On an analyticity property in (complex) phase space

Some consequences of one-valuedness on the real phase space for certain analytic functions over the complex phase space of a Hamiltonian system are demonstrated. The Bohr-Sommerfeld quantisation conditions are reformulated as one-valuedness conditions for these functions on the complex phase space.On leave of absence from Department of Physics, University of Khartoum, Sudan.     

Twistors and geometric quantisation theory

The basic geometry of twistors is developed as an application of geometric quantisation theory to the conformal group. It is found, however, that the Kähler form is not positive and that the quantised Hilbert space is trivial. This serves both to highlight difficulties in the quantisation theory for semi-simple Lie groups and to point out some of the obstacles in the way of developing a rigorous theory of twistors. It also suggests some areas in which the interplay between the two theories may be helpful in clarifying issues.     

On structure preserving quantisations

Maps of functions on classical phase space to quantum operators do not preserve the algebraic structure. After locating the algebraic reasons for it, the problem of quantisation is redefined and the Moyal bracket is discussed for its structure preservation. This quantisation entails the inclusion of Schwartz distributions to the space of classical functions.     

Semiclassical quantisation rules for the Dirac and Pauli equations

We derive explicit semiclassical quantisation conditions for the Dirac and Pauli equations. We show that the spin degree of freedom yields a contribution which is of the same order of magnitude as the Maslov correction in Einstein-Brillouin-Keller quantisation. In order to obtain this result a generalisation of the notion of integrability for a certain skew product flow of classical translational dynamics and classical spin precession has to be derived. Among the examples discussed is the relativistic Kepler problem with Thomas precession, whose treatment sheds some light on the amazing success of Sommerfeld’s theory of fine structure [Ann. Phys. (Leipzig) 51 (1916) 1].     

Constrained quantisation and θ-angles II

Certain aspects of the quantisation of 2D Yang-Mills theory are analysed with a new constrained quantisation technique, which quantises classical symplectic reduction. To apply this new method, it is necessary to take the gauge group as a Hilbert Lie group, and use the Wiener measure on a certain completion of the gauge group. The physical Hilbert space, the Hamiltonian and the Wilson loop observables for 2D Yang-Mills with a semisimple compact structure group are explicitly constructed using this technique. A new theory of θ-angle, not particular to 2D, emerges naturally. An interesting relation between Fock space coherent states and heat kernels on compact Lie groups is also found. All computations are explicit and simple in the sense that all results originate from the fundamental properties of Wiener measure. An example of SO(3) Yang-Mills is given. The possibility and difficulties of generalising our computations to higher dimensions by utilising the Levy measure (which generalises the Wiener measure) is briefly discussed.     

Quantum Circuits and Schr?dinger’s Cat States

Quantum systems that are confined to circuit geometries are called quantum circuits. Macroscopic superconducting circuits are quantum circuits which can be modelled using a Quantisation by Parts scheme based on the macroscopic wave function approach of Feynman. This paper studies the circuit composed of an input wire and an output plate. We find that in order to achieve a consistent theory of supercurrent flow we have to generalize the quantisation by parts scheme to quantise in a path space. The generalized theory predicts a current flow down the wire into the plane. In addition to a current flowing radially outwards in the plane, the theory allows a circulating current round the origin. Strikingly, the circulating current can flow clockwise or anti-clockwise in such a way as to generate a magnetic moment of magnitude half of a Bohr magneton for an orbiting electron in an atom and a magnetic flux half that of the magnetic flux quantum of a superconducting ring. There is also the possibility of a macroscopic superposition of the two states of opposing circulating currents resembling a Schr?dinger’s cat situation. Furthermore, we outline a setup involving an external magnetic field that may allow experimental tests of the theory.     

Circle actions in geometric quantisation

The aim of this article is to present unifying proofs for results in geometric quantisation with real polarisations by exploring the existence of symplectic circle actions. It provides an extension of Rawnsley’s results on the Kostant complex, and gives a partial result for the focus–focus contribution to geometric quantisation; as well as, an alternative proof for theorems of Śniatycki and Hamilton.     

