This thesis explores various perspectives on quantum phenomena, and how our understanding of these phenomena is informed by the study of general probabilistic theories. Particular attention is given to quantum nonlocality, and its interaction with areas of physical and mathematical interest such as entropy, reversible dynamics, information-based games and the idea of negative probability. We begin with a review of non-signaling distributions and convex operational theories, including “black box” descriptions of experiments and the mathematics of convex vector spaces. In Chapter 3 we derive various classical and quantum-like quasiprobabilistic representations of arbitrary non-signaling distributions. Previously, results in which the density operator is allowed to become non-positive [1] have proved useful in derivations of quantum theory from physical requirements [2]; we derive a dual result in which the measurement operators instead are allowed to become non-positive, and show that the generation of any non-signaling distribution is possible using a fixed separable state with negligible correlation. We also derive two distinct “quasi-local” models of non-signaling correlations. Chapter 4 investigates non-local games, in particular the game known as Information Causality. By analysing the probability of success in this game, we prove the conjectured tightness of a bound given in [3] concerning how well entanglement allows us to perform the task of random access coding, and introduce a quadratic bias bound which seems to capture a great deal of information about the set of quantum-achievable correlations. By reformulating Information Causality in terms of entropies, we find that a sensible measure of entropy precludes many general probabilistic theories whose non-locality is stronger than that of quantum theory. Chapter 5 explores the role that reversible transitivity (the principle that any two pure states are joined by a reversible transformation) plays as a characteristic feature of quantum theory. It has previously been shown that in Boxworld, the theory allowing for the full set of non-signaling correlations, any reversible transformation on a restricted class of composite systems is merely a composition of relabellings of measurement choices and outcomes, and permutations of subsystems [4]. We develop a tabular description of Boxworld states and effects first introduced in [5], and use this to extend this reversibility result to any composite Boxworld system in which none of the subsystems are classical.

This thesis contains a study of topological phases of matter, with a strong emphasis on symmetry as a unifying theme. We take the point of view that the "topology" in many examples of what is loosely termed "topological matter", has its origin in the symmetry data of the system in question. From the fundamental work of Wigner, we know that topology resides not only in the group of symmetries, but also in the cohomological data of projective unitary-antiunitary representations. Furthermore, recent ideas from condensed matter physics highlight the fundamental role of charge-conjugation symmetry. With these as physical motivation, we propose to study the topological features of gapped phases of free fermions through a Z₂-graded C*-algebra encoding the symmetry data of their dynamics. In particular, each combination of time reversal and charge conjugation symmetries can be associated with a Clifford algebra. K-theory is intimately related to topology, representation theory, Clifford algebras, and Z₂-gradings, so it presents itself as a powerful tool for studying gapped topological phases. Our basic strategy is to use various K</em-theoretic invariants of the symmetry algebra to classify symmetry-compatible gapped phases. The super-representation group of the algebra classifies such gapped phases, while its K-theoretic difference-group classifies the obstructions in passing between two such phases. Our approach is a noncommutative version of the twisted K-theory approach of Freed--Moore, and generalises the K-theoretic classification first suggested by Kitaev. It has the advantage of conceptual simplicity in its uniform treatment of all symmetries. Physically, it encompasses phenomena which require noncommutative algebras in their description; mathematically, it clarifies and provides rigour to the meaning of "homotopic phases", and easily explains the salient features of Kitaev's Periodic Table.

This thesis addresses the problem of a form of anomalous decoherence that sheds light into the spectroscopy of blinking quantum dots. The system studied is a two-state system, interacting with an external environment that has the effect of establishing an interaction between the two states, via a coherence generating coupling, called inphasing. The collisions with the environment produce also decoherence, named dephasing. Decoherence is interpreted as the entanglement of the coherent superposition of these two states with the environment. The joint action of inphasing and dephasing generates a Markov master equation statistically equivalent to a random walker jumping from one state to the other. This model can be used to describe intermittent fluorescence, as a sequence of "light on" and "light off" states. The experiments on blinking quantum dots indicate that the sojourn times are distributed with an inverse power law. Thus, a proposal to turn the model for Poisson fluorescence intermittency into a model for non-Poisson fluorescence intermittency is made. The collision-like interaction of the two-state system with the environment is assumed to takes place at random times rather than at regular times. The time distance between one collision and the next is given by a distribution, called the subordination distribution. If the subordination distribution is exponential, a sequence of collisions yielding no persistence is turned into a sequence of "light on" and "light off" states with significant persistence. If the subordination function is an inverse power law the sequel of "light on" and "light off" states becomes equivalent to the experimental sequences. Different conditions are considered, ranging from predominant inphasing to predominant dephasing. When dephasing is predominant the sequel of "light on" and "light off" states in the time asymptotic limit becomes an inverse power law. If the predominant dephasing involves a time scale much larger than the minimum time scale accessible to the experimental observation, thereby generating persistence, the resulting distribution becomes a Mittag-Leffler function. If dephasing is predominant, in addition to the inverse power law distribution of "light off" and "light on" time duration, a strong correlation between "light on" and "light off" state is predicted.

Many are beginning to see the promise that the quantum world has offered those who manage and lead organizations (Wheatley, 1992; Zohar, 1997). The Newtonian world is one in which all "things" are reduced to their smallest parts, separated, divided, and analyzed with predictability, with complete control being the ultimate goal. The quantum world is one of infinite possibilities, infinite fields of influence, and infinite relationships. The hallmark characteristics found in a manager who has been schooled in the quantum sciences are flexibility, responsiveness, synchronicity, serendipity, creativity, innovation, participation, and motivation. In a quantum organization there is the constant awareness of the whole system, but there is also diversity (wave or particle), which allows for self-organization that is based on the environment and its requirements. In the quantum world many paths lead from A to Z, and depending on the path chosen, numerous realities wait to unfold. It was the goal of this research to explore the changing of leader behaviors through exposure to the models and theories found in quantum physics. From a quantum perspective this behavior change is possible; the only question is the readiness, willingness, and ability of the leaders to allow their behaviors to be surfaced and challenged. These are indeed the greatest challenges for all people as they proceed through life and work - readiness for change, willingness to change, and ability to surface key areas where change is needed.

