A phenomenological theory of the isotropic to chiral smectic-C phase transition

In this paper we discuss the direct isotropic to chiral smectic-C phase transition on the basis of a phenomenological theory. The model free energy is written in terms of the coupled order parameters including the spontaneous polarization. We present a detailed analysis of the different phases that can occur and analyze the question under which conditions a direct isotropic to chiral smectic-C phase transition is possible when compared to other phase transitions. On the basis of this model the isotropic-smectic-C^* transition is always of first order. The theoretical predictions are compared with the available experimental results.-1     

On the asymptotic character of electromagnetic waves in a Friedmann–Robertson–Walker universe

Asymptotic properties of electromagnetic waves are studied within the context of Friedmann–Robertson–Walker (FRW) cosmology. Electromagnetic fields are considered as small perturbations on the background spacetime and Maxwells equations are solved for all three cases of flat, closed and open FRW universes. The asymptotic character of these solutions is investigated and their relevance to the problem of cosmological tails of electromagnetic waves is discussed.     

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.     

Evolution of Orbits and Quantum Correlation Functions by Quadratic Hamiltonians

Quantum systems with quadratic Hamiltonians are considered. Some results about the time evolution of homogeneous polynomials and of quantum correlation functions are given. The image of arbitrary orbit of Weyl–Heisenberg group under this time evolution is shown to be again an orbit of this group. For quantum free particle it is shown that its time evolution intersects arbitrary such orbit at most once. A result about existence of more orbits having the same dispersion of some quantum position is presented.PACS: 02.20.Qs, 02.30.Sa     

Hestenes' Tetrad and Spin Connections

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of SL(2,R)× U(1) gauge potentials F_α^K and a complex scalar field ρ. This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary linear connection acting on the tensor fields (F_α^K, ρ). We also derive spin connections for which Dirac's bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection.     

Invariant surfaces of a three-dimensional manifold with constant Gauss curvature

We give a reduction procedure to determine (locally) the surfaces with constant Gauss curvature in a three-dimensional manifold which are invariant under the action of a one-parameter subgroup of the isometry group of the ambient space. We apply this procedure to describe the invariant surfaces with constant Gauss curvature in $H^2\times R$ and in $H_3$.     

On splitting of totally singular quadratic forms

In this note we completely study the standard splitting of quasi-Pfister forms and their neighbors, and we include some general results on standard splitting towers of totally singular quadratic forms. The author was supported by the European research network HPRN-CT-2002-00287 “AlgebraicK-Theory, Linear Algebraic Groups and Related Structures”.     

Unital Groups and General Comparability Property

Pseudo-effect algebras are partial algebras (E;+,0,1) with a partially defined addition + which is not necessarily commutative and therefore with two complements, left and right. If they satisfy a special kind of the Riesz decomposition property, they are intervals in unital po-groups. The general comparability property in unital po-groups with strong unit (G,u), allows to compare elements of G in some intervals with Boolean ends. Such a po-group is always an -group admitting a state. We prove that every such (G,u) is a subdirect product of linearly ordered unital po-groups.     

Early Gravity-Wave Detection Experiments, 1960-1975

I discuss the experiments of Joseph Weber (1919-2000) of the University of Maryland between 1960 and 1973 that were aimed at the detection of gravity waves. He used throughout a quadratic detector to demodulate his antenna signal, which I analyze and compare to a linear detector.The latter was used by all of the other groups that entered the field. Of these, Richard Garwin and I at IBM were one of the first groups to publish, and I discuss in detail our experiments between 1973 and 1975. I then discuss the experiments that were carried out at Bell Labs-Rochester, Glasgow, Munich-Frascati,Moscow, and Tokyo. I compare the results, all of which were negative, with Webers claimed detection of large numbers of gravity-wave events, as many as seven per day. I conclude that these were not in fact gravity waves, but artifacts of his extremely hands-on data-analysis procedures, which I discuss in detail. Finally, I speculate on how this came about.     

On quadratic fields with large 3-rank

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the general cubic discriminants (real or imaginary) up to in time and space , or more generally in time and space for a freely chosen positive . A variant computes the -ranks of all quadratic fields of discriminant up to with the same time complexity, but using only units of storage. As an application we obtain the first real quadratic fields with , and prove that is the smallest imaginary quadratic field with -rank equal to .