272
Appendix F: U(1) Problem Solved by Gauge Group Topology
By a standard result (see Chap. 11), the invariance of ω 0 under G implies that, in the
representation of F W defined by ω 0 , G is implemented by unitary operators V (U),
which are determined up to elements of the commutant of F W , and therefore unique
(apart from multiples of the identity) if F W is irreducible in the Hilbert space H of
such a representation.
A crucial role in the standard approach and in particular in Jackiw analysis is
played by the so-called topological current C μ
C
μ
(x) = −(16π
2
)
−1
ε
μνρσ Tr [F νρ (x)A σ (x) −
2
3 A ν (x) A ρ (x) A σ (x)],
(F.17)
∂ μ C
μ
(x) = −(16π
2
)
−1 1
2 ε
μνρσ Tr [F μν (x) F ρσ (x)] ≡ P(x).
In the mathematical literature on classical fields, P is called the “Pontryagin density”
and C μ the “Chern–Simons secondary characteristic class”.
55 At the classical level,
one may easily prove that
α U (C 0 (x)) = C 0 (x) − (8π
2
)
−1
∂ i ε
i jk Tr [∂ j U(x) U(x)
−1 A k ] + n U (x).
(F.18)
The quantum version of the above equations requires some point splitting regularization for the products of fields at the same point x; it is reasonable to assume (as
implicit in the standard approach) that this can be done by preserving the transformation property of Eq. (F.18).
Moreover, an argument similar to that of Proposition F.1 excludes the possibility of defining C 0 ( f R α) (and a related topological charge) as an operator in
H, and, as elements of F W , one may rather consider only its formal exponentials
V
C
( f R α) ∼ exp iC 0 ( f R α), with gauge transformation properties which reflect those
of C 0 ( f R α).
56
This means that given a gauge transformation U, with space support K U , for any
f R such that f R (x) = 1 for x ∈ K U , so that f R ∂ j U = ∂ j U and ∂ i f R ∂ j U = 0, one
has
α U (V
C
( f R α)) = V (U) V
C
( f R α) V (U)
−1
= e
in U V
C
( f R α).
(F.19)
The above equation may suggest that the operators V
C
( f R α) are good candidates
for displaying the effects of the non-trivial topology of G on the physical states, as
argued by Jackiw. Unfortunately, as a consequence of the invariance of the physical
55 See, e.g. S. Coleman [1985], Chap. 7; R. Jackiw [1985], especially Sect. 3.
56 See G. Morchio and F.Strocchi, Ann. Phys. 324, 2236 (2009); the content of this Appendix relies
on this reference. See also F. Strocchi [13, 16], Chap. 8. The suggested escape of considering
non-normalizable states, in particular a non-normalizable vacuum vector, is incompatible with the
existence of a vacuum representation of the observable algebra which contains the identity; for
such mathematical inconsistencies, see also the discussion in 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
By a standard result (see Chap. 11), the invariance of ω 0 under G implies that, in the
representation of F W defined by ω 0 , G is implemented by unitary operators V (U),
which are determined up to elements of the commutant of F W , and therefore unique
(apart from multiples of the identity) if F W is irreducible in the Hilbert space H of
such a representation.
A crucial role in the standard approach and in particular in Jackiw analysis is
played by the so-called topological current C μ
C
μ
(x) = −(16π
2
)
−1
ε
μνρσ Tr [F νρ (x)A σ (x) −
2
3 A ν (x) A ρ (x) A σ (x)],
(F.17)
∂ μ C
μ
(x) = −(16π
2
)
−1 1
2 ε
μνρσ Tr [F μν (x) F ρσ (x)] ≡ P(x).
In the mathematical literature on classical fields, P is called the “Pontryagin density”
and C μ the “Chern–Simons secondary characteristic class”.
55 At the classical level,
one may easily prove that
α U (C 0 (x)) = C 0 (x) − (8π
2
)
−1
∂ i ε
i jk Tr [∂ j U(x) U(x)
−1 A k ] + n U (x).
(F.18)
The quantum version of the above equations requires some point splitting regularization for the products of fields at the same point x; it is reasonable to assume (as
implicit in the standard approach) that this can be done by preserving the transformation property of Eq. (F.18).
Moreover, an argument similar to that of Proposition F.1 excludes the possibility of defining C 0 ( f R α) (and a related topological charge) as an operator in
H, and, as elements of F W , one may rather consider only its formal exponentials
V
C
( f R α) ∼ exp iC 0 ( f R α), with gauge transformation properties which reflect those
of C 0 ( f R α).
56
This means that given a gauge transformation U, with space support K U , for any
f R such that f R (x) = 1 for x ∈ K U , so that f R ∂ j U = ∂ j U and ∂ i f R ∂ j U = 0, one
has
α U (V
C
( f R α)) = V (U) V
C
( f R α) V (U)
−1
= e
in U V
C
( f R α).
(F.19)
The above equation may suggest that the operators V
C
( f R α) are good candidates
for displaying the effects of the non-trivial topology of G on the physical states, as
argued by Jackiw. Unfortunately, as a consequence of the invariance of the physical
55 See, e.g. S. Coleman [1985], Chap. 7; R. Jackiw [1985], especially Sect. 3.
56 See G. Morchio and F.Strocchi, Ann. Phys. 324, 2236 (2009); the content of this Appendix relies
on this reference. See also F. Strocchi [13, 16], Chap. 8. The suggested escape of considering
non-normalizable states, in particular a non-normalizable vacuum vector, is incompatible with the
existence of a vacuum representation of the observable algebra which contains the identity; for
such mathematical inconsistencies, see also the discussion in F. Strocchi, Gauge Invariance and
Weyl-polymer Quantization, Springer Lecture Notes in Physics 904, 2016.