G. Longhi

Dynamics of Carroll particles

We investigate particles whose dynamics are invariant under the Carroll group. Although a single, free such Carroll particle has no non-trivial dynamics (the Carroll particle does not move), we show that non-trivial dynamics exists for a set of interacting Carroll particles. Furthermore, we gauge the Carroll algebra and couple the Carroll particle to these gauge fields. It turns out that for such a coupled system, even a single Carroll particle can have non-trivial dynamics.

A Lagrangian for two interacting relativistic particles: Canonical formulation

SummaryThe canonical formalism of a singular-Lagrangian model describing the interaction between two relativistic particles is studied. Instead of following the Dirac method, we make use of a canonical transformation that enables us to work in the complete phase space. The covariance and the quantization of the model are discussed.RiassuntoSi studia il formalismo canonico di un modello con lagrangiana singolare che descrive l’interazione di due particelle relativistiche. Invece di seguire il metodo di Dirac, si fa uso di una trasformazione canonica che ci permette di lavorare nello spazio delle fasi completo. La covarianza e la quantizzazione del modello sono studiate.РезюмеИсследуется канонический формализм для модели с сингулярным Лагранжианом, которая описывает взаимодействие между двумя редятивистскими частицами. Вместо того, чтобы следовать методу Дирака, мы используем каноническое преобразование, которое позволяет нам работать в фазовом пространстве. Обсуждаются ковариантность и квантование предложенной модели.

A Lagrangian for two interacting relativistic particles

SummaryIt is discussed a classical model for the interaction of two relativistic particles via an action at a distance. The model is based on a singular Lagrangian with multiplicative potential which allows us to discuss a scattering process as well as a bound state. It is discussed in some detail the case of a finite-range potential with respect to the possibility to get continuous solutions without limitations on the initial data.RiassuntoIn questo lavoro si studia un modello classico per l–interazione di due particelle relativistiche tramite un–azione a distanza. Il modello è derivato da una lagrangiana singolare con potenziale moltiplicativo che permette di discutere sia un processo di scattering che un processo di stato legato. Si discute il caso di un Potenziale a range finito, in particolare per quanto riguarda la possibilità di ottenere soluzioni continue senza limitazioni sui dati iniziali.РезюмеОбсуждается классическая модель для взаимодействия двух релятивистских частиц через действие на расстоянии. Предложенная модель основана на сингулярном Лагранжиане с мультипликативным потенциалом, который позволяет исследовать процесс рассеяния, а также связанное состояние. Подробно обсуждается случай потенциала с конечным радиусом действия относительно возможности получения непрерывных решений без ограничений на начальные данные.

The relativistic point revisited in the light of the string model

SummaryThe quantum mechanics of the relativistic material point is discussed in the light of some recent ideas, developed for the study of the interacting string model. In accordance with these ideas the interaction among the particles is introduced in a geometrical way, but we also show how to interpret such an interaction as due to an external field. The same considerations are briefly applied to the string model.RiassuntoNel presente lavoro si discute la meccanica quantistica del punto materiale relativistico, alla luce di alcune idee recentemente sviluppate in connessione con lo studio del modello della corda relativistica interagente. In accordo con queste idee si introduce in modo geometrico l’interazione tra le varie particelle, ma si mostra anche che può essere reinterpretata come dovuta ad un campo esterno. Le stesse considerazioni si applicano brevemente al modello della corda relativistica.РеэюмеОбсуждается квантовая механика релятивистской материальной точки в свете недавних идей, предложенных для исследования модели вэаимодействуюшей струны. В соответствии с зтими идеями, вэаимодействие между частицами вводится геометрическим обраэом. Также покаэывается, как интерпретировать такое вэаимодействие в виде внещнего поля. Укаэанные рассуждения применяются к модели струны.

Space-time vector supersymmetry and massive spinning particle

We construct the action of a relativistic spinning particle from a non-linear realization of a space-time odd vector extension of the Poincare group. For particular values of the parameters appearing in the lagrangian the model has a gauge world-line supersymmetry. As a consequence of this local symmetry there are BPS solutions in the model preserving 1/5 of the supersymmetries. A supersymmetric invariant quantization produces two decoupled 4d Dirac equations.

