Masujima M. Applied Mathematical Methods in Theoretical Physics
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368 10 Calculus of Variations: Applications
We have
[ˆπ
B
(x
1
),
ˆ
φ
B
(x
2
)] =
i
δ
σ
(x
1
− x
2
). (10.2.24F)
For quantum field theory of the Fermi field, for the same reason as in (10.2.25M),
we obtain
{ˆπ
F
(x
1
),
ˆ
φ
F
(x
2
)}=
i
δ
σ
(x
1
− x
2
). (10.2.25F)
Equations (10.2.24F) and (10.2.25F) are the natural consequences of the choice of
the momentum operator as the displacement operator, (10.2.15F).
From Feynman’s action principle, (10.2.1M,F), the assumptions (A.1), (A-2), and
(A-3), the path integral definition of the operator, (10.2.7M,F), and the definition of
the momentum operator as the displacement operator, (10.2.15M,F), we deduced
the following four statements:
(a) the (Euler-) Lagrange equation of motion, (10.2.13M,F),
(b) the definition of the time-ordered product, (10.2.14M,F),
(c) the definition of canonical conjugate momentum, (10.2.21M,F),
(d) the equal time canonical commutator, (10.2.24M), (10.2.24F), and (10.2.25M,F).
Thus we demonstrated the equivalence of canonical quantization and path
integral quantization for the nonsingular Lagrangian (density). In the next section,
we will establish the equivalence of Schwinger’s action principle and Feynman’s
action principle for the nonsingular Lagrangian (density).
10.3
Schwinger’s Action Principle in Quantum Theory
In this section, we will discuss Schwinger’s action principle in quantum theory
and demonstrate its equivalence to Feynman’s action principle. We first state
Schwinger’s action principle and then obtain the transition probability amplitude
in the form of Feynman’s action principle by the method of the functional Fourier
transform.
Schwinger’s action principle asserts that the variation of the transition probability
amplitude
∞
−∞
results from variation of action functional, I [
ˆ
φ
i
,
ˆ
ψ
i
], which
assumes the following form:
δ
∞
−∞
= i
∞
δI[
ˆ
φ
i
,
ˆ
ψ
i
]
−∞
. (10.3.1)
Here
−∞
(
∞
) is any eigenket (any eigenbra) of any dynamical quantity which
lies in the remote past (future),
ˆ
φ
i
and
ˆ
ψ
i
generically represent Boson variables
and Fermion variables, respectively, and the indices i and i represent both the
space–time degrees of freedom and the internal degrees of freedom, respectively.
10.3 Schwinger’s Action Principle in Quantum Theory 369
In order to visualize what we mean by the variation δI[
ˆ
φ
i
,
ˆ
ψ
i
] of the action functional
I[
ˆ
φ
i
,
ˆ
ψ
i
], it is convenient to introduce the external hook coupling as
I[
ˆ
φ
i
,
ˆ
ψ
i
] + J
i
ˆ
φ
i
+ J
i
ˆ
ψ
i
, (10.3.2)
and consider the variation of the c-number external hooks, J
i
and J
i
.
Applying Schwinger’s action principle to the action functional, (10.3.2), we have
the following equations for
∞
|
−∞
,
(δiδJ
i
)
∞
−∞
=
∞
ˆ
φ
i
−∞
, (10.3.3B)
(δiδJ
i
)
∞
−∞
=
∞
ˆ
ψ
i
−∞
. (10.3.3F)
We express the right-hand sides of (10.3.3B) and (10.3.3F) as
∞
ˆ
φ
i
−∞
=
#
∞
φ
i
φ
i
φ
i
−∞
,
∞
ˆ
ψ
i
−∞
=
#
∞
ψ
i
ψ
i
ψ
i
−∞
.
(10.3.4)
Here the summation is over the eigenvectors
φ
i
and
ψ
i
of complete sets
of commuting and anticommuting operators
ˆ
φ
i
and
ˆ
ψ
i
, respectively. Taking the
functional derivative once more, we have
(δiδJ
j
)(δiδJ
i
)
∞
−∞
=
∞
T(
ˆ
φ
j
ˆ
φ
i
)
−∞
,
(δiδJ
j
)(δiδJ
i
)
∞
−∞
=
∞
T(
ˆ
ψ
j
ˆ
φ
i
)
−∞
,
(δiδJ
j
)(δiδJ
i
)
∞
−∞
=
∞
T(
ˆ
ψ
j
ˆ
ψ
i
−∞
.
