Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

The present paper deals with the study of conditional entropy and its properties in a quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. First, we obtained a pseudo-metric on the family of all partitions of the couple (B,s), where B is a Boolean algebra and s is a state on B. This pseudo-metric turns out to be a metric (called the Rokhlin metric) by using a new notion of s-refinement and by identifying those partitions of (B,s) which are s-equivalent. The present theory has then been extended to the quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. Applying the theory of commutators and Bell inequalities, it is shown that the couple (L,s) can be equivalently replaced by a couple (B,s ₀), where B is a Boolean algebra and s ₀ is a state on B.     

The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.     

A grand superspace for unified field theories

Agrand superspace is proposed as the phase space for gauge field theories with a fixed structure groupG over a fixed space-time manifoldM. This superspace incorporatesall principal fiber bundles with these data. This phase space is the space of isomorphism classes ofall connections onall G-principal fiber bundles overM (fixedG andM). The justification for choosing this grand superspace for the phase space is that the space-time and the structure group are determinants of the physical theory, but the principal fiber bundle with the givenG andM is not. Grand superspace is studied in terms of a natural universal principal fiber bundle overM, canonically associated withM alone, and with a natural universal connection on this bundle. This bundle and its connection are universal in the sense that all connections on allG-principal fiber bundles (anyG) overM can be recovered from this universal bundle and its universal connection by a canonical construction. WhenG is Abelian, grand superspace is shown to be an Abelian group. Various subspaces of grand superspace consisting of the isomorphism classes of flat connections and of Yang-Mills connections are also discussed.     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.     

Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Field-theoretical Renormalization-Group approach to critical dynamics of crosslinked polymer blends

We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, ζ. First, by averaging over ζ, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q ⁻¹, a time length scale R(t) ∼ t ^1/z , and the mesh size ξ ^*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of ξ ^*, the dynamics is stopped. The final time, t ^*, then scales as t ^* ∼ ξ ^{* z}, with the dynamic exponent z = 6−η. Here, η is the usual Ising critical exponent. Since the final size of microdomains ξ ^* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.     

Chain tensor products and interval effect algebras

Weak and strongn-doublings (n∈N) are defined for an effect algebraP and the concept of a normal interval algebra is introduced. It is shown that the following statements are equivalent: (1) There is a morphism fromP into an interval algebra. (2)P admits a tensor product with every finite chain. (3)P has a weakn-doubling for everyn∈N. Moreover, the following are equivalent: (4)P is a normal interval algebra. (5)P admits a strong tensor product with every chain of length 2 ⁿ ,n∈N. (6)P has a strongn-doubling for everyn∈N. Finally, it is shown that ifP possesses an order-determining set of states, thenP is a normal interval algebra.     

The Spectrum of the Billiard Laplacian of a Family of Random Billiards

Random billiards are billiard dynamical systems for which the reflection law giving the post-collision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P. Billiards with microstructure are random billiards whose Markov operator is derived from a “microscopic surface structure” on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cellQ, the shape of which completely determines the operator P. This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P. We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. 2), that the billiard Laplacian P−I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P.     

Modular Categories and Orbifold Models

In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? ^G , is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module. Received: 27 August 2001 / Accepted: 1 March 2002     

The variable viscoelasticity oscillator

The complex dynamics of a variable viscoelasticity oscillator is studied using the novel concept of Variable‐Order (VO) Calculus. The damping force in the oscillator varies continuously between the elastic and viscous regimes depending on the position of the mass. The oscillator considered here is composed of a linear spring of stiffness k that inputs a restitutive force F_k = ‐k x, a VO damper of order q(x(t)) that generates a damping force F_q = ‐c_q ??^q(x(t)) x, and a mass m. A modified Runge‐Kutta method is used in conjunction with a trapezoidal numerical integration technique to yield a second‐order accurate method for the solution of the resulting VO Differential Equation (VODE). The VO oscillator is also modelled using a Constant Order (CO) formulation where a number of CO fractional order differentials are weighted to simulate the VO behavior. The CO formulation asymptotically approaches the VO results when a relatively large number of weights is used. For the viscoelastic range of 0 ≤ q ≤ 1, the dynamics of the oscillator is well approximated by the CO formulation when 5 or more fractional terms are included (e.g., 0, 1/4, 1/2, 3/4, and 1).     

On the Braided FRT-Construction

Abstract

A fully braided analog of the Faddeev-Reshetikhin-Takhtajan construction of a quasitriangular bialgebra A(X, R) is proposed. For a given pairing C, the factor-algebra A(X, R; C) is a dual quantum braided group. Corresponding inhomogeneous quantum group is obtained as a result of generalized bosonization. Construction of a first order bicovariant differential calculus is proposed.     

Lorentz cobordism

A Lorentz cobordism between two (in general nondiffeomorphic) 3-manifoldsM ₀,M ₁ is a pair (M,v), whereM is a differentiable 4-manifold andv is a differentiable vector field onM, such that 1) the boundary ofM is the disjoint union ofM ₀ andM ₁, 2)v is everywhere nonzero, 3)v is interior normal onM ₀ and exterior normal onM ₁. Such a manifoldM admits a Lorentz tensor with respect to whichM ₀ andM ₁ are spacelike hypersurfaces; thus a Lorentz cobordism is a model of a portion of a spacetime in which the topology of spacelike hypersurfaces is changing. We discuss the form that these changes can take, and give two methods for constructing a Lorentz cobordism between two nondiffeomorphic 3-manifolds. We comment on the possible relevance of Lorentz cobordism to the problem of gravitational collapse.     

