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12

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Sl(k0)

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U(1)

GR/Ap (z) GR/Ap ()

G

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c 3 108 m

s

t (x,y,z)

(x0

, x1

, x2

, x3

)

x0 ct, x1 x, x2 y, x3 z.

,

x = (x0, x) = (x0, x1, x2, x3).

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g =

1 0 0 00 1 0 00 0 1 00 0 0 1

.

x x

x =3

=0

gx x =3

=0

gx

x

x

x = gx

, = 0, 1, 2, 3

a, b = 1, 2, 3

a

b

ab = ab = a0b0 a b,

a b

a

aa =

a0

2 |a|2

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aa =

< 0 a

= 0 a

> 0 a

s2 = aa = aga = (ct)2 |a|2

=s

c=

(ct)2 |x|2

x = const.

0

S, S

= s

aa = 0

g

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x

ct

world line of (accelerated)

light cone

(world line of a photon)

x=ct

world line of a free, massive

particle: v=x/t1 always

L

x

S v vxL

s2 = x0

x1 x2 x3

=1

1 vc

2 , = vcL

(L) =

0 0

0 0

0 0 1 0

0 0 0 1

(L) = 1.

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(L) = +1 L00 1

(L) = 1 L00 1

(L) = +1 L00 1

(L) = 1 L00 1

i

t(x, t) =

2

2m + V(x)

(x, t)

x0 xa, a = 1, 2, 3

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(x)

E =

p2c2 + m2c4.

i

t =

2c2 2 + m2c4,

p = i

.

2 2

t2 =

2c2 2 + m2c4

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=

x

=

x ,

x = (ct, x1, x2, x3) x = (ct, x1, x2, x3) = (ct, x1, x2, x3)

i (ct)

= i x0

= i0

i xa

= ia, a = 1, 2, 3

p i = i

(ct)

2 2

(ct)2(x, t) = (2 2 + m2c2)(x, t)

+

mc

2(x) = 0

2

:=

=

2

0

2

mc

(x, t) = exp

i

(Et px)

= exp

i

px

,

E =

p2c2 + (mc2)2.

p

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k

= p

(x) =

d4k

24

4

kk

mc

2A(k)eikx

.

A(k) k0 < 0

E = +

p2c2 + (mc2)2 > 0

(+)(x, t) = ei

(Etpx)

()(x, t) = ei

(Etpx) = ei

(E(t)px)

E < 0

/x

O(p2) p

/(ct)

=

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=

2mi .

t + = 0.

+

mc

2 = 0.

+mc

2 = 0.

( ) = 0,

t i2mc2 t t

+

2mi

= 0.

(x, t)

(, x) =

mc2

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12

12

E = p2c2 + (mc2)2

E2 = c2p2 + (mc2)2!

= (c p + mc2)2,

= (x, y, z), ,

c2(p2x + p2y + p

2z) + m

2c4 = c2(2xp2x +

2yp

2y +

2zp

2z) +

2m2c4

+c2pxpy(xy + yx) + c2pypz(yz + zy)

+c2pzpx(zx + xz) + mc3[px(x+ x)

+py(y+ y) + pz(z+ z)]

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12

x, y, z,

2i = 2 = 1

ij + ji =: [i, j]+ = 0

i+ i =: [i, ]+ = 0

i = x,y,z

i, j = x,y,z

i = x,y,z

,

[M, M]+ = 21

M = , xyz.

x, y, z,

H = c p + mc2

M = 1

=

(M)2 = 1

M = 1

2

M

(M) = 0, = 0, 1, 2, 3

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=

MM = MM MM

1

M = MMM

(M) = (MMM)= (M MM

=1

)

= (M)

= 0

2

x, y, z,

0 = (M) =d

i=1i =

d

i=1(1)

2

x, y, z, d 4 d = 2

d = 4

=

0

0

, =

1 0

0 1

= (x, y, z)

x =

0 1

1 0

, y =

0 ii 0

, z =

1 0

0 1

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12

,

i

t(x) = (c p+mc2)(x)

p = i.

x, y, z, (x)

(x)

(x)

22 = 4 = d

d = 2 (2S+ 1)

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i

t = (ca p + mc2)

1c

(i0 + iii + mc) = 0

0 =

i = i, i = 1, 2, 3.

