208
29 Symmetry Breaking in Gauge Theories
< δ A >= lim
R→∞
< [L R , A] > = 0, L R ≡ ( j 0 − ∂
i F 0i )( f R α).
(29.6)
For the implication of the δ(k
2
) on the energy-momentum spectrum of the physical vectors, since the vacuum expectation of the commutator is proportional to
Im < j 0 ( f R α)A >, as in the standard proof one has to insert there a complete set
of vectors n (or better discuss a possible spectral representation of the space–time
translations).
This requires some care since, by the same argument of the GNS representation, the
vacuum expectations of the local field algebra F define a representation of the field
algebra in a vector space V = FΨ 0 , with Ψ 0 the vacuum vector, but the inner product
<, > defined by such expectations cannot be semi-definite and therefore V does not
have a pre-Hilbert space structure. However, under general conditions
199 one can
embed V into a Hilbert space K, with scalar product ( , ), such that ∀A, B ∈ F
< Ψ 0 , A
∗ BΨ 0 >=< AΨ 0 , BΨ 0 >= (AΨ 0 , η BΨ 0 ),
where η is the metric operator, satisfying η
∗
= η, η
2
= 1, ηΨ 0 = Ψ 0 .
200 Actually,
the conclusions of the Theorem are independent of the specific properties of such an
embedding; the only relevant property is that, in any case, the subspace K phys ⊂ K
of physical states must satisfy the subsidiary condition
< Ψ, L μ (x)) >= (Ψ, ηL μ (x))) = 0, ∀Ψ, , ∈ K phys ,
(29.7)
in order to ensure the validity of the Maxwell equations.
Then, for the proof of the last statement of the Theorem, it is convenient to choose the
complete set of intermediate states n according to the (orthogonal) decomposition
of K = K phys ⊕ K
⊥
phys . Hence, the generic insertion takes the following form:
(Ψ 0 , ηL R n )(( n , AΨ 0 ) =< Ψ 0 , L R n > (( n , AΨ 0 )
and by the weak Gauss law the physical vectors cannot contribute. Thus, the Goldstone modes associated with the δ(k
2
) singularity, which appears in the Fourier
transform of the two-point function < j 0 (x)A >, cannot be ascribed to physical
states.
199 G. Morchio and F. Strocchi, Ann. H. Poincareé, A 33, 251 (1980); F. Strocchi, Selected Topics
on the General Properties of Quantum Field Theory, World Scientific 1993, Chap. VI.
200 In the Feynman (Gupta–Bleuler) gauge of free QED, η = (−1) N0 , N 0 =
d 3 ka ∗
0 (k)a 0 (k) (the
number of “time-like photons”).
29 Symmetry Breaking in Gauge Theories
< δ A >= lim
R→∞
< [L R , A] > = 0, L R ≡ ( j 0 − ∂
i F 0i )( f R α).
(29.6)
For the implication of the δ(k
2
) on the energy-momentum spectrum of the physical vectors, since the vacuum expectation of the commutator is proportional to
Im < j 0 ( f R α)A >, as in the standard proof one has to insert there a complete set
of vectors n (or better discuss a possible spectral representation of the space–time
translations).
This requires some care since, by the same argument of the GNS representation, the
vacuum expectations of the local field algebra F define a representation of the field
algebra in a vector space V = FΨ 0 , with Ψ 0 the vacuum vector, but the inner product
<, > defined by such expectations cannot be semi-definite and therefore V does not
have a pre-Hilbert space structure. However, under general conditions
199 one can
embed V into a Hilbert space K, with scalar product ( , ), such that ∀A, B ∈ F
< Ψ 0 , A
∗ BΨ 0 >=< AΨ 0 , BΨ 0 >= (AΨ 0 , η BΨ 0 ),
where η is the metric operator, satisfying η
∗
= η, η
2
= 1, ηΨ 0 = Ψ 0 .
200 Actually,
the conclusions of the Theorem are independent of the specific properties of such an
embedding; the only relevant property is that, in any case, the subspace K phys ⊂ K
of physical states must satisfy the subsidiary condition
< Ψ, L μ (x)) >= (Ψ, ηL μ (x))) = 0, ∀Ψ, , ∈ K phys ,
(29.7)
in order to ensure the validity of the Maxwell equations.
Then, for the proof of the last statement of the Theorem, it is convenient to choose the
complete set of intermediate states n according to the (orthogonal) decomposition
of K = K phys ⊕ K
⊥
phys . Hence, the generic insertion takes the following form:
(Ψ 0 , ηL R n )(( n , AΨ 0 ) =< Ψ 0 , L R n > (( n , AΨ 0 )
and by the weak Gauss law the physical vectors cannot contribute. Thus, the Goldstone modes associated with the δ(k
2
) singularity, which appears in the Fourier
transform of the two-point function < j 0 (x)A >, cannot be ascribed to physical
states.
199 G. Morchio and F. Strocchi, Ann. H. Poincareé, A 33, 251 (1980); F. Strocchi, Selected Topics
on the General Properties of Quantum Field Theory, World Scientific 1993, Chap. VI.
200 In the Feynman (Gupta–Bleuler) gauge of free QED, η = (−1) N0 , N 0 =
d 3 ka ∗
0 (k)a 0 (k) (the
number of “time-like photons”).