Volume 52, Issue 12, December 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3671331View Description Hide Description
Let be the set of all bounded self-adjoint linear operators on a complex Hilbert space In 2006, Gudder [Math. Slovaca56, 573 (2006)] introduced a new order ≼ on Since then, the existence conditions and representations of the supremum and infimum of two elements in with respect to the order ≼ have been intensively studied. Specifically, Li and Sun [J. Math. Phys.50, 122107 (2009)]10.1063/1.3272542 obtained simpler representations of A ∧ P and A ∨ P, where and P is an orthogonal projection on In this note, we present more intuitive and concise results on A ∨ P and extend the results of Li and Sun to more general cases. Moreover, some applications of our results are given to show that our results are easier to deal with.
Extreme covariant quantum observables in the case of an Abelian symmetry group and a transitive value space52(2011); http://dx.doi.org/10.1063/1.3668317View Description Hide Description
We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.
52(2011); http://dx.doi.org/10.1063/1.3672208View Description Hide Description
52(2011); http://dx.doi.org/10.1063/1.3667207View Description Hide Description
In this paper, we consider quantization of powers of the ratio between the Hamiltonian coordinates for position and momentum in one-dimensional systems. The domain of the operators consists of square integrable functions over a finite real interval to ensure boundedness and self-adjointness. The spectral problems for the operators that result from using Weyl-ordering are discussed by introducing Fredholm integral operator forms in position representation, and the symmetry of the actions of the parity and time reversal operators on the kernels is discussed. Finally, the general structures and properties of the eigenfunctions and eigenvalues are also derived and analyzed.
52(2011); http://dx.doi.org/10.1063/1.3672196View Description Hide Description
We obtain exact analytical expressions for the total energy per particle and related quantities corresponding to a finite quantum Hall system of electrons in a disk geometry when filling factor of the Landau level is one. Such exact results apply to finite systems of electrons with an arbitrary number of particles. The reported calculations for finite systems of electrons in a disk geometry complement earlier calculations for finite systems of electrons in a simpler spherical geometry. The results we provide can serve as benchmarks to gauge the accuracy of various theoretical approximations and numerical methods used to study the properties of strongly correlated many-body systems.
52(2011); http://dx.doi.org/10.1063/1.3663431View Description Hide Description
We consider the quantum δ-Bose gas on the infinite line. For repulsive interactions, Tracy and Widom have obtained an exact formula for the quantum propagator. In our contribution we explicitly perform its analytic continuation to attractive interactions. We also study the connection to the expansion of the propagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide an independent proof of their completeness.
52(2011); http://dx.doi.org/10.1063/1.3671966View Description Hide Description
We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum, and Darboux-Bäcklund transformations.
52(2011); http://dx.doi.org/10.1063/1.3671640View Description Hide Description
A nonpolynomial one-dimensional quantum potential in the form of an isotonic oscillator (harmonic oscillator with a centripetal barrier) is studied. We provide the nonrelativistic bound state energy spectrum E n and the wave functions ψ n (x) in terms of the associated Laguerre polynomials in the framework of the Nikiforov-Uvarov method. Under the spin and pseudospin symmetric limits, the analytic eigenvalues and the corresponding two-component upper- and lower-spinors of the Dirac particle are obtained in closed form.
- Quantum Information and Computation
52(2011); http://dx.doi.org/10.1063/1.3663835View Description Hide Description
Let D and E be subspaces of the tensor product of the m- and n-dimensional complex spaces, with co-dimensions k and ℓ, respectively. In order to give upper bounds for ranks of entangled edge states with positive partial transposes, we show that if k + ℓ < m + n − 2, then there must exist a product vector in D whose partial conjugate lies in E. If k + ℓ = m + n − 2, then such a product vector may or may not exist depending on k and ℓ.
52(2011); http://dx.doi.org/10.1063/1.3663836View Description Hide Description
We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general unextendible product bases, nevertheless, they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's theorem from algebraic geometry.
52(2011); http://dx.doi.org/10.1063/1.3663837View Description Hide Description
It is known that some two-qutrit entangled states of rank 4 with positive partial transpose can be built from the unextendible product bases (UPB) [C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett.82, 5385 (1999)]. We show that this fact is indeed universal, namely, all such states can be constructed from UPB as conjectured recently by Leinaas, Myrheim, and Sollid. We also classify the five-dimensional subspaces of two qutrits which contain only finitely many product states (up to scalar multiple), and in particular those spanned by an UPB.
