Galliano Valent

Self-adjoint extensions of operators and the teaching of quantum mechanics

For the example of the infinite well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different settings. Additional physical requirements such as parity, time reversal, and positivity are used to restrict the large class of self-adjoint extensions of the Hamiltonian.

Linear birth and death models and associated Laguerre and Meixner polynomials

Abstract We study birth and death processes with linear rates λ n = n + α + c + 1, μ n + 1 = n + c , n ⩾ 0 and μ 0 is either zero or c . The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ 0 = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.

Classical and quantum structure of the compact kählerian sigma models

Abstract The kahlerian sigma models in two-dimensional space-time are investigated. A convenient parametrization is used and its classical properties, lagrangian and symmetries, are studied. Within dimensional regularization, quantum divergences are then shown to be proportional to the invariant lagrangian. A calculation of the one-loop divergences for a large class of kahlerian models is presented, and the result is used to compute the one-loop β-function.

Quantum integrability of quadratic Killing tensors

Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a “minimal” quantization scheme, quantum integrability is ensured for a large class of classic examples.

U(1)×U(1) quaternionic metrics from harmonic superspace

Abstract We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-Kahler extension of the most general two centres hyper-Kahler metric. It possesses U(1)×U(1) isometry, contains as special cases the quaternionic-Kahler extensions of the Taub-NUT and Eguchi–Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-Kahler limit. Some emphasis is put on the relation between this class of quaternionic-Kahler metrics and self-dual Weyl solutions of the coupled Einstein–Maxwell equations. The relation between our explicit results and the recent general ansatz of Calderbank and Pedersen for quaternionic-Kahler metrics with U(1)×U(1) isometries is traced in detail.

ON A CLASS OF COMPACT AND NON-COMPACT QUASI-EINSTEIN METRICS AND THEIR RENORMALIZABILITY PROPERTIES

We construct a family of Kahlerian quasi-Einstein metrics with an isometry group U(n) acting linearly on the holomorphic coordinates. Suitable restrictions on the parameters give rise to complete non-compact as well as compact metrics whose geometrical structure is studied in detail. The two-loop renormalizability properties are discussed according to whether one considers the bosonic σ-models or their (2,2) supersymmetric extension.

Quaternionic metrics from harmonic superspace: Lagrangian approach and quotient construction

Abstract Starting from the most general harmonic superspace action of self-interacting Q + hypermultiplets in the background of N =2 conformal supergravity, we derive the general action for the bosonic sigma model with a generic 4 n -dimensional quaternionic-Kahler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential and supplies an efficient systematic procedure of the explicit construction of QK metrics by the given QK potential. We find out this action to have two flat limits. One gives the hyper-Kahler (HK) sigma model with a 4 n -dimensional target manifold, while another yields a conformally-invariant sigma model with 4( n +1)-dimensional HK target. We work out the harmonic superspace version of the QK quotient construction and use it to give a new derivation of QK extensions of four-dimensional Taub–NUT and Eguchi–Hanson metrics. We analyze in detail the geometrical and symmetry structure of the second metric. The QK sigma model approach allows us to reveal the enhancement of its SU (2)⊗ U (1) isometry to SU (3) or SU (1,2) at the special relations between its free parameters: the Sp (1) curvature (“Einstein constant”) and the “mass”.

Local heterotic geometry in holomorphic coordinates

In the same spirit as for N=2 and N=4 supersymmetric nonlinear models in two spacetime dimensions by Zumino and by Alvarez-Gaume and Freedman, we analyse the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2,0) geometry is conformally equivalent to a (2,2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real four-dimensional case where we exhibit new examples. We show that, contrarily to Callan et als claim for real four-dimensional manifolds, (4,0) heterotic geometry is not necessarily conformally equivalent to a (4,4) Kahler--Ricci flat geometry. We rather prove that, whatever the real dimension is, they are special quasi-Ricci flat spaces, and we exemplify our results on Eguchi--Hanson and Taub-NUT metrics with torsion.

(4,0) and (4,4) sigma models with a tri-holomorphic Killing vector

Abstract We analyze (4,0) supersymmetric σ-models on a four dimensional target space which possess one tri-holomorphic Killing vector which is also assumed to leave invariant the torsion. The problem is reduced to two stages: first finding “special” three dimensional Einstein-Weyl spaces, and second solving a monopole-like equation on the special Einstein-Weyl space. A new class of examples is constructed using as Einstein-Weyl geometry the Berger sphere which includes the round three sphere as a particular case. When the Einstein-Weyl geometry is taken to be the round three sphere we show that the corresponding (4,0) geometries can be lifted to (4,4) geometries with two sets of non-commuting hyper-complex structures.

Multicentre metrics and harmonic superspace

We develop a procedure for constructing a possible harmonic superspace lagrangian for any d = 4 multicentre metric and show that the lagrangians leading to such metrics are characterized by the existence of a U(1) of Pauli-Gursey invariance. As an application, the mixed Eguchi-Hanson-Taub-NUT metric is shown, by explicit calculations, to be the double Taub-NUT metric.