S
(+, −, −, −)
 = c = 1
§
s
s
2
= (ct)
2
− (x
2
+ y
2
+ z
2
). (1.1.1)
t =
t

+
V
c
2
x


1 −
V
2
c

2
, x =
x

+ V t


1 −
V
2
c
2
, y = y

, z = z

, (1.1.2)
V
s (1.1.2)
x
0
= ct, x
1
= x, x
2
= y, x
3
= z, (1.1.3)
s
s

2
= (x
0
)
2
− [(x
1
)
2
+ (x
2
)
2
+ (x
3
)
2
], (1.1.4)
s
v
µ
v
0
, v
1
, v
2
, v
3
x

µ
µ 0, 1, 2, 3
v
µ
v
0
=
v
0
+
V
c
v
1

1 −
V
2
c
2
, v
1
=
v
1
+
V
c
v
0


1 −
V
2
c
2
, v
2
= v
2
, v
3
= v
3
. (1.1.5)
|v|
2
= (v
0
)
2
− [(v
1
)
2
+ (v
2
)
2
+ (v

3
)
2
] = (v
0
)
2
−
→
v
. (1.1.6)
v
µ
v
µ
v
0
= v
0
; v
1
= −v
1
; v
2
= −v
2
; v
3
= −v

3
. (1.1.7)
v
µ
v
µ
v
|v|
2
= v
0
v
0
+ v
1
v
1
+ v
2
v
2
+ v
3
v
3
= v
0
v
0
+ v

1
v
1
+ v
2
v
2
+ v
3
v
3
(1.1.8)
=
3

µ=0
v
µ
v
µ
=
3

µ=0
v
µ
v
µ
.
v

µ
v
µ
v
µ
v
µ
t
µν
v
µ
w
ν
t
µν
v
µ
w
ν
t
µ
ν
v
µ
w
ν
g
µν
= g
µν

=




1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1




(1.1.9)
v
µ
= g
µν
v
ν
; v
µ
= g
µν
v
ν
.
|v|
2
= v

µ
v
µ
= v
µ
v
µ
= g
µν
v
ν
v
µ
= g
µν
v
ν
v
µ
.
t
µ
ν
= g
µλ
t
λν
; t
µν
= g

µλ
g
νρ
t
λρ
.
t
µ
ν
= g
µλ
t
λν
; t
µν
= g
µλ
g
νρ
t
λρ
.
µ, ν, λ, ρ = 0, 1, 2, 3
g
µν
g
µν
•
v
0

= v
0
; t
0
0
= t
00
; t
0
1
= t
01
; t
30
= t
3
0
;
•
v
2
= −v
2
; t
01
= −t
0
1
; t
1

2
= −t
12
; t
23
= t
23
; t
13
= t
13
;
t
i
i
= t
0
0
+ t
1
1
+ t
2
2
+ t
3
3
t
i
i

= t
i
i
t
µν
t
µν
= t
νµ
t
µν
= −t
νµ
ϕ
∂ϕ
∂x
µ
=

1
c
∂ϕ
∂t
,
−→
∇ϕ

≡ ∂
µ
ϕ ≡ ϕ

;µ
. (1.1.10)
≡
ϕ
dϕ =
∂ϕ
∂x
µ
dx
µ
≡ ∂
µ
ϕdx
µ
≡ ϕ
;µ
dx
µ
. (1.1.11)
∂ϕ/∂x
µ
dx
µ
x
µ
∂ϕ
∂x
µ
=


1
c
∂ϕ
∂t
, −
−→
∇ϕ

≡ ∂
µ
ϕ ≡ ϕ
;µ
. (1.1.12)
∂v
µ
/∂x
µ
≡ ∂
µ
v
µ
∂v
µ
/∂x
µ
≡ ∂
µ
v
µ
•

dx
µ
•
df
ij
= dx
i
dx

j
−
dx
j
dx

i
; (i, j = 1, 2, 3),
−→
dS = (dS
i
), dS
i
=
(1/2)
ijk
df
jk
df
µν
= dx

µ
dx

ν
−dx
ν
dx

µ
−→
dS
df
∗µν
= (1/2)
µνλρ
df
λρ
(1.1.13)

ijk
i, j, k = 1, 2, 3 
123
= +1

123
+1
−1 
213
= −1, 
231

= +1
0 
113
= 0, 
133
= 0

µνλρ
µ, ν, λ, ρ = 0, 1, 2, 3

0123
= +1
•
dS
µνλ
dS
µνλ
=






dx
µ
dx
µ
dx
µ

dx
ν
dx
ν
dx
ν
dx
λ
dx
λ
dx
λ






. (1.1.14)
dS
µ
= −1/6
µνλρ
dS
νλρ
, dS
νλρ
= 
µνλρ
dS

µ
, (1.1.15)
dS
0
= dS
123
, dS
1
= dS
023
,
•
dΩ = dx
0
dx
1
dx
2
dx
3
= cdtdV.
d
4
x dΩ
d
3
x dV

