Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m

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1 Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m p µp ν {γ µ, γ ν } + m p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp ν γ ν + m η µν p µ p ν + m iγ µ µ + mψ ψ m {γ µ, γ ν } γ µ γ ν + γ ν γ µ

2 Dirac Lorentz 4 v µ v 0, v Lorentz v µ v µ Λ µ νv ν 5.6 v, w w µ v, w η µν v µ w ν η µν Λ µ σλ ν λv σ w λ 5.7 η µν Λ µ σλ ν λ η σλ 5.8 Lorentz Lorentz O3, Lorentz Lorentz Lorentz Lorentz facter γ 1/ 1 β 2 S β k S v 0 γv 0 + β k v k v 0 v 0 + β v v k γv k + β k v 0 v v + βv β 1 Λ µ ν ϵ 0 β 1 + t 1 iβ β k K k δ ν µ i ωρσ µ M ρσ ν 5.10 K 1 i , K 2 i , K 3 i L 1 i , L i, L 3 i O3, 1 SO3, 1 P,T

3 Dirac 40 L k K k [L i, L j ] iϵ ijk L k 5.13 [K i, K j ] iϵ ijk L k 5.14 [L i, K j ] iϵ ijk K k 5.15 L k K k i 0 0 t 0 L 3 k 0 e t k e k ϵ kijm ij 5.16 ˆM k0 M 0k 5.17 M µν M ij M 0i Λ µ ν ϵ v ν v µ i 1 2 ωρσ M ρσ µ ν v ν 5.18 Λ e i 1 2 ωµν M µν e iθi L i +β i K i 5.19 M σρ 12 4 v M ρσ µ ν iδ µ [ρ η σ]ν 5.20 δv µ 1 2 ωρσ M ρσ µ ν v ν i 1 2 ωρσ δ µ [ρ η σ]νv ν iω ρσ δ µ ρ η σν v ν iω µσ v σ 5.21 ω µν ω k0 β k ω 0k, ω ij ϵ ijk θ k 5.22 K i, L i η µν n 13 M [ ˆM ρσ, ˆM ρ σ ] iη ρ[ρ ˆMσσ ] η σ[ρ ˆMρσ ] i M i

4 Dirac 41 2 [ ˆM ρσ, ˆM ρ σ ] iη ρ[ρ ˆMσ ]σ η σ[ρ ˆMσ ]ρ 5.24 [, ] a a A [µ B ν] A µ B ν A ν B µ. M Proof : [M ρσ, M ρ σ ] δµ [ρ η σ]λδ λ [ρ η σ ]ν δ µ [ρ η σ ]λδ λ [ρ η σ]ν δ µ [ρ η σ][ρ η σ ]ν δ µ [ρ η σ ][ρη σ]ν δ ρ µ η σρ η σ ν δ µ ρ η σ ρη σν δ ση µ ρρ η σ ν + δ µ σ η ρ ρη σν δ ρ µ η σσ η ρ ν + δ µ ρ η σ ση ρν +δ ση µ ρσ η ρ ν δ µ σ η ρ ση ρν iη σρ M µ ρσ ν δ µ ρ η σ ρη σν }{{} IV η ρρ M µ σσ ν + δ µ σ η ρ ρη σν }{{} II η σσ M µ ρρ ν + δ µ ρ η σ ση ρν }{{} III +η ρσ M µ σρ ν δ µ σ η ρ ση ρν }{{} I iη ρ[ρ M σ ]σ µ ν η σ[ρ M σ ]ρ µ ν 5.25 M Lorentz [ ˆM ρσ, ˆM ρ σ ] iη ρ[ρ ˆMσ ]σ η σ[ρ ˆMσ ]ρ q.e.d Clifford d d {γ µ, γ ν } 2η µν 5.27 η µν γ µ Clifford 2 [ d 2 ] 2 [ d 2 ] 14 Clifford 14 [x] x γ 0 γ 0, γ k γ k 5.28

