SO(2)

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12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6 12.5 SO(3).......................... 7 12.6......................... 9 12.7 SO(4) SO(5)............................. 10 12.8................................ 11 12.9..................... 12 12.10 SU(3)......................... 13 2

12 12.1 G G G ( ) 0 n GL(n, C) 1 n SL(n, C) SL(n, C) GL(n, C) n : GL(n, C) n det A 0 2n 2 SL(n, C) n det A = 1 2n 2 2 U(n) n A A = δ n 2 SU(n) n A A = δ, det A = 1 n 2 1 n : GL(n, R) n det A 0 n 2 SL(n, R) n det A = 1 n 2 1 O(n) n A T A = δ n(n 1)/2 SO(n) n A T A = δ, det A = 1 n(n 1)/2 3

δ R C 12.2 G δ ɛ δ + ɛx G X <G> G 2 (X <G> a R) ax <G>. [ ] ɛ δ + ɛx G. ɛ = ɛ a ɛ δ + ɛ ax G. ax <G>. [ ] X, Y <G> X + Y <G>. [ ] G (δ + ɛx)(δ + ɛy ) G. ɛ δ + ɛ(x + Y ) G. X + Y <G>. [ ] <G> T a (a = 1, 2,, N) <G> θ a ( ) X = θ a T a (1) ( ) 2 X <G> e X G. (2) [ ] G n (δ + ɛx) n G. ɛ δ + ɛx = e ɛx e nɛx G. nɛ = 1 n [ ] X, Y <G> [X, Y ] <G>. (3) [ ] e ɛx e ɛy e ɛx e ɛy G. ɛ δ + ɛ 2 [X, Y ] G. [X, Y ] <G>. [ ] (3) T a [T a, T b ] = f abc T c 4

f abc (1)(2) A(θ) = e θ at a G. T a G 2 ( ) M M ( T a N) (* ) 2 G, G ab G (f(ab) = f(a)f(b)) f : G G f f ( ) G G 12.3 n O(n) (n 2) A T A = δ n A ( δ ) det A T = det A (det A) 2 = 1 det A = ±1. O(n) 2 det A = +1 n SO(n) O(n) SO(n) : P = diag(1,, 1, 1) SO(n) A = δ + ɛx ( ɛ ) 5

X T = X <SO(n)> n < SO(n) > n(n 1)/2 SO(n) 12.4 SO(2) n = 2 X T = X θ ( ) 0 1 X = θt, T = 1 0 T 1 0 SO(2) = { e θt θ R } [X, Y ] = 0 [e X, e Y ] = 0 SO(2) 1 SO(2) ( ) { ( ) } 0 θ iθ 0 A(θ) = e θt = exp = exp U U θ 0 0 iθ ( ) ( ) e iθ 0 cos θ sin θ = U 0 e iθ U =. U = 1 ( ) 1 1. sin θ cos θ 2 i i U 2 δ (n = 0 mod 4) T n T (n = 1 mod 4) = δ (n = 2 mod 4) T (n = 3 mod 4) A(θ) = e θt = n=0 1 (θt )n n! = δ + θt 1 2! θ2 δ 1 3! θ3 T + 1 4! θ4 δ + 1 5! θ5 T + ( ) cos θ sin θ = cos θ δ + sin θ T = sin θ cos θ 6

SO(2) 1 { 1 } e θ. R + { i } e iθ. 1 U(1) : U(1) = { e iθ θ R }. A(θ)A(φ) = A(θ + φ) 0 {δ}. 12.5 SO(3) n = 2 n = 3 n = 3 X T = X X = θ a T a, a = 1, 2, 3, 0 0 0 0 0 1 0 1 0 T 1 = 0 0 1, T 2 = 0 0 0, T 3 = 1 0 0 0 1 0 1 0 0 0 0 0 3 ɛ abc (T a ) bc = ɛ bac T a SO(3) SO(3) = { e θ at a θ a R, a = 1, 2, 3 } [T a, T b ] = ɛ abc T c <SO(3)> ɛ abc SO(3) 3 7

[J a, J b ] = iɛ abc J c ( ) ij a ɛ abc J a (R (l) a ) mm = <l, m J a l, m > a l ɛ abc {e iθ ar (l) a } SO(3) l (2l + 1) ir (l) J 1 = 1 2 (J + + J ), J 2 = 1 2i (J + J ), J ± l, m> = (l m)(l ± m + 1) l, m ± 1>, J 3 l, m> = m l, m>. l = 0 R (0) a = 0 1 {δ} l = 1/2 +> = 1 2, + 1 2 >, > = 1 2, 1 2 > <+ J 1 > = 1 2, < J 1 +> = 1 2, <+ J 2 > = i 2, < J 2 +> = i 2, <+ J 3 +> = 1 2, < J 3 > = 1 2 0 R a (1/2) = σ ( ) ( ) ( ) a 0 1 0 i 1 0 σ 1 =, σ 2 =, σ 3 = 2, 1 0 i 0 0 1 σ a : σ a = σ a, tr σ a = 0 8

