Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.Comment: 22 pages, 1 figure, LaTeX, partially presented at the Workshop "Nonlinearity and Coherence in Classical and Quantum Systems" held at the University "Federico II" in Naples, Italy on December 4, 2009 in honor of Prof. Margarita A. Man'ko in connection with her 70th birthday, minor misprints are corrected in the second versio

Exact Renormalization Group Equations. An Introductory Review

We critically review the use of the exact renormalization group equations (ERGE) in the framework of the scalar theory. We lay emphasis on the existence of different versions of the ERGE and on an approximation method to solve it: the derivative expansion. The leading order of this expansion appears as an excellent textbook example to underline the nonperturbative features of the Wilson renormalization group theory. We limit ourselves to the consideration of the scalar field (this is why it is an introductory review) but the reader will find (at the end of the review) a set of references to existing studies on more complex systems.Comment: Final version to appear in Phys. Rep.; Many references added, section 4.2 added, minor corrections. 65 pages, 6 fig

Derivative expansion of the renormalization group in O(N) scalar field theory

We apply a derivative expansion to the Legendre effective action flow equations of O(N) symmetric scalar field theory, making no other approximation. We calculate the critical exponents eta, nu, and omega at the both the leading and second order of the expansion, associated to the three dimensional Wilson-Fisher fixed points, at various values of N. In addition, we show how the derivative expansion reproduces exactly known results, at special values N=infinity,-2,-4, ... .Comment: 29 pages including 4 eps figures, uses LaTeX, epsfig, and latexsy

Theory of differential inclusions and its application in mechanics

The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load change is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems

Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras

As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^\alpha that are characterized by a root space of non-degenerate one-forms \alpha. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.Comment: LaTeX. 34 pages.v2. deleted some distracting paragraphs in the introduction to bring more out the main results of the paper. typos corrected and references adde

In situ evidence for the structure of the magnetic null in a 3D reconnection event in the Earth's magnetotail

Magnetic reconnection is one of the most important processes in astrophysical, space and laboratory plasmas. Identifying the structure around the point at which the magnetic field lines break and subsequently reform, known as the magnetic null point, is crucial to improving our understanding reconnection. But owing to the inherently three-dimensional nature of this process, magnetic nulls are only detectable through measurements obtained simultaneously from at least four points in space. Using data collected by the four spacecraft of the Cluster constellation as they traversed a diffusion region in the Earth's magnetotail on 15 September, 2001, we report here the first in situ evidence for the structure of an isolated magnetic null. The results indicate that it has a positive-spiral structure whose spatial extent is of the same order as the local ion inertial length scale, suggesting that the Hall effect could play an important role in 3D reconnection dynamics.Comment: 14 pages, 4 figure

On supersymmetric quantum mechanics

This paper constitutes a review on N=2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian can be associated with the algebra W_k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W_k, of the N=2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.Comment: Review paper (31 pages) to be published in: Fundamental World of Quantum Chemistry, A Tribute to the Memory of Per-Olov Lowdin, Volume 3, E. Brandas and E.S. Kryachko (Eds.), Springer-Verlag, Berlin, 200

Synthesis and solution properties of a temperature-responsive PNIPAM–b-PDMS–b-PNIPAM triblock copolymer

In this paper, we report the synthesis and self-assembly of a novel thermoresponsive PNIPAM60–b-PDMS70–b-PNIPAM60 triblock copolymer in aqueous solution. The copolymer used a commercially available precursor modified with an atom transfer radical polymerization (ATRP) initiator to produce an ABA triblock copolymer via ATRP. Small-angle neutron scattering (SANS) was used to shed light on the structures of nanoparticles formed in aqueous solutions of this copolymer at two temperatures, 25 and 40 °C. The poly(dimethylsiloxane) block is very hydrophobic and poly(N-isopropylacrylamide) (PNIPAM) is thermoresponsive. SANS data at 25 °C indicates that the solutions of PNIPAM–b-PDMS–b-PNIPAM copolymers form well-defined aggregates with presumably core–shell structures below cloud point temperature. The scattering curves originating from nanoparticles formed at 40 °C in 100% D2O or 100% H2O were successfully fitted with the Beaucage model describing aggregates with hierarchical structure