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.     

“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.     

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.     

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.     

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.     

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.     

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.     

Real spin-Clifford bundle and the spinor structure of space-time

By analyzing the conditions for the existence on a space-time of a global algebraic spinor field, we prove the following result, known as Geroch's theorem: A necessary and sufficient condition for to admit a spinor structure is that the orthonormal frame bundleF ₀() have a global section. Our proof, which does not use in any stage the complexification of _1,3 (the space-time Clifford algebra), is simple, requiring only the explicit construction of the algebraic spinor and the spinorial metric within _1,3 and elementary facts about associated bundles and the bundle reduction process. This is to be compared with the original proof, which uses the full algebraic topology machinery. We also clarify the relation of the covariant spinor structure and Graf'se-spinor structure.     

Noise Corrections to Stochastic Trace Formulas

We review studies of an evolution operator for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator in the weak noise limit can be computed in several ways. A method using a local matrix representation of the operator allows to push the corrections to the escape rate up to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actually, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators ⁿ. Using an integral representation of the evolution operator , we then investigate the high order corrections to the latter traces. Their asymptotic behavior is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of trace formula.     

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.     

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.     

The singular holonomy group

The fibre of the extension of the frame-bundle of a space-time over ab-boundary pointp is a homogeneous space /G _p . It is shown thatG _p can be found by a construction like that for a holonomy group, and that it contains a subgroup determined by the Riemann tensor. Near a curvature singularity one would expectG _p =      

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).     

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.     

Gravitation and universal Fermi coupling in general relativity

The generally covariant Lagrangian densityG = + 2K _matter of the Hamiltonian principle in general relativity, formulated by Einstein and Hilbert, can be interpreted as a functional of the potentialsg _ikand of the gravitational and matter fields. In this general relativistic interpretation, the Riemann-Christoffel form _kl ⁱ = _kl ⁱ for the coefficients _kl ⁱ of the affine connections is postulated a priori. Alternatively, we can interpret the LagrangianG as a functional of , g_ik, and the coefficients _kl ⁱ . Then the _kl ⁱ are determined by the Palatini equations. From these equations and from the symmetry _kl ⁱ = _lk ⁱ for all matter fields with /=0 the Christoffel symbols again result. However, for Dirac's bispinor fields, / becomes dependent on the Dirac current, essentially with a coupling factor Khc. In this case, the Palatini equations define a new transport rule for the spinor fields, according to which a second universal interaction results for the Dirac spinors, besides Einstein's gravitation. The generally covariant Dirac wave equations become the general relativistic nonlinear Heisenberg wave equations, and the second universal interaction is given by a Fermi-like interaction term of the V-A type. The geometrically induced Fermi constant is, however, very small and of the order 10^–81erg cm³     

A contribution to the theory of the triple crystal diffractometer

The first part of the paper gives a general equation for triple-crystal arrangement with perfect crystals on the assumption that the third crystal is rotated. It is shown that in the case of perfect crystals the shape of the reflection curve is practically independent of the vertical divergence. The case of mosaic crystals is also solved and the possibility of rotation by other than the third crystal is considered. A method is proposed for investigating the imperfection of a crystal which is different from methods used up to now. The paper is supplemented by some experimental results.

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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.     

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     

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 vector-tensor minimally coupled field theories

We consider vector-tensor minimally coupled Lagrangians, i.e., scalar densities of the form = g ^1/2 R +L(g _ij ; _i ; _i,j ). We prove that the gauge invariance of any of the sets of Euler-Lagrange expressions implies the gauge invariance of the Lagrangian itself forn even, and an almost gauge invariance forn odd. We also find those for whichE ⁱ () = 0 orE ^ij (L) = 0, generalizing well-known results by Lovelock and a result by the authors.