Volume 55, Issue 2, February 2014
Index of content:

The Dirac equation is solved for quaternionic potentials, i V 0 + j W 0 ( ). The study shows two different solutions. The first one contains particle and antiparticle solutions and leads to the diffusion, tunneling, and Klein energy zones. The standard solution is recovered taking the complex limit of this solution. The second solution, which does not have a complex counterpart, can be seen as a V 0antiparticle or W 0particle solution.
 ARTICLES


Partial Differential Equations

A nonlinear Bloch model for Coulomb interaction in quantum dots
View Description Hide DescriptionIn this paper, we first derive a Coulomb Hamiltonian for electron–electron interaction in quantum dots in the Heisenberg picture. Then we use this Hamiltonian to enhance a Bloch model, which happens to be nonlinear in the density matrix. The coupling with Maxwell equations in case of interaction with an electromagnetic field is also considered from the Cauchy problem point of view. The study is completed by numerical results and a discussion about the advisability of neglecting intraband coherences, as is done in part of the literature.

Representation Theory and Algebraic Methods

RotaBaxter operators on and solutions of the classical YangBaxter equation
View Description Hide DescriptionWe explicitly determine all RotaBaxter operators (of weight zero) on under the CartanWeyl basis. For the skewsymmetric operators, we give the corresponding skewsymmetric solutions of the classical YangBaxter equation in , confirming the related study by SemenovTianShansky. In general, these RotaBaxter operators give a family of solutions of the classical YangBaxter equation in the sixdimensional Lie algebra . They also give rise to threedimensional preLie algebras which in turn yield solutions of the classical YangBaxter equation in other sixdimensional Lie algebras.

Hermitian Young operators
View Description Hide DescriptionStarting from conventional Young operators, we construct Hermitian operators which project orthogonally onto irreducible representations of the (special) unitary group.

Hirota equations associated with simply laced affine Lie algebras: The cuspidal class E _{6} of
View Description Hide DescriptionIn this paper we derive Hirota equations associated with the simply laced affine Lie algebras , where is one of the simply laced complex Lie algebras or , defined by finite order automorphisms of which we call Lepowsky automorphisms. In particular, we investigate the Hirota equations for Lepowsky automorphisms of defined by the cuspidal class E 6 of the Weyl group W(E 6) of . We also investigate the relationship between the Lepowsky automorphisms of the simply laced complex Lie algebras and the conjugate canonical automorphisms defined by Kac. This analysis is applied to identify the canonical automorphisms for the cuspidal class E 6 of .

ManyBody and Condensed Matter Physics

Meanfield dynamics of fermions with relativistic dispersion
View Description Hide DescriptionWe extend the derivation of the timedependent HartreeFock equation recently obtained by Benedikter et al. [“Meanfield evolution of fermionic systems,” Commun. Math. Phys. (to be published)] to fermions with a relativistic dispersion law. The main new ingredient is the propagation of semiclassical commutator bounds along the pseudorelativistic HartreeFock evolution.

Geometry of matrix product states: Metric, parallel transport, and curvature
View Description Hide DescriptionWe study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e., the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a Kähler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a wellchosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the LeviCivita connection) and the Riemann curvature tensor.

Translationally invariant conservation laws of local Lindblad equations
View Description Hide DescriptionWe study the conditions under which one can conserve local translationally invariant operators by local translationally invariant Lindblad equations in onedimensional rings of spin1/2 particles. We prove that for any 1local operator (e.g., particle density) there exist Lindblad dissipators that conserve that operator, while on the other hand we prove that among 2local operators (e.g., energy density) only trivial ones of the Ising type can be conserved, while all the other cannot be conserved, neither locally nor globally, by any 2 or 3local translationally invariant Lindblad equation. Our statements hold for rings of any finite length larger than some minimal length determined by the locality of Lindblad equation. These results show in particular that conservation of energy density in interacting systems is fundamentally more difficult than conservation of 1local quantities.