Macroscopic Time Evolution and MaxEnt Inference for Closed Systems with Hamiltonian Dynamics

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set of phase space paths determined by solutions of Hamilton’s equations. The MaxEnt inference algorithm with Shannon’s concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes’ formulation of predictive statistical mechanics, and the concept of macroscopic reproducibility for time dependent processes. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation, and leads to corresponding loss of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system.     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

Additive Invariant Functionals for Dynamical Systems

We consider the problem of defining completely a class of additive conservation laws for the generalized Liouville equation whose characteristics are given by an arbitrary system of first-order ordinary differential equations. We first show that if the conservation law, a time-invariant functional, is additive on functions having disjoint compact support in phase space, then it is represented by an integral over phase space of a kernel which is a function of the solution to the Liouville equation. Then we use the fact that in classical mechanics phase space is usually a direct product of physical space and velocity space (Newtonian systems). We prove that for such systems the aforementioned representation of the invariant functionals will hold for conservation laws which are additive only in physical space; i.e., additivity in physical space automatically implies additivity in the whole phase space. We extend the results to include non-degenerate Hamiltonian systems, and, more generally, to include both conservative and dissipative dynamical systems. Some applications of the results are discussed.     

Memory function for dressed particles

This paper is concerned with the calculation of the memory function and derivation of a kinetic equation for one-body phase space correlation functions. The theory uses a one-body additive projection operator and a division of the Liouville operator with an unperturbed part that describes dressed particles. Binary collisions are neglected, for the theory aims at describing the screening and backflow effects of a type contained in the plasma kinetic theory of Balescu and Lenard. We obtain an explicit kinetic equation which is an improvement of these theories for the plasma case, and involves the exact equilibrium pair and triplet distributions. The equation also describes systems with strong short-range forces and shows how the screening effects occur in this case as well. The unifying function is the direct correlation function. The theory is meant to provide understanding for a more complete theory of fluids where a proper account is given of close collisions.Work supported by National Science Foundation, Grant No. GH 35691.     

A theory of extended quantisation

A new theory of quantisation is presented. After arguments are given indicating that mass-energy in the universe is quantised, this quantisation is mathematically related to the lifespan and maximum size of the universe. Various consequences are then deduced, such as the existence of a minimum force.     

Path algebras,wave-particle duality,and quantization of phase space

Semigroup algebras admit certain ‘coherent’ deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave length is that hypothesized by de Broglie’s wave-particle duality. This theory leads to a model of “physical” phase space of which mathematical phase space, the cotangent bundle of configuration space, is a projection. This space is singular, quantized at the Planck level, its structure implies the existence of spin, and the spread of a packet can be described as a random walk. The wavelength associated to a particle moving in this space need not be constant and its phase can change discontinuously.     

论动态信息理论

本文综述了作者的研究成果.近十年,作者将现有静态统计信息理论拓展至动态过程,建立了以表述动态信息演化规律的动态信息演化方程为核心的动态统计信息理论.基于服从随机性规律的动力学系统(如随机动力学系统和非平衡态统计物理系统)与遵守确定性规律的动力学系统(如电动力学系统)的态变量概率密度演化方程都可看成是其信息符号演化方程,推导出了动态信息(熵)演化方程.它们表明:对于服从随机性规律的动力学系统,动态信息密度随时间的变化率是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和耗损三者引起的,而动态信息熵密度随时间的变化率则是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和产生三者引起的.对于遵守确定性规律的动力学系统,动态信息(熵)演化方程与前者的相比,除动态信息(熵)密度在系统内部的态变量空间仅有漂移外,其余皆相同.信息和熵已与系统的状态和变化规律结合在一起,信息扩散和信息耗损同时存在.当空间噪声可略去时,将会出现信息波.若仅研究系统内部的信息变化,动态信息演化方程就约化为与表述上述动力学系统变化规律的动力学方程相对应的信息方程,它既可看成是表述动力学系统动态信息的演化规律,亦可看成是动力学系统的变化规律都可由信息方程表述.进而给出了漂移和扩散信息流公式、信息耗散率公式和信息熵产生率公式及动力学系统退化和进化的统一信息表述公式.得到了反映信息在传递过程中耗散特性的动态互信息公式和动态信道容量公式,它们在信道长度和信号传递速度之比趋于零的极限情况下变为现有的静态互信息公式和静态信道容量公式.所有这些新的理论公式和结果都是从动态信息演化方程统一推导出的.     

