量子力学A

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1 c S

2

3 S S, t = t = 0, S S x v S t, x, y, z, S t, x, y, z x, y, z, t x, y, z, t x = x vt, y = y, z = z, t = t x = x + vt, y = y, z = z, t = t, S c S c v,, x = α x vt, y = y, z = z 1.1 x = α x + vt, y = y, z = z 1., α t = t = 0 x S c, S c t x = ct S, t x = c t x = ct, x = c t c t = αct vt = αc vt, ct = αc t + vt = αc + vt cc tt = α c vc + vtt v = 0 α = 1 > 0 cc α = c vc + v c = c v α = 1, c = c c α = c vc + v = v /c 1.1 x 1. x = α αx vt + vt = α x α vt + αvt, t = α t + 1 α α v x α = 1 t = t, 1.3 t = t vx/c 1 v /c 1.4

4 1, S S x, y, z, t t = t vx/c 1 v /c, x vt x = 1 v /c, y = y, z = z, t ct ct = ct v/c x 1 v /c, x = x v/c ct 1 v /c, y = y, z = z ct x y z = ct x y z 1.6 x 0 = ct, x 1 = x, x = y, x 3 = z x µ µ 0, 1,, 3 1 0, 1,, 3, 1,, 3 x µ = x 0, x k = x 0, x 1, x, x 3, 0 3, 1 3, 1, µ = ν = 0 g µν = 1, µ = ν = 1,, 3 0, µ ν 1.7, 1.6 g µν x µ x ν = g µν x µ x ν 1.8, µ, ν g µν x µ x ν = g µν x µ x ν µ=0 ν=0 µ=0 ν=0, 1.5 x µ = a µ ν x ν a µ ν xµ, g σρ x σ x ρ = g σρ a σ µx µ a ρ νx ν = g σρ a σ µ a ρ ν x µ x ν = g µν x µ x ν g σρ a σ µ a ρ ν = g µν 1.10

5 1 3 µ, ν, µ = ν 4 µ ν 16 4/ = 6 10, 16 a µ ν 6, 1.8 a µν a µν a µ ν,,, a µ ν 4 4 A, A µν = a µ ν A Ã 1.10 Ã µν = A νµ = a ν µ 1.11 Ã µσ g σρ A ρν = ÃGA µν = G µν, 4 4 G G µν = g µν detã = deta detãga = detg deta = detg deta = deta µ ν = ± µ = ν = 0 3 a 0 0 a k 0 = 1, a 0 0 = ± 1 + k=1 1/ 3 a k 0 deta µ ν a 0 0 detaµ ν = 1, a proper Lorentz transformation 1.5 a 0 0 = a 1 1 = 1 1 v /c, a0 1 = a 1 0 = k=1 v/c 1 v /c, a = a 3 3 = 1, = 0 1.1, deta µ ν = 1, a deta µ ν = 1, a x 4 = ix x 1 = 1 1 v /c x1 + iv/c 1 v /c x4, x 4 = 1 1 v /c x4 iv/c 1 v /c x1 1 iv/c + = 1 1 v /c 1 v /c cos φ = 1 1 v /c, sin φ = iv/c 1 v /c φ x 1 = x 1 cos φ + x 4 sin φ, x 4 = x 4 cos φ x 1 sin φ x φ θ φ = i θ cos φ = cosh θ, sin φ = i sinh θ x 1 = x 1 cosh θ x 0 sinh θ, x 0 = x 0 cosh θ x 1 sinh θ 1.13 cosh θ = 1 1 v /c, sinh θ = v/c 1 v /c

6 1 4 1., 0 S F x 0, x 1, x, x 3 F x S, x µ = a µ νx ν F x F x = F x, F x F x = g µν x µ x ν 1 4 V µ x, x µ V µ x = a µ νv ν x 1.14 V µ x contravariant vector, V µ g µν V ν, V 0 = V 0, V k = V k V µ covariant vector, V µ V µ V µ = g µν V ν, g µν g µν V µ V µ,, 3, g µν, g µν V µ V µ = g µσ V σ = g µσ a σ ρv ρ = g µσ a σ ρ g ρν V ν = a ν µ V ν, a ν µ = g µσ a σ ρ g ρν , a ν µ µ aσ ρ σ g µσ ρ ν 1.10 g σρ a σ µ a ρ λ gλν = g µλ g λν a σ µ a ν σ = δ ν µ, δ ν µ = g µλ g λν = { 1, µ = ν 0, µ ν 1.16 δ ν µ 1.14, V µ aµ σ = a µ νaµ σ V ν = δν σ V ν = V σ, V µ a µ σ = a µ σaµ ν V ν = δσ ν V ν = V σ V µ = V ν a µ ν, V µ = V ν a ν µ 1.17

7 1 5 A µ, B µ A B A B = A µ B µ = A µ B µ = A 0 B 0 A k B k = A 0 B 0 A B A µ B µ = a µ σ a ρ µ A σ B ρ = δ ρ σ A σ B ρ = A σ B σ,,, A µ A A A : A = A 0 A A A > 0, time like vector, A < 0 space like vector, A = 0 null vector A, 3 x µ / x µ µ µ = 1 x µ = c t, 1.18 µ = xν = x µ x µ x ν 1.17 x ν = x µ aµ ν xν / x µ = aµ ν µ = aµ ν ν, µ µ µ = g µν ν = 1 c t, 1 c t = µ µ 1.19 µ µ = µ µ 1.18, 1.19 A µ = A 0, A, A µ = A 0, A T µν x = V µ xu ν x T µν x = V µ x U ν x = a µ σ a ν ρv σ xu ρ x = a µ σ a ν ρ T σρ x, T µν x = a µ σ a ν ρ T σρ x 1.0, T µν x, T ν µ x T µν λσ

8 1 6 contraction 1, contraction, T µ ν = A µ B ν 0 T µ µ = A µ B µ 1.3, 1.1 B = 0, E = 1 ε 0 ρ, B t B = A, 1. 1 c t A = A A 1 c t + E = B 1 c E t = µ 0j, c = 1 µ0 ε 0 1. E = A t φ 1.3 φ 1 φ t c t + A = 1 ρ 1.4 ε 0 A + 1 φ c t + A = µ 0 j 1.5 A 0 = 1 c φ, A1 = A x, A = A y, A 3 = A z A 0 j 0 = cρ, j 1 = j x, j = j y, j 3 = j z 1 φ A0 c + A = t x 0 + Ak x k = νa ν x 0 νa ν = A 0 0 ν A ν = 1 ε 0 c ρ = µ 0 cρ = µ 0 j A k + 1.6, k = = x k x k x k νa ν = A k k ν A ν = µ 0 j k 1.7 A µ µ ν A ν = µ 0 j µ 1.8, 1.

9 µ µ 0 µ j µ = µ A µ ν A ν = 0, µ j µ = ρ t + j = 0 j µ x, µ j µ, µ j µ = 0 µ j µ = 0, µ = a ν µ ν µ j µ x = a ν µ ν j µ x j µ x = a µ λ jλ x µ j µ x = a ν µ a µ λ νj λ x = δ ν λ ν j λ x = µ j µ x = 0 j µ x, µ j µ x = 0 µ j µ x = 0, j µ x ρ j cρ, j 1, φ A A µ, j µ, =, νa ν = ν A ν A µ µ νa ν µ 0 j µ = a µ λ A λ λ ν A ν µ 0 j λ, S 1.8 S,, ν ν A µ ν µ A ν = ν ν A µ µ A ν ν F νµ = µ 0 j µ, F µν = µ A ν ν A µ = F νµ F µν E B k = / x k = / x k B x = A z y A y z = A A = F 3, B y = F 13, B z = F 1 E x = A x t φ x = c 1 A 0 0 A 1 = cf 10, E y = cf 0, E z = cf 30 0 E x /c E y /c E z /c E F µν = x /c 0 B z B y E y /c B z 0 B x E z /c B y B x 0 E, B, F µν E B E B, F µν A µ χx Ā µ = A µ µ χ, φ = φ χ, Ā = A + χ 1.9 t

10 1 8 Āµ F µν F µν = µ Ā ν ν Ā µ = µ A ν ν A µ µ ν χ + ν µ χ = µ A ν ν A µ = F µν Āµ A µ Āµ µ ν Ā ν = A µ µ ν A ν µ χ + µ ν ν χ = A µ µ ν A ν, Ā µ A µ A µ,, µ A µ = 0 µ A µ 0 µ Ā µ = µ A µ χ χ = µ A µ χx 1.8 A µ = µ 0 j µ F µν, A F = AF Ã A 1.1 b 0 1 A =, b = β 1 β β 1, β = v c 0, 1,, F 4 E x = E x, E y = E y vb z, 1 β E z = E z + vb y 1 β 1.30 B x = B x, B y = B y + ve z /c 1 β, B z = B z ve y /c 1 β 1.31 S, B e, S, B x v,, E = 0 E = v B 1 β, B = B 1 β v B v 1 β v, v = v, 0, 0 S V, F F = ee + ev B = e v + V B 1 v B V v + e 1 1 β 1 β v

11 1 9 V = v F = 0 S, e v B ee, S, e v x, S E x = e x 4πε 0 r 3, E y = e y 4πε 0 r 3, E z = e z 4πε 0 r 3, B = 0, r = x + y + z S S x v 1.30, 1.31 v v, E, B E, B E x = e x 4πε 0 r 3, E y = e y 1 4πε 0 r 3, E z = e z 1 1 β 4πε 0 r 3 1 β B x = 0, B y = v c e 4πε 0 z r β, B z = v c e 4πε 0 y r β x = x vt/ 1 β, y = y, z = z E = 1 e x vt 1 β 4πε 0 r 3, B = 1 c v E, x vt r = 1 β + y + z 1.3 v = v, 0, 0 A µ A 0 = φ c = A 0 + βa x, A x = A x + βa 0, A y = A 1 β 1 β y, A z = A z φ = φ = e 4πε 0 r, A = 0 1 e 1 β 4πε 0 r, A = 1 µ 0 ev 1 β 4π r φx = r = x vt 1 β + y + z =, vt, 0, 0 x 1 β β 0 E e x vt 4πε 0 r 3, B µ 0 ev x vt 4π r 3, r = x vt + y + z 1/ E vt, 0, 0 e, B e v δx vt , 1.31 x = 0, y, β 1 v c,, E B

12 10, m v, E p p = mv 1 v /c, E = mc 1 v /c = c p + mc p µ = E/c, p p µ,, p 0 = E c i c p = p µ p µ = p 0 p = m c t = i x 0, pk i x k = + i x k = i k, p µ ˆp µ i µ.1 [ x k, ˆp l ] = i δ kl, [ x 0, ˆp 0 ] = i, [ x µ, ˆp ν ] = i g µν.1,, N ψ 1 t, x ψ t, x ψt, x =., x ρt, x ψ N t, x ρt, x = ψ t, x ψt, x = N ψ a t, x., ψt, x H a=1 i ψ t = Hψ, i ψ a = t N H ab ψ b.3 b=1, ψ, ϕ d 3 r ψ Hϕ = d 3 r ϕ Hψ, d 3 r ψah ab ϕ b = ab ab d 3 r ϕ a H ab ψ b i d d 3 r ψ ψ = d 3 r i ψ a dt t ψ a + ψa i ψ a = t a d 3 r ab ψ a H ab ψ b + ψ ah ab ψ b

