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Jan 2011

Volume 52, Issue 1, Articles (01xxxx)

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back to top Quantum Mechanics (General and Nonrelativistic)

Some remarks on assignment maps

F. Masillo, G. Scolarici, and L. Solombrino

J. Math. Phys. 52, 012101 (2011); http://dx.doi.org/10.1063/1.3525832 (13 pages)

Online Publication Date: 5 January 2011

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We study the properties of general linear assignment maps, showing that positivity axiom can be suitably relaxed, and propose a new class of dynamical maps (generalized dynamics). A puzzling result, arising in such a context in quantum information theory, is also discussed.
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03.67.-a Quantum information
03.65.Aa Quantum systems with finite Hilbert space
03.65.-w Quantum mechanics

Connes' embedding problem and Tsirelson's problem

M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, and R. F. Werner

J. Math. Phys. 52, 012102 (2011); http://dx.doi.org/10.1063/1.3514538 (12 pages) | Cited 1 time

Online Publication Date: 6 January 2011

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We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes' embedding problem asks whether any separable II1 factor is a subfactor of the ultrapower of the hyperfinite II1 factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely, a positive answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem.
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03.65.Aa Quantum systems with finite Hilbert space
02.10.-v Logic, set theory, and algebra
03.65.Fd Algebraic methods
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

On solvable Dirac equation with polynomial potentials

Tomasz Stachowiak

J. Math. Phys. 52, 012301 (2011); http://dx.doi.org/10.1063/1.3533946 (3 pages)

Online Publication Date: 3 January 2011

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One-dimensional Dirac equation is analyzed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.
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03.65.Pm Relativistic wave equations
02.10.De Algebraic structures and number theory

A Goldstone theorem in thermal relativistic quantum field theory

Christian D. Jäkel and Walter F. Wreszinski

J. Math. Phys. 52, 012302 (2011); http://dx.doi.org/10.1063/1.3526961 (14 pages)

Online Publication Date: 6 January 2011

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We prove a Goldstone theorem in thermal relativistic quantum field theory, which relates spontaneous symmetry breaking to the rate of spacelike decay of the two-point function. The critical rate of fall-off coincides with that of the massless free scalar field theory. Related results and open problems are briefly discussed.
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11.10.-z Field theory
11.30.Qc Spontaneous and radiative symmetry breaking

Gravitational repulsion within a black hole using the Stueckelberg quantum formalism

D. M. Ludwin and L. P. Horwitz

J. Math. Phys. 52, 012303 (2011); http://dx.doi.org/10.1063/1.3533399 (12 pages)

Online Publication Date: 18 January 2011

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We wish to study an application of Stueckelberg's relativistic quantum theory in the framework of general relativity. We study the form of the wave equation of a massive body in the presence of a Schwarzschild gravitational field. We treat the mathematical behavior of the wavefunction also around and beyond the horizon (r = 2M). Classically, within the horizon, the time component of the metric becomes spacelike and distance from the origin singularity becomes timelike, suggesting an inevitable propagation of all matter within the horizon to a total collapse at r = 0. However, the quantum description of the wavefunction provides a different understanding of the behavior of matter within the horizon. We find that a test particle can almost never be found at the origin and is more probable to be found at the horizon. Matter outside the horizon has a very small wavelength and therefore interference effects can be found only on a very small atomic scale. However, within the horizon, matter becomes totally “tachyonic” and is potentially “spread” over all space. Small location uncertainties on the atomic scale become large around the horizon, and different mass components of the wavefunction can therefore interfere on a stellar scale. This interference phenomenon, where the probability of finding matter decreases as a function of the distance from the horizon, appears as an effective gravitational repulsion.
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04.60.Bc Phenomenology of quantum gravity
04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics
03.65.Pm Relativistic wave equations
03.65.Ge Solutions of wave equations: bound states
04.20.Gz Spacetime topology, causal structure, spinor structure
back to top General Relativity and Gravitation

Unitary irreducible representations of SL (2,math) in discrete and continuous SU (1,1) bases

Florian Conrady and Jeff Hnybida

J. Math. Phys. 52, 012501 (2011); http://dx.doi.org/10.1063/1.3533393 (18 pages)

Online Publication Date: 18 January 2011

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We derive the matrix elements of generators of unitary irreducible representations of SL (2,math) with respect to basis states arising from a decomposition into irreducible representations of SU(1,1). This is done with regard to a discrete basis diagonalized by J3 and a continuous basis diagonalized by K1, and for both the discrete and continuous series of SU(1,1). For completeness, we also treat the more conventional SU(2) decomposition as a fifth case. The derivation proceeds in a functional/differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases. The states are defined explicitly and related to SU(1,1) and SU(2) matrix elements.
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11.30.Ly Other internal and higher symmetries

