Volume 48, Issue 5, May 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Quantum SU(3) Skyrme model with noncanonical embedded SO(3) soliton
View Description Hide DescriptionThe new ansatz which is the SO(3) group soliton was defined for the SU(3) Skyrme model. The model is considered in noncanonical bases for the state vectors. A complete canonical quantization of the model has been investigated in the collective coordinate formalism for the fundamental SU(3) representation of the unitary field. The independent quantum variable manifold covers all the eight dimension SU(3) group manifold due to the new ansatz. The explicit expressions of the Lagrangian and Hamiltonian densities are derived for this modified quantum skyrmion.

Geometry of quantum active subspaces and of effective Hamiltonians
View Description Hide DescriptionWe propose a geometric formulation of the theory of effective Hamiltonians associated with active spaces. We analyze particularly the case of the timedependent wave operator theory. This formulation is related to the geometry of the manifold of the active spaces, particularly to its Kählerian structure. We introduce the concept of quantum distance between active spaces. We show that the timedependent wave operator theory is, in fact, a gauge theory, and we analyze its relationship with the geometric phase concept.

Extended reduction criterion and lattice states
View Description Hide DescriptionWe study a particular class of states of a bipartite system consisting of two fourlevel parties. By means of an adapted extended reduction criterion we associate their entanglement properties to the geometric patterns of a planar lattice consisting of 16 points.

Evenly distributed unitaries: On the structure of unitary designs
View Description Hide DescriptionWe clarify the mathematical structure underlying unitary designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2designs are derived. We show how to turn mutually unbiased bases into approximate 2designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

Homological error correction: Classical and quantum codes
View Description Hide DescriptionWe prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension , while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension . We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor’s 9qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a selfcontained manner.

Mutually unbiased bases and Hadamard matrices of order six
View Description Hide DescriptionWe report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics
View Description Hide DescriptionThe knowledge of quantum phase flow induced under Weyl’s association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of firstorder ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phasespace analytic geometry based on the starproduct operation can hardly be visualized. The statement “quantum trajectory belongs to a constraint submanifold” can be changed, e.g., to the opposite by a unitary transformation. Some of the relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the starcomposition law and preserves Hamiltonian and constraint star functions.

Position and length operators in a theory with minimal length
View Description Hide DescriptionRelations between the notions of fundamental and minimal lengths, and duality, in a system with minimal length uncertainty relations are examined. Selfadjoint versions of operators relevant to the problem, and their spectra, are analyzed in detail.

Analytic solution of Hedin’s equations in zero dimensions
View Description Hide DescriptionFeynman diagrams for the manybody perturbational theory are enumerated by solving the system of Hedin’s equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.

Subnormalized states and tracenonincreasing maps
View Description Hide DescriptionWe investigate the set of completely positive, tracenonincreasing linear maps acting on the set of mixed quantum states of size . Extremal point of this set of maps are characterized and its volume with respect to the HilbertSchmidt (HS) (Euclidean) measure is computed explicitly for an arbitrary . The spectra of partially reduced rescaled dynamical matrices associated with tracenonincreasing completely positive maps belong to the cube inscribed in the set of subnormalized states of size . As a byproduct we derive the measure in induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in .

Quaternionic wave packets
View Description Hide DescriptionWe compare the behavior of a wave packet in the presence of a complex and a pure quaternionic potential step. This analysis, done for a Gaussian convolution function, sheds new light on the possibility to recognize quaternionic deviations from standard quantum mechanics.

AharonovBohm effect on the Poincaré disk
View Description Hide DescriptionWe consider formal quantum Hamiltonian of a charged particle on the Poincaré disk in the presence of an AharonovBohm magnetic vortex and a uniform magnetic field. It is shown that this Hamiltonian admits a fourparameter family of selfadjoint extensions. Its resolvent and the density of states are calculated for natural values of the extension parameters.