13 11 H d 3 r ab ψ a H ab ψ b = d 3 r ab ψ ah ab ψ b d d 3 r ψ ψ = 0.4 dt..3.3,.3,, 1, H = c α ˆp + mc β = i c α + mc β.5 α = α x, α y, α z β N N H d 3 r ψa β ab ϕ b = d 3 r ϕ a β ab ψ b = d 3 r ψaβ baϕ a ab ab ab β ab = βba = β ab, β i d 3 r ψa α ab ϕ b = d 3 r ϕ a i α ab ψ b ab ab d 3 r ϕ a i α ab ψ b = i ab ab d 3 r ψ b α ab ϕ a = i ab d 3 r ψ a α ba ϕ b α p µ p µ m c = 0 ψ p µ ˆp µ ˆp µ ˆp µ m c ψ = c t + m c ψ = ψ t = Hi ψ t = H ψ.7 H = c + mc α 1 = α x, α = α y, α 3 = α z,.5 H = c α α i m c 3 α β + βα + mc β = c 3 k,l=1.8 α k α l + α l α k x k i m c3 xl 3 α k β + βα k x k + mc β k=1 α k α l + α l α k = δ kl, α k β + βα k = 0, β = α k, β,, α k = β = 1,,,

14 1, N α k = β = 1 α k, β ± 1 β α k α k + βα k β = 0, β + α k βα k = 0, k, TrAB = TrBA Trα k = Trβα k β = Trα k β = Trα k = 0, Trβ = 0, β U b b 0 U 1 βu = 0 0 b 3, b k = β = ± n +, 1 n n + + n = N Trβ = TrU 1 βu = k b k = n + n = 0 N = n + = N N =, 4 N N = 4, 0 σ 1 0 α =, β =.10 σ σ σ = σ x =, σ = σ y = i i 0, σ 3 = σ z = , , 0 1, β N = ψ 4 ψ 1 t, x ψ t, x ψt, x = ψ 3 t, x ψ 4 t, x.11 i ψ t = Hψ, H = c α ˆp + mc β, ˆp = i.1 α β ,

15 13 A µ = φ/c, A, p µ p µ qa µ, q = < 0 ˆp µ ˆp µ qa µ, i t i qφ, ˆp ˆp qa t.1 i ψ t = Hψ, H = qφ + c α ˆp qa + mc β ψt, x = e iet/ ϕx, mc, E = mc + ε,.13 Hϕx = mc + ε ϕx.14 4 ϕx F x, Gx F x ϕx = Gx,.10 H mc V 0 c σ ˆπ ϕ = c σ ˆπ V 0 mc F G, V 0 x = qφx, ˆπ = ˆp qa.14 V 0 F + c σ ˆπ G = εf, V0 mc G + c σ ˆπF = εg.15 mc, mc V 0, ε c G = mc σ ˆπF = 1 1 ε V 0 + ε V 0 mc mc + σ ˆπ σ ˆπF mc F H NR F = εf, H NR = σ ˆπ σ ˆπ m + V 0 σ a σ b = a b + iσ a b

16 14 σ ˆπ σ ˆπ = ˆπ + iσ ˆπ ˆπ ˆπ ˆπ ˆπ = q ˆp A + A ˆp = i q A = i q B, H NR = 1 q ˆp qa g m m S B + V 0, S = σ, g = S,,,, g =.003 1/ 1, g g =,., A = 0 V 0 r = x.16 1 G 1 mc 1 ε V 0 mc σ ˆpF H NR + H F = εf H 1 4m c r dv 0 dr σ L L = x ˆp β/c γ µ = γ 0, γ, γ 0 = β, γ = βα i γ 0 ψ x 0 = qγ 0 A 0 + γ 0 α ˆp qa + mc ψ [ γ 0 i 0 qa 0 ] γ ˆp qa mc ψ = 0 γ µ i µ qa µ = γ 0 i 0 qa 0 γ k i k qa k = γ 0 i 0 qa 0 γ ˆp qa,.13 [ γ µ ] i µ qa µ mc ψ = 0.17

17 15, /a γ µ a µ = γ 0 a 0 γ a.18 i / q /A mc ψ = 0.9 γ µ α k = γ 0 γ k γ 0 γ k γ 0 γ l + γ 0 γ l γ 0 γ k = δ kl, γ 0 γ k γ 0 + γ 0 γ k = 0, γ 0 = 1 γ 0 γ 0 γ k + γ k γ 0 = 0 γ k γ 0, 1 γ k γ l + γ l γ k = δ kl 3 1 γ µ γ ν + γ ν γ µ = g µν, β γ 0 = γ 0 α k γ 0 γ k = γ k γ 0 = γ k γ 0 = γ 0 γ k γ k = γ 0 γ k γ 0 = γ k γ k γ 0 = γ 0 γ µ = γ 0 γ µ γ 0 γ µ γ ν + γ ν γ µ = g µν, γ µ = γ 0 γ µ γ 0.19 γ µ γ µ = g µν γ ν, γ 0 = γ 0, γ k = γ k γ 0 γ 0 = γ 0 = 1, γ k γ k = γ k = 1, k γ µ γ µ, γ 5 = γ 5 iγ 0 γ 1 γ γ 3.0 γ 5 γ 5 = 1, γ 5 = γ 5 γ 5 γ µ + γ µ γ 5 = σ γ =, γ 0 = σ , γ 5 = /a = a, γ µ /aγ µ = /a, γ µ /a/bγ µ = 4 a b

18 16.4 H.13 ψ t = c α ψ + 1 qφ cqα A + mc β ψ i ψ t = c ψ α 1 i ψ qφ cqα A + mc β, ρ = ψ ψ ρ t = ψ ψ ψ + ψ t t = c ψ α ψ c ψ α ψ = c ψ αψ ρ t + j = 0, j = c ψ αψ j ψ ψ γ 0. ρ = ψγ 0 ψ, j = c ψ γ 0 αψ = c ψ γψ j µ = cρ, j j µ = c ψγ µ ψ µ j µ = 0 j µ, γ µ ν.19 1 γ ν γ µ = γ ν γ µ γ ν, ν Tr γ µ = Tr γ µ γ ν γ ν = Tr γ µ = 0, Tr γ 5 = 0.3,.19 1 Tr γ µ γ ν = g µν Tr1 = 4 g µν.4 n Tr /a 1 /a n = Tr /a 1 /a n γ 5 γ 5 = Tr γ 5 /a 1 /a n γ 5 γ 5 γ µ γ 5 /a 1 /a n = 1 n /a 1 /a n γ 5.4 n Tr /a 1 /a n = Tr /a 1 /a n = 0.5 Tr /a /b = a µ b ν Tr γ µ γ ν = 4 a b.6 γ µ γ ν = g µν γ ν γ µ /a /b = a b /b /a, 4 /a /b /c /d = a b /b /a /c /d = a b /c/d /b a c /c/a /d = a b /c/d a c /b/d + /b/c a d /d /a

19 17 Tr /b /c /d /a = Tr /a /b /c /d Tr /a /b /c /d = a b Tr /c/d a c Tr /b/d + a d Tr /b/c = 4 a b c d a c b d + a d b c.7 n Tr /a 1 /a n = a 1 a Tr /a 3 /a n a 1 a 3 Tr /a /a 4 /a n + + a 1 a n Tr /a /a n 1.8, n n

20 = c = ,,, a µ ν a µ ν deta µ ν = ± 1, a a deta µ ν = 1, a 0 0 = 1, deta µ ν = 1, a 0 0 1, ω µ ν x µ = x µ + ω µ ν x ν = δ µ ν + ω µ ν x ν, a µ ν = δ µ ν + ω µ ν 3.1 a λ µa ν λ = δ ν µ ω 1 δ λ µ + ω λ µ δ ν λ + ωλ ν = δµ ν + ω ν µ + ωµ ν = δµ ν ω ν µ + ω ν µ = 0, ω µν + ω νµ = ω µν 6,,, x S x z x, x = x, x 3 = x 3, x 0 x 1 x 0 x 1 x = x 0 x 1, x = x 0 x x 0 x 1 = x 0 x 1 + ω 0 0 ω 0 1 ω 1 0 ω 1 1 x 0 x 1 = x 0 x 1 + ω 00 ω 01 ω 10 ω 11 x 0 x 1 3. ω 00 = ω 11 = 0, ω 01 = ω 10 ω 01 = θ 3.3 x = 1 θ ω tx x, ω tx =

21 3 19 θ θ = θ/n ω tx = 1 x = lim N 1 θ ω tx N x = exp θω tx x θ k exp θω tx = k! + ω θ k+1 tx k + 1! k=0 k=0 cosh θ = cosh θ ω tx sinh θ = sinh θ sinh θ cosh θ 3.4 x 0 = x 0 cosh θ x 1 sinh θ, x 1 = x 1 cosh θ x 0 sinh θ 1.13 cosh θ = 1, tanh θ = v v 3.4 a µ ν = cosh θ sinh θ 0 0 sinh θ cosh θ deta µ ν = cosh θ sinh θ = 1, a 0 0 = cosh θ 1 z x 1 x x 1 x x = x 1 x, x = x 1 x x 1 x = x 1 x + ω 1 1 ω 1 ω 1 ω x 1 x = x 1 x 0 ω 1 ω 1 0 x 1 x ω 1 = ω 1 ω 1 = θ x = 1 i θ ω xy x, ω xy =, θ ω xy = 1 x = exp i θ ω xy x 0 i i 0 exp i θω xy = k=0 iθ k k! i ω xy = cos θ i ω xy sin θ = k=0 i k θ k+1 k + 1! cos θ sin θ sin θ cos θ

22 3 0 x = x cos θ + y sin θ, y = x sin θ + y cos θ θ a µ ν = cos θ sin θ 0 0 sin θ cos θ deta µ ν = cos θ + sin θ = 1, a 0 0 = 1 3. S.17 [ γ µ ] i µ qa µ x m ψx = 0 3.6, S 4 ψ x x µ = a µ νx ν x = ax ax µ = a µ νx ν ψx ψ x 4 4 Ra ψ x = ψ ax = Raψx 3.7 ψx = R 1 aψ x 3.6 [ γ µ R 1 ] i µ qa µ x R 1 m ψ x = 0 R [ Rγ µ R 1 ] i µ qa µ x m ψ x = A µ x = A νx a ν µ, µ = ν a ν µ [ ] Rγ µ R 1 a ν µ i ν qa νx m ψ x = 0 a µ ν Rγ µ R 1 a ν µ = γ ν, a ν µγ µ = R 1 γ ν R 3.8 R [ γ µ ] i µ qa µx m ψ x = 0,, R R = 1 i 4 σ µν ω µν 3.9

