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1 : Coulomb Maxwell

2 ., 3,, 973.,,, 3.,,, ,,,,,, J.D. Bjoken, and S.D. Dell, Relativistic Quantum Mechanics, McGaw-Hill, J.D. Bjoken, and S.D. Dell, Relativistic Quantum Fields, McGaw-Hill, ψ. 3. Ω ω n Ωu n ω n u n. Ω Ω E p x n u n ψ Ω u n ψt, x n a n tu n x.3 a n

3 5. Ω ω n ω n u n a n a n t a n. n Ω < Ω > < Ω > t a n t ω n n dxψ x, tωψx, t t x ρt, x i h h ψt, x Hψt, x t m ψt, x + V t, xψt, x..5 ρt, x ψ t, xψt, x.6 t ψt, x ρt, x t j 3 + div jt, x, div j j.7 j h im ψ ψ ψ ψ c c cm/s.9 E p c E pc m c p µ p, p E, p. c 4 c 4 x µ x, x ct, x. 3

4 4 d 4 x 4 3 d 3 x d 4 x dx dx dx dx 3 cdtd 3 x..3 i,, 3 4 µ, ν,, µ, ν,,,, 3 4 A µ A, A i B µ B, B i A B A B A B A B A 3 B A µ A, A i A, A i.5 3 A B A µ B µ A B + A B + A B + A 3 B 3.6 µ 4 4 η µν η µν ηµν.7 A B A µ η µν B ν A µ η µν B ν i, j, k,,, 3 dx i dx i dx j a j kdx k.9 dx j dx k a j la j k δ lk. a t a deta ±. deta + SO3 SO3 θ j k a j k δ j k + θ j k. a j ka j l δ kl θ l k θ k l.3 4

5 3 θ 3 θ ω θ θ dθ.4 adθ j k + dθ.5 daθ dθ aθ.6 aθ j k cos θ sin θ sin θ cos θ.7 4 aθ µ ν cos θ sin θ sin θ cos θ dx µ dx µ dx µ a µ νdx ν.9 ds η µν dx µ dx ν.3 ds ds a µ λη µν a ν ρ η λρ.3 a t ηa η deta ±.3 + SO, dω ω..5,.6 5

6 3. ω aω µ ν coshω sinhω sinhω coshω.33 v tanhω β v c, coshω γ β.34 v c 4 p µ 4 A µ x v A µ 4 A γ γβ A A A γβ γ A A, β v c, γ β A 3 A V E p m + V.36 V E p E p + m c.37 c E hν i h t, p p j pj hk j h i p E c c i h t i h x, p j h i x j.38 x j i h x j.39 p µ i h x µ i h µ.4 V ν x ν x µ xν V ν V µ.4 6

7 .36 ψt, x.5.37 φt, x i h c t h φt, x i x j φt, x + m c φt, x.4 Klein Godon.4 ψ, φ ψ, φ i d 3 x [ψ t, x φt, x ψ t, xφt, x].43 4 j µ.3. j µ iψ µ φ µ ψ φ µ j µ d ψ, φ.46 dt.7.4 Diac N ψt, x ψ t, x ψ t, x. ψ N t, x.47 7

8 4 N N α j, j,, 3 β m i h ψt, x c t h i αj ψt, x + mcβψt, x.48 xj i h c t ψ h h i i αj x j h x j + mcβ i αk x k + mcβ ψ ψ + m c ψ..49 α j β α j α k + α k α j δ jk,.5 α j β + βα j, β..5 α j, β.4.. β, α i α j α j, β β j k,.5. Tα i Tβ.53.5 β cyclic α j β, α j N N / σ, σ i, σ i β, α j N N 4 N 4 α j σ j, β,.55 σ j 4 ρt, x ψ ψ.56 8