,

i + mc

= 0

mc

=

x

=

= L

u = 0u0 u =: /u

i/+ mc

= 0

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12

0 = (0)2 = 1

i, i = 1, 2, 3 (i)2 = 1

(i) = (i) = i = i = i(i)2 = ii = ii = 1

2

[, ]+ = 2g1

a,

0 = 1 0

0 1

, i =

0 ii 0

, i = 1, 2, 3

= AA1,

=

14

, = (1 , . . . , 4)

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= =4

i=1

i i,

t() =

t

+

t

.

i

t

= (ic + mc2)

i

t

= (ic + mc2).

i

t

= i

c + mc2

it = i(ca) () + ( ) (c)

= (ca)

t + = 0 j = 0,

(j

) = c , () =

(ct)

x .

=

= (c) = v

v := c

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12

j

j = Lj.

E2 = p2c2 + (mc2)2

E/c =

i/(ct) p = i E p

pp =

E

c

2 p2 = (mc)2,

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x = Lx x = L x

L

(x) = S(L)(x)

= S(L)(L1(x))

4 4

i + mc

(x) = 0 I

i +

mc

(x) = 0 I

=

x =

x.

= AA1 .

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12

= T(L)T1(L).

= A = T1(L)

2

= x

= x

x

x= L

x = Lx

x

x= L

S1(x) = (x).

iL +

mc

S1(x) = 0.

S

iSLS1(x) +mc

(x) = 0.

S(L)

SLS1 =

S1(L)S(L) = L

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S(L)

L =

+

,

L00 = 1

Lab = cos()

L = cosh()

L = 1 + O(2)

L sin()sinh() = O(), =

= O + O(2)

()

S(L)

S = 1 + S1 = 1

(1 )(1 + ) = + + O(2)= +

[, ] =

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12

1

1, 2

[, 1 2] = 0.

1 2 = 1 R

2

S(L)

(x)(x) =4

=1

(x)(x)

=

(x)(x)x

(S) = 1,

1 = (S) = (1 + ) = (1) + ()

= 1 + () + O(2)

() = 0

=1

8g

( ) = 18

g [, ]

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L =

+

S(L) = 1 +1

8g

[, ]

()

+

N

, N

L =

lim

N

1 +

NN

= (e )

= v/c

=

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

=: (01)x 101 :=

1 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

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12

(L) = 1 +

k=1

1

k! (x1)k

= 1 101 +

k=0

1

(2k)!2k101

+

k=0

1

(2k + 1)!2ki+1(01)x

(L) = 1 101 + cosh()101 + sinh()(01)x

= cosh() sinh() 0 0

sinh() cosh() 0 00 0 1 0

0 0 0 1

v/c

tanh() =v

c= ,

cosh() = =1

1 vc2 ,

sinh() = .

S(L) = limN

[1 +

N

1

8(g

[, ]

())]N

= exp 18

(g [, ])

44

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v/c x

g [, ] =

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 00 0 1 00 0 0 1

[, ]= 4

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 00 0 1 00 0 0 1

=

0 1 0 01 0 0 0

0 0 0 0

0 0 0 0

[, ] = [, ]+ 2= 2g

1

2

[0, 1] = 210

= 2

0 11 0

1 0

0 1

= 2

0 1

1 0

= 21

S(L)

S(L) = exp

21

=

k=0

1

(2k)!

2

2k+

k=0

1

(2k + 1)!

2

2k+11

S(L) = cosh(/2)1+ sinh(/2)1

S(L) =

cosh(/2) 0 0 sinh(/2)0 cosh(/2) sinh(/2) 0sinh(/2) 0 0 cosh(/2)

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12

x tanh() = v/c

j = c 0

= c

1

S

S1

S0

= b0

S1

b =

+1, L00 1

1, L00 1

:= 0,

j

= S

= S0 = b0S1 = bS1

j = c

j = cb S1S

=L

= cbL = bLj

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= 0

= bS1S = b

i

t = [c p + mc2

]

(x) (x) = ei(x)

(x)

(ct)

(ct)+ i

(ct)=: Dt

x

x+ i

x=: Dx

p = i

x p

x=

x = ctx .

