136
20 Constructive Symmetry Breaking
Now, since the Lagrangian, and therefore the action, is invariant under the symmetry g, by a change of variables in the functional integral, say ϕ
K ≡ ϕ
g
K , the right-hand
side of the above equation becomes
Z
−1
V,K ,ϕ
Dϕ
e
−(A V (ϕ
)+A ∂V (g
−1 ϕ
))
ϕ
K (x 1 ) . . . ϕ
K (x n ) =
< ϕ(x 1 ) . . . ϕ(x n ) > V,K ,g −1 ϕ .
(20.5)
Thus, the non-invariance of the above correlation functions in the thermodynamical
limit is equivalent to the dependence on the (non-symmetric) boundary conditions.
Clearly, if the chosen boundary conditions are symmetric (e.g. periodic boundary
conditions), the corresponding correlation functions are invariant, but this cannot
be taken as a criterion for absence of spontaneous symmetry breaking, because
the so constructed correlation functions may correspond to a mixed phase or to a
representation with more than one translationally invariant state as displayed by the
failure of the cluster property.
C. Bogoliubov Strategy
Another constructive way of obtaining symmetry breaking order parameters was
discussed by Bogoliubov
115 and exploited in particular in his treatment of superconductivity. The idea is to introduce a symmetry breaking interaction with an external
field, which is sent to zero at the very end. Such a prescription looks more physical,
since it reflects the operational way of producing, e.g., a ferromagnet, but does not
seem to be under the same rigorous mathematical control as is the Ruelle strategy.
In the Goldstone model discussed above, the idea of the Bogoliubov strategy can
be implemented by introducing in the (infrared and ultraviolet) regularized theory
an n-component external field h(x) which plays the role of the external magnetic
field for ferromagnets, linearly coupled to ϕ(x) (more generally one may modify
the coupling constant). Clearly, the volume interaction with the external field wins
over the surface terms due to the boundary conditions and the latter ones become
irrelevant.
Then, one computes the correlation functions in the thermodynamical limit and
finally one lets h → 0. Proceeding as in the above discussion of the Ruelle strategy, one easily gets the following relation between the infinite volume correlation
functions:
< ϕ
g
(x 1 ) . . . ϕ
g
(x m ) > K ,n =< ϕ(x 1 ) . . . ϕ(x m ) > K ,g −1 n ,
(20.6)
where n denotes the direction along which h is sent to zero.
115 N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 2, Gordon and Breach 1970, Part 1.
20 Constructive Symmetry Breaking
Now, since the Lagrangian, and therefore the action, is invariant under the symmetry g, by a change of variables in the functional integral, say ϕ
K ≡ ϕ
g
K , the right-hand
side of the above equation becomes
Z
−1
V,K ,ϕ
Dϕ
e
−(A V (ϕ
)+A ∂V (g
−1 ϕ
))
ϕ
K (x 1 ) . . . ϕ
K (x n ) =
< ϕ(x 1 ) . . . ϕ(x n ) > V,K ,g −1 ϕ .
(20.5)
Thus, the non-invariance of the above correlation functions in the thermodynamical
limit is equivalent to the dependence on the (non-symmetric) boundary conditions.
Clearly, if the chosen boundary conditions are symmetric (e.g. periodic boundary
conditions), the corresponding correlation functions are invariant, but this cannot
be taken as a criterion for absence of spontaneous symmetry breaking, because
the so constructed correlation functions may correspond to a mixed phase or to a
representation with more than one translationally invariant state as displayed by the
failure of the cluster property.
C. Bogoliubov Strategy
Another constructive way of obtaining symmetry breaking order parameters was
discussed by Bogoliubov
115 and exploited in particular in his treatment of superconductivity. The idea is to introduce a symmetry breaking interaction with an external
field, which is sent to zero at the very end. Such a prescription looks more physical,
since it reflects the operational way of producing, e.g., a ferromagnet, but does not
seem to be under the same rigorous mathematical control as is the Ruelle strategy.
In the Goldstone model discussed above, the idea of the Bogoliubov strategy can
be implemented by introducing in the (infrared and ultraviolet) regularized theory
an n-component external field h(x) which plays the role of the external magnetic
field for ferromagnets, linearly coupled to ϕ(x) (more generally one may modify
the coupling constant). Clearly, the volume interaction with the external field wins
over the surface terms due to the boundary conditions and the latter ones become
irrelevant.
Then, one computes the correlation functions in the thermodynamical limit and
finally one lets h → 0. Proceeding as in the above discussion of the Ruelle strategy, one easily gets the following relation between the infinite volume correlation
functions:
< ϕ
g
(x 1 ) . . . ϕ
g
(x m ) > K ,n =< ϕ(x 1 ) . . . ϕ(x m ) > K ,g −1 n ,
(20.6)
where n denotes the direction along which h is sent to zero.
115 N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 2, Gordon and Breach 1970, Part 1.