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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