Quantum Mechanics

A magnetic contribution to the Hardy inequality
View Description Hide DescriptionWe study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a nonnegative potential term depending on properties of the magnetic field.

The BenderDunne basis operators as Hilbert space operators
View Description Hide DescriptionThe BenderDunne basis operators, where and are the position and momentum operators, respectively, are formal integral operators in position representation in the entire real line for positive integers n and m. We show, by explicit construction of a dense domain, that the operators 's are densely defined operators in the Hilbert space .

Scalar spectral measures associated with an operatorfractal
View Description Hide DescriptionWe study a spectraltheoretic model on a Hilbert space L ^{2}(μ) where μ is a fixed Cantor measure. In addition to μ, we also consider an independent scaling operator U acting in L ^{2}(μ). To make our model concrete, we focus on explicit formulas: We take μ to be the Bernoulli infiniteconvolution measure corresponding to scale number . We then define the unitary operator U in L ^{2}(μ) from a scaleby5 operation. The spectraltheoretic and geometric properties we have previously established for U are as follows: (i) U acts as an ergodic operator; (ii) the action of U is not spatial; and finally, (iii) U is fractal in the sense that it is unitarily equivalent to a countable infinite direct sum of (twisted) copies of itself. In this paper, we prove new results about the projectionvalued measures and scalar spectral measures associated to U and its constituent parts. Our techniques make use of the representations of the Cuntz algebra on L ^{2}(μ).

invariance of type B 3fold supersymmetric systems
View Description Hide DescriptionType B 3fold supersymmetry is a necessary and sufficient condition for a quantum Hamiltonian to admit three linearly independent local solutions in closed form. We show that any such a system is invariant under homogeneous linear transformations. In particular, we prove explicitly that the parameter space is transformed as an adjoint representation of it and that every coefficient of the characteristic polynomial appeared in 3fold superalgebra is algebraic invariants. In the type A case, it includes as a subgroup the projective transformation studied in the literature. We argue that any fold supersymmetric system has a invariance for an arbitrary integral .

Maximal quantum mechanical symmetry: Projective representations of the inhomogeneous symplectic group
View Description Hide DescriptionA symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations that are fundamental to quantum mechanics must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the WeylHeisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups.

Quantum Information and Computation

Base norms and discrimination of generalized quantum channels
View Description Hide DescriptionWe introduce and study norms in the space of hermitian matrices, obtained from base norms in positively generated subspaces. These norms are closely related to discrimination of socalled generalized quantum channels, including quantum states, channels, and networks. We further introduce generalized quantum decision problems and show that the maximal average payoffs of decision procedures are again given by these norms. We also study optimality of decision procedures, in particular, we obtain a necessary and sufficient condition under which an optimal 1tester for discrimination of quantum channels exists, such that the input state is maximally entangled.

Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Dirac solutions for quaternionic potentials
View Description Hide DescriptionThe Dirac equation is solved for quaternionic potentials, i V 0 + j W 0 ( ). The study shows two different solutions. The first one contains particle and antiparticle solutions and leads to the diffusion, tunneling, and Klein energy zones. The standard solution is recovered taking the complex limit of this solution. The second solution, which does not have a complex counterpart, can be seen as a V 0antiparticle or W 0particle solution.

Quantum mechanical effects from deformation theory
View Description Hide DescriptionWe consider deformations of quantum mechanical operators by using the novel construction tool of warped convolutions. The deformation enables us to obtain several quantum mechanical effects where electromagnetic and gravitomagnetic fields play a role. Furthermore, a quantum plane can be defined by using the deformation techniques. This in turn gives an experimentally verifiable effect.

