Thermodynamic duality for classical systems of arbitrary spin

Given a classical spins system, namely, a set of spin sites of maximum spins inv-dimensional space along with a Hamiltonian defined on the possible spin configurations, a general method is described for constructing a large class of dual lattices of the same spin. The method utilizes the commutative group structure with which the configuration space is endowed.On leave from: Department of Mathematics, Indiana University, Bloomington, Indiana.     

A generalization of Fermat's principle for classical and quantum systems

The analogy between dynamics and optics had a great influence on the development of the foundations of classical and quantum mechanics. We take this analogy one step further and investigate the validity of Fermat's principle in many-dimensional spaces describing dynamical systems (i.e., the quantum Hilbert space and the classical phase and configuration space). We propose that if the notion of a metric distance is well defined in that space and the velocity of the representative point of the system is an invariant of motion, then a generalized version of Fermat's principle will hold. We substantiate this conjecture for time-independent quantum systems and for a classical system consisting of coupled harmonic oscillators. An exception to this principle is the configuration space of a charged particle in a constant magnetic field; in this case the principle is valid in a frame rotating by half the Larmor frequency, not the stationary lab frame.     

Remarks on identical particles in de Broglie-Bohm theory

It is argued that the topological approach to the (anti-)symmetrisation condition for the quantum state of a collection of identical particles, defined in the “reduced” configuration space, is particularly natural from the perspective of de Broglie-Bohm pilot-wave theory.     

Information-theoretical aspects of quantum measurement

We present criteria for comparing measurements on a given system from the point of view of the information they provide. These criteria lead to a concept ofinformational completeness of a set of observables, which generalizes the conventional concept of completeness. The entropy of a state with respect to an arbitrary sample space of potential measurement outcomes is defined, and then studied in the context of configuration space and fuzzy stochastic phase space.Supported in part by the National Research Council of Canada.     

Curvature of the energy landscape and folding of model proteins

We study the geometric properties of the energy landscape of coarse-grained, off-lattice models of polymers by endowing the configuration space with a suitable metric, depending on the potential energy function, such that the dynamical trajectories are the geodesics of the metric. Using numerical simulations, we show that the fluctuations of the curvature clearly mark the folding transition, and that this quantity allows to distinguish between polymers having a proteinlike behavior (i.e., that fold to a unique configuration) and polymers which undergo a hydrophobic collapse but do not have a folding transition. These geometrical properties are defined by the potential energy without requiring any prior knowledge of the native configuration.     

The Toy Top,an Integrable System of Rigid Body Dynamics

Abstract

A toy top is defined as a rotationally symmetric body moving in a constant gravitational field while one point on the symmetry axis is constrained to stay in a horizontal plane. It is an integrable system similar to the Lagrange top. Euler-Poisson equations are derived. Following Felix Klein, the special unitary group SU(2) is used as configuration space and the solution is given in terms of hyperelliptic integrals. The curve traced by the point moving in the horizontal plane is analyzed, and a qualitative classification is achieved. The cases in which the hyperelliptic integrals degenerate to elliptic ones are found and the corresponding solutions are given in terms of Weierstrass elliptic functions.     

Canonical variables for the Dirac theory

A new canonical structure for Dirac's theory is proposed. The new configuration space A is a real, four-dimensional subbundle of the spinor bundle. A Lagrangian defined on Q describes a theory equivalent to the Dirac one. In this way we obtain a theory without second-type constraints.     

Twistor integral representations of fundamental solutions of massless field equations

We consider the general dimensional (complex) Minkowski spaces and the extended twistor spaces. We show that the fundamental solutions of the complex wave or Laplace equations are explicitly represented by the integrals of some closed forms on the twistor spaces. The closed form is defined from labeled trees explained in graphs theory, and is written, as the cohomology class, by the linear combination of the logrithmic forms on some hyperplane configuration complement in some complex affine space.     

Force theory for multiphase bodies

We present a framework for the study of bodies wherein the deformation gradient may suffer a jump across an evolving nonmaterial interface. To formulate the kinematics relevant to such a situation, we use a global approach in which the configuration space has the structure of an infinite dimensional bundle. We show that a force, defined as an element of the cotangent bundle of the configuration manifold, may be represented by bulk and interfacial stress measures. The invariant decomposition of that force into bulk and interfacial components is discussed and we show that, in the case where the stress measures representing the force are given in terms of smooth densities, such a decomposition is determined by the average stress on the interface.     

Uniqueness and exponential decay of correlations for some two-dimensional spin lattice systems

We consider a two-dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice. The configuration space of the spin system is a direct product of mixing subshifts of finite type. The potential is defined on the set of all squares in Z² and decays exponentially with the linear size of the square. Via the polymer expansion technique we prove that for sufficiently high temperatures the limit Gibbs distribution is unique and has an exponential decay of correlations.     

Density of States for a specified correlation function and the energy landscape

The degeneracy of two-phase disordered microstructures consistent with a specified correlation function is analyzed by mapping it to a ground-state degeneracy. We determine for the first time the associated density of states via a Monte Carlo algorithm. Our results are explained in terms of the roughness of an energy landscape, defined on a hypercubic configuration space. The use of a Hamming distance in this space enables us to define a roughness metric, which is calculated from the correlation function alone and related quantitatively to the structural degeneracy. This relation is validated for a wide variety of disordered structures.     

Complexification of gauge theories

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification G_C of G. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.     

Integrality of the monopole number in SU(2) Yang-Mills-Higgs theory on ℝ3

We prove that in classical SU(2) Yang-Mills-Higgs theories on ³ with a Higgs field in the adjoint representation, an integer-valued monopole number (magnetic charge) is canonically defined for any finite-actionL _1,loc ² configuration. In particular the result is true for smooth configurations. The monopole number is shown to decompose the configuration space into path components.Research supported in part by NSF Grants 8120790 and PHY-03669     

空间羣不可约表示对称冪及反对称冪的简约

本文将作者们对空间羣不可约表示直接乘积的简约所发展的方法推广到空间羣不可约表示对称幂及反对称幂的简约,列出了对六角密积与纤维锌矿结构对称化平方的计算结果;利用这些结果,对组态不隐定性作了初步讨论。     

Moments of the reduced local energy

Compact measures of the local accuracy of a wavefunction are proposed. They are defined in terms of the reduced local energy. These compact measures are applied to examine the local accuracy in different regions of configuration space for some well known Hartree-Fock wavefunctions from helium through argon.