5 Dirac 42 k Σ µν 1 4i [γ µ, γ ν ] 5.29 M µν d 1 dd 1 Σ d SOd 2 2 [ d 2 ] Clifford 1 16 [[γ µ, γ ν ], [γ ρ, γ λ ]] 1 4 [γ µγ ν, γ ρ γ λ ] 1 4 γ µγ ν γ ρ γ λ 1 4 η µργ [ν γ λ] + η νρ 5.30 Clifford Σ µν i 4 [γ µ, γ ν ] [Σ µν, Σ ρλ ] iη µ[ρ Σ νλ] η ν[ρ Σ µλ] L i 1 2i γ jγ k cyclic, K i 1 2i γ iγ K i, L i ψ ψ ψ Uψ, ψ ψ ψu U e iθi L i +β i K i 5.34 ψ ψ γ 0

6 Dirac ψ ψ ψ ψu 1 γ 0 Σ ρσγ 0 Σ ρσ U 1 γ µ U Λ µ νγ ν 5.36 Σ µν ω ρσ [γ µ, Σ ρσ ] 1 2i ωρσ [{γ ρ, γ µ }, γ σ ] iω µσ γ σ ω M µ νγ ν Dirac Dirac iγ µ µ mψ Gamma 5.36 iγ µ µ mψ Uiγ µ µ mψ γ SL2, C K i, L i A ± j 1 2 L j ± ik j 5.40 A ± SU2 [A ± i, A± j ] iϵ ijka ± k and [A + i, A j ] SU2 SU2

7 Dirac 44 j +, j D j+,j n 2j j , 0 + 0, γ µ 0 σ µ σ µ σ µ α β 1, σ and σµ αβ 1, σ 5.44 Wess-Bagger 15 γ , γ k 0 σ k σ k L ψiγ µ µ mψ 5.46 ψ ψ γ ψ ψu ψ ψu γ 0 ψu Σ µν 1 4i [γ µ, γ ν ] 1 σ [µ σ ν] 0 i 4i 0 σ [µ σ ν] σ µν 0 0 σ µν L i 1 2i γ jγ k cyclic, K i 1 2i γ iγ WB p µ E, p σ µ p µ E + σ p ξ α

8 Dirac 45 L i σ i 0 0 σ i, K i i σ i 0 0 σ i γ iγ 0 γ 1 γ 2 γ P L γ5, P R γ5, ψ ψ L ψ R 5.53 L R L R i 0 σ ψ L 0 i 0 + σ ψ R σ µ α β 1, σ and σµ αβ 1, σ 17 iσ µ α β µ η β i 0 σ η 0 mξ 5.56 i σ µ αβ µ ξ β i 0 + σ ξ 0 m η 5.57 η β ξ β i 0 σ i 0 + σ σ Left-Right ψ L η i 0 σ ψ L E + σ pψ L 0 σ p E ψ L ψ L 5.59 σ µ W B σµ P S, σµ W B σµ P S 5.55

9 Dirac 46 iγ µ µ mψ m i 0 + σ i 0 σ m ψ 5.61 iγ µ µ + miγ µ µ mψ µ µ m 2 ψ Dirac Dirac ψ uke ik x, k 2 m γ µ k µ muk σ kσ k k 2 m σ k 0 u s α σ k s kα s 0 σ k α s σ kα s 5.66 α s 2 α 1 1, α

10 Dirac 47 Proof : m σ k m σ k uk σ k m σ k σ k σ k σ k m σ k α α α q.e.d. ψx vke ik x 5.69 γ µ k µ mvk σ kα v s s k σ kα s d 3 k 1 ψx b s 2π 3 2Ek ku s ke ik x + d s k vs ke k x s d ψx 3 k 1 d s 2π 3 2Ek k v s ke ik x + b s k ūs ke k x ψ γ s anticommutation relation {b s k, b s k } {d s k, d s k } δ ss 2π 3 δ 3 k k 5.73 {ψx, ψx } eq d 3 kd 3 k 1 2π 6 2 E k E k s,s {b s k, b s k }u s kū s k e ikx k x + {d s k, ds k }vs k v s k e ikx k x d 3 k 1 u s kū s ke ikx x + v s k v s ke ikx x 2π 3 2E k s d 3 k 1 e ikx x γ µ k + 2π 3 2E k µ γ 0 δ 3 x x 5.74 k k E k, k u s kū s k s m σk σk m γ µ k µ + m

