Masujima M. Applied Mathematical Methods in Theoretical Physics
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438 10 Calculus of Variations: Applications
with a
α
(x) arbitrary function independent of A
αµ
(x). By this consideration, we
obtain the second formula of Faddeev–Popov. We arrive at the covariant gauge.We
introduce the Faddeev–Popov ghost (fictitious scalar fermion) only in the internal
loop, which restores the unitarity. We discuss the various gauge-fixing conditions,
and the advantage of the axial gauge.
The First Formula of Faddeev–Popov: In Section 10.2, we derived the path
integral formula for the vacuum-to-vacuum transition amplitude
0, out
0, in
for
the nonsingular Lagrangian density L(φ
i
(x), ∂
µ
φ
i
(x)) as
0, out
0, in
=
D[φ
i
]exp
i
d
4
x L(φ
i
(x), ∂
µ
φ
i
(x))
. (10.10.1)
We cannot apply this path integral formula naively to the non-Abelian gauge field
theory. The kernel K
αµ,βν
(x − y) of the quadratic part of the action functional of the
gauge field A
γµ
(x),
I
gauge
[A
γµ
] =
d
4
x L
gauge
(F
γµν
(x)) =−
1
4
d
4
xF
γµν
(x)F
µν
γ
(x), (10.10.2a)
F
γµν
(x) = ∂
µ
A
γν
(x) −∂
ν
A
γµ
(x) − C
αβγ
A
αµ
(x)A
βν
(x), (10.10.2b)
is given by
I
quad
gauge
[A
γµ
] =−
1
2
d
4
xd
4
yA
µ
α
(x)K
αµ,βν
(x − y)A
ν
β
(y), (10.10.3a)
K
αµ,βν
(x − y) = δ
αβ
−η
µν
∂
2
+ ∂
µ
∂
ν
δ
4
(x − y), µ, ν = 0, 1, 2, 3. (10.10.3b)
K
αµ,βν
(x − y) is singular and noninvertible. We cannot define the ‘‘free’’ Green’s
function of A
αµ
(x) as it stands now, and cannot apply the Feynman path integral
formula, (10.10.1), for the path integral quantization of the gauge field. As a matter
of fact, this K
αµ,βν
(x − y) is the four-dimensional transverse projection operator for
arbitrary A
αµ
(x). The origin of such calamity lies in the gauge invariance of the
gauge field action functional I
gauge
[A
γµ
] under the gauge transformation,
A
γµ
(x) −→ A
g
γµ
(x). (10.10.4a)
A
g
γµ
(x)isgivenby
(
t
γ
A
g
γµ
(x) = U(g(x))
1
t
γ
A
γµ
(x) + U
−1
(g(x))
)
i∂
µ
U(g(x))
*2
U
−1
(g(x)),
U(g(x)) = exp
)
ig
γ
(x)t
γ
*
.
(10.10.4b)
10.10 Path Integral Quantization of Gauge Field I 439
In another words, it originates from the fact that the gauge field action functional
I
gauge
[A
γµ
] is constant on the orbit of the gauge group G. By the orbit here, we
mean the gauge equivalent class
{A
g
γµ
(x):A
γµ
(x)fixed,∀g(x) ∈ G}. (10.10.5)
Here A
g
γµ
(x) is defined by (10.10.4b). The naive expression for the vacuum-to-
vacuum transition amplitude
0, out
0, in
of the gauge field A
γµ
(x),
0, out
0, in
=
D[A
γµ
]exp
i
d
4
x L
gauge
(F
γµν
(x))
, (10.10.6)
is hence proportional to the orbit volume V
G
,
V
G
=
dg(x), (10.10.7)
which is infinite and independent of A
γµ
(x).
