Volume 52, Issue 2, February 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3545800View Description Hide Description
We formulate and examine a phase-space path integral representation of a quantum time evolution in terms of Fock–Bargmann spaces and stochastic line integrals. The Fock–Bargmann space is a closed subspace of , where is the phase space. Let E FB be the orthogonal projection onto , and h the classical Hamiltonian, then H = E FB hE FB is understood as a quantization of h. The quantum time evolution e −itH is asymptotically represented by the solution of a variant of diffusionequation with the imaginary potential ih. Thus the quantum time evolution can be represented stochastically, in terms of Brownian motions and stochastic integrals.
52(2011); http://dx.doi.org/10.1063/1.3548076View Description Hide Description
52(2011); http://dx.doi.org/10.1063/1.3549587View Description Hide Description
We consider a two-body quantum system in dimension one composed by a test particle interacting with a harmonic oscillator placed at the position a > 0. At time zero the test particle is concentrated around the position R 0 with average velocity ±v 0 while the oscillator is in its ground state. In a suitable scaling limit, corresponding for the test particle to a semiclassical regime with small energy exchange with the oscillator, we give a complete asymptotic expansion of the wave function of the system in both cases and .
52(2011); http://dx.doi.org/10.1063/1.3549586View Description Hide Description
We find a new highly symmetrical and very numerous class (millions of nonisomorphic sets) of 4-dim Kochen–Specker (KS) vector sets. Due to the nature of their geometrical symmetries, they cannot be obtained from previously known ones. We generate the sets from a single set of 60 orthogonal spin vectors and 75 of their tetrads (which we obtained from the 600-cell) by means of our newly developed stripping technique. We also consider critical KS subsets and analyze their geometry. The algorithms and programs for the generation of our KS sets are presented.
52(2011); http://dx.doi.org/10.1063/1.3549585View Description Hide Description
The Rényi and Shannon entropies are information-theoretic measures, which have enabled to formulate the position–momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density functionals of quantum systems. Here we derive upper bounds on these quantities in terms of the second-order moment 〈r 2〉 for general central potentials. This improves previous results of this type. The proof uses the Rényi maximization procedure with a covariance constraint due to Costa et al. [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A.Rangarajan, M. A. T.Figueiredo, and J.Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci.52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the d-dimensional (d ⩾ 3) hydrogenic and oscillator-like systems is provided.
52(2011); http://dx.doi.org/10.1063/1.3553396View Description Hide Description
Two-dimensional quantum models which obey the property of shape invariance are built in the framework of polynomial two-dimensional supersymmetric quantum mechanics. They are obtained using the expressions for known one-dimensional shape invariant potentials. The constructed Hamiltonians are integrable with symmetry operators of fourth order in momenta, and they are not amenable to the conventional separation of variables.
52(2011); http://dx.doi.org/10.1063/1.3549588View Description Hide Description
A global construction of a family of symplectic connections on a cotangent bundle compatible with some linear symmetric connection on the base space is proposed. Examples of the compatible symplectic connections are given. A detailed analysis of an Abelian connection and of flat sections for this kind of symplectic connection in the Fedosov algorithm are presented. Some properties of the *–product determined by the compatible symplectic connection are shown.
A special asymptotic limit of a Kampé de Fériet hypergeometric function appearing in nonhomogeneous Coulomb problems52(2011); http://dx.doi.org/10.1063/1.3554698View Description Hide Description
In the investigation of two-body Coulomb Schrödinger equations with some types of nonhomogeneities, the particular solution can be expressed in terms of a two-variable Kampé de Fériet hypergeometric function. The asymptotic limit of the latter—for both variables being large but their ratio being a bound constant—is required in order to extract relevant physical information from the solutions. In this report the mathematical limit is provided. For that purpose, a particular series representation of the hypergeometric function—in terms of products of Kummer and Gauss functions—is first derived.
52(2011); http://dx.doi.org/10.1063/1.3555432View Description Hide Description
In the single band tight-binding approximation, we consider the transport properties of an electron subject to a homogeneous static electric field. We show that repeated interactions of the electron with two-level systems in thermal equilibrium suppress the Bloch oscillations and induce a steady current, the statistical properties of which we study.
52(2011); http://dx.doi.org/10.1063/1.3544476View Description Hide Description
We study a C*-dynamical system describing a particle coupled to an infinitely extended heat bath at positive temperature. For small coupling constant we prove return to equilibrium exponentially fast in time. The novelty in this context is to model the particle by a harmonic or anharmonic oscillator, respectively. The proof is based on explicit formulas for the time evolution of Weyl operators in the harmonic oscillator case. In the anharmonic oscillator case, a Dyson's expansion for the dynamics is essential. Moreover, we show in the harmonic oscillator case that is the absolute continuous spectrum of the Standard Liouvillean and that zero is a unique eigenvalue.
