[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin + 8 5 Clifford Spin 10 A 12 B 17 1 Clifford Spin D Euclid Clifford Γ µ, µ = 1,, D {Γ µ, Γ ν } = 2η µν 1. (1) η µν 1 (µ = ν = 1, 2,, t), η µν = +1 (µ = ν = t + 1,, D), 0 (µ ν). t, s := D t (Γ µ ) 2 = 1, (µ = 1,, t), (Γ µ ) 2 = +1, (µ = t + 1,, D), Γ µ Γ ν = Γ ν Γ µ, µ ν Clifford 1
Clifford so(t, s) Clifford so(t, s) Γ µ Clifford 1 J µν J µν := 1 2 Γµν Γ µ 1µ 2...µ r := Γ [µ 1 Γ µ2 Γ µ r ] J µν [J µν, J ρσ ] = η νρ J µσ + η µσ J νρ η µρ J νσ η νσ J µρ so(t, s) 2 exp Spin(t, s) Clifford Γ µ U Γ µ := U Γ µ U 1 (1) Clifford Γ µ Γ µ 2 Euclid Clifford Euclid η µν = δ µν Clifford {Γ µ, Γ ν } = 2δ µν 1 (2) Euclid Clifford 2.1 D = 2n b 1 := 1 2 (Γ1 + iγ 2 ), b 1 := 1 2 (Γ1 iγ 2 ), (3) b n := 1 2 (Γ2n 1 + iγ 2n ), b n := 1 2 (Γ2n 1 iγ 2n ). 1 2 2
(3) {b A,b B } = δ AB, {b A,b B } = {b A,b B } = 0 + + + b A + + + = 0, A = 1,, n b 1 b 2 b 1 + + + =: + +, + + + =: + +, + =: + b 1 b 2 = b 2 b 1 b 2 + = + 2 n ± ± ± 2 n [1] Clifford 3 2 b 1 + = 0, b 1 + =:, b 1 = 0, b 1 = b 1 b 1 + = (1 b 1 b 1) + = + 3 Clifford Spin(2n) Clifford 3
+ = ( 1 0), = ( 0 1) b 1 = 0 0, b 1 = 0 1 1 0 0 0 2n Γ 1 = b 1 + b 1 = 0 1 =: σ 1, Γ 2 = 1 1 0 i (b 1 b 1 ) = 0 i =: σ 2 i 0 Γ 1 = σ 1 1 1, Γ 2 = σ 2 1 1, Γ 3 = σ 3 σ 1 1, Γ 4 = σ 3 σ 2 1, (4) 2.1.1 C Γ µ Clifford (Γ µ ) T Γ µ Clifford C η, η = ±1 C η C η Γ µ C 1 η = η Γ µt, C η C η = 1. (5) C + = σ 1 σ 2 σ 1, C = σ 2 σ 1 σ 2. (6) C A C ϵ (6) C T η = ϵ C η 4
n mod 4 1 2 3 4 ϵ when η = + + + ϵ when η = + + ϵ = ( 1) [n/2] η n [ ] Gauss 2.2 D = 2n + 1 D = 2n + 1 Γ 1 Γ 2n D = 2n Γ 2n+1 Γ 2n+1 = ( i) n Γ 1 Γ 2 Γ 2n (7) Clifford (2) Clifford (7) Γ 2n+1 Γ 2n+1 = ( i) n Γ 1 Γ 2 Γ 2n (8) (7) (8) Γ µ U Γ µ U 1 Clifford Γ µ Γ µ Γ µt Γ µ Γ µ η = ± ξ CΓ 2n+1 C 1 = ξ (Γ 2n+1 ) T ξ (7) C,C 1 CΓ 2n+1 C 1 = ( i) n CΓ 1 C 1 CΓ 2 C 1 CΓ 2n C 1 = ( i) n (Γ 1 ) T (Γ 2 ) T (Γ 2n ) T = ( i) n (Γ 2n Γ 1 ) T = ( 1) n ( i) n (Γ 1 Γ 2n ) T = ( 1) n (Γ 2n+1 ) T 5
