216
29 Symmetry Breaking in Gauge Theories
representation of the observable subalgebra F obs . In fact, Ψ 0 gives a reducible representation of F obs (as signalled by the failure of the cluster property by the corresponding vacuum expectations), with a non-trivial centre which is generated by
the large gauge transformations T n and is not pointwise invariant under U (1) axial
transformations.
Thus, the symmetry is broken in each pure physical phase (θ-vacuum sectors)
obtained by the diagonalization of the T n (in the technical terminology by a central
decomposition of the observables) in the subspace F obs Ψ 0 .
It should be stressed that the so obtained (gauge invariant) θ-vacua do not provide
well defined representations of the field algebra F, since the latter transforms nontrivially under T n . This is at the origin of the difficulties (and paradoxes) arising in
the discussion of the chiral Ward identities (corresponding to the conservation of J
5
μ )
in θ-vacua expectations.
211
In the θ sectors, a conserved axial current may be constructed as a non-local
operator, typically by using for J
5
μ its (non-local) expression in terms of the observable fields in a physical gauge. The above discussion, in particular the lack of time
independence of the charge density commutators, as a consequence of the failure
of relative locality between the current and the order parameter, applies to such
non-local currents.
The resulting mechanism for the solution of the U (1) problem can be made
explicit in the Coulomb gauge. In the Schwinger model, in the Coulomb gauge
one has K 0 = (e/π)A 1 = 0, K 1 = (e/π)A 0 , so that J
5
0 = j
5
0 and the (θ-)vacuum
expectations of the commutators [J
5
0 ( f R , t), A], [ j
5
0 ( f R , t), A], A ∈ F obs , coincide
and describe the same mass spectrum; however, the time dependence in the limit
R → ∞ in the first case can be ascribed to the non-locality of the conserved axial
current, whereas in the second case it reflects the non- conservation of j
5
μ . A more
detailed discussion of the U (1) problem and its solution is given in Appendix F below.
211 R.J. Crewter, in Field Theoretical methods in Particle Physics, W. Rühl ed. Reidel (1980), p. 529.
29 Symmetry Breaking in Gauge Theories
representation of the observable subalgebra F obs . In fact, Ψ 0 gives a reducible representation of F obs (as signalled by the failure of the cluster property by the corresponding vacuum expectations), with a non-trivial centre which is generated by
the large gauge transformations T n and is not pointwise invariant under U (1) axial
transformations.
Thus, the symmetry is broken in each pure physical phase (θ-vacuum sectors)
obtained by the diagonalization of the T n (in the technical terminology by a central
decomposition of the observables) in the subspace F obs Ψ 0 .
It should be stressed that the so obtained (gauge invariant) θ-vacua do not provide
well defined representations of the field algebra F, since the latter transforms nontrivially under T n . This is at the origin of the difficulties (and paradoxes) arising in
the discussion of the chiral Ward identities (corresponding to the conservation of J
5
μ )
in θ-vacua expectations.
211
In the θ sectors, a conserved axial current may be constructed as a non-local
operator, typically by using for J
5
μ its (non-local) expression in terms of the observable fields in a physical gauge. The above discussion, in particular the lack of time
independence of the charge density commutators, as a consequence of the failure
of relative locality between the current and the order parameter, applies to such
non-local currents.
The resulting mechanism for the solution of the U (1) problem can be made
explicit in the Coulomb gauge. In the Schwinger model, in the Coulomb gauge
one has K 0 = (e/π)A 1 = 0, K 1 = (e/π)A 0 , so that J
5
0 = j
5
0 and the (θ-)vacuum
expectations of the commutators [J
5
0 ( f R , t), A], [ j
5
0 ( f R , t), A], A ∈ F obs , coincide
and describe the same mass spectrum; however, the time dependence in the limit
R → ∞ in the first case can be ascribed to the non-locality of the conserved axial
current, whereas in the second case it reflects the non- conservation of j
5
μ . A more
detailed discussion of the U (1) problem and its solution is given in Appendix F below.
211 R.J. Crewter, in Field Theoretical methods in Particle Physics, W. Rühl ed. Reidel (1980), p. 529.