23 3 1 { σ µν i γ µ γ ν γ ν γ µ = 0, µ = ν iγ µ γ ν, µ ν 3.10 S = σ µν ω µν R = 1 i S/4 1 + i4 S R = 1 + i4 1 S i4 S = 1 + S 16 ω 1 S R 1 = 1 + i S/4 µ = ν ω µν = 0 i Sγ ν γ ν S 4 R 1 γ ν R = γ ν + i 4 Sγ ν γ ν S S = σ αβ ω αβ = iγ α γ β ω αβ i Sγ ν γ ν S = ωαβ 4 4 γ ν γ α γ β γ α γ β γ ν γ ν γ α + γ α γ ν = g αλ γ ν γ λ + γ λ γ ν = g αλ g λν = δ ν α = ωαβ 4 δ ν α γ β γ α γ ν γ β = ωαβ δα ν γ β δβ ν γ α = 1 ω νµ γ µ ω µν γ µ = ω νµ γ µ R 1 γ ν R = γ ν + ω νµ γ µ = δ ν µ + ω ν µ γ µ σ µν = i γ νγ µ γ µγ ν = i γ0 γ µ γ ν γ ν γ µ γ 0 = γ 0 σ µν γ 0 R = 1 + i 4 σ µν ω µν = γ i 4 σ µν ω µν γ 0 = γ 0 R 1 γ 0 γ 0 γ 0 R γ 0 = R , R x R R B ω 10 = ω R B = 1 i σ01 ω 01 + σ 10 ω 10 = 1 i 4 σ 01 ω 01 = 1 i σ 01 θ

24 3 θ, θ 3.5 R B = lim 1 i N N σ θ 01 = exp iθσ 01/ N α x = 1 σ 01 = iγ 0 γ 1 = iγ 0 γ 1 = iα x R B = exp iθσ 01 / = exp θα x / = cosh θ α x sinh θ v = vn, n R B = cosh θ α n sinh θ 3.1 R B R 1 B α n = v R B R 1 B α R B R 1 B = cosh θ + α n sinh θ = cosh θ α n sinh θ = cosh θ sinh θ = 1 R B = R B R 1 B R B R B = 1 + b b α v 1, b = 1 + b 1 v z.1 σ x σ y = iσ z σ 1 = iγ 1 γ = iγ 1 γ 0 σ x 0 σ y = i σ x 0 σ y 0 = σ z 0 0 σ z σ 3 = σ x 0 0 σ x, σ 31 = σ y 0 0 σ y σ 4 4 Σ.0 γ 5 Σ x = σ 3, Σ y = σ 31, Σ z = σ γ 5 γ 0 γ 1 = iγ γ 3 = σ 3, γ 5 γ 0 γ = iγ 3 γ 1 = σ 31, γ 5 γ 0 γ 3 = iγ 1 γ = σ 1

25 3 3 Σ = γ 5 γ 0 γ = γ 5 α 3.14, Σ 3.13 γ µ.19 Σ x Σ y Σ y Σ x = iσ z, Σ x = 1,, Σ σ, Σ A Σ B = A B + i Σ A B 3.15 R R R ω 1 = ω 1 0 θ R R = 1 i σ 1 ω 1 = 1 + i Σ z θ R R = lim 1 + i N N Σ θ z = exp iθσ z / N, θ z, n R R = exp iθσ n/ S, n ψ x = exp iθs n ψx ψ x = ψx,, 1/ 3.15 Σ n = n n = 1 S = 1 Σ R R = cos θ + iσ n sin θ 3.16 R R π = 1, 1, 1, ψ.19 Σ x = iγ 1 γ = iγ γ 1 = iγ 0γ γ 1 γ 0 = iγ γ 1 = iγ 1 γ = Σ x, Σ R R = exp iθσ n/ = R 1 R R R R B 3 R R x 0 = x 0, x k = x k, a µ ν = g µν

26 3 4 det a µ ν = 1 R P 3.8 a µ ν = g µν γ 0 = P 1 γ 0 P, γ k = P 1 γ k P P γ 0 γ k P = γ R B, 3.16 R R P ψ ψ = ψ γ 0 biliear form ψxγ ψx, Γ = 4 4 ψ x = Rψx 3.11 ψ x = ψ x γ 0 = ψ xr γ 0 = ψxγ 0 R γ 0 = ψxr 1 ψ x Γ ψ x = ψxr 1 Γ R ψx Γ = 1 ψ x ψ x = ψxψx ψxψx V µ x = ψxγ µ ψx 3.8 R 1 γ µ R = a µ νγ ν V µ x = ψ x γ µ ψ x = a µ νψxγ ν ψx = a µ νv ν x V µ x j µ = c ψγ µ ψ, µ j µ = 0 T µν x = ψxσ µν ψx, T µν x = a µ ρa ν λt ρλ x Sx = ψxγ 5 ψx, R 3.1, 3.16 γ 5 γ µ α = γ 0 γ Σ = γ 5 α R 1 B γ5 R B = cosh θ α n sinh θ γ 5 = cosh θ θ sinh γ 5 = γ 5 R 1 R γ5 R R = cos θ + Σ n sin θ γ 5 = cos θ + sin θ γ 5 = γ 5

27 3 5 S x = ψxr 1 γ 5 R ψx = ψxγ 5 ψx = Sx R = γ 0 R 1 γ 5 R = γ 5 S x = ψxr 1 γ 5 R ψx = Sx, pseudoscalar det a µ ν = 1 det a µ ν = 1 R 1 γ 5 R = det a µ ν γ 5, S x = det a µ ν Sx 3.3 V µ x = ψxγ 5 γ µ ψx V µ x = det a µ ν a µ νv ν x V µ x 3.4,,, S S S ψ L θ ψ θ ψ x x S ψx ψ L x, ψx x v,, z ψ L x S S x µ = a µ ν x ν, x = a x S ψx ψ x x x ψx = ψ L x ψx ψ x 3.7, 3.7 x x, x x, ψx ψ x, ψ x ψx

28 3 6 : ψx = Ra ψ x ψ L x = ψ x = Ra 1 ψx = Ra 1 ψa x ψ L x = Ra 1 ψax ψ x = Raψx = Raψa 1 x, a 1 x µ = x ν a µ ν 3.18 ψ L x = Ua ψx ψ L x = Ra 1 ψax = 1 + i4 σ µν ω µν ψx µ + ω µ νx ν ψx µ + ω µ νx ν = ψx + ω µ νx ν x µ ψx = ψx + ωµ νx ν µ ψx = ψx + ω µν x ν µ ψx ω µν ω µν = ω νµ ω µν x ν µ ψx = 1 ω µν x ν µ ω νµ x ν µ ψx = 1 ωµν x ν µ x µ ν ψx L µν = i x µ ν x ν µ = x µ ˆp ν x ν ˆp µ, ˆp µ = + i µ ψx µ + ω µ νx ν = 1 + i ωµν L µν ψx, ω µν 1 ψ L x = 1 + i4 σ µν ω 1 µν + i ωµν L µν ψx = 1 + i ωµν J µν ψx, J µν = L µν + 1 σ µν z ω 1 = ω 1 = θ 0 ψ L x = 1 i θj 1 ψx 3.13 x 1 = x 1 = x, x = y, x 3 = z L z = L 1 = i x 1 x 1 = i x y y x L y = L 31 = i z x x, L x = L 3 = i z y z z y

29 3 7 ψ L x = 1 i θj z ψx, J = L + 1 Σ n ψ L x = 1 i θ n J ψx, θ ψ L x = Uψx, U = exp iθ n J 4 J, L S = Σ/ 3.18 R = γ 0 P 0 ψ L x = γ 0 ψax, ax µ = x 0, x P 0 ψt, x = ψt, x ψ L x = P ψx, P = γ 0 P 0 P 0 ψx = ψx P 0 = 1 P P = γ 0 P 0 = 1 P ± 1 P P η η = 1 η = 1 ψ η x : P ψ η x = η ψ η x ψ η x F η x, G η x F η x ψ η x = G η x P ψ η x = γ 0 P 0 ψ η x = P 0 F η x P 0 G η x = P 0 F η x P 0 G η x P 0 F η x = η F η x, P 0 G η x = η G η x, P 0,, F η G η 1, F η x x, G η x, b µ x µ = x µ + b µ

30 3 8, ψx b µ ψ L x ψ L x = ψx b b µ b µ ψ L x = ψx b µ x µ ψx = 1 + i b µ ˆp µ ψx ψ L x = Uψx, U = exp i b µ ˆp µ b µ = 0, b U = exp ib ˆp, ˆp = i

31 i ψ t ˆp = Hψ, H = α ˆp + mβ, ˆp = i 4.1 ψt, x = exp ip x wp, p x = p µ x µ = p 0 x 0 p x = Et p x wp 4 ψt, x H D wp = E wp, H D = α p + mβ 4..9 HD = p + m HD w = E w E = ± E p, E p = p + m E p, E p 4 w χ ζ χp wp = ζp, α β m σ p χ = E σ p m ζ χ ζ E = ± E p ζ = χ = σ p E + m χ 4.3 σ p E m ζ 4.4 σ p σ p E + m E m = p E m = , p = 0 E = m ζ = 0, E = m χ = 0 E = E p 4.3, E = E p 4.4 E = ± E p w ± p χ w + p = N σ p E p + m χ, w p = N σ p E p + m χ χ N w w = wγ 0 w wγ µ w, w w w w w w 1, w w w ± w ± = ± N χ χ χ σ p E p + m χ = ± 1

32 4 30 σ p E p + m = p E p + m = E p m E p + m = E p m E p + m N m E p + m χ χ = 1 N = E p + m/m, χ s w + Ep + m p, s = m σ p E p + m χ 4.6 s w Ep + m p, s = σ p E p + m χ s m,, s = ± 1 : χ s χ s χ s = δ ss χ s w ± p, s H D χ s, E = E p E = E p Bjorken & Drell, up, s, vp, s 4.1 ψ + ps t, x = exp ie p t + ip x up, s = e ip x up, s, up, s = w + p, s 4.8 ψ ps t, x = exp ie p t ip x vp, s = e ip x vp, s, vp, s = w p, s 4.9, p 0 = E p > 0, χ s, σ z σ z χ s = s χ s, s = ± 1, w ± p, s p z Σ z w ± p, s = s w ± σ z 0 p, s, Σ z = 0 σ z, w ± p, s Σ z, s,, 3.14 Σ ph D H D Σ p = γ 5 α p α p α p γ 5 α p + m γ 5 α p β βγ 5 α p γ 5 γ µ α = γ 0 γ 1 0 β α γ 5 0, Σ p = Σ p p H D w, s = ± 1 H D σ p 0 Σ p =, σ p = σ p 0 σ p p

