Index of content:
Volume 57, Issue 9, September 2016
Many physical systems can be described by eigenvalues of nonlinear equations and bifurcation problems with a linear part that is non-selfadjoint, e.g., due to the presence of loss and gain. The balance of these effects is reflected in an antilinear symmetry, e.g., the -symmetry. Under the symmetry we show that the nonlinear eigenvalues bifurcating from real linear eigenvalues remain real and the corresponding nonlinear eigenfunctions remain symmetric. The abstract result is applied in a number of physical models of Bose-Einstein condensation, nonlinear optics, and superconductivity, and numerical examples are presented.
- Partial Differential Equations
The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution57(2016); http://dx.doi.org/10.1063/1.4962917View Description Hide Description
In this paper, we investigate the one-dimensional parabolic-parabolic Patlak-Keller-Segel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the one-soliton solution of the Korteweg-de Vries equation.
57(2016); http://dx.doi.org/10.1063/1.4963172View Description Hide Description
- Representation Theory and Algebraic Methods
57(2016); http://dx.doi.org/10.1063/1.4962392View Description Hide Description
Lepowsky and Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via vertex operator constructions of standard (i.e., integrable highest weight) representations of affine Kac-Moody Lie algebras. Meurman and Primc developed further this approach for by using vertex operator algebras and Verma modules. In this paper, we use the same method to construct combinatorial bases of basic modules for affine Lie algebras of type and, as a consequence, we obtain a series of Rogers-Ramanujan type identities. A major new insight is a combinatorial parametrization of leading terms of defining relations for level one standard modules for affine Lie algebra of type .
57(2016); http://dx.doi.org/10.1063/1.4962722View Description Hide Description
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
57(2016); http://dx.doi.org/10.1063/1.4963142View Description Hide Description
In this paper, the Lie group , of which the kinematical symmetry group G NC of noncommutative quantum mechanics (NCQM) is a special case due to fixed nonzero α, β, and γ, is three-parameter deformation quantized using the method suggested by Ballesteros and Musso [J. Phys. A: Math. Theor. 46, 195203 (2013)]. A certain family of QUE algebras, corresponding to with two of the deformation parameters approaching zero, is found to be in agreement with the existing results of the literature on quantum Heisenberg group. Finally, we dualize the underlying QUE algebra to obtain an expression for the underlying star-product between smooth functions on .
57(2016); http://dx.doi.org/10.1063/1.4963171View Description Hide Description
Irreducible sp(4)-module with highest weight, labeled by the azimuthal and magnetic quantum numbers l and m, is split into the direct sums of the irreducible su(2)- and su(1, 1)-submodules in four different ways: finite integer unitary irreducible subspaces corresponding to the orbital angular momentum algebra su(2), infinite positive discrete series of su(1, 1) with an arbitrary half-integer Bargmann index, and the positive and negative discrete series of su(1, 1) with both the Bargmann indices 1/4 and 3/4. Even and odd coherent states for the positive su(1, 1)-submodules with the Bargmann indices 1/4 and 3/4 are constructed and it is shown that they enjoy the property of completeness by two appropriate positive definite measures. We show that the even and odd coherent states themselves form the positive discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4, respectively. For these even and odd coherent states, we consider the uncertainty relations for the x- and y-components of the angular momentum as well as the generators of the negative discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4.
- Many-Body and Condensed Matter Physics
57(2016); http://dx.doi.org/10.1063/1.4962337View Description Hide Description
We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to , which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n −2). This contrasts with gapless frustrated systems where the gap can be Θ(n −1). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.
- Quantum Mechanics
57(2016); http://dx.doi.org/10.1063/1.4961317View Description Hide Description
We present a Fermi golden rule giving rates of decay of states obtained by perturbing embedded eigenvalues of a quantum graph. To illustrate the procedure in a notationally simpler setting, we first describe a Fermi golden rule for boundary value problems on surfaces with constant curvature cusps. We also provide a resonance existence result which is uniform on compact sets of energies and metric graphs. The results are illustrated by numerical experiments.
