A gauge field governing parallel transport along mixed states

At first, a short account is given of some basic notations and results on parallel transport along mixed states. A new connection form (gauge field) is introduced to give a geometric meaning to the concept of parallelity in the theory of density operators.     

The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension

In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This Letter describes a lattice version, namely a natural way of modifying the 2D Heisenberg model to achieve topological stability on the lattice.     

On quantum holonomy for mixed states

There is a natural connection and parallel transport on the Hilbert tensor product (or, equivalently, the space of Hilbert-Schmidt operators), the elements of which represent density matrices in up to unitary operators. We postulate a time evolution equation, which leads to this connection after extracting a proper dynamical unitary phase. As an example, we compute the holonomy of a loop of temperature states for the spin in a rotating magnetic field.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P5588.     

P-adic Schrödinger-type equation

A Schrödinger-type equation is considered in relation to p-adic quantum mechanics. We discuss the appropriate notion of differential operators. A solution of the Schrödinger-type equation is given and a new set of vacuum states for the p-adic quantum harmonic oscillator is presented. The correspondence principle with the standard quantum mechanics is also discussed.     

Universal deformations of reductive Lie algebras

We construct multiparameter quantizations of reductive Lie algebras which have the property of universality within a certain class of deformations. The universal deformations can be defined so that the algebra structure on each simple component is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear, as a special case of deformations of a semisimple algebra whose simple components remain classical. Deformations are defined as algebras over power series rings and it is essential to require them to be torsion free to secure the universality. The Poincaré-Birkhoff-Witt theorem and the torsion freeness are established for the universal deformation on the basis of results on the representation theory of the deformed algebras.     

Algebraic approach to Kaluza-Klein models

We propose a new method for the construction of injections into classical Lie algebras. This method is very useful for the analysis of vacuum solutions in Kaluza-Klein models. With the help of the proposed procedure, one can obtain all the known solutions as well as a new one explicitly constructed in this letter.     

The concept of a quantum semisimple group

We explore the possibility of introducing the concept of a quantum semisimple group by exhibiting a class of deformations of classical groups whose Dynkin diagrams are disconnected at the classical level, but become connected at the quantum level. The possibility of applications to the quantization of Lorentz and Poincaré groups are mentioned.     

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.     

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.     

On current superalgebras and super-Schwinger terms

We present a general construction of current superalgebras within the framework of quasi-free second quantization of bosons and fermions. Mathematically speaking, we given projective representations of certain Lie superalgebras realized as bounded operators on Z₂-graded Hilbert spaces and, more generally, on Grassmann algebra-modules. The super-Schwinger terms occurring correspond to Z₂-graded two-cocycles.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P5588.     

Quantization of a Hamiltonian system with an infinite number of degrees of freedom

We propose a method of quantization of a discrete Hamiltonian system with an infinite number of degrees of freedom. Our approach is analogous to the usual finite-dimensional quantum mechanics. We construct an infinite-dimensional Schrödinger equation. We show that it is possible to pass from the finite-dimensional quantum mechanics to our construction in the limit when the number of particles tends to infinity. Rigorous mathematical methods are used.     

Coulombic potentials in the semi-classical limit

This paper is devoted to Schrödinger operators in two dimensions with singular (Coulombic) potentials. We investigate the behaviour of the eigenvalues at the bottom of the spectrum in the semi-classical limit. To overcome the difficulties due to the singularities, we use some kind of generalisation of the Levi-Civita transform. After this regularisation, we apply the theory of Helffer and Sjöstrand to get the full asymptotics for the eigenvalues.     

Splitting of Landau levels in the presence of external potentials

We study the quantum mechanical problem of a charged particle moving in a constant magnetic field and an azimuthally symmetric external potential. We show that the lowest Landau level splits in a definite way if the external potential is a monotone function of the radial distance in the plane orthogonal to the magnetic field. For two-dimensional systems, we give sufficient conditions such that all Landau levels are completely ordered. For the lowest Landau level, this result holds for nonhomogeneous magnetic fields as long as they are cylindrically symmetric.Supported by the Federal Ministry of Science and Research, Austria. Part of Project P8916-PHY of the Fonds zur Förderung der wissenschaftliche Forschung in Österreich.     

A Remark on the Estimate of a Determinant by Minami

In the context of the Anderson model, Minami proved a Wegner type bound on the expectation of 2 × 2 determinant of Green’s functions. We generalize it so as to allow for a magnetic field, as well as to determinants of higher order.      

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model (A-model) in the case when the target space is an (m ₀|m ₁)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m ₀–m ₁)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.Research supported in part by NSF grant No. DMS-9201366.     

Constraints on two-dimensional conformal field theories from knot invariants

The connection between Witten's topological three-dimensional gauge theory and RCFTs provides a natural setting to study the interplay between surface diffeomorphisms and intertwining of Wilson loops. These considerations lead directly to constraints on RFCTs including those previously derived by Vafa.     

Supercurrent coupling in the Faddeev–Skyrme model

Motivated by the sigma model limit of multicomponent Ginzburg–Landau theory, a version of the Faddeev–Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space M$M$ is an oriented Riemannian manifold and the target space N$N$ is a Kähler manifold, and its properties are compared with the usual, uncoupled Faddeev–Skyrme model. In the case N=S²$N=S^2$, it is shown that supercurrent coupling destroys the familiar topological lower energy bound of Vakulenko and Kapitanski when M=R³$M={\mathbb{R}}^3$, and the less familiar linear bound when M$M$ is a compact 3-manifold. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. In particular, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius R$R$ is unstable   for all R$R$. This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the usual Faddeev–Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low energy regime of two-component superconductors.     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).