33 4 31 σ p χ s = s χ s, s = ± 1 χ s α p + mβ up, s = E p up, s, α p + mβ vp, s = E p vp, s γ 0 γ 0 E p γ p m up, s = 0, γ 0 E p γ p + m vp, s = 0 γ µ p µ m up, s = 0, γ µ p µ + m vp, s = 0 p 0 = p 0 = E p = p + m, p = p µ p µ = p 0 p = m.18 /p m up, s = 0, /p + m vp, s = u p, s /p m = 0, v p, s /p + m = 0 γ µ = γ 0 γ µ γ 0 /p = γ 0 /pγ 0 4.6, 4.7 up, s /p m = 0, vp, s /p + m = up, sup, s = δ ss, vp, svp, s = δ ss, up, svp, s = s s, u p, s p 0 u p, sγ 0 up, s p u p, sγup, s m u p, sup, s = 0 s s p 0 u p, sγ 0 up, s + p u p, sγup, s m u p, sup, s = 0 u p, s up, s = p0 m up, sup, s = E p m δ ss u p, s up, s = E p m δ ss, v p, s vp, s = E p m δ ss, u p, s v p, s =

34 4 3 Up, s = m m up, s, V p, s = vp, s 4.14 E p E p U p, s Up, s = δ ss, V p, s V p, s = δ ss, U p, s V p, s = up, sγ up, s, s vp, sγ vp, s, s Γ = 4 4 Λ + p αβ = s u α p, su β p, s, Λ p αβ = s v α p, sv β p, s Λ ± up, sγ up, s = s s u α p, s Γ αβ u β p, s = αβ αβ Γ αβ Λ + βα = Tr Γ Λ +.1 /p + m = E p γ 0 p γ + m = E p + m σ p σ p E p + m /p + m χ s 0 = E p + m χ s σ p χ s = E p + m χ s σ p E p + m χ s 4.6 up, s = w + /p + m p, s = m E p + m χ s 0, p 0 = E p Λ 0 = s χ s 0 χ s 0 Λ + = s up, s u p, sγ 0 = = 1 m E p + m /p + m Λ 0 /p + m γ 0 1 m E p + m /p + m Λ 0γ 0 /p + m

35 4 33 s χ s χ s = 1 Λ 0 = s χ s χ s = = 1 + γ0 Λ + = γ0 /p + m /p + m m E p + m γ 0 γ µ + γ µ γ 0 = g µ0 γ 0 /p = p 0 /p γ γ 0 /p + m = p 0 + m + /p m 1 γ0 Λ + = /p + m m /p + m /p m 1 γ 0 + m E p + m /p = p = m /p + m /p m = /p m = 0 Λ + p = s up, s up, s = /p + m m 4.16 vp, s = w /p + m p, s = m E p + m 0 χ s 4.17 Λ p = s vp, s vp, s = /p + m m 4.18 Λ ± Λ ± = Λ ±, Λ + Λ = Λ Λ + = 0, Λ + + Λ = ψ a s, b s ψ = ψ + + ψ, ψ + = s a s up, s, ψ = s b s vp, s Λ + ψ + = ss a s up, s up, s up, s = ss a s up, s δ ss = ψ + Λ ψ + = ss a s vp, s vp, s up, s = 0 Λ + ψ = 0, Λ ψ = ψ Λ ± ψ = ψ ± Λ + 4, Λ , 4.18, 4.19 Λ + + Λ = s up, sup, s vp, svp, s = 1 4.0

36 4 34, 4 ψ ψ = a s up, s + b s vp, s, a s = up, sψ, b s = vp, sψ s, 4.16, 4.18 s s up, su p, s = /p + m m γ0 = 1 E p p γ γ 0 + mγ 0 m vp, sv p, s = /p + m m γ0 = 1 E p p γ γ 0 mγ 0 m up, su p, s + v p, sv p, s s = E p m 4.14 U, V Up, su p, s + V p, sv p, s = s ψx p = mv 1 v, v = p E p v ψ L x p 3.18 Ra ψ L x = Ra 1 ψax 4. cosh θ = 1 1 v = E p m cosh θ = Ep + m m, sinh θ = Ep m m = p Ep + m E p + m m R 1 = cosh θ + α n sinh θ = Ep + m m ψ L x = Ep + m m 0 ψx 4.8, 4.9 p = p α E p + m 1 + p α ψax E p + m ψx = e iet w ± E = m : w + 0 = χ 0, E = m : w 0 = 0 χ

37 x µ a µ νx ν 4.3 t = x 0 a 0 µx µ ψ ± L x = Ep + m 1 + p α w ± 0 exp i m a 0 m E p + m µx µ x m a 0 µx µ = m 1 v x0 mv 1 v x1 = E p t p x x, m a 0 µx µ = E p t p x = p x, p 0 = E p ψ ± L x = e ip x Ep + m 1 + p α w ± 0 = m E p + m e ip x up, s, + e ip x vp, s, , ψ 1 x ψ x, ψ γ µ i γ ν ν m ψ 1 = 0, i γ ν ν m ψ = 0 ψ γ µ ψ 1 = i m ψ γ µ γ ν ν ψ γ 0 0 = i ν ψ γν m ψ γ 0 = i ν ψ γ ν m ψ ψ γ µ ψ 1 = i m νψ γ ν γ µ ψ σ µν = i γµ γ ν γ ν γ µ = i γ µ γ ν g µν = i g µν γ ν γ µ 4.5, 4.6 ψ γ µ ψ 1 = i m ψ µ ψ m ψ σ µν ν ψ 1, ψ γ µ ψ 1 = i m µ ψ ψ m νψ σ µν ψ 1 ψ γ µ ψ 1 = i ψ m µ ψ 1 µ ψ ψ m ν ψ σ µν ψ Gordon decomposition ψ γ µ ψ 1 1

38 4 36 ψ 1, ψ ψ 1 x = e ip x up, s, ψ x = e ip x up, s up, s γ µ up, s = 1 m up, s p + p µ + i σ µν p p ν up, s 4.8 vp, s γ µ vp, s = 1 m vp, s p + p µ + i σ µν p p ν vp, s 4.9 vp, s γ µ up, s = 1 m vp, s p p µ i σ µν p + p ν up, s i / m ψ = 0,, ψx = ψ + x + ψ x ψ + x = s ψ x = s d 3 p m π 3 ap, s up, se ip x E p d 3 p m π 3 b p, s vp, se ip x E p ap, s, b p, s ψ ± x J µ t = d 3 x ψxγ µ ψx = J +t µ + J t µ + K µ t J ±t µ = d 3 x ψ ± xγ µ ψ ± x K µ t = d 3 x = ψ + xγ µ ψ x + ψ xγ µ ψ + x d 3 x ψ xγ µ ψ x + + ψ xγ µ ψ + x = Re d 3 x ψ xγ µ ψ + x Re J µ +t = ss d 3 p d 3 p π 6 m a p, s ap, s up, s γ µ up, s e iep E p t d 3 x e ip p x Ep E p = ss d 3 p π 3 m E p a p, s ap, s up, s γ µ up, s

39 , p = p σ µν 0 J µ +t = ss d 3 p π 3 m E p a p, s ap, s pµ m up, s up, s = s d 3 p pµ ap, s π 3 E p J µ t = s d 3 p pµ bp, s π 3 E p d 3 x ψ xγ µ ψ + x = ss d 3 p π 3 m E p ap, sb p, s v p, s γ µ up, s e ie pt = d 3 p ap, sb p, s π 3 E p ss v p, s p p µ i σ µν p + p ν up, s e iept p µ = E p, p µ = 0 µ = k p p 0 i σ 0ν p + p ν = E p E p i σ 0k p k + p k = 0 p p k i σ kν p + p ν = p k i σ k0 p 0 + p 0 = p k ie p σ k0 = p k E p α k K 0 = 0, K = Re d 3 p p π 3 ap, sb p, s v p, s α up, s e ie pt E p ss J 0 t = d 3 x ψ xψx = s Jt = s d 3 p π 3 ap, s + bp, s d 3 p π 3 ap, s + bp, s p + Kt E p J 0 t = 1 J 0 t = d 3 x ψ xψx = s d 3 p π 3 ap, s = 1, Jt = s d 3 p π 3 ap, s p E p J p/e p,,, Kt e ie pt, π E p < π m 10 1 sec Zitterbewegung Kt, t = 0 d ψ0, x = 1 /d πd e x w, w = 3/4 χ 0

40 4 38 χ x ψx a, b ψ0, x d 3 x π 3 eip p x = e ie p E p t d 3 x d 3 x π 3 eip+p x = e iept δp + p d 3 x e ip x u p, sψx = s u, v 4.13 t = 0 ap, s = = b p, s = = m d 3 p E p π 3 e ip p x = e ie p E p t δp p = δp p d 3 x π 3 ap, s u p, sup, s e ip p x + b p, s u p, svp, s e ip+p x m ap, s u p, sup, s + b p, s u p, sv p, s e iept E p s m m E p E p d 3 x e ip x u p, sψx = ap, s 4.31 d 3 x e ip x v p, sψx = b p, s 4.3 m 1 E p πd 3/4 u p, sw d 3 x e x /d ip x m 4πd 3/4 e p d / u p, sw = E p m 4πd 3/4 e p d / v p, sw = E p b p, s ap, s p E p + m E p + m 4πd 3/4 e p d / χ E sχ p E p + m 4πd 3/4 e p d / χ s E p σ p E p + m χ p m bp, s ap, s, e p d / a, b p p 1/d, m 1/d, d 1/m /mc m,,, m, m,,, hole theory E = E p > 0, p, s ψ + ps t, x = e ip x up, s

41 E = E p < 0, p, s ψ ps t, x = e ip x vp, s,,,,,, Dirac sea 1,,,,,,,, +,,, ψ ps hole e > 0, E p = E p > 0, p = p, s = s, anti-particle ψ ps E p > 0, p, s, mc,, 1 1, pair creation,, pair annihilation, 1,, 1,,,,,,

42 α x p x + mβ + V x E ψx = 0, α x p x + mβ + V x E = m + V x E σ x p x p x = i d dx σ x p x m + V x E ψx = ϕ u x ϕ d x dϕ d m + V x E ϕ u x iσ x dx = 0, 1 ϕ d x = d 1 dx m V x + E iσ dϕ u x dx m V x + E ϕ d x = 0 iσ x dϕ u m V x + E dx dϕ u dx m + V x E ϕ u x = 0, ϕ u x x a ϕ u x = F x χ u χ u =, a, b = b d 1 dx m V x + E ϕ d x = Gx χ d, Gx = V x df dx m + V x E F x = d F dx + E V m F = 0 i df m V x + E dx, χ d = σ x χ u V x = { 0, x < 0 V 0, x > 0, V 0 x < 0 k = E m d F dx = k F, Gx = i df m + E dx

43 A, B F x = Ae ikx + Be ikx, Gx = x > 0 K = E V 0 m F x = Ce ikx + De ikx, Gx = k Ae ikx Be ikx m + E K Ce ikx De ikx m + E V 0 E > m k, x < 0 x > 0 K x e ikx, K K K = ik e ±ikx = e K x, x F, G e ikx = e K x D = 0 x = 0 F x Gx A + B = C, k K A B = C m + E m + E V 0 B A = 1 f 1 + f, C A = 1 + f, f = K k ψ α x ψ = F χ u G χ d χ d = σ x χ u, χ uχ u = 1 0 σ x σ x 0 F χ u Gχ d ψ α x ψ = F G + G F = Re F G J i F = Ae ikx, G = kae ikx /m + E J i = m + E m + E V = F Gχ uσ x χ d + G F χ d σ xχ u k m + E A 5.4 J r K J r = k m + E B F x = Ce ikx = C e ik x, Gx = F G = K C e ik K x m + E V 0 K m + E V 0 Ce ikx K e ik K x = e ikx F G, J t J t = k m + E f C, K 0, K 5.5