57(2016); http://dx.doi.org/10.1063/1.4962926View Description Hide Description
We study the time evolution of the uncertainties Δx and Δp in position and momentum, respectively, associated with the semiclassical propagation of certain Gaussian initial states. We show that these quantities behave generically as , where P 1 and P 2 are periodic in time with period that of an underlying classical trajectory. We also show that, despite the overall (generically quadratic) growth in time, the uncertainty product ΔxΔp achieves its minimum of ħ/2 at arbitrarily large times.
57(2016); http://dx.doi.org/10.1063/1.4962923View Description Hide Description
We calculate the Fisher information (F r and F p) and the Shannon entropies (S r and S p) of a spinless charged particle in three different configurations of time-varying magnetic field, B(t). To do so, we first obtain the exact wave functions for a general time-dependent system by using a dynamical invariant method. By considering the solutions for n = m = 0, we were able to obtain the expressions of F r, F p, S r, and S p in terms of a c-number quantity, ρ, which has to be a real solution of the Milne–Pinney equation. We observe that the inequality F r F p ≤ 16 holds for the systems considered. We also observed squeezing phenomenon in momentum or/and coordinate spaces with increasing time.
57(2016); http://dx.doi.org/10.1063/1.4962924View Description Hide Description
We introduce a Hartmann system in the generalized Taub-NUT space with Abelian monopole interaction. This quantum system includes well known Kaluza-Klein monopole and MIC-Zwanziger monopole as special cases. It is shown that the corresponding Schrödinger equation of the Hamiltonian is separable in both spherical and parabolic coordinates. We obtain the integrals of motion of this superintegrable model and construct the quadratic algebra and Casimir operator. This algebra can be realized in terms of a deformed oscillator algebra and has finite dimensional unitary representations (unirreps) which provide energy spectra of the system. This result coincides with the physical spectra obtained from the separation of variables.
- Quantum Information and Computation
57(2016); http://dx.doi.org/10.1063/1.4962339View Description Hide Description
We introduce a 3-parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking, or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglement-breaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglement-annihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations.
57(2016); http://dx.doi.org/10.1063/1.4962340View Description Hide Description
We study the possible difference between the quantum and the private capacities of a quantum channel in the zero-error setting. For a family of channels introduced by Leung et al. [Phys. Rev. Lett. 113, 030512 (2014)], we demonstrate an extreme difference: the zero-error quantum capacity is zero, whereas the zero-error private capacity is maximum given the quantum output dimension.
57(2016); http://dx.doi.org/10.1063/1.4961526View Description Hide Description
We study the sequential product the operation on the set of effects, [0, 1]𝒜, of a von Neumann algebra 𝒜 that represents sequential measurement of first p and then q. In their work [J. Math. Phys. 49(5), 052106 (2008)], Gudder and Latrémolière give a list of axioms based on physical grounds that completely determines the sequential product on a von Neumann algebra of type I, that is, a von Neumann algebra ℬ(ℋ) of all bounded operators on some Hilbert space ℋ. In this paper we give a list of axioms that completely determines the sequential product on all von Neumann algebras simultaneously (Theorem 4).
- Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory
57(2016); http://dx.doi.org/10.1063/1.4962800View Description Hide Description
We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick’s theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal parameter irrespective of convergence properties. We establish invariance properties of such a Wick expansion under coordinate changes and the action of a Lie group of symmetries, and we use this to study essential features of path integral manipulations, including coordinate changes, Ward identities, Schwinger-Dyson equations, Faddeev-Popov gauge-fixing, and eliminating fields by their equation of motion. We also discuss the asymptotic nature of the Wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively.
- General Relativity and Gravitation
57(2016); http://dx.doi.org/10.1063/1.4961929View Description Hide Description
We systematically analyze the nonlinear partial differential equation that determines the behaviour of a bounded radiating spherical mass in general relativity. Four categories of solution are possible. These are identified in terms of restrictions on the gravitational potentials. One category of solution can be related to the horizon function transformation which was recently introduced. A Lie symmetry analysis of the resulting Riccati equation shows that several new classes of exact solutions are possible. The relationship between the horizon function, Euclidean star models, and other earlier investigations is clarified.
