2 Spontaneous Symmetry Breaking
9
claimed simple mechanical examples of spontaneous symmetry breaking discussed
in the literature are equally misleading.
Even if the existence of non-symmetric minima is a rather peculiar phenomenon
which deserves special interest, it does not imply spontaneous symmetry breaking
in the radical sense of its realization in elementary particle physics, many-body
systems, statistical mechanics, etc., where a symmetry of the dynamics is not shared
by physical realizations or disjoint phases of the system. This is a much deeper
phenomenon than the mere existence of non-symmetric minima.
The relevance of the distinction between non-symmetric minima or ground states
and spontaneous symmetry breaking appears clear if one considers, e.g. a free particle on a line, where each configuration (q 0 ∈ R, p 0 = 0) is a minimum of the
Hamiltonian and it is not stable under translations, but nevertheless one does not
speak of symmetry breaking; in fact, according to our definition, there is only one
phase stable under translations.
The two concepts of symmetry breaking coincide for infinitely extended systems,
since in this case, as we shall see below, different ground states define different
phases or disjoint worlds; therefore their asymmetry necessarily leads to symmetry
breaking in the radical sense of a non-symmetric physical description (see Chap. 7
below).
Similar considerations apply to classical systems which exhibit bifurcation
2 for
which, strictly speaking, one does not have spontaneous symmetry breaking as long
as the multiple solutions are related by physically realizable operations. As we shall
see later, the latter property may fail if one considers the infinite volume (or thermodynamical) limit, and in this way spontaneous symmetry breaking may occur.
2 D.H. Sattinger, Spontaneous Symmetry Breaking: mathematical methods, applications and problems in the physical sciences, in Applications of Non-Linear Analysis, H. Amann et al. eds., Pitman
1981.
9
claimed simple mechanical examples of spontaneous symmetry breaking discussed
in the literature are equally misleading.
Even if the existence of non-symmetric minima is a rather peculiar phenomenon
which deserves special interest, it does not imply spontaneous symmetry breaking
in the radical sense of its realization in elementary particle physics, many-body
systems, statistical mechanics, etc., where a symmetry of the dynamics is not shared
by physical realizations or disjoint phases of the system. This is a much deeper
phenomenon than the mere existence of non-symmetric minima.
The relevance of the distinction between non-symmetric minima or ground states
and spontaneous symmetry breaking appears clear if one considers, e.g. a free particle on a line, where each configuration (q 0 ∈ R, p 0 = 0) is a minimum of the
Hamiltonian and it is not stable under translations, but nevertheless one does not
speak of symmetry breaking; in fact, according to our definition, there is only one
phase stable under translations.
The two concepts of symmetry breaking coincide for infinitely extended systems,
since in this case, as we shall see below, different ground states define different
phases or disjoint worlds; therefore their asymmetry necessarily leads to symmetry
breaking in the radical sense of a non-symmetric physical description (see Chap. 7
below).
Similar considerations apply to classical systems which exhibit bifurcation
2 for
which, strictly speaking, one does not have spontaneous symmetry breaking as long
as the multiple solutions are related by physically realizable operations. As we shall
see later, the latter property may fail if one considers the infinite volume (or thermodynamical) limit, and in this way spontaneous symmetry breaking may occur.
2 D.H. Sattinger, Spontaneous Symmetry Breaking: mathematical methods, applications and problems in the physical sciences, in Applications of Non-Linear Analysis, H. Amann et al. eds., Pitman
1981.