Volume 52, Issue 1, January 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

Some remarks on assignment maps
View Description Hide DescriptionWe 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 informationtheory, is also discussed.

Connes' embedding problem and Tsirelson's problem
View Description Hide DescriptionWe 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 II factor is a subfactor of the ultrapower of the hyperfinite II 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.

Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

On solvable Dirac equation with polynomial potentials
View Description Hide DescriptionOnedimensional Dirac equation is analyzed with regard to the existence of exact (or closedform) 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.

A Goldstone theorem in thermal relativistic quantum field theory
View Description Hide DescriptionWe prove a Goldstone theorem in thermal relativistic quantum field theory, which relates spontaneous symmetry breaking to the rate of spacelike decay of the twopoint function. The critical rate of falloff coincides with that of the massless free scalar field theory. Related results and open problems are briefly discussed.

Gravitational repulsion within a black hole using the Stueckelberg quantum formalism
View Description Hide DescriptionWe 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.

General Relativity and Gravitation

Unitary irreducible representations of in discrete and continuous bases
View Description Hide DescriptionWe derive the matrix elements of generators of unitary irreducible representations of 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 and a continuous basis diagonalized by , 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.

Some results concerning the representation theory of the algebra underlying loop quantum gravity
View Description Hide DescriptionImportant characteristics of the loop approach to quantum gravity are a specific choice of the algebra of (kinematical) observables and of a representation of 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. The content of the present work is the observation that under some mild assumptions, the mathematical structure of representations of 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.

When do measures on the space of connections support the triad operators of loop quantum gravity?
View Description Hide DescriptionIn 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 nonAbelian 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 “nogo theorems” asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces. The fluxobservables 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 nonAbelian gauge theories.

Dynamical Systems

The theory of wavelet transform method on chaotic synchronization of coupled map lattices
View Description Hide DescriptionThe 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.

Analytic integrability of Hamiltonian systems with a homogeneous polynomial potential of degree 4
View Description Hide DescriptionIn 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.

Classical Mechanics and Classical Fields

The Weitzenböck connection and time reparameterization in nonholonomic mechanics
View Description Hide DescriptionWe 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 nonabelian 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.

Existence of solutions for Hamiltonian field theories by the Hamilton–Jacobi technique
View Description Hide DescriptionThe 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.

Plus–minus construction leads to perfect invisibility
View Description Hide DescriptionRecent 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.

Noether symmetries, energy–momentum tensors, and conformal invariance in classical field theory
View Description Hide DescriptionIn 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.

Nonexistence of Skyrmion–Skyrmion and Skyrmion–antiSkyrmion static equilibria
View Description Hide DescriptionWe consider classical static Skyrmion–antiSkyrmion and Skyrmion–Skyrmion configurations, symmetric with respect to a reflection plane, or symmetric up to a Gparity 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 antiSkyrmions 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 multiSkyrmion configurations. Our results are consistent with existing analyses of Skyrmion forces at large separation, and with numerical results on Skymion–antiSkyrmion 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 selfgravitating Skyrmions or black holes with Skyrmion hair cannot rest in symmetric equilibrium with selfgravitating antiSkyrmions.

Selfforce via energy–momentum and angular momentum balance equations
View Description Hide DescriptionThe 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 HarishChandra 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 selfforce produces a timechanging inertial mass.

Statistical Physics

Phase transition in a lognormal Markov functional model
View Description Hide DescriptionWe derive the exact solution of a onedimensional Markovfunctional 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 generatingfunction in the complex plane, in analogy with the Lee–Yang theory of the phase transitions in condensed matter physics.

Periodic Ising Correlations
View Description Hide DescriptionIn 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 and 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.

Methods of Mathematical Physics

Diagonalization of infinite transfer matrix of boundary face model
View Description Hide DescriptionWe study infinitely many commuting operators , which we call infinite transfer matrix of boundary face model. We diagonalize the infinite transfer matrix by using free field realizations of the vertex operators of the elliptic quantum group.

Arbitrary decays in linear viscoelasticity
View Description Hide DescriptionIt 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.