(10.3.5)
Here the time-ordered products are defined by
T(
ˆ
φ
j
ˆ
φ
i
) ≡ θ (j, i)
ˆ
φ
j
ˆ
φ
i
+ θ(i, j)
ˆ
φ
i
ˆ
φ
j
,
T(
ˆ
ψ
j
ˆ
φ
i
) ≡ θ(j, i)
ˆ
ψ
j
ˆ
φ
i
+ θ(i,j)
ˆ
φ
i
ˆ
ψ
j
,
T(
ˆ
ψ
j
ˆ
ψ
i
) ≡ θ (j, i)
ˆ
ψ
j
ˆ
ψ
i
− θ(i, j)
ˆ
ψ
i
ˆ
ψ
j
.
(10.3.6)
The multiple time-ordered products are defined by the successive application of
the functional derivatives upon
∞
−∞
.Weemploytheabbreviation
T
ij···kl···
≡ T(
ˆ
φ
i
ˆ
φ
j
···
ˆ
ψ
k
ˆ
ψ
l
···). (10.3.7)
Then, for the matrix element of the multiple time-ordered products , we have
∞
T
ij···kl···
−∞
=
δ
iδJ
i
δ
iδJ
j
···
δ
iδJ
k
δ
iδJ
l
···
∞
−∞
. (10.3.8)
370 10 Calculus of Variations: Applications
In order to remove the operator ordering ambiguity, we write (10.3.1) as
δ
∞
−∞
= i
∞
T(δI[
ˆ
φ
i
,
ˆ
ψ
i
])
−∞
. (10.3.9)
Then the operator dynamical equations are written as
T
δI[
ˆ
φ
i
,
ˆ
ψ
i
]
δ
ˆ
φ
i
=−J
i
, (10.3.10)
T
δI[
ˆ
φ
i
,
ˆ
ψ
i
]
δ
ˆ
ψ
i
=−J
i
. (10.3.11)
We now take the matrix elements of (10.3.10) and (10.3.11) between the states,
−∞
and
∞
,withtheresults
δI[δiδJ]
δφ
i
∞
−∞
=−J
i
∞
−∞
, (10.3.12)
δI[δiδJ]
δψ
i
∞
−∞
=−J
i
∞
−∞
. (10.3.13)
We note that δI[δiδJ]δφ
i
and δI[δiδJ]δψ
i
are obtained by expanding
δI[φ
i
, ψ
i
]δφ
i
and δI[φ
i
, ψ
i
]δψ
i
in powers of the φ
i
and ψ
i
and replacing
the φ
i
and ψ
i
in the expansion with the δiδJ
i
and δiδJ
i
, respectively. In order to
solve (10.3.12) and (10.3.13) for the transition probability amplitude
∞
−∞
,we
introduce the functional Fourier transform of the transition probability amplitude
∞
−∞
as
∞
−∞
=
D [φ]D [ψ]F[φ, ψ]exp[i(J
i
φ
i
+ J
i
ψ
i
)]. (10.3.14)
Here D [φ]andD [ψ] are formally defined by
D [φ] ≡
$
i
dφ
i
, D [ψ] ≡
$
i
dψ
i
. (10.3.15)
Inserting (10.3.14) into (10.3.12) and (10.3.13), we get
0 =
D [φ]D [ψ]F[φ, ψ](δI[φ, ψ]δφ
i
+J
i
)exp[i(J
i
φ
i
+ J
i
ψ
i
)]
=
D [φ]D [ψ]F[φ, ψ](δI[φ, ψ]δφ
i
+ δiδφ
i
)exp[i(J
i
φ
i
+J
i
ψ
i
)],
(10.3.16)
0 =
D [φ]D [ψ]F[φ, ψ](δI[φ, ψ]δψ
i
+J
i
)exp[i(J
i
φ
i
+ J
i
ψ
i
)]
=
D [φ]D [ψ]F[φ, ψ](δI[φ, ψ]δψ
i
+ δiδψ
i
)exp[i(J
i
φ
i
+J
i
ψ
i
)].