In order to accomplish the path integral quantization of the gauge field A
γµ
(x)
correctly, we must extract the orbit volume V
G
from the vacuum-to-vacuum
transition amplitude,
0, out
0, in
. In the path integral,
3
D[A
γµ
], in (10.10.6), we
should not path-integrate over all possible configurations of the gauge field, but we
should path-integrate over only distinct orbit of the gauge field A
γµ
(x). We shall
introduce the hypersurface,
F
α
(A
γµ
(x)) = 0, α = 1, ..., N, (10.10.8)
in the manifold of the gauge field A
γµ
(x), which intersects with each orbit only
once. This implies that
F
α
(A
g
γµ
(x)) = 0, α = 1, ..., N, (10.10.9)
has the unique solution g(x) ∈ G for arbitrary A
γµ
(x). In this sense, the hypersur-
face, (10.10.8), is called the gauge-fixing condition.
Here we recall the group theory.
(1)
g(x), g
(x) ∈ G ⇒ (gg
)(x ) ∈ G. (10.10.10a)
(2)
U(g(x))U(g
(x)) = U((gg
)(x )). (10.10.10b)
(3)
dg
(x) = d(gg
)(x ), Invariant Hurwitz measure. (10.10.10c)
440 10 Calculus of Variations: Applications
We parametrize U(g(x)) in the neighborhood of the identity element of G as
U(g(x)) = 1 +it
γ
ε
γ
(x) + O(ε
2
), (10.10.11a)
$
x
dg(x) =
$
α,x
dε
α
(x)forg(x) ≈ Identity. (10.10.11b)
We define Faddeev–Popov determinant
F
[A
γµ
]by
F
[A
γµ
]
$
x
dg(x)
$
α,x
δ(F
α
(A
g
γµ
(x))) = 1. (10.10.12)
We show that
F
[A
γµ
] is gauge invariant under the gauge transformation
A
γµ
(x) −→ A
g
γµ
(x). (10.10.4a)
Under the gauge transformation, (10.10.4a), we have
F
[A
g
γµ
]
−1
=
$
x
dg
(x)
$
α,x
δ
F
α
A
gg
γµ
=
$
x
d(gg
)(x )
$
α,x
δ
F
α
A
gg
γµ
=
$
x
dg
(x)
$
α,x
δ
F
α
A
g
γµ
=
F
[A
γµ
]
−1
.
Namely, we have
F
[A
g
γµ
] =
F
[A
γµ
]. (10.10.13)
We substitute the defining equation of the Faddeev–Popov determinant into the
functional integrand on the right-hand side of the naive expression, (10.10.6), for
the vacuum-to-vacuum transition amplitude
0, out
0, in
.
0, out
0, in
=
$
x
dg(x)D[A
γµ
]
F
[A
γµ
]
$
α,x
δ
F
α
A
g
γµ
(x)
×exp
i
d
4
x L
gauge
F
γµν
(x)
. (10.10.14)
In (10.10.14), we perform the following gauge transformation:
A
γµ
(x) −→ A
g
−1
γµ
(x). (10.10.15)
10.10 Path Integral Quantization of Gauge Field I 441
Under this gauge transformation, we have
D[A
γµ
] −→ D[A
γµ
],
F
[A
γµ
] −→
F
[A
γµ
], δ
F
α
A
g
γµ
(
x
)
→ δ
F
α
A
γµ
(
x
)
,
d
4
x L
gauge
F
γµν
(
x
)
−→
d
4
x L
gauge
F
γµν
(
x
)
. (10.10.16)
Since the value of the functional integral remains unchanged under the change of
the function variable, from (10.10.14) and (10.10.16), we have
0, out
0, in
=
$
x
dg(x)
