Abstract

Introduction

The degrees of freedom in either the Lagrangian or the Hamiltonian formulation of a singular system are reduced in two stages: reduction from using the natural constraints (with consistency requirements) and reduction through a gauge fixing procedure. In general the number of natural constraints in the Lagrangian formulation differs from the number in the Hamiltonian formulation [1]. In this paper we show how to determine the dynamics and fix the gauge in both formalisms. We believe the Lagrangian discussion is new and useful. The result includes a proof that the final number of degrees of freedom in the two formalisms is the same. Important parts of our methods are the two steps of determining the dynamics and determining the independent initial data.

In section 2 we develop a detailed version of the Hamiltonian gauge fixing procedure. In section 3 we study the Lagrangian gauge fixing procedure and show that the number of degrees of freedom in the Hamiltonian and Lagrangian formalisms are equal.

Abstract

Introduction

The Einstein field equations for space-times that admit a two-dimensional Abelian group of isometries which acts orthogonally and transitively on non-null orbits are non-linear partial differential equations in two variables [Kramer et al. 1980]. Since the pioneering work of Geroch, it has been known that the field equations in the stationary axisymmetric case admit an infinite dimensional group of symmetry transformations [Geroch 1971, 1972]. This result has encouraged the research in solution-generating methods, the main idea being that the complete class of solutions can then be generated from a particular solution, such as flat space [Cosgrove, 1980-1982].

Introduction

The problem of N mutually interacting particles in direct interaction has received much attention in recent years [1].

The problem can be stated in the following form: to find N mass constraints which satisfy the following requirements

1. i) they must be first class constraints in the sense of Dirac [2],[3]

2. ii) they must have the cluster decomposition property (separability), defined as the possibility of splitting the system in subsystems (clusters) by switching off some of the interactions [5],[6],[7] and [8].

The two requirements (i) and (ii) are difficult to satisfy for more than two bodies. In particular the cluster decomposition property or separability requires the presence of genuine N body forces [9],[10] and [11].

In the present paper we start an analysis of the spinless case, with harmonic interaction. We have chosen this simplified case, since the classical solution is well known, in order to have some insight into the difficulties of the problem. A separable model for three fermions has been proposed in ref.[12].

We will show that it is possible to give the form of the first class constraints in the general case, but defined in implicit form. In our intentions this analysis would be the starting point of a relativistic theory of small vibrations about a stable configuration of some general potential.

In a particular configuration, when two over three of the coupling constants are equal, there is a great simplification, particularly if one uses a suitable gauge fixing. The analysis of this case is in progress.

Abstract

Introduction

The issue ‘time’ is perhaps the most crucial one in canonical - especially quantum - gravity and a number of different approaches have been pursued in recent years (see the excellent reviews[1, 2]). Whereas the discussions of general aspects are certainly essential, one might possibly learn a lot by the analysis of a single - even very simple - system for which the corresponding quantisation can be carried through completely and in which the quantity time appears as a classical and quantum gauge invariant ‘observable’.

Such a system is spherically symmetric pure gravity. In view of Birkhoff's theorem which seems to eliminate completely the notion of time for such systems this assertion may appear to be quite surprising.

In the hills above Florence lies Arcetri, where Galileo Galilei spent his years of house arrest after his trial in 1633. Galileo's statement of the law of inertia launched the study of dynamics on the voyage of discovery and invention on which most if not all of modern physics depends. We also remember Galileo when we speak of Galilean relativity, and it was therefore fitting that it was in Arcetri that the workshop on ‘Constraint's theory and relativistic dynamics’ was held in 1986. This workshop was organised by Giorgio Longhi, Luca Lusanna and Giuseppe Marmo “to examine the current situation of relativistic dynamics”, and I had the good fortune to be able to attend, meeting there many of those who were later to become the “Constraints Club”.