44 5 1 4, R T R = J r J i = B A = 1 f 1 + f, T = J t J i = f 1 + f, K 0, K 5.6, E m V 0 E > V 0 + m, K f 0 < R < 1 1 f 4f R = = f 1 + f = 1 T R + T = 1 E = m + ε, m ε > V 0 E + me V 0 m f = E me V 0 + m = m + εε V 0 ε V0 εm + ε V 0 ε R = ε ε V0 ε + ε V0,, E < V 0 + m, E m, R = 1, T = 0, K f, 5.6 R = 1, T = 0 K E V 0 m < 0, V 0 m < E < V 0 + m V 0 m < m, E > m m < E < V 0 + m, V 0 > m, m < V 0 m < E < V 0 + m R = 1, T = 0, m < E < V 0 m K, m+e V 0 < 0 f, J t < 0,, R > 1 E K E ± = V 0 ± K + m +, V 0 > m, m < E < V 0 m x < 0 m < k + m = V 0 K + m < V 0 m m V 0 + m V 0 V 0 m E + E,, x < 0 x > 0 0 m E d 1 dx D x df 1 dx D +xf 1 x = 0, d 1 dx D x df dx D +xf x = 0

45 D ± x = m ± V x E d F df 1 dx D dx = 1 df df 1 D dx dx + F d 1 df 1 dx D dx = 1 df D dx df 1 dx + D +F 1 F 1 d dx F df 1 D dx F 1 df D dx = 0, F D df 1 dx F 1 D df dx = x F 0 = 0 df 1 F dx F df 1 dx = 0 F F 1,, 1 V x = V x 5.1 F x F x F x = cf x F x = cf x = c F x c = ± 1, V x = V x, F x 5. F x Gx F x Gx 5. ϕ d x = i Gx χ d Gx = 1 df m V x + E dx, χ d = σ x χ u F x, Gx 0, x > a V x = V 0, x < a, V 0 > V x = V x, F x, Gx, x 0 x < a 5.1 d F dx + E + V 0 m F = 0 F x = A sinkx + B coskx, Gx = K A coskx B sinkx E + V 0 + m K = E + V 0 m x > a V 0 = 0 F x = Ce qx a + De qx a, Gx = q Ce qx a De qx a m + E q = m E x F 0 q D = 0 F x = Ce qx a, Gx = q m + E Ce qx a, E < m

46 x = a F x Gx K A sinka + B coska = C, A coska B sinka = q m + E + V 0 m + E C, A = 0 B coska = C, K q B sinka = m + E + V 0 m + E C tanka = q K m + E + V 0 m + E = m EE + V 0 + m m + EE + V 0 m 5.8 B = 0 cotka = q K m + E + V 0 m + E = m EE + V 0 + m m + EE + V 0 m 5.9 K K K = ik m < E < m K tanka = K 1 ek a 1 + e K a 1 + ek a < 0, K cotka = K 1 e K a < 0 q m + E + V 0 m + E 5.8, 5.9 K > 0 E + V 0 m > 0, E > m V 0 E < m V 0 E > m, m < E < m E > m V 0 0 < V 0 < m m V 0 > m m V 0 < E < m,, V 0 > m m V 0 < m m < E < m m V 0 < E < m, E K m V 0 m 0 m E E 1 E ± = V 0 ± K + m V 0 x < a E + V 0 > m m V 0 E < m E < m V 0 + K + m = m + k k, x > a V 0 > m E > m k E = E, E = E 1

47 , 5.9, E V 0 5.8, 5.9 E V 0,, V 0,, V 0 x < a, E m = V 0 + m nπ, n = 1,, 3, ma, E < m m 0 1 V 0 /m m mr = 5.0, V min < V 0 < V max, V min 5.8 E = m nπ tan a Vmin + mv min = 0, V min = m m + m = a V max E = m tan a Vmax mv max = + n + 1 V max = m + a π + m = m n ma π +, n + 1 V min = m + a n + 1 V max = m + a n = 0, 1,, E = m ε 0 < ε < V 0 m π m = 1 n + 1 m ma π + π + m = m n ma π + K = V 0 εm + V 0 ε mv 0 ε m EE + V 0 + m m + EE + V 0 m = εm + V 0 ε m εv 0 ε ε V 0 ε nπ + ma 5.8, 5.9 β = α tan α, β = α cot α

48 α = ma V 0 ε, β = ma ε, α + β = ma V 0, K B coskx E + V F x = Gx = 0 + m B sinkx Ce qx a q m + E Ce qx a 0 0 dx dx F = B dx G = ab cos Ka = F + G = ab tan Ka = m + EE + V 0 m mv 0 sinka coska a + + cos Ka = ab m + E K q ma m E E + V 0 m m E E + V 0 + m ma m + E m + E ma m E + E + V 0 m E + V 0 + m E m E = m + ε α a B, 1 + α m/v 0 α = ma ε m E = 1 ma m + E tanka + coska 0, coska 1 n α V0 m, Ka n + 1 π, n = 0, 1,, ma V 0 C B = coska α 1n ma V0 m V 0, K E + V 0 + m B V0 m B V 0 q m + E C = K V0 m B sinka 1n B E + V 0 + m V 0 ε 0 B, C 0, C/B α ε C B 0 E m B coskx F x 0 V0 m sinkx, Gx B V 0 1 n+1, x < a, x > a F x x < a, Gx

49 K A sinkx E + V F x = Gx = 0 + m A coskx Ce qx a q m + E Ce qx a 0 0 sin Ka = m + EE + V 0 m mv 0 dx F = aa dx G = aa m + E ma m E E + V 0 m m E E + V 0 + m ma m + E, E m A sinkx V0 m coskx, x < a F x Gx A 0 V 0 1 n+1, x > a F x, Gx a F x, a Gx n = 1, n = 0 V 0 /m =.363 E/m = V 0 /m =.1701 E/m = x/a x/a

50 ,,,, 1 V r, r = x,.13 qφ = V, A = 0 H ψx = E ψx, H = i α + βm + V r H J = L + Σ/ P = γ 0 P 0 L = i x mβ, V r L x = ix 3 + ix 3 [ H, L ] = i [ α, L ] = i α k [ k, L ], k = x k [ k, L x ] = i [ k, x ] 3 + i [ k, x 3 ] = i δ k 3 + i δ k3 [ H, L x ] = α 3 + α 3 = α x [ H, L ] = α 6.1 Σ x = iγ γ 3 = i γ 0 γ γ 0 γ 3 = i α α 3 Σ x mβ, V r [ H, Σ x ] = [ α k, α α 3 ] k α k α l = δ kl α l α k α k α α 3 = δ k α α k α 3 = δ k α 3 α δ k3 α 3 α k = δ k α 3 δ k3 α + α α 3 α k [ H, Σ x ] = α 3 α 3 = α x [ H, Σ ] = α 6. [ H, J ] = 0 H L, Σ J = L + Σ/ P 0 ψx = ψ x, P 0 V rψx = V rψ x HP ψx = i α + βm + V r γ 0 ψ x = γ 0 i α + βm + V r ψ x P Hψx = γ 0 P 0 i α + βm + V r ψx = γ 0 i α + βm + V r ψ x HP P H ψx = 0 H P

51 6 49 H, J, J z, P ψx ϕ u, ϕ d ϕ u x ψx = ϕ d x 6.3 j 0 J =, j = L + σ/, P = γ 0 P 0 = 0 j P P 0 J ψ = jj + 1ψ, J z ψ = m 3 ψ, P ψ = η ψ, η = ± 1 j ϕ u = jj + 1 ϕ u, j z ϕ u = m 3 ϕ u, P 0 ϕ u = η ϕ u j ϕ d = jj + 1 ϕ d, j z ϕ d = m 3 ϕ d, P 0 ϕ d = η ϕ d ϕ u ϕ d j, j z, L Y ljm3 θ, φ : Y ljm3 j Y ljm3 = jj + 1Y ljm3, j z Y ljm3 = m 3 Y ljm3, L Y ljm3 = ll + 1Y ljm3 Y lm j, l l ± = j ± 1/ P 0 Y ljm3 = 1 l Y ljm3 ϕ u x = F + ry l+jm 3 + F ry l jm 3 P 0 ϕ u = 1 l+ F + ry l+ jm l F ry l jm 3 = 1 l+ F + ry l+ jm 3 F ry l jm 3 ϕ u P 0 ϕ u x = F + ry l+jm 3, ϕ u x = F ry l jm 3 ϕ d ϕ u ϕ u x ψx =, ϕ u x = F r Y ljm3, ϕ d x = i Gr Y l jm ϕ d x r r 3 6.4, 1/r i j 1/, l = j + 1/ l 1, l = l = j + 1/, l = j 1/ l + 1, j = l 1/ j = l + 1/ 6.5, l = 0 j = l 1/ F r, Gr ψx J, J z, P ϕ u ϕ d L, ϕ u ϕ d l, ψx L, H L, H L H F r, Gr

52 6 50 H = i α + βm + V r = m + V r i σ i σ m + V r Hψ = E ψ m + V r E ϕ u i σ ϕ d = i σ ϕ u m V r + E ϕ d = σ x σ = x + i σ x = r r σ L = r r + K, K = 1 σ L K j = L + σ/ σ x σ x = r j = L + σ 4 + σ L = L σ L, K = L j 1 4 σ ϕ u x = σ x r σ x σ ϕ ux = σ x r r r + K ϕ u x σ ϕ u x = σ x r KY ljm3 = ϕ u x = F r Y ljm3 θ, φ r df dr Y ljm 3 + F r KY ljm 3 L j 1 Y ljm3 = κ Y ljm3 4 κ = ll + 1 jj = l + 1, l, = 1 j+l+1/ j + 1 j = l + 1 j = l σ ϕ u x = 1 r df dr + κ σ x r F r r Y ljm i df r dr + κ σ x r F r r Y ljm 3 m V r + E ϕ d = 0 σ x Y ljm3 /r r ϕ d x = i Gr σ x r r Y ljm 3

53 6 51 df dr + κ r F r m V r + E Gr = 0, 6.6 F r, Gr m + V r E F r r Y ljm3 σ σ x Gr r Y ljm 3 = 0 σ σ x = x + i σ x = 3 + r r + σ L = r r + 1 K σ σ x Gr d r Y G ljm 3 = dr r + 1 κ r G Y ljm3 = 1 dg r dr κ r G Y ljm3 dg dr κ r Gr m + V r E F r = 0, F r Y ljm3 θ, φ ψx = r i Gr σ x 6.9 r r Y ljm 3 θ, φ, F r Gr df dr = κ r F r + m + E V r Gr 6.10 dg dr = + κ r Gr + m E + V r F r 6.11 σ x r Y ljm 3 = Y l jm 3, E = m + ε 6.10 Gr = 1 d m + ε V r dr + κ F r r 6.11 d dr κ 1 d r m + ε V r dr + κ F r = ε + V r F r r m ε V r 1 m d dr κ d r dr + κ F r ε + V r F r r d dr κ d r dr + κ d κκ + 1 F r = r dr r F r 1 m d κκ dr mr + V r F r εf r