9 .4.. ψ ψ + t x k cψ α k ψ.57 4 µ j µ.58 j j k j k cψ α k ψ, j cψ ψ, j µ j, j, j, j m.5. ψt, x i h hc t i σj x j ψt, x σj p j cψt, x.6. σ j σ k δ jk + iϵ jkl σ l σ j σ j.6 ϵ jkl ϵ α j, β γ µ γ β, γ i βα i.. γ γ j γ γ, γ j γ j. γ µ α k.7 η µν γ µ γ ν + γ ν γ µ η µν.3 γ µ i h µ mc ψt, x.4 9

10 γ 5 γ 5 iγ γ γ γ 3.5 γ µ γ 5 γ µ + γ µ γ 5, µ,,, 3, γ 5.6 γ µ U γ µ DP U.7 γdp, γ j DP σ j σ j, j,, 3, γ σ σ, σ σ, σ 3 σ 3.9 σ σ σ 3 ρ I, ρ ii, ρ 3 I. I ii I γdp ρ 3, γ j DP iρ σ j, β DP ρ 3, α j DP ρ σ j.... ψx ψ x Saψx, x ax. ψx Sa ψ x, x a x.3 S Sa [Sa] S a.4 x µ x ν x µ x ν aν µ x ν.5 γ µ i h x µ mc ψx γ µ i h x µ mc ψ x.6 S a a ν µγ µ S aγ ν Sa.7 a µ ν δ µ ν + ω µ ν σ µν Sa i 4 σ µν ω µν, σ µν σ νµ.8

11 .7 iδ ν αγ β δ ν βγ α [γ ν, σ αβ ].9 σ µν i [γ µ, γ ν ] σ jk σ jk σ l ϵ jkl σ σ l j σ j iσ j iσ j... ω aω dω ω ω aω + dω adω aω. Saω + dω Sadω Saω.3 dsaω i dω σ Saω.4 i σ + Saω exp σ ω cos ω + iσ sin ω.5 3 S 3 S 3 h σ.6 π Sω π cos π.7 4π ψ, θ, φ + π Sω πψ, θ, φ ψ, θ, φ.8 dω ω ω dsaω i dω σ Saω.9 σ Saω exp i σ ω cosh ω iσ sinh ω.3 σ / Saω.4

12 γ 4 ψ ψ ψ γ ψ, ψ, ψ 3, ψ 4γ ψ, ψ, ψ 3, ψ 4.3 σ jk σ jk, σ j σ j, γ σ µνγ σ µν.3 S ψ γ S γ S.33 ψ x ψ x γ ψ xs γ ψ xγ S ψxs.34 4 ψ 4 ψ µ γ µ 4 ψ, ψ,, 4 A µ, B µ, ψ ψ A B, ψ γ µ ψ A µ, ψ γ µ γ ν ψ A µ B ν.35 ψ ψ ψ ψ S Diac d 4 x ψx i h µ γ µ mc ψx.36 ψ ψ ψ e iα ψ, ψ ψ e iα ψ.37 S Diac S Diac d 4 x ψ x i h µ γ µ mc ψ x S Diac.38 ψt, xψt, x.39 t

13 3 3. i h ψ t βmc ψ 3. 4 E p j,, 4 E p c + m c 4, ψt, x wp exp i Et h + ip x h. 3. wp ψ e i mc h t, ψ e i mc h t 3.3 ψ 3 ei mc h t, ψ 4 { ψ x w p e iϵ mc h t, ϵ ei mc h t 3.4 +, 3.5 3, 4 3 S z.6 σ l σ jk ϵ jkl 3.6 σ l 4 3 w, w 3, S z + h, w, w 4, S z h x µ v x µ v ψ x ψ x Sω vψx 3.8 mc exp iϵ h t exp iϵ p µ x µ h 3.9 Sω v e i ωσ, tanh ω v c 3. ψ x w pe iϵ pµx µ h, σ iα iρ σ 3. 3