(S)
−→

u d
−→
S =

(V )
div
−→
u dV, (1.1.16)
S V

(Σ)
A
µ
dS
µ
=

(Ω)
∂
µ
A
µ
dΩ, (1.1.17)
Σ Ω

(c)
−→
u d
−→
r =


(Σ)
rot
−→
u d
−→
S , (1.1.18)

(c)
A
µ
dx
µ
=

(S)
∂
µ
A
ν
df
µν
=
1
2

(S)
(∂
µ
A

ν
− ∂
ν
A
µ
) df
µν
. (1.1.19)
§
N
N 3N 3N
x
i
p
(t) x
i
p
(i = 1, 2, 3; p = 1, , N)
L = L(x
i
p
,
.
x
i
p
) =
N

p=1

3

i=1
m
p

.
x
i
p

2
2
− V (x) , (1.2.1)
m
p
p
.
x
i
p
x
i
p
I =
t
2

t
1

L

x,
.
x

dt (1.2.2)
x
i
p
(t)
t
1
≤ t
2
x
i
p
(t)
x
i
p
(t)
x
i
p
(t)
x
i
p

(t) −→ x

i
p
(t) = x
i
p
(t) + δx
i
p
(t), (1.2.3)
δx
i
p
(t) t δx
i
p
(t
1
) =
δx
i
p
(t
2
) = 0 x
i
p
(t)
δx

i
p
(t) = x

i
p
(t) − x
i
p
(t), δ
.
x
i
p
(t) =
.
x

i
p
(t)−
.
x
i
p
(t) =
d
dt
δx
i

p
(t). (1.2.4)
δI
δI = I

− I =
t
2

t
1

L

x

,
.
x


− L

x,
.
x


dt =
t

2

t
1


p,i
∂L
∂x
i
p
δx
i
p
+

p,i
∂L
∂
.
x
i
p
δ
.
x
i
p

dt.

∂L
∂
.
x
i
p
δ
.
x
i
p
=
d
dt

∂L
∂
.
x
i
p
δx
i
p

−
d
dt

∂L

∂
.
x
i
p

δx
i
p
,
δI =
t
2

t
1

p,i

∂L
∂x
i
p
−
d
dt
∂L
∂
.
x

i
p

δx
i
p
dt +

p,i
∂L
∂x
i
p
δx
i
p|
t
2
t
1
.
0 δI = 0
∂L
∂x
i
p
−
d
dt
∂L

∂
.
x
i
p
= 0. (1.2.5)
L
x
i
p
(t) N
m
p

x
i
p
= −
∂V
∂x
i
p
, (1.2.6)
N
L =
N

p=1
L


x
i
p
,
.
x
i
p

,
ϕ(x) x
ϕ(x)
ϕ
p
≡ ϕ(x
p
)
x
p
ϕ
p
, p = 1, 2, , N; N → ∞
L =
N

p=1
L(ϕ
p
, ∂ϕ
p

)
N→∞
−→

(∞)
L(ϕ(x), ∂ϕ(x)) d
3
x,
I =
t
2

t
1
Ldt =

(Ω)
L(ϕ, ∂ϕ) dΩ, (1.2.7)
dΩ = dx
0
d
3
x Ω
t = t
1
t = t
2
L
ϕ(x) → ϕ


(x) = ϕ(x) + δϕ(x),
∂ϕ(x) → ∂ϕ

(x) = ∂ϕ(x) + δ (∂ϕ(x)) ,
δϕ(x) |
t=t
1
= δϕ(x) |
t=t
2
= 0,
δ (∂ϕ(x)) |
t=t
1
= δ (∂ϕ(x)) |
t=t
2
= 0.
δI =

(Ω)

