Ideal,first-kind measurements in a proposition-state structure

Let be an orthomodular lattice and a strongly ordering set of probability measures on such that supports of measures exist in . Then we show the existence of a set of mappings of into that are physically interpretable as ideal, first-kind measurements.     

“Statistical” symmetry with applications to phase transitions

Hermann proposed that mesomorphic media should be classified by assigning certain statistical symmetry groups to each possible partially ordered array. Two translational groups introduced were called superordinate and subordinate. We find that the average density in such a partially ordered medium has the superordinate symmetry ₁, while the pair correlation function has the subordinate symmetry ₂. A complete listing is made of all compatible combinations of ₁ and ₂ in two and three dimensions. This leads to more possible symmetries than Hermann obtained, e.g., also to nonstoichiometric crystals. The order parameter space for the systems is found to be the quotient space ₁/₂. In most cases it is identical to the order parameter space of low-dimensionalXY spin systems. The Landau free energy is expanded as functional of the two-particle correlation functionK; the translation group is found to be ₁×₂. A Landau mean-field theory can then be carried out by expanding the system free energy into a series of invariants of the active irreducible representations ofK and mapping the free energy onto that for anXY planar spin system. We predict novel critical behavior for transitions between mesomorphic phases and go nogo selection rules for continuous transitions. We give the structure factors for X-ray scattering so changes in all such phase transitions are observable. The statistical symmetry groups, which describe point and translational symmetries of the mesophases, are classified. Proposals are made to include quasi-long-range or topological order in the classification scheme.This work supported in part by National Science Foundation (Division of International Programs), the PSC-BHE—Faculty Research Award CUNY and Deutsche Forschungsgemeinschaft.     

The projection approach to the Fokker-Planck equation. I. Colored Gaussian noise

It is shown that the Fokker-Planck operator can be derived via a projection-perturbation approach, using the repartition of a more detailed operator into a perturbation ₁ and an unperturbed part ₀. The standard Fokker-Planck structure is recovered at the second order in ₁, whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure. To get rid of these higher order terms, a key approximation, local linearization (LL), is made. In general, to evaluate at the second order in ₁ the exact expression of the diffusion coefficient which simulates the influence of a Gaussian noise with a finite correlation time, a resummation up to infinite order in must be carried out, leading to what other authors call the best Fokker-Planck approximation (BFPA). It is shown that, due to the role of terms of higher order in ₁, the BFPA leads to predictions on the equilibrium distributions that are reliable only up to the first order in t. The LL, on the contrary, in addition to making the influence of terms of higher order in ₁ vanish, results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the Fox theory. The corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady-state distributions. Therefore, over the whole range 0 the LL turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in ₁ vanish. In the short- region the LL leads to results virtually coincident with those of the BFPA. In the large- region the LL is a more accurate approximation than the BFPA itself. These theoretical arguments are supported by the results of both analog and digital simulation.     

Spectral properties of a class of operators associated with conformal maps in two dimensions

Iff is a rational map of the Riemann sphere, define the transfer operator by Let also be the Banach space of functions for which the second derivatives are measures. Ifg andg satisfies a simple integrability condition (implying thatg vanishes at critical points and multiple poles off) then is a bounded linear operator on . The essential spectral radius of can be estimated and, under suitable conditions, proved to be strictly less than the spectral radius. Similar estimates for more general operators are also obtained.     

p-adic Heisenberg group and Maslov index

A system of coordinates on a set of selfdual lattices in a two-dimensionalp-adic symplectic space (V,) is suggested. A unitary irreducible representation of the Heisenberg group of the space (V,) depending on a lattice (an analogue of the Cartier representation) is constructed and its properties are investigated. By the use of such representations for three different lattices one defines the Maslov index =(₁,₂,₃) of a triple of lattices. Properties of the index are investigated and values of in coordinates for different triples of lattices are calculated.     

Abstract scattering theory

The asymptotic condition is formulated for a system whose theory is more general than quantum mechanics. Its logic forms an orthocomplemented weakly modular -lattice. The set of states , consisting of all the probability measures on , is endowed with the most suitable metric physically, called here the natural one. In this space it is proved that the asymptotic condition implies the existence of two convex automorphisms _+- of which we call the wave-automorphisms. From these theS-automorphism _– ^–1 ₊ is defined and corresponds to the scattering operator in conventional quantum theory.     

Conserved moments in nonequilibrium field dynamics

We demonstrate with the example of Cahn-Hilliard dynamics that the macroscopic kinetics of first-order phase transitions exhibits an infinite number of constants of motion. Moreover, this result holds in any space dimension for a broad class of nonequilibrium processes whose macroscopic behavior is governed by equations of the form /t = W(), where is an order parameter,W is an arbitrary function of , and is a linear Hermitian operator. We speculate on the implications of this result.     

On the isomorphism of a ‘quantum logic’ with the logic of the projections in a hilbert space

A topology is introduced in a logic using the set of pure states of . It is shown that , equipped with this topology, under suitable conditions, determines the division ring , or 2e. With the continuity of the antiautomorphism of the division ring added, it is shown that these conditions are necessary and sufficient for the projective logic to be isomorphic with the projective logic of the projections in a Hilbert space over , or 2e.P. Cotta-Ramusino gratefully acknowledges a fellowship of the Consiglio Nazionale delle Ricerche.     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

The unified approach to spectral analysis

We develop a new, unified, method to construct a closed (selfad-joint in ²) extension of a partial differential operator in all the spaces ^p ( ⁿ ) 1p. Our method is not only an unified approach but it is also very efficient. We obtain very weak conditions on the potentials.     

