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7/30/2019 Relativistic Quantum Mechanics-
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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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(+)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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