Fermionic quantum stochastic flows

Existence, uniqueness and, using a higher-order Fermionic Itô formula, multiplicativity are proved for even quantum stochastic flows over ₂graded C ^*-algebras driven by Fermionic noise with bounded structure maps.Work supported by EU HCM Contract CHRX-CT93-QO94.     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

Symmetry transformations in Batalin-Vilkovisky formalism

Let us suppose that the functionalS on an odd symplectic manifold satisfies the quantum master equation e ^s = 0. We prove that in some sense every quantum observable (i.e. every functionH obeying _p (He ^s) = 0) determines a symmetry of the theory with the action functionalS. Research supported in part by NSF grant No. DMS-9201366.     

Weakly coupled states on branching graphs

We consider a Schrödinger particle on a graph consisting of N links joined at a single point. Each link supports a real locally integrable potential V _j ; the self-adjointness is ensured by the type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as x ^–1– along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrödinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the coupling constant may be interpreted in terms of a family of squeezed potentials.     

On the schwarz problem for the [^(\textsu)]\text2 \widehat{{\text{su}}}_{\text{2}} Knizhnik-Zamolodchikov equation

We study the monodromy representations ^k,Iof the mapping class group ₄ acting on 4-point blocks satisfying the Knizhnik-Zamolodchikov equation for the levelk su₂ current algebra. We classify all irreducible ^k,Iwhich are realized by finite groups; we also display finite irreducible components for the reducible representations corresponding tok = 10.Supported by the Federal Ministry of Science and Research, Austria.     

Spectral stochastic processes arising in quantum mechanical models with a non-L 2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non-L ² ground state. As a prototype, the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal Abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite-volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite-volume measures.Supported by DFG, Nr. Al 374/1-2     

Generalized measures in gauge theory

LetP M be a principalG-bundle. We construct well-defined analogs of Lebesgue measure on the spaceA of connections onP and Haar measure on the groupG of gauge transformations. More precisely, we define algebras of cylinder functions on the spacesA,G, andA/G, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures onA,G, andA/G in terms of graphs embedded inM. We use this characterization to construct generalized measures onA andG whenG is compact. The uniform generalized measure onA is invariant under the group of automorphisms ofP. It projects down to the generalized measure onA/G considered by Ashtekar and Lewandowski in the caseG = SU(n). The generalized Haar measure onG is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure onA against generalized Haar measure gives aG-invariant generalized measure onA.     

Continued fractions and fermionic representations for characters of M(p,p′) minimal models

We present fermionic sum representations of the characters _{, s} ^{(p, p)} of the minimal M(p,p) models for all relatively prime integers p>p for some allowed values of r and s. Our starting point is biomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1/2 chain of anisotropy –=–cos((p/p)). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of {p/p} (and p/p), where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the Hermitian irreducible representations of SU _q- (2) (and SU _q+ (2)) with q–=exp(i{p/p}) (and q+=exp(i(p/p))). We also establish the duality relation M(p,p) M(p–p,p) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.Dedicated to Prof. Vladimir Rittenberg on his 60th birthday     

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsF . We derive a topological bound on R⁸, , wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.     

Limits on stability of positive molecular ions

It is shown that a positive diatomic molecule consisting of N electrons and two nuclei with charges Z ₁ and Z ₂ is unstable with respect to breakup into two atomic subsystems if the nuclear charges are sufficiently large. Bounds on the critical charge are obtained in the limit as N .Research supported in part by NSF grants DMS-8709805 and DMS-8808112.     

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

Interrelations between quantum groups and reflection equation (braided) algebras

We show that the differential complex _B over the braided matrix algebra BM _q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra _A which is a differential extension of GL _q (N). On the other hand, the algebra _A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra _B . Geometrical aspects of these results are discussed.     

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.     

q-transformation theory

We obtain the complete set of states of theq-oscillator in both configuration space and momentum space as well as the transformation between these spaces. The states as well as the matrix elements lie in the SU _q (2) algebra. To obtain transition probabilities, one must take the Woronowicz square.     

Local quantum field theories involving the U(1) current algebra on the circle

The OPE algebra Q=Q(g ² ) generated by a pair of oppositely charged currents (z,±g)(|z|=1) of spin is specified by the leading terms in the small distance expansions of (z ₁,g)(z ₂, -g) and (z ₁,g)(z ₂,g). The current (z,g) splits into a product of a U(1)-Thirring field and a Zamolodchikov-Fattev parafermionic current. The quasilocal(i.e.single-or double-valued) representations of Q are classified. The level k states involve 2(k+1) (ks–k+1) lowest weights (dimensions). The results can be viewed as an extension of the (known) representation theory of the SU(2) current algebra in the bosonic case corresponding to even values of g ² and of the N=2 extended superconformal algebra in the fermionic case corresponding to odd g ².     

p-Adic string compactified on a torus

The open p-adic string world sheet is a coset space F=T/, where T is the Bruhat-Tits tree for the p-adic linear group GL(2, _p ) and PGL(2, _p ) is some Schottky group. The string dynamics is governed by the local action on F, with the fields taking values in a compact group G. We find the correlation functions and partition functions for the p-adic string surfaces of arbitrary genus and G=U(1)^xD (D-dimensional torus).     

Trace identities from tau functions

An approach to trace identities is described based on the-function formalism for mKdV and affine Toda systems. The set-up is given for discrete and continuous spectrum solutions by exploiting the relation between Jost solutions, diagonal entries of the monodromy matrix, and-functions. Examples for solutions with discrete scattering data are worked out.     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

The inverse scattering matrix for the Schrödinger equation when the potentialq(x) ∈L 1 1 with a singular term

When the potentialq(x) L ₁ ¹ with a singular term, the continuities of the scattering matrix of the Schrödinger equation are investigated. By means of the transformation approach, we arrive at the conclusion that the scattering matrix S(k) of such a potential is continuous for the wholek,- <k < .