(/x)

=p q

cA

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12

q

cA =

x

q

= g = p q

cA

c0 = i

t q

= i

x q

cA

i

t = [c

p q

cA

+mc2 + q]

v c = 0

E = c + mc2.(t) = e

i

Et

(x) =

(x)

(x)

,

, E mc2 c c E+ mc2

= 0

(E mc2) Es

c = 0

(E+ mc2)

Es+2mc2

c = 0

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Es mc2

E

v c qc A |p|

Es p2

2m mc2

|| m|v| mc

=

c

E+ mc2

E+ mc2 = Es + 2mc2

2mc2

c || mc|v|ii = 12 vc 1, i = 1, 2

= 12mc

( )

Es =1

2m( P i)( P i)

Es

i

t

i

t =

1

2m( )( )

( u)( v) = u v + i [u v]

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12

u v

[ ] = p qc A p qc A= [p p]

=0

qc

[ A p] qc

[p A] + q2

c2[ A A]

=0

= qc

[ A p] qc

[p A] + qc

[ A p]

=iq

c[ A]

=iq

cB

i

t =

12m

(p qc

A)2 qmc

=2B

12

B

12

qmc

= 2B = gB1

2 g = 2

B =q

mc

= c = 1

(

i + m) = 0

m

, c

m mc

=1

mc2 = E0

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p = 0

(i00 + m) = 0

1 0 0 0

0 1 0 0

0 0 1 00 0 0 1

(i0) +

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

m

= 0

(+)(x) = ur(E = m, p = 0)eimt r = 1, 2

()(x) = vr(E = m, p = 0)e+imt m =

u1(m,0) =

1

0

0

0

, u2(m,0) =

0

1

0

0

v1(m,0) =

0

0

1

0

, v2(m,0) =

0

0

0

1

(p) =

m0

E > 0(u1, u2) E < 0(v1, v2)

p

vxp px = p

p v

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12

Ep =

m

0 = m

m

=1

1 2 =

v

c= v

px = m

m

L x

=

cosh sinh() 0 0

sinh() cosh() 0 0

0 0 1 0

0 0 0 1

.

S(Lx

) = cosh(/2)1 + sinh(/2)1,

u1(E, p) =

cosh(/2)

0

0

sinh(/2)

, u2(E, p) =

0

cosh(/2)

sinh(/2)

0

v1(E, p) = 0

sinh(/2)cosh(/2)

0

, v2(E, p) = sinh(/2)

00

cosh(/2)

ui, v

i, i = 1, 2 E

p

cosh() = =E

msinh() = =

pxm

(E > 0)

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1/2

cosh(/2) = 12

(cosh() + 1) = 12m

(E+ m)

sinh(/2) = ()

1

2(cosh() 1) = (px)

1

2m(Em)

= (px)

1

2m

E2 m2E+ m

= px

1

2m(E+ m)

u1(E, px) =

1

2m(E+ m)1

px

1

2m(E+m)2

u2(E, px) =

12m (E+ m)2px

12m(E+m)

1

v1(E, px) =

px 12m(E+m) 21

2m(E+ m)1

v2(E, px) =

px

12m(E+m)

1

1

2m(E+ m)2

1 =

1

0

2 =

0

1

.

v c

u1, u2 1,u1, u2

v

2c 1,

v1, v2 v2c

1,v1, v2 1.

p

px2 p1px1 p2.

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12

px

eimt

= eip0x0 = eipx

= ei(Etpx)

e+imt

= e+ip0

x0 = e+ip

x

= ei(Et+px)

p

(+)p,r (x) = ur(E, p)e

i(Etpx)

()p,r (x) = vr(E, p)ei(Et+px)

E = +

p2 + m2

H2D = it HD

ur, vr

ur(k)us(k) = rs r, s = 1, 2

vr(k)vs(k) = rsur(k)vs(k) = 0

vr(k)us(k) = 0

L =

+

S(L) = 1 +1

8[, ]

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, || =ij = ijk k (0 = 0 = 0)

=i

2[, ]

ij = ij = ijk k

k = k 0

0

k k

S(L) = 1 i4

(x) = L {(x)} = S(x) = S L1x

1

E < 0

E > 0

E < 0

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12

(x) =

d4p

(2)4(p20 E2)

s=1,2

(2)2mb(p, s)Ws(p) eipx

d4p(2)4

(p20 E2) = d3pE

2m

b(p, s) E =

|p|2 + m2

Ws(p) =

us(p), p0 > 0

vs(p), p0 < 0.

(p20 E2) = 12p0 [(p0 E) + (p0 + E)]

(x) = d3p(2)3

mE

s=1,2

b(p, s)us(p)eipx + d(p, s)vs(p)e+ipx

b(p, s) = 2b(E, p, s)d(p, s) = 2

b(E, p, s).