Some results concerning the representation theory of the algebra underlying loop quantum gravity

Hanno Sahlmann

J. Math. Phys. 52, 012502 (2011); http://dx.doi.org/10.1063/1.3525705 (9 pages) | Cited 1 time

Online Publication Date: 20 January 2011

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Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra math of (kinematical) observables and of a representation of math on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra math. The content of the present work is the observation that under some mild assumptions, the mathematical structure of representations of math can be analyzed rather effortlessly, to a certain extent: each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions.
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04.60.-m Quantum gravity
02.40.-k Geometry, differential geometry, and topology
02.10.-v Logic, set theory, and algebra

When do measures on the space of connections support the triad operators of loop quantum gravity?

Hanno Sahlmann

J. Math. Phys. 52, 012503 (2011); http://dx.doi.org/10.1063/1.3525706 (14 pages) | Cited 2 times

Online Publication Date: 20 January 2011

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In this work we investigate the question under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux. We give the problem a precise mathematical formulation and start its investigation. For the technically simple case of U(1) as gauge group, we establish a number of “no-go theorems” asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces. The flux-observables we consider, play an important role in loop quantum gravity since they can be defined without recurse to a background geometry and they might also be of interest in the general context of quantization of non-Abelian gauge theories.
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04.60.-m Quantum gravity
03.65.Aa Quantum systems with finite Hilbert space
back to top Dynamical Systems

The theory of wavelet transform method on chaotic synchronization of coupled map lattices

Jonq Juang and Chin-Lung Li

J. Math. Phys. 52, 012701 (2011); http://dx.doi.org/10.1063/1.3525802 (14 pages)

Online Publication Date: 5 January 2011

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The wavelet transform method originated by Wei et al. [Phys. Rev. Lett. 89, 284103.4 (2002)] was proved [Juang and Li, J. Math. Phys. 47, 072704.16 (2006); Juang et al., J. Math. Phys. 47, 122702.11 (2006); Shieh et al., J. Math. Phys. 47, 082701.10 (2006)] to be an effective tool to reduce the order of coupling strength for coupled chaotic systems to acquire the synchrony regardless the size of oscillators. In Juang et al., [IEEE Trans. Circuits Syst., I: Regul. Pap. 56, 840 (2009)] such method was applied to coupled map lattices (CMLs). It was demonstrated that by adjusting the wavelet constant of the method can greatly increase the applicable range of coupling strengths, the parameters, range of the individual oscillator, and the number of nodes for local synchronization of CMLs. No analytical proof is given there. In this paper, the optimal or near optimal wavelet constant can be explicitly identified. As a result, the above described scenario can be rigorously verified.
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05.45.-a Nonlinear dynamics and chaos
02.30.Uu Integral transforms

Analytic integrability of Hamiltonian systems with a homogeneous polynomial potential of degree 4

Jaume Llibre, Adam Mahdi, and Claudia Valls

J. Math. Phys. 52, 012702 (2011); http://dx.doi.org/10.1063/1.3544473 (9 pages) | Cited 3 times

Online Publication Date: 25 January 2011

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In the analytic case we prove the conjecture of Maciejewski and Przybylska [J. Math. Phys. 46(6), 062901 (2005)] regarding Hamiltonian systems with a homogeneous polynomial potential of degree 4. The proof of the conjecture completes the characterization of all the analytic integrable Hamiltonian system with a homogeneous polynomial potential of degree 4.
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02.30.Cj Measure and integration
02.60.Jh Numerical differentiation and integration
02.10.De Algebraic structures and number theory
back to top Classical Mechanics and Classical Fields

The Weitzenböck connection and time reparameterization in nonholonomic mechanics

O. E. Fernandez and A. M. Bloch

J. Math. Phys. 52, 012901 (2011); http://dx.doi.org/10.1063/1.3525798 (18 pages) | Cited 1 time

Online Publication Date: 10 January 2011

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We show that the torsion of the Weitzenböck connection is responsible for the fictitious pseudogyroscopic force experienced by a general mechanical system in a noncoordinate moving frame. In particular, we show that for the class of mechanical systems subjected to nonintegrable constraints known as non-abelian nonholonomic Chaplygin systems, the constraint reaction force directly depends on this torsion. For these Chaplygin systems, we show how this torsional force can in some cases be removed by an appropriate choice of frame depending on a multiplier f(q), linking these results to the process of Chaplygin Hamiltonization through time reparameterization. Lastly, we show that the cyclic symmetries of f in some cases lead to the existence of momentum conservation laws for the original nonholonomic system and illustrate the results through several examples.
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04.50.-h Higher-dimensional gravity and other theories of gravity

