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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
7/30/2019 Relativistic Q
