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a
xk
xk
S
vc
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L2
(
p) = ( + 1) (
p)
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()
()
S
S
HI
A A
e e
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m = 0
C
P
T
W
W
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i
t=
2
2m2
E
p
E =p2
2 m
E
p
E E = i t
,
p p = i .
t+ = 0 ,
(t, x) =
|(t, x)
|2 ,
(t, x) = i2 m
.
V
t
V
dV =
V
tdV =
V
dV =
S
nS dS ,
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S
V
nS S
S
S
nS dS = 0
t dV = 0 ,
E2 = p2c2 + m2c4
= c = 1
pp = m2 ,
p p = i
= m2 .
p
p
= (E,p)
/x
+ m2 + m2
+ m2
= 0
p(x) = eipx
eipx
,
i
tp(x) = p0 p(x) =
p2 + m2 p(x) .
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j = 0 ,
j =i
2 m[ () () ] .
j0
j0 =i
2 m
t
t
(x) = 1, 2, . . . , N
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=N
=1
(x) =
1(x)
N(x)
.
=
i m = 0 .
iN
n=1
()n(n) m = 0 ( = 1, . . . , N ) ,
i + m .
0 = (i + m) (i m)
=
1
2( + ) + m
2
.
+ = 2 g .
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02
=
,
k2
=
(k = 1, 2, 3) .
m
i m = 0 , 0
0
i
0 () +
0
m
0 0
= 0 .
0
0
= 0 .
0
0 = 0
=
0
.
j
j = 0 .
j0 = ,
(0)2 =
= 00 ,
k
k
= k .
j
j = ,
0
i
+ m = 0 ,
0
(0)2 =
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[] = 0 .
k
0
k
k
=
00k
=
0k0
=
00k
=
k
k
= 0 .
N
0k = k0 = (
) k0 .
0
k = (1)N
k
0 .
0 = 0
k = 0
0
k
1 = (1)N = N
.
N = 4
N = 2
0
k
0
1
k
i
5
5 i 0 1 2 3
5 5 = 5
5,
= 0
52
=
5
= 5
5
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a
a
a = 1, 2, . . . , 16
1
2 5 6 11 i
2[, ]
12 15 5
16 5
a
1
i
a
b
a=b
c =
a b = c = 1, i .
a = b
a b
c =
a
(a)2 =
a =
b
ab = ba .
a
a > 1
[a] = 0
a > 1 .
b a
[a] =
a
b2
=
bab
=
b2
a
=
[a] .
a b
= 0
a=b .
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0 0 =
0 0 = ,
S
4
a
a = a ,
= S a = Sa S1
i
t= H
H = i + m .
k
k = 0 k
0
= 0
k
= ,
k
= k ,
k
i j + j i = 2 ij ,
i + i = 0 ,
2 =
.
k k xk
.
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0
0 =
00
, k =
0 kk 0
, 5 =
0
0
,
=
00
, k =
0 k
k 0
.
k
1 =
0 11 0
, 2 =
0 ii 0
, 3 =
1 00
1
.
x x = x
(i m) (x) = 0 , (i
m)
(x
) = 0 .
x
= x , x = x
,
=
, =
.
(x)
(x)
(x) = S(x) ,
i m S (x) = 0 .
S1 i S1 S m
(x) = 0 .
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S1 S = .
= g ,
S1 S = .
S =
+
=
+
=
( + )
= 2 g
= 2 g
.
=
S
= g +
,
|
| 1 .
= .
S = + a O ,
O 4 4 O a
S1 = a O ,
a O
+ a O
=
g +
.
+ a [ , O] = + .
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a [, O] = .
O = a = i/4
[,
]
= g i = i g ,
[A,BC] = {A, B} C B {A, C}
[, ] = i [, g ] = i [, ] = i {, } i {, }
= 2 i
g g
,
i4
[, ] =
.
S = i4
.
S = e i4 .
xk
x1
(R1)
() =
1 0 0 00 1 0 00 0 cos sin 0 0 sin cos
=
1 0 0 00 1 0 00 0 1 0 0 1
+ O(2)
= g + 0 0 0 0
0 0 0 00 0 0 10 0 1 0
+ O(2) = g + (r1) + O(2) ,
(r1)
=d(R1)
()
d
=0
=
0 0 0 00 0 0 00 0 0 10 0 1 0
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x1
(r1)
(r1)23 = +1 (r1)
32 = 1 (r1)23 = 1 (r1)32 = +1
(r1) = 23 + 32 = 2 23 ,
= 223 x1
SR1() = ei2
23 .
k
ijk
123 = +1
ij = ijk k k = 12
kij ij
23 = 1
SR1() = ei2
1 .
xk
SRk() = ei2
k =
cos
2+ i k sin
2.
ei
2 k
=
n=0
i2 kn
n! =
n=0
i2 k2n
(2n)! +
n=0
i2 k2n+1
(2n + 1)!
=
n=0
(1)n12
2n
(2n)!+ i k
n=0
(1)n12
2n+1
(2n + 1)!
=
cos
2+ i k sin
2,
k
2n
=
,
k
2n+1
= k , i2n = (1)n ,
cos x =
n=0
(1)n x2n(2n)!
, sin x =
n=0
(1)n x2n+1(2n + 1)!
.
SRk( + 2) = SRk() ,
SRk
k
k = 0k5
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xk
v
x1
x
0= cosh x0
sinh x1 ,
x1 = sinh x0 + cosh x1 ,x
2= x2 ,
x3
= x3 ,
cosh = 1 v21/2 ,sinh = v .
x1
(1)
() =
cosh sinh 0 0 sinh cosh 0 0
0 0 1 00 0 0 1
=
1 0 0 1 0 0
0 0 1 00 0 0 1
+ O(2)
= g +
0 1 0 01 0 0 00 0 0 00 0 0 0
+ O(2) = g + (1) + O(2) ,
(1)
=d(1)
()
d
=0
=
0 1 0 01 0 0 00 0 0 00 0 0 0
x1
(1)
(1)01 = 1 (1)10 = 1 (1)01 = 1 (1)10 = +1
(1) = 01 + 10 = 2 01 ,
= 201
x1
S1() = ei2
01 .
0k i2
0, k
= i 0 k = i k ,
xk
Sk() = e12
k
=
cosh
2
k sinh
2
.
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S
S
S = ei4
.
0 0 =
00 = i
20
,
0 = i2
[, ] = ,
0 S 0 = ei4
,
0 S 0 = S1 .
xk
SRk() = ei
2
k
.
k
SRk() = ei2
k
= ei2
k
= S1Rk ()
SRk() k 0 = 0 k
xk
Sk() = e 12 k .
k
Sk() = e12
k
= e12
k
= Sk()
Sk() k 0 = 0 k
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xP
x = (x0,
x)
(x)P (x) = SP (x)
i 0
t+ i m
(x) = 0
i 0 t i mSP (x) = 0 .
S1P i S1P 0 SP
t i S1P SP m
(x) = 0 .
S1P 0 SP = 0
S1P
SP =
= SP = P 0 ,
P
(t, x) (t, x) = (t, x) (t, x) = P (t, x) 0 P 0 (t, x)
= |P|2 (t, x) (t, x)
|P|2 = 1 P = ei P
1/2
P
P = +1 SP
= P =
1 0 0 00 1 0 00 0 1 00 0 0 1
.
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xT x = (x0, x)
(x)T (x) = B (x) ,
B
B
i 0
t+ i m
(x) = 0
i 0
t+ i m
B (x) = 0 .
i t
B 0 + i B m B = 0 .
B1
i tB 0 B1+ i B B1 m = 0 .
i
t0 + i + m = 0 ,
B0 B1 = 0 ,
B B1 = .
