Intuitionistic Quantum Logic of an <Emphasis Type="Italic">n</Emphasis>-level System

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M _n (?) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ _n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ?(M _n (?)) of projections in M _n (?) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice \(\mathcal{O}(\Sigma_{n})\) of functions from the poset \(\mathcal{C}(M_{n}(\mathbb{C}))\) of all unital commutative C*-subalgebras C of M _n (?) to ?(M _n (?)). The lattice \(\mathcal{O}(\Sigma_{n})\) is essentially the (pointfree) topology of the quantum phase space Σ _n , and as such defines a Heyting algebra. Each element of \(\mathcal{O}(\Sigma_{n})\) corresponds to a “Bohrified” proposition, in the sense that to each classical context \(C\in\mathcal{C}(M_{n}(\mathbb{C}))\) it associates a yes-no question (i.e. an element of the Boolean lattice ?(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.     

Polynomial Realization of <Emphasis Type="Italic">s</Emphasis>l<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2) and Fusion Rules at Exceptional Values of <Emphasis Type="Italic">q</Emphasis>

Representations of the sℓ_q(2) algebra are constructed in the space of polynomials of real (complex) variable for q^N=1. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.     

<Emphasis Type="Italic">q</Emphasis>-deformed Coherent States on a Circle

Coherent states for the q-deformed quantum mechanics on a circle are introduced.     

<Emphasis Type="Italic">J</Emphasis><Subscript><Emphasis Type="Italic">AW,WA</Emphasis></Subscript> functions in Passarino-Veltman reduction

We continue to study a special class of Passarino-Veltman functions J arising at the reduction of infrared divergent box diagrams. We describe a procedure of separation of two types of singularities, infrared and mass singularities, which are absorbed in simple C ₀ functions. The infrared divergences of C ₀’s can be regularized then by any method: photon mass, dimensionally or by the width of an unstable particle. Functions D ₀ are represented as certain linear combinations of the standard C ₀ Passarino-Veltman functions and infrared finite functions J. Then mass singularities are extracted from J to other combinations of C ₀. The rests are free of both types of singularities and are expressed as explicit and compact linear combination of logarithms and dilogarithm functions. The extensive comparison of numerical results with those obtained with the aid of the Loop Tools package is presented.     

Quantum secret sharing between <Emphasis Type="Italic">m</Emphasis>-party and <Emphasis Type="Italic">n</Emphasis>-party with six states

A quantum secret sharing scheme between an m-party group and an n-party group is proposed using three conjugate bases. A sequence of single photons, each of which is prepared in one of the six states, is used directly to encode classical information in the quantum secret sharing process. In this scheme, each of all m members in group 1 chooses randomly his/her own secret key individually and independently, and directly encodes his/her respective secret information on the states of single photons via unitary operations, then the last one sends 1/n of the resulting qubits to each member of group 2. By measuring their respective qubits, all members in group 2 share the secret information shared by all members in group 1. It renders impossible a Trojan horse attack with a multi-photon signal, a fake-signal attack with EPR pairs, an attack with single photons, and an attack with invisible photons. We give the upper bounds on the average success probabilities for dishonest agent eavesdropping encryption using the fake-signal attack with any two-particle entangled states. Supported by the National Natural Science Foundation of China (Grant No. 10671054), the Key Project of Science and Technology Research of Education Ministry of China (Grant No. 207011) and the Natural Science Foundation of Hebei Province, China (Grant Nos. 07M006 and F2009000311)     

<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Cosmology from <Emphasis Type="Italic">q</Emphasis>-theory

From a macroscopic theory of the quantum vacuum in terms of conserved relativistic charges (generically denoted by q ^(a) with label a), we have obtained, in the low-energy limit, a particular type of f(R) model relevant to cosmology. The macroscopic quantum-vacuum theory allows us to distinguish between different phenomenological f(R) models on physical grounds. The text was submitted by the authors in English.     

Energy levels of perturbed 6p<Emphasis Type="Italic">n</Emphasis>d <Emphasis Type="Italic">J</Emphasis>=2 Rydberg states of Pb I by multichannel quantum defect theory

Energy levels of the odd-parity 6pnd J=2 Rydberg states of atomic lead are analyzed by the multichannel quantum defect theory (MQDT) with previous experimental energy levels. The obtained. Using these parameters, admixture coefficients for each level are calculated to designate the level denotations. The results show that strong channel mixings exist for the levels near the 6pnd (3/2)[5/2]₂ ⁰ and 6p6d (3/2)[3/2]₂ ⁰ pertubers. The five-channel model different from that in literature is used to predict 21 energy positions of 6pnd (1/2)[3/2]₂ ⁰ levels and to determine the denotations of all the 6pnd J=2 Rydberg states under 59788 cm⁻¹ for Pb I.     

