270
Appendix F: U(1) Problem Solved by Gauge Group Topology
The occurrence of non-regular representations of Weyl/exponential algebras
should not be considered as an odd, or even pathological, feature, since it is an
inevitable consequence of a quantization defined by a gauge invariant ground/vacuum
state.
In fact, it occurs in the description of interesting systems, like the electron in a
periodic potential (Bloch electron), the Quantum Hall electron, the quantum particle
on a circle, etc.; it also characterizes the so-called polymer representations of Loop
Quantum Gravity defined by diffeomorphism invariant vacuum states.
53
Thus, the mathematically correct formulation of the temporal gauge should make
reference to the local field algebra F W , (briefly called local Weyl algebra), with the
chiral symmetry defined by Eq. (F.7) and consider its representations defined by a
Gauss invariant vacuum state ω 0
ω 0 (α U (g) (F)) = ω 0 (F), ∀ U(g) ∈ G 0 , ∀ F ∈ F W .
As discussed in Chap. 11, this implies that the Gauss transformations α U (λg) are
implemented by (local) one-parameter unitary operators V (U(λg)) (in the Hilbert
space H defined by the vacuum correlation functions of ω 0 ). Then, the subsidiary
condition which selects the subspace H
of physical state vectors reads
V (U(λg))Ψ = Ψ, ∀ U(λg) ∈ G 0 , Ψ ∈ H
⊂ H.
(F.15)
The operators V (U(λg)) are formally the exponentials of the Gauss operator
V (U(λg)) ∼ e
iλ G(g)
, G(g) ≡
a
G a (g a ), g a ∈ D(R
3
).
The characterization of the local gauge group in terms of space localized gauge
functions deserves special consideration, since it will allow to fully exploit the local
structure of the theory. In the standard approach, in order to account for the asymptotic
behaviour of the (non-local) instanton solutions one has to consider non-local gauge
functions.
F.4 The Topology of the Local Gauge Group
The topological classification of the C
∞ local gauge functions, which characterize
the residual gauge group, is much better founded and simpler than investing in the
analysis of the topology of the finite action configurations, which relies on the instanton semi-classical approximation and on assumptions on their asymptotic behaviour.
53 For a general account of non-regular representations of Weyl algebras, see F. Strocchi, Gauge
Invariance and Weyl-polymer Quantization, Springer Lecture Notes in Physics 904, 2016.
Appendix F: U(1) Problem Solved by Gauge Group Topology
The occurrence of non-regular representations of Weyl/exponential algebras
should not be considered as an odd, or even pathological, feature, since it is an
inevitable consequence of a quantization defined by a gauge invariant ground/vacuum
state.
In fact, it occurs in the description of interesting systems, like the electron in a
periodic potential (Bloch electron), the Quantum Hall electron, the quantum particle
on a circle, etc.; it also characterizes the so-called polymer representations of Loop
Quantum Gravity defined by diffeomorphism invariant vacuum states.
53
Thus, the mathematically correct formulation of the temporal gauge should make
reference to the local field algebra F W , (briefly called local Weyl algebra), with the
chiral symmetry defined by Eq. (F.7) and consider its representations defined by a
Gauss invariant vacuum state ω 0
ω 0 (α U (g) (F)) = ω 0 (F), ∀ U(g) ∈ G 0 , ∀ F ∈ F W .
As discussed in Chap. 11, this implies that the Gauss transformations α U (λg) are
implemented by (local) one-parameter unitary operators V (U(λg)) (in the Hilbert
space H defined by the vacuum correlation functions of ω 0 ). Then, the subsidiary
condition which selects the subspace H
of physical state vectors reads
V (U(λg))Ψ = Ψ, ∀ U(λg) ∈ G 0 , Ψ ∈ H
⊂ H.
(F.15)
The operators V (U(λg)) are formally the exponentials of the Gauss operator
V (U(λg)) ∼ e
iλ G(g)
, G(g) ≡
a
G a (g a ), g a ∈ D(R
3
).
The characterization of the local gauge group in terms of space localized gauge
functions deserves special consideration, since it will allow to fully exploit the local
structure of the theory. In the standard approach, in order to account for the asymptotic
behaviour of the (non-local) instanton solutions one has to consider non-local gauge
functions.
F.4 The Topology of the Local Gauge Group
The topological classification of the C
∞ local gauge functions, which characterize
the residual gauge group, is much better founded and simpler than investing in the
analysis of the topology of the finite action configurations, which relies on the instanton semi-classical approximation and on assumptions on their asymptotic behaviour.
53 For a general account of non-regular representations of Weyl algebras, see F. Strocchi, Gauge
Invariance and Weyl-polymer Quantization, Springer Lecture Notes in Physics 904, 2016.