E > 0

E > 0

d 0 p0 = E

(+)(x) =

d3p

(2)32

m

E

s=1,2

b(p, s)us(p)eipx

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J(+) =

d3x

(2)3 (+)(x)

= c

d3x

(2)32

(+)(x)(+)(x)

= c

s

d3p

(2)3m

E

b(p, s)2 pE

= p

E = vG

vG =Ep

=|p|2+m2

p= p

E.

u v

E > 0 E < 0

E > 0 E < 0

t = 0

E > 0 d

4d

(t = 0, x) =1

(2d2)34

eixk x2

(2d)2 w

w E > 0 w =

1

0

=

1

0

0

0

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12

4d

k

eixk x2

(2d2) =

d3p

(2)3

4d2

2

32

ed2(pk)2

p k

b(p, s) = 232 d3ed

2(pk)2us(vp)w = 0d(p, s) = 2

32 d3ed

2(pk)2vs(p)w = 0

(t, x) E > 0 E < 0

E < 0 E > 0

C

d C = mc

,c=1=

1

m.

w =

10

us vs

d(p)

b(p)=

|p k|E+ m

|k| 1

.

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d 1m

|p k| d1 m d

b 1.

d 1m

|p k| d1 m = |p k| E d

b 1

x

E > 0

x =

d3x (x)x(x)

d

dtx = d

dt

d3x (0, x)e+iHtxeiHt (0, x)

=

d3x (t, x)i [H, x]

ic

(t, x)

=

d3x (x)c(x) = J(t)

Ji(t) = d3p

(2)3

m

Epi

Es |b(p, s)|2 + |d(p, s)|2+ i

s,s

b(p, s)d(p, s)e2iEtus(p)i0vs(p)

b(p, s)d(p, s)e2iEtvs(p)i0us(p)

2E > 2mc2

= 2 1021s1

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12

I II

x1=x

V(x1)=q (x1)

V0

V(x1) = const

E

E+m

2m

E > 0

in(x) = eiEteipx

1

0

0p

E+m

refl(x) = eiEtaeipx

1

0

0pE+m

u1

+beipx

0

1

p

E+m

0

u2

E = +

p2 + m2 > m, p = +

E2 m2.

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refl

trans(x) = eiEt

ceiqx

1

0

0q

EV0+m

+ deiqx

0

1q

EV0+m0

q =

+

(E V0)2 m2, |E V0| mi

m2 (E V0)2, |E V0| m.

E V0 + mq = 0

E

q

Re(q)Re(q)

Im(q)

m+V0mV0 V0

E m + V0 > 0 E m + V0 ( E m : V0 2m)

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12

m + V

0< E < m + V

0

x

I(0)!

= II(0)

1 + a = c

b = d

b pE+m

= d qEV0+m

2. b = d = 0V0 = 0 , p = q

(1 a) = rc , r = qp

E+mEV0+m

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propagating

solutions

m0

E>m

E

x

EV0 m

|E

V0

|> m

E > m

V0 > 0

1) E V0 m V0 > 0 E V0 > 0

2)

E

V0

m

E m m E V0 mE0 > 2m E V0 < 0.E m V0 2m

V0

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12

jtrans jrefl

x = 0

c =2

1 + ra =

1 r1 + r

.

j = c = 0

0 c

j

jy = jz, jx = 0.jtrans

jin=

4r

(1 + r)2,

jrefl

jin=

1 r1 + r

2

jtransjin

+jrefljin

= 1.

q,p > 0, m < E < V0 m, i.e. V0 > 2m, E V0 + m < 0, r < 0

jrefljin

> 1,jtransjin

< 0

vtrans,x =dE

dq=

d

dq

q2 + m2

=

2q

+

q2 + m2=

2q

E V0 = 2q

|E V0|

vtrans

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E > m V0 > 2m

jtrans

E < 0

E < 0

E < 0

E < 0

E < 0E > 0

E < 0

e

e+

h

p

EE(p)

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12

E < 0

E < 0, p, q

E > 0, p, q;

> 2mc

2

h

p

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E < 0

-4 -2 0 2 4-3

-2

-1

0

1

x

E < 0

() 0

(+)(x) E > 0 p E > 0 p e

()(x) E < 0 p E > 0 p e

particle

particle

antiparticle

0

V

x

V

jin(+)

(+)

jrefl>jin

()

jtrans2m

m

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12

x

p x1

E c p cx

= x < 12

mc=

1

2C

E 2mc2

C

E < 0

E < 0 e = e0 < 0 E > 0e = e0 > 0

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[(i eA) m] = 0[(i + eA) m] C = 0

C C = C

i eA

C [(i + eA) m] C1C = 0. C

C C1 =

0 = =

1 0

0 1

, i =

0 i

i 0

, i = 1, 2, 3 ,

C

i

i (i = 1, 2, 3)

C =

0 0 0 1

0 0 1 00 1 0 01 0 0 0

= i2

C1 =

C .