On solvable models in classical lattice systems

We consider one dimensional classical lattice systems and an increasing sequence _n (n=1,2, ...) of subsets of the state space; _n takes into account correlations betweenn successive lattice points.If the interaction range of the potential is finite, we prove that the equilibrium states defined by the variational principle are elements of { _n } _n<. Finally we give a new proof of the fact that all faithful states of _n are DLR-states for some potential.Bevoegdverklaard navorser NFWO     

Measures on infinite-dimensional orthomodular spaces

We classify the measures on the lattice of all closed subspaces of infinite-dimensional orthomodular spaces (E, ) over fields of generalized power series with coefficients in . We prove that every -additive measure on can be obtained by lifting measures from the residual spaces of (E, ). The measures being lifted are known, for the residual spaces are Euclidean. From the classification we deduce, among other things, that the set of all measures on is not separating.Research supported by the Swiss National Science Foundation.     

On the relaxation time of Gauss' continued-fraction map. II. The Banach space approach (transfer operator method)

The spectrum of the transfer operator for the mapTx=1/x–[1/x] when restricted to a certain Banach space of holomorphic functions is shown to coincide with the spectrum of the adjointU* of Koopman's isometric operatorUf(x)=f·T(x) when the former is restricted to the Hilbert space () introduced in part I of this work. IfN denotes the operator –P ₁ withP ₁ the projector onto the eigenfunction to the dominant eigenvalue ₁ =1 of , then –N is au ₀-positive operator with respect to some cone and therefore has a dominant positive, simple eigenvalue – ₂. A minimax principle holds giving rigorous upper and lower bounds both for ₂ and the relaxation time of the mapT.     

The choice of test functions in gauge quantum field theories

We discuss the problem of the choice of test functions in gauge quantum field theories. Analysis of explicitly soluble models suggests that the test function spaces which are suitable for local and covariant formulation of gauge theories are the Gelfand and Shilov spaces , +>1. We also discuss a possible generalization of the spectral condition.     

Third-order geometrical null symmetries of gravitational fields

The first-, the second-, and the third-order null Killing vectors in a gravitational field are explored separately. When an algebraically special Petrov-type free gravitational field isaligned with a source-free (nonnull/null) electromagnetic field, their common propagation vectorl ^a is shown to be a third-order null Killing vector field ( _{l l l} g _ab = 0, _{l l} g _ab 0). When the two fields arenot aligned, the necessary and sufficient conditions for the existence of a third-order null symmetry are obtained in Newman-Penrose formalism.     

Infinitesimal holonomy groups of Einstein-Maxwell space-times

Some general results concerning the Petrov classification of the Weyl tensor of space-times whose infinitesimal holonomy group belongs to a given conjugacy class of the Lorentz group ₊ are established using the null tetrad notation. Those conjugacy classes, corresponding to proper subgroups of ₊ which contain the infinitesimal holonomy groups of Einstein-Maxwell space-times, are determined, and explicit examples of such space-times are given for each conjugacy class.     

Generalized two-level quantum dynamics. II. Non-Hamiltonian state evolution

A theorem is derived that enables a systematic enumeration of all the linear superoperators (associated with a two-level quantum system) that generate, via the law of motion = , mappings (0) (t) restricted to the domain of statistical operators. Such dynamical evolutions include the usual Hamiltonian motion as a special case, but they also encompass more general motions, which are noncyclic and feature a destination state (t ) that is in some cases independent of (0).     

The Classical Solutions of Two-dimensional Gravity

The solutions of two-dimensional gravityfollowing from a non-linear Lagrangian =f(R)g are classified, and their symmetry andsingularity properties are described. Then a conformaltransformation is applied to rewrite these solutions as analogoussolutions of two-dimensional Einstein-dilaton gravityand vice versa.     

Duality for dual covariance algebras

One way of generalizing the definition of an action of the dual group of a locally compact abelian group on a von Neumann algebra to non-abelian groups is to consider (G)-comodules, where (G) is the Hopfvon Neumann algebra generated by the left regular representation ofG. To a (G)-comodule we shall associate a dual covariance algebra and a natural covariant system ( , ,G), and in Theorem 1 the covariant systems coming from (G)-comodules are characterized. In [2] it was shown that the covariance algebra of a covariant system in a natural way is a (G)-comodule. Therefore one can form the dual covariance algebra of a covariance algebra and the covariance algebra of a dual covariance algebra. Theorems 2 and 3 deal with these algebras — generalizing a result by Takesaki. As an application we give a new proof of a theorem by Digernes stating that the commutant of a covariance algebra itself is a covariance algebra and prove the similar result for dual covariance algebras.     

Probabilistic formulation of classical mechanics

Starting axiomatically with a system of finite degrees of freedom whose logic _c is an atomic Boolean -algebra, we prove the existence of phase space _c, as a separable metric space, and a natural (weak) topology on the set of statesI (all the probability measures on _c) such that _c, the subspace of pure statesP, the set of atoms of _c and the spaceP( _c) of all the atomic measures on _c, are all homeomorphic. The only physically accessible states are the points of _c. This probabilistic formulation is shown to be reducible to a purely deterministic theory.