Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

Loop Observables for BF Theories in Any Dimension and the Cohomology of Knots

A generalization of Wilson loop observables for BF theories in any dimension is introduced within the Batalin–Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the Letter has only trivalent interactions and, irrespective of the actual dimension, looks like a three-dimensional Chern–Simons theory.     

Boundary G/G Theory and Topological Poisson–Lie Sigma Model

We study a boundary version of the gauged WZW model with a Poisson–Lie group G as the target. The Poisson–Lie structure of G is used to define the Wess–Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson–Lie group G ^* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov–Tian–Shansky.     

The Hyperbolic Volume of Knots from the Quantum Dilogarithm

The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.     

The gauging of BV algebras

A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending on the nature of the global symmetry, the gauging involves coupling to a pure ghost system with a varying amount of ghostly supersymmetry. Coupling to an N=0$N=0$ ghost system yields an ordinary gauge theory whose observables are appropriately classified by the invariant BV cohomology. Coupling to an N=1$N=1$ ghost system leads to a topological gauge field theory whose observables are classified by the equivariant BV cohomology. Coupling to higher N$N$ ghost systems yields topological gauge field theories with higher topological symmetry. In the latter case, however, problems of a completely new kind emerge, which call for a revision of the standard BV algebraic framework.     

Lagrangian for the t–J Model Constructed from the Generators of the Supersymmetric Hubbard Algebra

In order to describe the dynamics of the t–J model, two different families of first-order Lagrangians in terms of the generators of the Hubbard algebra are found. Such families correspond to different dynamical second-class constrained systems. The quantization is carried out by using the path-integral formalism. In this context the introduction of proper ghost fields is needed to render the model renormalizable. In each case the standard Feynman diagrammatics is obtained and the renormalized physical quantities are computed and analyzed. In the first case a nonperturbative large-N expansion is considered with the purpose of studying the generalized Hubbard model describing N-fold-degenerate correlated bands. In this case the 1/N correction to the renormalized boson propagator is computed. In the second case the perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator coming from our formalism is studied in details. As an example the thermal softening of the magnon frequency is computed. The antiferromagnetic case is also analyzed, and the results are confronted with previous one obtained by means of the spin-polaron theories.     

On the Topological Multivortex Solutions of the Self-Dual Maxwell-Chern-Simons Gauged <Emphasis Type="Italic">O</Emphasis>(3) Sigma Model

We consider the self-dual Maxwell-Chern-Simons gauged O(3) sigma model in the whole plane . After reducing to a system of semilinear elliptic equations, we construct a topological solution pair via iteration method and compute the conserved energy explicitly using the exponential decay property of the solution.Jongmin Han: Supported by grant No. R01-2002-000-00390-0 from the Basic Research Program of the Korea Science and Engineering Foundation.Hee-seok Nam: Supported by Korea Research Foundation Grant (KRF-2004-003-C00017).Mathematics Subject Classification: 81T13, 35B40.     

On the universality of string theory

String theory is accused by some of its critics to be a purely abstract mathematical discipline, having lost the contact to the simple yet deeply rooted questions which physics provided until the beginning of this century. We argue that, in contrary, there are indications that string theory might be linked to a fundamental principle of a quantum computational character. In addition, the nature of this principle can possibly provide some new insight into the question of universality of string theory (string theory as the “theory of everything”).     

Algebraic Geometry in Discrete Quantum Integrable Model and Baxter's T–Q Relation

We study the diagonalization problem of certain discrete quantum integrable models by the method of Baxter's T–Q relation from the algebraic geometry aspect. Among those the Hofstadter type model (with the rational magnetic flux), discrete quantum pendulum and discrete sine-Gordon model are our main concern in this report. By the quantum inverse scattering method, the Baxter's T–Q relation is formulated on the associated spectral curve, a high genus Riemann surface in general, arisen from the study of spectrum problem of the system. In the case of degenerated spectral curve where the spectral variables lie on rational curves, we obtain the complete and explicit solution of the T–Q polynomial equation associated to the model, and the intimate relation between the Baxter's T–Q relation and algebraic Bethe Ansatz is clearly revealed. The algebraic geometry of a general spectral curve attached to the model and certain qualitative properties of solutions of the Baxter's T–Q relation are discussed incorporating the physical consideration.     

On the observability of the fieldB (3): Relativistic effects in magneto-optics

It is shown that the novel vacuum fieldB ⁽³⁾ is an experimental observable, and several methods of observation are suggested: these include the pulsed microwave magnetization of a plasma, the optical Aharonov-Bohm effect, and the microwave frequency optical Faraday effect. The effect ofB ⁽³⁾ is presented in the form of relativistically corrected semi-classical theory.     