∂L
∂ϕ
δϕ +
∂L
∂(∂
µ
ϕ)
δ(∂

µ
ϕ)

dΩ. (1.2.8)
δ(∂
µ
ϕ) = ∂
µ
ϕ

− ∂
µ
ϕ = ∂
µ
(δϕ),
∂L
∂(∂
µ
ϕ)
δ(∂
µ
ϕ) = ∂
µ

∂L
∂(∂
µ
ϕ)
δϕ


− ∂
µ

∂L
∂(∂
µ
ϕ)

δϕ,
δI
δI =

(Ω)

∂L
∂ϕ
δϕ − ∂
µ

∂L
∂(∂
µ
ϕ)

δϕ + ∂
µ

∂L
∂(∂
µ

ϕ)
δϕ

dΩ. (1.2.9)

(Ω)
∂
µ

∂
µ
∂L
∂(∂
µ
ϕ)
δϕ

dΩ =

(Σ)
∂L
∂(∂
µ
ϕ)
δϕdS
µ
= 0
δϕ |
Σ
= 0 Σ t = t

1
, t = t
2
x = ∞, y = ∞, z = ∞ δI = 0
∂L
∂ϕ
− ∂
µ

∂L
∂(∂
µ
ϕ)

= 0. (1.2.10)
ϕ
1
(x), ϕ
2
(x),
∂L
∂ϕ
i
− ∂
µ

∂L
∂(∂
µ
ϕ

i
)

= 0, i = 1, 2, (1.2.11)
§
L
N
N
x
i
p
−→ x
i
p
= x
i
p
+ a
i
, i = 1, 2, 3; p = 1, 2, , N, (1.3.1)
a
i
p
L(x

,
.
x

) = L(x,

.
x
). (1.3.2)
a
i
= ε
i
L(x

,
.
x

) = L(x,
.
x
) +

i,p
ε
i
∂L
∂x
i
p
. (1.3.3)
ε
i

p

∂L
∂x
i
p
= 0, (i = 1, 2, 3). (1.3.4)

p
∂L
∂x
i
p
=
d
dt


p
∂L
∂
.
x
i
p

= 0.
P
i
≡

p

∂L
∂
.
x
i
p
, (i = 1, 2, 3) (1.3.5)
dP
i
/dt = 0
P
i
= const. (1.3.6)
N
P
i
=

p
m
p
.
x
i
p
= const. (1.3.7)
t −→ t

= t + τ. (1.3.8)
τ = ε

L(x

,
.
x

) = L(x,
.
x
) + ε
∂L
∂t
.
ε ∂L/∂t = 0
L t
dL
dt
=

i,p
∂L
∂x
i
p
dx
i
p
dt
+


i,p
∂L
∂
.
x
i
p
d
.
x
i
p
dt
=

i,p
d
dt

∂L
∂
.
x
i
p

.
x
i
p

+

i,p
∂L
∂
.
x
i
p
d
.
x
i
p
dt
=
d
dt


i,p
∂L
∂
.
x
i
p
.
x
i

p

.
d
dt


i,p
∂L
∂
.
x
i
p
.
x
i
p
−L

= 0. (1.3.9)
H =

i,p
∂L
∂
.
x
i
p

.
x
i
p
−L (1.3.10)
dH/dt = 0 ⇒ H = const
N
H =

i,p
m
p

.
x
i
p

2
2
+ V (x
i
p
). (1.3.11)
x
µ
−→ x
µ
+ a
µ

. (1.3.12)
ϕ(x)
ϕ

(x

) = ϕ(x), (1.3.13)
L(ϕ(x), ∂ϕ(x)) = L(ϕ

(x

), ∂

ϕ

(x

)) . (1.3.14)
a
µ
ε
µ
ϕ

(x) = ϕ(x − ε) = ϕ(x) − ε
µ
∂
µ
ϕ = ϕ(x) + δϕ(x),
δϕ(x) = −ε

µ
∂
µ
, (1.3.15)
L(ϕ

(x

), ∂

ϕ

(x

)) = L(ϕ

(x + ε), ∂ |
x+ε
ϕ

(x + ε))
= L(ϕ

(x), ∂ϕ

(x)) + ε
µ
∂
µ
L(ϕ(x), ∂ϕ(x)) , = L(ϕ(x), ∂ϕ(x)) ,

L(ϕ

(x), ∂ϕ

(x)) = L(ϕ(x), ∂ϕ(x)) − ε
µ
∂
µ
L(ϕ(x), ∂ϕ(x))
= L(ϕ(x), ∂ϕ(x)) + δL,
δL = −ε
µ
∂
µ
L = −∂
µ
(ε
µ
L). (1.3.16)
ϕ(x)
δL
δL =
∂L
∂ϕ
δϕ +
∂L
∂(∂
µ
ϕ)
δ(∂