Melting,freezing and other peculiarities in small systems

The theory of solid-liquid phase changes in small systems implies that such systems may—but need not—exhibit sharp but unequal freezing and melting temperatures. The origin of this conclusion is reviewed and its implications for the theory of first-order phase transitions in bulk matter are discussed. The logical separation is made of the two temperatures as limits of stable existence, each of its own ‘phase’; and the convergence, with increasing size of cluster, of the observable coexistence to a sharp transition temperature is discussed. The equilibrium ratios of concentrations for such a coexistence are discontinuous functions of temperature at the limits of stability. The possibility of observing coexisting forms in equilibrium depends on there being a time scale separability, who validity lies outside the realm of thermodynamics. It is conjectured that spinodals are the loci of the same kind of locally stable states responsible for coexisting solid and liquid forms of clusters, and that the limits of spinodals are the points of discontinuity in the equilibrium concentration ratios, the chemical ‘equilibrium constants’.     

Semiclassical Theory of Quasiparticles in the Superconducting State

We have developed a semiclassical approach to solving the Bogoliubov-de Gennes equations for superconductors. It is based on the study of classical orbits governed by an effective Hamiltonian corresponding to the quasiparticles in the superconducting state and includes an account of the Bohr-Sommerfeld quantisation rule, the Maslov index, torus quantisation, topological phases arising from lines of phase singularities (vortices), and semiclassical wave functions for multidimensional systems. The method is illustrated by studying the problem of an SNS junction and a single vortex.     

Relativistic rigid particles: Classical tachyons and quantum anomalies

Causal rigid particles whose action includes anarbitrary dependence on the world-line extrinsic curvature are considered. General classes of solutions are constructed, includingcausal tachyonic ones. The Hamiltonian formulation is developed in detail except for one degenerate situation for which only partial results are given and requiring a separate analysis. However, for otherwise generic rigid particles, the precise specification of Hamiltonian gauge symmetries is obtained with in particular the identification of the Teichmüller and modular spaces for these systems. Finally, canonical quantisation of the generic case is performed paying special attention to the phase space restriction due to causal propagation. A mixed Lorenz-gravitational anomaly is found in the commutator of Lorentz boosts with world-line reparametrisations. The subspace of gauge invariant physical states is therefore not invariant under Lorentz transformations. Consequences for rigid strings and membranes are also discussed.     

Principle of corresponding states for hard polyhedron fluids

ABSTRACT

We introduce an equation of state for hard polyhedra and validate our new theory with simulations performed for 22 different shapes spanning a range of particle anisotropy. The derived expression not only shows excellent agreement with simulations but also exhibits ‘corresponding states’-like behaviour across shape space, allowing for a systematic reduction of thermodynamic properties onto a single master curve. Additionally, we propose a scaling-type argument for predicting the order–disorder transition packing fraction for hard polyhedron fluids that accurately captures the observed transitions in simulations. Our works suggest that hard-polyhedron systems can be thought of as perturbations about a hard-sphere where corrections in the principle axes of inertia and excess volume account for differences in relative orientation between neighbouring particles and each polyhedron's intrinsic asphericity, respectively. Additionally, our theory greatly benefits from requiring only knowledge of the inherent geometry for a polyhedron of interest with no necessary fitting parameters and thus provides a good heuristic rule of thumb for targeting relevant regions of simulation interest for novel systems of hard polyhedra.     

On the role of frustration in excitable systems

We study the role of frustration in excitable systems that allow for oscillations either by construction or in an induced way. We first generalize the notion of frustration to systems whose dynamical equations do not derive from a Hamiltonian. Their couplings can be directed or undirected; they should come in pairs of opposing effects like attractive and repulsive, or activating and repressive, ferromagnetic and antiferromagnetic. As examples we then consider bistable frustrated units as elementary building blocks of our motifs of coupled units. Frustration can be implemented in these systems in various ways: on the level of a single unit via the coupling of a self-loop of positive feedback to a negative feedback loop, on the level of coupled units via the topology or via the type of coupling which may be repressive or activating. In comparison to systems without frustration, we analyze the impact of frustration on the type and number of attractors and observe a considerable enrichment of phase space, ranging from stable fixed-point behavior over different patterns of coexisting options for phase-locked motion to chaotic behavior. In particular we find multistable behavior even for the smallest motifs as long as they are frustrated. Therefore we confirm an enrichment of phase space here for excitable systems with their many applications in biological systems, a phenomenon that is familiar from frustrated spin systems and less known from frustrated phase oscillators. So the enrichment of phase space seems to be a generic effect of frustration in dynamical systems. For a certain range of parameters our systems may be realized in cell tissues. Our results point therefore on a possible generic origin for dynamical behavior that is flexible and functionally stable at the same time, since frustrated systems provide alternative paths for the same set of parameters and at the same "energy costs."