70
Part II: SYMMETRY BREAKING IN QUANTUM SYSTEMS
second quantization. As emphasized by Segal, Haag, Kastler and others, it is more
convenient, logically more economical and actually more general to formulate the
principles of quantum mechanics in terms of (the algebra of) observables and states
as positive linear functionals or expectations on the observables. This covers both
the case of finite degrees of freedom, where the von Neumann theorem selects the
Schrödinger representation in an essentially unique way, and the case of infinite
degrees of freedom, for which even the Fock representation is generically forbidden
(apart from the free field case).
These notes focus the attention on the mechanism of spontaneous symmetry breaking (SSB). It seems fair to say that the realization of such a possibility represented a
real breakthrough in the development of theoretical physics. In fact, it is at the basis
of most of the recent achievements in Many-Body Theory and in Elementary Particle
Theory.
In spite of the cheap explanations, the phenomenon of SSB is deep and subtle and
crucially involves the occurrence of infinite degrees of freedom. From elementary
quantum mechanics, one learns that the symmetries of the Hamiltonian are symmetries of the physical description of the system, which does not mean that the ground
state is symmetric, but rather that the symmetry transformations commute with the
time evolution. Thus, whenever the symmetry can be implemented by a (physically
realizable) correspondence between the states of the systems, no symmetry breaking
can be observed.
The way out of this obstruction is the realization that for infinitely extended systems, the algebra of observables, which define a given system, and its time evolution
do not select a unique realization of the system, but rather one has more than one
“physical world” or (infinite volume thermodynamical) “phase”, which are physically disjoints in the sense that no physically realizable operation can lead from
one to the other. Technically this corresponds to the existence of inequivalent representations of the algebra of observables. The occurrence of spontaneous symmetry
breaking in a given world is then related to its instability with respect to the symmetry
transformations. Thus, the lack of symmetry is due to the impossibility of comparing
the properties of a state with those of its transformed one, since the latter belongs
to a physically disjoint world. The necessary localization in space (and time) of any
physically realizable operation and the infinite extension of the system are crucial
ingredients for such a phenomenon.
The occurrence of inequivalent representations of the algebra of canonical variables or more generally of observables, for systems with infinite degrees of freedom
(briefly infinite systems), is briefly reviewed in Chaps. 11–13.
A general formulation of quantum mechanics of infinitely extended systems is
made possible by exploiting the localization properties of the algebra of canonical
variables or of observables. As emphasized by Haag, the local structure is the key
property and together with the related asymptotic abelianess and cluster property
plays a crucial role for the identification of the physically relevant representations
and of the “pure” phases. A clear discussion of spontaneous symmetry breaking
could not be done without the realization of these points (Chaps. 14–17).
Part II: SYMMETRY BREAKING IN QUANTUM SYSTEMS
second quantization. As emphasized by Segal, Haag, Kastler and others, it is more
convenient, logically more economical and actually more general to formulate the
principles of quantum mechanics in terms of (the algebra of) observables and states
as positive linear functionals or expectations on the observables. This covers both
the case of finite degrees of freedom, where the von Neumann theorem selects the
Schrödinger representation in an essentially unique way, and the case of infinite
degrees of freedom, for which even the Fock representation is generically forbidden
(apart from the free field case).
These notes focus the attention on the mechanism of spontaneous symmetry breaking (SSB). It seems fair to say that the realization of such a possibility represented a
real breakthrough in the development of theoretical physics. In fact, it is at the basis
of most of the recent achievements in Many-Body Theory and in Elementary Particle
Theory.
In spite of the cheap explanations, the phenomenon of SSB is deep and subtle and
crucially involves the occurrence of infinite degrees of freedom. From elementary
quantum mechanics, one learns that the symmetries of the Hamiltonian are symmetries of the physical description of the system, which does not mean that the ground
state is symmetric, but rather that the symmetry transformations commute with the
time evolution. Thus, whenever the symmetry can be implemented by a (physically
realizable) correspondence between the states of the systems, no symmetry breaking
can be observed.
The way out of this obstruction is the realization that for infinitely extended systems, the algebra of observables, which define a given system, and its time evolution
do not select a unique realization of the system, but rather one has more than one
“physical world” or (infinite volume thermodynamical) “phase”, which are physically disjoints in the sense that no physically realizable operation can lead from
one to the other. Technically this corresponds to the existence of inequivalent representations of the algebra of observables. The occurrence of spontaneous symmetry
breaking in a given world is then related to its instability with respect to the symmetry
transformations. Thus, the lack of symmetry is due to the impossibility of comparing
the properties of a state with those of its transformed one, since the latter belongs
to a physically disjoint world. The necessary localization in space (and time) of any
physically realizable operation and the infinite extension of the system are crucial
ingredients for such a phenomenon.
The occurrence of inequivalent representations of the algebra of canonical variables or more generally of observables, for systems with infinite degrees of freedom
(briefly infinite systems), is briefly reviewed in Chaps. 11–13.
A general formulation of quantum mechanics of infinitely extended systems is
made possible by exploiting the localization properties of the algebra of canonical
variables or of observables. As emphasized by Haag, the local structure is the key
property and together with the related asymptotic abelianess and cluster property
plays a crucial role for the identification of the physically relevant representations
and of the “pure” phases. A clear discussion of spontaneous symmetry breaking
could not be done without the realization of these points (Chaps. 14–17).