D[A
γµ
]
F
[A
γµ
]
$
α,x
δ
F
α
A
γµ
(
x
)
×exp
i
d
4
x L
gauge
F
γµν
(x)
= V
G
D[A
γµ
]
F
[A
γµ
]
$
α,x
δ
F
α
A
γµ
(x)
× exp
i
d
4
x L
gauge
F
γµν
(x)
. (10.10.17)
In this manner, we extract the orbit volume V
G
, which is infinite and is independent
of A
γµ
(x), and at the expense of the introduction of
F
[A
γµ
], we perform the path
integral of the gauge field
3
D[A
γµ
] on the hypersurface,
F
α
(A
γµ
(x)) = 0, α = 1, ..., N, (10.10.8)
which intersects with each orbit of the gauge field only once. From now on,
we drop V
G
and we write the vacuum-to-vacuum transition amplitude under the
gauge-fixing condition, (10.10.8), as
0, out
0, in
F
=
D[A
γµ
]
F
[A
γµ
]
$
α,x
δ
F
α
A
γµ
(x)
×exp
i
d
4
x L
gauge
F
γµν
(x)
. (10.10.18)
In (10.10.18), the Faddeev–Popov determinant
F
[A
γµ
]getsmultipliedby
:
α,x
δ(F
α
(A
γµ
(x))). We calculate
F
[A
γµ
]forA
γµ
(x) which satisfies the gauge-fixing
condition, (10.10.8). Mimicking the parameterization of U(g(x)), we parameterize
F
α
(A
g
γµ
(x)) as
F
α
A
g
γµ
(x)
= F
α
A
γµ
(x)
+
N
β=1
d
4
yM
F
αx,βy
(A
γµ
)ε
β
(y) + O
ε
2
. (10.10.19)
442 10 Calculus of Variations: Applications
The matrix M
F
αx,βy
(A
γµ
) is the kernel of a linear response with respect to ε
β
(y)of
the gauge-fixing condition under the infinitesimal gauge transformation,
δA
g
γµ
(x) =−
D
adj
µ
ε(x)
γ
. (10.10.20)
Thus, from (10.10.19) and (10.10.20), we have
M
F
αx,βy
A
γµ
=
δF
α
A
g
γµ
(x)
δε
β
(y)
g=1
=
δF
α
A
γµ
(x)
δA
γµ
(x)
−D
adj
µ
γβ
δ
4
(x − y)
≡ M
F
α,β
A
γµ
(x)
δ
4
(x − y). (10.10.21)
Here, we have
D
adj
µ
αβ
= ∂
µ
δ
αβ
+ C
αβγ
A
γµ
(x). For those A
γµ
(x) which satisfy the
gauge-fixing condition, (10.10.8), we have
F
[A
γµ
]
−1
=
$
α,x
dε
α
(x)
$
α,x
δ
M
F
α,β
A
γµ
(x)
ε
β
(x)
=
DetM
F
A
γµ
−1
.
Namely, we have
F
[A
γµ
] = DetM
F
A
γµ
. (10.10.22)
We introduce the Faddeev–Popov ghost (fictitious scalar fermion), c
α
(x)andc
β
(x),
in order to exponentiate the Faddeev–Popov determinant
F
[A
γµ
]. Then the
determinant
F
[A
γµ
] can be written as
F
[A
γµ
] = DetM
F
A
γµ
=
D[
c
α
]D[c
β
]exp
i
d
4
x c
α
(x)M
F
α,β
A
γµ
(x)
c
β
(x)
. (10.10.23)
Thus, we obtain the first formula of Faddeev–Popov:
0, out
0, in
F
=
D[A
γµ
]
F
[A
γµ
]
$
α,x
δ
F
α
A
γµ
(x)
×exp
i
d
4
x L
gauge
F
γµν
(x)
(10.10.24a)
=
D[A
γµ
]D[c
α
]D[c
β
]
$
α,x
δ
F
α
A
γµ
(x)
× exp
i
d
4
x
1
L
gauge
F
γµν
(x)
+
c
α
(x)M
F
α,β
A
γµ
(x)
c
β
(x)
0
. (10.10.24b)
10.10 Path Integral Quantization of Gauge Field I 443
Faddeev–Popov determinant
F
[A
γµ
] plays the role of restoring the unitarity.
