Part I
SYMMETRY BREAKING
IN CLASSICAL SYSTEMS
Introduction
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).
Besides its extraordinary success, the idea of SSB also deserves being discussed
because of its innovative philosophical content, and in our opinion it should be part of
the background knowledge for mathematical and theoretical physics students, especially those who are interested in questions of principle and in general mathematical
structures.
By the general wisdom of Classical Mechanics, codified in the classical Noether
theorem, one learns that the symmetries of the Hamiltonian or of the Lagrangian
are automatically symmetries of the physical system described by it, which does not
mean that the (equilibrium) solutions are symmetric, but rather that the symmetry
transformation commutes with time evolution and hence is a symmetry of the physical
behaviour of the system. This belief therefore precludes the possibility of describing
systems with different dynamical properties in terms of the same Hamiltonian. The
realization that this obstruction does not a priori exist and that one may unify the
description of apparently different systems in terms of a single Hamiltonian and
account for the different behaviours by the mechanism of SSB, is a real revolution in
the way of thinking in terms of symmetries and corresponding properties of physical
systems. It is, in fact, non-trivial to understand how the conclusions of the Noether
theorem can be evaded and how a symmetry of the dynamics cannot be realized as
a mapping of the physical configurations of the system, which commutes with the
time evolution.
The standard folklore explanations of SSB, which one often finds in the literature,
is partly misleading, because it does not emphasize the crucial ingredient underlying
the phenomenon, namely the role of infinite degrees of freedom. Despite the many
popular accounts, the phenomenon of SSB is deep and subtle and it is not without
reasons that it has been fully understood only in recent times. The standard cheap
SYMMETRY BREAKING
IN CLASSICAL SYSTEMS
Introduction
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).
Besides its extraordinary success, the idea of SSB also deserves being discussed
because of its innovative philosophical content, and in our opinion it should be part of
the background knowledge for mathematical and theoretical physics students, especially those who are interested in questions of principle and in general mathematical
structures.
By the general wisdom of Classical Mechanics, codified in the classical Noether
theorem, one learns that the symmetries of the Hamiltonian or of the Lagrangian
are automatically symmetries of the physical system described by it, which does not
mean that the (equilibrium) solutions are symmetric, but rather that the symmetry
transformation commutes with time evolution and hence is a symmetry of the physical
behaviour of the system. This belief therefore precludes the possibility of describing
systems with different dynamical properties in terms of the same Hamiltonian. The
realization that this obstruction does not a priori exist and that one may unify the
description of apparently different systems in terms of a single Hamiltonian and
account for the different behaviours by the mechanism of SSB, is a real revolution in
the way of thinking in terms of symmetries and corresponding properties of physical
systems. It is, in fact, non-trivial to understand how the conclusions of the Noether
theorem can be evaded and how a symmetry of the dynamics cannot be realized as
a mapping of the physical configurations of the system, which commutes with the
time evolution.
The standard folklore explanations of SSB, which one often finds in the literature,
is partly misleading, because it does not emphasize the crucial ingredient underlying
the phenomenon, namely the role of infinite degrees of freedom. Despite the many
popular accounts, the phenomenon of SSB is deep and subtle and it is not without
reasons that it has been fully understood only in recent times. The standard cheap