14 w p e i ωσ w p cosh ω α sinh ω w p tanh ω cosh ω tanh ω tanh ω tanh ω w p 3. tanh ω tanh ω + tanh ω p mvc v/c mc + tanh ω mc pc E + mc 3.3 mv v/c, E mc v/c mc + pc 3.4 cosh ω cosh ω + E + mc mc 3.5 ψ x w pe iϵ p µ x µ / h 3.6 p µ γ µ ϵ mcw p, w pp µ γ µ ϵ mc 3.7 w pw p δ ϵ 3.8 ψ w ϵ p E w ϵ p δ mc 3.9 p µ 4 4 ϵ wαp w βp δ αβ 3. p Sωp Sωp p S S 3.3 P P P P δ 3. p µ γ µ mcp µ γ µ + mc p µ γ µ mcp µ γ µ + mc p mc 3. 4

15 ±p µ γ µ + mc±p µ γ µ + mc p µ γ µ ± p µ γ µ mc + mc p ± p µ γ µ mc + mc mc ±p µ γ µ + mc p µ Λ + p Λ + p Λ ± p ±p µγ µ + mc mc 3.4 z +σ z 4 s µ,,, p µ E/c,,, 4 s µ p µ 3.5 s µ 4 Σs + σ z + iγ γ + γ 5s µ γ µ γ + γ 5γ 3 γ Σs, s µ p µ e e m e mev/c m e E p E p c + m c p E E p c + m c 4, E p c + m c hole 5

16 / p E p c + m c 4 e p E + p c + m c 4 +e m E p c /c q S z 93 C.D.Andeson 3.5 C Minimal inteaction e > e Q Q Q Minimal inteaction p µ i h x µ p µ ea µ i h x µ ea µ 3.3 Minimal inteaction [ γ µ i h ] [ x µ mc ψ γ µ i h ] x µ ea µ mc ψ 3.3 A µ Q [ i h µ i eq h ] A µ γ µ mc ψx 3.3 C ψ c C ψ T Cγ T ψ

17 ψ ψ γ T [ i h µ + i eq h ] A µ γ µ mc ψ c x ψ 3.33 C C γ µ C γ µt 3.35 C C C 3.36 C C iγ γ ψ c ψ T C ψ cψc ψ T ψ T 3.39 ψ cγµ ψ c +ψ T γ µt ψ T 3.4 ψ cγµ γ ν ψ c ψ T γ µt γ νt ψ T 3.4 ψ cγ5 ψ c ψ T γ 5T ψ T p µ i h x µ p µ + ea µ i h x µ + ea µ 4. [ g νµ i h ] x ν + ea ν i h x µ + ea µ mc ψ 4. 7

18 [ γ µ i h ] x µ + ea µ mc ψ 4.3 [ γ ν i h x ν + ea ν + mc ][γ µ i h ] x µ + ea µ mc ψ 4.4 γ µ γ ν η µν iσ µν 4.5 [ g νµ i h x ν + ea ν i h x µ + ea µ mc iσ νµ i h x ν + ea ν i h x µ + ea ] µ ψ 4.6 iσ νµ i h x ν + ea ν i h x µ + ea σ νµ [ µ i i h x ν + ea ν, i h ] x µ + ea µ i σνµ ie h Aµ x ν A ν x µ e hσνµ F νµ F µν F µν A µ x ν A ν x µ 4.8 / [ g νµ i h x ν + ea ν i h x µ + ea µ mc + ] e hσνµ F νµ ψ Et i ψt, x uxe h, ux φ χ 4. [ E + eφ c p + ea mc + ] e hσνµ F νµ u, A Φ c 4. W W E mc 4. W mc, W + eφ mc 4.3 [m W + eφ + W + eφ c p + ea ] e hσνµ F νµ u 4.4 8

19 W + eφ W + eφ [ p + ea W u m ] eφ e h 4m σνµ F νµ u 4.5 σ νµ. F j σ j φ χ F jk σ jk φ χ σ jk F jk ϵ jkl σ l F jk σ l ϵ ljk j A k σ l ϵ ljk j A k σ l ota l σ l B l 4.6 [ p + ea W φ eφ + e h ] m m σj B j φ 4.7 B ota, A B x 4.8 p + ea p + epa + Ap 4.9 pa + Ap p B x B x p B L 4. S j h σj 4. [ p W φ m + ] el + S B φ 4. m / µ g- g g- g µ j e h m σj e m Sj e m Sj g, g E Φ Ȧ, B A 5. A µ Φ, A, Φ Φ, A 5. c H α j p j c + βmc eφ H + V, V eφ 5.3 H ρ σ j p j c + ρ 3 mc, H mc + V σ j p j c σ j p j c mc + V 5.4 9

