Quantum Groups GLp,q(2)- and $\mathit{SU}_{q_{1}/q_{2}}(2)$ -Invariant Bosonic Gases: A Comparative Study

We discuss the algebras, representations, and thermodynamics of quantum group bosonic gas models with two different symmetries: GL _p,q (2) and . We establish the nature of the basic numbers which follow from these GL _p,q (2)- and -invariant bosonic algebras. The Fock space representations of both of these quantum group invariant bosonic oscillator algebras are analyzed. It is concisely shown that these two quantum group invariant bosonic particle gases have different algebraic and high-temperature thermo-statistical properties.     

Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice , which is a sublattice of the lattice of all quantum propositions and is determined by a quantum state e and a preferred determinate observable R. Topos-theoretic extension is made in the functor category of which base category is determined by R. Each true atom, which determines truth values, true or false, of all propositions in , generates also a multi-valued valuation function of which domain and range are and a Heyting algebra given by the subobject classifier in , respectively. All true propositions in are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category . Here, the base category includes all ’s as subcategories. Although has a structure apparently different from , a subobject semi-classifier of gives valuations completely equivalent to those in ’s.     

The Invariant Eigen-Operator Method for?N?Harmonically Coupled Identical Oscillators

In this paper, we apply the method of “invariant eigen-operator” to study the Hamiltonian of arbitrary number of coupled identical oscillators and derive their invariant eigen-operator. The results show that, (1) for the system of arbitrary number of identical harmonic oscillators by coordinate coupling or momentum coupling, the invariant eigen operator of system always has the form of or ; (2) the energy level gap of the system has two kinds of possibilities: one is that gap only related to ω that the frequency of oscillators; another one is that gap not only related to ω that the frequency of oscillators, but also related to the number of the coupling oscillators.     

A Note on the Extreme Points of Positive Quantum Operations

In this paper, an error in the proof of Theorem 4.9 in Gudder’s paper (Int. J. Theor. Phys. 47(1):268–279, 2008) is pointed out and it is proved that if such that E _i ∈ℂI∖{0} and E _j ∉ℂI for some i,j in {1,2,…,n}, then . This subject is supported by the NNSF of China (No. 10571113, 10871224).     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Lower Bound of Minimal Time Evolution in Quantum Mechanics

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S ², must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2×2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector P of the evolving quantum state with the vector of the 2×2 hermitian Hamiltonians and it is shown that the Hamiltonian H can be parameterized by two independent parameters and Θ.     

Scenario of Inflationary Cosmology from the Phenomenological Λ Models

Choosing the three phenomenological models of the dynamical cosmological term Λ, viz., , and Λ∼ρ where a is the cosmic scale factor, it has been shown by the method of numerical analysis for the considered non-linear differential equations that the three models are equivalent for the flat Universe k=0 and for arbitrary non-linear equation of state. The evolution plots for dynamical cosmological term Λ vs. time t and also the cosmic scale factor a vs. t are drawn here for k=0,+1. A qualitative analysis has been made from the plots which supports the idea of inflation and hence expanding Universe.     

Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Let be the spectrum of in L ²(ℝ), where q is an even almost-periodic complex-valued function with bounded primitive and derivative. It is known that , where is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to the first asymptotic correction is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of Δμ _n .      

The Invariant Eigen-Operator Method for Hamiltonians with Coordinates-Coordinates Coupling Terms

In this paper, we apply the method of “invariant eigen-operator” to study the Hamiltonian of harmonic oscillator with couplings and derive their invariant eigen-operator. We first discuss decoupling of coupled harmonic oscillators with the two different quality and frequencies. And then, we propose an operator Hamiltonian to describe the linear lattice chain with Born–von Karman boundary condition. The vibrating spectrum is thus obtained. The results show that, for the system of coupled harmonic oscillators by coordinate coupling or momentum coupling, the invariant eigen operator of system always has the form of or .     

Preparation,structure and properties of La<Subscript>0.67</Subscript>Pb<Subscript>0.33</Subscript>(Mn<Subscript>1-x</Subscript>Co<Subscript>x</Subscript>)O<Subscript>3-δ</Subscript>

La_0.67Pb_0.33(Mn_1-xCo_x)O_3-δ ceramics with x=0, 0.01, 0.03, 0.06, 0.1 and 0.15 have been prepared in a two-step procedure. Precursor gels were made by the wet chemical malic acid method. The gels were calcined and then converted into ceramics by heat treatment at 950 °C and 1000 °C in air. X-ray diffraction showed that the compounds were phase pure. The crystal structure symmetry of the compounds was confirmed to be rhombohedral (space group R3̄c) for the whole investigated range of x. All compounds undergo a paramagnetic–ferromagnetic phase transition between 335 K and 225 K. The basic magnetic characteristics such as the Curie temperature , the paramagnetic Curie temperature θ, the effective magnetic moment and the saturated magnetization decrease with increasing Co doping. The ferromagnetic transition is accompanied by an anomaly in the electrical resistance for all compounds. The high-temperature insulator–metal transitions () do not coincide with the relevant . A large magnetoresistance peak of about 15% was observed for all compounds at . PACS 72.80.Ga; 75.47.Lx; 75.60.Ej     

Near-flat space limit of strings on AdS 4×ℂℙ3

The non-linear nature of string theory on non-trivial backgrounds, related to the AdS/CFT correspondence, force one to look for simplifications. Two such simplifications proved to be useful in studying string theory. These are the pp-wave limit, which describes point-like strings, and the so-called “near-flat space” limit which connects two different sectors of string theory—pp-wave and “giant magnons”. Recently another example of AdS/CFT duality emerged—AdS ₄/CFT ₃, which suggests duality between CS theory and superstring theory on . In this paper we study the “near-flat space” limit of strings on an background and discuss possible applications of the limiting theory. R.C. Rashkov is on leave from Department of Physics, Sofia University, Bulgaria.     

