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

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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) (7.1) E = m 2 c 4 + p 2 c 2 = mc 2 1+ p2 m 2 c 2 (7.5) 123

2 124 7 p 2 /(mc) 2 E = mc p 2 2 m 2 c 2 1 ( ) p m 2 c ( ) p m 2 c 2 + = mc [ 2 mv2 1 1 ( ) v ( ) (7.6) v 4 ] 4 c 8 c p = mv (7.4) v/c p p = p 0 p 0 (p 1 p 1 + p 2 p 2 + p 3 p 3 )= E2 c 2 p2 (7.7) ( ) E p µ =(p 0,p 1,p 2,p 3 )= c, p (7.8) a b = 3 a µ b µ = µ=0 3 a µ b µ = a 0 b 0 a b (7.9) µ=0 contravariant vector covariant vector x µ 3 3 x x = x µ x µ = x µ x µ =(ct) 2 x 2 (7.10) µ=0 µ=0 Minkowski g µν = g µν = (7.11) a b = µν g µν a µ b ν = g µν a µ b ν (7.12) µν

3 a µ = g µν a ν a µ = g µν a ν (7.13) ν=0 Einstein ν=0 ν 0 3 a µ = g µν a ν a µ = g µν a ν (7.14) Lorentz Newton Galilei Newton a µ z v a µ a 0 a 1 a 2 a 3 1/ 1 β β/ 1 β 2 = β/ 1 β / 1 β 2 a 0 a 1 a 2 a 3 (7.15) β = v c (7.16) cosh 2 θ sinh 2 θ = 1 β = tanh θ (7.17) 1 1 β 2 = cosh θ β 1 β 2 tanh θ = sinh θ cosh θ (7.18) = sinh θ (7.19) Lorentz (7.15) a 0 a 1 a 2 a 3 = cosh θ 0 0 sinh θ sinh θ 0 0 cosh θ a 0 a 1 a 2 a 3 (7.20)

4 126 7 x µ x µ Lorentz ct x y z = 1/ 1 β β/ 1 β β/ 1 β / 1 β 2 ct x y z (7.21) z x y z t = z = t 1 β 2 + v 1 β 2 + vz/c2 1 β 2 z 1 β 2 (7.22) β = v/c 0 t x y z = v t x y z (7.23) Galilei p µ Lorentz E /c p 1 p 2 p 3 = 1/ 1 β β/ 1 β β/ 1 β / 1 β 2 E/c p 1 p 2 p 3 (7.24) p =0 E = mc 2 Lorentz E = mc2, 1 β 2 p 1 = p 2 =0, p 3 mv = 1 β 2 (7.25) E v m E 2 = p 2 c 2 + m 2 c 4 = m2 c 4 1 v2 c 2 (7.26)

5 p = Ev c 2 (7.27) Lorentz a µ b µ Lorentz (7.15) a µ b µ a b = a 0 b 0 a 1 b 1 a 2 b 2 a 3 b 3 = a0 βa 3 1 β 2 b 0 βb 3 1 β 2 a1 b 1 a 2 b 2 βa0 + a 3 1 β 2 βb 0 + b 3 1 β 2 (7.28) = a 0 b 0 a 1 b 1 a 2 b 2 a 3 b 3 = a b

6 Dirac m E = p2 2m (7.29) Schrödinger E p E i h t p h i (7.30) (7.29) Hamiltonian H = p2 2m (7.31) Schrödinger i h ψ(t, x) t = h2 2m 2 ψ(t, x) (7.32) ψ(t, x) (7.32) ψ (7.32) ψ ρ t + j = 0 (7.33) ρ = ψ ψ = ψ 2 j = 1 h 2m i [ψ ( ψ) ( ψ ) ψ] (7.34) (7.33) ρ Klein-Gordon Schrödinger (7.32) E 2 = p 2 c 2 + m 2 c 4 (7.35) Lorentz (7.35)