(10.3.17)
10.4 Schwinger–Dyson Equation in Quantum Field Theory 371
Integrating by parts, we obtain the differential equation for the functional Fourier
transform F[φ, ψ]as
δI[φ, ψ]
δφ
i
−
δ
iδφ
i
F[φ, ψ] = 0, (10.3.18)
δI[φ, ψ]
δψ
i
−
δ
iδψ
i
F[φ, ψ] = 0. (10.3.19)
Differential equations, (10.3.18) and (10.3.19), can be immediately integrated with
the result,
F[φ, ψ] =
1
N
exp[iI[φ, ψ]], (10.3.20)
where N is the normalization constant. Hence finally we have
∞
−∞
=
1
N
D [φ]D [ψ]exp[i(I[φ, ψ] +J
i
φ
i
+ J
i
ψ
i
)]. (10.3.21)
This is Feynman’s action principle. If we apply the identity, (10.3.8), to (10.3.21),
we have
∞
T
ij···kl···
−∞
=
1
N
D [φ]D [ψ]φ
i
φ
j
···ψ
k
ψ
l
···
× exp[i(I[φ, ψ] +J
i
φ
i
+ J
i
ψ
i
)]. (10.3.22)
We thus have established the equivalence of Schwinger’s action principle and
Feynman’s action principle, at least for the nonsingular system in the abstract
notation without committing ourselves to quantum mechanics or quantum field
theory.
When the infinite-dimensional invariance group is present, our discussion
becomes invalid, since the action functional I[φ, ψ ] remains constant under the
action of the infinite-dimensional invariance group. The appearance of the external
hooks, J
i
and J
i
, coupled linearly to the dynamical variables, (φ
i
, ψ
i
), violates the
infinite-dimensional group invariance .
The systems with the infinite-dimensional invariance group , for example, the
Abelian and non-Abelian gauge fields and the gravitational field , are the subject
matter of Section 10.9.
10.4
Schwinger–Dyson Equation in Quantum Field Theory
In quantum field theory, we also have the notion of Green’s functions, which is
quite distinct from the ordinary Green’s functions in mathematical physics in one
important aspect: the governing equation of motion of the connected part of the
two-point ‘‘full’’ Green’s function in quantum field theory is the closed system
372 10 Calculus of Variations: Applications
of the coupled nonlinear integro-differential equations. We illustrate this point in
some detail for the Yukawa coupling of the fermion field and the boson field.
We shall establish the relativistic notation. We employ the natural unit system
in which
= c = 1. (10.4.1)
We define the Minkowski space–time metric tensor η
µν
by
η
µν
≡ diag(1;−1, −1, −1) ≡ η
µν
, µ, ν = 0, 1, 2, 3. (10.4.2)
We define the contravariant components and the covariant components of the
space–time coordinates x by
x
µ
≡ (x
0
, x
1
, x
2
, x
3
), (10.4.3a)
x
µ
≡ η
µν
x
ν
= (x
0
, −x
1
, −x
2
, −x
3
). (10.4.3b)
We define the differential operators ∂
µ
and ∂
µ
by
∂
µ
≡
∂
∂x
µ
=
∂
∂x
0
,
∂
∂x
1
,
∂
∂x
2
,
∂
∂x
3
, ∂
µ
≡
∂
∂x
µ
= η
µν
∂
ν
. (10.4.4)
We define the four-scalar product by
x · y ≡ x
µ
y
µ
= η
µν
x
µ
y
ν
= x
0
· y
0
−
x ·
y. (10.4.5)
We adopt the convention that the Greek indices µ, ν,..., run over 0, 1, 2, and 3, the
Latin indices i, j, ..., run over 1, 2, and 3, and the repeated indices are summed
over.