The Second Formula of Faddeev–Popov: We now generalize the gauge-fixing
condition, (10.10.8):
F
α
A
γµ
(x)
= a
α
(x), α = 1, ..., N; a
α
(x) independent of A
αµ
(x).
(10.10.25)
According to the first formula of Faddeev–Popov, (10.10.24a), we have
Z
F
(a(x)) ≡
0, out
0, in
F,a
=
D[A
γµ
]
F
[A
γµ
]
$
α,x
δ
F
α
A
γµ
(x)
−a
α
(x)
×exp
i
d
4
x L
gauge
F
γµν
(x)
. (10.10.26)
We perform the nonlinear infinitesimal gauge transformation g
0
,inthepath
integrand of (10.10.26), parametrized by λ
β
(y),
ε
α
x ;M
F
A
γµ
=
M
F
A
γµ
−1
αx,βy
λ
β
(y), (10.10.27)
where λ
β
(y) is an arbitrary infinitesimal function, independent of A
αµ
(x). Under
this g
0
, we have the following three statements:
(1) The gauge invariance of the gauge field action functional,
d
4
x L
gauge
F
γµν
(x)
is invariant under g
0
.
(2) The gauge invariance of the integration measure,
D[A
γµ
]
F
[A
γµ
] = invariant measure under g
0
. (10.10.28)
(3) The gauge-fixing condition F
α
(A
γµ
(x)) is transformed into the following
form:
F
α
A
g
0
γµ
(x)
= F
α
A
γµ
(x)
+λ
α
(x) +O
λ
2
α
(x)
. (10.10.29)
Since the value of the functional integral remains unchanged under the change
of the function variable, the value of Z
F
(a(x)) remains unchanged under the
nonlinear gauge transformation g
0
. When we choose the infinitesimal parameter
λ
α
(x)as
λ
α
(x) = δa
α
(x), α = 1, ..., N, (10.10.30)
we have from the above stated (1), (2), and (3),
Z
F
(a(x)) = Z
F
a(x ) +δa(x)
or
d
da(x )
Z
F
(a(x)) = 0. (10.10.31)
444 10 Calculus of Variations: Applications
Namely, Z
F
(a(x)) is independent of a(x). Introducing an arbitrary weighting
functional H[a
α
(x)] for Z
F
(a(x)), and path-integrating with respect to a
α
(x), we
obtain the weighted Z
F
(a(x)),
Z
F
≡
$
α,x
da
α
(x)H[a
α
(x)]Z
F
(a(x))
=
D
)
A
γµ
*
F
)
A
γµ
*
H
)
F
α
(A
γµ
(x))
*
exp
i
d
4
x L
gauge
F
γµν
(x)
.
(10.10.32)
As the weighting functional, we choose the quasi-Gaussian functional,
H
)
a
α
(x)
*
= exp
−
i
2
d
4
xa
2
α
(x)
. (10.10.33)
Then we obtain as Z
F
,
Z
F
=
D[A
γµ
]
F
[A
γµ
]exp
i
d
4
x
L
gauge
F
γµν
(x)
−
1
2
F
2
α
A
γµ
(x)
=
D[A
γµ
]D[c
α
]D[c
β
]exp
i
d
4
x
L
gauge
F
γµν
(x)
−
1
2
F
2
α
A
γµ
(x)
+
c
α
(x)M
F
α,β
A
γµ
(x)
c
β
(x)
. (10.10.34)
Thus we obtain the second formula of Faddeev–Popov:
0, out
0, in
F
=
D[A
γµ
]D[c
α
]D[c
β
]exp
)
iI
eff
[A
γµ
, c
α
, c
β
]
*
, (10.10.35a)
with the effective action functional I
eff
[A
γµ
, c
α
, c
β
],
I
eff
[A
γµ
, c
α
, c
β
] =
d
4
x L
eff
, (10.10.35b)
where the effective Lagrangian density L
eff
is given by
L
eff
= L
gauge
F
γµν
(x)
−
1
2
F
2
α
A
γµ
(x)
+ c
α
(x)M
F
α,β
A
γµ
(x)
c
β
(x).
