Catálogo de publicaciones - libros

These notes essentially reproduce lectures given at the International School for Advanced Studies (Trieste) and at the Scuola Normale Superiore (Pisa) on various occasions. The scope of the short series of lectures, typically a fraction of a one-semester course, was to explain on general grounds, also to mathematicians, the phenomenon of Spontaneous Symmetry Breaking (SSB), a mechanism which seems at the basis of most of the recent developments in theoretical physics (from Statistical Mechanics to Many-Body theory and to Elementary Particle theory).

The realization of symmetries in physical systems has proven to be of help in the description of physical phenomena: it makes it possible to relate the behaviour of similar systems and therefore it leads to a great simpli.cation of the mathematical description of Nature.

One of the most powerful ideas of modern theoretical physics is the mechanism of spontaneous symmetry breaking. It is at the basis of most of the recent achievements in the description of phase transitions in Statistical Mechanics as well as of collective phenomena in solid state physics. It has also made possible the unification of weak, electromagnetic and strong interactions in elementary particle physics. Philosophically, the idea is very deep and subtle (this is probably why its exploitation is a rather recent achievement) and the popular accounts do not fully do justice to it.

As the previous discussion indicates, it is impossible to realize the phenomenon of (spontaneous) breaking of a continuous symmetry in classical mechanical systems with a finite number of degrees of freedom. We are thus led to consider infinite dimensional systems, like classical fields.

The first basic question is to identify the possible configurations of the systems (3.1), namely the set of initial data for which the time evolution is well defined and which is mapped onto itself by time evolution. In the mathematical language, one has to find the functional space for which the Cauchy problem is well posed. In order to see this, one has to give conditions on ′ ( ) and to specify the class of initial data or, equivalently, the class of solutions one is interested in. Here one faces an apparently technical mathematical problem, which has also deep physical connections.

The mathematical investigation of the existence of solutions for the non-linear (4.6) does not exhaust the problem of the physical interpretation of the corresponding classical field theory. For infinitely extended systems, in general not every solution is physically acceptable; one has to supplement the analysis of the possible solutions by a list of mathematical properties which the solutions must share in order to allow a physical interpretation.

We shall now discuss the requirement III of finite energy-momentum, briefly mentioned in the previous section.

Clearly, the possibility of using solutions of non-linear field equations for the description of physical systems requires that such solutions have finite energy-momentum, and the localization properties of the physical measurements requires the existence of an energy-momentum density.

The existence of sectors, i.e. of “closed worlds” in the set of solutions of the non-linear equation (4.6), provides the mathematical and physical basis for the mechanism of spontaneous symmetry breaking briefly discussed in Chap. 2. We can now understand the relation between the Noether theorem, the existence of conserved currents and the occurrence of spontaneous symmetry breaking which, among other things, imply the lack of existence of the corresponding charges. As shown by the following Proposition, the mechanism of spontaneous symmetry breaking is related to the instability of a closed world under a symmetry operation.

The model describes the simplest non-linear field theory and it can be regarded as a prototype of field theories in one space dimension (=1). The model can also be interpreted as a non-linear generalization of the wave equation. The interest of the model is that, even at the classical level, it has stable solutions with a possible particle interpretation.

The mechanism of SSB does not only provide a general strategy for unifying the description of apparently different systems, but it also provide information on the energy spectrum of an infinite dimensional system, by means of the so-called , according to which to each broken generator of a continuous symmetry there corresponds a massless mode, i.e. a free wave. The quantum version of such a statement has been turned into a theorem, whereas, as far as we know, no analogous theorem has been proved for classical (infinite dimensional) systems and the standard accounts seem to rely on heuristic arguments.