Quantum Approach to Cournot-type Competition

The aim of this paper is to investigate Cournot-type competition in the quantum domain with the use of the Li-Du-Massar scheme for continuous-variable quantum games. We derive a formula which, in a simple way, determines a unique Nash equilibrium. The result concerns a large class of Cournot duopoly problems including the competition, where the demand and cost functions are not necessary linear. Further, we show that the Nash equilibrium converges to a Pareto-optimal strategy profile as the quantum correlation increases. In addition to illustrating how the formula works, we provide the readers with two examples.     

Non-traditional Approach to Quantum Computing

The N-qubit system characterized by an effective spin \(S = 2^{N - 1} - {1/2}\) is carried out in the representation of two coupled harmonic oscillators. It is shown that quantum computing results obtained with spinor algebra can be obtained also using the algebra of two coupled harmonic oscillators which is a convenient formalism, especially in the case of large number of qubits. In this formalism the non-abelian and abelian groups of the order of 16 related to one- and two-qubit systems were found. The structure of Cayley tables of those groups is different due to different commutation (anticommutation) relations for operators forming each group.     

Homotopy Approach to Quantum Gravity

Gravity may be a quantum-space-time effect. General relativity is quantized by small generic changes in its commutation relations that make its Lie algebras simple on all levels, positing extra variables frozen by self-organization as needed. This quantizes space-time coordinates as well as fields and eliminates physical singularities. Fermi statistics and sl (nℝ) Lie algebras are assumed for all levels. Spin 1/2 is taken to be anomalous, arising from vacuum organization; the spin-statistics relation is incorporated. The gravitational field is quartic in Fermi variables. Einstein’s non-commutativity of parallel transport emerges as a vestige of Heisenberg’s quantum non-commutativity near the classical limit.     

Macroscopic Approach to Quantum Optics

This paper investigates how an approach alternative to the canonical quantization schemes may be used to describe Quantum Optics phenomena. By utilizing the approach pioneered by Keldysh, we derive equations for the time dependent correlation functions of the quantized optical fields. These contain the coupling to matter in linear and nonlinar response functions which replace the material parameters of phenomenological macroscopic theories. We present these results as alternatives to existing theoretical methods in Quantum Optics. The paper presents the general formulation of the theory, derives the equations in some specific cases relating to non‐linear optics, and solves some illustrative special cases. We regain known results but also some additional terms deriving from the quantum fluctuations of the material media.     

Topological Quantum Field Theory for the Universal Quantum Invariant

We extend the universal quantum invariant defined in [15] to an invariant of 3-manifolds with boundaries, and show that the invariant satisfies modified axioms of TQFT. Received: 22 August 1996 / Accepted: 17 January 1997     

Quantum Group Approach to q-Special Functions

Quantum groups are used to define q-special functions. The Casimir operatorsof a variation of SU _q(2) and E _q(2) are derived. The proposed q-associated Legendreand q-Bessel functions are the eigenfunctions of the Casimirs. The results differfrom ordinary q-special functions, but this is expected since the q-generalizationis not unique.     

Quantum Approach to Fast Protein-Folding Time

In the traditional random-conformational-search model,various hypotheses with a series of meta-stable intermediate states were proposed to resolve the Levinthal paradox in protein-folding time.Here we introduce a quantum strategy to formulate protein folding as a quantum walk on a definite graph, which provides us a g'eneral framework without making hypotheses.Evaluating it by the mean of first passage time,we find that the folding time via our quantum approach is much shorter than the one obtained via.classical random walks.This idea is expected to evoke more insights for future studies.     