7 7.2 Dirac 129 (7.30) [ ( ) ] h 2 2 t 2 c2 2 + m 2 c 4 ψ = 0 (7.36) Klein-Gordon ( ) ( 1 1 c t, = c t, x 1, x 2, ) x 3 (7.37) Lorentz ( ct, x 1,x 2,x 3 ) ( ) ( ) 1 1 c t, = c t, x 1, x 2, x 3 (7.38) ( ct, x 1, x 2, x 3 )=(ct, x 1,x 2,x 3 ) p µ p µ p µ = i h µ = i h x µ = p µ = i h µ = i h x µ = ( i h c ( i h c t, h i Klein-Gordon t, h i ) x x ) (7.39) p µ p µ m 2 c 2 = 0 (7.40) [ h 2 µ µ + m 2 c 2 ] ψ = 0 (7.41) Schrödinger (7.33) Klein-Gordon ρ = i h 2mc 2 j = 1 2m ρ t + j = 0 (7.42) [ ( ) ( ψ ψ ψ t t h i [ψ ( ψ) ( ψ ) ψ] ) ] ψ (7.43)

8 130 7 j ρ ρ ρ Klein-Gordon Klein-Gordon ψ = N exp [ īh ] (Et p x) (7.44) (7.43) ρ = 1 mc 2 N 2 E (7.45) E = ± p 2 c 2 + m 2 c 4 (7.46) ρ Klein-Gordon Klein-Gordon Dirac P.A.M. Dirac Lorentz E 2 = p 2 c 2 + m 2 c 4 Schrödinger Dirac Hψ = i h ψ t = hc ( α 1 ψ ) ψ ψ + α2 + α3 i x1 x2 x 3 + βmc 2 ψ ( hc = ) i αk x k + βmc2 ψ (7.47) k =1, 2, 3 ( ) i h 2 ( hc )( hc ψ = c t i αj x j + ) βmc2 i αk x k + βmc2 ψ 3 = ( hc) 2 (α k ) 2 2 ψ 3 ( x k ) 2 ( hc)2 (α j α k + α k α j ) + hcm i k=1 3 k=1 1=j<k (α k β + βα k ) ψ x k + β2 m 2 c 4 ψ 2 ψ x j x k (7.48)

9 7.2 Dirac 131 (7.35) Klein-Gordon ( ) i h 2 ) ( hc 2 ψ = c t i x k ψ + m 2 c 4 ψ (7.49) α k β α j α k + α k α j =2δ ik α k β + βα i = 0 (7.50) β 2 = 1 (7.51) Hamiltonian Hamiltonian Hermite α k β Hermite α k = α k β = β (7.52) α k β ψ α k β γ 0 = β γ k = βα k ( k =1, 2, 3 ) (7.53) α k β (7.50), (7.51), (7.52) γ µ γ ν + γ ν γ µ =2g µν (7.54) γ 0 = γ 0 γ k = γ k (7.55) g µν γ 0 Hermite γ k Hermite ψ Pauli (7.54) µ = ν =0 (γ 0 ) 2 =1 γ 0 ±1 k = µ = ν 0 (γ k ) 2 = 1 γ k ±i µ ν (γ ν ) 2 = ±1 Tr (γ µ γ ν γ ν )= Tr (γ ν γ µ γ ν )= Tr (γ µ γ ν γ ν ) = 0 (7.56) Tr (γ µ ) = 0 (7.57)

