String vertices,overlap equations, τ functions and the Hirota equation

String vertices,V, are shown to satisfy a new type of overlap equation of the form as well as corresponding equations forA _n andB _n cycles. A special case of such an equation, when integrated, is shown to be the Hirota equation for the K–P hierarchy.     

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

We develop a technique for the construction of integrable models with a ₂ grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group , with a matrix deformation parameter q such that (q )² = q ². The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of _q (gl(N)) with a Cartan generator related to deformation parameter –1.     

Gauge covariant theory of the generating operator. I

A gauge covariant formulation of the generating operator (-operator) theory for the Zakharov-Shabat system is proposed. The operator , corresponding to the gauge equivalent system in the pole gauge is explicitly calculated. Thus the unified approach to the nonlinear Schrödinger-type equations based on is automatically reformulated with the help of for the Heisenberg ferromagnet-type equations. Consequently, it is established that the conserved densities for the Heisenberg-ferromagnet-type equations are polynomial inS(x) and itsx-derivatives. Special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge-equivalent equations. It is shown that the geometrical properties of the conjugated operator * are gaugeindependent.     

Complements to various Stone-Weierstrass theorems forC *-algebras and a theorem of Shultz

J. Glimm's Stone-Weierstrass theorem states that ifA is aC ^*-algebra,P(A) is the set of pure states ofA, andB is aC ^*-subalgebra which separates , thenB=A. We show that ifB is aC ^*-subalgebra ofA andx an element ofA such that any two elements of which agree onB agree also onx, thenxB. Similar complements are given to other Stone-Weierstrass theorems. A theorem of F. Shultz states that ifxA ^**, the enveloping von Neumann algebra ofA, and ifx, x ^*, x, andxx ^* are uniformly continuous onP(A){0}, then there is an element ofA which agrees withx onP(A). We show that the hypotheses onx ^*x andxx ^* can be dropped.     

The Relation Between Two Types of Characteristic Equations for Quantum Matrices

A definition (modification) of the power of quantum matrices using the -matrix has recently been proven useful to obtain generalizations of many well known theorems from linear algebra to the quantum case, among which are the Cayley–Hamilton theorem and the Newton identities. A separate effort has provided another generalization of the Cayley–Hamilton theorem for GL _q (n), which uses usual matrix powers but diagonal matrices as coefficients.We show that the latter generalization can be derived in the aforementioned more general framework and it is the expression of the modified quantum power in terms of the usual ones that accounts for the appearance of diagonal matrices.     

Finite Range of Large Perturbations in Hamiltonian Dynamics

The dynamics defined by the Hamiltonian , where the _m are fixed random phases, is investigated for large values of A, and for . For a given P ^* and for , this Hamiltonian is transformed through a rigorous perturbative treatment into a Hamiltonian where the sum of all the nonresonant terms, having a Q dependence of the kind cos(kQ – nt + _m) with \Delta \upsilon$$ " align="middle" border="0"> , is a random variable whose r.m.s. with respect to the _m is exponentially small in the parameter . Using this result, a rationale is provided showing that the statistical properties of the dynamics defined by H, and of the reduced dynamics including at each time t only the terms in H such that , can be made arbitrarily close by increasing . For practical purposes close to 5 is enough, as confirmed numerically. The reduced dynamics being nondeterministic, it is thus analytically shown, without using the random-phase approximation, that the statistical properties of a chaotic Hamiltonian dynamics can be made arbitrarily close to that of a stochastic dynamics. An appropriate rescaling of momentum and time shows that the statistical properties of the dynamics defined by H can be considered as independent of A, on a finite time interval, for A large. The way these results could generalize to a wider class of Hamiltonians is indicated.     

Limiting behavior of asymptotically flat gravitational fields

Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that , (d/dr) and (/x ^A ) , wherex ^A (A = 2, 3) are angular coordinates, they show that , where ₁ 2 and ₁<₀; , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption , (d/dr) , and (/x ^A ) does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

NMRON studies of MnBr2 \cdot 4H2O in applied magnetic fields

NMRON studies for the ⁵⁴Mn transitions in antiferromagnetic MnBr₂ 4H₂O, in the millikelvin regime, are presented and discussed. New values are given for (i) the sum of the effective molecular exchange and magnetic anisotropy fields acting on the Mn²⁺ ions, =2.23(2) T, and (ii) the magnetic dipole hyperfine splitting, A=-201.99(1) MHz, electric quadrupole hyperfine splitting P=0.049(8) MHz and pseudoquadrupole splitting =1.63(2) MHz for the ⁵⁴Mn nuclei.     

