scholarly journals CONNES–LOTT MODEL BUILDING ON THE TWO-SPHERE

2001 ◽  
Vol 13 (01) ◽  
pp. 1-28 ◽  
Author(s):  
J. A. MIGNACO ◽  
C. SIGAUD ◽  
F. J. VANHECKE ◽  
A. R. DA SILVA
Keyword(s):  
Hilbert Space ◽  
Projective Module ◽  
Topological Structure ◽  
Model Building ◽  
Finite Algebra ◽  
Magnetic Monopoles ◽  
Spectral Triple ◽  
Spinor Bundle ◽  
Full Algebra ◽  
The Continuum

In this work we examine generalized Connes–Lott models, with C⊕C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over S2. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields.

Correlations in a General Theory of Quantum Measurement

2007 ◽  
Vol 14 (04) ◽  
pp. 445-458 ◽  
Author(s):  
Hanna Podsędkowska
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
General Theory ◽  
Quantum Measurement ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Von Neumann ◽  
Full Algebra ◽  
Algebraic Formalism ◽  
Algebra Of Operators

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states — by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.

scholarly journals The Behavior of Basic Fields

2018 ◽  
Vol 1 (3) ◽  
Keyword(s):  
Mathematical Modeling ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Differential Calculus ◽  
Separable Hilbert Space ◽  
Floating Platform ◽  
Basic Field ◽  
Floating Platforms ◽  
The Continuum

A basic field is defined in the realm of a mathematical modeling platform that is based on a collection of floating platforms and an embedding platform. Each floating platform is represented by a quaternionic separable Hilbert space. The embedding platform is a non-separable Hilbert space. A basic field is a continuum eigenspace of an operator that resides in the non-separable embedding Hilbert space. The continuum can be described by a quaternionic function, and its behavior is described by quaternionic differential calculus. The separable Hilbert spaces contain the point-like artifacts that trigger the basic field. The floating platforms possess symmetry, which in combination with the background platform generates the sources of symmetry related fields.

Quantum Blackwell–Sherman–Stein Theorem and Related Results

2013 ◽  
Vol 20 (04) ◽  
pp. 1350017 ◽  
Author(s):  
Krzysztof Kaniowski ◽  
Katarzyna Lubnauer ◽  
Andrzej Łuczak
Keyword(s):  
Hilbert Space ◽  
Quantum Information ◽  
Statistical Models ◽  
Finite Dimension ◽  
Arbitrary Dimension ◽  
Information Structures ◽  
General Operator ◽  
Quantum Statistical ◽  
Full Algebra ◽  
Algebra Of Operators

We investigate the problem of comparing quantum statistical models in the general operator algebra framework in arbitrary dimension, thus generalizing results obtained so far in finite dimension, and for the full algebra of operators on a Hilbert space. In particular, the quantum Blackwell–Sherman–Stein theorem is obtained, and informational subordination of quantum information structures is characterized.

scholarly journals Some aspects of quantum sufficiency for finite and full von Neumann algebras

2021 ◽  
Vol 111 (4) ◽  
Author(s):  
Andrzej Łuczak
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Pure States ◽  
Normal States ◽  
Full Algebra

AbstractSome features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.

GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY

2006 ◽  
Vol 17 (04) ◽  
pp. 477-491 ◽  
Author(s):  
ESTEBAN ANDRUCHOW ◽  
LÁZARO RECHT
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Finite Algebra ◽  
Strong Operator Topology ◽  
Smooth Curve ◽  
Minimal Length ◽  
Type Ii ◽  
Von Neumann ◽  
Space Norm ◽  
Minimal Geodesic

If [Formula: see text] is a type II1 von Neumann algebra with a faithful trace τ, we consider the set [Formula: see text] of self-adjoint projections of [Formula: see text] as a subset of the Hilbert space [Formula: see text]. We prove that though it is not a differentiable submanifold, the geodesics of the natural Levi–Civita connection given by the trace have minimal length. More precisely: the curves of the form γ(t) = eitxpe-itx with x* = x, pxp = (1 - p)x(1 - p) = 0 have minimal length when measured in the Hilbert space norm of [Formula: see text], provided that the operator norm ‖x‖ is less or equal than π/2. Moreover, any two projections which are unitary equivalent are joined by at least one such minimal geodesic, and only unitary equivalent projections can be joined by a smooth curve. Finally, we prove that these geodesics have also minimal length if one measures them with the Schatten k-norms of τ, ‖x‖k = τ((x* x)k/2)1/k, for all k ∈ ℝ, k ≥ 0. We also characterize curves of unitaries which have minimal length with these k-norms.

scholarly journals The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

2020 ◽  
Author(s):  
Giuseppe De Nittis ◽  
◽  
Maximiliano Sandoval ◽  
◽  
◽  
...  
Keyword(s):  
Dirac Operator ◽  
Hall Effect ◽  
Noncommutative Geometry ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Spectral Triple ◽  
C Algebra ◽  
Compact Resolvent ◽  
Landau Hamiltonian ◽  
The Continuum

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the C∗-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

scholarly journals Topology Structure Implied in β-Hilbert Space, Heisenberg Uncertainty Quantum Characteristics and Numerical Simulation of the DE Algorithm