0 5
0
0 51
= 0 ,
0 5
0 51
= ,
B
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C
C C1 = ,
B = T 0 5 C ,
T C
(x) a (x) ,
a =
, , , 5 , 5 ,
(x) =
S (x)
(x) (x) 0 = (x) S 0 = (x) 0 S 0 .
0 S 0 = S1 ,
(x) = (x) S1 ,
(x) = B (x) 0 = B 0 (x) 0 = (x) 0 B 0 .
B
0 B 0 = B1
(x) = (x) B1 .
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a =
(x) (x) = (x) S1 S (x) = (x) (x) .
a =
(x) (x) = (x) S1 S (x) = (x) (x) .
a =
(x) (x) = (x) S1 S (x) = (x) (x) .
a = 5
(x) 5 (x) = (x) S1 5 S (x) = (x)
S1 5 S
(x) .
S = e i4 5,
= 0
= S1 5 S = 5 .
SP = 00, 5
= 0
= S1P 5 SP = 5 .
(x)5(x)
(x) 5 (x) = (x) 5 (x) ,
(x) 5 (x) = (x) 5 (x) = (x) 0 5 (x)
= 0 ,
+ (x) k 5 (x)
= k .
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a = 5
(x) 5 (x) = (x) S1 5 S (x) .
(x) 5 (x) = (x) 5 (x) ,
(x) 5 (x) = (x) 5 (x) .
5
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(i m) (x) = 0
p p0 > 0
p(x)
ei px ei p x
u(p)
p,+(x) = ei px u(p)
p0 > 0 .
u(p)
p,+(x) x
p,+(x) = i p p,+(x) ,
(p m) u(p) = 0 ,
( p m) u(p) = 0 .
u(p) (
p m) = 0 .
v v v
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m=0
p = (m,0) 0
u(0) = 0 .
u(0) =+(0)
+(0)
0
0 =
00
,
0 00 +(0)+(0) = 0 .
(1)+ (0) =
10
(2)+ (0) =
01
,
+(0) =
00
,
u(1)(0) =
1000
u
(2)(0) =
0100
.
u(r)(0) (r = 1, 2)
i
t p,(x) = |p0| p,(x) .
p,(x) = ei px v(p)
p0 > 0 .
(
p + m) v(p) = 0 .
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(
p m) (
p + m) u(r)(0)
= 0 ,
r = 1, 2 .
u(r)(p) = C( p + m) u(r)(0) ,
C
u(r)(p) = C(
p + m) u(r)(0)
= C
00
E
0
0
p +
00
m
(r)+ (0)
0
= C
E+ m p
p E+ m
(r)+ (0)
0
= C(E+ m) (r)+ (0) p (r)+ (0)
= C(E+ m)
(r)+ (0) pE+ m
(r)+ (0)
.
C
u(r)(p) u(s)(p) = rs .
u(r)(p) u(s)(p) = |C|2 (E+ m)2
(r)+ (0)
(r)+ (0) p
E+ m
(s)+ (0) pE+ m
(s)+ (0)
= |C|2 (E+ m)2
1 p
2
(E+ m)2
rs
= |C|2 2 m (E+ m) rs ,
C =12 m (E+ m) ,
u(r)(p) =
p + m2 m (E+ m)
u(r)(0) ,
u(r)(p) = u(r)(0)
p + m2 m (E+ m)
.
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u(r)(p)
u(r)(p)
u(r)(p)
u(r)(p) (r)+ (p)(r)+ (p)
=E+ m
2 m
(r)+ (0) pE+ m
(r)+ (0)
.
u(1)(p) =
E+ m
2 m
10
p
E+ m
10
,
u(2)(p) =E+ m
2 m
01
pE+ m
01
.
v(r)(p) = C(
p + m) v(r)(0) .
v(r)(p) = C(
p + m) u(r)(0)
= C 0
0
E+
0 0
p + 00
m 0
(r) (0)
= C(E+ m)
pE+ m (r) (0)(r) (0)
.
v(r)(p) v(s)(p) =
|C
|2 (E+ m)2
(r) (0)
p
E+ m (r) (0)
p
E+ m(s) (0)
(s) (0)
= |C|2 (E+ m)2
p2
(E+ m)2 1
rs
= |C|2 2 m (E+ m) rs .
v(r)(p) v(s)(p) = rs ,
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C
v(r)(p) =
p + m
2 m (E+ m)
v(r)(0) ,
v(r)(p) = v(r)(0) p + m2 m (E+ m)
.
v(r)(p)
v(r)(p)
(r) (p)
(r) (p)
=
E+ m
2 m
pE+ m (r) (0)(r) (0)
.
v(1)(p) =
E+ m
2 m
p
E+ m
10
10
,
v(2)(p) =
E+ m
2 m
p
E+ m
01
01
.
u(r)(p) v(s)(p) = 0 ,
v(r)(p) u(s)(p) = 0 .
uu uu
pu = mu
u(r)(p) u(s)(p) = u(r)(p) 0 u(s)(p) = u(r)(p)
p 0 + 0
p
2 mu(s)(p)
= u(r)(p)g0p
mu(s)(p) =
E
mu(r)(p) u(s)(p) =
E
mrs .
p,+(x) p,+(x) d3x
d3x =1 v2/c2 d3x
= p,+(x) p,+(x) = u(p) u(p)
= /
1 v2/c2 = E/m
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V
d3x (r)
p,+
(x)
(s)p,+(x) = |N|2 u(r)
(p) u(s)(p) p,p
V
d3x = |N|2 Em
V rs p,p ,
N =1V
m
E.
d3x (r)p,+
(x) (s)p,+(x) = |N|2 u(r)
(p) u(s)(p)
d3x ei(pp
)x = |N|2 Em
rs (2 )3 3(p p) ,
N =1
(2 )3/2
mE
.
(r)p,(x) =1V
m
Eei px v(r)(p)
(r)p,(x) = 1(2 )3/2
mE
ei px v(r)(p)
(r)p,+(x) (r)
p,(x) d3x
(r)p,+
(x)
(s)p,(x) = 0 .
u(r)(p) = u(r)(0) p + m2 m (E+ m) ,
v(r)(p) = v(r)(0) p + m
2 m (E+ m),
p = (p0,p) = (p0, p)
u(r)(p) v(s)(p) = 0 ,
v(r)(
p) u(s)(p) = 0 .
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(r)
p,+(x) (r)
p,(x)
(x) = r
d3p b(r)p (r)p,+(x) + d(r)p (r)p,(x)=
1
(2)3/2
r
d3p
m
E
b(r)p u
(r)(p) eipx + d(r)p
v(r)(p) eipx
.
b(r)p d
(r)p
1
(2)3/2
d3p 1
V
p
.
(x)
d3x (x) (x) = 1 ,
r
d3p
|b(r)p |2 + |d(r)p |2
= 1 .
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+(p)
r=1,2
u(r)(p) u(r)(p) ,
(+(p))
r=1,2
u(r) (p) u(r) (p) .
+(p)
(+(p)) u(s) (p) =
r=1,2
u(r) (p)
u(r) (p) u
(s) (p) = u
(s) (p)
+(p) v(s)(p) = 0 .
+(p)
(+(p))2 = +(p)
r=1,2
u(r) (p) u(r) (p)
s=1,2
u(s) (p) u(s) (p) =
r,s
u(r) (p)
u(r) (p) u
(s) (p)
rs
u(s) (p)
=
r=1,2
u(r) (p) u(r) (p) .
(p) r=1,2
v(r)
(p) v(r)
(p) ,
((p)) v(s) (p) = v
(s) (p) ,
(p) u(s)(p) = 0 ,
((p))2 = (p) .
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+(p)
+(p) =1
2 m (E+ m)(
p + m)
r=1,2
u(r)(0) u(r)(0)
+(0)(
p + m) ,
+(0)
0
+(0) = 0 ,
+(0)
0
= 0 .