Nonadditive entropy: The concept and its use

The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to construct the exact differential dS = Q/T , where Q is the heat transfer and the absolute temperature T its integrating factor. A few years later, in the period 1872-1877, it was shown by Boltzmann that this quantity can be expressed in terms of the probabilities associated with the microscopic configurations of the system. We refer to this fundamental connection as the Boltzmann-Gibbs (BG) entropy, namely (in its discrete form) , where k is the Boltzmann constant, and {p _i} the probabilities corresponding to the W microscopic configurations (hence ∑^W _i=1 p _i = 1 . This entropic form, further discussed by Gibbs, von Neumann and Shannon, and constituting the basis of the celebrated BG statistical mechanics, is additive. Indeed, if we consider a system composed by any two probabilistically independent subsystems A and B (i.e., , we verify that . If a system is constituted by N equal elements which are either independent or quasi-independent (i.e., not too strongly correlated, in some specific nonlocal sense), this additivity guarantees S_BG to be extensive in the thermodynamical sense, i.e., that in the N ≫ 1 limit. If, on the contrary, the correlations between the N elements are strong enough, then the extensivity of S_BG is lost, being therefore incompatible with classical thermodynamics. In such a case, the many and precious relations described in textbooks of thermodynamics become invalid. Along a line which will be shown to overcome this difficulty, and which consistently enables the generalization of BG statistical mechanics, it was proposed in 1988 the entropy . In the context of cybernetics and information theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form S_q is, for any q 1 , nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies . This form will turn out to be extensive for an important class of nonlocal correlations, if q is set equal to a special value different from unity, noted q_ent (where ent stands for entropy . In other words, for such systems, we verify that , thus legitimating the use of the classical thermodynamical relations. Standard systems, for which S_BG is extensive, obviously correspond to q _ent = 1 . Quite complex systems exist in the sense that, for them, no value of q exists such that S_q is extensive. Such systems are out of the present scope: they might need forms of entropy different from S_q, or perhaps --more plainly-- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with S_q, the q -generalizations of the Central Limit Theorem and of its extended Lévy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of q -exponentials, q -Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations --in high-energy physics and elsewhere-- are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms versus distinct regimes of a single physical mechanism. This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.     

(1+1)-Dirac Particle with Position-Dependent Mass in Complexified Lorentz Scalar Interactions: Effectively $\mathcal{PT}$ -Symmetric

The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the “quasi-parity” on the Dirac particles’ spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S(z)=S(x−ib) (an inversely linear plus linear, leading to a symmetric oscillator model), and S(x)=S _r (x)+iS _i (x) (a -symmetric Scarf II model) are considered. Moreover, a first-order intertwining differential operator and an η-weak-pseudo-Hermiticity generator are presented and a complexified -symmetric periodic-type model is used as an illustrative example.     

First-Order Intertwining Operators with Position Dependent Mass and <Emphasis Type="Italic">η</Emphasis>-Weak-Pseudo-Hermiticity Generators

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM—Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding η-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models ( -symmetric and non- -symmetric) are given as products of the reference-target map.     

Primordial <Superscript>4</Superscript>He Abundance Constrains the Possible Time Variation of the Higgs Vacuum Expectation Value

We constrain the possible time variation of the Higgs vacuum expectation value (v) by recent results on the primordial ⁴He abundance (Y _P ). For that, we use an analytic approach which enables us to take important issues into consideration, that have been ignored by previous works, like the v-dependence of the relevant cross sections of deuterium production and photodisintegration, including the full Klein–Nishina cross section. Furthermore, we take a non-equilibrium Ansatz for the freeze-out concentration of neutrons and protons and incorporate the latest results on the neutron decay. Finally, we approximate the key-parameters of the primordial ⁴He production (the mean lifetime of the free neutron and the binding energy of the deuteron) by terms of (where v ₀ denotes the present theoretical estimate). Eventually, we derive the relation and the most stringent limit on a possible time variation of v is given by: .     

Dilatonic Cosmology Model in the <Emphasis Type="Italic">ω</Emphasis>–<Emphasis Type="Italic">ω</Emphasis>′ Plane

In dilatonic cosmology model, we study the behavior of attractor solution in ω–ω′ plane, which is defined by the equation of state parameter for the dark energy and its derivative with respect to N (the logarithm of the scale factor a). This is a good method which is useful to the study of classifying the dynamical dark energy models including “freezing” and “thawing” model. We find that our model belongs to “freezing” type model classified in ω–ω′ plane. We show mathematically the property of attractor solutions which correspond to ω _σ =−1, Ω _σ =1. The present values of energy density parameter , and are 0.715001, 0.284972 and 0.00002706 respectively, which meet the current observations well. Finally, we can obtain that the coupling between dilaton and matter affects the evolutive process of the Universe, but not the fate of the Universe.     

Quantum Computer Algorithm for Parity Determination Based on Quantum Counting

A new quantum computer algorithm is proposed for determining the parity of function f(x) by using quantum counting algorithm. The parity of function f(x) can be determined by counting exactly the number of satisfying f(x)=−1, which is equivalent to determine the number of solutions, M, to an N item search problem. The algorithm can be accomplished in time of order .     

Wavelet Transformation and Wigner-Husimi Distribution Function

We find that the optical wavelet transformation can be used to study the Husimi distribution function in phase space theory of quantum optics. We prove that the Husimi distribution function of a quantum state |ψ〉 is just the modulus square of the wavelet transform of with ψ(x) being the mother wavelet up to a Gaussian function. Thus a convenient approach for calculating various Husimi distribution functions of miscellaneous quantum states is presented.