Existence of solutions for Hamiltonian field theories by the Hamilton–Jacobi technique

Danilo Bruno

J. Math. Phys. 52, 012902 (2011); http://dx.doi.org/10.1063/1.3533762 (12 pages)

Online Publication Date: 11 January 2011

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The paper is devoted to prove the existence of a local solution of the Hamilton–Jacobi equation in field theory, whence the general solution of the field equations can be obtained. The solution is adapted to the choice of the submanifold where the initial data of the field equations are assigned. Finally, a technique to obtain the general solution of the field equations, starting from the given initial manifold, is deduced.
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03.65.Ge Solutions of wave equations: bound states
02.30.Jr Partial differential equations

Plus–minus construction leads to perfect invisibility

J. C. Nacher and T. Ochiai

J. Math. Phys. 52, 012903 (2011); http://dx.doi.org/10.1063/1.3533937 (17 pages)

Online Publication Date: 11 January 2011

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Recent theoretical advances applied to metamaterials have opened new avenues to design a coating that hides objects from electromagnetic radiation and even the sight. Here, we propose a new design of cloaking devices that creates perfect invisibility in isotropic media. A combination of positive and negative refractive indices, called plus–minus construction, is essential to achieve perfect invisibility (i.e., no time delay and total absence of reflection). Contrary to the common understanding that between two isotropic materials having different refractive indices the electromagnetic reflection is unavoidable, our method shows that surprisingly the reflection phenomena can be completely eliminated. The invented method, different from the classical impedance matching, may also find electromagnetic applications outside of cloaking devices, wherever distortions are present arising from reflections.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation
42.70.-a Optical materials

Noether symmetries, energy–momentum tensors, and conformal invariance in classical field theory

Josep M. Pons

J. Math. Phys. 52, 012904 (2011); http://dx.doi.org/10.1063/1.3532941 (21 pages)

Online Publication Date: 18 January 2011

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In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. With this baggage on board, we next discuss in detail, for Poincaré invariant theories in flat spacetime, the differences between the Belinfante energy–momentum tensor and a family of Hilbert energy–momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of Poincaré symmetries and it is a basic ingredient of the generators of other eventual spacetime symmetries which may happen to exist. Instead, Hilbert tensors are the means to test whether a theory contains other spacetime symmetries beyond Poincaré. We discuss at length the case of scale and conformal symmetry, of which we give some examples. We show, for Poincaré invariant Lagrangians, that the realization of scale invariance selects a unique Hilbert tensor which allows for an easy test as to whether conformal invariance is also realized. Finally we make some basic remarks on metric generally covariant theories and classical field theory in a fixed curved background.
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03.50.-z Classical field theories
11.10.Lm Nonlinear or nonlocal theories and models
11.30.Cp Lorentz and Poincaré invariance
11.25.Hf Conformal field theory, algebraic structures
11.15.-q Gauge field theories
11.30.Ly Other internal and higher symmetries

Nonexistence of Skyrmion–Skyrmion and Skyrmion–anti-Skyrmion static equilibria

G. W. Gibbons, C. M. Warnick, and W. W. Wong

J. Math. Phys. 52, 012905 (2011); http://dx.doi.org/10.1063/1.3523469 (9 pages)

Online Publication Date: 19 January 2011

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We consider classical static Skyrmion–anti-Skyrmion and Skyrmion–Skyrmion configurations, symmetric with respect to a reflection plane, or symmetric up to a G-parity transformation, respectively. We show that the stress tensor component completely normal to the reflection plane, and hence its integral over the plane, is negative definite or positive definite, respectively. Classical Skyrmions always repel classical Skyrmions and classical Skyrmions always attract classical anti-Skyrmions and thus no static equilibrium, whether stable or unstable, is possible in either case. No other symmetry assumption is made and so our results also apply to multi-Skyrmion configurations. Our results are consistent with existing analyses of Skyrmion forces at large separation, and with numerical results on Skymion–anti-Skyrmion configurations in the literature which admit a different reflection symmetry. They also hold for the massive Skyrme model. We also point out that reflection symmetric self-gravitating Skyrmions or black holes with Skyrmion hair cannot rest in symmetric equilibrium with self-gravitating anti-Skyrmions.
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11.10.Lm Nonlinear or nonlocal theories and models
03.65.Pm Relativistic wave equations

