Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.affp.mics.msu.su/spec/formulas.pdf Äàòà èçìåíåíèÿ: Wed Sep 11 00:56:35 2013 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 22:49:27 2016 Êîäèðîâêà:

A

A.1
, , . . . µ, , . . . 0, 1, 2, 3. 4- : aµ = {a0 , a}, : g
µ

aµ gµ a = {a0 , -a}.

(A.1.1)

= diag(1, -1, -1, -1).


(A.1.2)

eµ e
0123

: (A.1.3)

= 1.

: e e e e
µ µ µ

e

µ

= -24, = -6 , (A.1.4)

µ

e

µ

eµ

= -2[ - ],

eµ = - [ - ]+

+ [ - ] - [ - ]. i j 12 = 1. (A.1.5)




2

A.2
eµ : i eµ = {1, 0} eµ = {0, ei }. i 0 Liµ :
µ L0 = uµ ,

(A.2.1)

Liµ = (uei ) , ei +

(uei )u 1 + u0

(A.2.2)

-- eµ . i : u u L= u u
2 0 1 2 3

u

u

u

1

u1 u1 1+ 1+u u2 u1 1 + u0 u3 u1 1 + u0
-1

0

u1 u2 1 + u0 u2 u2 1+ 1+u u3 u2 1 + u0
0

u1 u3 1 + u0 u2 u3 1 + u0 1+ u3 u3 1+u
0

(A.2.3)

3

L

u -u. (a1,2 u) -n , (nu)
µ

µ : i µ 0 =u ,
µ



µ 1,2

=a

µ 1,2

nµ = -u (nu)
µ 3

µ

(A.2.4)

-- eµ . i : u
0

u1 u- 1 0 u1 u-

u2 1 -u u- u- 0 1 -u -u
1

0

u1 = u u
2

(A.2.5)
2

3

u2 1 -u u- u-

3




3

Niµ :
µ N0 = uµ , µ N1 = aµ (a1 u) - aµ (a2 u) u-1 , 2 1 -1/2

µ N2 = uµ u - aµ 1

(a1 u) (a2 u) (1 + u2 ) - aµ 2 u u µ N3 = nµ (n+ u) - nµ (nu) (1 + u2 )-1/2 +

,

(A.2.6)

-- eµ . i : u
0

0 u2 u u1 u

u0 u

2 2

u

3 2

1+u u N= u u
2 1

1+u 0

1+u 1+u u3 u
2

u1 u2 - u u 0

(A.2.7)
2

0 u
0 2

3

1+u

1+u

, , : nµ n jµ = gi j , i e
µ

gi j nµ n = gµ , ij e
i jkl

niµ n j nk nl = ei jkl ,


niµ n j nk nl = e e
i jkl µ ni n j nk nl

µ

,

(A.2.8)

1 = Det nµ n n n = - e 0123 4!

µ

.

eµ n j nk nl = e ei jkl n j nk nl = e eµ nk nl = e eµ nl = e ei jkl nl = e

i jkl

gim nµ , m gim nµ , m (A.2.9)

µ

i jkl

gim g jn nµ n , mn

ei jkl nk nl = eµ gim g jn nµ n , mn
i jkl

gim g jn gk p nµ n n , mnp gim g jn gk p nµ n n . mnp

µ




4

: nµ = (1, n) , aµ 1 = (0, a1 ) , 1 (1, - n) , 2 aµ = (0, a2 ) . 2 nµ = + (nn+ ) = 1, -a
µ 2 a2 µ

(A.2.10)

n2 = n2 = (nai ) = (n+ ai ) = 0, + nµ n + +n
µ +n

(ai a j ) = -i j ,

-a

µ 1 a1

=g .

(A.2.11)

(A.2.10) pµ pµ = nµ p+ + nµ p- - aµ pi , + i p- = (n p),
0

(A.2.12)

p+ = (n+ p),

pi = (ai p).

(A.2.13)

= Det nµ n a a = -e + 12

µ µ n+ n a1 a2

,

(A.2.14)

= nµ n - nµ n + + + nµ n - nµ n + + a a - a 21
0 e 1 a2 µ

=
µ



+ n nµ - n n + +

a a - a a + 21 12 aµ a - aµ a + 21 12 aµ a - aµ a . 21 12



a a - a a + n n - n n + + 21 12



(A.2.15)

+ n n - n n + + 1 2 1 - 2 1 - 2 1 - 2 -

aµ a - aµ a + n n - n n + + 21 12 n a - a n 1 1 n a - a n 2 2 n
n+

0 eµ 0 eµ 0 eµ 0 eµ

= - nµ a2 - a2µ n , = nµ a1 - a1µ n , = a2µ a1 - a1µ a2 , = - nµ n
+

-n

+n 2

(A.2.16)

a a - a a 21 1

- n+µ n .