µ
ϕ)
= ∂

∂L
∂(∂
µ
ϕ)

δϕ +
∂L
∂(∂
µ
ϕ)
δ(∂
µ
ϕ)
= ∂
µ

∂L
∂(∂
µ
ϕ)
δϕ

= ∂
µ

∂L

∂(∂
µ
ϕ)
ε
ν
∂
ν
ϕ

. (1.3.17)
ε
µ
∂
µ

∂L
∂(∂
µ
ϕ)
∂
ν
ϕ − δ
µ
ν
L

= 0. (1.3.18)
T
µ
ν

T
µ
ν
=
∂L
∂(∂
µ
ϕ)
∂
ν
ϕ − δ
µ
ν
L, (1.3.19)
T
µ
ν
∂
µ
T
µ
ν
= 0. (1.3.20)
∂
0
T
0
ν
− ∂
i

T
iν
= 0.
V
∂
0

(V )
T
0
ν
d
3
x −

(V )
∂
i
T
iν
d
3
x = 0.
S V

(V )
∂
i
T
i

ν
d
3
x =

(S)
T
i
ν
dS
i
.
V
S
∂
∂x
0




(V )
T
0
ν
d
3
x




= 0. (1.3.21)
P
ν
=

(V )
T
0
ν
d
3
x (1.3.22)
∂
∂x
0
P
ν
= 0
P
ν
P
0
=


∂L
∂(∂
0
ϕ)

∂
0
ϕ − L

d
3
x, (1.3.23)
P
i
=

∂L
∂(∂
0
ϕ)
∂
i
ϕ, i = 1, 2, 3. (1.3.24)
P
0
P
i
§
x
µ
= Λ
µ
ν
x
ν

, (1.4.1)
Λ
µ
ν
Λ
µ
ν
x
µ
x

µ
= Λ
µ
ν
Λ
ρ
µ
x
ν
x
ρ
= x
µ
x
µ
,
Λ
µ
ν

Λ
µ
ν
Λ
ρ
µ
x
ν
x
ρ
= x
µ
x
µ
. (1.4.2)
Λ
µ
ν
= δ
µ
ν
+ ω
µ
ν
x
µ
= (δ
µ
ν
+ ω

µ
ν
)x
ν
= x
µ
+ δx
µ
, (1.4.3)
δx
µ
= ω
µ
ν
x
ν
,
x
µ
x

µ
= (x
µ
+ ω
µ
ν
)(x
µ
+ ω

λ
µ
x
λ
) = x
µ
x
µ
+ ω
µ
ν
x
ν
x
µ
+ x
µ
ω
λ
µ
x
λ
+
ω
µ
ν
ω
µν
x
µ

x
ν
= 0,
1/2(ω
µν
+ ω
νµ
)x
µ
x
ν
= 0,
ω
µν
= −ω
νµ
.
ω
µν
ϕ

(x

) = ϕ(x),
L(ϕ

(x

), ∂


ϕ

(x

)) = L(ϕ(x), ∂ϕ(x)) . (1.4.4)
ϕ(x) ϕ

(x)
ϕ

(x) = ϕ(x − δx) = ϕ(x) − δx
µ
∂
µ
ϕ(x) = ϕ + δϕ,
δϕ(x) = −δx
µ
∂
µ
ϕ(x). (1.4.5)
δL = L(ϕ

(x), ∂ϕ

(x)) − L(ϕ(x), ∂ϕ(x))
L(ϕ

(x

), ∂


ϕ

(x

)) − δx
µ
∂
µ
L(ϕ(x), ∂ϕ(x)) − L(ϕ(x), ∂ϕ(x)) .
δL = δx
µ
∂
µ
L(ϕ(x), ∂ϕ(x)) . (1.4.6)
∂ (δx
µ
L) = (∂
µ
(δx
µ
)) L + δx
µ
∂
µ
L,
∂(δx
µ
) = ∂
µ

(ω
µ
ν
x
ν
) = ω
µ
ν
∂
µ
x
ν
= ω
µ
ν
δ
ν
µ
= ω
µ
µ
= 0,
δL = −∂
µ
(δx
µ
L(ϕ(x), ∂ϕ(x))) . (1.4.7)
δL
δL = ∂


