124
18 Symmetry Breaking in Quantum Systems
To quickly see when the phases can be eliminated, we note that the set of the
C ab defines a (real valued) antisymmetric bilinear form C(t a , t b ) ≡ C ab satisfying,
by (18.9),
C([t a , t b ], t c ) + C([t b , t c ], t a ) + C([t c , t a ], t b ) = 0.
(18.10)
If the (simply connected) group G has the property that any bilinear form with the
above properties can be written in terms of a linear form ω, in the sense that C(t a , t b ) =
ω([t a , t b ]), (technically this means that the second cohomology group H
2
(G, R) of
L G , with coefficients in R, is trivial), then the phases can be eliminated.
108 In fact,
this is obtained by the following redefinition of the generators: T a = t a + ω(t a ).
18.2 Spontaneous Symmetry Breaking
The exploitation of symmetries for the description of quantum systems has played an
important role in obtaining information without having to solve the full dynamical
problem. It also proved useful in the case in which the symmetry is not exact by
offering the possibility of unifying the description of systems related by an approximate symmetry, in terms of a “small” symmetry breaking term in the Hamiltonian
in order to account for their “small” differences. Such a strategy has been successful
when applied to quantum systems with a finite number of degrees of freedom, but
it showed practical and conceptual difficulties when applied to infinitely extended
systems.
First, the viability of such a strategy is restricted to the case of “small” symmetry breaking and therefore does not allow to unify the description of systems with
rather different physical behaviour (e.g. the electromagnetic and weak interactions
of elementary particles, or different thermodynamical phases in many-body theory).
Second, renormalization problems require an independent renormalization of the
basic physical parameters, with the result of nullifying some of the possible predictions of the symmetry breaking (e.g. the electromagnetic mass differences due to
isospin breaking in elementary particle theory).
From this point of view, the realization of the mechanism of spontaneous symmetry
breaking represented a real breakthrough in the development of theoretical physics,
108 The triviality of the second cohomology group allows the construction of a Lie algebra homomorphism α : L G → L Gω of the form α (A) = (A, ξ(A)) (and therefore of a homomorphism α :
G → G ω as discussed in the previous footnote). In fact, any linear map β(A) = (A, λ β (A)) ∈ L Gω ,
A ∈ L G , defines a real-valued antisymmetric bilinear form C β (A, B)
C β (A, B) ≡ [β(A), β(B)] − β([A, B]),
which satisfies (18.10) and is therefore of the form ω([A, B]). Then, α (A) ≡ β(A) + ω(A) yields
the desired Lie algebra homomorphism.
18 Symmetry Breaking in Quantum Systems
To quickly see when the phases can be eliminated, we note that the set of the
C ab defines a (real valued) antisymmetric bilinear form C(t a , t b ) ≡ C ab satisfying,
by (18.9),
C([t a , t b ], t c ) + C([t b , t c ], t a ) + C([t c , t a ], t b ) = 0.
(18.10)
If the (simply connected) group G has the property that any bilinear form with the
above properties can be written in terms of a linear form ω, in the sense that C(t a , t b ) =
ω([t a , t b ]), (technically this means that the second cohomology group H
2
(G, R) of
L G , with coefficients in R, is trivial), then the phases can be eliminated.
108 In fact,
this is obtained by the following redefinition of the generators: T a = t a + ω(t a ).
18.2 Spontaneous Symmetry Breaking
The exploitation of symmetries for the description of quantum systems has played an
important role in obtaining information without having to solve the full dynamical
problem. It also proved useful in the case in which the symmetry is not exact by
offering the possibility of unifying the description of systems related by an approximate symmetry, in terms of a “small” symmetry breaking term in the Hamiltonian
in order to account for their “small” differences. Such a strategy has been successful
when applied to quantum systems with a finite number of degrees of freedom, but
it showed practical and conceptual difficulties when applied to infinitely extended
systems.
First, the viability of such a strategy is restricted to the case of “small” symmetry breaking and therefore does not allow to unify the description of systems with
rather different physical behaviour (e.g. the electromagnetic and weak interactions
of elementary particles, or different thermodynamical phases in many-body theory).
Second, renormalization problems require an independent renormalization of the
basic physical parameters, with the result of nullifying some of the possible predictions of the symmetry breaking (e.g. the electromagnetic mass differences due to
isospin breaking in elementary particle theory).
From this point of view, the realization of the mechanism of spontaneous symmetry
breaking represented a real breakthrough in the development of theoretical physics,
108 The triviality of the second cohomology group allows the construction of a Lie algebra homomorphism α : L G → L Gω of the form α (A) = (A, ξ(A)) (and therefore of a homomorphism α :
G → G ω as discussed in the previous footnote). In fact, any linear map β(A) = (A, λ β (A)) ∈ L Gω ,
A ∈ L G , defines a real-valued antisymmetric bilinear form C β (A, B)
C β (A, B) ≡ [β(A), β(B)] − β([A, B]),
which satisfies (18.10) and is therefore of the form ω([A, B]). Then, α (A) ≡ β(A) + ω(A) yields
the desired Lie algebra homomorphism.