5

A.3
: 0 Aµ = (p, a) = -p
1 2 3

p a

1

p

2 3

p3 a2 -a 0 -p a
2 1 1

0
3 2

-a 0 a

-p -p

-a

1

0 - p1 - p Aµ = (-p, a) = p p p
1 2 3

2

3

0 a
3 2

-a 0 a p a
3

3

-a 0

-a p

1

(A.3.1) 0 Aµ = p p
1 2 3 1 2

p a

3 2

0 -a a

3

-a 0

0 -a
1

1

p

2

0 Aµ = -p -p -p
1 2 3

-p 0 -a a

1

-p a
3

2

-p -a a 0

3

2

3

0 -a
1

1

2

: 1 µ e A , 2 : Aµ = -
def

Aµ = -Aµ .

(A.3.2)

F

µ

= µ A - Aµ = (E, H).

(A.3.3)




6

: 0 Aµ = (a, -p) = -a -a
1 2 3

a

1

a

2 3

a

3

0 -p p
3

p

- p2 p
1

0 -p
1

-a

2

0

0 -a1 -a2 -a Aµ = (-a, -p) = a
1 2 3

3 2

0 - p3 p
2

p

3

-p p 0

a a

0 -p a
1

1

(A.3.4) 0 Aµ = a
1

a p

1

2

a

3 2 1

0
3 2

- p3 0 p
1

p

a2

-p 0

a3 - p 0 Aµ = -a -a -a
1 2 3

-a 0 p

1

-a 0 p

2 3

-a p

3

-p

2 1

3 2

-p 0

-p

1

: 1 1 I1 = Aµ Aµ = - Aµ A 4 4
µ

1 = (a2 - p2 ), 2

1 I2 = Aµ A µ = -pa. 4 : AA
µ

(A.3.5)

=

p2

-[p â a]k

-[p â a]i ik a2 - ( pi pk + ai ak )

(A.3.6)




7

A

µ

A =



a

2

-[p â a]

k

-[p â a]i ik p2 - ( pi pk + ai ak ) Aµ A - Aµ A = µ (a2 - p2 ).


(A.3.7)

Aµ A = µ (pa), Aµ A A Aµ A


(A.3.8) (A.3.9) (A.3.10) (A.3.11)

= -Aµ (a2 - p2 ) + Aµ (pa).

A = Aµ (a2 - p2 ) + Aµ (pa).

Aµ Aµ = 2(aa - pp ). Aµ = (aµ b - bµ a ) + (cµ d - dµ c ). 4- bµ , dµ , (bd) Aµ (bd) = - bµ A d - Aµ d b + , bµ Aµ A b d2 + bµ Aµ d =b Aµ = aµ b - bµ a , .
µ µ A 2

(A.3.12)

0: dµ A b - Aµ b d . (A.3.13)

- dµ Aµ A d b2 - bµ Aµ d
µ µ A A d 2



2

=

A d

(A.3.14)

(bd) - b

(bd) = 2(bd) I1 .

(A.3.15)

, I2 = 0. (A.3.16)

:
2 4 + 2I1 2 - I2 = 0.

(A.3.17)




8

: 1,2 = ± p,
2 2 p = (I1 + I2 ) 1/2



3,4

= ±ia,

(A.3.18)

-I

1/2 1

,

2 2 a = (I1 + I2 )1/2 + I1

1/2

.

(A.3.19)

pa = |pa|, a2 - p2 = a2 - p2 . (A.3.20)

, , , . : , . Xiµ = {Xi0 , Xi }: pXi = -i Xi0 , : Xi = - X X
1,2

pXi0 + [a â Xi ] = -i Xi .

(A.3.21)

X

0 i 2

2 + a i

i p +

(pa)a + [p â a] . i

(A.3.22)

- 0 = -X+ 1 [p â a] ± ( p p - a a signI2 ) X1,2 ,

3,4

= -X

( X i X i ) = 0,

0 [p â a] ± i(a p + p a signI2 ) X3,4 , 1 X± = [p2 + a2 ± ( p2 + a2 )]. 2 (X1,2 X3,4 ) = 0, (X1 X2 ) = 1, (X3 X4 ) = -1.

