Group Theory and Quasiprobability Integrals of Wigner Functions

The integral of the Wigner function of a quantum mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0,1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric disks and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in Hilbert space carrying the positive discrete series representations of the algebra su(1,1)or so(2,1). The explicit relation between the spectra of operators associated with disks and circles with proportional radii, is given in terms of the dicrete variable Meixner polynomials.Comment: 11 pages, latex fil

Homogeneous Field and WKB Approximation In Deformed Quantum Mechanics with Minimal Length

In the framework of the deformed quantum mechanics with minimal length, we consider the motion of a non-relativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up to $\mathcal{O}(\beta)$. We also show that, if the slope of the potential at a turning point is too steep, the WKB connection formula fall apart around the turning point.Comment: 31 pages; v2: a new subsection about applications of WKB approximation and references added, published versio

Dobrushin-Kotecky-Shlosman theorem for polygonal Markov fields in the plane

We consider the so-called length-interacting Arak-Surgailis polygonal Markov fields with V-shaped nodes - a continuum and isometry invariant process in the plane sharing a number of properties with the two-dimensional Ising model. For these polygonal fields we establish a low-temperature phase separation theorem in the spirit of the Dobrushin-Kotecky-Shlosman theory, with the corresponding Wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model. As an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process, following the ideas of Fernandez, Ferrari and Garcia.Comment: 59 pages, new version revised according to the referee's suggestions and now publishe

Accurate video object tracking using a region-based particle filter

Usually, in particle filters applied to video tracking, a simple geometrical shape, typically an ellipse, is used in order to bound the object being tracked. Although it is a good tracker, it tends to a bad object representation, as most of the world objects are not simple geometrical shapes. A better way to represent the object is by using a region-based approach, such as the Region Based Particle Filter (RBPF). This method exploits a hierarchical region based representation associated with images to tackle both problems at the same time: tracking and video object segmentation. By means of RBPF the object segmentation is resolved with high accuracy, but new problems arise. The object representation is now based on image partitions instead of pixels. This means that the amount of possible combinations has now decreased, which is computationally good, but an error on the regions taken for the object representation leads to a higher estimation error than methods working at pixel level. On the other hand, if the level of regions detail in the partition is high, the estimation of the object turns to be very noisy, making it hard to accurately propagate the object segmentation. In this thesis we present new tools to the existing RBPF. These tools are focused on increasing the RBPF performance by means of guiding the particles towards a good solution while maintaining a particle filter approach. The concept of hierarchical flow is presented and exploited, a Bayesian estimation is used in order to assign probabilities of being object or background to each region, and the reduction, in an intelligent way, of the solution space , to increase the RBPF robustness while reducing computational effort. Also changes on the already proposed co-clustering in the RBPF approach are proposed. Finally, we present results on the recently presented DAVIS database. This database comprises 50 High Definition video sequences representing several challenging situations. By using this dataset, we compare the RBPF with other state-ofthe- art methods

Representation of a complex Green function on a real basis: I. General Theory

When the Hamiltonian of a system is represented by a finite matrix, constructed from a discrete basis, the matrix representation of the resolvent covers only one branch. We show how all branches can be specified by the phase of a complex unit of time. This permits the Hamiltonian matrix to be constructed on a real basis; the only duty of the basis is to span the dynamical region of space, without regard for the particular asymptotic boundary conditions that pertain to the problem of interest.Comment: about 40 pages with 5 eps-figure

Form factors of exponential fields for two-parametric family of integrable models

A two-parametric family of integrable models (the SS model) that contains as particular cases several well known integrable quantum field theories is considered. After the quantum group restriction it describes a wide class of integrable perturbed conformal field theories. Exponential fields in the SS model are closely related to the primary fields in these perturbed theories. We use the bosonization approach to derive an integral representation for the form factors of the exponential fields in the SS model. The same representations for the sausage model and the cosine-cosine model are obtained as limiting cases. The results are tested at the special points, where the theory contains free particles.Comment: 37 pages, 3 figures; some misprints corrected; Eq. (B.12b) correcte