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 330 ◽  
Author(s):  
Kaiguang Wang ◽  
Yuelin Gao
Keyword(s):  
Numerical Simulation ◽  
Hilbert Space ◽  
Topological Structure ◽  
Single Point ◽  
Inverse Correlation ◽  
Quantum States ◽  
Iterative Sequence ◽  
Optimal Point ◽  
Global Optimal ◽  
Changing Rate

The differential evolutionary ( D E ) algorithm is a global optimization algorithm. To explore the convergence implied in the H i l b e r t space with the parameter β of the D E algorithm and the quantum properties of the optimal point in the space, we establish a control convergent iterative form of a higher-order differential equation under the conditions of P - ε and analyze the control convergent properties of its iterative sequence; analyze the three topological structures implied in H i l b e r t space of the single-point topological structure, branch topological structure, and discrete topological structure; and establish and analyze the association between the H e i s e n b e r g uncertainty quantum characteristics depending on quantum physics and its topological structure implied in the β -Hilbert space of the D E algorithm as follows: The speed resolution Δ v 2 of the iterative sequence convergent speed and the position resolution Δ x β ε of the global optimal point with the swinging range are a pair of conjugate variables of the quantum states in β -Hilbert space about eigenvalues λ i ∈ R , corresponding to the uncertainty characteristics on quantum states, and they cannot simultaneously achieve bidirectional efficiency between convergent speed and the best point precision with any procedural improvements. Where λ i ∈ R is a constant in the β -Hilbert space. Finally, the conclusion is verified by the quantum numerical simulation of high-dimensional data. We get the following important quantitative conclusions by numerical simulation: except for several dead points and invalid points, under the condition of spatial dimension, the number of the population, mutated operator, crossover operator, and selected operator are generally decreasing or increasing with a variance deviation rate + 0 . 50 and the error of less than ± 0 . 5 ; correspondingly, speed changing rate of the individual iterative points and position changing rate of global optimal point β exhibit a inverse correlation in β -Hilbert space in the statistical perspectives, which illustrates the association between the H e i s e n b e r g uncertainty quantum characteristics and its topological structure implied in the β -Hilbert space of the D E algorithm.

C. The Continuous Spectrum of Pulsating Variable Stars

1967 ◽  
Vol 28 ◽  
pp. 177-206
Author(s):  
J. B. Oke ◽  
C. A. Whitney
Keyword(s):  
Continuous Spectrum ◽  
Variable Stars ◽  
Velocity Fields ◽  
The Other ◽  
Line Spectrum ◽  
Direct Information ◽  
Thermodynamic Structure ◽  
Diagnostic Interpretation ◽  
Pulsating Variable Stars ◽  
The Continuum

Pecker:The topic to be considered today is the continuous spectrum of certain stars, whose variability we attribute to a pulsation of some part of their structure. Obviously, this continuous spectrum provides a test of the pulsation theory to the extent that the continuum is completely and accurately observed and that we can analyse it to infer the structure of the star producing it. The continuum is one of the two possible spectral observations; the other is the line spectrum. It is obvious that from studies of the continuum alone, we obtain no direct information on the velocity fields in the star. We obtain information only on the thermodynamic structure of the photospheric layers of these stars–the photospheric layers being defined as those from which the observed continuum directly arises. So the problems arising in a study of the continuum are of two general kinds: completeness of observation, and adequacy of diagnostic interpretation. I will make a few comments on these, then turn the meeting over to Oke and Whitney.

Photo-electric Hβ and uvby photometry

1966 ◽  
Vol 24 ◽  
pp. 170-180
Author(s):  
D. L. Crawford
Keyword(s):  
Narrow Band ◽  
Line Strength ◽  
Interference Filter ◽  
B Stars ◽  
Relative Line ◽  
The Mean ◽  
F Stars ◽  
Two Parameters ◽  
Filter Photometry ◽  
The Continuum

Early in the 1950's Strömgren (1, 2, 3, 4, 5) introduced medium to narrow-band interference filter photometry at the McDonald Observatory. He used six interference filters to obtain two parameters of astrophysical interest. These parameters he calledlandc, for line and continuum hydrogen absorption. The first measured empirically the absorption line strength of Hβby means of a filter of half width 35Å centered on Hβand compared to the mean of two filters situated in the continuum near Hβ. The second index measured empirically the Balmer discontinuity by means of a filter situated below the Balmer discontinuity and two above it. He showed that these two indices could accurately predict the spectral type and luminosity of both B stars and A and F stars. He later derived (6) an indexmfrom the same filters. This index was a measure of the relative line blanketing near 4100Å compared to two filters above 4500Å. These three indices confirmed earlier work by many people, including Lindblad and Becker. References to this earlier work and to the systems discussed today can be found in Strömgren's article inBasic Astronomical Data(7).

Observations of the continuum near the galactic centre

1967 ◽  
Vol 31 ◽  
pp. 405
Author(s):  
F. J. Kerr
Keyword(s):  
Galactic Centre ◽  
The Past ◽  
Centre Region ◽  
The Continuum

A continuum survey of the galactic-centre region has been carried out at Parkes at 20 cm wavelength over the areal11= 355° to 5°,b11= -3° to +3° (Kerr and Sinclair 1966, 1967). This is a larger region than has been covered in such surveys in the past. The observations were done as declination scans.

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