54 6 5 j = l ± 1/ κκ + 1 = ll m d ll dr mr + V r F r εf r, l j V r, j, 6.10, 6.11 κ, j = l + 1/ j = l 1/ , c Y = σ x r Y ljm3 j Y = jj + 1Y, j z Y = m 3 Y, P 0 Y = 1 l+1 Y, N Y = NY l jm 3 N = 1 π dθ π 0 0 dφ sin θ Y ljm 3 Y ljm3 = 1 Y ljm3 N = d 3 x ψ xψx = 0 dr F r + G r 6. V r = Gr = 1 E + m df dr + κ r F r 6.11 d κκ + 1 dr r + E m F r = κκ + 1 = ll + 1 q = E m d ll + 1 dr r + q F r = 0 rj l qr rn l qr, F r = Nrj l qr, N = G = 1 df E + m dr + κ r F = N κ + 1 j l qr + qr j E + m lqr, j lx = dj l dx

55 6 53 j = l 1/ 6.5, 6.8 l = l 1, κ = l j l G = j = l + 1/ l = l + 1, κ = l 1 j l 1 ρ = j lρ + l + 1 ρ j lρ Nq E + m rj l 1qr = Nq E + m rj l qr j l+1 ρ = j lρ + l ρ j lρ G = Nq E + m rj l qr j = l 1/ F r Y ljm3 ψx = r j l qry ljm3 igr = N ± i q Y l r jm 3 E + m j l qr Y l jm 3 z z sin z lπ/ j l z z E < m q j l qr r E m E m E = m + q m q q E + m = q 1 q m 4m + ψx N j l qry ljm3 ± i q m j l qr Y l jm 3 N j l qry ljm E m, q m ψx 6.3 Ze, V r = Ze 4πr = Zα r, α = e 4π = , 6.11 df dr = κ r F + m + E + Zα G, r dg dr = κ r G + m E Zα F r

56 6 54 r m E > 0 df dr = m + E G, d F dr dg dr = m E F = m + EdG dr = m E F F r, Gr e m E r, r, r 0 df dr + κ r F Zα r G = 0, dg dr κ r G + Zα r F = 0 F = c f r λ, G = c g r λ, λ > 0 λ κ c g + Zα c f = 0, λ + κ c f Zα c g = 0, λ = κ Zα Z < 137 Zα Z/137 < 1, κ 0 λ Z > 137 λ κ λ = i λ r λ = e i λ log r = cos λ log r + i sin λ log r r 0 1, λ κ, m Z < 137 dg dρ κ ν ρ G Zα F = 0, ρ ρ dg λ dρ + κ ρ g + ν = ρ = m E r m E m E m E = m + E F = ρ λ e ρ/ f, G = Zα ν ρ f = 0, f 1 = f + g f = f g df dρ + κ 1 ρ F ν + Zα G = 0 ρ m E m + E ρλ e ρ/ g ρ df λ dρ + + κ ρ f Zαν + ρ g = 0 ρ df 1 dρ + λ + ν 1 ρ f 1 + κ + ν f = 0, ρ df dρ + λ ν 1 f + κ ν f 1 = ν 1 = Zα 1 ν ν ZαE = m E, ν = Zα 1 ν + ν Zαm = m E

57 ρ d f dρ + df λ + 1 ρ dρ f 1 = 1 ρ df ν κ dρ + λ ν 1 f λ ν 1 λ κ + ν ν 1 ρ, ν ν 1 = Zα, λ κ = Zα ρ d f dρ + df λ + 1 ρ dρ λ ν 1 f = 0 f = 0, f f ρ d f 1 dρ + df1 λ + 1 ρ dρ λ ν f 1 = 0 z d dz + b z d dz a wz = 0, Ma, b, z Ma, b, z = 1 + a b aa + 1 z z + bb + 1! aa + 1a + z 3 + bb + 1b + 3! + f 1 ρ = C 1 Mλ ν 1 + 1, λ + 1, ρ, f ρ = C Mλ ν 1, λ + 1, ρ 6.1 ρ = 0 λ + ν 1 C 1 + κ + ν C = 0, C 1 = κ + ν λ + ν 1 C 6.13 Ma, b, z a = n, n = 0, 1,, n, a n z Ma, b, z e z r F, G 0 λ ν 1 = λ ZαE m E = n r, n r = 0, 1,, 3, E n r + λ E = m nr + λ + Zα, λ = κ Zα 6.14, λ ν 1 = n r = 0, f 1 ρ = C 1 M 1, λ + 1, ρ C 1 = 0 ν 1 = λ ν = Zα + ν1 = Zα + λ = κ, C 1 = κ + ν C = κ + κ C λ + ν 1 λ, n r = 0 κ < 0 n r + λ E, n r = 0, κ = 1 E = m 1 Zα = m 1 Zα Zα4 8

58 6 56 mzα /, n = n r + κ = n r + j + 1, δ j = κ κ Zα = Zα j Zα4 j Zα 1 1/ E = m 1 + Zα n δ j = m 1 1 Zα n δ j + 3 Zα 4 8 n δ j 4 + = m 1 1 Zα n 1 + δ j + 3 Zα 4 n 8 n 4 + = E NR mzα4 n 4 n j , E NR = m mzα n 6.16 Zα 4 E NR E NR n, j n, E j fine structure nl j n r = 0, κ < 0 n = n r + l + 1, n r = 0, 1,, κ < 0 κ = l , κ > 0 κ = l 6.15 n n = n r + l n = l, κ > 0 n r = 0 n n r κ l j / 1S 1/ / 1P 1/ / S 1/ } / P 1/ 0 1 3/ P 3/ 0 3/ D 3/ n n r κ l j / 3S 1/ } 1 1 1/ 3P 1/ 1 1 3/ 3P 3/ } 1 3/ 3D 3/ 0 3 5/ 3D 5/ / 3F 5/, 6.14 i f E = E i E f 10 5 ev, E NR ev i f 6.14 E NR S 1/ P 1/ P 3/ S 1/ S 1/ 3P 1/ P 3/ 3S 1/ D 3/ 3P 1/ D 5/ 3D 3/ Phys. Rev. Lett , Phys. Rev. Lett

59 6 57,, S 1/ P 1/, 3S 1/ 3P 1/, 3P 3/ 3D 3/ 6.14, Lamb shift, n r = 0, κ = 1 F = ρ λ e ρ/ f 1 + f, G = m E m + E ρλ e ρ/ f 1 f E = m 1 Zα, n r = 0 f 1 = 0, M0, λ + 1, ρ = 1 F = Cρ λ e ρ/, λ = 1 Zα, G = C 1 λ Zα ρλ e ρ/ ρ = m E r = Zαm r = Z a B r a B = 1/αm 6.9 F r Y ljm3 θ, φ ψx = r λ 1 Zr i Gr σ x = C e Zr/a B r r Y a B ljm 3 θ, φ χ ± i 1 λ σ x Zα r χ ± l = 0, j = 1/ Y ljm3 χ ± σ z χ ± = ± χ ± Zα 1 λ 1 = Zα / 0 ψx C χ e Zr/aB ± 0,, 1 < λ 1 < 0 ψ Zr a B λ 1 e, Zr e 1/λ 1 e /Zα = /Z a B,,,

60 ,, L q i p i = L/ q i, q i p i [ q i, q j ] = [ p i, p j ] = 0, [ q i, p j ] = i δ ij, = 1, q, φt, x,, x i, x i + dx q i t = φt, x i d 3 x 1,, q i L/ q i L L = d 3 x L, L φx = φt, x µ φx, φ I = t t 1 dt L = t t 1 dt d 3 x L φx φx + εx, εt 1, x = εt, x = 0 δi = 0 δi = t t 1 Lφ + ε, µ φ + ε = Lφ, µ φ + L φ ε + L µ φ µ ε L dt d 3 x φ ε + L µ φ µ ε = t, µ t 1 L µ φ = L φ L dt d 3 x ε φ L µ µ = 0 φ φx L L L = i Lφ i, µ φ i d 3 x, φ i = φt, x i q i = φ i d 3 x p i H H = p i q i L = i i p i = L q i = L φ i p i φi d 3 x L = d 3 x H, H = px φx L

61 7 59 [ q i, p j ] = [ φ i, p j ] d 3 x = i δ ij dx 0 δ ij /d 3 x δx i x j [ φt, x, φt, x ] = [ pt, x, pt, x ] = 0, [ φt, x, pt, x ] = i δx x 7.1 px = L φx, φx 7.1, { φt, x, φt, x } = { pt, x, pt, x } = 0, {φt, x, pt, x } = i δx x 7. { A, B } AB + BA 7., 0, 0, m 0 φx Klein-Gordon µ µ + m 0 φx = L L = 1 µ φ µ φ m 0φ = 1 φ φ m 0φ L µ φ = µφ, L φ = m 0φ, 7.3 px px = L = φ φ, [ ] [ φt, x, φt, x ] = φt, x, φt, x = 0, [ ] φt, x, φt, x = i δx x 7.4, e ip x 7.3 H = p φ L = 1 φ + φ + m 0φ µ µ + m 0 e ip x = p + m 0 e ip x

62 7 60 p 0 = ± ω p, ω p = p + m 0 e ip x 7.3 p 0 = ω p > 0 e ± ip x 7.3, 7.3 d 3 p 1 φx = c π 3/ p e ip x + c p e ip x 7.5 ωp φx = i d 3 p π 3/ ωp c p e ip x + c p e ip x 7.6, d 3 x e iq x φ0, x = 1 c π 3/ q + c d 3 x q, e iq x ωq φ0, x = i c ωq π 3/ q + c q [ cp, c ] 1 q = ω p ω q = ω p + ω q ω p ω q = ω p + ω q ω p ω q c q = d 3 x d 3 x π 3 d 3 x d 3 x π 3 d 3 x e iq x ω π 3/ q φ0, x + i φ0, x ωq [ e ip x+iq x ω p φ0, x + i φ0, x, ω q φ0, x i φ0, x ] e ip x+iq x δx x d 3 x π 3 e ip q x = δp q [ cp, c ] q = δp q, [ cp, c q ] = [ c p, c ] q = H c, c d 3 x φx ωp = d 3 p d 3 ω q q = 1 d 3 x π 3 c p e ip x c p e ip x c q e iq x c q e iq x d 3 p ω p c pc p + c p c p c p c p e iωpt c pc pe iωpt d 3 x φ = d 3 p d 3 q = 1 p q ω p ω q d 3 x π 3 c p e ip x c p e ip x c q e iq x c q e iq x d 3 p p ω p c pc p + c p c p + c p c p e iωpt + c pc pe iωpt d 3 x φ = d 3 p d 3 1 q ω p ω q = 1 d 3 x π 3 c p e ip x + c p e ip x c q e iq x + c q e iq x d 3 p 1 ω p c pc p + c p c p + c p c p e iω pt + c pc pe iω pt H = 1 d 3 x φ + φ + m 0φ = 1 d 3 p ω p c pc p + c p c p = d 3 p ω p c pc p + 1 δp p