20 V L L x p, [L, V ] [L j, H] J L + h σ, [J, H] 5.6. σ kl, k, l,, σ j [σ j, H] σ kl ϵ klj σ j 5.7 Hψ Eψ J L+S ψ φ χ ψ x, x P ψx, x 5.8, σ j σ j P η 5.9 σ j Jφ Jχ, J z φ J z χ, P φ P χ : L L x + L y + L z h [ sin θ L Y m l sin θ + θ θ sin θ ] φ. 5. L z i h φ. 5. θ, φ h ll + Y m θ, φ, l L z Yl m θ, φ hmyl m θ, φ 5.3 Y m l θ, φ ϵl, m l + l m! 4π l + m! P l m cos θe imφ 5.4 { m, m >, ϵl, m, m. 5.5 J L + S 5.6

21 l / l > l + l l l + l 5.7 l j / z z φ l+,l+ Y l l Y l l φ + j,j, j l J L + S h z J φ jjz h j + j z j + j z φ jjz 5.9 L Y m l h l + ml + my m l 5. S h 5. j z 5.8 J j l + z h φ l+,l l + J h L l + h Y l l l + ly l l φ l+,l+ L + S l + h + Yl l + Y l l Yl l ly l l l + Y l l 5. z l l + z l + l φ l,l Y l l + l lyl l l + Y l l ly l l φ j,j, j l 5.3 l + z 5.8 l z 5.3 J L + S j l +, l, φ+ j,m m j + my j j m+ 5.4 j my j j l, l, φ j,m j + j + my m j+ j + + my m+ j+ 5.5

22 φ ± j,m π dφ π J φ ± j,m jj + h φ ± j,m 5.6 J z φ ± j,m m hφ± j,m 5.7 dθ sin θφ α j m φα jm δ j jδ m mδ α α, α, α ± 5.8 hσ k L k φ ± j,m [ J L h ] σ k σ k φ ± j,m [ h jj + ll + 3 ] φ ± j,m 4 κ { h + κφ ± j,m 5.9 l + j + j l + l j + j l 5.3 σ k x k / [L j, x k ] [ϵ jlm x l p m, x k ] i hϵ jlk x l, [σ j, σ k ] iϵ jkl σ l, [J j, σk x k ] σk σk [ L j, x k] + h [ σ j, σ k] x k i hϵjlk x l + h iϵjkl σ l h J σk x k φ ± j,m h jj + σk x k φ ± j,m, J z σ k x k φ ± x k 5.3 j,m hmσk x k φ ± j,m 5.3 z j, m φ + jm φ jm σk x k σk x k P φ + jm j φ + jm P φ jm j+ φ jm 5.33 x x, p p, L x p +L 5.34 σk x k σ k σ k, σ k x k σk x k 5.35 σ k x k φ j,m c j,m φ+ j,m 5.36 m j J ± z m m j θ c j,m φ + j,m σk x k φ j,m, φ j,m σk x k φ + j,m 5.37