∂L
∂(∂
µ
ϕ)
δϕ

= −∂
µ

δx
ν
∂L
∂(∂
µ
ϕ)
δ
ν
ϕ

. (1.4.8)
∂
µ

δx
ν

∂L
∂(∂
µ
ϕ)

δ
ν
ϕ

− δx
ν
(δ
µ
ν
L)

= 0. (1.4.9)
ω
ν
ρ
∂
µ

x
ρ

∂L
∂(∂
µ
ϕ)
δ
ν
ϕ − δ
µ
ν

L)

= 0. (1.4.10)
T
µ
ν
ω
νρ
∂
µ
(x
ρ
T
µ
ν
) = 0. (1.4.11)
ω
µν

ν,ρ
ω
νρ
∂
µ
(x
ρ
T
µ
ν
) =


ν>ρ
+

ν=ρ
+

ν<ρ

=

ν>ρ
ωνρ∂
µ
(x
ρ
T
µ
ν
) +

ν<ρ
ωνρ∂
µ
(x
ρ
T
µ
ν
)


ν>ρ
ωνρ∂
µ

x
ρ
T
µ
ν
− x
ν
T
µ
ρ

= 0. (1.4.12)
M
µ
ρν
= x
ρ
T
µ
ν
− x
ν
T
µ
ρ

, (1.4.13)
∂
µ
M
µ
ρν
= 0, (1.4.14)
V
∂
0

(V )
M
0
ρν
d
3
x −

(V )
∂
i
M
i
ρν
d
3
x = 0.
S V V
S

∂
0

M
0
ρν
d
3
x = 0. (1.4.15)
M
ρν
=

M
0
ρν
, (1.4.16)
M
µ
ρν
§
ϕ(x)
ϕ(x) ϕ
∗
(x)
L = L(ϕ(x), ∂ϕ(x), ϕ
∗
(x), ∂ϕ
∗
(x)) . (1.5.1)

∂L
∂ϕ
− ∂
µ

∂L
∂(∂
µ
ϕ)

= 0, (1.5.2)
∂L
∂ϕ
∗
− ∂
µ

∂L
∂(∂
µ
ϕ
∗
)

= 0. (1.5.3)
ϕ(x) −→ ϕ

(x) = e
iΛ
ϕ(x), (1.5.4)

Λ
L(ϕ

, ∂ϕ

, ϕ
∗
, ∂ϕ
∗
) = L(ϕ, ∂ϕ, ϕ
∗
, ∂ϕ
∗
) . (1.5.5)
Λ
ϕ

(x) = ϕ(x) + δϕ(x), δϕ(x) = iΛϕ(x),
ϕ
∗
(x) = ϕ
∗
(x) + δϕ
∗
(x), δϕ
∗
(x) = −iΛϕ
∗
(x).
δL = 0 δL

δL =
∂L
∂ϕ
δϕ +
∂L
∂(∂ϕ)
δ(∂ϕ) +
∂L
∂ϕ
∗
δϕ
∗
+
∂L
∂(∂ϕ
∗
)
δ(∂ϕ
∗
).
δL = ∂
µ

∂L
∂(∂
µ
ϕ)

δϕ +
∂L

∂(∂ϕ)
δ(∂ϕ) + ∂
µ

∂L
∂(∂
µ
ϕ
∗
)

δϕ
∗
+
∂L
∂(∂ϕ
∗
)
δ(∂ϕ
∗
)
= ∂
µ

∂L
∂(∂
µ
ϕ)
δϕ +
∂L

∂(∂
µ
ϕ
∗
)
δϕ
∗

= iΛ∂
µ

∂L
∂(∂
µ
ϕ)
ϕ(x) −
∂L
∂(∂
µ
ϕ
∗
)
ϕ
∗
(x)

= 0.
J
µ
(x) = −i


∂L
∂(∂
µ
ϕ)
ϕ(x) −
∂L
∂(∂
µ
ϕ
∗
)
ϕ
∗
(x)

, (1.5.6)
∂
µ
J
µ
(x) = 0. (1.5.7)
J
µ
(x)
∂
0

J
0

(x)d
3
x −

div
−→
J d
3
x = 0,
∂
0

J
0
(x)d
3
x = 0.
Q =

J
0
(x)d
3
x, (1.5.8)
∂
0
Q = 0. (1.5.9)
Q Q
Q

§
δL = ∂
µ
B
µ
, (1.6.1)
B
µ
= −a
µ
L B
µ
=
−ω
µ
ν
x
ν
L B
µ
= 0
δL =
∂L
∂ϕ
δϕ +
∂L
∂(∂ϕ)
δ(∂ϕ) = ∂
µ


