Real Polarization Research Articles

Reversals in the large-scale αΩ-dynamo with memory

Abstract. Inversion of the magnetic field in a model of large-scale αΩ-dynamo with α-effect with stochastic memory is under investigation. The model allows us to reproduce the main features of the geomagnetic field reversals. It was established that the polarity intervals in the model are distributed according to the power law. Model magnetic polarity timescale is fractal. Its dimension is consistent with the dimension of the real geomagnetic polarity timescale.

Inglés

Using standard techniques from geometric quantization, we re- derive the product of functions on R2 which was first introduced by von Neumann and later reintroduced by Groenewold and which is the integral version of the Moyal product. More specifically, by pairing the diagonal real polarization on the pair groupoid R2 × R2 with its standard holomorphic polarization, we obtain the well-known Segal-Bargmann transform in a ro tated and scaled (and half-conjugated) form. Together with a convolution of functions in the Segal-Bargmann space, which is a natural deformation of the usual convolution of functions on the pair groupoid, this defines the Groenewold-von Neumann product on L2(R2). Keywords : Geometric quantization, star product, Segal-Bargmann trans form, Fock spaces. To cite this article: J.B. Moreno, Groenewold-von Neumann product via Segal-Bargmann transform, Rev. Integr. Temas Mat. 33 (2015), No. 2, 135–144.

Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings

We study the Maslov correction to semiclassical states by using a Kähler regularized BKS pairing map from the energy representation to the Schrödinger representation. For general semiclassical states, the existence of this regularization is based on recently found families of Kähler polarizations degenerating to singular real polarizations and corresponding to special geodesic rays in the space of Kähler metrics. In the case of the one-dimensional harmonic oscillator, we show that the correct phases associated with caustic points of the projection of the Lagrangian curves to the configuration space are correctly reproduced.

Zeno inhibition of polarization rotation in an optically active medium

We describe an experiment in which the rotation of the polarization of light propagating in an optically active water solution of D-fructose tends to be inhibited by frequent monitoring whether the polarization remains unchanged. This is an example of the Zeno effect that has remarkable pedagogical interest because of its conceptual simplicity, easy implementation, low cost, and because the same the Zeno effect holds at classical and quantum levels. An added value is the demonstration of the Zeno effect beyond typical idealized assumptions in a practical setting with real polarizers.

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

Let $$P$$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $$X_{P}$$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $$t = {\sqrt{-1}}s$$ . We relate the quantization of $$X_{P}$$ in two different toric Kähler polarizations by taking the time- $${\sqrt{-1}}s$$ Hamiltonian “flow” of strongly convex functions on the moment polytope $$P$$ . By taking $$s$$ to infinity, we obtain the quantization of $$X_{P}$$ in the (singular) real toric polarization. Recall that $$X_{P}$$ has an open dense subset which is biholomorphic to $$({\mathbb {C}}^{*})^{n}$$ . The quantization of $$X_{P}$$ in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu–Guillemin symplectic potential $$g$$ , at time $$t={\sqrt{-1}}$$ , to an appropriate finite-dimensional subspace of quantum states in the quantization of $$T^{*}{\mathbb {T}}^{n}$$ in the vertical polarization. By taking other imaginary times, $$t= k {\sqrt{-1}}, k\in {\mathbb {R}}$$ , we describe toric Kähler metrics with cone singularities along the toric divisors in $$X_{P}$$ . For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of $$X_{P}$$ . We show that the pointwise and $$L^2$$ -norms of quantum states are asymptotically vanishing on negative-definite regions.

Degenerating Kähler structures and geometric quantization

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

Coherent state transforms and the Mackey-Stone-Von Neumann theorem

Mackey showed that for a compact Lie group K, the pair (K, C 0(K)) has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of K × K invariant polarizations on T*K. The Kähler polarizations in the family are generated by (complex) time-τ Hamiltonian flows applied to the (Schrödinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of T*K is then studied by considering covariant pairs of representations of K defined by geometric prequantization and of representations of C 0(K) defined via Heisenberg time-(−τ) evolution followed by time-(+τ) geometric-quantization-induced evolution. We show that in the semiclassical and large imaginary time limits, the unitary transform whose existence is guaranteed by Mackey's theorem can be approximated by composition of the time-(+τ) geometric-quantization-induced evolution with the time-(−τ) evolution associated with the momentum space [W. D. Kirwin and S. Wu, “Momentum space for compact Lie groups and the Peter-Weyl theorem” (unpublished)] quantization of the Hamiltonian function generating the flow. In the case of quadratic Hamiltonians, this asymptotic result is exact and unitary equivalence between quantizations is achieved by identifying the Heisenberg imaginary time evolution with heat operator evolution, in accordance with the coherent state transform of Hall.