ξ = ( 1) n C C ξ 2.3 Clifford Clifford D = 2n D = 2n + 1 D = 2n + 1 Γ µ Γ 1 Γ 2n+1 = ±i n C CΓ µ C 1 = η (Γ µ ) T, C C = 1 D = 2n D = 2n + 1 ξ = ( 1) n η η = ξ C C T = ϵ C ϵ D mod 8 D mod 8 1 2 3 4 5 6 7 8 ξ + + + + η + + + + + + ϵ + + + + + + 1 3 Euclid Spin 3.1 Dirac Weyl D = 2n Γ µ Clifford J µν = 1 2 Γµν so(2n) ψ Dirac (7) Γ 2n+1 6
[Γ 2n+1, J µν ] = 0 Γ 2n+1 Γ 2n+1 ψ = ±ψ so(2n) so(2n) Weyl Γ 2n+1 chirality D = 2n + 1 Clifford so(2n + 1) Dirac D = 2n + 1 Clifford so(2n + 1) 3.2 D = 2n D = 2n + 1 C C ψ Dirac ψ c ψ c = C 1 ψ (9) ψ c Dirac so(d) (J µν ψ ) c = J µν ψ c (5) Γ µ (ψ c ) c = (C 1 ψ ) c = C 1 C 1 ψ = ϵ C 1 C 1 ψ = ϵ ψ ϵ = +1 Dirac ψ c = ψ Majorana C χ,ψ Dirac C αβ χ α ψ β = χ T Cψ = χ c ψ so(d) 7
Majorana ϵ = +1 Dirac C = 1 A.3 Γ µ ϵ = 1 Dirac Dirac ϵ Dirac Weyl 3.3 Weyl D = 2n ψ Γ 2n+1 ψ = aψ, a = ± Weyl (9) ψ c Γ 2n+1 Γ 2n+1 ψ c = C 1 CΓ 2n+1 C 1 ψ = C 1 ξ Γ 2n+1 ψ = ξ ac 1 ψ = ξ aψ c ψ c chirality ξ a ξ = 1 D = 4l + 2 ψ ψ c Weyl ξ = +1 D = 4l ψ ψ c D = 4l ϵ = +1 C 1 D = 8k ϵ = +1 Weyl C = 1 Γ µ Γ 2n+1 Weyl ψ c = ψ Majorana-Weyl D = 8k + 4 ϵ = 1 Weyl 3.4 Spin(D) 2 4 Euclid Pin + Clifford Pin + (D) Pin + (D) O(D) Spin(D) + R R 2 = 1 8
D mod 8 1 2 3 4 5 6 7 8 ξ + + + + η + + + + + + ϵ + + + + + + M M,W W M,W M M,W,MW C, R, PR R C PR PR PR C R R 2 Spin(D) M: Majorana, W: Weyl, MW: Majorana-Weyl C, R, PR Spin(D) Dirac, Weyl C: R: PR: Clifford Spin(D) exp ( θ µν Γ µν ) µ Γ µ 4.1 D = 2n chirality Dirac Clifford Pin + (2n) Pin + (2n) Γ µ C = C + Pin + (2n) д CдC 1 = д ϵ = + ϵ = 4.2 D = 2n + 1 Dirac Clifford Pin + (2n + 1) CΓ µ C 1 = ξ Γ µ ξ = 1 D = 4l + 3 ξ = +1 D = 4l + 1 ϵ = +1 ϵ = 1 4.3 Pin + (D) Pin + (D) C η = +1 9
D mod 8 1 2 3 4 5 6 7 8 ξ + + + + ϵ + + + + C, R, PR R R C PR PR PR C R 3 Pin + (D) η = +1 5 Clifford Spin Γ µ E, µ = 1,, 2n (4) Euclid Clifford D = 2n + 1 (7) (8) Γ 2n+1 E Γ µ {Γ µ E, Γν E } = 2δ µν 1 Γ µ iγ µ E, µ = 1,, t, := Γ µ E, µ = t + 1,, D, (1) Clifford C (6) C η i CΓ µ C 1 = η (Γ µ ) T η = ξ 5.1 Dirac ψ Dirac ψ Dirac ψ ψ := ψ Γ 1 Γ t 10
