268
Appendix F: U(1) Problem Solved by Gauge Group Topology
of the gauge group rather than the topological classification of the instanton solutions.
52
In the sequel, we shall follow Jackiw’s proposal, with attention to the mathematical
points needed for improving his approach.
F.3 Gauss Constraint and Field Algebra Representations
A non-perturbative derivation of the vacuum structure in QCD and the related solution
of the U (1) problem hinge on the classification of the vacuum representations of the
observable fields. As discussed before, this is conveniently dealt with the use of a
field algebra and the discussion of its vacuum representations.
To this purpose the temporal gauge is particularly suitable because (in its standard
formulation), it satisfies locality of the fields and positivity of the vacuum representation; moreover, see below, the Gauss law constraint for the states is equivalent to
local gauge invariance.
Formally, the temporal gauge is defined by the gauge condition A
a
0 = 0; correspondingly, one may consider the local field algebra F generated by the fields A
a
i , ψ,
including their polynomial functions at a point x, defined through a suitable point
splitting procedure.
The formal Lagrangian is (sum over repeated indices understood)
L =
1
2
a
(E
2
a − B
2
a ) + ¯
ψiγ
μ D μ ψ,
(F.11)
E a = − ˙
A a , B a = ∇ × A a −
1
2 f abc A b × A c .
The residual local gauge group, still denoted by G, is parametrized by G-valued
C
∞ unitary functions U(x), differing from the identity only on a compact set, K U ⊂
R
3 ; the correspondent (time-independent) localized gauge transformations α U of the
fields are
α U (A(f)) = A(U f U
−1
) + U ∂ U
−1
(f), α U (ψ(x)) = U(x) ψ ψ(x),
U ∂ U
−1
(f) =
d
4 x Tr [ U(x)∂
i
U
−1
(x) f
i
(x) ],
(F.12)
52 R. Jackiw, Topological Investigations of Quantized Gauge theories, in Current Algebra and
Anomalies, S.B. Treiman, R. Jackiw, B. Zumino and E. Witten eds., World Scientific 1985, pp. 211–
359, hereafter referred to as R. Jackiw [1985]. The idea of exploiting the topology of the gauge
group has been proven fruitful also for the treatment of the Schwinger model and the derivation of
its vacuum structure; for a review, see, e.g. F. Strocchi, Selected topics on the general properties of
quantum field theory, World Scientific 1993, Sect. 7.4.
Appendix F: U(1) Problem Solved by Gauge Group Topology
of the gauge group rather than the topological classification of the instanton solutions.
52
In the sequel, we shall follow Jackiw’s proposal, with attention to the mathematical
points needed for improving his approach.
F.3 Gauss Constraint and Field Algebra Representations
A non-perturbative derivation of the vacuum structure in QCD and the related solution
of the U (1) problem hinge on the classification of the vacuum representations of the
observable fields. As discussed before, this is conveniently dealt with the use of a
field algebra and the discussion of its vacuum representations.
To this purpose the temporal gauge is particularly suitable because (in its standard
formulation), it satisfies locality of the fields and positivity of the vacuum representation; moreover, see below, the Gauss law constraint for the states is equivalent to
local gauge invariance.
Formally, the temporal gauge is defined by the gauge condition A
a
0 = 0; correspondingly, one may consider the local field algebra F generated by the fields A
a
i , ψ,
including their polynomial functions at a point x, defined through a suitable point
splitting procedure.
The formal Lagrangian is (sum over repeated indices understood)
L =
1
2
a
(E
2
a − B
2
a ) + ¯
ψiγ
μ D μ ψ,
(F.11)
E a = − ˙
A a , B a = ∇ × A a −
1
2 f abc A b × A c .
The residual local gauge group, still denoted by G, is parametrized by G-valued
C
∞ unitary functions U(x), differing from the identity only on a compact set, K U ⊂
R
3 ; the correspondent (time-independent) localized gauge transformations α U of the
fields are
α U (A(f)) = A(U f U
−1
) + U ∂ U
−1
(f), α U (ψ(x)) = U(x) ψ ψ(x),
U ∂ U
−1
(f) =
d
4 x Tr [ U(x)∂
i
U
−1
(x) f
i
(x) ],
(F.12)
52 R. Jackiw, Topological Investigations of Quantized Gauge theories, in Current Algebra and
Anomalies, S.B. Treiman, R. Jackiw, B. Zumino and E. Witten eds., World Scientific 1985, pp. 211–
359, hereafter referred to as R. Jackiw [1985]. The idea of exploiting the topology of the gauge
group has been proven fruitful also for the treatment of the Schwinger model and the derivation of
its vacuum structure; for a review, see, e.g. F. Strocchi, Selected topics on the general properties of
quantum field theory, World Scientific 1993, Sect. 7.4.