Guy Bonneau

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.

Consistency in dimensional regularization with γ5

Abstract Some general remarks about dimensional regularization are offered. Then we show that among the schemes so far proposed for theories involving γ 5 and (or) ϵ μνϱσ only the original one with a non-anticommuting γ 5 is consistent.

Trace and axial anomalies in dimensional renormalization through Zimmermann-like identities

Abstract The problem of anomalies is solved in dimensional renormalization in two steps. Firstly one shows that trace and γ5 anomalies can be expressed as the anomalous normal product N [ g μϱ O μϱλ… (x)] where g μϱ is the metric tensor in D-4 dimensions (D being the space-time dimension) and O μϱλ …(x) a monomial in the fields and their derivatives. Then, with techniques similar to those used in a previous work to study N [(4 − D) O μϱλ (x)] , we prove a Zimmermann-like identity that gives the decomposition of such anomalous normal product on “usual” normal products, the coefficients being explicitly given as residues of the simple pole in ν = 4 − D of definite proper Green functions where the overall substraction has not been done. We apply the above formalism to obtain the renormalization group as a consequence of trace anomalies in the dilatation current and to derive the Adler-Bardeen theorem for massive QED.

Zimmermann identities and renormalization group equation in dimensional renormalization

In the framework of dimensional renormalization, Zimmermann-like identities are shown: D being the space-time dimension and O(x) a monomial in the fields and their derivatives, they give the decomposition of the “oversubtracted” normal product N[(4 - D)O(x)] on “usual” normal products. These relations are expected to be essential in the study of anomalies in the dimensional scheme as will be shown in a further publication. Moreover, using similar techniques, a rigorous proof of the renormalization group equation in the minimal dimensional renormalization scheme is also given.

S-matrix properties versus renormalizability in two-dimensional O(N) symmetric models

Abstract The S -matrix property of factorizability, when applied to O( N ) symmetric models, is shown to select N = 2 and, at the classical level, two well-known complexifications of sine-Gordon theory. Factorizability remains true at the one-loop level only when some finite quantum counterterms are added to the lagrangians. On the other hand, when dimensionally regularized the models are one-loop renormalizable and asymptotically scale invariant but at two loops order multiplicative renormalizability is lost. However we have been able to exhibit the complete expression of the finite quantum corrections that restore renormalizability. Moreover, a simultaneous request for one-loop non-production properties and two-loop renormalizability fixes all physical parameters and gives as an extra bonus two-loop vanishing of the β -function.

Some consequences of analyticity and unitarity for the pion form factor

Abstract Proved results of local quantum field theory are used — with a few reasonable additional hypotheses — to derive a modulus representation of the pion electromagnetic form factor. Then, unitarity conditions, such as a P-wave behaviour for the phase of the pion form factor near threshold, lead to two new sum rules and to new optimal bounds on the spacelike values of the form factor and on its phase above threshold. These sum rules and bounds are evaluated with the help of timelike data from electron-positron storage rings experiments; in particular for the pion electromagnetic radius we obtain: 0.24 fm ⩽ r π ⩽ 0.78 fm. Our predictions agree with spacelike data.

Asymptotic scale invariance in two-dimensional theories: A massive Thirring model with bosons

Abstract The exact vanishing of all β functions of the Callan-Symanzik equation is proved in a two-dimensional model with Thirring and gradient couplings: ( ψ γ μ ψ) 2 , ( ψ γ 5 γ μ ψ)ϱ μ π and ( ψ γ μ ψ)ϱ μ σ , where ψ, π and σ are fermion, pseudoscalar and scalar massive fields, respectively. The anomalous dimensions of scalar and pseudoscalar fields are also shown to be zero. Then we demonstrate that in two-dimensional models including at most one fermion field, only these three couplings can give asymptotic scale invariance.

CONSTRUCTION AND PROPERTIES OF QUASI RICCI FLAT SPACES

We look for the necessary and sufficient conditions for a generalized torsion-free nonlinear σ-model to be one-loop finite. The corresponding metrics are not only Ricci flat ones, but also a larger class we call “quasi Ricci flat” spaces. We give expressions for the corresponding Lagrangian densities in the real and Kahler cases. In the latter, the manifold is shown to be proper, complete and nonhomogeneous. Unfortunately, in the compact case, relevant for string theory, these quasi Ricci flat manifolds become Ricci flat ones.

Scattering lengths and baryon magnetic moments in a gauge model of strong and electromagnetic interactions

Abstract We present a broken SU(3) gauge model of strong and electromagnetic interactions with the usual vector mesons. All particles (9 vectors, 8 baryons and 9 pseudoscalars) have the right masses by means of the Higgs mechanism. We study the consequences of Sakurais idea that strong and electromagnetic interactions are mediated by vector mesons universally coupled to nearly conserved currents: one finds encouraging values for scattering lengths except for P-wave parameters in the meson-baryon sector that are too small. The calculation of the baryon anomalous magnetic moments also gives too-small numbers.