6
1 Symmetries of a Classical System
2) does not change the dynamical behaviour,
1 namely
α
t gS γ = α
t S gγ ≡ S (gγ)(t) = S gγ(t) = gα
t S γ .
(1.3)
The above condition states that the symmetry transformation commutes with
time evolution. For classical canonical systems, this amounts to the invariance of the
Hamiltonian under the symmetry g (symmetric Hamiltonian).
The realization of a symmetry which relates (the configurations of) two seemingly
different systems clearly leads to a unification of their description. In particular, the
solution of the dynamical problem for one configuration automatically gives the
solution for the symmetry related configuration (see (1.3)).
Example 1.1, double well potential. Consider a particle moving on a line, subject
to a double well potential, i.e. described by the following Hamiltonian
H =
1
2
p
2
+
1
4
λ(q
2
− a
2
)
2
,
(1.4)
with q, p the canonical coordinates which label the configurations of the particle.
The reflection g : q → −q, p → −p leaves the Hamiltonian invariant and is a
symmetry of the system; obviously, it maps solutions (of the Hamilton equations)
into solutions.
Now, consider the two classes ± of solutions corresponding to initial conditions
in the neighbourhoods of the two absolute minima q 0 = ±a, with p 0 <
√
λa
2
/2,
respectively, and suppose that by some (artificial) ansatz, in the preparation of the
initial configurations one cannot dispose of energies greater than λa
4
/4. This means
that the two classes of solutions describe two disjoint realizations of the system, in the
sense that by fiat no physically realizable operation allows to change a configuration
from one class to the other. In this way, one gets a picture similar to the case of
the thermodynamical phases, which are physically disjoint in the thermodynamical
limit, but nevertheless described by the same Hamiltonian and related by a symmetry
which is not implementable in each phase. Clearly, the existence of such a symmetry,
even if devoid of physical operational meaning, provides a unified description of the
two “phases”.
For a particle moving on a plane, the analog of the double well potential defines a
Hamiltonian which is invariant under rotations around the axis (through the origin)
orthogonal to the plane and one has a continuous group of symmetries. There is
a continuous family of absolute minima lying on the circle |q 0 |
2
= a
2 . Since such
minima are not separated by any energy barrier, one cannot associate with them
different systems by some artificial ansatz as above.
1 To simplify the discussion, here we do not consider the more general case in which the dynamics
transform covariantly under g (like, e.g. in the case of Lorentz transformations). For a general
discussion of symmetries and of their relevance in physics, see R.M.F. Houtappel, H. Van Dam and
E.P. Wigner, Rev. Mod. Phys. 37, 595 (1965).
1 Symmetries of a Classical System
2) does not change the dynamical behaviour,
1 namely
α
t gS γ = α
t S gγ ≡ S (gγ)(t) = S gγ(t) = gα
t S γ .
(1.3)
The above condition states that the symmetry transformation commutes with
time evolution. For classical canonical systems, this amounts to the invariance of the
Hamiltonian under the symmetry g (symmetric Hamiltonian).
The realization of a symmetry which relates (the configurations of) two seemingly
different systems clearly leads to a unification of their description. In particular, the
solution of the dynamical problem for one configuration automatically gives the
solution for the symmetry related configuration (see (1.3)).
Example 1.1, double well potential. Consider a particle moving on a line, subject
to a double well potential, i.e. described by the following Hamiltonian
H =
1
2
p
2
+
1
4
λ(q
2
− a
2
)
2
,
(1.4)
with q, p the canonical coordinates which label the configurations of the particle.
The reflection g : q → −q, p → −p leaves the Hamiltonian invariant and is a
symmetry of the system; obviously, it maps solutions (of the Hamilton equations)
into solutions.
Now, consider the two classes ± of solutions corresponding to initial conditions
in the neighbourhoods of the two absolute minima q 0 = ±a, with p 0 <
√
λa
2
/2,
respectively, and suppose that by some (artificial) ansatz, in the preparation of the
initial configurations one cannot dispose of energies greater than λa
4
/4. This means
that the two classes of solutions describe two disjoint realizations of the system, in the
sense that by fiat no physically realizable operation allows to change a configuration
from one class to the other. In this way, one gets a picture similar to the case of
the thermodynamical phases, which are physically disjoint in the thermodynamical
limit, but nevertheless described by the same Hamiltonian and related by a symmetry
which is not implementable in each phase. Clearly, the existence of such a symmetry,
even if devoid of physical operational meaning, provides a unified description of the
two “phases”.
For a particle moving on a plane, the analog of the double well potential defines a
Hamiltonian which is invariant under rotations around the axis (through the origin)
orthogonal to the plane and one has a continuous group of symmetries. There is
a continuous family of absolute minima lying on the circle |q 0 |
2
= a
2 . Since such
minima are not separated by any energy barrier, one cannot associate with them
different systems by some artificial ansatz as above.
1 To simplify the discussion, here we do not consider the more general case in which the dynamics
transform covariantly under g (like, e.g. in the case of Lorentz transformations). For a general
discussion of symmetries and of their relevance in physics, see R.M.F. Houtappel, H. Van Dam and
E.P. Wigner, Rev. Mod. Phys. 37, 595 (1965).