Quantum simulation of the t–J model

Computer simulation of a many-particle quantum system is bound to reach the inevitable limits of its ability as the system size increases. The primary reason for this is that the memory size used in a classical simulator grows polynomially whereas the Hilbert space of the quantum system does so exponentially. Replacing the classical simulator by a quantum simulator would be an effective method of surmounting this obstacle. The prevailing techniques for simulating quantum systems on a quantum computer have been developed for purposes of computing numerical algorithms designed to obtain approximate physical quantities of interest. The method suggested here requires no numerical algorithms; it is a direct isomorphic translation between a quantum simulator and the quantum system to be simulated. In the quantum simulator, physical parameters of the system, which are the fixed parameters of the simulated quantum system, are under the control of the experimenter. A method of simulating a model for high-temperature superconducting oxides, the t–J model, by optical control, as an example of such a quantum simulation, is presented.     

On a Trialgebraic Deformation of the Manin Plane

Quantum groups (or more precisely, function algebras on quantum groups), i.e. bialgebras with certain additional properties, can be gained by deforming an appropriate function algebra on a group. In a similar way, we show that a polynomial-like algebra on the (function algebra of the) Manin plane leads to a so-called trialgebra (as suggested by Crane and Frenkel), i.e. an algebraic structure possessing a coproduct and two products in a compatible way. We show how to deform this trialgebra to a noncocommutative and totally noncommutative (i.e. in both products) one. Trialgebras are of interest for various reasons, e.g. the search for topological invariants for four manifolds or the duality operation for non-Abelian lattice gauge theories, recently suggested by the authors.     

The Equations of Grand Unified Field Theory in Terms of the Maurer-Cartan Structure Relations of Differential Geometry

The first and second Maurer-Cartan structure relations are combined with the Evans field equation [1] for differential forms to build a grand unified field theory based on differential geometry. The tetrad or vielbein plays a central role in this theory, and all four fields currently thought to exist in nature can be described by the same equations, the tangent space index of the tetrad in general relativity being identified with the tetrad's internal (gauge group) index guage theory.     

Vortex pinning properties of –Ba–Cu–O and –Ba–Cu–O superconducting bulks

We have studied the effect of a small amount of Y-site substitution by La or Pr ions on the vortex pinning in the Y–Ba–Cu–O system. (Y_1-xLa_x)–Ba–Cu–O and (Y_1-xPr_x)–Ba–Cu–O bulks were fabricated by the melt-textured growth, in which x was varied from 0 to 0.01. The critical current density J_c at 77 K is improved in magnetic fields parallel to the c-axis above 2–4.5 T and the corresponding irreversibility field, H_irr, shifts to the higher value in both bulks.     

Far-field properties and beam quality of vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

The far-field properties and beam quality of vectorial nonparaxial Hermite–Laguerre–Gaussian (HLG) beams are studied in detail, where, instead of the second-order-moments-based M² factor, the extended power in the bucket (PIB) and βparameter are used to characterize the beam quality in the far field and the intensity in the formulae is replaced by the z component of the time-averaged Poynting vector S_z. It is found that the S_z PIB and βparameter of vectorial nonparaxial HLG beams depend on the mode indices n, m, αparameter and waist-width-to-wavelength ratio w₀/λ and the PIB and βparameter are additionally dependent on the bucket's size taken.     

The factor of nonparaxial Hermite–Gaussian beams and related problems

On the basis of the second-order moment of the power density and in the use of the series expansion, the expressions for the beam width, far-field divergence angle and M² factor of nonparaxial Hermite–Gaussian (H–G) beams are derived and expressed in a sum of the series of the Gamma function. The theoretical results are illustrated with numerical examples. The M² factor of nonparaxial H–G beams depends not only on the beam order m, but also on the waist-width to wavelength ratio w₀/λ. The far-field divergence angles of nonparaxial H–G beams with even and odd orders approach their upper limits θ_max=63.435 and 73.898, respectively, which results in M²<1 as w₀/λ→0. For the special case of m=0 our results reduce to those of nonparaxial Gaussian beams. Some problems related to the characterization of the nonparaxial beam quality are also discussed.     

Aspects of Generalized Calogero Model

A multispecies model of Calogero type in D 1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analyzing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior.     

Derivation of O(3) Electrodynamics from the Einstein–Sachs Theory of General Relativity

General relativity is reduced to O(3) electrodynamics by consideration of the irreducible representations of the Einstein group and through a particular choice of basis. The photon is shown always to possess a scalar curvature R, and so the origin of quantization is found in general relativity.