The Effect of low Momentum Quantum Fluctuations on a Coherent Field Structure

In the present work the evolution of a coherent field structure of the Sine-Gordon equation under quantum fluctuations is studied. The basic equations are derived from the coherent state approximation to the functional Schr\"odinger equation for the field. These equations are solved asymptotically and numerically for three physical situations. The first is the study of the nonlinear mechanism responsible for the quantum stability of the soliton in the presence of low momentum fluctuations. The second considers the scattering of a wave by the Soliton. Finally the third problem considered is the collision of Solitons and the stability of a breather. It is shown that the complete integrability of the Sine-Gordon equation precludes fusion and splitting processes in this simplified model. The approximate results obtained are non-perturbative in nature, and are valid for the full nonlinear interaction in the limit of low momentum fluctuations. It is also found that these approximate results are in good agreement with full numerical solutions of the governing equations. This suggests that a similar approach could be used for the baby Skyrme model, which is not completely integrable. In this case the higher space dimensionality and the internal degrees of freedom which prevent the integrability will be responsable for fusion and splitting processes. This work provides a starting point in the numerical solution of the full quantum problem of the interaction of the field with a fluctuation.Comment: 15 pages, 9 (ps) figures, Revtex file. Some discussion expanded but conclusions unchanged. Final version to appear in PR

Quantum Collapse of a Small Dust Shell

The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by H\'aj{\'\i}\v{c}ek and Kucha\v{r}. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self adjoint momentum operator in the space of the shell motion which is confined to a half line. The first problem is solved by identifying and neglecting orbits of small action in order to obtain a single valued Hamiltonian. The second problem is solved by introducing an appropriate lapse function. The resulting quantum mechanics is then studied by means of analytical and numerical techniques. We find that the region of total collapse has very small probability. We also find that the solution concentrates around the classical Schwarzschild radius. The present work obtains from first principles a quantum mechanics for the shell and provides numerical solutions, whose behavior is explained by a detailed WKB analysis for a wide class of collapsing shells.Comment: 23 pages, 8 figures, Revtex4 fil

Additional restrictions on quasi-exactly solvable systems

In this paper we discuss constraints on two-dimensional quantum-mechanical systems living in domains with boundaries. The constrains result from the requirement of hermicity of corresponding Hamiltonians. We construct new two-dimensional families of formally exactly solvable systems and applying such constraints show that in real the systems are quasi-exactly solvable at best. Nevertheless in the context of pseudo-Hermitian Hamiltonians some of the constructed families are exactly solvable.Comment: 11 pages, 3 figures, extended version of talk given at the International Workshop on Classical and Quantum Integrable Systems "CQIS-06", Protvino, Russia, January 23-26, 200

Variational Approach to Gaussian Approximate Coherent States: Quantum Mechanics and Minisuperspace Field Theory

This paper has a dual purpose. One aim is to study the evolution of coherent states in ordinary quantum mechanics. This is done by means of a Hamiltonian approach to the evolution of the parameters that define the state. The stability of the solutions is studied. The second aim is to apply these techniques to the study of the stability of minisuperspace solutions in field theory. For a $\lambda \varphi^4$ theory we show, both by means of perturbation theory and rigorously by means of theorems of the K.A.M. type, that the homogeneous minisuperspace sector is indeed stable for positive values of the parameters that define the field theory.Comment: 26 pages, Plain TeX, no figure

New Algebraic Quantum Many-body Problems

We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to root systems, in some cases with an additional external field. The quasi-exactly solvable models can be considered as deformations of the previous ones which share their algebraic character.Comment: LaTeX 2e with amstex package, 36 page

From nonassociativity to solutions of the KP hierarchy

A recently observed relation between 'weakly nonassociative' algebras A (for which the associator (A,A^2,A) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus A' of A) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with A' a matrix algebra, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.Comment: 7 pages, 4 figures, International Colloquium 'Integrable Systems and Quantum Symmetries', Prague, 15-17 June 200

Controls on Microbial and Oolitic Carbonate Sedimentation and Stratigraphic Cyclicity Within a Mixed Carbonate-Siliciclastic System: Upper Cambrian Wilberns Formation, Llano Uplift, Mason County, Texas, USA

The upper Cambrian Wilberns Formation in central Texas records deposition on a low-gradient shelf within a mixed carbonate–siliciclastic tidal-flat system that changes offshore to subtidal shelf and open-marine oolitic skeletal shoals with large microbial mounds. Siliciclastic sediment is interpreted to have been delivered to the tidal flat by aeolian processes because of the narrow range in grain size and paucity of clay. Tidal influence is dominant as evidenced by reversing currents and desiccation on the tidal flat, and megaripples with reversing current indicators in offshore shoals. Intraclastic conglomerates were deposited in broad channels on the tidal flats during storm surges. Microbialite deposition is interpreted to be controlled by accommodation favouring amalgamated thin biostromes developed in the tidal flat vs. larger mounds with greater synoptic relief in the offshore, and current energy resulting in preferential elongation of offshore mounds in a NE–SW orientation. Intertidal mounds and biostromes grew in the presence of significant siliciclastic flux and trapped it within their structure, whereas offshore large buildups incorporated little siliciclastic component. Oolite and skeletal grainstone formed in tide agitated shoals associated with large subtidal microbial mounds. Storms extensively recycled and redistributed skeletal and oolitic sands from the offshore shoals across the shelf as thin sand sheets. Spatial mixing of siliciclastic and carbonate sediment occurred across the tidal flat and shelf. Low-frequency and intermediate-frequency stratigraphic cycles were driven by shifts in the shoreline and changes in rate of siliciclastic flux in response to relative sea-level fluctuation. Random facies stacking and the lack of metre-scale cyclicity are interpreted to reflect stratigraphic incompleteness and an episodic signal introduced by storms