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12

C = C = i2

+ = 2g1

(i

2

)

(i2

) = [(i

2

2 ) =i2

(i2

)]

= [22]= [2(2g21 2)]= [222 ]

=

, = 222 2 = 2 , = 2

(2)2 = 1

C E E ( )

p p

(C) e e

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=

1

0p

E+m

0

ei(Etpz)

C =

0 0 0 1

0 0 1 00

1 0 0

1 0 0 0

1

0p

E+m

0

e+i(Etpz)

=

0pEm0

1

ei(Et(p)z)

E > 0 p||z pz = p E < 0 p ||z pz = p

A

T : t t

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12

T : p p ( )L = [r p] L ( )

S S ( L)(x, t) (x, t) = (x, t)A(x, t) A(x, t) = A(x, t)

e e ( )

(1.264),(1.265)= A(x, t) A(x, t) ( )

[(i eA) m] = 0

[(i egA) m] T = 0

T = T

T = T = i2

i2 =

0 1 0 0

1 0 0 00 0 0 1

0 0 1 0

= i

2 0

0 2

.

C T

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P : x x t t , E Ep pL +LS +S

A A

A A

eipx

P : p0 p0 , x0 x0pi pi , xi xi , i = 1, 2, 3.

P (x, t) = 0 (x, t) 0 =

1 0

0 1

.

= 2

0

0

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12

(+)p=0 , =

1

0

0

0

eimt , (+)p=0 , =

0

1

0

0

eimt , E = m > 0

()p=0 , =

0

0

1

0

e+imt , ()p=0 , =

0

0

0

1

e+imt , E = m < 0

z =

2

1 0 0 0

0 1 0 00 0 1 0

0 0 0 1

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z

(+)p|| bz , (x) =

E+ m

2m

1

0pz

E+m

0

eipx

(+)

p||bz ,

(x) = E+ m2m

0

1

0pz

E+m

eipx

()p|| bz , (x) =

E+ m

2m

pz

E+m

0

1

0

e+ipx

()p|| bz , (x) = E+ m2m 0

pz

E+m0

1

e+ipx

z||p^ ^

z

z s = 12

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12

z

(+)p|| bx , (x) =E+ m

2m

1

0

0pz

E+m

eipx

(+)p|| bx , (x) =

E+ m

2m

0

1pz

E+m

0

eipx

()p|| bx , (x) =

E+ m

2m

0

pzE+m

1

0

e+ipx

()p|| bx , (x) =

E+ m

2m

pzE+m

0

0

1

e+ipx

z

x||p^ ^

^

1 i

t =

m + e

m + e

= H .

= A A1

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E < 0

E < 0

z z

z = mE

2!

|v| c E =

p2 + m2

zpx = 0 ypx = 0.|v| c

x

v=c

(+)

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12

(+)p|| bx , (+)p|| bx , =E+ m

2m 1 0 0px

E+m

1

0

0pxE+m

=E+ m

2m

(E2 + m2 + 2Em) +

E2m2p2

(E+ m)2

=E+ m

2m

2E2 + 2Em

(E+ m)2=

E

m

d4x

z =normalization

m

E

(+)p||bx , z

(+)p||bx ,

=m

E

E+ m

2m

(E2 + m2 + 2Em) p2(E+ m)2

=m

E

E+ m

2m

2m2 + 2Em

(E+ m)2=

m

E.

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(+)p , (x) =

E+ m

2m

1

0pz

E+mpx+ipy

E+m

eipx

(+)p , (x) = E+ m2m 0

1px+ipyE+m

pzE+m

eipx

()p , (x) =

E+ m

2m

pz

E+mpx+ipy

E+m

1

0

e+ipx

()p , (x) =

E+ m

2m

px+ipy

E+mpzE+m

0

1

e+ipx

Emm

A1 =: eiS eiS

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12

= e+iS

i

teiSe+iS

1

= H eiSe+iS 1

i

t

eiS

i( t eiS)+ieiSt

= H eiS e+iS

i

t = e+iS

H i

t

eiS

H.

