Appendix F: U(1) Problem Solved by Gauge Group Topology
269
where the matrix notation has been used also for the C
∞ test functions of fast decrease
( f
i
(x) = f
i
a (x) T
a ) and U(x) ψ denotes the matrix representation of G on ψ(x, t).
We then define the Gauss subgroup G 0 ⊂ G as the group generated by the oneparameter subgroups parametrized by unitary gauge functions
U(λg) = e
iλ g(x)
, λ ∈ R, g(x) = g a (x) T
a
, g a ∈ D(R
3
),
continuously connected to the identity, (in the following, for simplicity, we shall
often adopt the short-hand notation U(g), or U g ).
The infinitesimal transformations of the Gauss subgroup are formally generated
by the Gauss operators
G
a
≡ (D · E)
a
− j
a
0 ,
j
a
μ = i ¯
ψ γ μ t
a
ψ,
(F.13)
in the sense that δ
g a F = i [ G
a
(g
a
), F ].
The next delicate step is to analyse the representations of the field algebra defined
by a vacuum state ω 0 satisfying the Gauss law constraint (Gauss invariant vacuum).
Here, a subtle, but crucial, mathematical obstruction appears, since the Gauss constraint excludes the possibility of a representation of F by Hilbert space operators
(see below). The way out is to consider the exponential algebra F W associated with
F, generated by the formal exponentials of the local fields: W (λf) ∼ e
iA(λf) , λ ∈ R,
and the formal exponentials of the compound local fields, like the Gauss field G
a ,
the currents j
5
μ , J
5
μ , etc. The algebraic structure of F W is defined by the algebraic
relations which reflect those of the corresponding formal exponentials.
Proposition F.1 The representation of F W (in a Hilbert space H) defined by (the
correlation functions of) a Gauss invariant vacuum state ω 0 is non-regular, since
ω 0 (W (λ f)) = 0, if λf = 0,
(F.14)
so that the expectations of the Weyl exponentials W (λf), λ ∈ R, are not weakly
continuous in λ. Hence, the fields A(f), the formal generators of W (λf), cannot be
represented by operators in H.
Proof. For each given f, f
i
=
a f
i
a T
a , one may find a one-parameter subgroup
U(λg) ∈ G 0 , such that U(λg) f U(−λg) = f, as well as exp i (U(λg) ∂ U(−λg)(f))
= 1. Then, one has
ω 0 (W (λf)) = ω 0 (V (U(λg))W (λf)V (U(−λg))) =
= ω 0 (W (λf)) exp i(U(λg)∂U(−λg)(f)),
and Eq. (F.14) follows.
This fact, far from being a mere technical subtlety, will be shown to play a crucial
role for the derivation of relevant physical properties.
269
where the matrix notation has been used also for the C
∞ test functions of fast decrease
( f
i
(x) = f
i
a (x) T
a ) and U(x) ψ denotes the matrix representation of G on ψ(x, t).
We then define the Gauss subgroup G 0 ⊂ G as the group generated by the oneparameter subgroups parametrized by unitary gauge functions
U(λg) = e
iλ g(x)
, λ ∈ R, g(x) = g a (x) T
a
, g a ∈ D(R
3
),
continuously connected to the identity, (in the following, for simplicity, we shall
often adopt the short-hand notation U(g), or U g ).
The infinitesimal transformations of the Gauss subgroup are formally generated
by the Gauss operators
G
a
≡ (D · E)
a
− j
a
0 ,
j
a
μ = i ¯
ψ γ μ t
a
ψ,
(F.13)
in the sense that δ
g a F = i [ G
a
(g
a
), F ].
The next delicate step is to analyse the representations of the field algebra defined
by a vacuum state ω 0 satisfying the Gauss law constraint (Gauss invariant vacuum).
Here, a subtle, but crucial, mathematical obstruction appears, since the Gauss constraint excludes the possibility of a representation of F by Hilbert space operators
(see below). The way out is to consider the exponential algebra F W associated with
F, generated by the formal exponentials of the local fields: W (λf) ∼ e
iA(λf) , λ ∈ R,
and the formal exponentials of the compound local fields, like the Gauss field G
a ,
the currents j
5
μ , J
5
μ , etc. The algebraic structure of F W is defined by the algebraic
relations which reflect those of the corresponding formal exponentials.
Proposition F.1 The representation of F W (in a Hilbert space H) defined by (the
correlation functions of) a Gauss invariant vacuum state ω 0 is non-regular, since
ω 0 (W (λ f)) = 0, if λf = 0,
(F.14)
so that the expectations of the Weyl exponentials W (λf), λ ∈ R, are not weakly
continuous in λ. Hence, the fields A(f), the formal generators of W (λf), cannot be
represented by operators in H.
Proof. For each given f, f
i
=
a f
i
a T
a , one may find a one-parameter subgroup
U(λg) ∈ G 0 , such that U(λg) f U(−λg) = f, as well as exp i (U(λg) ∂ U(−λg)(f))
= 1. Then, one has
ω 0 (W (λf)) = ω 0 (V (U(λg))W (λf)V (U(−λg))) =
= ω 0 (W (λf)) exp i(U(λg)∂U(−λg)(f)),
and Eq. (F.14) follows.
This fact, far from being a mere technical subtlety, will be shown to play a crucial
role for the derivation of relevant physical properties.