We consider the quantum field theory described by the total Lagrangian density
L
tot
of the form
L
tot
=
1
4
%
&
ψ
α
(x), D
αβ
(x)
ˆ
ψ
β
(x)
'
+
1
4
%
D
T
βα
(−x)
&
ψ
α
(x),
ˆ
ψ
β
(x)
'
+
1
2
ˆ
φ(x)K(x)
ˆ
φ(x) +L
int
(
ˆ
φ(x),
ˆ
ψ(x),
&
ψ(x)), (10.4.6)
where we have
D
αβ
(x) = (iγ
µ
∂
µ
− m + iε)
αβ
, D
T
βα
(−x) = (−iγ
T
µ
∂
µ
− m + iε)
βα
,
(10.4.7a)
K(x) =−∂
2
− κ
2
+ iε, (10.4.7b)
{γ
µ
, γ
ν
}=2η
µν
,(γ
µ
)
†
= γ
0
γ
µ
γ
0
,
&
ψ
α
(x) = (
ˆ
ψ
†
(x)γ
0
)
α
, (10.4.8)
I
tot
[
ˆ
φ,
ˆ
ψ,
&
ψ] =
d
4
x L
tot
((10.4.9)), I
int
[
ˆ
φ,
ˆ
ψ,
&
ψ] =
d
4
x L
int
((10.4.6)).
(10.4.9)
10.4 Schwinger–Dyson Equation in Quantum Field Theory 373
We have Euler–Lagrange equations of motion for the field operators:
ˆ
ψ
α
(x):
δ
ˆ
I
tot
δ
&
ψ
α
(x)
= 0orD
αβ
(x)
ˆ
ψ
β
(x) +
δ
ˆ
I
int
δ
&
ψ
α
(x)
= 0, (10.4.10a)
&
ψ
β
(x):
δ
ˆ
I
tot
δ
ˆ
ψ
β
(x)
= 0or − D
T
βα
(−x)
&
ψ
α
(x) +
δ
ˆ
I
int
δ
ˆ
ψ
β
(x)
= 0, (10.4.10b)
ˆ
φ(x):
δI
tot
δ
ˆ
φ(x)
= 0orK(x)
ˆ
φ(x) +
δ
ˆ
I
int
δ
ˆ
φ(x)
= 0. (10.4.10c)
We have the equal time canonical (anti-)commutators:
δ(x
0
− y
0
){
ˆ
ψ
β
(x),
&
ψ
α
(x)}=γ
0
βα
δ
4
(x − y), (10.4.11a)
δ(x
0
− y
0
){
ˆ
ψ
β
(x),
ˆ
ψ
α
(y)}=δ(x
0
− y
0
){
&
ψ
β
(x),
&
ψ
α
(x)}=0, (10.4.11b)
δ(x
0
− y
0
)[
ˆ
φ(x), ∂
y
0
ˆ
φ(y)] = iδ
4
(x − y), (10.4.11c)
δ(x
0
− y
0
)[
ˆ
φ(x),
ˆ
φ(y)] = δ(x
0
− y
0
)[∂
x
0
ˆ
φ(x), ∂
y
0
ˆ
φ(y)] = 0, (10.4.11d)
and the rest of the equal time mixed canonical commutators are equal to 0. We
define the generating functional of the full Green’s functions by
Z [J, η, η]
≡0, out
|
T(exp[i{(J
ˆ
φ) + (
η
ˆ
ψ) +(
&
ψη)}])
|
0, in
≡
∞
l,m,n=0
i
l+m+n
l!m!n!
0, out
|
T((
η
ˆ
ψ)
l
(J
ˆ
φ)
m
(
&
ψη)
n
)
|
0, in
≡
∞
l,m,n=0
i
l+m+n
l!m!n!
J(y
1
) ···J(y
m
)η
α
l
(x
l
) ···η
α
1
(x
1
)
×0, out
|
T{
ˆ
ψ
α
1
(x
1
) ···
ˆ
ψ
α
l
(x
l
)
ˆ
φ(y
1
) ··
ˆ
φ(y
m
)
×
&
ψ
β
1
(z
1
) ···
&
ψ
β
n
(z
n
)}
|
0, in η
β
n
(z
n
) ···η
β
1
(z
1
)
≡
∞
n=0
i
n
n!
0, out
|
T(
η
ˆ
ψ + J
ˆ
φ +
&
ψη)
n
|
0, in . (10.4.12)
Here the repeated continuous space–time indices x
1
through z
n
are to be integrated
over and we introduced the abbreviations in Eq. (10.4.12),
J
ˆ
φ ≡
d
4
yJ(y)
ˆ
φ(y), η
ˆ
ψ ≡
d
4
x η(x)
ˆ
ψ(x),
&
ψη ≡
d
4
z
&
ψ(z)η(z).
