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In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector-fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie-derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original non-supersymmetric model. Besides the supersymmetric extension of the original Hamiltonian, we also provide the extension of the other conformal generators present in the original system. These extensions have also a supersymmetric character being the square of some Grassmannian charge. We build the whole superalgebra of these charges and analyze their closure. The representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character., Comment: Superfield re-defined

A new class of multimeron solutions is obtained in the framework of a generally invariant gauge theory.

We review classical solutions, both of the instanton and meron type, for the nonlinear σ-model in two dimensions and discuss their generalization to the four-dimensional case, which involves a fourth-order derivative Lagrangian.

In this paper it is shown how to get, through the use of suitable commutation relations, sum rules for the renormalization ratios between bare and dressed coupling constants. The method is completely general and it can be applied to any broken symmetry. As an illustration we have considered the cases of the weak vector current, both strangeness-conserving and strangeness-changing.

The equations of the renormalization group are applied to the perturbative expansion of the photon propagator in Q.E.D. They are shown to correspond to a «coherence principle in the renormalization of the single graphs» and for this reason they cannot give, by themselves, any improvement of the perturbative expansion. This is no more the case if they are used together with an explicit assumption on the form of the propagator. Two conditions of this kind, which has been used in the literature, are discussed.

The properties of the bound-state solutions of the BetheSalpeter equation in the ladder approximation are investigated in the case of potentials strongly singular at the origin. The analogy with potential scattering and the connexion to field theory are discussed. A method is given which allows the reduction of the singular problem to a regular one and explicit solutions for eigenvalue and eigenfunctions are presented in a particular case.

We consider different examples of field theories, superrenormalizable, renormalizable and nonrenormalizable ones, in order to evaluate the renormalization constantZ of the vertex between a bound state and the component particles. In ladder approximation, some properties of theZ’s are examined, with particular regard to the possibility of the cut-off factorization.

Starting from the Callan-Treiman relation between the K→πμν and K→μν amplitudes, we analyse the problem of the extrapolation from the soft-pion limit, where we have the prescription of current algebra, to the physical region for a general vertex function using a sum rule formulation. The simplest analyticity properties are those exhibited in the pion energy at fixed three-momentum. The corresponding dispersive path, in the plane of the squared pion mass and of the momentum transfer, is a parabola depending on the external variables. The advantages of choosing the parabola, of maximum curvature are discussed. We study in particular the case of Kl3 decay and we find three possible independent sum rules, one of which represents a generalization of the Callan-Treiman result. The effect of a possible strange scalar meson is explicitly considered.

Using a dispersive representation for the equal-time commutators of theSU2⊗SU2 algebra we get a sum rule for the axial vector form factorG(Δ2), which can be expressed in terms of the vector form factorF1V(Δ2) and of the imaginary part of the electroproduction amplitude.

We study classical solutions of the meron type in the framework of N = 3 and N = 4 extended supergravity.

We offer a systematic treatment of the connection between stochastic quantization and supersymmetry. This is done in the original Nicolai-Parisi-Sourlas approach and by use of the “stochastic identities” which can be derived as a consequence of the supersymmetry invariance both of the action and of the vacuum. We deal extensively with the cases of quantum mechanics, scalar field theory and gauge theory. One of the main results is the presence of a reduced set of stochastic identities even when a local Nicolai transformation does not exist.

The correspondence between stochastic quantization and supersymmetry is extended to N =1 gauge supersymmetry coupled with matter. This applies to N =2 extended gauge supersymmetry and also to N =1 SUSY in six dimensions. The special role of the light plane gauge is emphasized.

We present a preliminary discussion of some consequences of a fully dilatational invariant formulation of the Einstein theory of gravitation. Arguments are given to show that the small distance behaviour of the theory can be greatly improved, leading to a situation very similar to usual renormalizable theories.

The authors discuss in detail the classical solutions of two field theoretical models invariant under general variable transformations. In particular they examine the case of a Yang-Mills theory and of a four-dimensional nonlinear sigma model, both coupled to 'strong gravitation'. Instanton, meron and multimeron configurations are obtained and their properties discussed. (25 refs).

The validity of the renormalization group equations used to obtain Regge pole behavior is discussed. It is shown that the renormalization group equations contain a certain amount of arbitrariness when a perturbation theory amplitude is used as input. (C.E.S.)