52(2011); http://dx.doi.org/10.1063/1.3668081View Description Hide Description
The space of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in , the algebra of linear operators acting on a d-dimensional Hilbert space, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra into . This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps . The extremal and C*-extremal points of are determined.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
52(2011); http://dx.doi.org/10.1063/1.3664760View Description Hide Description
We construct the graviton propagator on de Sitter background in exact de Donder gauge. We prove that it must break de Sitter invariance, just like the propagator of the massless, minimally coupled scalar. Our explicit solutions for its two scalar structure functions preserve spatial homogeneity and isotropy so that the propagator can be used within the larger context of inflationary cosmology; however, it is simple to alter the residual symmetry. Because our gauge condition is de Sitter invariant (although no solution for the propagator can be) renormalization should be simpler using this propagator than one based on a noncovariant gauge. It remains to be seen how other computational steps compare.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3670398View Description Hide Description
We analyze structures of covering space over a Lorentzian manifold. By use of this we show that, if a Lorentzian manifold is globally hyperbolic then for any two causally related points p and q, the number of homotopy classes of causal curves from p to q is finite and each of its homotopy classes has a causal geodesic from p to q.
- Dynamical Systems
52(2011); http://dx.doi.org/10.1063/1.3666020View Description Hide Description
A technique of dynamically defined measures is developed and its relation to the theory of equilibrium states is shown. The technique uses Carathéodory's method and the outer measure introduced in a previous work by I. Werner [Math. Proc. Camb. Phil. Soc.140(2), 333–347 (2006)10.1017/S0305004105009072]. As an application, equilibrium states for contractive Markov systems [I. Werner, J. London Math. Soc.71(1), 236–258 (2005)10.1112/S0024610704006088] are obtained.
52(2011); http://dx.doi.org/10.1063/1.3672193View Description Hide Description
In this paper, we consider the three-wave interaction system where γ, δ are real parameters. Our main results are the complete characterization of all values of the parameters γ, δ for which the three-wave interaction system admits either invariant algebraic surfaces or algebraic (polynomial or rational) first integrals.
- Classical Mechanics and Classical Fields
Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors52(2011); http://dx.doi.org/10.1063/1.3664754View Description Hide Description
Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdička-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher rank Stäckel-Killing tensors and that in the case of the pp-wave spacetimes, the symmetry group of the Stäckel-Killing tensors is the well-known Newton-Hooke group.
- Statistical Physics
52(2011); http://dx.doi.org/10.1063/1.3669485View Description Hide Description
We have studied the specific heat of the (N S + N B ) model for an N S -body harmonic oscillator (HO) system which is strongly coupled to an N B -body HO bath without dissipation. The system specific heat of C S (T) becomes N S k B at T → ∞ and vanishes at T = 0 in accordance with the third law of thermodynamics. The calculated C S (T) at low temperatures is not proportional to N S and shows an anomalous temperature dependence, strongly depending on N S , N B , and the system-bath coupling. In particular at very low (but finite) temperatures, it may become negative for a strong system-bath coupling, which is in contrast with non-negative specific heat of a HO system with N S = 1 reported by Ingold, Hänggi, and Talkner [Phys. Rev. E79, 061105 (2005)]. Our calculation indicates an importance of taking account of finite N S in studying open quantum systems which may include an arbitrary number of particles in general.
- Methods of Mathematical Physics
Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics. II. The no-pair operator52(2011); http://dx.doi.org/10.1063/1.3658863View Description Hide Description
We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of a no-pair operator acting in the positive spectral subspace of the free Dirac operator minimally coupled to the quantized vector potential. We prove that the infimum of the spectrum of the no-pair operator is an evenly degenerate eigenvalue. In particular, we show that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the fine-structure constant and the ultraviolet cut-off and for all Coulomb coupling constants less than the critical one of the Brown-Ravenhall model, 2/(2/π + π/2). For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the no-pair operator is unbounded below. Along the way we discuss the domains and operator cores of the semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling constants less than or equal to the critical ones.
52(2011); http://dx.doi.org/10.1063/1.3653197View Description Hide Description
As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.