4
Part I: SYMMETRY BREAKING IN CLASSICAL SYSTEMS
a matter of fact, such sets of initial data do correspond to representations of a group
of transformations which is isomorphic to G, but which does not commute with the
dynamics, and therefore the above form of the initial data does not extend to arbitrary
times; thus the above identification of multiplets at the initial time is not stable under
time evolution. As a matter of fact, the group of transformations which commute
with the time evolution corresponds to ¯
ϕ + χ → g ¯
ϕ + gχ, g ∈ G, which, however,
does not leave H ¯
ϕ stable.
Within this approach, it is possible to prove a classical counterpart of the so-called
Goldstone theorem, according to which there are massless modes (i.e. solutions of
the free wave equation) associated to each broken generator. The theorem proved
here provides a mathematically acceptable substitute of the heuristic arguments and
improves the conclusions based on the quadratic approximation of the potential
around an absolute minimum.
Explicit examples which illustrate how these ideas work in concrete models are
discussed in Chap. 8.
The discussion of symmetry breaking in classical systems relies, with some additions, on papers written jointly with Cesare Parenti and Giorgio Velo, to whom I
am greatly indebted (see the references at the relevant points). An attempt is made
to reduce the mathematical details to the minimum required to make the arguments
self-contained and also convincing for a mathematically minded reader. The required
background technical knowledge is kept to a rather low level, in order that the lectures be accessible also to undergraduate students with a basic knowledge of Hilbert
space structures.
Part I: SYMMETRY BREAKING IN CLASSICAL SYSTEMS
a matter of fact, such sets of initial data do correspond to representations of a group
of transformations which is isomorphic to G, but which does not commute with the
dynamics, and therefore the above form of the initial data does not extend to arbitrary
times; thus the above identification of multiplets at the initial time is not stable under
time evolution. As a matter of fact, the group of transformations which commute
with the time evolution corresponds to ¯
ϕ + χ → g ¯
ϕ + gχ, g ∈ G, which, however,
does not leave H ¯
ϕ stable.
Within this approach, it is possible to prove a classical counterpart of the so-called
Goldstone theorem, according to which there are massless modes (i.e. solutions of
the free wave equation) associated to each broken generator. The theorem proved
here provides a mathematically acceptable substitute of the heuristic arguments and
improves the conclusions based on the quadratic approximation of the potential
around an absolute minimum.
Explicit examples which illustrate how these ideas work in concrete models are
discussed in Chap. 8.
The discussion of symmetry breaking in classical systems relies, with some additions, on papers written jointly with Cesare Parenti and Giorgio Velo, to whom I
am greatly indebted (see the references at the relevant points). An attempt is made
to reduce the mathematical details to the minimum required to make the arguments
self-contained and also convincing for a mathematically minded reader. The required
background technical knowledge is kept to a rather low level, in order that the lectures be accessible also to undergraduate students with a basic knowledge of Hilbert
space structures.