The time-ordered product is defined by
T{
ˆ
(x
1
) ···
ˆ
(x
n
)}
≡
all possible
permutations P
δ
P
θ(x
0
P1
− x
0
P2
) ···θ (x
0
P(n−1)
− x
0
Pn
)
ˆ
(x
P1
) ···
ˆ
(x
Pn
),
374 10 Calculus of Variations: Applications
with
δ
P
=
1 P even and odd for Boson,
1 P even for Fermion,
−1 P odd for Fermion.
We observe that
δ
δη
β
(x)
(
η
ˆ
ψ)
l
= l
ˆ
ψ
β
(x)(η
ˆ
ψ)
l−1
, (10.4.13a)
δ
δη
α
(x)
(
&
ψη)
n
=−n
&
ψ
α
(x)(
&
ψη)
n−1
, (10.4.13b)
δ
δJ(x )
(J
ˆ
φ)
m
= m
ˆ
φ(x)(J
ˆ
φ)
m−1
. (10.4.13c)
We also observe from the definition of Z[J, η, η],
1
i
δ
δη
β
(x)
Z [J,
η, η] =
0, out
T
ˆ
ψ
β
(x)exp
i
d
4
z{J(z)
ˆ
φ(z)
+
η
α
(z)
ˆ
ψ
α
(z) +
&
ψ
β
(z)η
β
(z)}
0, in
, (10.4.14a)
i
δ
δη
α
(x)
Z [J,
η, η] =
0, out
T
&
ψ
α
(x)exp
i
d
4
z{J(z)
ˆ
φ(z)
+
η
α
(z)
ˆ
ψ
α
(z) +
&
ψ
β
(z)η
β
(z)}
0, in
, (10.4.14b)
1
i
δ
δJ(x )
Z [J,
η, η] =
0, out
T
ˆ
φ(x)exp
i
d
4
z{J(z)
ˆ
φ(z)
+
η
α
(z)
ˆ
ψ
α
(z) +
&
ψ
β
(z)η
β
(z)}
0, in
. (10.4.14c)
From the definition of the time-ordered product and the equal time canonical
(anti-)commutators, Eqs. (10.4.11a)–((10.4.11d), we have at the operator level,
Fermion:
D
αβ
(x)T(
ˆ
ψ
β
(x)exp[i{J
ˆ
φ + η
α
ˆ
ψ
α
+
&
ψ
β
η
β
}])
= T(D
αβ
(x)
ˆ
ψ
β
(x)exp[i{J
ˆ
φ + η
ˆ
ψ +
&
ψη}])
− η
α
(x)T(exp[i{J
ˆ
φ + η
ˆ
ψ +
&
ψη}]), (10.4.15)
Antifermion:
− D
T
βα
(−x)T(
&
ψ
α
(x)exp[i{J
ˆ
φ + η
α
ˆ
ψ
α
+
&
ψ
β
η
β
}])
= T(−D
T
βα
(−x)
&
ψ
α
(x)exp[i{J
ˆ
φ + η
ˆ
ψ +
&
ψη}])
+
η
β
(x)T(exp[i{J
ˆ
φ + η
ˆ
ψ +
&
ψη}]), (10.4.16)
10.4 Schwinger–Dyson Equation in Quantum Field Theory 375
Boson:
K(x)T(
ˆ
φ(x)exp[i{J
ˆ
φ + η
α
ˆ
ψ
α
+
&
ψ
β
η
β
}])
= T(K(x)
ˆ
φ(x)exp[i{J
ˆ
φ +
η
ˆ
ψ +
&
ψη}])
− J(x)T(exp[i{J
ˆ
φ +
η
ˆ
ψ +
&
ψη}]). (10.4.17)
Applying Euler–Lagrange equations of motion, Eqs. (10.4.10a through c), to the
first terms of the right-hand sides of Eqs. (10.4.15), (10.4.16), and (10.4.17), and
taking the vacuum expectation values , we obtain the equations of motion of the
generating functional Z[J,
η, η] of the ‘‘full’’ Green’s functions as
(
D
αβ
(x)
1
i
δ
δη
β
(x)
+
δI
int
)
