276
Appendix F: U(1) Problem Solved by Gauge Group Topology
that
T
n = P 0 V (U
n ) P 0 = P 0 V (U n ) V (U(g)) P 0 = P 0 V (U n ) P 0 = T n .
b) T n T m = T n+m , T 0 = P 0 .
In fact, for any U(g) ∈ G 0 , V (U n )
∗ V (U(g))V (U n ) = V (U(g
)) implies
V (U(g)) V (U n ) P 0 Ψ = V (U n ) V (U(g
)) P 0 Ψ = V (U n ) P 0 Ψ , i.e. V (U n )P 0 H ∈
H
, P 0 V (U n ) P 0 = V (U n ) P 0 , and similarly for V (U n )
∗ , so that
P 0 V (U) = (V (U)
∗ P 0 )
∗
= (P 0 V (U)
∗ P 0 )
∗
= P 0 V (U)P 0 = V (U) P 0 .
Then, one has
T n T m = P 0 V (U n ) P 0 V (U m ) P 0 = P 0 V (U n ) V (U m ) P 0 =
= P 0 V (U
n+m ) P 0 = T n+m .
Furthermore, since T
∗
n = T −n , one has T
∗
n T n = T n T
∗
n = P 0 and therefore T n
reduces to a unitary operator in the physical space.
c) V (U) T n V (U)
−1
= T n , i.e. the T n are gauge invariant.
In fact, since [ V (U), P 0 ] = 0, one has
V (U) T n V (U)
−1
= P 0 V (U) V (U n ) V (U)
−1 P 0 = P 0 V (U
n ) P 0 = T n .
In conclusion, the topological group T is represented by the gauge invariant operators T n , (therefore belonging to the weak closure of the algebra of observables),
with the identity represented by T 0 .
d) The T n belong to the centre Z of the observables.
Since the observable fields commute with the gauge transformations and with
P 0 , (a spectral projection of the Gauss group), by Eq. (F.23) they commute with
the T n ; then, the T n belong to the centre Z of the observables.
A final basic issue is the spectrum of T ; in order to derive a non-trivial θ structure
of the physical states, one must exclude that in any representation of the local field
algebra F W defined by a Gauss invariant vacuum the T n reduce to one on the corresponding subspace H
of physical states. To this purpose, the presence of fermions
and the related chiral symmetry plays a crucial role.
59
59 This issue is not taken care of by Jackiw analysis, which claims a non-trivial θ structure even in
the absence of fermions.
Appendix F: U(1) Problem Solved by Gauge Group Topology
that
T
n = P 0 V (U
n ) P 0 = P 0 V (U n ) V (U(g)) P 0 = P 0 V (U n ) P 0 = T n .
b) T n T m = T n+m , T 0 = P 0 .
In fact, for any U(g) ∈ G 0 , V (U n )
∗ V (U(g))V (U n ) = V (U(g
)) implies
V (U(g)) V (U n ) P 0 Ψ = V (U n ) V (U(g
)) P 0 Ψ = V (U n ) P 0 Ψ , i.e. V (U n )P 0 H ∈
H
, P 0 V (U n ) P 0 = V (U n ) P 0 , and similarly for V (U n )
∗ , so that
P 0 V (U) = (V (U)
∗ P 0 )
∗
= (P 0 V (U)
∗ P 0 )
∗
= P 0 V (U)P 0 = V (U) P 0 .
Then, one has
T n T m = P 0 V (U n ) P 0 V (U m ) P 0 = P 0 V (U n ) V (U m ) P 0 =
= P 0 V (U
n+m ) P 0 = T n+m .
Furthermore, since T
∗
n = T −n , one has T
∗
n T n = T n T
∗
n = P 0 and therefore T n
reduces to a unitary operator in the physical space.
c) V (U) T n V (U)
−1
= T n , i.e. the T n are gauge invariant.
In fact, since [ V (U), P 0 ] = 0, one has
V (U) T n V (U)
−1
= P 0 V (U) V (U n ) V (U)
−1 P 0 = P 0 V (U
n ) P 0 = T n .
In conclusion, the topological group T is represented by the gauge invariant operators T n , (therefore belonging to the weak closure of the algebra of observables),
with the identity represented by T 0 .
d) The T n belong to the centre Z of the observables.
Since the observable fields commute with the gauge transformations and with
P 0 , (a spectral projection of the Gauss group), by Eq. (F.23) they commute with
the T n ; then, the T n belong to the centre Z of the observables.
A final basic issue is the spectrum of T ; in order to derive a non-trivial θ structure
of the physical states, one must exclude that in any representation of the local field
algebra F W defined by a Gauss invariant vacuum the T n reduce to one on the corresponding subspace H
of physical states. To this purpose, the presence of fermions
and the related chiral symmetry plays a crucial role.
59
59 This issue is not taken care of by Jackiw analysis, which claims a non-trivial θ structure even in
the absence of fermions.