omega_{k,a} lifts to a unitary representation of the universal covering of SL(2,R), and can even be extended to a holomorphic semigroup \Omega_{k,a}. In the k\equiv 0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup by the second author with G. Mano (a=1). One boundary value of our semigroup \Omega_{k,a} provides us with (k,a)-generalized Fourier transforms F_{k,a}, which includes the Dunkl transform D_k (a=2) and a new unitary operator H_k (a=1), namely a Dunkl-Hankel transform. We establish the inversion formula, and a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty inequality for F_{k,a}. We also find kernel functions for \Omega_{k,a} and F_{k,a} for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.Comment: final version (some few typos, updated references

Twisted Gauge Theory Model of Topological Phases in Three Dimensions

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a $4$-cocycle $\omega$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the $3$-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasi-excitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $\omega$ and quantities derived from $\omega$. Our model fulfills a Hamiltonian extension of the $3+1$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.Comment: 37 pages, 9 figures, 4 tables; revised to improve the clarity; references adde

LQG for the Bewildered

We present a pedagogical introduction to the notions underlying the connection formulation of General Relativity - Loop Quantum Gravity (LQG) - with an emphasis on the physical aspects of the framework. We begin by reviewing General Relativity and Quantum Field Theory, to emphasise the similarities between them which establish a foundation upon which to build a theory of quantum gravity. We then explain, in a concise and clear manner, the steps leading from the Einstein-Hilbert action for gravity to the construction of the quantum states of geometry, known as \emph{spin-networks}, which provide the basis for the kinematical Hilbert space of quantum general relativity. Along the way we introduce the various associated concepts of \emph{tetrads}, \emph{spin-connection} and \emph{holonomies} which are a pre-requisite for understanding the LQG formalism. Having provided a minimal introduction to the LQG framework, we discuss its applications to the problems of black hole entropy and of quantum cosmology. A list of the most common criticisms of LQG is presented, which are then tackled one by one in order to convince the reader of the physical viability of the theory. An extensive set of appendices provide accessible introductions to several key notions such as the \emph{Peter-Weyl theorem}, \emph{duality} of differential forms and \emph{Regge calculus}, among others. The presentation is aimed at graduate students and researchers who have some familiarity with the tools of quantum mechanics and field theory and/or General Relativity, but are intimidated by the seeming technical prowess required to browse through the existing LQG literature. Our hope is to make the formalism appear a little less bewildering to the un-initiated and to help lower the barrier for entry into the field.Comment: 87 pages, 15 figures, manuscript submitted for publicatio

Boundary Integral Operators in Linear and Second-order Nonlinear Nano-optics

Recent advances in the fabrication of nanoscale structures have enabled the production of almost arbitrarily shaped nanoparticles and so-called optical metamaterials. Such materials can be designed to have optical properties not found in nature, such as negative index of refraction. Noble metal nanostructures can enhance the local electric field, which is beneficial for nonlinear optical effects. The study of nonlinear optical properties of nanostructures and metamaterials is becoming increasingly important due to their possible uses in nanoscale optical switches, frequency converters and many other devices.The responses of nanostructures depend heavily on their geometry, which calls for versatile modeling methods. In this work, we develop a boundary element method for the modeling of surface second-harmonic generation from isolated nanoparticles of very general shape. The method is also capable of modeling spatially periodic structures by the use of appropriate Green’s function. We further show how to utilize geometrical symmetries to lower the computational time and memory requirements in the boundary element method even in cases where the incident field is not symmetrical.We validate the boundary element approach by the calculation of second-harmonic scattering from gold spheres of different radii. Comparison to analytical solution reveals that under one percent relative error is easily achieved. The method is then applied to model second-harmonic microscopy of single gold nanodots and second-harmonic generation from arrays of L- and T-shaped gold particles. The agreement between the calculations and measurements is shown to be excellent.To provide a more intuitive understanding of the optical response of nanostructures, we develop a full-wave spectral approach, which is based on boundary integral operators. We present a theory which proves that the resonances of a smooth scatterer are isolated poles that occur at complex frequencies. Other types of singularities, such as branch-cuts, may occur only via the fundamental Green function or material dispersion. We propose a definition of an eigenvalue problem at fixed real frequencies which gives rise to modes defined over the surface of the scatterer. We illustrate that these modes accurately describe the optical responses that are usually seen for certain particle shapes when using plane-wave excitations. With the spectral approach, the resonance frequencies and the modal responses of a scatterer can be found as intrinsic properties independent of any incident field. We show that the spectral theory is compatible with the Mie theory for pherical particles and with a previously studied quasi-static theory in the limit of zero frequency

Automated Transverse Momentum Resummation for Electroweak Boson Production

The production of electroweak bosons, followed by leptonic decays, is among the most basic hard-scattering processes studied at hadron colliders. Such processes provide backgrounds to new physics searches, enable the study of possible anomalous gauge couplings and provide spectra for determining the W boson mass and the angle θW. At small transverse momentum qT , the electroweak boson production processes involve disparate scales, namely the small qT and the large mass M of the bosonic states. Fixed-order perturbative results suffer from large logarithms of the ratio of these scales and hence become unreliable. The appropriate treatment of these logarithms is their resummation. This thesis presents a framework for transverse momentum resummation for quark-induced boson production processes with arbitrary electroweak final states. The resummation is performed in an automated way and is based on reweighting events generated using a tree-level event generator. The kinematics of the electroweak final states are accessible, and this allows for the analysis of general observables in the small transverse momentum region. Making use of the event generator MadGraph5_aMC@NLO, the resummation is implemented at next-to-next-to-leading logarithmic accuracy and matched to next-to-leading fixed-order results. Results for Z and W boson production with leptonic decay as well as for WZ production are presented. The predictions are validated using an existing resummation code and compared to experimental measurements

Exploring Quantum Computation Through the Lens of Classical Simulation

It is widely believed that quantum computation has the potential to offer an ex- ponential speedup over classical devices. However, there is currently no definitive proof of this separation in computational power. Such a separation would in turn imply that quantum circuits cannot be efficiently simulated classically. However, it is well known that certain classes of quantum computations nonetheless admit an efficient classical description. Recent work has also argued that efficient classical simulation of quantum circuits would imply the collapse of the Polynomial Hierarchy, something which is commonly invoked in clas- sical complexity theory as a no-go theorem. This suggests a route for studying this ‘quantum advantage’ through classical simulations. This project looks at the problem of classically simulating quantum circuits through decompositions into stabilizer circuits. These are a restricted class of quantum computation which can be efficiently simulated classically. In this picture, the rank of the decomposition determines the temporal and spatial complexity of the simulation. We approach the problem by considering classical simulations of stabilizer circuits, introducing two new representations with novel features compared to previous meth- ods. We then examine techniques for building these so-called ‘stabilizer rank’ decom- positions, both exact and approximate. Finally, we combine these two ingredients to introduce an improved method for classically simulating broad classes of circuits using the stabilizer rank method