This thesis examines the quantum dynamics of electrons in periodic semiconductor superlattices in the presence of electric fields, especially uniform static fields. Chapter 1 is an introduction to this vast and active field of research, with an analysis and suggested solutions to the fundamental theoretical difficulties. Chapter 2 is a detailed historical review of relevant theories, and Chapter 3 is a historical review of experiments. Chapter 4 is devoted to the time-independent quantum mechanical study of the electric-field-induced changes in the transmission properties of ballistic electrons, using the transfer matrix method. In Chapter 5, a new time-dependent quantum mechanical model free from the fundamental theoretical difficulties is introduced, with its validity tested at various limiting cases. A simplified method for calculating field-free bands of various potential models is designed. In Chapter 6, the general features of "Shifting Periodicity", a distinctive feature of this new model, is discussed, and a "Bloch-Floquet Theorem" is rigorously proven. Numerical evidences for the existence of Wannier-Stark-Ladders are presented, and the conditions for its experimental observability is also discussed. In Chapter 7, an analytical solution is found for Bloch Oscillations and Wannier-Stark-Ladders at low electric fields. In Chapter 8, a new quantum mechanical interpretation for Bloch Oscillations and Wannier-Stark-Ladders is derived from the analytical result. The extension of this work to the cases of time-dependent electric fields is also discussed.

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials.

Recently Wulfman found great merit in Barut's idea on atomic super-multiplets, and he introduced the concept of the generalized Hamiltonian that is the Hamiltonian of all atoms. 24 Investigating the Schrodinger equation with this generalized Hamiltonian, it should be possible to relate the properties of different atoms and find the structure of the periodic chart from fundamental principles of dynamics and group theory. One can can use the same kinds of methods for relating the properties of different states of a single hydrogen atom with the aid of the degeneracy group S0 (4) and dynamical group SO (4, 2). These groups represent the symmetries of the time-independent and time-dependent Schrodinger equations with ordinary Hamiltonian.(25,26) The idea then is to apply these methods to the system defined by a generalized Hamiltonian. In chapter II of this thesis, we will consider the classification of chemical elements, in the light of the concept of the generalized Hamiltonian. We will make a group theoretical classification based on the characteristics of the outermost electrons in the central-field model of atomic ground states. We conclude that the classification group may be SO (p,q) with p+q≧, p ≧4. In chapter III of this thesis, we will review Wulfman's work briefly and consider an application of his idea to the ground state of helium making use of the group SO (4,1) xSO (4,1). We arrive at the conclusion that we can obtain physically significant configuration mixing using SO (4,1) xSO (4,1) or SO (4,2) xSO (4,2) in a manner analogous to the way in which SO (4) xSO (4) is used to determine configuration mixing in doubly excited states of helium-like systems.

The properties of a physical system are determined by its equation of motion, and every such equation admits one-parameter groups which keep the equation invariant. Thus, for a particular system, if one can find the generator of a one-parameter group which keeps the equation and some further function or functional invariant, then one can change this system into others by changing the parameter, while keeping some properties constant. In this way, one can tell why different systems have some common properties. More importantly, one can use this method to find relationships between the physical properties of different systems. In the next section, we will illustrate the group theoretic approach by applying it to systems of two coupled oscillators and the hydrogen molecular ion. In section III of this thesis, we will investigate the helium atom system, considering both classical and quantum cases. In the quantum case our attention will be concentrated on the Schrodinger equation in matrix form. We will use a finite set of wavefunctions as our basis. Hence the results obtained will be approximate.

In this thesis, I study the group theoretic structure of the Schrodinger equations of simple systems by making use of a new systematic method. Group theoretic analysis of Schrodinger equations have been made previously by numerous physicists. The groups found may be classified as: a) geometrical groups; b) dynamical degeneracy groups; c) dynamical groups The geometrical group arises simply from the spatial symmetry of the system. Although the geometrical groups are very useful, they are not very interesting from the physical viewpoint. On the other hand, the study of the dynamical degeneracy groups and the dynamical group is very attractive because it reflects the dynamic of the system. Extensive studies have previously been made by other authors on systems which exhibit nontrivial degeneracy (accidental degeneracy). It turns out that all the states which belong to the same energy level provide a basis for a unitary irreducible representation of some compact group, and the group itself is generated by a set of constants of the motion. These groups are called “dynamical degeneracy groups”. Detailed discussion on degeneracy groups will be found in the paper by McIntosh alluded to above.

A group theoretic approach to Layzer's 1/2 expansion method is explored. In part this builds on earlier work of Wulfman(2), of Moshinsky et al(l4), and of Sinanoglu, Herrick(lS), and Kellman (16) on second row atoms. I investigate atoms with electrons in the 3s-3p-3d shell and find: 1. Wulfman's constant of motion accurately predicts configuration mixing for systems with two to eight electrons in the 3s-3p subshell. 2. The same constant of motion accurately predicts configuration mixing for systems with two electrons in the 3s-3p-3d shell. 3. It accurately predicts configuration mixing in systems of high angular momentum L and of high spin angular momentum S containing three electrons in the 3s-3p-3d shell, but gives less accurate results when L and S are both small. I also show how effective nuclear charges may be calculated by a group theoretical approach. In addition I explore several new methods for expressing electron repulsion operators in terms of operators of the 80(4,2) dynamical group of one - electron atoms.