Effects of deposition temperature on characteristics of ferroelectric Sr2Bi4Ti5O18 nanoplates fabricated by RF sputtering

Sr2Bi4Ti5O18 (SBTO) films with a- and b-axis orientations, and thicknesses of 0.9–1.2 µm were sputter-deposited on conductive Nb:TiO2(101) substrates containing 0.79 mass % Nb. The deposition temperature was varied from 575 to 700 °C under a fixed gas pressure of 0.4 Pa, and the structural and ferroelectric characteristics of the films were investigated. SBTO films deposited at 625–700 °C had a mostly single-phase orthorhombic structure, with a high degree of a- and b-axis orientations [α(h00)/(0k0)] of 99.0–99.8%. In addition, the full width at half maximum of the (200) diffraction peak was 0.69–0.86°, which indicated good crystallinity. SBTO films deposited at 625–650 °C had a nanoplate-like microstructure with the plates aligned along the [010] direction. The real room-temperature remanent polarization , taking the porosity between the nanoplates into account, exhibited a maximum of 40 µC/cm2 at 650 °C. Thus, the optimal deposition temperature for heteroepitaxial growth of SBTO nanoplates with a high α(h00)/(0k0) of ≥99.0% and excellent ferroelectric properties on conductive Nb:TiO2 substrates is 650 °C under a gas pressure of 0.4 Pa.

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of K\"ahler structure on toric varieties by symplectic potentials.

Effects of Eu3+ Doping on Characteristics of (Bi3.25Nd0.75)Ti3O12 Nanoplates

a- and b-axis-oriented (Bi3.25Nd0.75 - xEux)Ti3O12 (BNEuT, x = 0–0.75) films of 3.0 µm thickness were fabricated on conductive Nb:TiO2(101) substrates containing 0.79 mass % Nb by high-temperature sputtering at 650 °C, and their structural and ferroelectric characteristics were investigated. All the films had a mostly single-phase orthorhombic structure, with high degrees of a- and b-axis orientations of 99.0–99.8%. The lattice parameters (a-, b-, and c-axis lengths) and the calculated orthorhombic lattice distortion decreased monotonically with increasing Eu content. The microstructure of BNEuT films with x = 0–0.50 was nanoplate-like, whereas that of films with x≥0.60 was significantly more bulk-like. The real room-temperature remanent polarization (2Pr*), taking the porosity between the nanoplates into account, had a maximum value of 2Pr* = 87 µC/cm2 at x = 0.10, which was approximately 1.3 times larger than that (65 µC/cm2) of the nondoped BNT film. It is shown that lattice distortion caused by rotation of octahedra in the a–b plane due to the Eu substitution plays a significant role in the improvement of ferroelectricity.

Parametric inversion of spin currents in semiconductor microcavities

The optical spin-Hall effect results in the formation of an antisymmetric real space polarization pattern forming spin currents. In this paper, we show that the exciton-polariton parametric scattering allows us to reverse the sign of these currents. We describe the pulsed resonant excitation of a strongly coupled microcavity with a linearly polarized pump at normal incidence. The energy of the pulse is set to be close to the inflexion point of the polariton dispersion and the focusing in real space populates the reciprocal space on a ring. For pumping powers below the parametric scattering threshold, the propagation of the injected polaritons in the effective magnetic field induced by the TE and TM splitting produce the normal optical spin-Hall effect. Keeping the same input polarization but increasing the pump intensity, the parametric scattering towards an idler and a signal state is triggered on the whole elastic circle. The injected particles are scattered toward these states while propagating radially all over the plane, gaining a cross linear polarisation with respect to the pump during the nonlinear process. Eventually, the propagation of the polaritons in the effective field results in the optical spin Hall-effect, but this time with inverted polarization domains.