General Relativity and Gravitation

Breaking generalized covariance, classical renormalization, and boundary conditions from superpotentials
View Description Hide DescriptionSuperpotentials offer a direct means of calculating conserved charges associated with the asymptotic symmetries of spacetime. Yet superpotentials have been plagued with inconsistencies, resulting in nonphysical or incongruent values for the mass, angular momentum, and energy loss due to radiation. The approach of Regge and Teitelboim, aimed at a clear Hamiltonian formulation with a boundary, and its extension to the Lagrangian formulation by Julia and Silva have resolved these issues, and have resulted in a consistent, welldefined and unique variational equation for the superpotential, thereby placing it on a firm footing. A hallmark solution of this equation is the KBL superpotential obtained from the firstorder Lovelock Lagrangian. Nevertheless, here we show that these formulations are still insufficient for Lovelock Lagrangians of higher orders. We present a paradox, whereby the choice of fields affects the superpotential for equivalent onshell dynamics. We offer two solutions to this paradox: either the original Lagrangian must be effectively renormalized, or that boundary conditions must be imposed, so that spacetime be asymptotically maximally symmetric. Nonmetricity is central to this paradox, and we show how quadratic nonmetricity in the bulk of spacetime contributes to the conserved charges on the boundary, where it vanishes identically. This is a realization of the gravitational Higgs mechanism, proposed by Percacci, where the nonmetricity is the analogue of the Goldstone boson.

What happens to Petrov classification, on horizons of axisymmetric dirty black holes
View Description Hide DescriptionWe consider axisymmetric stationary dirty black holes with regular nonextremal or extremal horizons, and compute their onhorizon Petrov types. The Petrov type (PT) in the frame of the observer crossing the horizon can be different from that formally obtained in the usual (but singular in the horizon limit) frame of an observer on a circular orbit. We call this entity the boosted Petrov type (BPT), as the corresponding frame is obtained by a singular boost from the regular one. The PT offhorizon can be more general than PT onhorizon and that can be more general than the BPT on horizon. This is valid for all regular metrics, irrespective of the extremality of the horizon. We analyze and classify the possible relations between the three characteristics and discuss the nature and features of the underlying singular boost. The three Petrov types can be the same only for spacetimes of PT D and O offhorizon. The mutual alignment of principal null directions and the generator in the vicinity of the horizon is studied in detail. As an example, we also analyze a special class of metrics with utraextremal horizons (for which the regularity conditions look different from the general case) and compare their offhorizon and onhorizon algebraic structure in both frames.

Dynamical Systems

Generalised Eisenhart lift of the Toda chain
View Description Hide DescriptionThe Toda chain of nearest neighbour interacting particles on a line can be described both in terms of geodesic motion on a manifold with one extra dimension, the Eisenhart lift, or in terms of geodesic motion in a symmetric space with several extra dimensions. We examine the relationship between these two realisations and discover that the symmetric space is a generalised, multiparticle Eisenhart lift of the original problem that reduces to the standard Eisenhart lift. Such generalised Eisenhart lift acts as an inverse KaluzaKlein reduction, promoting coupling constants to momenta in higher dimension. In particular, isometries of the generalised lift metric correspond to energy preserving transformations that mix coordinates and coupling constants. A byproduct of the analysis is that the lift of the Toda Lax pair can be used to construct higher rank Killing tensors for both the standard and generalised lift metrics.

Classical Mechanics and Classical Fields

On the zero modes of the FaddeevPopov operator in the Landau gauge
View Description Hide DescriptionFollowing Henyey procedure [Phys. Rev. D20, 1460 (1979)], we construct examples of zero modes of the FaddeevPopov operator in the Landau gauge in Euclidean space in D dimensions, for both SU (2) and SU (3) groups. We obtain gauge field configurations which give rise to a field strength, , whose nonlinear term, , turns out to be nonvanishing. To our knowledge, this is the first time where such a nonabelian configuration is explicitly obtained in the case of SU (3) in 4D.

On fractional differential inclusions with the Jumarie derivative
View Description Hide DescriptionIn the paper, fractional differential inclusions with the Jumarie derivative are studied. We discuss the existence and uniqueness of a solution to such problems. Our study relies on standard variational methods.