Another proof of GellMann and Low’s theorem
View Description Hide DescriptionThe theorem by GellMann and Low is a cornerstone in quantum field theory and zerotemperature manybody theory. The standard proof is based on Dyson’s timeordered expansion of the propagator; a proof based on exact identities for the time propagator is here given.

Exact solvability of superintegrable Benenti systems
View Description Hide DescriptionWe establish quantum and classical exact solvability for two large classes of maximally superintegrable Benenti systems in dimensions with arbitrarily large . Namely, we solve the HamiltonJacobi and Schrödinger equations for the systems in question. The results obtained are illustrated for a model with the cubic potential.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Generalized path dependent representations for gauge theories
View Description Hide DescriptionA set of differential operators acting by continuous deformations on path dependent functionals of open and closed curves is introduced. Geometrically, these path operators are interpreted as infinitesimal generators of curves in the base manifold of the gauge theory. They furnish a representation with the action of the group of loops having a fundamental role. We show that the path derivative, which is covariant by construction, satisfies the Ricci and Bianchi identities. Also, we provide a geometrical derivation of covariant Taylor expansions based on particular deformations of open curves. The formalism includes, as special cases, other path dependent operators such as end point derivatives and area derivatives.

Lie algebraic noncommuting structures from reparametrization symmetry
View Description Hide DescriptionWe extend our earlier work of revealing both spacespace and spacetime noncommuting structures in various models in particle mechanics exhibiting reparametrization symmetry. We show explicitly (in contrast to the earlier results in our paper [R. Banerjee et al.J. Phys. A38, 957 (2005), eprint hepth∕0405178]) that for some special choices of the reparametrization parameter , one can obtain spacespace noncommuting structures which are Lie algebraic in form even in the case of the relativistic free particle. The connection of these structures with the existing models in the literature is also briefly discussed. Further, there exists some values of for which the noncommutativity in the spacespace sector can be made to vanish. As a matter of internal consistency of our approach, we also study the angular momentumalgebra in details.

Relativistic quaternionic wave equation. II
View Description Hide DescriptionFurther results are reported for the onecomponent quaternionic wave equation recently introduced. A Lagrangian is found for the momentumspace version of the free equation, and another, nonlocal in time, is found for the complete equation. Further study of multiparticle systems has us looking into the mathematics of tensor products of Hilbert spaces. The principles of linearity and superposition are also clarified to good effect in advancing the quaternionic theory.

Simple spacetime symmetries: Generalizing conformal field theory
View Description Hide DescriptionWe study simple spacetime symmetry groups which act on a spacetime manifold which admits a invariant global causal structure. We classify pairs which share the following additional properties of conformal field theory. (1) The stability subgroup of is the identity component of a parabolic subgroup of , implying factorization , where generalizes Lorentz transformations, dilatations, and special conformal transformations. (2) Special conformal transformations act trivially on tangent vectors . The allowed simple Lie groups are the universal coverings of , and and are particular maximal parabolic subgroups. They coincide with the groups of fractional linear transformations of Euclidean Jordan algebras whose use as generalizations of Minkowski spacetime was advocated by Günaydin [Mod. Phys. Lett. A8, 1407 (1993)]. All these groups admit positive energy representations. It will also be shown that the classical conformal groups are the only allowed groups which possess an automorphism with properties appropriate for a time reflection.

Minkowski representations on Hilbert spaces
View Description Hide DescriptionThe algebra of functions on Minkowski noncommutative spacetime is studied as algebra of operators on Hilbert spaces. The representations of this algebra are constructed and classified. This new approach leads to a natural construction of integration in Minkowski spacetime in terms of the usual trace of operators.

Maximally symmetric vector propagator
View Description Hide DescriptionWe derive the propagator for a massive vector field on a de Sitter background of arbitrary dimension. This propagator is de Sitter invariant and possesses the proper flat spacetime and massless limits. Moreover, the retarded Green’s function inferred from it produces the correct classical response to a test source. Our result is expressed in a tensor basis which is convenient for performing quantumfieldtheory computations using dimensional regularization.