Geometric Quantization of real polarizations via sheaves

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization for Gelfand-Cetlin systems is based on a theorem due to \'Sniatycki for fibrations which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and K\"unneth formulae. As a consequence, a new short proof of this classical result for fibrations due to \'Sniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations.

Imaging VLBI polarimetry data from Active Galactic Nuclei using the Maximum Entropy Method

Mapping the relativistic jets emanating from AGN requires the use of a deconvolution algorithm to account for the effects of missing baseline spacings. The CLEAN algorithm is the most commonly used algorithm in VLBI imaging today and is suitable for imaging polarisation data. The Maximum Entropy Method (MEM) is presented as an alternative with some advantages over the CLEAN algorithm, including better spatial resolution and a more rigorous and unbiased approach to deconvolution. We have developed a MEM code suitable for deconvolving VLBI polarisation data. Monte Carlo simulations investigating the performance of CLEAN and the MEM code on a variety of source types are being carried out. Real polarisation (VLBA) data taken at multiple wavelengths have also been deconvolved using MEM, and several of the resulting polarisation and Faraday rotation maps are presented and discussed.

The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory

We establish a precise isomorphism between the Schrödinger representation and the holomorphic representation in linear and affine field theory. In the linear case, this isomorphism is induced by a one-to-one correspondence between complex structures and Schrödinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schrödinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any Kähler polarization.

Platelet uptake of GABA and glutamate in patients with bipolar disorder

Gamma aminobutyric acid (GABA) and glutamate (Glu) are the major neurotransmitters of the human central nervous system, and their actions are determined by specific transporters. Several studies suggest that GABA- and Glu-uptake mechanisms are modified in patients with bipolar disorder (BD). We explored the functionality of the GABA and Glu transporters in three groups of patients with BD, each with a different polarity of index episode (manic, depressive, or euthymic) at the time of blood draw. Forty patients with a diagnosis of BD, according to DSM-IV-TR criteria, and 15 healthy subjects were enrolled in the study. GABA and Glu uptake were evaluated in freshly prepared platelets using [(3) H]GABA or [(3) H]glutamate. Compared to controls, GABA uptake was significantly increased in patients with depressive episodes and significantly decreased in subjects with manic episodes. Glu uptake was significantly increased in patients with index manic episodes and in euthymic patients compared to healthy controls. Moreover, a positive correlation was found between GABA platelet uptake and Hamilton Depression Rating Scale scores and between Glu platelet uptake and Young Mania Rating Scale scores in patients with manic episodes. We found a relationship between GABA- and Glu-uptake levels and the polarity of episodes in patients with BD. Our data suggest that the functionality of both GABA and Glu transporters could represent a useful neurobiological marker to characterize the real polarity of an index episode of illness in patients with BD.

The quantization of the symplectic groupoid of the standard Podle s ̀ sphere

We give an explicit form of the symplectic groupoid G(S2,π) that integrates the semiclassical standard Podles̀ sphere (S2,π). We show that Sheu’s groupoid GS, whose convolution C∗-algebra quantizes the sphere, appears as the groupoid of the Bohr–Sommerfeld leaves of a (singular) real polarization of G(S2,π). By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.

Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when “looking at it from infinity” (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kähler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.

Analysis of protein interactions involved in real polarization of rat artery smooth muscle cells in atherosclerosis

Analysis of protein interactions involved in real polarization of rat artery smooth muscle cells in atherosclerosis

Quantization of abelian varieties: Distributional sections and the transition from Kähler to real polarizations

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil–Brezin expansion to describe distributional sections, we give a unified analytical description of the quantization spaces for all non-negative polarizations. The Blattner–Kostant–Sternberg (BKS) pairing maps between half-form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of Sp ( 2 g , R ) . Moreover, these maps are shown to be unitary.

Locally toric manifolds and singular Bohr-Sommerfeld leaves

A real polarization modelled on fibres of the moment map. The author computes the results directly and obtains a theorem similar to Sniatycki's, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kahler polarization.