Appendix F: U(1) Problem Solved by Gauge Group Topology
277
Since the large gauge transformations are implemented by unitary operators
V (U n ), Eq. (F.22) implies
V (U n ) V
5
R (λ) = e
i2nλ V
5
R (λ)V (U n )
and since the operators V (U n ) are local operators, Eq. (F.7) applies and gives
β
λ
(V (U n )) = e
−i2nλ V (U n ).
(F.24)
Moreover, P 0 is a spectral projection and therefore a weak limit of the Gauss operators V (U(g)), which are invariant under chiral transformations; then, it is natural
to extend the chiral transformations to P 0 , by putting
60
β
λ
(P 0 ) = P 0 .
(F.25)
Hence, by the definition of the T n , Eqs. (F.23) and (F.24), one has
β
λ
(T n ) = e
−i2nλ T n ,
(F.26)
which shows that the centre Z of the observables is not pointwise invariant under
chiral transformations. Furthermore, the spectrum of T n is non-trivial, {e
i2nθ
; θ ∈
[ 0, π) } and the angle θ labels the factorial representations of the local algebra of
observables.
Proposition F.5 The factorial sub-representations of A contained in H
are labelled
by an angle θ ∈ [ 0, π), θ sectors.
Since the T n are not invariant under chiral transformations, in each factorial
representation of A chiral symmetry is spontaneously broken.
If a Gauss invariant vacuum vector Ψ 0 defines an irreducible representation of
the local field algebra F W , then it selects a definite value of θ, and represents a θ
vacuum state ω θ :
T n Ψ 0 = e
i2nθ
Ψ 0 .
(F.27)
Proof. In a factorial representation π θ of A, labelled by θ, the chiral transformations cannot be implemented by unitary operators, since they would commute with
the multiples of the identity which represent the centre Z, contrary to the nontrivial transformation of the T n , Eq. (F.24). Actually, by Eq. (18.11) of Chap. 18 and
Eq. (F.24) one has
60 Such an extension of the definition of the chiral transformations may be further justified by
considering a reducible representation of F W defined by a chiral invariant vacuum, as may be
obtained by using chiral invariant boundary conditions in the functional integral in finite volume
(see J. Löffelholz, G. Morchio and F. Strocchi, Ann. Phys. 250, 367 (1996); B. Booß-Bavnbeck,
G. Morchio, F. Strocchi and K.P. Wojciechowski, Jour. Geom. Phys. 22, 219 (1997)). In such a
representation, chiral symmetry is implemented by unitary operators U 5 (λ) and one has β λ (P 0 ) =
P 0 . For details, see G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009).
277
Since the large gauge transformations are implemented by unitary operators
V (U n ), Eq. (F.22) implies
V (U n ) V
5
R (λ) = e
i2nλ V
5
R (λ)V (U n )
and since the operators V (U n ) are local operators, Eq. (F.7) applies and gives
β
λ
(V (U n )) = e
−i2nλ V (U n ).
(F.24)
Moreover, P 0 is a spectral projection and therefore a weak limit of the Gauss operators V (U(g)), which are invariant under chiral transformations; then, it is natural
to extend the chiral transformations to P 0 , by putting
60
β
λ
(P 0 ) = P 0 .
(F.25)
Hence, by the definition of the T n , Eqs. (F.23) and (F.24), one has
β
λ
(T n ) = e
−i2nλ T n ,
(F.26)
which shows that the centre Z of the observables is not pointwise invariant under
chiral transformations. Furthermore, the spectrum of T n is non-trivial, {e
i2nθ
; θ ∈
[ 0, π) } and the angle θ labels the factorial representations of the local algebra of
observables.
Proposition F.5 The factorial sub-representations of A contained in H
are labelled
by an angle θ ∈ [ 0, π), θ sectors.
Since the T n are not invariant under chiral transformations, in each factorial
representation of A chiral symmetry is spontaneously broken.
If a Gauss invariant vacuum vector Ψ 0 defines an irreducible representation of
the local field algebra F W , then it selects a definite value of θ, and represents a θ
vacuum state ω θ :
T n Ψ 0 = e
i2nθ
Ψ 0 .
(F.27)
Proof. In a factorial representation π θ of A, labelled by θ, the chiral transformations cannot be implemented by unitary operators, since they would commute with
the multiples of the identity which represent the centre Z, contrary to the nontrivial transformation of the T n , Eq. (F.24). Actually, by Eq. (18.11) of Chap. 18 and
Eq. (F.24) one has
60 Such an extension of the definition of the chiral transformations may be further justified by
considering a reducible representation of F W defined by a chiral invariant vacuum, as may be
obtained by using chiral invariant boundary conditions in the functional integral in finite volume
(see J. Löffelholz, G. Morchio and F. Strocchi, Ann. Phys. 250, 367 (1996); B. Booß-Bavnbeck,
G. Morchio, F. Strocchi and K.P. Wojciechowski, Jour. Geom. Phys. 22, 219 (1997)). In such a
representation, chiral symmetry is implemented by unitary operators U 5 (λ) and one has β λ (P 0 ) =
P 0 . For details, see G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009).