1
i
δ
δJ
,
1
i
δ
δη
, i
δ
δη
*
δ
i
δ
δη
α
(x)
+ η
α
(x)
+
Z [J, η, η] = 0, (10.4.18a)
−D
T
βα
(−x)i
δ
δη
α
(x)
+
δI
int
)
1
i
δ
δJ
,
1
i
δ
δη
, i
δ
δη
*
δ
1
i
δ
δη
β
(x)
−
η
β
(x)
Z [J,
η, η] = 0,
(10.4.18b)
(
K(x)
1
i
δ
δJ(x )
+
δI
int
)
1
i
δ
δJ
,
1
i
δ
δη
, i
δ
δη
*
δ
1
i
δ
δJ(x)
+ J(x)
+
Z [J,
η, η] = 0. (10.4.18c)
Equivalently, from Eqs. (10.4.10a), (10.4.10b), and (10.4.10c), we can write, respec-
tively, Eqs. (10.4.18a), (10.4.18b), and (10.4.18c) as
(
δI
tot
)
1
i
δ
δJ
,
1
i
δ
δη
, i
δ
δη
*
δ
i
δ
δη
α
(x)
+η
α
(x)
+
Z [J, η, η] = 0, (10.4.19a)
δI
tot
)
1
i
δ
δJ
,
1
i
δ
δη
, i
δ
δη
*
δ
1
i
δ
δη
β
(x)
−
η
β
(x)
Z [J,
η, η] = 0, (10.4.19b)
(
δI
tot
)
1
i
δ
δJ
,
1
i
δ
δη
, i
δ
δη
*
δ
1
i
δ
δJ(x)
+J (x)
+
Z [J,
η, η] = 0. (10.4.19c)
We note that the coefficients of the external hook terms, η
α
(x), η
β
(x), and J(x), in
Eqs. (10.4.19a)–(10.4.19c) are ±1, which is the reflection of the fact that we are
dealing with canonical quantum field theory and originates from the equal time
canonical (anti-)commutators .
With this preparation, we shall discuss the Schwinger theory of Green’s func-
tion with the interaction Lagrangian density L
int
(
ˆ
φ(x),
ˆ
ψ(x),
&
ψ(x)) of the Yukawa
coupling in mind,
L
int
(
ˆ
φ(x),
ˆ
ψ(x),
&
ψ(x)) =−G
0
&
ψ
α
(x)γ
αβ
(x)
ˆ
ψ
β
(x)
ˆ
φ(x), (10.4.20)
376 10 Calculus of Variations: Applications
with
G
0
=
g
0
f
e
, γ (x) =
iγ
5
iγ
5
τ
k
γ
µ
,
ˆ
ψ(x) =
ˆ
ψ
α
(x)
ˆ
ψ
N,α
(x)
ˆ
ψ
α
(x)
,
φ(x) =
ˆ
φ(x)
ˆ
φ
k
(x)
ˆ
A
µ
(x)
. (10.4.21)
We define the vacuum expectation values , F
J,η,η
and F
J
, of the operator function
F(
ˆ
φ(x),
ˆ
ψ(x),
&
ψ(x)) in the presence of the external hook terms {J, η, η} by
F
J,η,η
≡
1
Z [J, η, η]
F
1
i
δ
δJ(x )
,
1
i
δ
δη(x)
, i
δ
δη(x)
Z [J,
η, η], (10.4.22a)
F
J
=F
J,η,η
η=η=0
. (10.4.22b)
We define the connected parts of the two-point ‘‘full’’ Green’s functions in the
presence of the external hook J(x )byFermion:
S
J
F,αβ
(x
1
, x
2
) ≡
1
i
δ
δη
α
(x
1
)
i
δ
δη
β
(x
2
)
1
i
ln Z[J,
η, η]
η=η=0
=
1
i
1
i
δ
δη
α
(x
1
)
