(eBook) - Fisica Teorica

8/14/2019 (eBook) - Fisica Teorica

1/214


2/214


3/214

a

xk

xk

S

vc


4/214

L2

(

p) = ( + 1) (

p)


5/214

()

()

S

S

HI

A A

e e


6/214

m = 0

C

P

T

W

W


7/214


8/214


9/214

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 ,


10/214

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) .


11/214

j = 0 ,

j =i

2 m[ () () ] .

j0

j0 =i

2 m

t

t

(x) = 1, 2, . . . , N


12/214

=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 .


13/214

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 =


14/214

[] = 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


15/214

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 .


16/214


17/214

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

.


18/214

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 .


19/214

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] = + .


20/214

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


21/214

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


22/214

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

.


23/214

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


24/214

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

.


25/214

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


26/214

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 .


27/214

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 .


28/214

a = 5

(x) 5 (x) = (x) S1 5 S (x) .

(x) 5 (x) = (x) 5 (x) ,

(x) 5 (x) = (x) 5 (x) .

5


29/214

(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


30/214

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 .


31/214


32/214

(

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)

.


33/214

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 ,


34/214

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


35/214


36/214

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 .


37/214

(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 .


38/214

+(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) .


39/214

+(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 .


40/214

(p)

(p) =

p + m

2 m.

(p)

((p)) = 0 (p)

0 .

+(p) + (p) = ,

+(p) (p) = (p) +(p) = 0 .


41/214


42/214

(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


43/214

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 ,


44/214

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


45/214

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)


46/214

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 .


47/214

(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) .


48/214


49/214

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

+ . . .

.


50/214


51/214

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

.


52/214

(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) .


53/214

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.


54/214

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 .


55/214


56/214

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 .


57/214

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 ,


58/214

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)

,


59/214

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 ,


60/214

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 .


61/214

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 .


62/214


63/214

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

,


64/214

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


65/214

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


66/214


67/214

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


68/214

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


69/214

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|


70/214

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 .


71/214

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 ,


72/214


73/214

=

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) .


74/214


75/214

(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)


76/214

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 .


77/214

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


78/214

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) .


79/214

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) .


80/214


81/214

c

[

] = [E t] ,

[c] =

t1 .

= c = 1 .

[E] = t1

= 1

.

E2 = p2 + m2

[E] = [|p|] = [m] .

c = 197

,

197

= 1 .


82/214


83/214

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 .


84/214

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 ,


85/214

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 .


86/214

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) .


87/214

{ , } = 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

,


88/214

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 .


89/214

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

.


90/214

0C =

0

0

, kC =

0 k

k 0

,

5C =

00

, kC =

k 00 k

,

CC = i

2 00 2

, kC =

k 00 k

.


91/214

(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 .


92/214

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) .


93/214

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 ,


94/214

p L = 0

L

(1 )

L2

,

p = 2 2 ( p) .

( + 1) + 2 = ( + 1)

L2

( p) = ( + 1) ( p) .


95/214


96/214


97/214

(x)

F A


98/214

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 .


99/214

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


100/214

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) .


101/214


102/214

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 , . . . .


103/214

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 ,


105/214

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


106/214


107/214

(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)


108/214

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

()

.


109/214

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 ,


110/214

(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 ,


111/214

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) ,


112/214

[(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


113/214

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)


114/214


115/214

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) .


116/214


118/214

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 .


119/214

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.


120/214

(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

.


121/214

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.


122/214

(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) .


123/214

(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


124/214


125/214

+ 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


126/214


127/214

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


128/214

()

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) .


129/214

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, )


130/214

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 .


131/214

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() ,


132/214

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 .


133/214

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

()()

= ()() ,


134/214

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 ,


135/214


136/214


137/214

e2/4

1 137

T[A(x1)B(x2)] = (x01 x02)A(x1)B(x2) (x02 x01)B(x2)A(x1)

=

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