10 132 7 ψ(t, x) = ψ 1 (t, x) ψ 2 (t, x) ψ 3 (t, x) ψ 4 (t, x) (7.58) ( ) ( ) I 0 0 σ γ 0 = γ k k = 0 I σ k 0 (7.59) I σ k Pauli ( ) ( ) i σ 1 = σ 2 = σ 3 = 1 0 i 0 ( ) (7.60) Dirac-Pauli unitary ( ) ( ) 0 I 0 σ γ 0 = γ k k = I 0 σ k (7.61) 0 Wyle Chiral Dirac (7.47) ( i h γ 0 x 0 + γ1 x 1 + γ2 x 2 + ) γ3 x 3 ψ mc ψ = 0 (7.62) 1 2 Dirac Dirac ( i hγ µ µ mc ) ψ = 0 (7.63) Klein-Gordon Dirac (7.47) Hermite (7.47) Hermite i h ψ t ( = ψ hc i α ) + βmc 2 (7.64)

11 7.2 Dirac 133 ψ x ρ = ψ ψ ψ x = ψ x (7.65) ρ t + j = 0 (7.66) j = cψ α ψ = c( ψ α 1 ψ, ψ α 2 ψ, ψ α 3 ψ, ) ρ ρ =(ψ1,ψ2,ψ3,ψ4 ) (7.67) j µ x µ = µj µ =0 j µ =(cρ, j ) (7.68) ψ 1 ψ 2 ψ 3 ψ 4 4 = ψ k 2 > 0 (7.69) k= Dirac Dirac E p ψ(t, x) =w exp [ īh ] (Et p x) (7.70) w w = u 1 u 2 u 3 (7.71) u 4 (7.71) Dirac (7.62) Dirac-Pauli (7.59) (7.60) u k (p) k =1, 2, 3, 4 E mc 2 0 cp 3 c(p 1 ip 2 ) u 1 0 E mc 2 c(p 1 + ip 2 ) cp 3 u 2 cp 3 c(p 1 ip 2 ) E + mc 2 0 u = 0 (7.72) 3 c(p 1 + ip 2 ) cp 3 0 E + mc 2 u 4

12 134 7 u k 0 ( E 2 (pc) 2 m 2 c 4) 2 = 0 (7.73) p E ± = ± p 2 c 2 + m 2 c 4 (7.74) Klein- Gordon E + w = N 1 0 cp 3 E + + mc 2 c(p 1 + ip 2 ) E + + mc 2 w = N 0 1 c(p 1 ip 2 ) E + + mc 2 cp 3 E + + mc 2 E cp 3 c(p 1 ip 2 ) E mc 2 E c(p 1 + ip 2 mc 2 ) cp 3 w = N E mc 2 w = N E mc (7.75) (7.76) N = [ 1+ ] p 2 c 2 1/2 ( E ± + mc 2 ) 2 (7.77) w ( ) u1 ( ) φ = φ u w = ( 2 ) (7.78) χ u3 χ = (7.70) Dirac (7.62) (7.59) ( )( ) ( )( ) ( )( ) E I 0 φ 0 σ p φ I 0 φ + mc = 0 (7.79) c 0 I χ σ p 0 χ 0 I χ u 4

13 7.2 Dirac 135 φ χ (E mc 2 ) φ c σ p χ = 0 (E + mc 2 ) χ c σ p φ = 0 (7.80) χ w w = N φ c σ p E + mc 2 φ (7.81) (7.80) χ Pauli (σ p)(σ p) =p 2 I (7.82) (E mc 2 )(E + mc 2 ) φ =(pc) 2 φ (7.83) (7.73)

14 Dirac Lorentz Lorentz Lorentz Lorentz x µ x µ x µ = Λ µ ν x ν (7.84) x x = x µ x µ (7.85) x µ x µ = x µ x µ (7.86) (7.86) g µν x µ x µ x µ (7.84) (7.86) x µ x µ = g µν x µ x ν = gµν (Λ µ ρx ρ )(Λ ν σx σ )=g µν Λ µ ρλ ν σ x ρ x σ (7.87) x ρ x ρ = g ρσ x ρ x σ (7.88) Lorentz Λ µ ν g µν Λ µ ρλ ν σ = g ρσ (7.89) g µν Λ µ ν (7.89) (7.89) ( det Λ ) 2 = 1 (7.90) det Λ = ±1 (7.91) z Lorentz (7.15) 1/ 1 β β/ 1 β 2 det Λ = = 1 (7.92) β/ 1 β / 1 β 2