(+(0))2 = +(0)
+(0) =1 + 0
2.
+(0) =1+0
2+ B
B = 0
+(p) =1
2 m (E+ m)(
p + m)1 + 0
2(
p + m) .
(
p + m) 0 (
p + m) = (
p + m)
0p0 kpk + m
0
= (p + m)
p + m + 2 0p0
0
= 2 E(
p + m) ,
(
p + m) (
p + m) = (
p + m) (
p m + 2 m)= 2 m (
p + m) ,
(p + m)
1 + 0
2(
p + m) = (E+ m) ( p + m) ,
+(p) =
p + m
2 m.
+(p)
(+(p)) =
p + m
2 m= 0
p + m
2 m0 = 0 +(p)
0 .
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(p)
(p) =
p + m
2 m.
(p)
((p)) = 0 (p)
0 .
+(p) + (p) = ,
+(p) (p) = (p) +(p) = 0 .
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(x)
(x) =
1 + i J3
(x) ,
J = L +1
2 .
L
/2
/2
/2
J = L +1
2 .
3
=
00
3 u(r)(m,0) = 3 00 3
(r)+ (0)0
= 3 (r)+ (0)0
= (1)+ (0)
0 ,(2)+ (0)
0
.
3 v(r)(m,0) =
3 00 3
0
(r) (0)
=
0
3 (r) (0)
=
0
(1) (0)
,
0
(2) (0)
.
3 u(1)(m,0) = +u(1)(m,0)
3 u(2)(m,0) = u(2)(m,0)
3 v(1)(m,0) = +v(1)(m,0)
3 v(2)(m,0) = v(2)(m,0)
u(r)(m,0)
v(r)(m,0)
r = 1, 2
i/t
3
p0 1/2
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j , k = 2 i
jk ,
j=k
j , k
=
0 j 5 , 0 k 5
= j , k = 2 j k .
j
j =1
2jmn mn =
i
2jmn m n ,
jk j
jk j =i
2
jk jmn m n =i
2 km n
kn m
m n =i
2 k
k = i
k ,
j k = i jk
(H)
d(H)
dt= i
H , (H)
+
(H)
t.
(H)
(S)
(H)(t) = eiH t (S) eiH t .
H = p + m .
L
Lj
dLj
dt= i
H , Lj
= i
p , Lj = i k pk , Lj
= jk kp = ( p)j ,
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dL
dt= p .
L
/2
j
dj
dt= i
p + m , j = i p , j ,
,
= 0
[ p , j]
= 5
i kpk , j = i pk k 5 , j = i pk k , j 5 = 2pk kj 5= 2 kjpk = 2 ( p)j .
d
dt= 2 p
p
[H , p] = 0
d
dt p = i H , p = i H , p .
H ,
= 2 i p
d
dt
p
= 0 .
s p|p|
J
dJ
dt=
d
dt
L +
1
2
= 0 ,
J
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E
B
A
E = A
t A0 ,
B = A .
F
F A A ,
F =
0 E1 E2 E3E1 0 B3 B2E2 B3 0 B1E3
B2 B1 0
.
A
E
B
F
F
A(x) A(x) + (x) .
A(x)
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E = , B
E
t= j ,
= F = j ,
B = 0 , E+
B
t= 0 ,
= F + F + F = 0 .
=e2
4c=
e2
4
.
F
j
j = F
= 0 .
F A A
A
A (A) = j .
p p ec
A ,
e
p p ec
A ,
+ i e A .
i
m
i m i e A m
(i
e
A m) (x) = 0 .
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(x)
(x)
(x)
(x) = ei
(x)
(x)
j =
(x) (x) = ei(x) (x) .
(x)
ei(x) (x)
= iei(x) (x) (x) + ei(x) (x) ,
[i (x) m] (x) = 0 .
A
A (x) = (x)/e
A A = A 1
e(x) .
(x)
A(x)
D
D = + ieA .
(x)
ei(x) (x) ,
A(x) A(x) 1e
(x) ,
D(x) ei(x) D(x) .
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p e A + m + e A0 = E
(x) =
(x)(x)
,
0 0 p e
A(x)(x) = E e A
0
00
m
00
(x)(x) ,
p e A
(x) =
E e A0 m(x) ,
p e A
(x) =
E e A0 + m(x) . p e A
(x) =1
E
e A0 + m
p e A
(x) 1
E
e A0 + m
(x) .
(x)
1E e A0 + m (x) =
E e A0 m(x) .
v/c 1
E m e A0 m .
E(nr) E m m ,
1
E e A0 + m =1
2 m + E(nr) e A0
=1
2 m
1 E
(nr) e A02 m
+ . . .
.
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B ,
e 2 m c
2 e2 m c
1
2
= g
e
2 m cs ,
s =1
2 ,
,
g = 2 ,
.
B
B |e| 2 m c
= 0.579 1014 MeV gauss1 .
B
Hint
g = 2
g = 2
g = 2
1 +
2
.
(x)
p e A
2 m(x) .
A = 0
1
2m p
1 E
(nr) e A02 m
p + e A0
(x) = E(nr)(x) .
v2/c2
E(nr) e A0 mv2/2
E(nr) e A0m c2
= O
v2
c2
.
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(v2/c2)2
v2/c2
=
d3x = 1
1 =
d3x
+
d3x1 + p24 m2
.
A = 0
(x) p2 m
(x) = (x) vc
(x) .
v2/c2
X = ,
= 1 + p2
8 m2.
d3x XX =
d3x
1 +
p2
4 m2
= 1 + O
v2
c2
2.
1 = 1X
1
1
2m p
1 E
(nr) e A02 m
p + e A0
1X(x) = E(nr)2X(x) .
1 1 p2
8 m2,
1 p
2
8 m2
1
2m p
1 E
(nr) e A02 m
p + e A0
1 p
2
8 m2
X(x)
= E(nr)
1 p2
4 m2
X(x) ,
1 p28 m2( p)22 m + e A01 p28 m2 p E(nr) e A04 m2 pX(x)= E(nr)
1 p
2
4 m2
X(x) ,
p2
2 m+ e A0 (p
2)2
8 m3
p2
8 m2, e A0
p E
(nr) e A04 m2
p
X(x)
= E(nr)
1 p2
4 m2
X(x) .
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E(nr)p2
E(nr) , p2
/2
p2
2 m+ e A0 (p
2)2
8 m3+
1
8 m2
p2 , E(nr) e A0 2 p E(nr) e A0 pX(x)= E(nr) X(x) .
{A2 , B} 2 A B A = [A , [A , B]] p2 , E(nr) e A0 2 p E(nr) e A0 p = p , p , E(nr) e A0
= p , p , e A0 .
p , e A0
= i e E ,
p , E = j k pj , Ek+ j , kEkpj=
jk + i jk
pj , Ek
+ 2 i jk Ekpj
= p E+ i
p E
2 i
Ep
= i E 2 i
Ep
.
p E
p
E =
i
E = i
A0 = 0 .
p2
2 m+ e A0 (p
2)2
8 m3 c2 e
4 m2 c2
Ep
e2
8 m2 c2 E
X(x) = E(nr) X(x) ,
e A0
p2
2 m (p2)2
8 m3 c2
p2 c2 + m2 c4 m c2 = m c2
1 +
p2 c2
m2 c4
1/2 m c2
m c2
1 +p2
2 m2 c2 (p
2)2
8 m4 c4
m c2
p2
2 m (p
2)2
8 m3 c2.
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e4 m2 c2
Ep
B = E
p
m c
=e
2 m c ,
E = rr
dA0
dr,
Ep = 1r
dA0
dr (r p) = 2
1r
dA0
dr1
2 (r p)
= 2
1
r
dA0
drs ,
e4 m2 c2
Ep
=e
2 m2 c21
r
dA0
drs .
e2
8 m2 c2 E
A0(r + r) = A0(r) + r A0 + 12
xi xj2A0
xi xj,
A0(r + r)
= A0(r) +1
2
2A0
xi xj
xi xj
,
xi xj = 13 (r)2 ij 13 m c2
ij ,
(eA0) e 2
6 m2 c2A0 = e
2
6 m2 c2 E .
