Volume 51, Issue 8, August 2010
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
51(2010); http://dx.doi.org/10.1063/1.3466798View Description Hide Description
This paper reveals the relation between the canonical coset decomposition of unitary matrices and the corresponding decomposition via Householder reflections. These results can be used to parametrize unitary matrices via Householder reflections.
51(2010); http://dx.doi.org/10.1063/1.3464492View Description Hide Description
In this study, the effect of time fractionalization on the development of quantum systems is taken under consideration by making use of fractional calculus. In this context, a Mittag–Leffler function is introduced as an important mathematical tool in the generalization of the evolution operator. In order to investigate the time fractional evolution of the quantum (nano) systems, time fractional forms of motion are obtained for a Schrödinger equation and a Heisenberg equation. As an application of the concomitant formalism, the wave functions, energy eigenvalues, and probability densities of the potential well and harmonic oscillator are time fractionally obtained via the fractional derivative order , which is a measure of the fractality of time. In the case , where time becomes homogenous and continuous, traditional physical conclusions are recovered. Since energy and time are conjugate to each other, the fractional derivative order is relevant to time. It is understood that the fractionalization of time gives rise to energy fluctuations of the quantum (nano) systems.
51(2010); http://dx.doi.org/10.1063/1.3466802View Description Hide Description
We propose proper quantization rule, , where . The and are two turning points determined by , and is the number of the nodes of wave function. We carry out the exact solutions of solvable quantum systems by this rule and find that the energy spectra of solvable systems can be determined only from its ground state energy. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of the rule come from its meaning—whenever the number of the nodes of or the number of the nodes of the wave function increases by 1, the momentum integral will increase by . We apply this proper quantization rule to carry out solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization, the Hulthén potential, the Scarf II potential, the asymmetric trigonometric Rosen–Morse potential, the Pöschl–Teller type potentials, the Rosen–Morse potential, the Eckart potential, the harmonic oscillator in three dimensions, the hydrogen atom, and the Manning–Rosen potential in dimensions.
51(2010); http://dx.doi.org/10.1063/1.3467214View Description Hide Description
We present a deformation of a class of elementary classical integrable systems using stochastic diffusion processes. This deformation applies to the solution of the associated classical Newtonian, Hamiltonian, Lagrangian, and variational problems and to the Hamilton–Jacobi method of characteristics. The underlying stochastic action functionals involve dual random times, whose expectations are connected to the new variables of the system after a canonical transformation.
51(2010); http://dx.doi.org/10.1063/1.3479748View Description Hide Description
It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement and (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.
51(2010); http://dx.doi.org/10.1063/1.3474600View Description Hide Description
One of the crucial differences between mathematical models of classical and quantum mechanics (QM) is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems, one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and the quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein dreamed of. QM is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem. We present the basic physical interpretation of prequantum classical statistical field theory in Sec. II. However, this is only the first step toward real physical theory.
Dynamical typicality: Convergence of time evolved macro-observables to their mean values in random matrix models51(2010); http://dx.doi.org/10.1063/1.3374669View Description Hide Description
Here we analyze the notion of dynamical typicality in large quantum random matrix models. By dynamical typicality we mean that different Hamiltonian systems evolve in time in a practically indistinguishable manner. We prove dynamical typicality for a Hamiltonian belonging to the Gaussian unitary ensemble (GUE) ensemble and argue in the conclusion why the same result is valid for more general Hamiltonians.
- Quantum Information and Computation
51(2010); http://dx.doi.org/10.1063/1.3477319View Description Hide Description
Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols, they remain ill understood. Here, we construct special sets of up to mutually unbiased bases (MUBs) in dimension , which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to prove a lower bound for min-entropic entropic uncertainty relations for any set of MUBs and show that symmetry plays a central role in obtaining tight bounds. For example, we obtain for the first time a tight bound for four MUBs in dimension , which is attained by an eigenstate of our complementarity transform. Finally, we discuss the relation to other symmetries obtained by transformations in discrete phase space and note that the extrema of discrete Wigner functions are directly related to min-entropic uncertainty relations for MUBs.