15 7.3 Dirac 137 Lorentz det Λ =+1 det Λ = 1 Lorentz Λ P, T P = T = (7.93) (7.94) (7.89) g τρ τ ρ g τρ g µν Λ µ ρλ ν σ = g τρ Λ νρ Λ ν σ = Λ τ ν Λ ν σ (7.95) g τρ g ρν = δ τ σ (7.96) Λ τ ν Λν σ = δτ σ (7.97) δ τ σ Kronecker (7.97) Λµ ν Λ ν µ Λ (Λ 1 ) T Λ ν µ =(Λ 1 ) ν µ (7.98) x µ x µ x µ = g µν x ν = gµν Λ ν ρ xρ = g µν Λ ν ρ gρσ x σ = Λ σ µ x σ (7.99) Λ (7.98) x µ x µ = Λ ν µ x ν = x ν ( Λ 1 ) ν µ (7.100)

16 138 7 Lorentz T µν g µν T µν T µν = Λ µ ρ Λ ν σt ρσ (7.101) g µν g µν = Λ µ ρ Λ ν σg ρσ = Λ µσ Λ ν σ = g µρ Λ σ µ Λ ν σ = g µρ δ ν ρ = g µν (7.102) Lorentz Dirac Lorentz k k =1, 2, 3 θ k θ s J k 2s +1 ψ s ψ s = S(θ)ψ s (7.103) S(θ) = exp (ij k θ) (7.104) 1 2 Pauli J k = 1 2 σk ( ) S(θ) = exp i σk 2 θ (7.105) σ k exp(iσ k θ/2) ( ) ( ) exp i σk n 2 θ 1 = i σk n! 2 θ (7.106) Pauli n=0 ( ) exp i σk 2 θ = I cos θ 2 + iσk sin θ 2 ( σ k ) 2 = I (7.107) (7.108)

17 7.3 Dirac 139 Dirac 1 2 Dirac ψ ψ(x) ψ (x )=S(θ) ψ(x) (7.109) S(θ) S(θ) = ( I 0 0 I ) exp ( ) i σk 2 θ (7.110) σ k k Hermite unitary Lorentz x v x 0 x 1 β = v/c = tanh θ Lorentz (7.20) x 0 x 0 =+x 0 cosh θ x 1 sinh θ x 1 x 0 = x 0 sinh θ + x 1 cosh θ (7.111) Lorentz ψ(x) ψ(x) ψ (x )=S(θ) ψ(x) (7.112) S(θ) ( S(θ) = exp i σ ) 01 2 θ (7.113) σ 01 = i 0 0 i 0 0 i 0 0 i (7.114) σ 01 1 S(θ) =I 4 cosh θ 2 iσ 01 sinh θ 2 (7.115) I 4 σ 01 Hermite unitary

18 140 7 Lorentz Dirac ψ x µ x µ = Λ µ ν x ν (7.116) ψ ψ = Sψ (7.117) Lorentz (7.111) Dirac (7.112) unitary ψ Hermite ψ Lorentz Dirac Hermite Hermite γ 0 ψ ψ = ψ γ 0 (7.118) ψ =(ψ 1,ψ 2,ψ 3,ψ 4 ) γ 0 =(ψ 1,ψ 2, ψ 3, ψ 4 ) (7.119) Lorentz S(θ) ψ(x) ψ (x )=ψ(x) S(θ) 1 (7.120) (7.120) ψ 1 ψ 2 ψ 1 ψ 2 ψ 1ψ 2 = ψ 1 S 1 Sψ 2 = ψ 1 ψ 2 (7.121) Lorentz 1 2 Lorentz γ µ ψ 1 γ µ ψ 2 ψ 1 γ µ ψ 2 ψ 1γ µ ψ 2 = ψ 1 S 1 γ µ Sψ 2 (7.122) S S 1 γ µ S = Λ µ νγ ν (7.123) ψ 1 γ µ ψ 2 ψ 1γ µ ψ 2 = Λ µ ν ψ 1 γ µ ψ 2 (7.124)

