Quantum motion on two planes connected at one point

We study the free motion of a particle on the manifold which consists of two planes connected at one point. The four-parameter family of admissible Hamiltonians is constructed by self-adjoint extensions of the free Hamiltonian with the singular point removed. The probability of penetration between the two parts of the configuration manifold is calculated. The results can be used as a model for quantum point-contact spectroscopy.     

On the Aharonov–Bohm Hamiltonian

Using the theory of self-adjoint extensions, we construct all the possible Hamiltonians describing the nonrelativistic Aharonov–Bohm effect. In general, the resulting Hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.     

Norm Convergence of Self-Adjoint Extensions of the Aharonov–Bohm Hamiltonian as Solenoid Length Goes to Infinity

With respect to the Aharonov-Bohm effect, we consider only solenoids of zero radius and their action on its plane of symmetry. We find all self-adjoint extensions in case of finite length solenoids and compare them with the case of infinitely long one. It is then shown a convergence in the uniform resolvent sense for each angular momentum sector of such self-adjoint extensions as the solenoid length goes to infinity.     

Towards the construction of N = 2 supersymmetric integrable hierarchies

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super-W_n second Hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct Hamiltonians we are led to a new definition of super-residues for degenerate N = 2 supersymmetric pseudo-differential operators. We have found a new non-polynomials Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.     

Unboundedness of Adjacency Matrices of Locally Finite Graphs

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note, we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium.     

Infra-red stable fixed points of Yukawa couplings in supersymmetric models

We provide the solutions of the fixed point conditions of the Yukawa sector for a large class of N = 1 supersymmetric theories including the minimal and next-to-minimal supersymmetric standard models and their grand unified and other extensions. We also introduce a test which can discriminate between infra-red stable, infra-red unstable and saddle point solutions, and illustrate our methods with the example of the next-to-minimal supersymmetric standard model. We show that in this case, the fixed point prediction of the top quark mass is equivalent to that of the minimal supersymmetric standard model, supporting previous numerical analyses.     

Local nets and self-adjoint extensions of quantum field operators

We consider Wightman fields having the property that some closed extensions of the field operators generate locally commuting von Neumann algebras. We show that for such fields the hermitian field operators have self-adjoint extensions, possibly in an enlarged Hilbert space, such that bounded functions of the self-adjoint operators commute locally.     

Free Fermions and the Classical Compact Groups

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: (i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; (ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions.     

Spectral properties in supersymmetric matrix models

We formulate a general sufficiency criterion for discreteness of the spectrum of both supersymmmetric and non-supersymmetric theories with a fermionic contribution. This criterion allows an analysis of Hamiltonians in complete form rather than just their semiclassical limits. In such a framework we examine spectral properties of various (1+0) matrix models. We consider the BMN model of M-theory compactified on a maximally supersymmetric pp-wave background, different regularizations of the supermembrane with central charges and a non-supersymmetric model comprising a bound state of N D2 with m D0. While the first two examples have a purely discrete spectrum, the latter has a continuous spectrum with a lower end given in terms of the monopole charge.     

A simple model of thin-film point contact in two and three dimensions

We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two-parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admissible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films.On leave of absence fromNuclear Physics Institute, Czechosl. Acad. Sci., 250 68 e near Prague, Czechoslovakia.On leave of absence fromNuclear Centre, Charles University, V Holeovikách 2, 180 00 Prague 8, Czechoslovakia.     

Second Order Perturbation Theory for Embedded Eigenvalues

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling.     

Particle Creation at a Point Source by Means of Interior-Boundary Conditions

We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cut-off (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC.     

Quantum States of a Neutral Massive Fermion with an Anomalous Magnetic Moment in an External Electric Field

The quantum dynamics of a nonrelativistic neutral massive fermion with an anomalous magnetic moment (AMM) is examined in the external electric field of an infinitely long thin homogeneously charged thread in the plane with a normal directed along the thread. The Hamiltonian of the Dirac–Pauli equation for a neutral fermion with AMM is essentially singular in the considered external field and requires a supplementary extension of the definition in order for it to be treated as a self-adjoint quantum-mechanical operator. All one-parameter self-adjoint extensions of the Hamiltonian of the Dirac–Pauli equation in the considered external field are found in the nonrelativistic approximation. The corresponding Hilbert space of squareintegrable functions, including a singularity point of the Hamiltonian, is specified for each self-adjoint extension of the Hamiltonian. The wave functions of free and bound states, as well as discrete energy levels, are determined by the self-adjoint extension method and their correspondence with similar quantities obtained by the physical regularization procedure is discussed. It is shown that energy levels of bound states are simple poles of the scattering amplitude, which should be extended in definition by introducing the self-adjoint extension parameter into it. Expressions for the scattering amplitude and cross-section, depending on the orientation of the initial-state spin of fermion, are obtained.     

The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation     

New avenues in supersymmetry and supergravity

We analyze the problem of constructing supersymmetric versions of gauge theories of particles and of gravity which have a closed supersymmetric algebra. Inparticular we present the basic no-go theorems that indicate that in four dimensions it is not possible to construct suitably extended supersymmetric versions of the above theories without drastic modification of the supersymmetric algebra. Two ways past the“N=3” barrier are discussed; that of central charges involved highly constrained versions which appearn difficult to quantize effectively, while the use of light-cone variables seems to be the most promising. We give light-cone gauge versions of supersymmetric Yang-Mills theories for all extended cases of interest and briefly consider their ultraviolet divergence properties.     

Supersymmetric quantum-Hall effect on a fuzzy supersphere

Supersymmetric quantum-Hall liquids are constructed on a supersphere in a supermonopole background. We derive a supersymmetric generalization of the Laughlin wave function, which is a ground state of a hard-core OSp(1/2) invariant Hamiltonian. We also present excited topological objects, which are fractionally charged deficits made by super Hall currents. Several relations between quantum-Hall systems and their supersymmetric extensions are discussed.     

Is there a unique consistent theory of quantum gravity?

Superstrings have been proposed as a quantum-theoretical framework for unifying all the fundamental forces, including gravity. We consider the question of whether there might be more general supersymmetric possibilities, based on higher extended objects such as membranes, jellies, etc. We argue that all the possible extended objects in all possible spacetime dimensions are quantummechanically inconsistent except for the 10-dimensional superstring and the 11-dimensional supermembrane. These are also the only two such theories that contain massless gravitons and, thus, that can describe gravity at low energies. It is remarkable that the range of possibilities can be narrowed down to this extent. Whether these can be further narrowed down to just one consistent theory remains open to further research.This essay received the first award from the Gravity Research Foundation for the year 1988.-Ed.     

Contraction-based classification of supersymmetric extensions of kinematical lie algebras

We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry—Lévy-Leblond classification.     

Supersymmetric quantum mechanics

We give a general construction for supersymmetric Hamiltonians in quantum mechanics. We find that N-extended supersymmetry imposes very strong constraints, and for N > 4 the Hamiltonian is integrable. We give a variety of examples, for one-particle and for many-particle systems, in different numbers of dimensions.