11 Dirac 48 v s k v s k s m σk σk m γ µ k µ m 5.75 {ψx, ψ x } eq δ 3 x x Propagator S ret x y 0 {ψx, ψy} 0 x 0 >y 0 iγµ x µ + md ret x y 5.77 Proof : 2 0 ψx ψy 0 d 3 kd 3 k 1 2π 6 2 u s kū s k e ikx k y 2π 3 δ 3 k k E k E k s,s d 3 k 1 γ µ k 2π 3 µ + me ikx y 2E k d iγ µ µ x 3 k 1 + m e ikx y π 3 2E k 0 ψyψx 0 d 3 kd 3 k 1 2π 6 2 v s k v s k e iky k x 2π 3 δ 3 k k E k E k s,s d 3 k 1 γ µ k 2π 3 µ me iky x 2E k d iγ µ 3 k 1 µ + m e iky x π 3 2E k S F x y 0 T ψx ψy 0 θx 0 y 0 0 ψx ψy 0 θy 0 x 0 0 ψyψx 0 i / x + md F x y 5.80 { ψxψy x 0 > y 0 T {ψxψy} 5.81 ψyψx y 0 > x 0 18

12 Dirac 49 S F x y d 4 k i γ µ k µ + m 2π 4 k 2 m 2 + iϵ eikx y d 4 k i 2π 4 k/ m + iϵ eikx y 5.82 k/ γ µ k µ 5.83 k/ m k/ + m k 2 m i / x ms F x y iδ 4 x y ψ π ψ iψ 19 L ψiγ µ µ mψ ψ γ 0 iγ iγ k k mψ 5.86 H dx 3 ψ iγ k k + mψ 5.88 H dx 3 ψ iα + βmψ 5.89 α k γ 0 γ k β γ 0 H α p + mβ 5.90 Dirac i t ψ Hψ H π ψ L 5.87

13 Dirac 50 H d 3 k 2π 3 2 E k b s k b s k + d s k d s k 5.92 P s1 d 3 xψ i ψ d 3 k 2π 3 2 kb s k b s k + d s k d s k 5.93 s1 j µ e ψγ µ ψ 5.94 Q d 3 k 2π 3 2 b s k b s k d s k d s k 5.95 s1 b 0 d Pauli γ 20 γ and γ k 0 σ k σ k cp µ γ µ + Mc 2 ψ Eγ 0 cp k γ k + Mc 2 ψ γ 0 γ Eψ + cp k α k + Mc 2 βψ α γ 0 γ k 0 σ k σ k 0 and β γ U

14 Dirac 51 H c α p + Mc 2 β Mc 2 c σ p c σ p Mc Mc 2 E c σ p Hψ Eψ ψ c σ p Mc 2 E Mc 2 E det c σ p c σ p Mc 2 E E 2 M 2 c 4 c σ p 2 E 2 M 2 c 4 c 2 p E ±E p ± M 2 c 4 + c 2 p on shell E E p E E p ϕ ψ 2 χ Mc 2 Eϕ + c σ pχ 0 and c σ pϕ Mc 2 + Eχ E E p > 0 χ c σ p E p + Mc ϕ χ ϕ ϕ Large component, χ small component E p Mc 2 ϕ c2 σ p 2 E p + Mc Eϕ p2 2M ϕ + O p 2 M 2 c

15 Dirac Discrete Symmetries 1. Parity: ψt, x γ 0 ψt, x P Pb s k P η b b s k, Pds k P η b d s k PP PP 1, ηb 2, η2 d ±1 d 3 k 1 PψxP b s 2π 3 2Ek ku s ke ik x + d s k vs ke k x s d 3 k 1 η 2π 3 b b s 2Ek ku s ke i k x + ηdd s k vs ke k x s k k k 0, k k σα k σα uk γ k σα k 0 u k σα vk k σα k σα k σα γ k 0 v k σα d PψxP γ 0 3 k 1 η 2π 3 b b s ku s ke i k x η 2Ek dd s k vs ke k x ψt, x η d η b Parity bilinear s P ψψp η 2 b ψψt, x bilinear parity η b 1 bilinear Pψt, xp γ 0 ψt, x P ψγ 5 ψp ψγ 5 ψt, x P ψγ µ ψp ψγ { ψγ 0 γ µ γ 0 µ ψt, x for µ 0 ψt, x ψγ µ ψt, x for µ k 5.117

16 Dirac Time reversal:c number T ct c anti-linear involution, ψ T ψt, xt γ 1 γ 3 ψ t, x T b s kt b s k, T ds kt d s k CP T CPT

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