Phase diagram of the extended hubbard model with pair hopping interaction

A one-dimensional model of interacting electrons with on-site U, nearest-neighbor V, and pair-hopping interaction W is studied at half-filling using the continuum limit field theory approach. The ground state phase diagram is obtained for a wide range of coupling constants. In addition to the insulating spin-density wave (SDW) and charge-density wave (CDW) phases for large U and V, respectively, we identify a bond-charge-density-wave (BCDW) phase W < 0, | U - 2V| < | 2W| and a bond-spin-density-wave (BSDW) for W > 0, | U - 2V| < W. The possibility of bond-located ordering results from the site-off-diagonal nature of the pair-hopping term and is a special feature of the half-filled band case. The BCDW phase corresponding to an enhanced Peierls instability in the system. The BdSDW is an unconventional insulating magnetic phase, characterized by a gapless spin excitation spectrum and a staggered magnetization located on bonds between sites. The general ground state phase diagram including insulating, metallic, and superconducting phases is discussed. A transition to the η-superconducting phase at | U - 2V| ≪ 2t?W is briefly discussed. Received 20 February 2002 / Received in final form 11 April 2002 Published online 19 July 2002     

A universal phase space for particles in Yang-Mills fields

Given a principal G-bundle P over M and a Hamiltonian G-space Q, one may construct the reduced symplectic manifold (T*P x Q)₀. When a connection on P is chosen, this manifold becomes a bundle over T*M with fibre Q. It is shown that this bundle is precisely the phase space constructed by Sternberg for a classical particle in a Yang-Mills field.Research partially supported by NSF Grant MCS 74-23180.A01.     

Fluctuation of inverse compressibility for electronic systems with random capacitive matrices

Abstract

This article is concerned with the statistics of the addition spectra of certain many-body systems of identical particles. In the first part, the pertinent system consists of N identical particles distributed among K<N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it with random coefficients. On a large scale, the ground-state energy E(N) of the whole system grows quadratically with N, but in general there is no simple relation such as E_N = aN+bN ². The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) X_N ≡E(N+1)?2E(N)+E(N?1) is a fluctuating quantity. Regarding the numbers X_N as values assumed by a certain random variable X, we obtain a closed-form expression for its distribution F _(X). Its main feature is that the corresponding density P _(X)=dF _(X)/d _X has a maximum at the point X=0. As K→∞ the density is Poissonian, namely, P(X)→e^?X

This result serves as a starting point for the second part, in which coupling between subsystems is included. More generally, a classical model is suggested in order to study fluctuations of Coulomb blockade peak spacings in large two-dimensional semiconductor quantum dots. It is based on the electrostatics of several electron islands among which there are random inductive and capacitive couplings. Each island can accommodate electrons on quantum orbitals whose energy depends also on an external magnetic field. In contrast to a single-island quantum dot, where the spacing distribution between conductance peaks is close to Gaussian, here the distribution has a peak at small spacing value. The fluctuations are mainly due to charging effects. The model can explain the occasional occurrence of couples or even triples of closely spaced Coulomb blockade peaks, as well as the qualitative behaviour of peak positions with the applied magnetic field.     

Axiomatic Geometric Formulation of Electromagnetism with Only One Axiom: The Field Equation for the Bivector Field <Emphasis Type="Italic">F</Emphasis> with an Explanation of the Trouton-Noble Experiment

In this paper we present an axiomatic, geometric, formulation of electromagnetism with only one axiom: the field equation for the Faraday bivector field F. This formulation with F field is a self-contained, complete and consistent formulation that dispenses with either electric and magnetic fields or the electromagnetic potentials. All physical quantities are defined without reference frames, the absolute quantities, i.e., they are geometric four-dimensional (4D) quantities or, when some basis is introduced, every quantity is represented as a 4D coordinate-based geometric quantity comprising both components and a basis. The new observer-independent expressions for the stress-energy vector T(n) (1-vector), the energy density U (scalar), the Poynting vector S and the momentum density g (1-vectors), the angular momentum density M (bivector) and the Lorentz force K ((1-vector) are directly derived from the field equation for F. The local conservation laws are also directly derived from that field equation. The 1-vector Lagrangian with the F field as a 4D absolute quantity is presented; the interaction term is written in terms of F and not, as usual, in terms of A. It is shown that this geometric formulation is in a full agreement with the Trouton-Noble experiment.     

A study on the determination of mechanical properties of a power law material by its indentation force–depth curve

In this study a novel optimization approach is proposed to extract mechanical properties of a power law material whose stress–strain relationship may be expressed as a power law from its given experimental indentation P–h curve. A set of equations have been established to relate the P–h curve to mechanical properties E, σ _y and n of the material. For the loading part of a P–h curve this approach is based on the assumption that the indentation response of an elastic–plastic material is a linear combination of the corresponding elastic and elastic–perfect plastic materials. For the unloading part of the P–h curve it is based on the assumption that the unloading response of the elastic–plastic material is a linear combination of the full contact straight line and the purely elastic curve. Using the proposed optimization approach it was found that the mechanical properties of an elastic–plastic material usually cannot be decided uniquely by using only a single indentation P–h curve of the material. This is because in general a few matched sets of mechanical properties were found to produce a given P–h curve. It is however possible to identify the best matched set of mechanical properties by knowing some background information of the material. If the best matched material is identified, the predictions of mechanical properties are quite accurate.     

The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK _ij =A+B(i+j)+C _ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN _k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN _kl, also determined by a recursion relation. The physical meaning ofN _kl and the recursion relation is given. A method to get explicit expressions forN _k andN _kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.