Part II
SYMMETRY BREAKING
IN QUANTUM SYSTEMS
Introduction
These notes arose from courses given at the International School for Advanced Studies (Trieste) and at the Scuola Normale Superiore (Pisa) in various years, with the
purpose of discussing the structural features and collective effects which distinguish
the quantum mechanics of systems with infinite degrees of freedom from ordinary
quantum mechanics.
The motivations for considering systems with infinite degrees of freedom are many.
Historically, the first and one of the most important ones came from the problem of
describing particle interactions consistently with the principles of special relativity.
As it is well known, the concept of force as “action at a distance” between particles
involves the concept of simultaneity and it does not fit into the framework of relativity,
unless one is ready to accept highly non-local actions. This is the reason why so far
special relativity has provided a beautiful kinematics but no relativistically invariant
theory of (classical) particle (action at a distance) interactions. The transmission of
energy and momentum by local (or contact) actions leads to the concept of “medium”
or field as the carrier of the transmitted energy and momentum and therefore to a
system with infinite degrees of freedom.
Another important class of physical phenomena, whose description involves infinite degrees of freedom, are those related to the bulk properties of matter. In fact,
the intensive properties of systems consisting of a large number N ∼ 10
27 of constituents are largely independent of N and of the occupied volume V , for given fixed
density n = N /V ; therefore, their description greatly simplifies by taking the socalled thermodynamical limit N → ∞, V → ∞ with n fixed. In this way one passes
to the limit of infinite degrees of freedom. Collective phenomena, phase transitions,
thermodynamical properties, etc., could hardly have a simple treatment without such
a limit.
The quantization of systems with infinite degrees of freedom started being investigated soon after the birth of quantum mechanics and it was soon realized that new
theoretical structures were involved. In particular, the states of an infinite system
cannot be described by a single wave function (of an infinite number of variables) as
in ordinary quantum mechanics, i.e. the standard Schrödinger representation is not
possible. The changes involved were regarded so substantial to deserve the name of
SYMMETRY BREAKING
IN QUANTUM SYSTEMS
Introduction
These notes arose from courses given at the International School for Advanced Studies (Trieste) and at the Scuola Normale Superiore (Pisa) in various years, with the
purpose of discussing the structural features and collective effects which distinguish
the quantum mechanics of systems with infinite degrees of freedom from ordinary
quantum mechanics.
The motivations for considering systems with infinite degrees of freedom are many.
Historically, the first and one of the most important ones came from the problem of
describing particle interactions consistently with the principles of special relativity.
As it is well known, the concept of force as “action at a distance” between particles
involves the concept of simultaneity and it does not fit into the framework of relativity,
unless one is ready to accept highly non-local actions. This is the reason why so far
special relativity has provided a beautiful kinematics but no relativistically invariant
theory of (classical) particle (action at a distance) interactions. The transmission of
energy and momentum by local (or contact) actions leads to the concept of “medium”
or field as the carrier of the transmitted energy and momentum and therefore to a
system with infinite degrees of freedom.
Another important class of physical phenomena, whose description involves infinite degrees of freedom, are those related to the bulk properties of matter. In fact,
the intensive properties of systems consisting of a large number N ∼ 10
27 of constituents are largely independent of N and of the occupied volume V , for given fixed
density n = N /V ; therefore, their description greatly simplifies by taking the socalled thermodynamical limit N → ∞, V → ∞ with n fixed. In this way one passes
to the limit of infinite degrees of freedom. Collective phenomena, phase transitions,
thermodynamical properties, etc., could hardly have a simple treatment without such
a limit.
The quantization of systems with infinite degrees of freedom started being investigated soon after the birth of quantum mechanics and it was soon realized that new
theoretical structures were involved. In particular, the states of an infinite system
cannot be described by a single wave function (of an infinite number of variables) as
in ordinary quantum mechanics, i.e. the standard Schrödinger representation is not
possible. The changes involved were regarded so substantial to deserve the name of