51(2010); http://dx.doi.org/10.1063/1.3459068View Description Hide Description
We consider a family of vector and operator norms defined by the Schmidt decomposition theorem for quantum states. We use these norms to tackle two fundamental problems in quantum informationtheory: the classification problem for -positive linear maps and entanglement witnesses, and the existence problem for nonpositive partial transpose bound entangled states. We begin with an analysis of the norms, showing that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We then use the norms to establish what appears to be the most general spectral test for -positivity currently available, showing how it implies several other known tests as well as some new ones. Building on this work, we frame the nonpositive partial transpose bound entangled problem as a concrete problem on a specific limit, specifically that a particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
Minimal unitary representation of and its deformations as massless conformal fields and their supersymmetric extensions51(2010); http://dx.doi.org/10.1063/1.3447773View Description Hide Description
We study the minimal unitary representation (minrep) of over a Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of , which coincides with the minrep of similarly constructed, corresponds to a massless conformal scalar in four space-time dimensions. There exists a one-parameter family of deformations of the minrep of . For positive (negative) integer values of the deformation parameter , one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in representation of the subgroup. We construct the supersymmetric extensions of the minrep of and its deformations to those of . The minimal unitary supermultiplet of , in the undeformed case, simply corresponds to the massless Yang–Mills supermultiplet in four dimensions. For each given nonzero integer value of , one obtains a unique supermultiplet of massless conformal fields of higher spin. For , these supermultiplets are simply the doubleton supermultiplets studied in the work of Gunaydin et al. [Nucl. Phys. B534, 96 (1998); e-print arXiv:hep-th/9806042].
51(2010); http://dx.doi.org/10.1063/1.3449321View Description Hide Description
We give a coalgebra structure on 1-vertex irreducible graphs which is that of a cocommutative coassociative graded connected coalgebra. We generalize the coproduct to the algebraic representation of graphs so as to express a bare 1-particle irreducible -point function in terms of its loop order contributions. The algebraic representation is so that graphs can be evaluated as Feynman graphs.
51(2010); http://dx.doi.org/10.1063/1.3467037View Description Hide Description
We consider the pseudorelativistic Chandrasekhar/Herbst operator for the description of relativistic one-electron ions in a locally bounded magnetic field. We show that for Coulomb potentials of strength , the spectrum of is discrete below (the electron mass). For magnetic fields in the class , the ground-state energy of decreases according to as for , where is some critical value, depending on .
51(2010); http://dx.doi.org/10.1063/1.3449328View Description Hide Description
We explore the connections between three classes of theories: quiver matrix models, conformal Toda field theories, and supersymmetric conformal quiver gauge theories. In particular, we analyze the quiver matrix models recently introduced by Dijkgraaf and Vafa (unpublished) and make detailed comparisons with the corresponding quantities in the Toda field theories and the quiver gauge theories. We also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions.
51(2010); http://dx.doi.org/10.1063/1.3456122View Description Hide Description
We study the interaction of mutually noninteracting Klein–Gordon particles with localized sources on stochastically complete Riemannian surfaces. This asymptotically free theory requires regularization and coupling constant renormalization.Renormalization is performed nonperturbatively using the orthofermion algebra technique and the principal operator is found. The principal operator is then used to obtain the bound state spectrum, in terms of binding energies to single Dirac-delta function centers. The heat kernel method allows us to generalize this procedure to compact and Cartan–Hadamard type Riemannian manifolds. We make use of upper and lower bounds on the heat kernel to constrain the ground state energy from below, thus confirming that our neglect of pair creation is justified for certain ranges of parameters in the problem.