&
ψ
β
(x
2
)
J,η,η
η=η=0
=
1
i
ˆ
ψ
α
(x
1
)
&
ψ
β
(x
2
)
J,η,η
η=η=0
−
ˆ
ψ
α
(x
1
)
J,η,η
&
ψ
β
(x
2
)
J,η,η
η=η=0
=
1
i
ˆ
ψ
α
(x
1
)
&
ψ
β
(x
2
)
J
≡
1
i
0, out
|
T(
ˆ
ψ
α
(x
1
)
&
ψ
β
(x
2
))
|
0, in
J
C
, (10.4.23)
ˆ
ψ
α
(x
1
)
J,η,η
η=η=0
=
&
ψ
β
(x
2
)
J,η,η
η=η=0
= 0, (10.4.24)
and
Boson:
D
F
J(x
1
, x
2
) ≡
1
i
δ
δJ(x
1
)
1
i
δ
δJ(x
2
)
1
i
ln Z[J,
η, η]
η=η=0
=
1
i
1
i
δ
δJ(x
1
)
ˆ
φ(x
2
)
J
=
1
i
{
ˆ
φ(x
1
)
ˆ
φ(x
2
)
J
−
ˆ
φ(x
1
)
J
ˆ
φ(x
2
)
J
}
≡
1
i
0, out
T(
ˆ
φ(x
1
)
ˆ
φ(x
2
))
0, in
J
C
, (10.4.25)
ˆ
φ(x)
J
J=0
= 0. (10.4.26)
10.4 Schwinger–Dyson Equation in Quantum Field Theory 377
We have the equations of motion of Z[J, η, η], Eqs. (10.4.18a)–(10.4.18c), when the
interaction Lagrangian density L
int
(
ˆ
φ(x),
ˆ
ψ(x),
&
ψ(x)) is given by Eq. (10.4.20) as
(
D
αβ
(x)
1
i
δ
δη
β
(x)
− G
0
γ
αβ
(x)
1
i
δ
δη
β
(x)
1
i
δ
δJ(x )
+
Z [J,
η, η]
=−η
α
(x)Z[J, η, η], (10.4.27a)
−D
T
βα
(−x)
i
δ
δη
α
(x)
+ G
0
i
δ
δη
α
(x)
γ
αβ
(x)
1
i
δ
δJ(x )
Z [J,
η, η]
=+
η
β
(x)Z[J, η, η], (10.4.27b)
(
K(x)
1
i
δ
δJJ(x )
− G
0
i
δ
δη
α
(x)
γ
αβ
(x)
1
i
δ
δη
β
(x)
+
Z[J,
η, η]
=−J(x)Z[J,
η, η]. (10.4.27c)
Dividing Eqs. (10.4.27a)–(10.4.27c) by Z[J, η, η], and referring to Eqs. (10.4.22a)
and (10.4.22b), we obtain the equations of motion for
ˆ
ψ
β
(x)
J,η,η
,
&
ψ
α
(x)
J,η,η
and
ˆ
φ(x)
J,η,η
,
as
D
αβ
(x)
ˆ
ψ
β
(x)
J,η,η
− G
0
γ
αβ
(x)
ˆ
ψ
β
(x)
ˆ
φ(x)
J,η,η
=−η
α
(x), (10.4.28)
−D
T
βα
(−x)
&
ψ
α
(x)+G
0
γ
αβ
(x)
&
ψ
α
(x)
ˆ
φ(x)
J,η,η
=+η
β
(x), (10.4.29)
K(x)
ˆ
φ(x)
J,η,η
− G
0
γ
αβ
(x)
&
ψ
α
(x)
ˆ
ψ
β
(x)
J,η,η
=−J(x). (10.4.30)
We take the following functional derivatives:
i
δ
δη
ε
(y)
Eq. (10.4.28)
η=η=0
:
D
αβ
(x)
&
ψ
ε
(y)
ˆ
ψ
β
(x)
J
−G
0
γ
αβ
(x)
&
ψ
ε
(y)
ˆ
ψ
β
(x)
ˆ
φ(x)
J
=−iδ
αε
δ
4
(x − y),
1
i
δ
δη
ε
(y)
Eq. (10.4.29)
η=η=0
:
− D
T
βα
(−x)
ˆ
ψ
ε
(y)
&
ψ
α
(x)
J
+ G
0
γ
αβ
(x)
ˆ
ψ
ε
(y)
&
ψ
α
(x)
ˆ
φ(x)
J
=−iδ
βε
δ
4
(x − y),
1
i
δ
δJ(y)
Eq. (10.4.30)
η=η=0
:
K(x){
ˆ
φ(y)
ˆ
φ(x)
J
−φ(y)
J
φ(x)
J
}
− G
0
γ
αβ
(x)
1
i
δ
δJ(y)
&
ψ
α
(x)
ˆ
ψ
β
(x)
J
= iδ
4
(x − y).