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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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    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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