Spin(t, s) д ψ дψ ψ ψд 1 ψ, χ Dirac ψ χ Spin(t, s) 5.2 B Dirac ψ ψ c ψ c := C 1 ψ T ψ c Dirac B B B 1 = C 1 (Γ 1 Γ t ) T (10) ψ c := B 1 ψ B BΓ µ B 1 = ηγ µ, B T = ϵb. (11) ξ = ( 1) [ s t 2 ], ϵ = ( 1) [ s t 4 ] η [ s t 2 ] (12) η = ( 1) t η ± η = ξ B Euclid C, η, ξ, ϵ B, η, ξ, ϵ Dirac ψ (ψ c ) c = (B 1 ψ ) c = B 1 B 1 ψ = ϵb 1 B 1 ψ = ϵψ ϵ = 1 ψ c = ψ Majorana B = 1 D = 2n Weyl chirality Γ 2n+1 = ( i) n t Γ 1 Γ 2n s 2n + 1 Γ 2n+1 BΓ 2n+1 B 1 = ξ Γ 2n+1 11
ΓE 2n+1 ψ = aψ, a = ± Γ 2n+1 ψ c = B 1 BΓ 2n+1 E B 1 ψ = ξb 1 Γ 2n+1 ψ = ξab 1 ψ = ξaψ c ξ = 1 ξ = +1 ξ = +1 ϵ = +1 Majorana-Weyl Spin(s, t) B B Majorana(-Weyl) ψ c = ψ B = 1 B C 4 (s t) mod 8 1 2 3 4 5 6 7 8 ξ + + + + η + + + + + + ϵ + + + + + + M M,W W M,W M M,W,MW C, R, PR R C PR PR PR C R R E 4 Spin(s, t) A A.1 C U Γ µ Clifford Γ µ Γ µ = U Γ µ U 1 (13) Γ µ Clifford C 1 12
1 ϵ (5) C CΓ µ C 1 = η Γ µt (13) U T CU Γ µ U 1 CU 1T = η Γ µt C := U T CU (14) C Γ µ C 1 = η Γ µt C Γ µ C (14) C C C T = (U T CU ) T = U T C T U = ϵ U T CU = ϵ C C ϵ B C ϵ A.2 C C Clifford C 2 η C C CΓ µ C 1 = η(γ µ ) T, C Γ µ C 1 = η(γ µ ) T (15) a C = ac V V := C 1 C (15) V Γ µ V 1 = C 1 CΓ µ C 1 C = ηc 1 (Γ µ ) T C = Γ µ 13
V Γ µ Schur 4 V V = a1, (a ) C = ac C C a = 1 a A.3 C C = 1 [3] 3 C N N C = U T U U C A, B C = A + ib C 1 = C C = (A ib)(a + ib) = A 2 + B 2 + i(ab BA) [A, B] = 0, A 2 + B 2 = 1 (16) A B O a 1 A = O T a 2 a N b 1 O, B = O T b 2 b N O (16) a 2 i + b 2 i = 1 θ i 4 Schur Clifford Clifford Pin + Schur 14
a i = cos θ i, b i = sin θ i a 1 + ib 1 C = A + ib = O T a 2 + ib 2 a 1 + ib 1 = O T a 2 + ib 2 = O T e iθ 1/2 e iθ 2/2 e iθ N /2 O a N + ib N O = O T a N + ib N e iθ 1/2 e iθ 2/2 e iθ 1 e iθ 2 e iθ N /2 O e iθ N O U = e iθ 1/2 e iθ 2/2 e iθ N /2 O C = U T U A.4 C C (2K) (2K) Ω 0 1 1 0 Ω = 0 1 1 0 0 1 1 0 15