∂L
∂(∂
µ
ϕ)
δϕ

. (1.6.2)
∂
µ

∂L
∂(∂
µ
ϕ)
δϕ − B
µ

= 0. (1.6.3)
C
µ
∂
0

C
0
d
3
x −

div

−→
C d
3
x = 0,
D =

C
0
d
3
x, ∂
0
D = 0. (1.6.4)
§
ϕ(x) (ϕ
∗
(x) = ϕ(x))
0
0
(✷ + m
2
)ϕ(x) = 0, (2.1.1)
✷ ≡ ∂
µ
∂
µ
=
∂
2
∂x

02
− ∇
2
.
m
2
ϕ(x) = ϕ
0
e
−iqx
, qx = q
µ
x
µ
= q
0
x
0
−
→
q
.
→
x
. (2.1.2)
−q
0
2
+
→

q
2
+ m
2
= 0,
q
0
2
=
→
q
2
+ m
2
. (2.1.3)
q
0
→
q
m
m
2
ϕ(x)
L = 1/2

∂
µ
ϕ(x)∂
µ
ϕ(x) − m

2
ϕ
2
(x)

. (2.1.4)
P
0
≡ H = 1/2


(∂
0
ϕ)
2
+ (
→
∇
ϕ)
2
+ m
2
ϕ
2

d
3
x, (2.1.5)
→
P

= −

∂
0
ϕ
→
∇
ϕd
3
x. (2.1.6)
ϕ(x)
q
0
→
q
ϕ(q) = ϕ(q
0
,
→
q
),
q
ϕ(x) = ϕ(x
0
,
→
x
) =
1
(2π)

3/2

ϕ(x
0
,
→
q
)e
i
→
q
.
→
x
d
3
q. (2.1.7)
x
0

d
2
dx
02
+ (
→
q
2
+ m
2

)

ϕ(
→
q
, x
0
) = 0. (2.1.8)
ϕ(
→
q
, x
0
) =
a(q
0
,
→
x
)

2q
0
e
−iq
0
x
0
+
a(−q

0
,
→
x
)

2q
0
e
iq
0
x
0
. (2.1.9)
ϕ(x) =

N
q

a(q
0
,
→
q
)e
−iqx
+ a(−q
0
, −
→

q
)e
iqx

d
3
q. (2.1.10)
ϕ(x) = ϕ
(+)
(x) + ϕ
(−)
(x), (2.1.11)
ϕ
(+)
(x) =

N
q
a(q)e
−iqx
d
3
q
ϕ
(−)
(x) =

N
q
a(q)e

−iqx
d
3
q
N
q
=
1
(2π)
3/2

2q
0
a(q)
a(−q)
a(q) a(−q) ϕ(x)
a(q) = i

N
q
e
iqx
↔
∂
0
ϕ(x)d
3
x, (2.1.12)
a(−q) = −i


N
q
e
−iqx
↔
∂
0
ϕ(x)d
3
x, (2.1.13)
↔
∂
0
x
0
f(x)
↔
∂
0
g(x) ≡ f(x)
∂g(x)
∂x
0
−
∂f (x)
∂x
0
g(x). (2.1.14)
i


N
q
e
iqx
↔
∂
0
ϕ(x)d
3
x = i

N
q
e
iqx
∂
0
ϕ(x)d
3
x −i

N
q
∂
0
(e
iqx
)ϕ(x)d
3
x

= i
 

N
q
N
q

a(q

)(−iq

0
)e
−i(q

−q)x
+ N
q
N
q

a(−q

)(iq

0
)e
i(q


+q)x

d
3
q

d
3
x
−i
 

N
q
N
q

a(q

)(iq
0
)e
−i(q

−q)x
+ N
q
N
q


a(−q

)(iq
0
)e
i(q

+q)x

d
3
q

d
3
x
=
 
N
q
N
q

a(q

)(q
0
+ q

0

)e
−i(q
0
−q

0
)x
0
e
i(
→
q
−
→
q

)
→
x
d
3
q

d
3
x.