23 5.3 P P ψ jm ig + j F + j ig j F j φ+ jm φ jm φ jm φ+ jm ψ l jm ig + j F + j φ+ jm + ig j φ jm φ jm + F j j j+ iglj φl jm F lj σ k x k φ l jm φ+ jm ig + j F + j ig j F j 5.38 φ+ jm φ jm φ jm φ+ jm, P ψ l jmx, x l ψ l jmx, x 5.4 σ k a k σ l b l a k b k + iσ k a b k 5.43 σ k p k σl x l σ m x m σ k p k σl x l x k p k + iσ k x p k σl x l h i xk k + iσ k L k σl x l h i + iσk L k 5.44 σ k p k f φl jm σl x l h i + iσk L k f φl jm σl x l h i i h + κ f φl jm h f i h + κ f σ l x l i φl jm 5.45 mc + V σ k p k c Eψ Hψ ψ 5.46 σ k p k c mc + V E mc V G lj df lj d + κ F lj hc 5.47 E + mc dglj V F lj d + κ G lj hc Coulomb V eφ Ze 4πϵ Zα hc α 37 γ G lj γ a + a +, 5.5 F lj γ b + b

24 γ, γ g, f Zαa γ κb, Zαb γ + κa 5.53 γ κ Zα F lj γ κ Zα 5.54 j + Zα 5.55 E mc G lj hc df lj, d E + mc F lj hc dg lj, d 5.56 m c 4 E G lj hc d G lj d, 5.57 m c λ E /c 5.58 h e ±λ G, F e λ 5.59 F lj hλ lim G lj E/c + mc mc E mc E G lj γ e λ w 5.6 hλ F lj E/c + mc γ e λ w 5.6 x λ 5.63 dw dx w w + γ + κw + Aw x 5.64 dw dx + w w + γ κw + Bw x 5.65 A Zαλ h E/c + mc, B Zαλ h E/c mc 5.66 w a n x n w b n x n 5.67 n n na n a n b n + γ + κa n + Ab n 5.68 nb n + a n b n + γ κb n Ba n 5.69 a n b n n + A + γ κ n B + γ + κ c n 5.7 c n n + γ + A B/ n + a c n nn + γ nn + b 5.7 4

25 c n n + a c n nn + b 5.7 n cn x n F a; b; x + a c b n x + aa + bb + x + Γa + nγb n!γaγb + n xn 5.73 c n c n n 5.74 x w a n x n n + A + γ κc n x n n n x A+γ κ d x A+γ κ F a; b; x dx 5.75 w b n x n n B + γ + κc n x n n n x B+γ+κ d x B+γ+κ F a; b; x dx 5.76 G lj, F lj c cn x n x n n! ex e λ 5.77 G lj, F lj e λ n. n + a, c n, c n +, n,, 5.79 n a γ A B γ Zα mc E mc + E mc + E mc E 5.8 mc E Zα n +γ γ j + Zα 5.8 F a n ; b; x Γbn! Γb + n Lb n x

26 n n + κ n + j mc E n,j Zα n j+ + j+ Zα n,,,, j n l n E n,j mc Zα [ n + Zα n j + 3 ] + OZα 6 4n E mc Zα 5.88 u γ,, γ Zα φt, x φt, x φt, x x p [x, p] h q i, i,, N p i, i,, N x φt, x x.3.4 S S scala dtd 3 x c φx φx t mc φx h dtl dtd 3 xl scala 6. L L c t φt, x + φt, x mc φt, x 6. h L q,,, N q dq /dt,,, N Lq, q 6.3 p p L q 6.4 6

27 [q, p s ] i hδs, [q, q s ] [p, p s ] 6.5 q p H n Hq, p p q Lq, q 6.6 φt, x πt, x x πt, x L φt, x φt, x c t 6.7 [φt, x, πt, y] i hδ 3 x y, [φt, x, φt, y] [πt, x, πt, y] 6.8 H d 3 xπt, x φt, x L d 3 x c π + φ + mc φ h R φt, x, y, z φt, x + R, y, z, 6. y, z x k x e ikxx 7.34 k x R π R x, y, z R 3 V V / eikx, k π R n, n i Z, i,, 3 6. φt, x k d c dt φ k t k φ k t ω k c V / eikx φ k t 6. k + mc φk t 6.3 h mc 6.4 h e ic k t e ic k t a k φ k t hc ωk a+ ke iω kt + a ke iω kt 6.5 7