57(2016); http://dx.doi.org/10.1063/1.4962724View Description Hide Description
It is known that spherically symmetric static solutions of the Einstein equations with a positive cosmological constant for the energy-momentum tensor of a barotropic perfect fluid are governed by the Tolman-Oppenheimer-Volkoff-de Sitter equation. Some sufficient conditions for the existence of monotone-short solutions (with finite radii) of the equation are given in this article. Then we show that the interior metric can extend to the exterior Schwarzschild-de Sitter metric on the exterior vacuum region with twice continuous differentiability. In addition, we investigate the analytic property of the solutions at the vacuum boundary. Our result (Theorem 1) can be considered as the de Sitter version of the result by Rendall and Schmidt [Classical Quantum Gravity 8, 985-1000 (1991)]. Furthermore, one can see that there are different properties of the solutions with those of the Tolman-Oppenheimer-Volkoff equation (with zero cosmological constant) in certain situation.
57(2016); http://dx.doi.org/10.1063/1.4963144View Description Hide Description
The natural topology on the space of causal paths of a space-time depends on the topology chosen on the space-time itself. Here we consider the effect of using the path topology on space-time instead of the manifold topology, and its consequences for how properties of space-time are reflected in the structure of the space of causal paths.
57(2016); http://dx.doi.org/10.1063/1.4962723View Description Hide Description
Spin-weighted spherical functions provide a useful tool for analyzing tensor-valued functions on the sphere. A tensor field can be decomposed into complex-valued functions by taking contractions with tangent vectors on the sphere and the normal to the sphere. These component functions are usually presented as functions on the sphere itself, but this requires an implicit choice of distinguished tangent vectors with which to contract. Thus, we may more accurately say that spin-weighted spherical functions are functions of both a point on the sphere and a choice of frame in the tangent space at that point. The distinction becomes extremely important when transforming the coordinates in which these functions are expressed, because the implicit choice of frame will also transform. Here, it is proposed that spin-weighted spherical functions should be treated as functions on the spin or rotation groups, which simultaneously tracks the point on the sphere and the choice of tangent frame by rotating elements of an orthonormal basis. In practice, the functions simply take a quaternion argument and produce a complex value. This approach more cleanly reflects the geometry involved, and allows for a more elegant description of the behavior of spin-weighted functions. In this form, the spin-weighted spherical harmonics have simple expressions as elements of the Wigner 𝔇 representations, and transformations under rotation are simple. Two variants of the angular-momentum operator are defined directly in terms of the spin group; one is the standard angular-momentum operator L, while the other is shown to be related to the spin-raising operator ð.
57(2016); http://dx.doi.org/10.1063/1.4963143View Description Hide Description
We consider the construction of gauge theories of gravity, focussing in particular on the extension of local Poincaré invariance to include invariance under local changes of scale. We work exclusively in terms of finite transformations, which allow for a more transparent interpretation of such theories in terms of gauge fields in Minkowski spacetime. Our approach therefore differs from the usual geometrical description of locally scale-invariant Poincaré gauge theory (PGT) and Weyl gauge theory (WGT) in terms of Riemann–Cartan and Weyl–Cartan spacetimes, respectively. In particular, we reconsider the interpretation of the Einstein gauge and also the equations of motion of matter fields and test particles in these theories. Inspired by the observation that the PGT and WGT matter actions for the Dirac field and electromagnetic field have more general invariance properties than those imposed by construction, we go on to present a novel alternative to WGT by considering an “extended” form for the transformation law of the rotational gauge field under local dilations, which includes its “normal” transformation law in WGT as a special case. The resulting “extended” Weyl gauge theory (eWGT) has a number of interesting features that we describe in detail. In particular, we present a new scale-invariant gauge theory of gravity that accommodates ordinary matter and is defined by the most general parity-invariant eWGT Lagrangian that is at most quadratic in the eWGT field strengths, and we derive its field equations. We also consider the construction of PGTs that are invariant under local dilations assuming either the “normal” or “extended” transformation law for the rotational gauge field, but show that they are special cases of WGT and eWGT, respectively.