Arlen Anderson

Information-theoretic measure of uncertainty due to quantum and thermal fluctuations

We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information I of the phase-space probability distribution , where |z> are coherent states and ρ is the density matrix. As shown by Lieb I ≥ 1, and this bound represents a strengthened version of the uncertainty principle. For a harmonic oscillator in a thermal state, I coincides with von Neumann entropy, −Tr(ρlnρ), in the high-temperature regime, but unlike entropy, it is nonzero (and equal to the Lieb bound) at zero temperature. It therefore supplies a nontrivial measure of uncertainty due to both quantum and thermal fluctuations

Complex random surfaces

Abstract Several infinite sets of models of random surfaces, defined by means of integrals over matrix ensembles, are solved in a double-scaling limit. These models are exactly soluble in at least two distinct large N limits. The triangulated surfaces are complicated due to the existence of two distinct kinds of vertices in the triangulations. In one limit, the matrices possess a finite and fixed number of degrees of freedom as N becomes large-nevertheless, these models possess a nontrivial double-scaling limit. A special case of the other limit is known to describe two-dimensional quantum gravity.

Intertwining Operators for Solving Differential Equations, with Applications to Symmetric Spaces

The use of intertwining operators to solve both ordinary and partial differential equations is developed. Classes of intertwining operators are constructed which transform between Laplacians which are self-adjoint with respect to different non-trivial measures. In the two-dimensional case, the intertwining operator transforms a non-separable partial differential operator to a separable one. As an application, the heat kernels on the rank 1 and rank 2 symmetric spaces are constructed.

Phase space path integration of integrable quantum systems

Abstract A new method for exact evaluation of phase space path integrals for integrable quantum systems is presented. By making use of point canonical and other transformations to bring the Hamiltonian to a form linear in the coordinates, the path integral is changed so that the functional q integration may be done. This produces a momentum delta functional which can be evaluated to give an ordinary integral. This procedure is applied to find an expression for the exact propagator for a particle in the harmonic oscillator potential, the inverse quadratic potential, the Coulomb potential, the Morse potential, the 1 cosh 2 (x) potential, and for a free particle propagating on the 2-sphere. The latter two problems are solved for the first time wholly within the path integral formalism.

Unitarity restoration in the presence of closed timelike curves.

A proposal is made for a mathematically unambiguous treatment of evolution in the presence of closed timelike curves. In constrast to other proposals for handling the naively nonunitary evolution that is often present in such situations, this proposal is causal, linear in the initial density matrix and preserves probability. It provides a physically reasonable interpretation of invertible nonunitary evolution by redefining the final Hilbert space so that the evolution is unitary or equivalently by removing the nonunitary part of the evolution operator using a polar decomposition.