Self-force via energy–momentum and angular momentum balance equations

Yurij Yaremko

J. Math. Phys. 52, 012906 (2011); http://dx.doi.org/10.1063/1.3531986 (20 pages)

Online Publication Date: 24 January 2011

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The radiation reaction for a pointlike charge coupled to a massive scalar field is considered. The retarded Green's function associated with the Klein–Gordon wave equation has support not only on the future light cone of the emission point (direct part) but extends inside the light cone as well (tail part). Dirac's scheme of decomposition of the retarded electromagnetic field into the “mean of the advanced and retarded field” and the “radiation” field is adapted to theories where Green's function consists of the direct and the tail parts. The Harish-Chandra equation of motion of radiating scalar charge under the influence of an external force is obtained. This equation includes effect of particle's own field. The self-force produces a time-changing inertial mass.
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11.10.Cd Axiomatic approach
02.30.-f Function theory, analysis
back to top Statistical Physics

Phase transition in a log-normal Markov functional model

Dan Pirjol

J. Math. Phys. 52, 013301 (2011); http://dx.doi.org/10.1063/1.3526679 (14 pages)

Online Publication Date: 21 January 2011

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We derive the exact solution of a one-dimensional Markov functional model with log normally distributed interest rates and constant volatility in discrete time. The model is shown to have two distinct limiting states, corresponding to small and asymptotically large volatilities, respectively. These volatility regimes are separated by a phase transition at some critical value of the volatility, at which certain expectation values display nonanalytical behavior as a function of volatility. We investigate the conditions under which this phase transition occurs and show that it is related to the position of the zeros of an appropriately defined generating function in the complex plane, in analogy with the Lee–Yang theory of the phase transitions in condensed matter physics.
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02.50.Ga Markov processes
02.50.Ng Distribution theory and Monte Carlo studies
05.70.Fh Phase transitions: general studies
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

Periodic Ising Correlations

Grethe Hystad

J. Math. Phys. 52, 013302 (2011); http://dx.doi.org/10.1063/1.3517425 (33 pages) | Cited 2 times

Online Publication Date: 24 January 2011

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In this paper, we first rework B. Kaufman's 1949 paper [Phys. Rev. 76, 1232 (1949)] by using representation theory. Our approach leads to a simpler and more direct way of deriving the spectrum of the transfer matrix for the finite periodic Ising model. We then determine formulas for the spin correlation functions that depend on the matrix elements of the induced rotation associated with the spin operator in a basis of eigenvectors for the transfer matrix. The representation of the spin matrix elements is obtained by considering the spin operator as an intertwining map. We exhibit the “new” elements V+ and V− in the Bugrij–Lisovyy formula [Phys. Lett. A 319, 390 (2003)] as part of a holomorphic factorization of the periodic and antiperiodic summability kernels on the spectral curve associated with the induced rotation for the transfer matrix.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
back to top Methods of Mathematical Physics

Diagonalization of infinite transfer matrix of boundary Uq,p(AN−1(1)) face model

Takeo Kojima

J. Math. Phys. 52, 013501 (2011); http://dx.doi.org/10.1063/1.3521604 (26 pages) | Cited 1 time

Online Publication Date: 4 January 2011

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We study infinitely many commuting operators TB(z), which we call infinite transfer matrix of boundary Uq,p(AN−1(1)) face model. We diagonalize the infinite transfer matrix TB(z) by using free field realizations of the vertex operators of the elliptic quantum group Uq,p(AN−1(1)).
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.20.-a Group theory
03.65.-w Quantum mechanics

Arbitrary decays in linear viscoelasticity

Nasser-eddine Tatar

J. Math. Phys. 52, 013502 (2011); http://dx.doi.org/10.1063/1.3533766 (12 pages) | Cited 1 time

Online Publication Date: 4 January 2011

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It is by now well known that a necessary condition for the exponential (polynomial) decay of the energy of a problem arising in viscoelasticity is that the kernel (which appears in the memory term) itself be of exponential (polynomial) type. By a kernel of exponential (polynomial) type we mean that the product of this kernel with an exponential (polynomial) function is summable. Some researchers have started from this condition to seek other (sufficient) conditions ensuring exponential or polynomial decay of the energy. In this work we generalize these works to an arbitrary decay not necessarily of an exponential or polynomial rate.
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46.35.+z Viscoelasticity, plasticity, viscoplasticity
02.60.-x Numerical approximation and analysis
02.10.De Algebraic structures and number theory

