2
Part I: SYMMETRY BREAKING IN CLASSICAL SYSTEMS
explanation identifies the phenomenon with the existence of a degenerate ground (or
equilibrium) state, unstable under the symmetry operation, (ground state asymmetry), a feature often present even in simple mechanical models (as, for example, a
free particle on a plane, each point of which defines a ground state unstable under
translations), but which is usually not accompanied by a non-symmetric behaviour.
As it will be discussed in these lectures, the phenomenon of spontaneous symmetry
breaking in the radical sense of non-symmetric behaviour is rather related to the fact
that, for non-linear infinitely extended systems (therefore involving infinite degrees
of freedom), the solutions of the dynamical problem generically fall into classes
or “islands” or “phases”, each stable under time evolution and characterized by
the same behaviour at infinity of the corresponding solutions. Since all physically
realizable operations have an inevitable localization in space they cannot change such
a behaviour at infinity and therefore starting from the configurations of a given island
one cannot reach the configurations of a different island by physically realizable
modifications. The different islands can then be interpreted as describing physically
disjoint realizations or different phases, or disjoint physical worlds associated with
the given dynamics.
The spontaneous breaking of a symmetry (of the dynamics) in a given phase or
physical world can then be explained as the result of the instability of the given
island under the symmetry operation. In fact, in this case one cannot realize the
symmetry within the given island, namely one cannot operationally associate with
each configuration the one obtained by the symmetry operation.
The existence of such structures is not obvious and in general it involves a mathematical control of the non-linear time evolution of systems with infinite degrees of
freedom and the mathematical formalization of the concept of physical disjointness
of different islands. For quantum systems, where the mathematical basis of SSB has
mostly been discussed, the physical disjointness has been ascribed to the existence
of inequivalent representations of the algebra of local observables.
The scope of Part I of these lectures is to discuss the general mechanism of SSB
within the framework of classical dynamical systems, so that no specific knowledge
of quantum mechanics of infinite systems is needed and the message may also be
suitable for mathematical students. More specifically, the discussion will be based on
the mathematical control of the non-linear evolution of classical fields, with locally
square integrable initial data which may possibly have non-vanishing limits at infinity.
The mathematical formalization of physical disjointness relies on the constraint
of essential localization in space of any physically realizable operation. One can
in fact show that an island can be characterized by some bounded (locally “regular”) reference configuration, having the meaning of the “ground state”, and its H
1
perturbations. Each island is therefore isomorphic to a Hilbert space (Hilbert space
sector).
The stability under time evolution is guaranteed by the condition that the reference
configuration satisfies a generalized stationarity condition, i.e. it solves some elliptic
problem. Such a condition is in particular satisfied by the time-independent solutions
and a fortiori by the minima ¯
ϕ of the potential which define Hilbert space sectors
H ¯
ϕ of the form { ¯
ϕ + χ, χ ∈ H
1
}. The existence of minima of the potential unstable
Part I: SYMMETRY BREAKING IN CLASSICAL SYSTEMS
explanation identifies the phenomenon with the existence of a degenerate ground (or
equilibrium) state, unstable under the symmetry operation, (ground state asymmetry), a feature often present even in simple mechanical models (as, for example, a
free particle on a plane, each point of which defines a ground state unstable under
translations), but which is usually not accompanied by a non-symmetric behaviour.
As it will be discussed in these lectures, the phenomenon of spontaneous symmetry
breaking in the radical sense of non-symmetric behaviour is rather related to the fact
that, for non-linear infinitely extended systems (therefore involving infinite degrees
of freedom), the solutions of the dynamical problem generically fall into classes
or “islands” or “phases”, each stable under time evolution and characterized by
the same behaviour at infinity of the corresponding solutions. Since all physically
realizable operations have an inevitable localization in space they cannot change such
a behaviour at infinity and therefore starting from the configurations of a given island
one cannot reach the configurations of a different island by physically realizable
modifications. The different islands can then be interpreted as describing physically
disjoint realizations or different phases, or disjoint physical worlds associated with
the given dynamics.
The spontaneous breaking of a symmetry (of the dynamics) in a given phase or
physical world can then be explained as the result of the instability of the given
island under the symmetry operation. In fact, in this case one cannot realize the
symmetry within the given island, namely one cannot operationally associate with
each configuration the one obtained by the symmetry operation.
The existence of such structures is not obvious and in general it involves a mathematical control of the non-linear time evolution of systems with infinite degrees of
freedom and the mathematical formalization of the concept of physical disjointness
of different islands. For quantum systems, where the mathematical basis of SSB has
mostly been discussed, the physical disjointness has been ascribed to the existence
of inequivalent representations of the algebra of local observables.
The scope of Part I of these lectures is to discuss the general mechanism of SSB
within the framework of classical dynamical systems, so that no specific knowledge
of quantum mechanics of infinite systems is needed and the message may also be
suitable for mathematical students. More specifically, the discussion will be based on
the mathematical control of the non-linear evolution of classical fields, with locally
square integrable initial data which may possibly have non-vanishing limits at infinity.
The mathematical formalization of physical disjointness relies on the constraint
of essential localization in space of any physically realizable operation. One can
in fact show that an island can be characterized by some bounded (locally “regular”) reference configuration, having the meaning of the “ground state”, and its H
1
perturbations. Each island is therefore isomorphic to a Hilbert space (Hilbert space
sector).
The stability under time evolution is guaranteed by the condition that the reference
configuration satisfies a generalized stationarity condition, i.e. it solves some elliptic
problem. Such a condition is in particular satisfied by the time-independent solutions
and a fortiori by the minima ¯
ϕ of the potential which define Hilbert space sectors
H ¯
ϕ of the form { ¯
ϕ + χ, χ ∈ H
1
}. The existence of minima of the potential unstable