8
2 Spontaneous Symmetry Breaking
purpose of the following discussion is to make such a rather vague and intuitive
picture more precise.
For a classical finite-dimensional dynamical system, two configurations may be
said to be related by physically realizable operations if there is no physical obstruction
for operationally changing one into the other, e.g. if they are connected by a continuous path of configurations, all with finite energy. In this way, one gets a partition of
the configurations into classes and given a configuration S, the set of configurations
which can be reached from it, by means of physically realizable operations, will be
called the phase Γ S , or the “physical world”, to which S belongs.
A symmetry g will be said to be physically realized (or implementable or unbroken), in the phase Γ , if it leaves Γ stable.
In the mechanical example of the double well potential discussed above, there
is no natural and physically reasonable way of isolating the solutions in the neighbourhoods of the two minima, since an artificial limitation of the available energies
looks rather unphysical. Actually, according to the above definitions, there is only
one phase and the reflection symmetry is physically implementable or unbroken.
In order to further illustrate the above definitions, we consider a particle moving on
a line, subject to a deformed double well potential, still invariant under the reflection
g : q → −q, with two absolute minima at q 0 = ±a, but going to infinity as q → 0.
Consider now two kinds of (one-dimensional) creatures, one living in the valley
with bottom q 0 = a and the other in the valley with bottom q 0 = −a. The infinite
potential barrier prevents going from one valley to the other (tunnelling is impossible); then, e.g. the people living in the r.h.s. valley do not have access to the l.h.s.
valley, neither by action on the initial conditions of the particle nor by time evolution. Thus, the operations which are physically realizable (by each of the two kinds
of people) cannot make the transition from one valley to the other and the particle
configurations get divided into two phases, labelled by the two minima Γ a , Γ −a ,
respectively.
The reflection symmetry is not physically realized in each of the two phases. As a
matter of fact, even if the particle motion is described by a symmetric Hamiltonian, the
particle physical world will look asymmetric to each kind of creatures: the symmetry
is spontaneously broken.
The somewhat artificial example of spontaneous symmetry breaking discussed
above is made possible by the infinite potential barrier between the two absolute
minima. Clearly, such a mechanism is not available in the case of a continuous
symmetry, since then the (absolute) minima are continuously related by the symmetry
group and no potential barrier can occur between them (for a concrete example see the
two-dimensional double well discussed above). Thus, for finite-dimensional classical
dynamical systems, a continuous symmetry of the Hamiltonian is always unbroken
(even if the ground state is degenerate and non-symmetric).
The often-quoted example of a particle in a two-dimensional double well potential
is a somewhat misleading example of spontaneous breaking of continuous symmetry
(it is also an incorrect example in one dimension, unless the potential is so deformed
to produce an an infinite barrier between the two minima). Actually, most of the
2 Spontaneous Symmetry Breaking
purpose of the following discussion is to make such a rather vague and intuitive
picture more precise.
For a classical finite-dimensional dynamical system, two configurations may be
said to be related by physically realizable operations if there is no physical obstruction
for operationally changing one into the other, e.g. if they are connected by a continuous path of configurations, all with finite energy. In this way, one gets a partition of
the configurations into classes and given a configuration S, the set of configurations
which can be reached from it, by means of physically realizable operations, will be
called the phase Γ S , or the “physical world”, to which S belongs.
A symmetry g will be said to be physically realized (or implementable or unbroken), in the phase Γ , if it leaves Γ stable.
In the mechanical example of the double well potential discussed above, there
is no natural and physically reasonable way of isolating the solutions in the neighbourhoods of the two minima, since an artificial limitation of the available energies
looks rather unphysical. Actually, according to the above definitions, there is only
one phase and the reflection symmetry is physically implementable or unbroken.
In order to further illustrate the above definitions, we consider a particle moving on
a line, subject to a deformed double well potential, still invariant under the reflection
g : q → −q, with two absolute minima at q 0 = ±a, but going to infinity as q → 0.
Consider now two kinds of (one-dimensional) creatures, one living in the valley
with bottom q 0 = a and the other in the valley with bottom q 0 = −a. The infinite
potential barrier prevents going from one valley to the other (tunnelling is impossible); then, e.g. the people living in the r.h.s. valley do not have access to the l.h.s.
valley, neither by action on the initial conditions of the particle nor by time evolution. Thus, the operations which are physically realizable (by each of the two kinds
of people) cannot make the transition from one valley to the other and the particle
configurations get divided into two phases, labelled by the two minima Γ a , Γ −a ,
respectively.
The reflection symmetry is not physically realized in each of the two phases. As a
matter of fact, even if the particle motion is described by a symmetric Hamiltonian, the
particle physical world will look asymmetric to each kind of creatures: the symmetry
is spontaneously broken.
The somewhat artificial example of spontaneous symmetry breaking discussed
above is made possible by the infinite potential barrier between the two absolute
minima. Clearly, such a mechanism is not available in the case of a continuous
symmetry, since then the (absolute) minima are continuously related by the symmetry
group and no potential barrier can occur between them (for a concrete example see the
two-dimensional double well discussed above). Thus, for finite-dimensional classical
dynamical systems, a continuous symmetry of the Hamiltonian is always unbroken
(even if the ground state is degenerate and non-symmetric).
The often-quoted example of a particle in a two-dimensional double well potential
is a somewhat misleading example of spontaneous breaking of continuous symmetry
(it is also an incorrect example in one dimension, unless the potential is so deformed
to produce an an infinite barrier between the two minima). Actually, most of the