Appendix F: U(1) Problem Solved by Gauge Group Topology
273
states under the full local gauge group G (by Proposition F.2), the operators V
C
( f R α)
have vanishing matrix elements on the physical states.
57
Since in the standard approach the instanton winding number n is related to the
integral of ∂
μ C μ (see eqs. (F.9), (F.10)), the vanishing of the matrix elements of V
C
on the physical states seriously questions the physical relevance of the instanton
winding number.
As we shall see below, the physical topological effects are related to the topology
of G and may be rather derived through the use of the unitary operators V
5
R (λ),
λ ∈ R, formally the exponentials, e
i λ J
5
0 ( f R α) , J
5
0 = j
5
0 + 2C 0 , which generate the
chiral transformations of the observables, Eq. (F.7). The presence of fermions is
therefore crucial for such a derivation.
F.5 Solution of U(1) Problem Without Instantons
As stressed in Chaps. 25 and 27, a crucial assumption for the proof of the Goldstone
theorem, with the derivation of the massless Goldstone bosons, is that the infinitesimal transformations of the spontaneously broken continuous symmetry be generated
by a local charge associated with a conserved (space-time covariant) current.
As argued before, by its very constructive definition, the chiral symmetry β
λ
is locally generated by the unitary operators V
5
R (λ), according to Eq. (F.7) for
any F ∈ F W . The implication of the symmetry breaking on the Goldstone spectrum requires to relate < δ
5 A > = 0 to the two-point function < J
5
0 (x) A >, (as in
Chap. 25, Sect. 25.3, Chap. 27, Theorem 27.1):
< δ
5 A >= i lim
R→∞
< J
5
0 ( f R α)A − A J
5
0 ( f R α) >=
= −2 Im lim
R→∞
< J
5
0 ( f R α) A >,
(F.20)
(for A Hermitian). The non-trivial topology of the local gauge group G plays a
decisive role in excluding Eq. (F.20).
Proposition F.3 The Goldstone theorem is evaded by the spontaneous breaking of
the chiral U (1) A symmetry β
λ in QCD, because the expectation < δ
5 A > = 0, with
A an observable (Hermitian) field, is not related to the two-point function of A and
a (local conserved) current.
Proof. By Eq. (F.7) one has
< δ
5 A >=
d
dλ
< β
λ
(A) > | λ=0 =
d
dλ
lim
R
< V
5
R (λ) A V
5
R (−λ) > | λ=0 ,
57 G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009), Proposition 3.2.
273
states under the full local gauge group G (by Proposition F.2), the operators V
C
( f R α)
have vanishing matrix elements on the physical states.
57
Since in the standard approach the instanton winding number n is related to the
integral of ∂
μ C μ (see eqs. (F.9), (F.10)), the vanishing of the matrix elements of V
C
on the physical states seriously questions the physical relevance of the instanton
winding number.
As we shall see below, the physical topological effects are related to the topology
of G and may be rather derived through the use of the unitary operators V
5
R (λ),
λ ∈ R, formally the exponentials, e
i λ J
5
0 ( f R α) , J
5
0 = j
5
0 + 2C 0 , which generate the
chiral transformations of the observables, Eq. (F.7). The presence of fermions is
therefore crucial for such a derivation.
F.5 Solution of U(1) Problem Without Instantons
As stressed in Chaps. 25 and 27, a crucial assumption for the proof of the Goldstone
theorem, with the derivation of the massless Goldstone bosons, is that the infinitesimal transformations of the spontaneously broken continuous symmetry be generated
by a local charge associated with a conserved (space-time covariant) current.
As argued before, by its very constructive definition, the chiral symmetry β
λ
is locally generated by the unitary operators V
5
R (λ), according to Eq. (F.7) for
any F ∈ F W . The implication of the symmetry breaking on the Goldstone spectrum requires to relate < δ
5 A > = 0 to the two-point function < J
5
0 (x) A >, (as in
Chap. 25, Sect. 25.3, Chap. 27, Theorem 27.1):
< δ
5 A >= i lim
R→∞
< J
5
0 ( f R α)A − A J
5
0 ( f R α) >=
= −2 Im lim
R→∞
< J
5
0 ( f R α) A >,
(F.20)
(for A Hermitian). The non-trivial topology of the local gauge group G plays a
decisive role in excluding Eq. (F.20).
Proposition F.3 The Goldstone theorem is evaded by the spontaneous breaking of
the chiral U (1) A symmetry β
λ in QCD, because the expectation < δ
5 A > = 0, with
A an observable (Hermitian) field, is not related to the two-point function of A and
a (local conserved) current.
Proof. By Eq. (F.7) one has
< δ
5 A >=
d
dλ
< β
λ
(A) > | λ=0 =
d
dλ
lim
R
< V
5
R (λ) A V
5
R (−λ) > | λ=0 ,
57 G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009), Proposition 3.2.