H

H = p + m =

1m p p 1m

= m + O

odd

B =

Bx0Bz

H = xBx + z Bz

z

x

y

0

e i2y0 = e 12zx0

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eiS = e

z

x

( p ) (p) = 1 cos ( p )sin = eO

|p | ( )

p = p|p |

H = eiS

H eiS

= e( bp )( p + m)(1 cos ( p )sin )= e( bp ) (1 cos + ( p )sin )

e(bp )

( p + m)= (cos 2 + ( p )sin2)( p + m)= p

cos2 m|p |

!

= 0

+m

cos2 +

|p |m

sin2

= tan2 = |p|m

sin2 =p

m2 + p2=

p

E, cos2 =

mm2 + p2

=m

E

[ , ]+ = 0

H

H = m mE + |p |2mE = 1E|p |2 + m2E p

|p | m

iS = O |p | O1

2m

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12

e( bp ) =

k=0

1

(2k)![(

p )]2k +

k=0

1

(2k + 1)![(

p )]2k+1

( p )2 = ipijpj = 12{i, j}

ij

pipj = |p |2

H =

p e A

+ m + e = m + + O

= 1e =

O = (p e A) O = O

m O(m)1m

iS =

2mO O =

p e A

.

|pe A|m

H = eiSH i t eiSt

eiS

eABeA = B+[A , B]+1

2[A , [A , B]]+. . .+

1

k![A , [A , . . . , [A , B] . . .]]

+ . . .

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H = H + i[S , H] +i2

2 [S , [S , H]]

+i3

6[S , [S , [S , H]]] +

i4

24

O( 1m3

) [S , [S , [S , [S , H]]]]

S i2

[S , S] i2

6[S , [S , S]] + . . .

1m

i[S , H] = O + 2m

[O , ] + 1m

O2

O O 1m0

S = i 2m

O 0

H = m +

O22m O48m2+ 1

8m2[O , [O , ]] +

2m[O , ] O

3

3m2 O

= m + + O

O O 1m

O 1miS =

2mO O

1

m2

.

H = m + +

2m[O , ] + O 1

m4

= m + + O.

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12

iS = 2m

O O 1m3

O

1m3

H =

m +O22m

O4

8m3

+ 1

8m2

O , [O , ] + iO

+ O

1m4

O H O 1

m3

O H

H = m + p e A2

2m 1

8m3 p e A2 e B2 + e

e2m

B e8m2

E

e4m2

E

p e A

e8m2

E .

H

E > 0 = 0

i

t=

m + e +

1

2m

p e A

2 e

2m B

(|p |2)2

8m3 e

4m2

E

p e A

e8m2

E

H1 = (|p |2)2

8m3

|p |22

m2 + p2

=|p |22m

(|p |2)2

8m3+ . . .

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E = (r) = 1r

rr , A = 0

1

r

rr p

= 1

r

r

L

H2 =

e

4m21

r

r

L

r

H3 = e8m2

E

H3

Compton =

mc

m = 0

p = 0

1p0 = c

p

m = 0

{i, j} = 2i ijk k

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E =

p2c2 + (mc2)2 m = 0

E > 0

vp =Ep

m=0=

p

(|p |c) = c p|p | ,

|p |E

= 1

p = 0 m = 0

2 > 0 z

x y

m = 0 p

h(p ) = p|p |h(p )

p|p |2

2=

3i,j=1

ijpipj|p |2

=

3i=1

2i

=1

p2i|p |2 = 1,

pipj =

0 , i = jp2i , i = j

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h(p ) h = 1

p

p helicity h=+1

helicity h=1

p

p , (c p + mc2) H

= 0

p

p

p = 0

p = 0p0 = 0 0 p =

0

0

0 11 0

0 11 0

= 50 , 5 i0123 = 0 11 0 m = 0 50

p = 5p0

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p , 5

= 0

5 = i0123

ch = U U =1

21 + 5

ch = U U1 ,

p0 p

ch1 = 0

p0 + p

ch2 = 0

p|p |

ch

=1

21 + 5 p P p

h = +1 h = 1

h = 1h = +1

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0

k

0 = 0 2

2 0 ,1 = i

0 1

1 0

,

2 = i1 0

01

3 = i

0 3

3 0

i m

= 0 .

C = ,

(i eA) m

= 0 | (i eA) m = 0

(i + eA) m

= 0 = C

=

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