266
Appendix F: U(1) Problem Solved by Gauge Group Topology
gauge symmetries of the standard model, being generated by gauge dependent currents. Given the existence of the U (1) A transformations of the observable fields, a
gauge independent fact, no matter how its actual existence is proved, the real issue
is the mechanism for evading the Goldstone theorem, for which the non-abelianess
of the gauge group should play a decisive role.
F.2 The Semi-classical Instanton Approximation
The standard solution of the U (1) problem is based on the assumed instanton dominance of the functional integral and on the topological classification of the instanton
solutions.
49 The basic assumptions are
i) the structure of the Euclidean functional integral is governed by Euclidean field
configurations with finite Euclidean action;
ii) the relevant configurations are assumed to be continuous functions with the
following asymptotic behaviour, when the Euclidean radial variable r → ∞:
A μ ∼ g ∂ μ g
−1
+ O(1/r
2
),
(F.8)
where g is a pure gauge configuration depending only on the Euclidean angular
variables Ω, g = g(Ω);
iii) the instanton configurations are classified by their winding number n, which is
a topological invariant,
n = −(24π
2
)
−1
dθ 1 dθ 2 dθ 3 ε
i jk Tr[ g∂ i g
−1 g∂ j g
−1 g∂ k g
−1
],
(F.9)
where θ i , i = 1, 2, 3 are three angles which parametrize the 3-sphere S
3 and ∂ i
the corresponding partial derivatives;
iv) the functional integral may be evaluated by first integrating over the class of
Euclidean configurations with given winding number n and then by summing
over n, with a weighting factor e
i θ n , where θ is a free parameter, the so-called
θ angle.
The presence of such a factor may be interpreted as the addition of the following
term to the classical Euclidean action density
L θ = i(16π
2
)
−1
θ ε
μνρσ Tr [ F μν F ρσ ],
(F.10)
(the so-called θ term or topological term) .
49 For excellent reviews, see S. Coleman, Aspects of Symmetry, Cambridge Univ. Press 1985, Chap. 7,
hereafter referred to as S. Coleman [1985] and S. Weinberg [1996], Chap. 23. A brief account of
the main ideas is given in F. Strocchi [13, 16].
Appendix F: U(1) Problem Solved by Gauge Group Topology
gauge symmetries of the standard model, being generated by gauge dependent currents. Given the existence of the U (1) A transformations of the observable fields, a
gauge independent fact, no matter how its actual existence is proved, the real issue
is the mechanism for evading the Goldstone theorem, for which the non-abelianess
of the gauge group should play a decisive role.
F.2 The Semi-classical Instanton Approximation
The standard solution of the U (1) problem is based on the assumed instanton dominance of the functional integral and on the topological classification of the instanton
solutions.
49 The basic assumptions are
i) the structure of the Euclidean functional integral is governed by Euclidean field
configurations with finite Euclidean action;
ii) the relevant configurations are assumed to be continuous functions with the
following asymptotic behaviour, when the Euclidean radial variable r → ∞:
A μ ∼ g ∂ μ g
−1
+ O(1/r
2
),
(F.8)
where g is a pure gauge configuration depending only on the Euclidean angular
variables Ω, g = g(Ω);
iii) the instanton configurations are classified by their winding number n, which is
a topological invariant,
n = −(24π
2
)
−1
dθ 1 dθ 2 dθ 3 ε
i jk Tr[ g∂ i g
−1 g∂ j g
−1 g∂ k g
−1
],
(F.9)
where θ i , i = 1, 2, 3 are three angles which parametrize the 3-sphere S
3 and ∂ i
the corresponding partial derivatives;
iv) the functional integral may be evaluated by first integrating over the class of
Euclidean configurations with given winding number n and then by summing
over n, with a weighting factor e
i θ n , where θ is a free parameter, the so-called
θ angle.
The presence of such a factor may be interpreted as the addition of the following
term to the classical Euclidean action density
L θ = i(16π
2
)
−1
θ ε
μνρσ Tr [ F μν F ρσ ],
(F.10)
(the so-called θ term or topological term) .
49 For excellent reviews, see S. Coleman, Aspects of Symmetry, Cambridge Univ. Press 1985, Chap. 7,
hereafter referred to as S. Coleman [1985] and S. Weinberg [1996], Chap. 23. A brief account of
the main ideas is given in F. Strocchi [13, 16].