Non-vacuum Bianchi types <Emphasis Type="Italic">I</Emphasis> and <Emphasis Type="Italic">V</Emphasis> in <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity

In a recent paper (Sharif and Shamir in Class. Quantum Grav. 26:235020, 2009), we have studied the vacuum solutions of Bianchi types I and V spacetimes in the framework of metric f (R) gravity. Here we extend this work to perfect fluid solutions. For this purpose, we take stiff matter to find energy density and pressure of the universe. In particular, we find two exact solutions in each case which correspond to two models of the universe. The first solution gives a singular model while the second solution provides a non-singular model. The physical behavior of these models has been discussed using some physical quantities. Also, the function of the Ricci scalar is evaluated.     

Spin excitations and thermodynamics of the <Emphasis Type="Italic">t</Emphasis>-<Emphasis Type="Italic">J</Emphasis> model on the honeycomb lattice

We present a spin-rotation-invariant Green-function theory for the dynamic spin susceptibility in the spin-1/2 antiferromagnetic t-J Heisenberg model on the honeycomb lattice. Employing a generalized mean-field approximation for arbitrary temperatures and hole dopings, the electronic spectrum of excitations, the spin-excitation spectrum and thermodynamic quantities (two-spin correlation functions, staggered magnetization, magnetic susceptibility, correlation length) are calculated by solving a coupled system of self-consistency equations for the correlation functions. The temperature and doping dependence of the magnetic (uniform static) susceptibility is ascribed to antiferromagnetic short-range order. Our results on the doping dependencies of the magnetization and susceptibility are analyzed in comparison with previous results for the t-J model on the square lattice.     

Control of optical properties of polymethine dye <Emphasis Type="Italic">J</Emphasis>-aggregates using different additives

The effect of various additives (NaCl, DNA, and CPB surfactant) on the structure and optical properties of luminescent molecular nanoclusters, J-aggregates of L-21 polymethine dye, has been investigated using stationary and time-resolved optical spectroscopy. It has been shown that all investigated additives enhance the dye aggregation and change its aggregate structure as manifested in a change of tilt angle between molecules in the molecular chain. Salt causes the biggest change of tilt angle whereas surfactant causes the smallest. It has been found that a decrease of tilt angle leads to an increase of J-aggregate luminescence intensity. The addition of CPB surfactant is shown to modify the exciton dynamics in J-aggregates whereas the other additives affect only their structure. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 76, No. 2, pp. 250–255, March–April, 2009.     

<Emphasis Type="Italic">V</Emphasis>-<Emphasis Type="Italic">A</Emphasis> sum rules with <Emphasis Type="Italic">D</Emphasis> = 10 operators

The difference of vector and axial-vector charged current correlators is analyzed by means of QCD sum rules. The contribution of 10-dimensional 4-quark condensates is calculated and its value is estimated within the framework of the factorization hypothesis. It is compared to the result obtained from an operator fit of Borel sum rules in the complex q ²-plane, calculated from experimental data on hadronic -decays. This fit gives accurate values of the light quark condensate and the quark-gluon mixed condensate. The size of the high-order operators and the convergence of the operator series are discussed.Received: 10 May 2004, Revised: 7 September 2004, Published online: 18 November 2004     

Ferromagnetism and ferroelectricity in Eu<Emphasis Type="Italic">X</Emphasis> (<Emphasis Type="Italic">X</Emphasis> = <Emphasis Type="Italic">O</Emphasis>, <Emphasis Type="Italic">S</Emphasis>): pressure effects

Ferromagnetism and ferroelectricity in Eu monochalcogenides have been investigated by ab initio density functional theory in the DFT+U approach. Exchange interaction parameters and Curie temperatures under pressure are studied and discussed using Heisenberg Hamiltonian with first and second-nearest-neighbor interactions. The calculations showed that the hydrostatic pressure perfectly improves the Curie temperature (EuO: T _C = 175 K; EuS: T _C = 33.8 K) and in the other hand it cannot induce the spontaneous polarization (P _s ). The effect of uniaxial and biaxial pressure is also studied. Although the uniaxial strains slightly increases the Curie temperature, it ensures the ferrolectricity in these systems by producing a spontaneous polarization of the order of P _s (EuO) = 57.50 μC/cm² and P _s (EuS) = 42.86 μC/cm² with pressures of 5% and 4%, respectively. The search for new model systems is a necessity to better understand the physics related to multiferroïc materials and to consider possible applications.     