e
i
→

p
.
→
x
d
3
x = (2π)
3
δ(
→
p
)
d
3
x d
3
q

a(−q) = a
∗
(q). (2.1.15)
P
0
≡ H = 1/2

[a(q)a(−q) + a(−q)a(q)] q
0
d
3
q, (2.1.16)

→
P
= 1/2

[a(q)a(−q) + a(−q)a(q)]
→
q
d
3
q. (2.1.17)
x
0
∂
0
ϕ(x) = ∂
0

N
q

a(q)e
−iq
0
x
0
+i
→
q
→
x

+ a(−q)e
iq
0
x
0
−i
→
q
→
x

d
3
q
=

N
q

a(q)(−iq
0
)e
−iqx
+ a(−q)e
iqx

d
3
q.


(∂
0
ϕ)
2
d
3
x
=
  
N
q
N
q

a(q)a(q

)(−iq
0
)(−iq

0
)e
−i(q
0
+q

0
)x
0
+i(

→
q
+
→
q

)
→
x
d
3
qd
3
q

d
3
x
+2
  
N
q
N
q

a(q)a(−q

)(−iq
0
)(iq


0
)e
−i(q
0
−q

0
)x
0
+i(
→
q
−
→
q

)
→
x
d
3
qd
3
q

d
3
x
+

  
N
q
N
q

a(−q)a(−q

)(iq
0
)(iq

0
)e
i(q
0
+q

0
)x
0
−i(
→
q
+
→
q

)
→

x
d
3
qd
3
q

d
3
x.

e
i
→
p
.
→
x
= (2π)
3
δ(
→
p
)
d
3
x

(∂
0

ϕ)
2
d
3
x =

1
2q
0
a(q
0
,
→
q
)a(q
0
, −
→
q
)(−q
0
2
)e
−2iq
0
x
0
d
3
q

+2

1
2q
0
a(q
0
,
→
q
)a(−q
0
, −
→
q
)(q
0
2
)d
3
q
+

1
2q
0
a(−q
0
, −
→

q
)a(−q
0
,
→
q
)(−q
0
2
)e
2iq
0
x
0
d
3
q.

(
→
∇
ϕ)
2
d
3
x

m
2
ϕ

2
d
3
x
P
0
=

a
∗
(q)a(q)q
0
d
3
q, (2.1.18)
→
P
=

a
∗
(q)a(q)
→
q
d
3
q. (2.1.19)
a
∗
(q)a(q)q

0
a
∗
(q)a(q)
→
q
a
∗
(q)a(q)
§
0
ϕ
1
(x) ϕ
2
(x)
ϕ(x) =
1
√
2
[ϕ
1
(x) + ϕ
2
(x)] ,
ϕ(x) ϕ
∗
(x)
(✷ + m
2

)ϕ(x) = 0, (2.1.20)
(✷ + m
2
)ϕ
∗
(x) = 0. (2.1.21)
L = ∂
µ
ϕ
∗
(x)∂
µ
ϕ(x) − m
2
ϕ
∗
(x)ϕ(x). (2.1.22)
P
0
≡ H =


∂
0
ϕ
∗
∂
0
ϕ+
→

∇
ϕ
∗
→
∇
ϕ + m
2
ϕ
∗
ϕ

d
3
x, (2.1.23)
→
P
= −


∂
0
ϕ∗
→
∇
ϕ + ∂
0
ϕ
→
∇
ϕ∗


d
3
x. (2.1.24)
ϕ(x)
a(q)
a(−q) = a
∗
(q).
P
0
≡ H =

[a
∗
(q)a(q) + a
∗
(−q)a(−q)] q
0
d
3
q, (2.1.25)
→
P
=

[a(q)a(−q) + a(−q)a(q)]
→
q
d

3
q. (2.1.26)
S
J
µ
(x) = −i

∂L
∂(∂
µ
ϕ)
ϕ(x) −
∂L
∂(∂
µ
ϕ
∗
)
ϕ
∗
(x)

= i(ϕ(x)
∗
∂
µ
ϕ(x) − ϕ(x)∂
µ
ϕ
∗

(x)). (2.1.27)
Q
Q =

J
0
d
3
x = i

(ϕ(x)
∗
∂
0
ϕ(x) − ϕ(x)∂
0
ϕ
∗
(x)). (2.1.28)
Q a(q) a
∗
(q)
Q =

[a
∗
(q)a(q) − a
∗
(−q)a(−q)] d
3

q. (2.1.29)