A possible approach to the two-body relativistic problem

SummaryIn this paper a model which describes a relativistic interaction between two point particles via an action at a distance is derived from a set of hypotheses on the relativistic dynamics. From this set of hypotheses a singular Lagrangian is obtained. The aim of this paper is to find a link between the singular-Lagrangian approach and other approaches to the relativistic dynamics of two particles. The connection of this Lagrangian model with the predictive approach of the relativistic mechanics is studied, by showing that it is possible to calculate the instantaneous forces, at least in principle. An explicit canonical transformation is given, such that a subset of the new canonical variables becomes free of constraints. In this way the instant form of the relativistic dynamics found by Bakamjan and Thomas and by Foldy is recovered.RiassuntoIn questo lavoro si deriva, da un insieme di ipotesi sulla dinamica relativistica, un modello che descrive un’interazione tra due particelle puntiformi relativistiche tramite un’azione a distanza. Da questo insieme di ipotesi si ottiene una lagrangiana singolare. Lo scopo di questo lavoro è di trovare un legame tra lo schema delle lagrangiane singolari e altri schemi di approccio alla dinamica relativistica di due particelle. La connessione di questo modello con lo schema predittivo è studiata dimostrando che è possibile, almeno in via di principio, calcolare le forze istantanee. Si recupera inoltre la forma istante della dinamica relativistica di Bakamjan e Thomas e di Foldy per mezzo di una trasformazione canonica tale che un sottoinsieme delle nuove variabili canoniche è libero da vincoli.РеэюмеВ зтой статье иэ системы гипотеэ относительно релятивистской динамики выводится модель, которая описывает релятивистское вэаимодействие между двумя точечными частицами череэ действие на расстоянии. Иэ зтой системы гипотеэ получается сингулярный Лагранжиан. Цель зтой статьи — найти свяэь между подходом с испольэованием сингулярного Лагранжиана и другими подходами к релятивистской динамике двух частиц. Исследуется свяэь зтой модели Лагранжиана с предскаэываюшим подходом в релятивистское механике, отмечая, что можно вычислить, по крайней мере, в принципе, мгновенные силы. Приводится такое явное каноническое преобраэование, что подмножество новых канонических переменных становится свободными от ограничений. Таким способом восстанавливается мгновенная форма релятивистской динамики, найденная Бакамяном и Томасом и Фолди.

Canonical transformations and quantization of singular Lagrangian systems

SummaryIn this work we propose a method of quantization for models described by singular Lagrangians. A set of variables, suitable for the quantization, is obtained by means of a canonical transformation. The quantum theory is carried out by working in the complete phase space; in other words, we do not eliminate the variables corresponding to the second-class constraints, as suggested by Dirac. The expectation values of the constraints are required to be zero and the problem of the consistency of these conditions is studied.RiassuntoIn questo lavoro si espone un metodo di quantizzazione per modelli descritti da lagrangiane singolari. Mediante una trasformazione canonica si ottiene un gruppo di variabili appropriate per la quantizzazione; questa è eseguita nel nuovo spazio delle fasi complete: non vengono cioè eliminate le variabili corrispondenti ai vincoli di seconda classe, come suggerito da Dirac. Si richiede che siano nulli i valori di aspettazione dei vincoli e si studia il problema della consistenza di tali condizioni.

On the second quantization of a composite model for nonhadrons

SummaryThe field-theoretical version of a previously proposed composite model for nonhadrons is presented. Since the model describes an integer-spin tower of massless particles, by selecting the state of helicity one a nonlocal description of the photon may be got. The problem of the interaction of the system with itself is discussed, and it is shown how to construct the interaction Lagrangian in order to reproduce the results obtained in a previous work.RiassuntoSi presenta la seconda quantizzazione di un modello composto per non hadroni precedentemente proposto. Il modello descrive una torre di stati a spin intero senza massa; selezionando lo stato di elicità uno si ottiene una descrizione non locale del fotone. Si discute inoltre l’interazione del sistema con se stesso e si mostra come costruire una lagrangiana d’interazione per riprodurre i risultati trovati in (1).РеэюмеРассматривается вариант ранее предложенной составной модели адронов. Так как зта мпдель описывает бащню частиц с нулевой массой и с целыми спинами, то выбирая состояние со спиральностью единица, может быть получено нелокальное описание фотона. Обсуждается проблема собственного вэаимодействия системы. Покаэывается, как следует сконструировать лагранжиан вэаимодействия, чтобы воспроиэвести полученные реэультаты.

Canonical realization of Bondi-Metzner-Sachs symmetry: Quadratic Casimir

We study the canonical realization of Bondi-Metzner-Sacks symmetry for a massive scalar field introduced by Longhi and Materassi [J. Math. Phys. 40, 480 (1999)]. We construct an invariant scalar product for the generalized momenta. As a consequence we introduce a quadratic Casimir with the supertranslations.

Dynamical sectors of a relativistic two particle model

We reconsider a model of two relativistic particles interacting via a multiplicative potential, as an example of a simple dynamical system with sectors, or branches, with different dynamics and degrees of freedom.The presence or absence of sectors depends on the values of rest masses. Some aspects of the canonical quantization are described. The model could be interpreted as a bigravity model in one dimension.