63 H = c p 0 = 0 d 3 p ω p c pc p + 0 H 0 H,, 0 H 0 H c p, c p H = d 3 p ω p c pc p [ H, c p ] = d 3 p ω p c p [ c p, c p ] = ω p c p, [ H, c p ] = ω p c p 7.8, φt, x H 7.5 [ H, φt, x ] = α b d 3 p 1 ω π 3/ p c p e ip x + ω p c p e ip x = i φt, x ωp exp ib P α P exp ib P α φt, x, α φt, x b α expib P φt, x exp ib P α = α φt, x b α b expib P φt, x exp ib P = φt, x b i [ b P, φt, x ] = b φt, x [ P, φt, x ] = i φt, x, [ P, pt, x ] = i pt, x P = d 3 x px φx P c, c P = 1 d 3 p p c pc p + δp p + c p c p e iωpt + c pc pe iω pt p p, 1 0 P = d 3 p p c pc p α H P H α = E α α, P α = p α α

64 [ H, c q ] = ωq c q, [ H, c q ] = ω q c q, [ P, c q ] = q c q, [ P, c q ] = q c q Hc q α = [ H, c q ] + c q H α = E α + ω q c q α, P c q α = p α + q c q α Hc q α = E α ω q c q α, P c q α = p α q c q α c q ω q = q + m 0 q, c q m 0 q, c q q 0 c q 0 1 time-ordered product T T φx φx θt t φx φx + θt tφx φx 7.9 i F x x = 0 T φx φx 0 = θt t 0 φx φx 0 + θt t 0 φx φx F x x F x x 7.5 d 0 φx φx 3 p d 3 p 1 0 = π 3 0 c p e ip x + c p e ip x c p e ip x + c eip x p 0 ω p ω p d 3 p d 3 p = π 3 e ip x+ip x 0 c p c p ω p ω 0 = p d 3 p e p x x π 3 ω p i F x x = i F x = d 3 p π 3 1 ω p θt t e ip x x + θt t e ip x x d 3 p 1 π 3 θt e ip x + θ t e ip x 7.11 ω p R iε R R iε R t > 0 t < 0

65 7 63 θt = dk, ε πi k iε e ikt t > 0, k, R k R k = R e iϕ 0 ϕ π C + 1/k iε k = iε dk e ikt C + πi k iε = eikt k=iε = e εt 1, ε +0 k = R e iϕ e ikt = e tr sin ϕ e itr cos ϕ tr sin ϕ = e t > 0 sin ϕ > 0 e ikt 0, R, R 0 t > 0 e ikt dk πi k iε = 1 t < 0 C k = iε C e ikt dk C πi k iε = 0, e ikt t < 0 sin ϕ < 0 R 0, t < F x = 1 dk πi k iε = 0 e ikt d 3 p dk 1 π 4 exp iω p kt + ip x + exp iω p kt ip x ω p k iε 1 q 0 = ω p k, q = p, q 0 = ω p + k, q = p, k, p q 0, q F x = 1 d 4 q 1 π 4 ω q 1 ω q q 0 iε + 1 ω q + q 0 iε e iq x = d 4 q π 4 e iq x q 0 ω q iε q 0 ω q iε = q 0 q m 0 + i ω q ε = q m 0 + i ω q ε, q = q µ q µ = q 0 q, ω q > 0 ω q ε ε F x = d 4 q π 4 Fq e iq x, F q = q 0 q 0 = ω q F q m 0 + iε 7.13 t θt t = δt t, t θt t = δt t

66 7 64 i t Fx x = δt t 0 [ φt, x, φt, x ] T φx φx i t Fx x = δt t 0 [ φt, x, φt, x ] T φx φx 0 = iδt t δx x + 0 T φx φx 0 + m 0 F x x = δ 4 x x i 0 T + m 0 φx φx 0 = δ 4 x x 7.14 F x x m d 0 F x x 4 q m 0 q = e iq x x π 4 q m 0 + iε 1 x + iε = P 1 x iπδx, x P 1 x = 1, x δx = 0 + m d 0 F x x 4 q = e iq x x π 4 = δ 4 x x 7.3 L L = ψx i/ m ψx ψ 4 4 ψ α x, ψ α x,,, ψ ψ ψ ψ L = i/ m ψx ψ, α α, µ L µ ψ α = L ψ α L µ ψ α = 0 i/ m ψx = 0, ψ L ψ α = m ψ α, L µ ψ α = i ψγ µ α µ L µ ψ α L = i µ ψγ µ + mψ ψ α = 0 α

67 7 65 γ 0 i/ m ψx = 0, ψ α, p α = L ψ α = i ψγ 0 α = iψ α {ψ αt, x, ψ β t, x } = δ αβ δx x, {ψ α t, x, ψ β t, x } = ψx,, second quantization,, ψx ψx = s d 3 p m ap, s up, se ip x + cp, s vp, se ip x π 3/ E p p 0 = E p = p + m, ap, s, cp, s ψx ψx 4.31, 4.3 m d 3 x ap, s = eip x u p, sψx π 3/ E p m d 3 x cp, s = e ip x v p, sψx π 3/ E p, 7.15 a, a, c, c t = t { ap, s, a p, s } m d 3 x d 3 x = e ip x+ip x Ep E p π 3 e ip 0 p 0 t { } u αp, s u β p, s ψ α t, x, ψ β t, x = αβ m u p, sup, s e ip 0 p 0 d t 3 x Ep E p π 3 e ip p x = m E p u p, sup, s δp p 4.13 { ap, s, a p, s } = δ ss δp p { cp, s, c p, s } = δ ss δp p 0, { a p, s, a p, s } = 0, a p, s = 0, 1

68 7 66 H H = p ψ L = iψ 0 ψ ψ γ 0 iγ iγ m ψ = ψ iγ + m ψ H H = d 3 x ψx iγ + m ψx = d 3 x ψ x iα + mγ 0 ψx iα + mγ 0, H a, c iγ + m ψx = s d 3 p π 3/ m E p ap, s e ip x γ p + m up, s + cp, s e ip x γ p + m vp, s 4.10 γ p + m up, s = E p γ 0 up, s, γ p + m vp, s = E p γ 0 vp, s iγ + m ψx = s d 3 p mep γ 0 ap, s e ip x up, s cp, s e ip x vp, s π 3/ H = m ss d 3 p d 3 p E p E p d 3 x π 3 a p, s u p, s e ip x + c p, s v p, s e ip x ap, s up, s e ip x cp, s vp, s e ip x = m ss d 3 p u p, s up, s a p, s ap, s + v p, s up, s c p, s ap, s e ip0 t u p, s vp, s a p, s cp, s e ip0t v p, s vp, s c p, s cp, s u, v 4.13 H = s d 3 p E p a p, sap, s c p, scp, s ap, s 0 = 0, cp, s 0 = 0 a p, s ap, s a p, s = a p, s δ ss δp p a p, sap, s Ha p, s 0 = E p a p, s 0 Hc p, s 0 = E p c p, s 0

69 7 67 a, c, a E p, c E p 0,, c p, s 0 = 0,,, cp, s 0 b p, s = cp, s, ap, s 0 = 0, bp, s 0 = ap, s, bp, s ψx ψx = d 3 p m ap, s up, se ip x + b p, s vp, se ip x 7.18 π 3/ E s p, { ap, s, a p, s } = { bp, s, b p, s } = δ ss δp p, = c b, c b H = d 3 p E p a p, sap, s bp, sb p, s s = s d 3 p E p a p, sap, s + b p, sbp, s + 0 H 0 0 H 0 = s d 3 p E p δp p Ha p, s 0 = E p + 0 H 0 a p, s 0, Hb p, s 0 = E p + 0 H 0 b p, s 0, a, b H 0 H 0, 0 H 0,, 0 H 0 H = d 3 p E p a p, sap, s + b p, sbp, s s, µ j µ = 0, j µ x = ψxγ µ ψx,,, Q q Q = q d 3 x ψxγ 0 ψx

70 7 68 a, b Q = q d 3 p d 3 p m d 3 x Ep E ss p π 3 a p, s u p, se ip x + bp, s v p, se ip x ap, s up, s e ip x + b p, s vp, s e ip x = q s = q s d 3 p a p, sap, s + bp, sb p, s d 3 p a p, sap, s b p, sbp, s + 0 Q 0 0 Q 0 = q s d 3 p δp p H Q = q d 3 p a p, sap, s b p, sbp, s s Q a p, s 0 = q a p, s 0, Q b p, s 0 = q b p, s 0 q ,, ψx, T ψ α x ψ β x θt t ψ α x ψ β x θt tψ β x ψ α x S F i S F x x 0 T ψ α xψ β x 0 αβ i S F x x = d 3 p d 3 p m αβ π 3 Ep E ss p θt t 0 ap, s a p, s 0 u α p, su β p, s e ip x+ip x θt t 0 bp, s b p, s 0 v α p, sv β p, s ip x ip x e t < t, t t, t < t, t t 0 ap, s a p, s 0 = 0 δ ss δp p a p, s ap, s 0 = δ ss δp p

71 7 69 i S F x x = αβ s d 3 p m π 3 θt t u α p, su β p, s e ip x x E p θt t v α p, sv β p, s e ip x x = d 3 p m θt π 3 t Λ + p e ip x x + θt tλ p e ip x x E p αβ Λ ± p , 4.18 d 3 p /p + m S F x = i π 3 θte ip x /p m θ te ip x E p E p 7.1 d 3 p dk Ep γ 0 p γ + m exp ie p kt + ip x S F x = π 4 E p k iε E pγ 0 p γ m exp ie p kt ip x E p k iε F d 4 q 1 S F x = π 4 E q = = d 4 q π 4 Eq γ 0 q γ + m E q q 0 iε q 0 γ 0 q γ + m E q q 0 iε E q + q 0 iε e iq x d 4 q π 4 S Fq e iq x, S F q = E qγ 0 + q γ m E q + q 0 e iq x iε /q + m q m + iε 7.1 z /q + z/q z = q z /q + z q z = 1 /q z 4 4 /q z z = m iε S F q = /q + m q m + iε = 1 /q m + iε 7. e iq x x µ i µ e iq x = q µ e iq x d 4 q i/ + m d S F x = e iq x 4 q e iq x π 4 q m = i/ + m + iε π 4 q m + iε = i/ + m Fx i/ m i/ + m = / / m = m 7.14 i/ m S F x = + m F x = δ 4 x x S F x i/ m ψ = 0