28 a + k, a k φ a k a + k ak 6.6 φt, x k hc ake ikµ x µ + a ke ikµ x µ V ωk k µ 4 p µ k µ c k, k, p µ hk µ 6.8 πt, k πt, x i k h c ωk ake ikµ x µ a ke ikµ x µ V [ak, a l] δ k,l, [ak, al] [a k, a l] 6. ak, a k n R eiπnx y/r δx y H k hω k a kak + aka k k hω k a kak Maxwell E B Maxwell divb, 7. ote + B, t 7. E B D H D ϵe, B µh 7.3 ϵ, µ ϵ, µ 8

29 D H divd ρ, 7.4 oth D t j 7.5 ρt, x vt, x j ρv 7.5 div 7.4 divj + ρ t div oth div D + t t divd div oth 7.6 ot div 7. B div A ot B ota. 7.7 A 7. A ot E + A. 7.8 t ot φ gad E gadφ A t. 7.9 φ ϵ div gadφ t diva ρ, 7. ot ota + ϵ µ t gadφ + A t j. 7. φ, A E B A A gadλ, φ φ + Λ t. 7. E B A φ Bohm-Ahaonov 9

30 4 A µ φ c, A, jµ cρ, j, 7.3 F µν E /c E /c E 3 /c F µν µ A ν µ A ν E /c B 3 B E /c B 3 B E 3 /c B B 7.4 F µν ε µνστ µ F νσ 7.5 ε µνστ ε ε µνστ µ F νσ ε µνστ µ ν A σ σ A ν 7.6 Bianchi µ F µν µ µ A ν ν A µ c ϵ jν 7.7 A µ A µ + µ Λ, 7.8 E S em d 4 x cϵ H c d 4 x cϵ 4 F µνf µν d 4 x cϵ 4 µa ν ν A µ µ A ν ν A µ 7.9 A µ Λx F µν 7. A µ A µ diva. 7. 3

31 µ A µ φ 7. a, b, c, diva 7. ϵ div gadφ 7.3 φ 7.4 A ot ota + ϵ. 7.5 µ t a b c a bc + a cb a bc + ba c 7.6 A A + A A + diva c A + A c t, c 7.8 ϵ µ 7. E B A E A, B ota. 7.9 t S em dtd 3 x ϵ E µ B dtd 3 x A ϵ ota t µ dtd 3 xl em 7.3 Ct, x L em Ȧt, x ϵ A t 7.3 3

32 [A i t, x, C j t, y] i hδ ij δ 3 x y, [A i t, x, A j t, y] [C i t, x, C j t, y] 7.3 E em d 3 x ϵ E + µ B d 3 x A ϵ + ota t µ 7.33 R At, x, y, z At, x + R, y, z, 7.34 y, z x k x k x e ikxx 7.34 k x R π R x, y, z R 3 V V / eikx, k π L n, n i Z, i,, At, x k V / eikx A k t 7.36 A k t 7.8 ω k k A k t + c A k t t 7.37 ω k c k 7.38 e iω kt e iω kt h A k t a+ ke iωkt + a ke iω kt 7.39 ϵ ω k a + k, a k A a k a + k ak 7.4 At, x k V / h ake ikµ x µ + a ke ikµ x µ ϵ ω k k µ ω k, k 7.4 3

33 3 A C 7.3 ak [a i k, a j l] δ ij δ k,l, [a i k, a j l] [a i k, a j l], i, j,, a i a i 7. k ak [a i k, a j l] δ ij δ k,l, [a i k, a j l] [a i k, a j l], i, j, [A i t, x, C j t, y] i h δ ij i j ak E A t i k B ota i k δ 3 x y, [A i t, x, A j t, y] [C i t, x, C j t, y] 7.46 dive, divb 7.47 h ω V / k ake ipµ x µ/ h a ke ipµ x µ/ h 7.48 ϵ ω k V / h ϵ ω k k ake ipµ x µ / h a ke ipµ x µ / h 7.49 k 7.33 a i k, a i k E em k k hω k a i ka i k + a i ka i k i, hω k a i ka i k i, 33

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)