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K
K =
L +1
2
1
2
=
L +
1
2
23 1
2
=
L +
.
J
K
J , K
= 0 .
H
J2
J3
K
H = E ,J2
= 2j (j + 1) ,J3 = j3 ,K = .
K2
J2
K2
K2 =
L +
L +
=
L +
L +
.
a b = a b + i a b
L
L
= L2
+ i
L L
iL
= L2 L ,
K2 = L2
L + 2
L + 2 = L
2+
L + 2 .
J2
=
L +
1
2
2= L
2+ L + 3
42 .
K2 = J2
+1
42 .
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2 2 = 2j (j + 1) +
1
42 = 2
j +
1
2
2,
=
j +1
2
.
K
K =
L +
=
00
L + 00
L +
=
L + 00 L
.
K =
=
,
L + 0
0 L = ,
L +
= , L +
= ,
L = (1 + ) , L = (1 ) .
J2
L L2
J2
L2
J2
=
L +
1
2
2= L
2+ L + 3
42 ,
L2
= J2 L 3
42 ,
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L2
=
J2 L 3
42
=
2j (j + 1) + 2 (1 + ) 3
42
= 2 j (j + 1) + + 14
2 ( + 1) ,
L2
=
J2 L 3
42
=
2j (j + 1) + 2 (1 ) 3
42
= 2j (j + 1) + 1
4
2 ( + 1) .
L2
j
=
L2
( + 1) = j (j + 1) + +
14
,
( + 1) = j (j + 1) + 14 .
= +j + 12 = = j + 12 , = j 12 ,
=
j +1
2
=
= j 12 , = j +
12
,
c p + m c2 + V(r)(x) = E (x) ,
(x) =
(x)(x)
,
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c p (x) = E V m c2(x) ,c p (x) =
E V + m c2
(x) .
(x) (x) = g(r) Yjj3 , = i f(r) Yjj3 ,
Yjj3 J2
J3
L2
S2
Yjj3 = m,ms m s ms |j j3 Y,m s,ms .
p
p = rr2
( r) ( p) ,
( r) ( p) = r p + i (r p) = i r r
+ i L ,
p =
rr2 i r
r
+ i
L .
p = i rr2
r
r+ L
= i
rr2
r
r (1 )
= r
rYjj3
df
dr+
(1 )r
f
.
r
Yj
j3
J2
J3
L2
2j(j + 1)
j3 2( + 1)
Yjj3 J2
J3 L2
J2
J3 Yjj3
r =
rr
Yjj3 = 1 Yjj3 ,
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1 Yjj3
rr
2= 1
rr
Yjj3 = 2 Yjj3 .
12 = +1
Yjj3 Yjj3
1 = 2 = 1 r
rYjj3 = Yjj3 ,
rr
Yjj3 = Yjj3 .
p = dfdr + 1 r fYjj3 .
p = i
dg
dr+
1 +
rg
Yjj3 .
c
df
dr+
1 r
f=
E V m c2 g ,
cdg
dr+
1 + r
g
=
E V + m c2 f . f(r) g(r)
V(r)
F(r) = r f(r) , G(r) = r g(r) ,
c
dF
dr
rF
= E V m c2G ,
c
dG
dr+
rG
=
E V + m c2F .
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V(r) = 14
Z e2
r.
z1 =m c2 + E
c, z2 =
m c2 E c
,
=Z e2
4 c Z ,
=
z1 z2 r .
z1 z2 =(m c2)
2 E2( c)2
0
dF
d
F = z2
z1
G ,dG
d+
G =
z1z2
+
F .
dF
d=
z2z1
G ,
dG
d = z1z2 F ,=
d2F
d2= F ,
d2G
d2 = G ,
F() me , G() me ,
m
m
F() me , G() me .
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bn =
z1z2
an .
n an
an1=
2
n,
n
an n e2 ( ) .
n bn n e2 ( ) .
n an
n
n bn
n
F()
G()
F() s e
G() s e
an+1
an = 0 , an+1 = 0 .
n = n + 1
an = z2z1
bn bn+1 ,
(s + n + 1 + ) bn+1 bn =
z1z2
an .
bn+1 = 0
an =
z2z1
bn .
n = n bn
2
z1 z2 (s + n) = (z1 z2) .
z1 z2
z1 z2 =
1
c
(m c2)2 E ,
z1 z2 = 2 E c
,
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n j n
1
1S1/2
2
2P3/2
3 3D5/2
1
2S1/2
+1
2P1/2
2
3P3/2
+2
3D3/2
1
3S1/2
+1
3P1/2
E2
1 +2
(s + n)2
=
m c22
,
E = m c2
1 + (Z )2
n+
(j+ 12)2(Z )2
2.
E
n
|| = j + 1/2
= (j + 1/2)
= + (j + 1/2) = = j + 1/2 ,
= (j + 1/2) = = j 1/2 .
n = 0
< 0
n = 0
(s + ) b0 = a0 ,
a0 =
z2z1
b0 ,
= s + =
z2z1
.
s > 0
< 0
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n
j n
n = n + || = n +
j +1
2.
=
2
E = m c2
1 12 (Z )2n2
12
(Z )4
n3
1
j + 1/2 3
4 n
+O
6 .
0 E mc2
4
n
j
n
j
Z = 1
E = m c21
2
4
n3 1jmin + 1/2 1jmax + 1/2= m c2
1
2
4
n3jmax jmin
(jmin + 1/2) (jmax + 1/2).
E(2P) E2P3/2 E2P1/2 = 425
m c2
= 8.87 1011 m c2 = 4.53 105 eV ,
(2P) = 12
E
= E2 6.58 1022 MeV sec = 11 GHz .
V = Ze24r
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TE
0
m c2 eee
eee
eee
eee
eee
eee
eee
eee
eee
eee
eee
eee
+ m c2
Hhf = 23
e p |n(0)|2 ,
n(0) e
p
p = (1 + ap)|e|
2 mp c
p .
mp ap = 1.79
N
N |e| 2 mp c
= 3.15 1018 MeV gauss1 .
n
|n(0)|2 = 1 a30 n
3, a0 =
1
m c
Hhf =2
3(1 + ap)
1
n3m
mpm c2 4 e p .
e p =
1
3
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Ehfn=1( ) =8
3(1 + ap)
m
mpm c2 4
= 5.88 106 eV ,
=1
2
E
=
E
2 6.58 1016 eV secGHz
109 sec1= 1.42 GHz .
m/mp
n
j
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mc2
mc2 Eleg
E(E> 0)
E = Evuoto (E) = Evuoto + E,
Evuoto
Eoss = E
Evuoto =
E.
Q = Qvuoto e = Qvuoto + |e| , Qoss = Q Qvuoto = |e| .
E
E
e = |e|
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m c2e
ee
eee
eee
eee
eee
eee
eee
eee
eee
eee
eee
eee
+ m c2
m
T
E1
E2
E1
E2 E = E2 + E1
E2 E1
e + e+ .
e + e+ . . . ,
e + e+ 2 .
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e
e+
e+
e
e+
e
e+
e
p
e
S
E(2P1/2)
E(2S1/2) = 27MHz
2S1/2
2P1/2
(i
e
A m) (x) = 0 .
(i
+ e
A m) c(x) = 0 .
c(x)
i e A m = 0 ,
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=
0 =
1 00 1
001
0
eimt = eimt
00
10
,
c = iceimt
0 0 0 i0 0 i 00 i 0 0i 0 0 0
00
10
= ceimt
0100
.
c
|e|
(r)+ (x) = e
imt
(r)
0
,
(1) = 10 ,
(2) =
01
;
(r) (x) = e
+imt
0
(r)
.
