Appendix F: U(1) Problem Solved by Gauge Group Topology
275
F.6 Topological Group, Its Spectrum and θ Vacua
As discussed before, an important result of the standard approach is the derivation of
a θ angle structure of the vacuum; in this section, we shall derive a θ vacuum structure
merely from the topology of the gauge group, rather than from the topology of the
instanton solutions.
As a first step, we show that the topology of the gauge group is described by an
abelian group T , with elements T n classified by the (topological) winding number
n, and that the T n commute with the gauge transformations.
The group T is defined as the quotient T ≡ G/G 0 , which is a well defined group,
since G 0 is a normal subgroup of G, i.e. invariant under conjugation by elements of G:
g G 0 g
−1
= G 0 , ∀g ∈ G; elements of T are the equivalence classes T n , labelled by the
topological winding number n and the product is defined by the coset multiplication
gG 0 hG 0 = ghG 0 .
Clearly, T n T m = T n+m , so that T is an abelian group which encodes the topology
of G (topological abelian group).
Since the conjugation g
n → g g
n g
−1 , g, g
n ∈ G, does not change the winding number n, the elements T n are invariant under gauge transformations (technically under the left/right action of the group G on the left/right coset space [g
G 0 ]:
g [g
G 0 ] g
−1
= [g g
g
−1
G 0 ] = [g
G 0 ]).
In conclusion, the topology of the gauge group G is described by an abelian group
T , pointwise invariant under the gauge group.
The next step is the proof that, in the Hilbert space H (of the temporal gauge), the
topological group is represented by gauge invariant operators T n , which reduce to
unitary operators in the physical space H
. The proof of the existence of such a representation relies on the localization of the gauge transformations which characterizes
our analysis. This innocent-looking derivation, not available in Jackiw analysis, plays
a substantial role since the conclusions will hinge on the non-trivial spectrum of such
(existing) operators.
Proposition F.4 In a representation π of F W defined by a Gauss invariant vacuum,
the topological abelian group T is implemented by gauge invariant operators T n ,
labelled by the topological winding number n. The T n commute with the observable
fields, belong to the centre of the algebra of observables and reduce to unitary
operators on the physical space H
.
Proof. By Proposition F.2, G is implemented in π by unitary operators V (U) and,
denoting by P 0 the projection on the subspace of Gauss invariant states (i.e. on the
physical subspace H
), we define
T n ≡ P 0 V (U n ) P 0 .
(F.23)
a) T n depends only on the equivalence class of V (U n ).
In fact, given V (U n ) any other gauge function U
n with winding number n may
be written as U
n = U n U(g), with U(g) an element of the Gauss subgroup, so
275
F.6 Topological Group, Its Spectrum and θ Vacua
As discussed before, an important result of the standard approach is the derivation of
a θ angle structure of the vacuum; in this section, we shall derive a θ vacuum structure
merely from the topology of the gauge group, rather than from the topology of the
instanton solutions.
As a first step, we show that the topology of the gauge group is described by an
abelian group T , with elements T n classified by the (topological) winding number
n, and that the T n commute with the gauge transformations.
The group T is defined as the quotient T ≡ G/G 0 , which is a well defined group,
since G 0 is a normal subgroup of G, i.e. invariant under conjugation by elements of G:
g G 0 g
−1
= G 0 , ∀g ∈ G; elements of T are the equivalence classes T n , labelled by the
topological winding number n and the product is defined by the coset multiplication
gG 0 hG 0 = ghG 0 .
Clearly, T n T m = T n+m , so that T is an abelian group which encodes the topology
of G (topological abelian group).
Since the conjugation g
n → g g
n g
−1 , g, g
n ∈ G, does not change the winding number n, the elements T n are invariant under gauge transformations (technically under the left/right action of the group G on the left/right coset space [g
G 0 ]:
g [g
G 0 ] g
−1
= [g g
g
−1
G 0 ] = [g
G 0 ]).
In conclusion, the topology of the gauge group G is described by an abelian group
T , pointwise invariant under the gauge group.
The next step is the proof that, in the Hilbert space H (of the temporal gauge), the
topological group is represented by gauge invariant operators T n , which reduce to
unitary operators in the physical space H
. The proof of the existence of such a representation relies on the localization of the gauge transformations which characterizes
our analysis. This innocent-looking derivation, not available in Jackiw analysis, plays
a substantial role since the conclusions will hinge on the non-trivial spectrum of such
(existing) operators.
Proposition F.4 In a representation π of F W defined by a Gauss invariant vacuum,
the topological abelian group T is implemented by gauge invariant operators T n ,
labelled by the topological winding number n. The T n commute with the observable
fields, belong to the centre of the algebra of observables and reduce to unitary
operators on the physical space H
.
Proof. By Proposition F.2, G is implemented in π by unitary operators V (U) and,
denoting by P 0 the projection on the subspace of Gauss invariant states (i.e. on the
physical subspace H
), we define
T n ≡ P 0 V (U n ) P 0 .
(F.23)
a) T n depends only on the equivalence class of V (U n ).
In fact, given V (U n ) any other gauge function U
n with winding number n may
be written as U
n = U n U(g), with U(g) an element of the Gauss subgroup, so