- Quantum Information and Computation
52(2011); http://dx.doi.org/10.1063/1.3534799View Description Hide Description
Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code and every codeword stabilized code can be described by a graph and a classical code. For the construction of good quantum codes of relatively large block length, concatenated quantum codes and their generalizations play an important role. We develop a systematic method for constructing concatenated quantum codes based on “graph concatenation,” where graphs representing the inner and outer codes are concatenated via a simple graph operation called “generalized local complementation.” Our method applies to both binary and nonbinary concatenated quantum codes as well as their generalizations.
52(2011); http://dx.doi.org/10.1063/1.3555805View Description Hide Description
Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra. In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of . We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
52(2011); http://dx.doi.org/10.1063/1.3545970View Description Hide Description
We develop the off-shell nilpotent finite field dependent Becchi–Rouet–Stora–Tyutin (BRST) transformations and show that for different choices of the finite field dependent parameter these transformations connect the generating functionals corresponding to different effective theories. We also construct both on-shell and off-shell finite field dependent anti-BRST transformations for Yang–Mills theories and show that these transformations play the similar role in connecting different generating functionals of different effective theories. Analogous to the finite field dependent BRST transformations, the nontrivial Jacobians of the path integral measure which arise due to the finite field dependent anti-BRST transformations are responsible for the new results. We consider several explicit examples in each case to demonstrate the results.
52(2011); http://dx.doi.org/10.1063/1.3526966View Description Hide Description
For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.
Cauchy problem and Green's functions for first order differential operators and algebraic quantization52(2011); http://dx.doi.org/10.1063/1.3530846View Description Hide Description
Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.
52(2011); http://dx.doi.org/10.1063/1.3528673View Description Hide Description
We discover the unique, simple Lie algebra of the third kind, or LATKe, that stems from codimension 6 orbifold singularities and gives rise to a new kind of Yang–Mills theory which simultaneously is pure and contains matter. The root space of the LATKe is one-dimensional and its Dynkin diagram consists of one point. The uniqueness of the LATKe is a vacuum selection mechanism.
52(2011); http://dx.doi.org/10.1063/1.3553186View Description Hide Description
In this paper, the Hamiltonian of quantum electrodynamics with spatial cutoffs is investigated. A scaled total Hamiltonian is introduced and its asymptotic behavior is investigated. In the main theorem, it is shown that the scaled total Hamiltonian converges to a self-adjoint operator in the strong resolvent sense, and the effective potential of the Dirac field is derived.
52(2011); http://dx.doi.org/10.1063/1.3546030View Description Hide Description
Borrowing ideas from tight binding model, we propose a board class of lattice field models that are classified by non simply laced Lie algebras. In the case of A N − 1 ≃ su(N) series, we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice are governed by the complex fundamental representations and of su(N) and the second nearest neighbor interactions are described by its adjoint . The lattice models associated with the leading su(2), su(3), and su(4) cases are explicitly studied and their fermionic field realizations are given. It is also shown that the su(2) and su(3)models describe the electronic properties of the acetylene chain and the graphene, respectively. It is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the A N roots through the typical dependence with the wave vector.Other features such as the SO(2N) extension and other applications are also discussed.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3545578View Description Hide Description
We provide a geometric explanation for the existence of magnification relations for the family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any noncaustic target point, the total signed magnification of the corresponding preimages vanishes. As an application to gravitational lensing, it was also shown that, independent of the choice of a lens model, the total signed magnification vanishes for a light source anywhere in the four-image region close to elliptic and hyperbolic umbilic caustics. This is a more global and higher order analog of the well-known fold and cusp magnification relations. We now extend each of these mappings to weighted projective space, which is a compact orbifold, and show that magnification relations translate into a statement about the behavior of these extended mappings at infinity. This generalizes multidimensional residue techniques developed in previous work, and introduces weighted projective space as a new tool in the theory of caustic singularities and gravitational lensing.
52(2011); http://dx.doi.org/10.1063/1.3553185View Description Hide Description
Regge calculus minisuperspace action in the connection representation has the form in which each term is linear over some field variable (scale of area-type variable with sign). We are interested in the result of performing integration over connections in the path integral (now usual multiple integral) as function of area tensors. We study analytical properties of this function in the extended (as compared to physical one) region when its arguments (area tensors) are allowed to be independent of each other. To find this function (or distribution), we compute its moments, i.e., integrals with monomials over area tensors. Then the function of interest should be recovered from the moments. Calculation proceeds through intermediate appearance of δ-functions and integrating them out. Up to a singular part with support on some discrete set of physically unattainable points, the function of interest has finite moments. This function in physical region should therefore exponentially decay at large areas and it really does being restored from moments. This gives for gravity a way of defining such nonabsolutely convergent integral as path integral.