P = 0 =
(r)+ (x)
P +(r)+ (x) ,(r) (x)
P (r) (x) .
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(i
m) (x) = e
A(x) (x)
(x) = 0(x) +
d4x K(x x) e
A(x) (x) ,
0(x) K(x x)
4 4
(i
m) K(x x) = 4(x x) .
i
m
(x) = 0(x) + d4xK(x x)e A(x) 0(x) + d4xK(x x)e A(x) [0(x) + . . . ]= 0(x) +
d4xK(x x)e A(x)0(x)
+
d4xd4xK(x x)e A(x)K(x x)e A(x)0(x) + . . . .
K(x x)
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T
ERep0
Imp0
r rE+E
E E E ECF
CF
K(x
x)
K(xx)
K(x x) = 1(2)4
d4p K(p) eip(xx
)
(i
m) K(x x) = 1(2)4
d4p (
p m) K(p) eip(xx) .
4(x x) = 1(2)4
d4p eip(xx
) .
(
p m) K(p) =
,
K(p) =
p + m
p2
m2
,
(
p + m) (
p m) = p2 m2
K(x x) = 1(2)4
d4p
p + m
p2 m2 eip(xx) .
K(p)
p20 p2 m2 = 0 .
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p0
p0 =
p2 + m2 E .
p0
p0
K(p)
K(x x)
CF
p0
C
f(z) dz = 2 i
n
limzan
(z an) f(z)
,
z = an f(z) C
eip0(x0x0) = ei (x0x
0)Re(p0) e(x0x
0)Im(p0)
x0 > x0
x0 < x0
KF
KF(x x) = 1(2)4
CF
d4p
p + m
p2 m2 eip(xx) ,
KF(x
x) =
2i(x0
x0)
1
(2)4 d
3p lim
p0E(p0 E)p + m
(p0 E) (p0 + E)eip(xx
)
+ 2i(x0 x0)
1
(2)4
d3p lim
p0E
(p0 + E)
p + m
(p0 + E) (p0 E) eip(xx)
= i
d3p
(2)32E
(x0 x0) ( p + m) eip(xx
) (x0 x0) ( p m) eip(xx)
p0=E.
p0
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T
ERep0
Imp0
x0 > x0
s
r rE+E
E E E E
T
ERep0
Imp0
x0 < x0
s
r rE+E
E E E E
E E+ i +E +E i
KF(x x) = 1(2)4
d4p
p + m
p2 m2 + i eip(xx) ,
(p0 + E i ) (p0 E+ i ) = p2 m2 + i .
j(x)
+ m2(x) = j(x) .
(x) = 0(x) +
d4x G(x x)j(x) ,
G(x x)
+ m2
G(x x) = 4(x x) .
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G(x x) = 1(2)4
d4p G(p) eip(xx
) ,
+ m2
G(x x) = 1
(2)4
d4p
p2 + m2G(p) eip(xx) .
4(xx)
G(p) = 1p2 m2 .
GF(x) = 1(2)4
d4p
1
p2 m2 + i eipx ,
GF(x) KF(x)
(i
+ m) GF(x) = 1(2)4
d4p
p + m
p2 m2 + i eipx = KF(x) .
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c
[
] = [E t] ,
[c] =
t1 .
= c = 1 .
[E] = t1
= 1
.
E2 = p2 + m2
[E] = [|p|] = [m] .
c = 197
,
197
= 1 .
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v v =
(v0
, v1
, v2
, v3
)
v0
v1
v2
v3
v = (v1, v2, v3)
v = (v0, v1, v2, v3) v
v = g v ,
g
g
g =
1 0 0 00 1 0 00 0 1 00 0 0 1
.
v
v0 = v0 , vk = vk (k = 1, 2, 3) .
(g)
g g = g ,
g 1 0 0 0
0 1 0 00 0 1 00 0 0 1
.
. . .
k
i
j
. . .
u v = u0 v0 + u
1 v1 + u2 v2 + u
3 v3 .
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g = g =
1 0 0 00 1 0 00 0 1 00 0 0 1
.
u
v
u v = u v = u v = g u v = g u v = u0 v0 u v .
v2 v v
v2 > 0
v2 = 0
v2 < 0
L()
v
v
v
= v ,
v2
= v2 g v v = g v v .
g
= g .
v = v ,
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Tr(1 . . . n) = Tr
52
1 . . . n
= Tr
5 1 . . . n 5
= (1)nTr
52
1 . . . n
= (1)nTr(1 . . . n) ,
5
k
Tr() = 0 ,
Tr
5
= 0 .
n
Tr( ) =1
2Tr( + ) = g Tr(
) = 4 g .
Tr( ) = g Tr( ) g Tr( ) + g Tr( )= 4 (g g g g + g g) .
n
n
n 2 Tr(123 n) = g12Tr[34 n ] g13Tr[24 n ] +
+ g1nTr[23 n1 ] .
Tr
5
= 0 ,
Tr
5
= 0 .
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Tr
()2 5
= Tr 5 = Tr()2 5
()2
Tr(5) =
Tr(5) = 0
5
Tr
5
= i Tr 0 1 2 3= i
2Tr
0 1 2 3 i
2Tr
0 1 2 3
= i g3 Tr 0 1 2+ i g2 Tr 0 1 3 i g1 Tr 0 2 3+ i g0 Tr
1 2 3
i g Tr0 1 2 3= 0 .
5
5
Tr
5
= 4 i ,
0123 = +1
Tr
0 1 2 3 5
= i Tr
55
= 4 i = 4 i 0123 .
a
1
n
Tr(1 2 n1 n) = Tr(n n1 2 1) .
C1C =
C
CC1 =
n
Tr(1 2 n1 n) = TrC 1 C1 C 2 C1 C1C n1 C1 C n C1= (1)n Tr(
1
2
n1
n) = Tr
[n n1 2 1 ]T
= Tr(n n1
2 1) .
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{ , } = 2 g
,
0 0 = .
44
S
S1
= S S1 .
S
0D =
00
, kD =
0 k
k 0
,
5D =
0
0
, kD =
0 k
k 0
,
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CD = i 2D 0D = i 2D = i
0 2
2 0
,
0kD = i 0 k
k 0 , ijD = ijk k 00 k ,
0D 5D =
0
0
, kD
5D =
k 00 k
.
2
D
D
t+ + i m
= 0 ,
k
2D M D 2M
M = SM D S
1M
SM =1
2
D +
2D
=
12
0D +
0D
2D
=
12
2
2
.
SM = S1M = S
M .
M = SM D S1M =
2D ,
2M = SM 2D S
1M = D .
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0M =
0 2
2 0
, 1M =
i3 0
0 i3
,
2M = 0 2
2 0
, 3M =i1 0
0 i1
,
1M =
0 1
1 0
, 2M =
00
,
3M =
0 3
3 0
, 5M =
2 00 2
,
CM = i 0M = i 0 2
2 0
.
5
5C = 0D
0C = 5D
C = SC D S
1C ,
SC =1
2
+ 0D 5D
=
12
SC = SC = S
1C =
12
0D 5D=
12
.
k
kC = SC kD S
1C =
kD .
CC = SC CD SC = i2 00 i2
.
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0C =
0
0
, kC =
0 k
k 0
,
5C =
00
, kC =
k 00 k
,
CC = i
2 00 2
, kC =
k 00 k
.
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(x) a (x) = (x) B1 a B (x) = (x) B1 a B (x)T= (x) B a B1 (x) .