A few years later there arose an opportunity to establish this more formally as an association of researchers active in the area of constrained dynamical systems. Under the European Communities' science programme funds were available for “networks”, and an application was duly prepared to support young postdoctoral fellows to work within the “Constraints Club”, which at that time was a rather loosely coordinated group of five laboratories. This application was unsuccessful, but in 1992 an enlarged (and improved!) application was submitted to the Human Capital and Mobility programme, the successor to science. It too failed, but on resubmission in 1993 was at last accepted, and the network in “Constrained Dynamical Systems” came officially into being on 1 January 1994.

Abstract

Introduction

There are two main methods to quantize the Hamiltonian systems with first class constraints: the Dirac quantization [1] and the reduced phase space quantization [2], whereas two other methods, the path integral method [3, 2] and the BRST quantization [4] being the most popular method for the covariant quantization of gauge-invariant systems, are based on and proceed from them [2, 5]. The basic idea of the Dirac method consists in imposing quantum mechanically the first class constraints as operator conditions on the states for singling out the physical ones [1]. The reduced phase space quantization first identifies the physical degrees of freedom at the classical level by the factorization of the constraint surface with respect to the action of the gauge group, generated by the constraints. Then the resulting Hamiltonian system is quantized as a usual unconstrained system [2]. Naturally, the problem of the relationship of these two methods arises. It was discussed in different contexts in literature [6], and there is an opinion that the differences between the two quantization methods can be traced out to a choice of factor ordering in the construction of various physical operators.

Abstract

Introduction

One of the distinctive properties of the non-linear σ-model [1], is that its dynamical variables belong to a non-linear manifold [2], thus realising the corresponding symmetry group in a non-linear fashion [3]. Whence either the Lagrangian becomes non-polynomial in terms of unconstrained variables, or it becomes polynomial but in variables which satisfy a non-linear constraint. It is often convenient to work in a polynomial or ‘linearized’ representation of the model, where the symmetry is linearly realised.

Abstract

Introduction

The geometrical interpretation of W-type symmetries has attracted the attention of many mathematical physicists in recent years. Although a plethora of interesting results are now at our disposal it is commonly agreed that we have not yet a complete understanding of the underlying geometry. It is clear that simple mechanical systems enjoying W symmetry could be an unvaluable tool in this difficult task. On one hand they could provide us with some geometrical and/or physical interpretation for W-transformations (W-morphisms), while on the other hand they could give us some hints about which are the relevant structures associated with W-gravity - the paradigmatic example being provided by the standard relativistic particle and difTeomorphism invariance (W2).

It is well known by now the connection between W-morphisms and the extrinsic geometry of curves and surfaces [1]. Therefore, it seems natural to look for a W-particle candidate among the geometrical actions depending on the extrinsic curvature.

Abstract

Introduction

As is well-known, for over 40 years, quantum field theory has remained in a somewhat peculiar situation. On the one hand, perturbative treatments of realistic field theories in four space-time dimensions have been available for a long time and their predictions are in excellent agreement with experiments. It is clear therefore that there is something “essentially right” about these theories. On the other hand, their mathematical status continues to be dubious in all cases (with interactions), including QED. In particular, it is generally believed that the perturbation series one encounters here can be at best asymptotic. However, it is not clear what exactly they are asymptotic to.

This overall situation is in striking contrast with, for example, non-relativistic quantum mechanics. There, we know well at the outset what the Hilbert space of states is and what the observables are. In physically interesting models, we can generally construct the Hamiltonian operator and show that it is self-adjoint. We take recourse to perturbation theory mainly to calculate its eigenvalues and eigenvectors.

Abstract

Introduction

Vortices, topologically nontrivial localized structures, lie at the heart of all theories of particles with fractional statistics. It is these collective excitations of the field quanta that are considered as candidates for anyonic objects in the quasi-planar condensed matter physics. More precisely, in the case of the charged matter interacting with the Maxwell field, the anyon is a bound state of an (electrically neutral) vortex and a field quantum, a “flux” and a “charge”. If the gauge field is of the Chern-Simons type, the vortex is no more electrically neutral and behaves as an anyon itself.