Chapter 2
Spontaneous Symmetry Breaking
One of the most powerful ideas of modern theoretical physics is the mechanism of
spontaneous symmetry breaking. It is at the basis of most of the recent achievements
in the description of phase transitions in Statistical Mechanics as well as of collective
phenomena in solid state physics. It has also made possible the unification of weak,
electromagnetic and strong interactions in elementary particle physics. Philosophically, the idea is very deep and subtle (this is probably why its exploitation is a rather
recent achievement) and the popular accounts do not fully do justice to it.
Roughly, spontaneous symmetry breaking is said to occur when a symmetry of
the Hamiltonian, which governs the dynamics of a physical system, does not lead
to a symmetric description of the physical properties of the system. At first sight,
this may look almost paradoxical. From elementary courses on mechanical systems,
one learns that the symmetries of a system are displayed by the symmetries of the
Hamiltonian, which describes its time evolution; how can it then be that a symmetric
Hamiltonian gives rise to an asymmetric physical description of a dynamical system?
The cheap standard explanation is that such a phenomenon is due to the existence
of a non-symmetric absolute minimum or “ground state”, but the mechanism must
have a deeper explanation, since the symmetry of the Hamiltonian implies that an
asymmetric stable point cannot occur by alone, (the action of the symmetry on it will
produce another stable point). Now, the existence of a set of absolute minima related
by a symmetry (or “degenerate ground states”) does not imply a non-symmetric physical description. One actually gets a symmetric picture, if the correct correspondence
is made between the configurations of the system (and their time evolutions), and
such a correspondence is physically implementable if for any physically realizable
configuration its transformed one is also realizable.
The way out of this argument is to envisage a mechanism by which, given a nonsymmetric absolute minimum (or “ground” state) S 0 , there are physical obstructions
to reach its transformed one, g S 0 , by means of physically realizable operations, so
that effectively one gets confined to an asymmetric realization of the system. The
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_2
7
Spontaneous Symmetry Breaking
One of the most powerful ideas of modern theoretical physics is the mechanism of
spontaneous symmetry breaking. It is at the basis of most of the recent achievements
in the description of phase transitions in Statistical Mechanics as well as of collective
phenomena in solid state physics. It has also made possible the unification of weak,
electromagnetic and strong interactions in elementary particle physics. Philosophically, the idea is very deep and subtle (this is probably why its exploitation is a rather
recent achievement) and the popular accounts do not fully do justice to it.
Roughly, spontaneous symmetry breaking is said to occur when a symmetry of
the Hamiltonian, which governs the dynamics of a physical system, does not lead
to a symmetric description of the physical properties of the system. At first sight,
this may look almost paradoxical. From elementary courses on mechanical systems,
one learns that the symmetries of a system are displayed by the symmetries of the
Hamiltonian, which describes its time evolution; how can it then be that a symmetric
Hamiltonian gives rise to an asymmetric physical description of a dynamical system?
The cheap standard explanation is that such a phenomenon is due to the existence
of a non-symmetric absolute minimum or “ground state”, but the mechanism must
have a deeper explanation, since the symmetry of the Hamiltonian implies that an
asymmetric stable point cannot occur by alone, (the action of the symmetry on it will
produce another stable point). Now, the existence of a set of absolute minima related
by a symmetry (or “degenerate ground states”) does not imply a non-symmetric physical description. One actually gets a symmetric picture, if the correct correspondence
is made between the configurations of the system (and their time evolutions), and
such a correspondence is physically implementable if for any physically realizable
configuration its transformed one is also realizable.
The way out of this argument is to envisage a mechanism by which, given a nonsymmetric absolute minimum (or “ground” state) S 0 , there are physical obstructions
to reach its transformed one, g S 0 , by means of physically realizable operations, so
that effectively one gets confined to an asymmetric realization of the system. The
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_2
7