BaB1
B
5 B1 = 5 ,
B 5 B1 = B B1 5 = +0 5
= 0 ,
k 5 = k . ,
B B1 = + = 0, =0 =0, = 0 ,
=0, =0 . .
a =
(x) (x) = (x) B B1 (x) = (x) (x) .
a =
(x) (x) = (x) B B1 (x) = + (x) 0 (x) = 0 , (x) k (x)
= k .
a =
(x) (x) = (x) B B1 (x)=
+(x) (x)
= 0, =0
=0, = 0 ,(x) (x)
=0, =0 .
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a = 5
(x) 5 (x) = (x) B
5
B1 (x) =
+(x) 0 5 (x)
= 0 ,
(x) k 5 (x)
= k .
a = 5
(x) 5 (x) = (x) B 5 B1 (x) = (x) 5 (x) .
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L2
( p) =
+ 1
( p)
L2
2( + 1) 2( + 1)
p
L2
2( + 1) p L2
2( + 1)
L2
p
L2
( p) = ( p) L2 +
L2
, p
= 2
( + 1) ( p) + L2 , p .
[LkLk, pj] = [Lk, [Lk, pj]] +
2[Lk, pj]Lk
[Lk, pj] = km[rm, pj]p = i
kjp
L2
, p
=
k,j
j
Lk ,
Lk , pj
+ 2
Lk , pj
Lk
=
k,jj
2 kj kmpm + 2 i kjp Lk
= 2 2 p + 2 ( p) L .
k,
kj km = 2 jm ,
i
j
j jk = k k ,
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p L = 0
L
(1 )
L2
,
p = 2 2 ( p) .
( + 1) + 2 = ( + 1)
L2
( p) = ( + 1) ( p) .
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(x)
F A
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A(x) A(x) = A(x) + (x) .
A(x) A(x)
A(x)
A(x)
A = 0 .
A
A(x) = A(x) + (x) A
(x) = 0 (x)
(x) = A(x) .
A(x)
(x) = 0
A = 0 .
j = 0
A(x) = 0 ,A0(x) = 0 .
A(x) A(x) = A(x) (x) ,A0(x) A0(x) = A0(x) + 0(x)
(x)
(x) = A(x) = A(x) = 0 ,0(x) = A0(x) = A0(x) = 0 .
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A(x)
A (A) = j
j
= 0
= 0
A0 0
0A0 +
A
= 0 .
A0 = 0 0
A
= 0
A
A0 = 0
j = 0
A0 = 0
B =
A ,
E = A
t,
A
A = 0 .
A0 = 0
A
t = 0
L
V
A
A(0, x) =1V
k
=1,2
ck,(0)
()(k) eikx
uk,(x)
+ck,(0)()(k) ei
kx uk,
(x)
,
(1)(k)
(2)(k)
k
()(k)
k
(1)(k) (2)(k)
()(k) ()(k) = , ()(k) k = 0 .
uk,(x)
1
V
d3x uk,(x) uk,(x) = k, k
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Ek
T(1)
(2)
(1)(k)
(2)(k)
Lk = 2 (n1, n2, n3) , ni = 1, 2, . . .
A ck,(0)
ck,(t) = ck,(0) ei t ,
ck, + 2 ck, = 0 ,
=
2
T = |k|c .
A(t, x) =1V
k
=1,2
ck,(0)
()(k) eikx + ck,(0)()(k) eikx
,
k0 =
H =1
2
d3x
| E|2 + | B|2
H =
k
=1,2
2
c
2ck,(t) ck,(t) .
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A(t, x) =1V k =1,2 c
2 ak,()(k) eikx + ak,
()(k) eikx .
A
E
B
E(t, x) =iV
k
=1,2
c
2
ak,
()(k) eikx ak,()(k) eikx
,
B(t, x) = iV k =1,2
c
2k
ak,
()(k) eikx ak,()(k) eikx
.
H =1
2
k
=1,2
ak,ak, + ak,ak,
=
k =1,2
ak,ak, +
1
2
.
ak,ak,
k
Nk,
Nk, ak,ak, .
nk1,1 (k1, 1) nk2,2
(k2, 2)
|nk1,1 , nk2,2 , . . . , nki,i , . . . .
Nki,i nki,i
Nki,i|nk1,1 , nk2,2 , . . . , nki,i , . . . = nki,i |nk1,1 , nk2,2 , . . . , nki,i , . . . .
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ak, ak,
Nk,
[ak, , Nk,] = k,k ak, ,
[ak, , Nk,] = k,k ak, .
(k, )
k,
N a|n = a N + a |n = (n + 1) a|n ,
N a|n = (a N a) |n = (n 1) a|n .
a
a
a
,
a
.
a|n = c+|n + 1 , a|n = c|n 1 ,
c+ c
n |n = 1
|c+|2
= n|a a
|n = n| a a
a
a [a,a]=1
+ a
aN
|n = n + 1 ,
|c|2 = n| a aN
|n = n .
c+ c
a|n = n + 1|n + 1 , a|n = n|n 1 .
|0
(k, )
ak,|0 = 0 (k, ) .
ak,
|1k, = ak,|0 ,
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|1k, , 1 k, = ak,ak,
|0 .
(k, ) (k, )
H
H =
k
=1,2
Nk, +
1
2
.
1
2
k
=1,2
H|0 = 12 k =1,2
|0 .
H =
k
=1,2
Nk, = H|0 = 0 .
P =1
c
d3x ( E B) .
P
E
B
P =
k
=1,2
k
Nk, +
1
2
=
k
=1,2
kNk, .
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H ak,|0 = ak,
|0 ,
P a
k,|0 = k a
k,|0 .
E = = |k|c
p = k
m =1
c2
E2 p2c2 = 1
c2
2 2 2k2c2 = 0 .
()
(1)
(2)
() = (1)
i(2)
2
.
Tx Ty Tz T1,m m = 0, 1 T1,1 =
Tx i Ty2
,
T1,0 = Tz ,
()
m = 1 m = 0
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(x)
= 1, . . . , N
A(x)
= 1, . . . , 4
L = L(, )
(x)
x
x(s) x(s)
qs (t) = (t, s) (s = 1, 2, . . . ) ,
(t, s) = (t, x(s)) L(t) =
d3x L((x), (x))
L(t) =
s
x(s) L(s) ,
L(s) s qs (t)
ps (t) =L
qs (t)=
L
(t, s)=
L(s)(t, s)
x(s)
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H =
s
ps qs L .
(t, s) L(s)(t, s)
ps (t) = (t, s) x(s)
H
H =
s
x(s)
(t, s) (t, s) L(s)
.
x(s) 0
(t, s) (x)
(t, s) L(x)
(x) ,
H(t) =
d3x H(t, x) ,
H(x) = (x) (x) L(x) .
(x)
(x)
I()
d4x L(, )
(x) (x) +(x)
I() = 0 .
I() =
d4x
L
+
L()
()
()
=
d4x
L
+
L
()
L
()
.
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d4x
L
()
=
dSL
() = 0 ,
= 0
0 = I() =
d4x
L
L
()
.
L
() L
= 0 ( = 1, . . . , N ) .
(x)
L= a
+ b ,
a
b
a () b = 0 .
+ m2
= 0 ,
b/a = m2 L
a = 1/2
L = 12
1
2m2 .
N
(x)
= 1, . . . , N
L = 12
1
2m2 ,
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(x)
(+ m2) = 0 .
(x)
(x) =L
(0 )= 0
(x) ,
H = 0 L= ()2 1
2
+1
2m2
=1
2 ()2 + ()2 + m2()2 .
L = 12
a A
A + b A A
+ c (A)2 + d AA
.
L
A
A (A) = 0 .
L(A)
= a A + b A + c g A
,
L
(A)= aA + (b + c) (A
) ,
LA = d A .
aA + (b + c) (A) d A = 0 .
d = 0
b + c
a= 1 ,
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L = 12
a A
A a A A c A A + c (A)2
.
a
L = 14
FF +
c
2a
A
A (A)2
[A(Ag A)]
.