19 7.3 Dirac 141 (7.123) ψ 1 γ µ γ ν ψ 2 ψ 1γ µ γ ν ψ 2 = Λ µ ρλ ν σψ 1 γ ρ γ σ ψ 2 (7.125) γ µ γ ν σ µν σ µν = i 2 [ γµ,γ ν ] (7.126) σ 01 (7.113) Dirac j µ =(cρ, j )=(cψ ψ, cψ αψ ) (7.127) ψ β 2 = I β = γ 0 ψ = ψ γ 0 γ k = βα k j µ = cψγ µ ψ (7.128) j µ Lorentz { t + t x x (7.129) Lorentz Λ µ ν gµν { + γ P 1 γ µ P = g µν γ ν 0 (µ =0) = γ k (7.130) (µ =1, 2, 3) P P =e iϕ γ 0 (7.131) P P 1 =e iϕ (γ 0 ) 1 =e iϕ γ 0 (7.132) P 1 γ µ P = γ 0 γ µ γ 0 (7.133)

20 142 7 µ =0 ( γ 0) 3 = γ 0 µ =1, 2, 3 γ 0 γ k γ 0 = γ k γ 0 γ 0 = γ k (7.134) unitary P P = γ 0 e iϕ e iϕ γ 0 =(γ 0 ) 2 = I (7.135) P 1 = P (7.136) x p x p γ 5 ( ) 0 I γ 5 = γ 5 = iγ 0 γ 1 γ 2 γ 3 = (7.137) I 0 Dirac-Pauli ψ 1 γ 5 ψ 2 Lorentz ψ 1 ψ 2 ψ 1 γ µ ψ 2 ψ 1 γ 5 γ µ ψ 2 Lorentz Dirac ψ ψ = ψ γ 0 ψγψ 16 Γ 16 Lorentz S ψψ 1 P ψγ 5 ψ 1 V ψγ µ ψ 4 A ψγ 5 γ µ ψ 4 T ψσ µν ψ 6 16

21 7.3 Dirac 143 (7.137) γ 5 ( γ 5 ) 2 = ( I 0 0 I ) (7.138) ±1 Dirac +1 1 P + = 1+γ5 2 P = 1 γ5 2 (7.139) (7.138) ( P + ) 2 = P + ( P ) 2 = P P + P = 0 (7.140) P + + P = 1 (7.141) P ± P + γ 5 +1 P 1 ψ R = P + ψ ψ = ψ R + ψ L ψ L = P ψ (7.142) right-handed left-handed R L Hole Dirac Klein-Gordon p E E = ± p 2 c 2 + m 2 c 4 E =0 mc 2 <E<mc 2 Dirac Dirac Fermi Pauli

22 144 7 mc 2 mc mc 2 mc 2 7.1: hole hole hole 7.1 e + e + γ + γ (7.143) hole 7.1 Feynman Feynman e Feynman e

23 7.3 Dirac 145 t t e e e + e e e x x 7.2: e e + e e E e E e +e e + e + e e + E e + hole Dirac Dirac Hole Anderson positron ψ ψ Lorentz C Dirac

24 146 7 ψ c = C ψ T (7.144) ψ =(ψ ) T γ 0 ψ γ 0 C Dirac C C 1 γ µ C = (γ µ ) T (7.145) C unitary Dirac-Pauli C = iγ 2 γ 0 = (7.146) ψ ψ = ψ 1 ψ 2 ψ 3 (7.147) ψ 4 ψ ψ c = ψ 4 ψ 3 ψ 2 ψ 1 (7.148) (7.145) C γ 5 C 1 γ 5 C =(γ 5 ) T (7.149) ψ ψ c = C ψ T = C ( ψ γ 0) T = C (γ 0 ) T ψ (7.150) P L = 1 γ5 2 P R = 1+γ5 2 (7.151)