(10.10.35c)
There are the other methods to arrive at this second formula, due to Vassiliev.
Choice of Gauge Fixing Condition: We consider the following gauge-fixing
conditions:
10.10 Path Integral Quantization of Gauge Field I 445
(1) Axial gauge:
F
α
A
γµ
(x)
= n
µ
A
αµ
(x) = 0, α = 1, ..., N, (10.10.36a)
n
µ
= (0;0, 0, 1), n
2
= n
µ
n
µ
=−1, nk = n
µ
k
µ
=−k
3
. (10.10.36b)
(2) Landau gauge:
F
α
A
γµ
(x)
= ∂
µ
A
αµ
(x) = 0, α = 1, ..., N. (10.10.37)
(3) Covariant gauge:
F
α
A
γµ
(x)
=
9
ξ∂
µ
A
αµ
(x), α = 1, ..., N,0<ξ<∞. (10.10.38)
We obtain Green’s functions D
(A)
αµ,βν
(x − y)ofthegaugefieldA
γµ
(x)andD
(C)
α,β
(x − y)
of the Faddeev–Popov ghost fields,
c
α
(x)andc
β
(x), and the effective interaction
Lagrangian density L
int
eff
.
(1) Axial gauge: We use the first formula of Faddeev–Popov, (10.10.24a). Kernel
K
(axial)
αµ,βν
(x − y) of the quadratic part of the gauge field action functional I
gauge
[A
γµ
]
in the axial gauge, (10.10.36a) and (10.10.36b), is given by
I
quad
gauge
[A
γµ
] =−
1
2
d
4
xd
4
yA
µ
α
(x)K
(axial)
αµ,βν
(x − y)A
ν
β
(y)
K
(axial)
αµ,βν
(x − y) = δ
αβ
−η
µν
∂
2
+∂
µ
∂
ν
δ
4
(x − y)
µ, ν = 0, 1, 2; α, β = 1, ..., N. (10.10.39)
At first sight, this axial gauge kernel K
(axial)
αµ,βν
(x − y) appears to be the noninvertible
kernel. Since the Lorentz indices, µ and ν, run through 0,1,2 only, this axial gauge
kernel K
(axial)
αµ,βν
(x − y) is invertible. This has to do with the fact that the third spatial
component of the gauge field A
γµ
(x) gets killed by the gauge-fixing condition,
(10.10.36a) and (10.10.36b).
The ‘‘free’’ Green’s function D
(axial)
αµ,βν
(x − y) of the gauge field in the axial gauge
satisfies the following equation:
δ
αβ
η
µν
∂
2
− ∂
µ
∂
ν
D
(axial)
βν,γλ
(x − y) = δ
αγ
η
µ
λ
δ
4
(x − y). (10.10.40)
Upon Fourier transforming the ‘‘free’’ Green’s function D
(axial)
αµ,βν
(x − y),
D
(axial)
αµ,βν
(x − y) = δ
αβ
D
(axial)
µ,ν
(x − y)
= δ
αβ
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
D
(axial)
µ,ν
(k), (10.10.41)
446 10 Calculus of Variations: Applications
we have the momentum space Green’s function D
(axial)
µ,ν
(k) satisfying
−η
µν
k
2
+ k
µ
k
ν
D
(axial)
νλ
(k) = η
µ
λ
. (10.10.42a)
We introduce the transverse projection operator
µν
(n),
µν
(n) = η
µν
− n
µ
n
ν
/n
2
= η
µν
+ n
µ
n
ν
,
n(n) = (n)n = 0,
2
(n) = (n),
(10.10.43)
and write the momentum space Green’s function D
(axial)
νλ
(k)as
D
(axial)
νλ
(k) =
νσ
(n)
η
σρ
A(k
2
) +k
σ
k
ρ
B(k
2
)
ρλ
(n). (10.10.44)
On multiplying by
τµ
(n) on the left-hand side of (10.10.42a), we have
τµ
(n)
−η
µν
k
2
+k
µ
k
ν
D
(axial)
νλ
(k) =
τλ
(n). (10.10.42b)
Substituting (10.10.44) into (10.10.42b) and making use of the identities,
k(n)
µ
=
(n)k
µ
= k
µ
−
(nk)
n
2
n
µ
, k(n )k = k
2
−
(nk)
2
n
2
, (10.10.45)
we have
−k
2
A(k
2
)
τλ
(n) +
A(k
2
) −
(nk)
2
n
2
B(k
2
)
(n)k
τ
k(n)
λ
=
τλ
(n).