51(2010); http://dx.doi.org/10.1063/1.3475536View Description Hide Description
We study the two dimensional vector mesonmodel introduced by Thirring and Wess, that is to say the Schwinger model with massive photon and massless fermion. We prove, with a renormalization group approach, that the vector and axial Ward identities are broken by the Adler–Bell–Jackiw anomaly; and we rigorously establish three widely believed consequences: (a) the interacting meson-meson correlation equals a free boson propagator, although the mass is additively renormalized by the anomaly; (b) the anomaly is quadratic in the charge, in agreement with the Adler–Bardeen formula; (c) the fermion-fermion correlation has an anomalous long-distance decay.
- General Relativity and Gravitation
Binary spinning black hole Hamiltonian in canonical center-of-mass and rest-frame coordinates through higher post-Newtonian order51(2010); http://dx.doi.org/10.1063/1.3448924View Description Hide Description
The recently constructed Hamiltonians for spinless binary black holes through third post-Newtonian order and for spinning ones through formal second post-Newtonian order, where the spins are counted of zero post-Newtonian order, are transformed into fully canonical center-of-mass and rest-frame variables. The mixture terms in the Hamiltonians between center-of-mass and rest-frame variables are in accordance with the relation between the total linear momentum and the center-of-mass velocity as demanded by global Lorentz invariance. The various generating functions for the center-of-mass and rest-frame canonical variables are explicitly given in terms of the single-particle canonical variables. The no-interaction theorem does not apply because the world-line condition of Lorentz covariant position variables is not imposed.
51(2010); http://dx.doi.org/10.1063/1.3473786View Description Hide Description
In this work, we study the Hilbert space space of -valent SU(2) intertwiners with fixed total spin, which can be identified, at the classical level, with a space of convex polyhedra with faces and fixed total boundary area. We show that this Hilbert space provides, quite remarkably, an irreducible representation of the group. This gives us therefore a precise identification of as a group of area-preserving diffeomorphisms of polyhedral spheres. We use this result to get new closed formulas for the black holeentropy in loop quantum gravity.
51(2010); http://dx.doi.org/10.1063/1.3460856View Description Hide Description
We present an interesting connection between Einstein–Friedmann equations for the models of universe filled with scalar field and the special form of Abel equation of the first kind. This connection works in both ways: first, we show how, knowing the general solution of the Abel equation (corresponding to the given scalar field potential), one can obtain the general solution of the Friedman equation (and use the former for studying such problems as the existence of inflation with exit for particular models). On the other hand, one can invert the procedure and construct the Bäcklund autotransformations for the Abel equation.
51(2010); http://dx.doi.org/10.1063/1.3475798View Description Hide Description
One of the effects of noncommutative coordinate operators is that the delta function connected to the quantum mechanical amplitude between states sharp to the position operator gets smeared by a Gaussian distribution. Although this is not the full account of the effects of noncommutativity, this effect is, in particular, important as it removes the point singularities of Schwarzschild and Reissner–Nordström solutions. In this context, it seems to be of some importance to probe also into ringlike singularities which appear in the Kerr case. In particular, starting with an anisotropic energy-momentum tensor and a general axisymmetric ansatz of the metric together with an arbitrary mass distribution (e.g., Gaussian), we derive the full set of Einstein equations that the noncommutative geometry inspired Kerr solution should satisfy. Using these equations we prove two theorems regarding the existence of certain Kerr metrics inspired by noncommutative geometry.
- Dynamical Systems
51(2010); http://dx.doi.org/10.1063/1.3456080View Description Hide Description
In this paper, it is discovered that the statistical property of the consensus and synchronization of the small-world networks, that is, the Cheeger constant, is a major determinant to measure the convergence rate of the consensus and synchronization of the small-world networks. Further, we give a mathematical rigorous estimation of the lower bound for the algebraic connectivity of the small-world networks, which is much larger than the algebraic connectivity of the regular circle. This result explains why the consensus problems on the small-world network have an ultrafast convergence rate and how much it can be improved. Moreover, it also characterizes quantitatively what kind of the small-world networks can be synchronized.