25 7.3 Dirac 147 γ 5 Dirac-Pauli (ψ L ) c = C P L ψ T = C (γ 0 ) T (P L ψ) (7.150) = C (γ 0 ) T P L ψ P L = P L = CP R (γ 0 ) T ψ γ 0 γ 5 = γ 5 γ 0 (7.152) = P R C (γ 0 ) T ψ = P R ψ c (7.150) γ 5 (ψ L ) c =(ψ c ) R (ψ R ) c =(ψ c ) L (7.153)

26 Maxwell Maxwell E B ρ em j em E = ρ em (7.154) E = B (7.155) t B = 0 (7.156) B = j em + E (7.157) t ρ em t + j em = 0 (7.158) φ A E = φ A (7.159) t B = A (7.160) (7.155) (7.156) 0 Lorentz A µ A µ ( ) ( ) φ φ A µ = c, A A µ = c, A (7.161) ρ em j em j µ em =(cρ em, j em ) j em µ =(cρ em, j em ) (7.162) µ j em µ = j em µ x µ = 0 (7.163)

27 F µν = µ A ν ν A µ (7.164) F µν = F νµ (7.165) F µµ =0 µ =0,ν =1, 2, 3 F 0k = 1 A k 1 φ = 1 ( A k + φ ) c t c x k c t x k = 1 c Ek (7.166) E k 1 µ<ν 3 ( ) F jk = Ak Aj A k = x j x k x j Aj x k (7.167) B 0 E 1 /c E 2 /c E 3 /c F µν = E 1 /c 0 B 3 B 2 E 2 /c B 3 0 B 1 (7.168) E 3 /c B 2 B 1 0 F µν (7.164) Lorentz F µν = g µρ g νσ F ρσ (7.169) F µν = 0 E 1 /c E 2 /c E 3 /c E 1 /c 0 B 3 B 2 E 2 /c B 3 0 B 1 E 3 /c B 2 B 1 0 (7.170) 1 4 F µν F µν = 1 ( ) E 2 2 c 2 B2 (7.171)

28 150 7 Lorentz Maxwell (7.154) (7.157) µ F µν = Fµν x µ = j ν em (7.172) (7.164) A µ µ µ A ν ν ( µ A µ )=j ν em (7.173) E B φ A χ(t, x) A A = A + χ (7.174) 0 B E φ φ = φ χ (7.175) t A µ A µ = A µ µ χ (7.176) (7.174) (7.175) (7.176) χ F µν F µν (7.176) S em = cε d 4 x ( 1 ) 4 F µν F µν (7.177) A µ Maxwell F µν

29 Maxwell S em Lorentz µ A µ = 1 c φ t c + A = 0 (7.178) Lorentz Lorentz χ Lorentz A 1 2 χ c 2 ( χ) = 0 (7.179) t2 A = 0 (7.180) Coulomb ρ em = j em =0 φ =0 ( ) 1 2 c 2 t 2 A = 0 (7.181) A Coulomb (7.180)

30 L.I. Schiff, 3. Advanced Quantum Mechanics, J.J. Sakurai, (Addison-Wesley Publishing Company, 1967) 4. Relativistic Quantum Mechanics, J.D. Bjorken and S.D. Drell, (McGrow-Hill, Inc., New York, 1964) 5. Relativistic Quantum Field Theory, J.D. Bjorken and S.D. Drell, (McGrow-Hill, Inc., New York, 1964) 6. Gauge Theories in Particle Physics, I.J.R. Aitchison and A.J.G. Hey, Graduate Student Series in Physics, (Adam Hilger, Bristol, Philadelphia, 1989) 7. I II

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