[A (A
gA)]
L = 14
FF .
A F
A(x) A(x) = A(x) + (x)
L
L
AA
d = 0
A A
L = 14
FF +
1
2m2 AA
(+ m2) A (A) = 0 .
m2A
12
m2 AA
A
qs
ps
[qs , ps] = i ss ,
[qs , qs ] = [p
s , p
s] = 0 .
qs = (t, s)
ps = (t, s) x(s) ,
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[(t, s) , (t, s)] = i ss
x(s),
[(t, s) , (t, s)] = [(t, s) , (t, s)] = 0 .
x(s) 0 ss
x(s)
x(s)03(x x) ,
x = x(s) x = x(s)
fs =
sx(s)
ss
x(s)fs
x(s)0f(x) =
d3x 3(x x) f(x) .
[(t, x) , (t, x)] = i 3(x x) ,
[(t, x) , (t, x)] = [(t, x) , (t, x)] = 0 .
(x)
(x)
(x) = 0 (x) k0 = k =
+k2 + m2
(x) =1V
k
12k
ak e
ikx + ak eikx
k0=k,
(x) =i
V
k
k2
ak e
ikx ak eikx
k0=k.
ak
ak ak
ak
3(x x) = 1V
k e
ik(xx)
[ak , ak] = k,k ,
[ak , ak] = [ak
, ak ] = 0 .
ak ak
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H =1
2
d3x
()2 + ()2 + m2()2
t=0
= k kak ak +1
2 ,
H =
k
k Nk , Nk ak ak .
(x) (+)(x) + ()(x) ,
(+)(x)
,
()(x)
.
((+)(x)) = ()(x) .
[(x) , (x)] =1
V
k,k
12k
2k
ak e
ikx + ak eikx
,
ak eikx + a
k eik
x
k0=kk0=k
=1
V
k,k
12k
2k
[ak , a
k ] ei(kxk
x) + [ak , ak] ei(kxk
x)
k0=kk0=k
=1
V
k
1
2k
eik(xx
) eik(xx)
k0=k
V1
(2)3
d3k
2k eik(xx)
eik(xx
)k0=k .
(x) i(2)3
d3k
2k
eikx eikx
k0=k= 1
(2)3
d3k
ksin(kx)|k0=k ,
[(x) , (x)] = i (x x) .
(x)
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T
ERe k0
Im k0
k +k'& $%E
'
C
C
(x)
C
k
1(2)4
C
d4keikx
k2 m2 = 2i
(2)4
d3k
lim
k0k
(k0 k) e
ikx
(k0 k)(k0 + k)
+ limk0k
(k0 + k)
eikx
(k0 k)(k0 + k)
=i
(2)3
d3k
2k
eikx eikx
k0=k (x) .
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b(r)p , b
(r)
p
= r,r p,p , b(r)p , b
(r)
p
=
b(r)p
, b(r)
p
= 0 ,
r
b(r)p
|0
,
b(r)p
b(r)
p
|0 . r = r p = p
b(r)p
b(r)p
|0 = 12
b(r)p
, b
(r)p
|0 = 0 ,
r,p=r,p
b(r)p
b(r)
p
|0 = b(r)p
b(r)p
|0 ,
r,p r,p
N(r)
p b(r)p
b(r)p ,
N(r)p |0 = b(r)p
b(r)p |0 = 0 , N
(r)p (b
(r)p
|0) = b(r)p
b(r)p b
(r)p
|0 = b(r)p
(1 b(r)p
b(r)p ) |0 = b(r)p
|0 ,
N(r)
p
2= b
(r)p
b(r)p b
(r)p
b(r)p = b
(r)p
(1 b(r)p
b(r)p ) b
(r)p = b
(r)p
b(r)p = N
(r)p .
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N(r)
p (N(r)
p 1) = 0
N(r)p
L(x) = (x) (i c
m c2) (x)
= (x) (i c
m c2) (x)
L
L
L
L
= (i c
mc2) (x) , L()
= 0 ,
(i c mc2) (x) = 0 ,
(i
m) (x) = 0 .
L
L
= (x) (i
m) , L()
= 0 ,
(x) (i
+m) = 0 .
L
(x) =L
= i 0 = i .
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L
= 0
H = i 0 (i 00 + i kk m) = (
i
+ m)
= (i + m) .
H =
d3x (i + m) .
(x) =1V
r=1,2
p
m
E
b(r)p u
(r)(p) eipx + d(r)p v
(r)(p) eipx
p0=E.
E +
p2 + m2 (x) b d
b
d
b(r)p p r
d(r)p p r
b
d
b(r)p , b
(r)
p
=
d(r)p , d
(r)
p
= r,r p,p ,
b(r)p , b
(r)
p
=
b(r)p , b
(r)
p
=
d(r)p , d
(r)
p
=
d(r)p , d
(r)
p
= 0 ,
b(r)p , d
(r)
p
=
b(r)p , d
(r)
p
=
b(r)p , d
(r)
p
=
b(r)p , d
(r)
p
= 0 .
(x)
(x) = 1V
r=1,2
p
mEd(r)p v(r)(p) eipx + b(r)p u(r)(p) eipx
p0=E.
H =1
V
d3x
r,r
p,p
mEE
d(r)p v
(r)(p) eipx + b(r)p u
(r)(p) eipx
p0=E
(i + m)
b(r)
pu(r
)(p) eipx + d(r
)
pv(r
)(p) eipx
p0=E.
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(i + m) = i 0 i 0 eipx
k0=E
= E eipx
k0=E
,
H = 1Vr,r
p,p
mEE
E d(r)p d(r)p v(r)(p) v(r)(p) d3xei(pp)x V p,p
+ b(r)p b
(r)
pu(r)(p) u(r
)(p)
d3xei(p
p)x V p,p
p0=Ep0=E
u(r)(p) u(r)(p) = Em rr = v(r)(p) v(r)(p) ,
H
H =p,r
E
b(r)p b
(r)p d(r)p d(r)p
=p,r
E
b(r)p b
(r)p + d
(r)p d
(r)p 1
.
b(r)p b
(r)p N
(r)p (e)
d(r)p d
(r)p N(r)p (e+)
E
p,r E
H =p,r
E
N(r)
p (e) + N
(r)p (e
+)
.
Q = e
d3x
= ep,r
b(r)p b
(r)p + d
(r)p d
(r)p
= e
p,r
b(r)p b
(r)p d(r)p d(r)p + 1
.
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e = |e|
Q = e
p,r N
(r)p (e
) N(r)p (e+)
.
P = i
d3x
=p,r
p
b(r)p b
(r)p d(r)p d(r)p
= p,r pb
(r)p b
(r)p + d
(r)p d
(r)p 1 .
p p = 0
P =p,r
p
N(r)
p (e) + N
(r)p (e
+)
.
d(r)p |0 +p
b(r)p |0
{(x) , (x)} = (x) , (x) = 0 , (x) , (x
)
= i S(x x) ,
S(x) =i
(2)3
d3p
2E
( p + m) eipx + ( p m) eipx
p0=E.
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(x) , (x
)
=
=1
V p,r p,rm
EEu
(r) (p) u
(r) (p
)b(r)p , b
(r)
p r,r p,pei(pxp
x)
+ v(r) (p) v(r) (p
)
d(r)p , d(r)
p
r,r p,p
ei(pxpx)
p0=Ep0=E
=1
V
p
m
E
r
u(r) (p) u(r) (p)
(p+m)2m
eip(xx) +
r
v(r) (p) v(r) (p)
(pm)2m
eip(xx)
p0=E
=
1
V p
1
2E( p + m) eip(xx) + ( p m) eip(xx)p0=E
V
1
(2)3
d3p
2E
(
p + m) eip(xx) + (
p m) eip(xx)
p0=E
i S(x x) .