Spherical Symmetric Perfect Fluid Collapse in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) Gravity

This paper contains the study of spherically symmetric perfect fluid collapse in the frame work of f(R, T) modified theory of gravity. We proceed our work by considering the non-static spherically symmetric background in the interior and static spherically symmetric background in the exterior regions of the star. The junction conditions between exterior and interior regions are presented by matching the exterior and interior regions. The field equations are solved by taking the assumptions that the Ricci scalar as well as the trace of energy-momentum tensor are to be constant, for a particular f(R, T) model. By inserting the solution of the field equations in junction conditions, we evaluate the gravitational mass of the collapsing system. Also, we discuss the apparent horizons and their time formation for different possible cases. It is concluded that the term f(R ₀, T ₀) behaves as a source of repulsive force and that’s why it slowdowns the collapse of the matter.     

Energy dependence of Cronin momentum in saturation model for <Emphasis Type="Italic">p</Emphasis> + <Emphasis Type="Italic">A</Emphasis> and <Emphasis Type="Italic">A</Emphasis> + <Emphasis Type="Italic">A</Emphasis> collisions

The energy dependence of the Cronin momentum for p + A and A + A collisions in the saturation model are calculated. This dependence is consistent with simple dimensional considerations and can be used to test the validity of the saturation model. It gives the possibility to distinguish the different variants of the saturation model with precise experimental data and to measure the x dependence of the saturation momentum.     

Reconstruction of some cosmological models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>,<Emphasis Type="Italic">T</Emphasis>) cosmology

In this paper, we reconstruct cosmological models in the framework of f(R,T) gravity, where R is the Ricci scalar and T is the trace of the stress-energy tensor. We show that the dust fluid reproduces ΛCDM, phantom–non-phantom era and phantom cosmology. Further, we reconstruct different cosmological models, including the Chaplygin gas, and scalar field with some specific forms of f(R,T). Our numerical simulation for the Hubble parameter shows good agreement with the BAO observational data for low redshifts, z<2.     

Fermion Localization on the <Emphasis Type="Italic">orbifold</Emphasis><Emphasis Type="Italic">S</Emphasis><Subscript>1</Subscript>/<Emphasis Type="Italic">Z</Emphasis><Subscript>2</Subscript>

The hierarchical structure of fermion masses of the Standard Model is explained in split fermion models by localizing the fermions at different points in an extra dimension. We consider split fermion models with two bulk scalars compactified on an orbifold. In the static case we find analytical expression for the localizer. We also address the issue of stability of the localizer. We also find exact solutions for the fermion zero modes. We explore the parameter space of the model. We find ample opportunity for construction of phenomenologically viable theories exist.     

Collisions between molecules in the <Emphasis Type="Italic">J</Emphasis> = 0 state

Collisions between electrostatically trapped molecules in the J = 0 state are theoretically investigated. Molecules in this state are advantageous for the evaporative cooling, because inelastic collision is impossible at ultra-low temperature. However, inelastic collision is more significant than elastic collision at high temperature, because the off-diagonal dipole matrix elements are much larger than the diagonal elements. The elastic and inelastic collision cross-sections are calculated under the condition that the collisional kinetic energy is larger than the energy gap between the rotational states. The dependencies of the collision cross-sections on the collisional kinetic energy and molecular constants are examined.Received: 29 July 2004, Published online: 21 September 2004PACS: 31.15.Qg Molecular dynamics and other numerical methods - 33.80.Ps Optical cooling of molecules; trapping - 33.90. + h Other topics in molecular properties and interactions with photons     

<Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) black holes

We study the f (R)-Maxwell black hole imposed by constant curvature and its all thermodynamic quantities, which may lead to the Reissner-Nordström-AdS black hole by redefining Newtonian constant and charge. Further, we obtain the f (R)-Yang-Mills black hole imposed by constant curvature, which is related to the Einstein-Yang-Mills black hole in AdS space. Since there is no analytic black hole solution in the presence of Yang-Mills field, we obtain asymptotic solutions. Then, we confirm the presence of these solutions in a numerical way.     

<Emphasis Type="Italic">O</Emphasis>(1) eV sterile neutrino in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity

We refer [1] to the role of an additional O(1) eV sterile neutrino in modified gravity models. We find parameter constraints in particular f(R) gravity model using following up-to-dated cosmological data: measurements of the cosmic microwave background (CMB) anisotropy, the CMB lensing potential, the baryon acoustic oscillations (BAO), the cluster mass function and the Hubble constant. It was obtained for the sterile neutrino mass 0.47 eV < m _ν,sterile < 1 eV (2σ) assuming that the sterile neutrinos are thermalized and the active neutrinos are massless, not significantly larger than in the standard cosmology model within the same data set: 0.45 eV < m _ν,sterile < 0.92 eV (2σ). But, if the mass of sterile neutrino is fixed and equals ≈ 1.5 eV according to various anomalies in neutrino oscillation experiments, f(R) gravity is much more consistent with observation data than the CDM model.