Eli Hawkins

On the nonlocality of the fractional Schrödinger equation

A number of papers over the past eight years have claimed to solve the fractional Schrodinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrodinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrodinger equation for the general fractional parameter α. On a more positive note, we present a solution to the fractional Schrodinger equation for the one-dimensional harmonic oscillator with α=1.

QUANTIZATION OF EQUIVARIANT VECTOR BUNDLES

Abstract:The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (A coadjoint orbit is a symplectic manifold with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail.This subject should not be confused with the quantization of the total space of a vector bundle such as the cotangent bundle.

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

Abstract: I repeat my definition for quantization of a vector bundle. For the cases of the Toeplitz and geometric quantizations of a compact Kähler manifold, I give a construction for quantizing any smooth vector bundle, which depends functorially on a choice of connection on the bundle.Using this, the classification of formal deformation quantizations, and the formal, algebraic index theorem, I give a simple proof as to which formal deformation quantization (modulo isomorphism) is derived from a given geometric quantization.

Evolution in quantum causal histories

We provide a precise definition and analysis of quantum causal histories (QCHs). A QCH consists of a discrete, locally finite, causal pre-spacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the Wheeler–De Witt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.

Hamiltonian Gravity and Noncommutative Geometry

Abstract:A version of foliated spacetime is constructed in which the spatial geometry is described as a time-dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action.

Fredholm modules for quantum euclidean spheres

The quantum Euclidean spheres, SqN−1, are (noncommutative) homogeneous spaces of quantum orthogonal groups, SOq(N). The ∗-algebra A(SN−1q) of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres SqN−1. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i.e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra A(SN−1q).

Comment on “On the consistency of solutions of the space fractional Schrödinger equation” [J. Math. Phys. 53, 042105 (2012)]

In Bayins paper [J. Math. Phys. 53, 042105 (2012)]10.1063/1.4705268, he claims to prove the consistency of the purported piece-wise solutions to the fractional Schrodinger equation for an infinite square well. However, his calculation uses standard contour integral techniques despite the absence of an analytic integrand. The correct calculation is presented and supports our earlier work proving that the purported piece-wise solutions do not solve the fractional Schrodinger equation for an infinite square well [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, J. Math. Phys. 51, 062102 (2010)]10.1063/1.3430552.

An Obstruction to Quantization of the Sphere

In the standard example of strict deformation quantization of the symplectic sphere S2, the set of allowed values of the quantization parameter ħ is not connected; indeed, it is almost discrete. Li recently constructed a class of examples (including S2) in which ħ can take any value in an interval, but these examples are badly behaved. Here, I identify a natural additional axiom for strict deformation quantization and prove that it implies that the parameter set for quantizing S2 is never connected.

A Cohomological Perspective on Algebraic Quantum Field Theory

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model.This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited.To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

ON THE RELATIONSHIP BETWEEN SIGMA MODELS AND SPIN CHAINS

We consider the two-dimensional