Section A: Physique theorique.

Conservation laws

by

A. JADCZYK (**)

lnstitut fuer Theoretische Physik der Universitaet Goettingen,

Bunsenstrasse 9, D 3400 Goettingen

RESUME. On obtient par la methode de Souriau des equations de mouvement pour des distributions singulieres de matiere, telles que des particules ponctuelles, des cordes, des membranes et des sacs. On incorpore des interactions avec le tenseur metrique, avec des champs de jauge non abeliens et des G-structures. On examine aussi le cas de particules portant des charges spinorielles dans un champ de supergravite.

1. INTRODUCTION

It is well known that the geodesic principle of general relativity can be derived from energy-momentum conservation, the latter being in turn a consequence of general covariance of the theory. Some authors (see e. g. [1] [2]) claim that also the equations of motion of thc Nambu string can be derived by this method. We study singular distributions of matter, like point particles, strings, membranes, bags, etc. in a field of geometrical objects like metric tensor, gauge fields, G-structures. Instead of deriving relevant equations from conservation laws we follow a much simpler method of Souriau [3] which allows us to proceed directly from invariance to equations of motion. Nevertheless we have found it more convenient to change philosophy and appeal to Aristotel's Golden Rule of Mechanics rather than to "general covariance" . After formulation of a general framework in Sec. 2 we proceed to consider motions of charged singular distributions of matter in gravitational and non-Abelian gauge fields. For l-dimensional distributions we get equations of Kerner and Wong [4] [5] [6] [7], and for 2-dimensional ones our equations contain those derived by Nielsen [8] from an action principle. In fact, as is discussed in Sec. 3 and 4, Nielsen's equations are stronger than ours since they specify internal energy-momentum tensor of the string in terms of its geometry and its current. Our analysis is to be compared with that givcn in [1] [2] where (apart of the fact that we include gauge fields not discussed there) the authors overlooked the fact that conservation laws do not determine string's dynamics unless its internal dynamics is specified so that Cauchy data's constraints become explicit and an appropriate phase space can be defined.

In Sec. 6 we discuss a wide class of theories where geometry is described in terms of a G-structure. It is found that a possibility of deriving a full dynamics from conservation laws, even for point particles, depends on the group G. Orthogonal groups are the best in this respect what distinguishes field theories based on multi-dimensional Riemannian geometries (endowed with any set of covariant constraints like e. g. Kaluza-Klein theories). Supergravity [9], considered as a constrained Lorentz structure on super-manifold, seems to have too poor a structure group to give deterministic equations of motion for a point particle endowed with mass and spinorial charge. Much better in this respect is metric supergravity [10] [11] although it may cause some other problems [12].

2. The Golden Rule
3. Gauge geometries
4. Comparison with Kaluza-Klein approach
5. G-structures