Paul Bracken

By Paul Bracken

Part of the book: Theoretical Concepts of Quantum Mechanics

By Paul Bracken

Part of the book: Advances in Quantum Mechanics

By Paul Bracken

Several subjects which reside in the overlap area of quantum mechanics, statistical physics and thermodynamics are investigated in depth. This collection of subjects shares a common domain which is referred to as Maxwell’s demon. The classical version of this idea is introduced, and then, the contribution made by Szilard to the subject is presented. Several demons are considered, and it is shown that to best understand this area, quantum mechanics and the role information plays in it must be appreciated deeply.

Part of the book: Research Advances in Quantum Dynamics

By Paul Bracken

Differential operators that are defined on a differentiable manifold can be used to study various properties of manifolds. The spectrum and eigenfunctions play a very significant role in this process. The objective of this chapter is to develop the heat equation method and to describe how it can be used to prove the Hodge Theorem. The Minakshisundaram‐Pleijel parametrix and asymptotic expansion are then derived. The heat equation asymptotics can be used to give a development of the Gauss‐Bonnet theorem for two‐dimensional manifolds.

Part of the book: Manifolds

By Paul Bracken

The structure equations for a two‐dimensional manifold are introduced and two results based on the Codazzi equations pertinent to the study of isometric surfaces are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem due to Bonnet is obtained. A transformation for the connection forms is developed. It is proved that the angle of deformation must be harmonic, and that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations, and these results can be used to characterize Bonnet surfaces.

Part of the book: Manifolds

By Paul Bracken

In this chapter, some recent advances in the area of generalized Weierstrass representations will be given. This is an approach to the theory of surfaces in Euclidean three space. Weierstrass representations permit the explicit construction of surfaces in the designated space. The discussion proceeds in a novel and introductory manner. The inducing formulas for the coordinates of a surface are derived and important conservation laws are formulated. These lead to the inducing mechanism of a surface in terms of solutions to a system of two-dimensional Dirac equations. A set of fundamental forms as well as expressions for the mean and Gaussian curvatures are derived. The Cartan moving frame picture is also formulated to put everything in a broader perspective. A connection with the nonlinear sigma model is presented, which has important applications in physics. Some relationships are established between integrable systems and geometry by way of conclusion.

Part of the book: Manifolds II

By Paul Bracken

Part of the book: Research Advances in Chaos Theory

By Paul Bracken

Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in such a way as to illustrate many of the subjects that have appeared.

Part of the book: Quantum Mechanics

By Paul Bracken

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.

Part of the book: A Collection of Papers on Chaos Theory and Its Applications

By Paul Bracken

Gauge theories make up a class of physical theories that attempt to describe the physics of particles at a fundamental level. The purpose here is to study Yang-Mills theory at the classical level in terms of the geometry of fiber bundles and differentiable manifolds. It is shown how fundamental particles of bosonic and fermionic nature can be described mathematically. The Lagrangian for the basic interactions is presented and then put together in a unified form. Finally, some basic theorems are proved for a Yang-Mills on compact four-dimensional manifolds.

Part of the book: Manifolds III