A new integrable two-component system with cubic nonlinearity

Junfeng Song, Changzheng Qu, and Zhijun Qiao

J. Math. Phys. 52, 013503 (2011); http://dx.doi.org/10.1063/1.3530865 (9 pages)

Online Publication Date: 5 January 2011

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In this paper, a new integrable two-component system, mt = [m(uxvx−uv+uvx−uxv)]x,nt = [n(uxvx − uv + uvx − uxv)]x, where m  =  u − uxx and n = v−vxx, is proposed. Our system is a generalized version of the integrable system mt = [m(ux2−u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained.
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05.45.Yv Solitons
02.40.-k Geometry, differential geometry, and topology
02.30.Cj Measure and integration
02.60.Jh Numerical differentiation and integration

Embeddings of the Euclidean algebra math(3) into math(4,math) and restrictions of irreducible representations of math(4,math)

Andrew Douglas and Joe Repka

J. Math. Phys. 52, 013504 (2011); http://dx.doi.org/10.1063/1.3531984 (9 pages) | Cited 2 times

Online Publication Date: 6 January 2011

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The Euclidean group E(3) is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. It is the noncompact, semidirect product group E(3) ≅ SO(3)⋉math3. The Euclidean algebra math(3) is the complexification of the Lie algebra of E(3). We classify the embeddings of the Euclidean algebra math(3) into the simple Lie algebra math(4,math) and, as an application of this classification, discuss the restriction to various embeddings of math(3) of certain irreducible representations of math(4,math). In particular, we consider which of these restrictions are math(3)-indecomposable.
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02.20.Sv Lie algebras of Lie groups
02.10.Ud Linear algebra
02.20.Bb General structures of groups

Wave operator bounds for one-dimensional Schrödinger operators with singular potentials and applications

Vincent Duchêne, Jeremy L. Marzuola, and Michael I. Weinstein

J. Math. Phys. 52, 013505 (2011); http://dx.doi.org/10.1063/1.3525977 (17 pages) | Cited 1 time

Online Publication Date: 7 January 2011

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Boundedness of wave operators for Schrödinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.
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03.65.Ge Solutions of wave equations: bound states

Solution of Dirac equation with spin and pseudospin symmetry for an anharmonic oscillator

H. Goudarzi, M. Sohbati, and S. Zarrin

J. Math. Phys. 52, 013506 (2011); http://dx.doi.org/10.1063/1.3532930 (7 pages) | Cited 2 times

Online Publication Date: 7 January 2011

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We present the exact solutions of Dirac equation with anharmonic oscillator potential using the Nikiforov–Uvarov method. Taking into account potentials of vector field V(r) and scalar field S(r) in Dirac Hamiltonian, the bound state energy eigenvalues and the corresponding upper and lower two-component spinors of fermion are obtained. These solutions are considered in the framework of the spin and pseudospin symmetry concept.
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03.65.Pm Relativistic wave equations
03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods

Simple unified proofs of four duality theorems

D. J. Rowe, J. Repka, and M. J. Carvalho

J. Math. Phys. 52, 013507 (2011); http://dx.doi.org/10.1063/1.3525978 (24 pages)

Online Publication Date: 7 January 2011

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Duality relationships between the irreps (irreducible representations) of pairs of distinct commuting groups, G1 and G2, on Hilbert spaces of interest have long played important roles in the atomic and nuclear shell models. In addition to the well-known Schur–Weyl duality, the most widely used duality relationships are the so-called: unitary–unitary, orthogonal–symplectic (i.e., noncompact symplectic), symplectic–symplectic (compact symplectics), and orthogonal–orthogonal dualities. Proofs of these dualities exist in the literature. But most of them are not readily accessible to physicists or give little insight into how they might be used in practice. This paper presents unified proofs of the above-mentioned dualities based on the explicit construction of states which are simultaneously of extreme weight for the actions of both G1 and G2. The proofs expressed in language familiar to physicists are simple, systematic, and provide useful insights.
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12.40.Nn Regge theory, duality, absorptive/optical models
11.30.Na Nonlinear and dynamical symmetries (spectrum-generating symmetries)
03.65.Aa Quantum systems with finite Hilbert space
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