(10.10.46)
We obtain A(k
2
), B(k
2
)andD
(axial)
νλ
(k)as
A(k
2
) =−
1
k
2
, B(k
2
) =−
1
k
2
n
2
(nk)
2
, (10.10.47)
D
(axial)
νλ
(k) =−
1
k
2
η
νλ
−
n
ν
n
λ
n
2
+
n
2
(nk)
2
k
ν
−
(nk)
n
2
n
ν
k
λ
−
(nk)
n
2
n
λ
(10.10.48a)
=−
1
k
2
η
νλ
+
n
2
(nk)
2
k
ν
k
λ
−
1
(nk)
n
ν
k
λ
−
1
(nk)
k
ν
n
λ
(10.10.48b)
=−
1
k
2
η
νλ
−
k
ν
k
λ
k
2
3
+
n
ν
k
λ
k
3
+
k
ν
n
λ
k
3
. (10.10.48c)
10.10 Path Integral Quantization of Gauge Field I 447
Here in (10.10.48c) for the first time in this derivation, the following explicit
representations for the axial gauge are used,
n
2
=−1, nk =−k
3
. (10.10.36b)
Next we demonstrate that the Faddeev–Popov ghost does not show up in the
axial gauge. The kernel of the Faddeev–Popov ghost Lagrangian density is given
by
M
(axial)
αx,βy
A
γµ
=
δF
α
A
γµ
(x)
δA
γµ
(x)
−D
adj
µ
γβ
δ
4
(x − y)
=−n
µ
δ
αβ
∂
µ
+ C
αβγ
A
γµ
(x)
δ
4
(x − y) = δ
αβ
∂
3
δ
4
(x − y).
(10.10.49)
So the Faddeev–Popov determinant
F
[A
γµ
] is an infinite number indepen-
dent of the gauge field A
γµ
(x), and the Faddeev–Popov ghost does not show
up.
In this way, we note that we can carry out the canonical quantization of the
non-Abelian gauge field theory in the axial gauge, starting out with the separation
of the dynamical variable and the constraint variable. Thus we can apply the
phase space path integral formula with the minor change of the insertion of the
gauge-fixing condition
:
α,x
δ(n
µ
A
αµ
(x)) in the functional integrand. In the phase
space path integral formula, with the use of the constraint equation, we perform
the momentum integration and obtain
0, out
0, in
=
D[A
γµ
]
$
α,x
δ
n
µ
A
αµ
(x)
exp
i
d
4
x L
gauge
F
γµν
(x)
.
(10.10.50)
In (10.10.50), the manifest covariance is miserably destroyed. This point will be
overcome by the gauge transformation from the axial gauge to some covariant
gauge, say Landau gauge, and the Faddeev–Popov determinant
F
[A
γµ
]will
show up again as the Jacobian of the change of the function variable (the gauge
transformation) in the delta function,
F
(axial)
α
A
γµ
(x)
−→ F
(covariant)
α
A
γµ
(x)
. (10.10.51)
The effective interaction Lagrangian density L
int
eff
is given by
L
int
eff
= L
eff
−L
quad
eff
. (10.10.52)