Emanuele Sorace

Three‐dimensional quantum groups from contractions of SU(2)q

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

The quantum Heisenberg group H(1)q

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

The three‐dimensional Euclidean quantum group E(3)q and its R‐matrix

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three‐dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R‐matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R‐matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two‐dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

Quantized Grassmann variables and unified theories

Abstract The color and flavor degrees of freedom are described in terms of Fermi oscillators (quantized Grassmann variables). The unified theories constructed in this way are vector-like. The fundamental fermions come out to be classified in the spinorial representations of the orthogonal groups.

States of the Dirac equation in confining potentials.

We study the Dirac equation in confining potentials with pure vector coupling, proving the existence of metastable states with longer and longer lifetimes as the nonrelativistic limit is approached and eventually merging with continuity into the Schrödinger bound states. The existence of these states could concern high energy models and possible resonant scattering effects in systems like graphene. We present numerical results for the linear and the harmonic cases and we show that the density of the states of the continuous spectrum is well described by a sum of Breit-Wigner lines. The width of the line with lowest positive energy well reproduces the Schwinger pair production rate for a linear potential: this gives an explanation of the Klein paradox for bound states and a new concrete way to get information on pair production in unbounded, nonuniform electric fields, where very little is known.

Relativistic action at a distance through singular Lagrangians with multiplicative potentials and its relation to the nonrelativistic two-body problem

SummaryRecent models of relativistic action at a distance through singular Lagrangians with multiplicative potentials, describing two-point bound states, are re-examined. They are reformulated in such a way to be well suited to the study of extended bodies; we introduce a set of vierbeins, attached to the barycentric co-ordinates, which connect the Minkowski space with an inner relative space, and we define new relative co-ordinates in it. By using the irreducible representation theory of the Poincaré group, we show that this relative space is the natural relativistic generalization of the nonrelativistic relative one. The nonrelativistic limit of these models is exhibited, by recovering the Newtonian two-body problem with central forces.RiassuntoSi riesaminano alcuni recenti modelli di azione a distanza relativistici, che descrivono stati legati a due particelle partendo da lagrangiane singolari con potenziali moltiplicativi. Essi sono riformulati con un metodo suggerito dalla trattazione dei sistemi estesi e orientato alla loro descrizione. A tal fine si introduce un sistema di tetradi, attaccato al baricentro, che connette lo spazio di Minkowski a uno spazio relativo interno, ed in esso si definiscono nuove coordinate relative. Usando la teoria delle rappresentazioni irriducibili del gruppo di Poincaré, si mostra che questo spazio relativo è la naturale generalizzazione relativistica dello spazio relativo non relativistico. Si valuta inoltre il limite non relativistico di tali modelli, ritrovando, come atteso, il problema a due corpi con forze centrali della meccanica di Newton.РезюмеЗаново исследуются недавние модели релятивистского действия на расстоянии через сингулярные Лагранжианы с мультипликативными потенциалами, описывающими двухточечные связанные состояния. Модели переформулируются в виде удобном для изучения протяженных тел. Мы связываем пространство Минковского с внутренним относительным пространством. Мы определяем новые относительные координаты в этом пространстве. Используя теорию неприводимых представлений группы Пуанкаре, мы показываем, что это относительное пространство представляет естественное релятивистское обобщение нерелятивистского относительного пространства. Рассматривается нерелятивистский предел этих моделей, который соответствует ньютоновской проблеме двух тел с центральными силами.

Tetrad fields and metric tensor in the gauge theory of gravitation

The most relevant geometrical aspects of the gauge theory of gravitation are considered. A global definition of the tetrad fields is given and emphasis is placed on their role in defining an isomorphism between the tangent bundle of space-time and an appropriate vector bundle B associated to the gauge bundle. It is finally shown how to construct the fundamental geometrical objects on space-time, starting from B.

An independence theorem for Lagrangian systems on graded manifolds

Given a Lagrangian system on a graded manifold, we prove that the invariance of the action under independent reparametrizations of two subsystems implies the dynamical independence of those sybsystems.

Hamiltonian formulation for the gauge theory of the gravitational coupling

This paper studies the Hamiltonian mechanics of a relativistic particle interacting with a gravitational field considered as a gauge field of the Poincare group. We follow a general method developed by Sternberg for the case of internal symmetries, that describes the interaction by a suitable modification of the symplectic form. This approach is reviewed and the explicit examples of the electromagnetic and Yang–Mills gauge interactions are widely explained in local coordinates. The peculiar features of a gauge theory of the Poincare group are then discussed and the geometrical picture that emerges suggests the way of modifying the symplectic form for a correct description of the gravitational coupling.

A classical particle in a Poincar gauge field as a model for the gravitational interaction

The geometrical and mechanical aspects of a particle interacting with a Poincaré gauge field are considered and the relation with a gravitational interaction is studied.