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy , we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) . We define a Hamiltonian linearization of the theory, i.e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's , which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.     

Exact monopole solutions in gauge theories for an arbitrary semisimple compact group

We construct an exact n-parametric monopole and dyon solutions for an arbitrary compact gauge group G of rank n by using the symmetry between cylindrically symmetric instanton equations in Euclidean space R ₄ and monopole equations in Minkowski space R _3,1 (with Higgs scalar field in adjoint representation). The solutions are spherically symmetric with respect to the total momentum operator represents the minimal embedding of SU(2) in G. Explicit expressions for the monopole magnetic charge and mass matrices are obtained. The remarkable aspect of our results is the existence of discrete series of the monopole solutions, which are labelled by n quantum numbers and degenerated in the latter ones at a fixed monopole mass matrix.     

An electrical test particle in Einstein's unified field theory

Previous work on a class of exact solutions to the field equations of Einstein's unified field theory has shown that some of these solutions acquire an immediate physical meaning as soon as one allows for external sources, as it occurs in the general theory of relativity. It is evident that a four-current density j ⁱ , appended to the right-hand side of the field equation , has a fundamental role: in some solutions, a string built with this current density gives rise to partons, mutually interacting with forces that do not depend on distance, like the ones invoked to explain the confinement of quarks. In other solutions, for which obeys Maxwell's equations, jⁱ clearly displays electrical behavior. In the present paper it is shown under what conditions the electrical behavior of a charged test particle can be extracted from the field equations and from conservation identities related to the theory, when sources are appended in the way proposed by Borchsenius and Moffat.     

Canonical quantization of a nonrelativistic singular quasilinear system

Following Dirac's generalized canonical formalism, we develop a quantization scheme for theN-dimensional system described by the Lagrangian which is supposed to be invariant under the gauge transformation . The gauge invariance necessarily implies that the Lagrangian is singular. The identities imposed by the gauge invariance are enumerated and reduced to simpler forms. There are primary and secondary constraints, both of which are of first class. The reduced identities are solved explicitly for the case where the secondary constraints constitute the generators of the groupSO(M), and thus an explicit expression for the manifestly gauge-invariant Lagrangian is obtained. By fixing the gauge appropriately, the unphysical variables are eliminated and a quantization is achieved using only physical variables. Our formulation is covariant under an arbitrary point transformation of physical variables. The problem of formulating a quantum action principle is also commented on briefly.     

A note on the covariant anomaly as an equivariant momentum mapping

We show that there is a natural gauge invariant presymplectic structure on the spaceA of all vector potentials. The covariant axial anomaly is found to be the essentially unique infinitestimally equivariant momentum mapping for the action of the group of gauge transformations on (A,). The infinitesimal equivariance of is shown to be equivalent to the Wess-Zumino consistency condition for the consistent axial anomalyG. We also show that theX operator of Bardeen and Zumino, which relatesG and , corresponds to the one-form (onA) of the presymplectic structure.Research supported in part by NSF grant MCS 81-08814(A03)Research supported by the U.S. Department of Energy under contract number DE AC02 76 ER 02220     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

\left\langle {:\bar \psi \psi :} \right\rangle and three-point functions in QCD

We calculate the coefficients of to leading order in _s in the operator product expansion of the fundamental three-point functions of QCD in the deep Euclidean region. We demonstrate that these coefficients satisfy the Ward identities.Supported by BMFT 0233 REB4     

Bound States in a Locally Deformed Waveguide: The Critical Case

We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies for all sufficiently small.     

General two-mode squeezed states

The states |A ₁ A ₂ are considered, where the operators are associated with a unitary representation of the groupSp(4,), and the two-mode Glauber coherent states |A ₁ A ₂> are joint eigenstates of the destruction operatorsa ₁ anda ₂ for the two independent oscillator modes. We show that they are ordinary coherent states with respect to new operatorsb ₁ andb ₂, which are themselves general linear (Bogolibov) transformations of the original operatorsa ₁,a ₂ and their hermitian conjugatesa ¹ ,a ² . We further show how they may be regarded as the most general two-mode squeezed states. Most previous work on two-mode squeezed states appears to be based on more restrictive definitions than our own, and thereby reduces to special cases which are unified within our treatment.     

On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, , , with any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of H(x,u) with x at infinity determines many properties of geodesics. Essentially, a subquadratic growth of H ensures geodesic completeness and connectedness, while the critical situation appears when H(x,u) behaves in some direction as , as in the classical model of exact gravitational waves.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.