4 C (2K) (2K) C = U T ΩU 5 A N N O 0 a 1 a 1 0 A = O T 0 a 2 a 2 0 0 a K a K 0 0 O 0 A.5 5 A 2 v j c 2 j, c j 0 A 2 v i = c j 2 v i c j u 1,,u N a 1, a 2,, a K, K N /2 c 1 = 0 Av 1 = 0 Av 1 Av 1 2 = v T 1 AT Av 1 = v T 1 A2 v 1 = 0 c j c 1 c j 0 Av j = 0 u j = v j 16
c 1 > 0 u 1 := v 1 u 2 = Au 1 /c 1 A 2 u 2 = c 2 1 u 2, u 2 u 1 = 0, u 2 2 = 1 u 1,u 2 a 1 = c 1 Au 1 = a 1 u 2, Au 2 = a 1 u 1 A(u 1 u 2 ) = (u 1 u 2 ) 0 a 1 a 1 0 u 1,u 2 A 2 A 2 v 3,,v N u 1,u 2 c 2 j v 3 c 3 = 0 u j = v j c 3 > 0 u 3 = v 3, u 4 = Av 3 /c 3, a 2 = c 3 0 a 1 a A(u 1 u 2 u 3 u 4 ) = (u 1 u 2 u 3 u 4 ) 1 0 0 a 2 a 2 0 u 1,,u N a 1, a 2,, a K 0 a 1 a 1 0 A(u 1 u 2 u N ) = (u 1 u 2 u N ) 0 a 2 a 2 0 0 a K a K 0 0. 0 O = (u 1 u 2 u N ) T 5 17
B (11), (12) (10) B = (Γ 1 Γ t ) 1T C = ( 1) t (Γ 1 ) T (Γ t ) T C = ( 1) t η t CΓ 1 Γ t =: bcγ 1 Γ t b (11) m B.1 D = 2n µ = 1,, t ( 1) m = ( 1) m, ( 1) m(m 1)/2 = ( 1) [ m 2 ] BΓ µ B 1 = CΓ 1 Γ t Γ µ (Γ t ) 1 (Γ 1 ) 1 C 1 = ( 1) t 1 CΓ µ C 1 = ( 1) t 1 η (Γ µ ) T = ( 1) t η Γ µ Γ µ µ = t + 1,, D BΓ µ B 1 = ( 1) t η Γ µ η = ( 1) t η (11) B T = b(cγ 1 Γ t ) T = b(γ t ) T (Γ 1 ) T C T = ϵ b(γ t ) T (Γ 1 ) T C = ϵ η t bcγ t Γ 1 = ϵ η t ( 1) t(t 1)/2 bcγ 1 Γ t = ϵ η t ( 1) t(t 1)/2 B ϵ = ϵ η t ( 1) t(t 1)/2 = ( 1) n(n 1)/2 η n η t ( 1) t(t 1)/2 = ( 1) n(n 1)/2 t(n+t) t(t 1)/2 η n t ( 1) n(n 1)/2 t(n+t)+t(t 1)/2 = ( 1) 1 2 (n t)(n t 1) = ( 1) [ s t 4 ] ϵ = ( 1) [ s t 4 ] η [ s t 2 ] (11) (12) 18
B.2 D = 2n + 1 s = 0 B = (phase)c η = η, ξ = ξ, ϵ = ϵ (11)(12) s 1 Γ 2n+1 BΓ 2n+1 B 1 = ξ Γ 2n+1 ξ BΓ 2n+1 B 1 CΓ 1 Γ t Γ 2n+1 (Γ t ) 1 (Γ t ) 1 C 1 = ( 1) t CΓ 2n+1 C 1 = ( 1) t ( 1) n Γ 2n+1 ξ = ( 1) n t = ( 1) [n t+ 1 2] = ( 1) [ s 1+t 2 t+ 1 2] = ( 1) [ s t 2 ] (11) η = ξ ϵ s = s 1 s t 4 ϵ = ( 1) (11) (12) [ ] s t [ ] s t 4 η 2 = ( 1) [ s t 4 ] η [ s t 2 ] [1], Web. [2] J. Polchinski, String Theory vol. 2 Appendix. [3] B. Zumino, Normal Forms of Complex Matrices. J. Math. Phys. 3(1962)1055 1057 19