278
Appendix F: U(1) Problem Solved by Gauge Group Topology
(β
λ
)
∗
ω θ (F) = ω θ (β
λ
(F)) = ω θ+λ (F).
(F.28)
The operators V(U n ) defined by
V(U n ) FΨ 0 = α U n (F) Ψ 0 , ∀F ∈ F W , V(U n ) Ψ 0 = Ψ 0 ,
are densely defined unitary operator satisfying
V(U n ) F V(U n )
∗
= α U n (F),
as may be easily checked by applying the left-hand side to a generic vector F
Ψ 0 ,
F
∈ F W . Hence, the τ n ≡ V (U n ) V(U n )
∗ commute with F W , and with the V (U), so
that τ n τ m = τ n+m . If the representation is irreducible τ n must be a multiple of the
identity, say exp inη, actually a point of the spectrum of T n since
e
inη
Ψ 0 = V (U n ) V(U n )
∗
Ψ 0 = V (U n ) Ψ 0 = T n Ψ 0 .
Thus, Eq. (F.27) follows.
As it is standard in the case of spontaneous symmetry breaking, the θ vacua define
isomorphic representations of the observables, physically indistinguishable from the
θ = 0 representation, the common physically relevant property being that in each of
them chiral symmetry is spontaneously broken.
As in the case of spontaneous magnetization, the representations become physically distinct by the introduction of an “external field”; in the standard model this
role is played by the quark mass matrix induced by the electroweak interaction. The
physical parameter is then ˜
θ ≡ θ − θ F , with θ F the quark mass matrix angle, and
˜
θ = 0 implies strong C P violation (the so-called strong C P problem).
61
F.7 Conclusions
The above approach to the U (1) problem involves two different steps, both exploiting
the non-trivial topology of the local gauge group, with no reference to the problematic
instanton semi-classical approximation.
On one side, given the existence of the time-independent chiral symmetries as
automorphisms of the algebra of observable fields, one has to prove that the breaking
U (1) A is compatible with the absence of Goldstone bosons. This is achieved by recognizing that the non-trivial topology of the gauge group precludes the local generation
of the infinitesimal chiral transformations by a local current, a crucial assumption
for the proof of the Goldstone theorem (as discussed at length in Chap. 25); such
61 S. Weinberg, [1996], Sect. 23.6; for a possible solution of the strong C P problem see J. Löffelholz,
G. Morchio and F. Strocchi, Ann. Phys. 250, 367 (1996).
Appendix F: U(1) Problem Solved by Gauge Group Topology
(β
λ
)
∗
ω θ (F) = ω θ (β
λ
(F)) = ω θ+λ (F).
(F.28)
The operators V(U n ) defined by
V(U n ) FΨ 0 = α U n (F) Ψ 0 , ∀F ∈ F W , V(U n ) Ψ 0 = Ψ 0 ,
are densely defined unitary operator satisfying
V(U n ) F V(U n )
∗
= α U n (F),
as may be easily checked by applying the left-hand side to a generic vector F
Ψ 0 ,
F
∈ F W . Hence, the τ n ≡ V (U n ) V(U n )
∗ commute with F W , and with the V (U), so
that τ n τ m = τ n+m . If the representation is irreducible τ n must be a multiple of the
identity, say exp inη, actually a point of the spectrum of T n since
e
inη
Ψ 0 = V (U n ) V(U n )
∗
Ψ 0 = V (U n ) Ψ 0 = T n Ψ 0 .
Thus, Eq. (F.27) follows.
As it is standard in the case of spontaneous symmetry breaking, the θ vacua define
isomorphic representations of the observables, physically indistinguishable from the
θ = 0 representation, the common physically relevant property being that in each of
them chiral symmetry is spontaneously broken.
As in the case of spontaneous magnetization, the representations become physically distinct by the introduction of an “external field”; in the standard model this
role is played by the quark mass matrix induced by the electroweak interaction. The
physical parameter is then ˜
θ ≡ θ − θ F , with θ F the quark mass matrix angle, and
˜
θ = 0 implies strong C P violation (the so-called strong C P problem).
61
F.7 Conclusions
The above approach to the U (1) problem involves two different steps, both exploiting
the non-trivial topology of the local gauge group, with no reference to the problematic
instanton semi-classical approximation.
On one side, given the existence of the time-independent chiral symmetries as
automorphisms of the algebra of observable fields, one has to prove that the breaking
U (1) A is compatible with the absence of Goldstone bosons. This is achieved by recognizing that the non-trivial topology of the gauge group precludes the local generation
of the infinitesimal chiral transformations by a local current, a crucial assumption
for the proof of the Goldstone theorem (as discussed at length in Chap. 25); such
61 S. Weinberg, [1996], Sect. 23.6; for a possible solution of the strong C P problem see J. Löffelholz,
G. Morchio and F. Strocchi, Ann. Phys. 250, 367 (1996).