29.3 Higgs Mechanism in the Coulomb Gauge
211
exist but are α dependent and therefore the time-independent U (1) global gauge
group cannot be generated by such integrals of the charge density.
Proof. The convergence and time dependence of the charge density commutators
[ j 0 ( f R , x 0 ), ϕ(y)] = [∂
i F 0i ( f R , x 0 ), ϕ(y)]
are governed by the large (space-like) distance behaviour of the commutator
[F 0 j , ϕ(y)], i.e. of [F 0 j (x), e
−ie(−
−1 ∂ i A
i )(y)
], as a consequence of (29.13).
Now, for space-like separations |x| → ∞, one has the following general estimate,
(obtained by expanding the exponential and by exploiting the cluster property of the
correlation functions of the FGB fields),
[F μν (x), ϕ(y)] ∼
ie
4π
d
3 z∂
j
z
1
|y − z|
< [F μν (x), A j (z, y 0 )] > ϕ(y), (29.15)
the correction being at least O(|x|
−4
).
204 This and all the following equations are
understood to hold in matrix elements of Coulomb states Ψ, , ∈ F C Ψ 0 .
Since < [F μν (x), A j (z)] >= (∂ ν g μ j − ∂ μ g ν j )F(x − z), where F is the Lorentz
invariant distribution which characterizes the vacuum expectation of the electromagnetic field commutator
< [F μν (
1
2
x), F λσ (−
1
2
x)] > 0 = id μνλσ
dρ(m
2
))(x; m
2
) ≡ d μνλσ F(x),
(29.16)
d μνλσ ≡ g νλ ∂ μ ∂ σ + g μσ ∂ ν ∂ λ − g νσ ∂ μ ∂ λ − g μλ ∂ ν ∂ σ , one has, for R → ∞,
[ j 0 ( f R , x 0 ), ϕ(y)] = [∂
i F 0i ( f R , x 0 ), ϕ(y)] ∼
∼ −ie∂ 0
d
3 x f R (x)F(x − y)ϕ(y).
(29.17)
204 The point is that, by locality of the FGB fields, the commutator [F μν (x + a), A j (z, y 0 )] has a
compact support in z and therefore, for |a| → ∞, the convolution with ∂
j
z
1
|y−z| decreases at least
as |a| −2 . By the same reasons,
d
3 z ∂
j
z
1
|y − z|
[F μν (x + a), A j (z, y 0 )]
commutes with the other factors (− −1 ∂ j A j )(y), in the expansion of the exponential, apart from
terms decreasing at least as |a| −4 . Thus, for |a| → ∞
[F 0k (x + a), e
−ie(− −1 ∂ i A i )(y) ] ∼
∼
d
3 z ∂
j
z
1
|y − z|
[F 0k (x + a), A j (z, y 0 )]e
−ie(− −1 ∂ i A i )(y) ,
and, by the cluster property of the FGB correlation functions, the vacuum insertion gives the leading
contribution. For a proof of this behaviour, which takes into account the need of a UV regularization
of (29.13) and exploits the locality of the charged fields in the FGB gauge and the cluster property,
see D. Buchholz, S. Doplicher, G. Morchio, J.E. Roberts and F. Strocchi, Ann. Phys. 290, 53 (2001).
211
exist but are α dependent and therefore the time-independent U (1) global gauge
group cannot be generated by such integrals of the charge density.
Proof. The convergence and time dependence of the charge density commutators
[ j 0 ( f R , x 0 ), ϕ(y)] = [∂
i F 0i ( f R , x 0 ), ϕ(y)]
are governed by the large (space-like) distance behaviour of the commutator
[F 0 j , ϕ(y)], i.e. of [F 0 j (x), e
−ie(−
−1 ∂ i A
i )(y)
], as a consequence of (29.13).
Now, for space-like separations |x| → ∞, one has the following general estimate,
(obtained by expanding the exponential and by exploiting the cluster property of the
correlation functions of the FGB fields),
[F μν (x), ϕ(y)] ∼
ie
4π
d
3 z∂
j
z
1
|y − z|
< [F μν (x), A j (z, y 0 )] > ϕ(y), (29.15)
the correction being at least O(|x|
−4
).
204 This and all the following equations are
understood to hold in matrix elements of Coulomb states Ψ, , ∈ F C Ψ 0 .
Since < [F μν (x), A j (z)] >= (∂ ν g μ j − ∂ μ g ν j )F(x − z), where F is the Lorentz
invariant distribution which characterizes the vacuum expectation of the electromagnetic field commutator
< [F μν (
1
2
x), F λσ (−
1
2
x)] > 0 = id μνλσ
dρ(m
2
))(x; m
2
) ≡ d μνλσ F(x),
(29.16)
d μνλσ ≡ g νλ ∂ μ ∂ σ + g μσ ∂ ν ∂ λ − g νσ ∂ μ ∂ λ − g μλ ∂ ν ∂ σ , one has, for R → ∞,
[ j 0 ( f R , x 0 ), ϕ(y)] = [∂
i F 0i ( f R , x 0 ), ϕ(y)] ∼
∼ −ie∂ 0
d
3 x f R (x)F(x − y)ϕ(y).
(29.17)
204 The point is that, by locality of the FGB fields, the commutator [F μν (x + a), A j (z, y 0 )] has a
compact support in z and therefore, for |a| → ∞, the convolution with ∂
j
z
1
|y−z| decreases at least
as |a| −2 . By the same reasons,
d
3 z ∂
j
z
1
|y − z|
[F μν (x + a), A j (z, y 0 )]
commutes with the other factors (− −1 ∂ j A j )(y), in the expansion of the exponential, apart from
terms decreasing at least as |a| −4 . Thus, for |a| → ∞
[F 0k (x + a), e
−ie(− −1 ∂ i A i )(y) ] ∼
∼
d
3 z ∂
j
z
1
|y − z|
[F 0k (x + a), A j (z, y 0 )]e
−ie(− −1 ∂ i A i )(y) ,
and, by the cluster property of the FGB correlation functions, the vacuum insertion gives the leading
contribution. For a proof of this behaviour, which takes into account the need of a UV regularization
of (29.13) and exploits the locality of the charged fields in the FGB gauge and the cluster property,
see D. Buchholz, S. Doplicher, G. Morchio, J.E. Roberts and F. Strocchi, Ann. Phys. 290, 53 (2001).