Appendix F: U(1) Problem Solved by Gauge Group Topology
279
a decisive role of the group topology distinguishes the abelian and the non-abelian
cases, even if in both cases the axial anomaly is present.
The second step is the derivation of the θ vacuum structure and the inevitable
breaking of chiral symmetry in each factorial representation of the observable algebra.
The final picture is similar to that derived by the standard instanton semi-classical
approximation, but the origin and meaning of the θ vacuum angle are very different:
i) it does not hinge on the classification of the finite action Euclidean configurations,
which have zero functional measure, ii) it does not imply that chiral symmetry does
not exist as a group of time-independent transformations of the observables (as
incorrectly claimed in the literature, in particular by G. t’Hooft), iii) its observable
content follows from its being the spectrum of the centre of the observables, and
the algebraic inequivalence of different θ sectors is implied by the pointwise noninvariance of the centre of the observable algebra under chiral transformations.
More generally, the derived connection between the θ angle and the topology of
the gauge group disentangles it from the existence of instanton solutions, opening
the possibility of similar realizations in much more general contexts, whenever there
is a local gauge group with non-trivial topology.
To support the above analysis with examples under full mathematical control, it
is instructive to work out the bosonized Schwinger model in the temporal gauge.
The QCD structure and mechanisms discussed above are reproduced in all details
(the existence of time-independent “chiral" transformations of the observable fields,
generated by local unitary operators, the failure of local generation at the infinitesimal
level by a local current, the non-trivial topology of the local gauge group and the θ
angle given by the spectrum of the centre of the observable algebra).
62
Since the model has been regarded as a prototype of QDC structures, its full
mathematical control confirms the general features discussed above and disproves
opposite claims for QCD that appeared in the literature.
Another instructive realization of the non-perturbative mechanisms discussed in
this Appendix is provided by the simple model of a quantum particle on a circle (of
unit radius).
63
The dictionary for the strict analogy is the following: the exponential field algebra F W is now given by the Weyl algebra A W (generated by the formal exponentials
U (α) ∼ e
iαq
, V (β) ∼ e
iβ p , α, β ∈ R of the canonical variables q, p); the gauge
transformations are the rotations of angle 2π, labelled by the topological winding
number n and generated by the unitary operators V n ≡ V (2πn); the observable algebra A is the subalgebra of A W invariant under the gauge group and it is generated
by U (n), V (β), n ∈ Z, β ∈ R.
The analog of the chiral transformations is the one-parameter group
β
λ
(U (α)) = U (α) β
λ
(V (γ)) = e
−i γλ/π V (γ);
62 G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009).
63 F. Strocchi, An introduction to the mathematical structure of quantum mechanics, 2nd edition,
second expanded printing, World Scientific 2010, Sect. 6.8.
279
a decisive role of the group topology distinguishes the abelian and the non-abelian
cases, even if in both cases the axial anomaly is present.
The second step is the derivation of the θ vacuum structure and the inevitable
breaking of chiral symmetry in each factorial representation of the observable algebra.
The final picture is similar to that derived by the standard instanton semi-classical
approximation, but the origin and meaning of the θ vacuum angle are very different:
i) it does not hinge on the classification of the finite action Euclidean configurations,
which have zero functional measure, ii) it does not imply that chiral symmetry does
not exist as a group of time-independent transformations of the observables (as
incorrectly claimed in the literature, in particular by G. t’Hooft), iii) its observable
content follows from its being the spectrum of the centre of the observables, and
the algebraic inequivalence of different θ sectors is implied by the pointwise noninvariance of the centre of the observable algebra under chiral transformations.
More generally, the derived connection between the θ angle and the topology of
the gauge group disentangles it from the existence of instanton solutions, opening
the possibility of similar realizations in much more general contexts, whenever there
is a local gauge group with non-trivial topology.
To support the above analysis with examples under full mathematical control, it
is instructive to work out the bosonized Schwinger model in the temporal gauge.
The QCD structure and mechanisms discussed above are reproduced in all details
(the existence of time-independent “chiral" transformations of the observable fields,
generated by local unitary operators, the failure of local generation at the infinitesimal
level by a local current, the non-trivial topology of the local gauge group and the θ
angle given by the spectrum of the centre of the observable algebra).
62
Since the model has been regarded as a prototype of QDC structures, its full
mathematical control confirms the general features discussed above and disproves
opposite claims for QCD that appeared in the literature.
Another instructive realization of the non-perturbative mechanisms discussed in
this Appendix is provided by the simple model of a quantum particle on a circle (of
unit radius).
63
The dictionary for the strict analogy is the following: the exponential field algebra F W is now given by the Weyl algebra A W (generated by the formal exponentials
U (α) ∼ e
iαq
, V (β) ∼ e
iβ p , α, β ∈ R of the canonical variables q, p); the gauge
transformations are the rotations of angle 2π, labelled by the topological winding
number n and generated by the unitary operators V n ≡ V (2πn); the observable algebra A is the subalgebra of A W invariant under the gauge group and it is generated
by U (n), V (β), n ∈ Z, β ∈ R.
The analog of the chiral transformations is the one-parameter group
β
λ
(U (α)) = U (α) β
λ
(V (γ)) = e
−i γλ/π V (γ);
62 G. Morchio and F. Strocchi, Ann. Phys. 324, 2236 (2009).
63 F. Strocchi, An introduction to the mathematical structure of quantum mechanics, 2nd edition,
second expanded printing, World Scientific 2010, Sect. 6.8.