-1 -

(A.3.23) (A.3.24)

A.4
: µ + µ = 2gµ . : 0 -- , -- .
µ

(A.4.1)

= 0 µ 0 = µ ,

µ = gµ ,

5 = -i0 1 2 3 ,



5

= 5 .

(A.4.2)




9

aµ : a = µ aµ . ^ 1 = (µ - µ ) = (, i). 2 0 = , = 5 0 , = -5 .
µ

(A.4.3) (A.4.4) (A.4.5)

: (1) = E
µ (2) = µ (3) = µ µ µ (1) = E µ (2) = - µ (3) = (5) = µ µ µ

(1) = 4(1)
µ (2) = 0 µ (3) = -2 µ (3)

(A.4.6)

µ (4) = 5

µ (4) = -5 5

µ µ (4) = 2(4)

(5) = Sp
(1)

5

(5) = -4 i 1;

(5)

= 4;

Sp(ai) = 0,

(ai) (ai = E . )

(A.4.7)

: A=
i ,a

A(i)a (ai) ,

A(

i) a

1 = Sp A(ai . ) 4

(A.4.8)

µ = -2µ , µ = 4gµ , µ = -2 µ , µ = 2 µ + µ . 1 Spµ = gµ , 4 1 Sp5 µ = ieµ , 4 1 µ Sp = gµ g + gµ g - gµ g . 4 i eµ = - 5 [µ - µ ], 2 µ e = 2i5 µ , e e
µ

(A.4.9)

(A.4.10)

= 6i ,

5µ

(A.4.11)

µ

µ = -24i5 .




10

µ = gµ + µ , µ = gµ + g µ - gµ + i5 e
µ

, .

µ = gµ g + gµ g - gµ g + gµ + g µ + +g µ - gµ + gµ - g µ + i5 e µ = gµ g - gµ g + +g µ - gµ + gµ - g µ + i5 e
µ µ

(A.4.12)

.

(A.4.13) (A.4.14) (A.4.15) (A.4.16)

[µ , ] = 2{g µ - gµ + gµ - g µ }. [µ , ]+ = 2{gµ g - gµ g + i5 e µ = g µ - gµ + i5 e
µ µ

}.

, .

µ = -g µ + gµ + i5 e

µ

[µ , ] = 2{g µ - gµ }, [µ , ]+ = 2i5 e
µ

.

(A.4.17)

Aµ , Bµ -- . Aµ Aµ µ B
µ

= i5 Aµ µ ,
µ

= -2Aµ Bµ - 4Aµ Bµ + i5 e

Aµ B

(A.4.18)

-2Aµ Bµ - 4Aµ Bµ - 2i5 Aµ Bµ . , Aµ µ A
µ

= -2Aµ Aµ - 2i5 Aµ Aµ -8 I1 + i5 I2 , = -2 Aµ A + 2i Aµ A
5 µ µ

Aµ A

- 8 I 2 - i I 1 .
5

(A.4.19)

(A.3.13) , ^^ Aµ µ (bd) = A d b - bA d + +i
5

(A.4.20)

^^ A b d - d A b .



X, Y , ¯¯ , X , Y ,




11

- , ¯ ¯ Sp{ X }Sp{ Y } = Sp{(X + X )(Y + Y )}- -Sp{X }Sp{Y } - Sp{ X }Sp{ Y }.
5 5

(A.4.21)

A.5

2

J0 =

1 dO = (lu)2
0

d
0

4 sin d =µ . (u0 - |u| cos )2 u uµ

(A.5.1)

, J0 ­ , uµ uµ = 1 J0 = 4. (A.5.2)

(A.5.1) 4-, l dO = 4uµ , 3 (lu) µ 4 ll 1 dO = 4 u uµ - gµ , 3 3 (lu)4 µ lll 1 dO = 4 2u uµ u - gµ u + g uµ + gµ u 3 (lu)5 . . l0 = 1, , = 0, 1 = 4u0 , (lu)3 1 lµ 4 J2 = d O = 4 u0 uµ - g0µ , 4 3 3 (lu) lµ l 1 dO = 4 2u0 uµ u - g0µ u + g0 uµ + gµ u 5 3 (lu) J1 = dO

(A.5.3)


(A.5.4)
0

J3 = . .