凝聚体中的拓扑量子态

探寻拓扑上非平庸的凝聚体物质状态,特别是其电子结构和输运性质,是当前凝聚体物理 学领域非常重要的前沿研究方向。本文讨论的大多数主题都与电子波函数的拓扑性质有关。全文 除简短的引言外,包括拓扑量子现象、各种拓扑相、拓扑性准粒子的异常输运性质、拓扑性集 体激发和耦合激发,以及继续发展的拓扑量子态研究五个章节。这些章节着重反映拓扑量子态研 究的各个侧面,汇总起来方可以凸显凝聚体中拓扑量子态的全貌。     

Topological Knots in Quantum Spin Systems

Knot theory provides a powerful tool for understanding topological matters in biology, chemistry, and physics.Here knot theory is introduced to describe topological phases in a quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other, forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

Density-Functional Fidelity Approach to Quantum Phase Transitions

We propose a new approach to quantum phase transitions in terms of the density-functional fidelity, which measures the similarity between density distributions of two ground states in parameter space. The key feature of the approach is such that the density-functional fidelity can be measured easily in experiments. Both the validity and versatility of the approach are checked by the Lipkin-Meshkov-Glick model and the one-dimensional Hubbard model.     

Modified De Broglie-Bohm Approach to Quantum Mechanics

A modified de Broglie-Bohm (dBB) approach to quantum mechanics is presented. In this new deterministic theory, which uses complex methods in an intermediate step, the problem of zero velocity for bound states encountered in the dBB formulation does not appear. Also, this approach is equivalent to standard quantum mechanics when averages of observables like position, momentum and energy are taken.     

Quantum Information Approach to the Ultimatum Game

The paper is devoted to quantization of extensive games with the use of both the Marinatto-Weber and the Eisert-Wilkens-Lewenstein concept of quantum game. We revise the current conception of quantum ultimatum game and we show why the proposal is unacceptable. To support our approach, we present a new idea of the quantum ultimatum game. Our scheme also makes a point of departure for a general protocol for quantizing extensive games.     

A Unified Algebraic Approach to Quantum Theory

Conventional approaches to quantum mechanics are essentially dualistic. This is reflected in the fact that their mathematical formulation is based on two distinct mathematical structures: the algebra of dynamical variables (observables) and the vector space of state vectors. In contrast, coherent interpretations of quantum mechanics highlight the fact that quantum phenomena must be considered as undivided wholes. Here, we discuss a purely algebraic formulation of quantum mechanics. This formulation does not require the specification of a space of state vectors; rather, the required vector spaces can be identified as substructures in the algebra of dynamical variables (suitably extended for bosonic systems). This formulation of quantum mechanics captures the undivided wholeness characteristic of quantum phenomena, and provides insight into their characteristic nonseparability and nonlocality. The interpretation of the algebraic formulation in terms of quantum process is discussed.     

几个P-ylide反应机理的量子拓扑研究

采用MP2（FC）／6－311＋＋G（d,p) 对磷叶立德和类磷叶立德自由基反应机理进行了 探讨。优化了中间体、过渡态和产物的几何构型,并采用内禀 反应进行追踪。侧重从量子拓扑学的角度,对反应过程中各点进行电子密度拓扑分析,讨论了反应过程中化学键的断裂、生成和化学键的变化规律。上述两个反应都经功三员环过渡态,找到了这类反应的能量过渡态和结构过渡态。     

A Process Algebra Approach to Quantum Electrodynamics

The process algebra program is directed towards developing a realist model of quantum mechanics free of paradoxes, divergences and conceptual confusions. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM) without the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting, and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. Computations are carried out using the configuration space process covering map, although the connection to second quantization has not been fully explored.     

On Non-conservative Forces and Topological Quantum Phases

A generic non-conservative force, applied to an interferometer particle for a period in the past and then turned off, leaves stationary phase and fringe shifts in the now force-free interferometer. Both Aharonov-Bohm and Aharonov-Casher topological phase and fringe shifts can be created in this way. The specific sources of the non-conservative forces behind Aharonov-Bohm and Aharonov-Casher stationary fringe shifts are, respectively, Faraday induction fields and Maxwell displacement currents, now off.