72 S H H 0 V t t S i t t S = H t S, H = H 0 + V t S t I e ih 0t t S, V I t = e ih 0t V e ih 0t i t t I = e ih0t H 0 + i t S = e ih0t V t S = e ih0t V e ih0t t I = V I t t I 8.1 t t I,,,, H 0, V I V ϕ i ϕ i ϕ i x V = V [ϕ i x], V I t = V [ϕ i t, x], ϕ i t, x = e ih 0t ϕ i xe ih 0t 8. i t ϕ it, x = H 0 e ih 0t ϕ i xe ih 0t + e ih 0t ϕ i xe ih 0t H 0 = [ ϕ i t, x, H 0 ] 8.3 H 0, ϕ i t, x V [ϕ i ] L int V [ϕ i ] = d 3 x L int L int g 4! φ4 x, g ψxψxφx, g ψxγ µ ψxa µ x φ, ψ, A µ t S = e iht t 0 t 0 S t I = Ut, t 0 t 0 I t Ut, t 0 = i V I t Ut, t t I = e ih 0t e iht t 0 t 0 S = e ih 0t e iht t 0 e ih 0t 0 t 0 I Ut, t 0 = e ih0t e iht t0 e ih0t0 8.5

73 8 71 Ut, t Ut 0, t 0 = 1 Ut, t 0 = 1 + i T Ut, t 0 = 1 + i t t 0 dt 1 V I t 1 Ut 1, t 0 t = 1 + i dt 1 V I t 1 t 0 t1 1 + i t = 1 + i dt 1 V I t 1 + i t 0 t t 0 dt 1 V I t 1 + i t1 t t t t 0 dt 1 dt V I t Ut, t 0 t 0 t 0 dt 1 t1 t1 t 0 dt V I t 1 V I t Ut, t 0 t 0 dt V I t 1 V I t t + i 3 dt 1 dt t 0 t 0 t 0 dt 3 V I t 1 V I t V I t 3 + t t t = 1 + i dt 1 V I t 1 + i t 0 dt 1 t 0 dt θt 1 t V I t 1 V I t t 0 + i 3 t t 0 dt 1 t t 0 dt t t 0 dt 3 θt 1 t θt t 3 V I t 1 V I t V I t T V I t 1 V I t = θt 1 t V I t 1 V I t + θt t 1 V I t V I t 1 T V I t 1 V I t V I t 3 = 3 i,j,k=1 i j k θt i t j θt j t k V I t i V I t j V I t k n ϕ,, V I ϕ ϕ, V I 8.6 S Ut, t 0 = 1 + i = + i3 3! t t t dt 1 V I t 1 + i dt 1 dt T V I t 1 V I t t 0! t 0 t 0 t t dt 1 dt dt 3 T V I t 1 V I t V I t 3 + t 0 t 0 t 0 t i n n=0 n! t t 0 dt 1 t dt n T V I t 1 V I t n t 0 t = T exp i dt V I t t 0, t i t f,,, V I t e ε t, ε + 0 t e ε t = 1, t ± e ε t = 0 e ε t,, H 0 i = E i i, H 0 f = E f f

74 8 7 E i = E f t S = e ih0t t I = e ih0t Ut, t 0 t 0 I = e ih0t Ut, t 0 e ih0t0 t 0 S, t 0 i t 0 f f e ih 0t U, e ih 0t 0 i = e ie 0t t 0 f U, i f S i S = U, = T exp i dt V I t S Dyson S V I L int S = T exp i d 4 x L int x 8.7 L int ϕ i x, 8.3 ϕ i x, F, S F, H 0 H 0 0, 8.3 ϕx, H Φ i Φx = [ Φx, H ] t 0 T ΦxΦx 0 Φx = e iht Φ0, x e iht, t = 0 Φ0, x = ϕx 8. Φx = e iht e ih0t ϕxe ih0t e iht Φx Φx = e iht e ih0t ϕxe ih0t e iht t e ih0t ϕx e ih0t e iht 8.5 Ut, t = e ih0t e iht t e ih0t, U0, t = e iht e ih0t, Ut, 0 = e ih0t e iht Φx Φx = U0, tϕxut, t ϕx Ut, 0, 0 0 H 0 0 V I H 0 0 = e iθ U0, 0

75 8 73, V I 0 = e iθ U, 0 0, 0 = e iθ U 1, U = U 1 0 = e iθ 0 U, 0 1 = 0 0 = e iθ+θ 0 U, 0U0, 0 = e iθ+θ 0 S 0 0 ΦxΦx 0 = e iθ+θ 0 U, 0U0, tϕxut, t ϕx Ut, 0U0, 0 = 1 0 S 0 0 U, tϕxut, t ϕx Ut, 0 V I 1 U, t, Ut, t, Ut, i t dt 1 V I t 1 ϕxϕx + i t t dt 1 ϕxv I t 1 ϕx + i t dt 1 ϕxϕx V I t 1 t > t i dt 1 T ϕxϕx V I t 1 0 T ΦxΦx 0 = = 1 0 S 0 i n n=0 n! dt S 0 0 T ϕxϕx S 0 dt n 0 T ϕxϕx V I t 1 V I t n T Φx 1 Φx n 0 = 0 S 0 0 T ϕx 1 ϕx n S , 8.8 exp, L int,, 8. S f T L int x 1 L int x n i L int x ϕ i x f T ϕ 1 x 1 ϕ n x n i

76 8 74 Wick normal product ϕ 1 x 1 ϕ n x n,, 1 normal product : ϕ 1 x 1 ϕ n x n : ϕ i x ϕ + i x ϕ i x, φx : φx 1 φx : = : φ + x 1 + φ x 1 φ + x + φ x : ψx : ψ α x 1 ψ β x : = : = φ + x 1 φ + x + φ + x 1 φ x + φ + x φ x 1 + φ x 1 φ x 8.9 ψ + α = ψ + α x 1 ψ + ψ + β x 1 + ψ α x 1 β x + ψ α + x ψ α ψ + β x + ψ β x : x 1 ψ β x x 1 + ψ α x 1 ψ β x ψ α x 1 ψ + x 0, β ϕ i x 0 = 0, 0 ϕ + i x = 0 0 : ϕ 1 x 1 ϕ n x n : 0 = 0 [ φ x 1, φ + x ] x 1, x Cx 1, x 8.9 φx 1 φx : φx 1 φx : = [ φ x 1, φ + x ] = Cx 1, x, 0 0 φx 1 φx 0 = Cx 1, x φx 1 φx = : φx 1 φx : + 0 φx 1 φx 0 ψ α x 1 ψ β x : ψ α x 1 ψ β x : = {ψ α x 1, ψ + β x } = 0 ψ α x 1 ψ β x 0 ψ α x 1 ψ β x = : ψ α x 1 ψ β x : + 0 ψ α x 1 ψ β x 0

77 8 75 T φx 1 φx = θt 1 t φx 1 φx + θt t 1 φx φx 1 = θt 1 t : φx 1 φx : + 0 φx 1 φx 0 + θt t 1 : φx φx 1 : + 0 φx φx 1 0 : φx 1 φx : = : φx φx 1 : T φx 1 φx = : φx 1 φx : + 0 T φx 1 φx 0 = : φx 1 φx : + i F x 1 x T ψ α x 1 ψ β x = θt 1 t ψ α x 1 ψ β x θt t 1 ψ β x ψ α x 1 : ψ α x 1 ψ β x : = : ψ β x ψ α x 1 : = θt 1 t : ψ α x 1 ψ β x : θt t 1 : ψ β x ψ α x 1 : + 0 T ψ α x 1 ψ β x 0 T ψ α x 1 ψ β x = : ψ α x 1 ψ β x : + 0 T ψ α x 1 ψ β x 0 ϕ i x i ϕ i, = : ψ α x 1 ψ β x : + i S F x 1 x αβ ϕ 1 ϕ = : ϕ 1 ϕ : + ϕ 1 ϕ, T ϕ 1 ϕ = : ϕ 1 ϕ : + ϕ 1 ϕ 8.11 ϕ 1 ϕ 0 ϕ 1 ϕ 0, ϕ 1 ϕ 0 T ϕ 1 ϕ 0 contraction 8.11 n ϕ 1 ϕ ϕ n ϕ 1 ϕ ϕ n = : ϕ 1 ϕ ϕ n : + : ϕ 1 ϕ ϕ n : + : ϕ 1 ϕ ϕ 3 ϕ n : + + : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ n : + : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ n : + + : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ n : + T ϕ 1 ϕ ϕ n = : ϕ 1 ϕ ϕ n : + : ϕ 1 ϕ ϕ n : + : ϕ 1 ϕ ϕ 3 ϕ n : + + : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ n : + : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ n : + + : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ n : +

78 8 76,,, ϕ : ϕ 1 ϕ ϕ 3 ϕ 4 ϕ n := ϕ 1 ϕ 3 : ϕ ϕ 4 ϕ n := ϕ 1 ϕ 3 ϕ ϕ 4 : ϕ 5 ϕ n :, : ϕ 1 ϕ : : ϕ 3 ϕ 4 : = ϕ 1 ϕ ϕ 1 ϕ ϕ 3 ϕ 4 ϕ 3 ϕ 4 ϕ 1 ϕ ϕ 3 ϕ 4 = : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : : ϕ 1 ϕ : : ϕ 3 ϕ 4 : = : ϕ 1 ϕ ϕ 3 ϕ 4 : + ϕ 1 ϕ : ϕ 3 ϕ 4 : ϕ 3 ϕ 4 + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + ϕ 3 ϕ 4 : ϕ 1 ϕ : ϕ 1 ϕ + ϕ 1 ϕ ϕ 3 ϕ 4 + ϕ 1 ϕ ϕ 3 ϕ 4 : ϕ 1 ϕ : ϕ 1 ϕ = ϕ 1 ϕ : ϕ 1 ϕ : : ϕ 3 ϕ 4 : = : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + : ϕ 1 ϕ ϕ 3 ϕ 4 : + ϕ 1 ϕ ϕ 3 ϕ 4

79 L int = g : ψxψx φx : p 1, s 1, p, s p 1, s 1, p, s i = a p 1, s 1 a p, s 0, f = a p, s a p 1, s S fi = f S i = f i + i d 4 x f L int x i 1 d 4 x d 4 x f T L int xl int x i + i, f f φ i = Wick ψ = ψx, ψ = ψx 7.18 ψψ a + b a + b ψψ ψ ψ 0 f : φ φ : i = 0 f T L int xl int x i = g f T : ψψφ : : ψ ψ φ : i = g φ φ f : ψψψ ψ : + : ψψψ ψ : + : ψψψ ψ : + ψψψ ψ i = g φ φ f : ψψψ ψ : i + ψ α ψ β f : ψ α ψ β : i + ψ α ψ β f : ψ α ψ β : i + ψψψ ψ f i f : ψ α ψ β : i 0 ap 1, s 1ap, s a p, sap, s a p 1, s 1 a p, s 0, p 1, s 1 = p 1, s , 1 1, S fi = g d 4 x d 4 x f : ψxψx ψx ψx : i φxφx ψ a, ψ + b 9. f : ψxψx ψx ψx : i = f : ψ + xψ x ψ + x ψ x : i = αβ f ψ + α xψ + β x ψ α xψ β x i

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)

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