Discrete-Time Path Distributions on Hilbert Space

We construct a path distribution representing the kinetic part of the Feynman path integral at discrete times similar to that defined by Thomas [1], but on a Hilbert space of paths rather than a nuclear sequence space. We also consider different boundary conditions and show that the discrete-time Feynman path integral is well-defined for suitably smooth potentials

Three-dimensional Quantum Slit Diffraction and Diffraction in Time

We study the quantum slit diffraction problem in three dimensions. In the treatment of diffraction of particles by a slit, it is usually assumed that the motion perpendicular to the slit is classical. Here we take into account the effect of the quantum nature of the motion perpendicular to the slit using the Green function approach [18]. We treat the diffraction of a Gaussian wave packet for general boundary conditions on the shutter. The difference between the standard and our three-dimensional slit diffraction models is analogous to the diffraction in time phenomenon introduced in [16]. We derive corrections to the standard formula for the diffraction pattern, and we point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity

Renormalization Group Analysis of a Simple Hierarchical Fermion Model

A simple hierarchical fermion model is constructed which gives rise to an exact renormalization transformation in a 2-dimensional parameter space. The behaviour of this transformation is studied. It has two hyperbolic fixed points for which the existence of a global critical line is proven. The asymptotic behaviour of the transformation is used to prove the existence of the thermodynamic limit in a certain domain in parameter space. Also the existence of a continuum limit for these theories is investigated using information about the asymptotic renormalization behaviour. It turns out that the “trivial” fixed point gives rise to a two-parameter family of continuum limits corresponding to that part of parameter space where the renorma]ization trajectories originate at this fixed point. Although the model is not very realistic it serves as a simple example of the application of the renormalization group to proving the existence of the thermodynamic limit and the continuum limit of lattice models. Moreover, it illustrates possible complications that can arise in global renormalization group behaviour, and that might also be present in other models where no global analysis of the renormalization transformation has yet been achieved

Probabilistic derivation of a noncommutative version of Varadhan’s Theorem

We give a simple probabilistic derivation of a special case of a noncommutative version of Varadhan’s theorem, first proved by Petz, Raggio and Verbeure. It is based on a Feynman-Kac representation combined with a standard large deviation argument. In the final section, this theorem is then extended to a more difficult situation with Bose-symmetry

The Statistical Mechanics of a Bethe-Ansatz Soluble Model

The non-linear Schroedinger model is considered as an example of a model that is exactly soluble by means of the Bethe Ansatz. The theory of large deviations is applied to give a rigorous derivation of the thermodynamic formalism for this model, first proposed by Yang and Yang (1969) on heuristic grounds

Renormalization and the Continuum Limit

It is explained how the renormalization transformation can be used to take the continuum limit of a lattice field. It is shown that, by rescaling, the problem can be formulated on a fixed lattice Z^d. The procedure is illustrated by two examples: the one-dimensional Euclidean free field and a hierarchical model with ϕ^4-interaction

Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schro\"dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman-Molchanov method to prove localization.Comment: To appear in Commun. Math. Phys. (1999

Two Order Parameters in Quantum XZ Spin Models with Gibbsian Ground States

We describe a family of quantum spin models which are generators of a discrete Markovian process. We show that that there exists an explicit expression for the ground state of such models and give a simple argument for the existence of two types of long-range order in such systems. Two special examples of these systems are analysed in detail

On the Generalised Random Energy Model

The Generalised Random Energy Model is a generalisation of the Random Energy Model introduced by Derrida to mimic the ultrametric structure of the Parisi solution of the Sherrington-Kirkpatrick model of a spin glass. It was solved exactly in two special cases by Derrida and Gardner. A rigorous analysis by Capocaccia et al. claimed to give a complete solution for the thermodynamics of the model in the general case. Here we use Large Deviation Theory to analyse the model along the lines followed by Dorlas and Wedagedera for the Random Energy Model. The resulting variational expression for the free energy is the same as that found by Capocaccia et al. We show that it can be evaluated in a very simple way. We find that the answer given by Capocaccia et al. is incorrect