S(x)
S(x)
S(x) = i(2)3
d4p (
p + m) eipx (p0) (p2 m2) .
i
(2)3
d4p (
p + m) eipx (p0) (p2 m2) =
=i
(2)3
d4p (
p + m) eipx (p0)1
2E[(p0 E) + (p0 + E)]
=
i
(2)3d3p
2E ( p + m) eipxp0=E d3p
2E ( p + m) eipxp0=E d3p2E
(
p+m) eipxp0=E
S(x) .
S(x x)
S(x x) = (i
x + m) (x x) .
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(i
x + m) (x x) ==
i
(2)3 d4p (i x + m) e
ip(xx) (p0) (p2 m2)
=i
(2)3
d4p (
p + m) eip(xx) (p0) (p
2 m2) S(x x) .
(x x)
S(x x) = 0
(x) , (x)
= 0
(x x)2 < 0 .
(x) (x)
[L] =
E
3
[m4] .
1/2
[E] [ ] =
E3
[] = 13/2
[m3/2] .
u
v
b
d
L = 12
1
c2
t
2
2 1
2
m c
22 ,
[2] [2] = E3
[] =
E1/2
1/2
[m] .
ak
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+ i e A .
LD
LD = (i m) +i e A L = (i e A m) LD + LI ,
LI = e A = e j A
LI
HI =
LI
+ LI
+LI
A
A
LI .
LI HI = LI = e A = e j A .
Lem = 14
F F
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HI = e
(+)
+ ()
(+) + ()
A
= e ()
(+) A rrrre
e
+ e (+)
() A rrrr
e+ e+
+ e (+)
(+) A
rrrre
e+
+ e ()
() A
rrrre+
e
HI
e
e+
e
e+
(+)
u(r)(p)
sp
()
v(r)(p)
sp
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()
u(r)(p)
s
p
(+)
v(r)(p)
sp
() ()
()
()
(+) (x) , () (x) = 1V p
12E
(
p + m) eip(xx)
p0=E
V
1
(2)3
d3p
2E(
p + m) eip(xx)
p0=E
i S(+) (x x) ,
() (x) ,
(+)
(x)
=1
V
p
1
2E(
p m) eip(xx)p0=E
V1
(2)3 d3p
2E
(
p
m) e
ip(xx)p0=E i S() (x x) .
(x) , (x
)
=
(+) (x) , ()
(x)
+
() (x) , (+)
(x)
= i
S(+) (x x) + S() (x x)
= i S(x x) .
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S
|n1, n2, . . . , ni, . . . (n1, n2, . . . , ni, . . . ) ,
n1, n2, . . . , ni, . . . |n1, n2, . . . , ni, . . . = ((n1, n2, . . . , ni, . . . ) , (n1, n2, . . . , ni, . . . ))
=
i
ni,ni .
t =
t = +
H0
t =
t = +
HI
&&
&&
&&
&&
&b
~
e(pi, r) e(pf, s)
(ki, ) (kf, )
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HI = e j
A .
S(t)
i
tS(t) = HS S(t) = (HS0 + H
SI )
S(t) .
(t)
(t)
S(t)
S
(t) = eiHS0 t S(t) ,
(t) = eiHS0 t S eiH
S0 t .
(t)
i
t(t) = HS0 eiH
S0 t S(t) + eiH
S0 t i
tS(t)
= HS0 eiHS0 t S(t) + eiH
S0 t (HS0 + H
SI )
S(t)
= eiHS0 t HSI e
iHS0 t eiHS0 t S(t) HI(t) (t) ,
(t)
d(t)
dt= i HS0 e
iHS0 t S eiHS0 t i eiHS0 t S eiHS0 t HS0
= i [HS0 , (t)] = i [H0 , (t)] ,
HI(t) = eiHS0 t HSI e
iHS0 t ,
H0 = eiHS0 t HS0 e
iHS0 t = HS0 .
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S
U
(t) = U(t, t0) (t0) , U(t0, t0) = 1 ,
i
tU(t, t0)
(t0) = HI(t) U(t, t0) (t0) .
(t0)
U(t, t0) = 1 it
t0
dt HI(t) U(t, t0) .
U(t, t0) = 1 it
t0
dt1 HI(t1)
1 i
t1t0
dt2 HI(t2) U(t2, t0)
= 1 i
tt0
dt1 HI(t1) + (i)2t
t0
dt1
t1t0
dt2 HI(t1) HI(t2) + . . .
. . . + (i)n tt0
dt1 t1t0
dt2 . . .tn1t0
dtn HI(t1) HI(t2) . . . H I(tn) + . . . .
i t0 (t) t
(t) = U(t, t0) i .
t
f
(f , (t)) = (f , U(t, t0) i) Ufi(t, t0) .
i
f
Pf i =|Ufi(t, t0) fi|2
t t0 .
()
(+)
t =
t = +
()
(+)
U
(+) = U(+, ) () S() ,
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S U(+, )
S= 1 i+
dt1 HI(t1) + (i)2+
dt1
t1
dt2 HI(t1) HI(t2) + . . .
1 +
S(1) +
S(2) + . . .
=
n=0
(i)n+
dt1t1
dt2 . . .tn1
dtn HI(t1) HI(t2) . . . H I(tn) .
S
S(2)
S(2) = (i)2+
dt1
t1
dt2 HI(t1) HI(t2) ,
S(2)
S(2) = (i)2 +
dt1 +t1
dt2 HI(t2) HI(t1) .
HI
H0
t1 t2
S(2) = (i)2+
dt2
t2
dt1 HI(t2) HI(t1) .
t1 t2 t2 > t1
S(2)
S(2) = (i)2+
dt1
+t1
dt2 HI(t2) HI(t1) ,
S(2)
S(2) = (i)2
2 +
dt1
t1
dt2 (HI(t1) HI(t2)) +
+
t1
dt2 (HI(t2) HI(t1))
=
(i)22
+
dt1+
dt2 P[HI(t1) HI(t2)]
=(i)2
2
d4x1
d4x2 P[HI(x1) HI(x2)] ,
P[(x) (x)] =
(x) (x)
x0 > x0 ,
(x) (x)
x0 > x0 .
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T
Et1
t2
T
Et1
t2
S(2)
P
S(n)
S(n) = (i)n
n!
+
dt1+
dt2 . . .+
dtn P[HI(t1) HI(t2) . . . H I(tn)]
=(i)n
n!
d4x1
d4x2 . . .
d4xn P[HI(x1) HI(x2) . . . HI(xn)] .
S
S=
n=0
(i)nn!
d4x1
d4x2 . . .
d4xn P[HI(x1) HI(x2) . . . HI(xn)] .
S
S S = S S=
,
n
Sfn Sin =
n
Snf Sni = fi ,
N
(+)()
= ()(+) , N
()(+)
= ()(+) ,
N
(+)(+)
= (+)(+) , N
()()
= ()() ,
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N[] = N
(+) + ()
(+) + ()
= (+)(+) + 2()(+) + ()() .
N
(+) ()
= () (+) , N
(+)
()
= () (+) ,
N
(+)
()
= ()
(+)
, N
(+) ()
= () (+) ,
() ()
N = N(+)
+ ()
(+) +
()
= (+)
(+) +
()
(+) ()
(+)
+ ()
() .
N
() (+)
(+)
+
TT
()
=
()
()
(+)
(+)
= ()
()
(+)
(+) .
N () (+) (+) JJ
RR
() = () () (+) (+) .
N[A1A2 . . . An] = (1)PB1B2 . . . Bn ,
P
B1B2 . . . Bn
A1A2 . . . An
: AB : N[AB] .
0|N[A1A2 . . . An]|0 = 0 ,
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e2/4
1 137
T[A(x1)B(x2)] = (x01 x02)A(x1)B(x2) (x02 x01)B(x2)A(x1)
=