M det e M = e trm ( ) {e iθ aσ a /2 } 1 2 SU(2) : SU(2) = { e iθ aσ a /2 θ a R, a = 1, 2, 3 }. SO(3) 2 l = 1 R a (1) = τ a, τ 1 = 1 0 1 0 1 0 1, τ 2 = i 0 1 0 1 0 0 1 0 1, τ 3 = 0 0 0 2 2 0 1 0 0 1 0 0 0 1 {e iθ aτ a } 3 4 5 a 3 (z ) (R (l) 3 ) mm = mδ mm R (l) 12.6 SO(3) e θ at a θ a = θn a, n a n a = 1 n a θ n a T a n a T a = U diag(0, i, i) U, n 1 γ(n 2 1 1) γ(n 2 1 1) 1 U = n 2 γ(n 1 n 2 + in 3 ) γ(n 1 n 2 in 3 ), γ = 2(1 n 2 n 3 γ(n 1 n 3 in 2 ) γ(n 1 n 3 + in 2 ) 1 ) e θ at a = e θn at a = U diag(1, e iθ, e iθ ) U cos θ + n 2 1(1 cos θ) n 1 n 2 (1 cos θ) n 3 sin θ n 1 n 3 (1 cos θ)+n 2 sin θ = n 2 n 1 (1 cos θ)+n 3 sin θ cos θ + n 2 2(1 cos θ) n 2 n 3 (1 cos θ) n 1 sin θ n 3 n 1 (1 cos θ) n 2 sin θ n 3 n 2 (1 cos θ)+n 1 sin θ cos θ + n 2 3(1 cos θ) 3 9

12.7 SO(4) SO(5) n = 4 X T = X 0 θ 3 θ 2 φ 1 θ X = 3 0 θ 1 φ 2 θ 2 θ 1 0 φ 3 = θ at a + φ a S a φ 1 φ 2 φ 3 0 (T a ) ij = ɛ iaj4, (S a ) ij = δ ai δ 4j δ aj δ 4i, a = 1, 2, 3 <SO(4)> 6 : SO(4) = { e θ at a +φ a S a θ a, φ a R, a = 1, 2, 3 }. [T a, T b ] = ɛ abc T c, [T a, S b ] = ɛ abc S c, [S a, S b ] = ɛ abc T c T a i 2 σ a, S a ± i 2 σ a ( σ a ) { e (i/2)(θ a φ a )σ a } SO(4) 2 n = 5 (T a ) ij = ɛ iaj45, (S a ) ij = δ ai δ 4j δ aj δ 4i, (S a) ij = δ ai δ 5j δ aj δ 5i, (R) ij = δ 4i δ 5j δ 4j δ 5i SO(5) = { e θ at a +φ a S a +φ as a+ωr θ a, φ a, φ a, ω R, a = 1, 2, 3 }. [T a, T b ] = ɛ abc T c, [S a, S b ] = ɛ abc T c, [S a, S b] = ɛ abc T c, [T a, S b ] = ɛ abc S c, [T a, S b] = ɛ abc S c, [T a, R] = O, [S a, S b] = δ ab R, [S a, R] = S a, [S a, R] = S a 10

( O ) 2 T a i ( ) σa 0, S a i ( ) 0 σa, 2 0 σ a 2 σ a 0 S a 1 ( ) 0 σa, R i ( ) δ 0 2 σ a 0 2 0 δ <SO(5)> 4 [σ a, σ b ] = 2iɛ abc σ c, {σ a, σ b } = 2δ ab δ {A, B} = AB + BA 12.8 : [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0. [T a, T b ] = f abc T c f bcd f ade + f cad f bde + f abd f cde = 0 f bac = f abc f cad f dbe f cbd f dae = f abd f cde (F a ) bc = f bac F a ( ) 3 SO(3) T a SU(2) SO(3) [ ] SO(4) = { e θ AT A θ A R, A = 1, 2,, 6 } (T a ) ij = ɛ iaj4, (T a ) ij = δ ai δ 4j δ aj δ 4i, a = 1, 2, 3, a = a + 3 = 4, 5, 6. [ ] [T a, T b ] = ɛ abc T c, [T a, T b ] = ɛ abc T c, [T a, T b ] = ɛ abc T c 11

f abc = f ab c = f a bc = f a b c = ɛ abc ( 0 ). (F A ) BC = f BAC ( ) ( ) Ea O O Ea F a =, F a =, (E a ) bc = ɛ bac O O E a E a { e θ AF A } SO(4) (6 ) [ ] 12.9 n U(n) A A = δ n A A = δ + ɛx ( ɛ ) X = X X n T µ n X = iθ µ T µ T µ n n 2 T µ n 2 0 n 2 1 U(n) U(n) = {e iθ µt µ θ µ R, µ = 0, 1,, n 2 1 } n 2 T µ µ ν (T µ) ij (T ν ) ij = 0 (T µ ) ji (T ν ) ij = 0 tr(t µ T ν ) = 0. T 0 U(n) = {e iθ 0 e iθ at a } = U(1) {e iθ at a } a = 1, 2,, n 2 1 T 0, T a tr T a = 0 e iθ at a 1 SU(n) n 2 U(n) = U(1) SU(n), 12

SU(n) = {e iθ at a θ a R, a = 1,, n 2 1} T a tr(t a T b ) = 1 2 δ ab SU(2) T a = σ a /2 <SU(n)> f abc [ it a, it b ] = f abc ( it c ) [T a, T b ] = if abc T c. tr ([T a, T b ] T c ) = i 2 f abc f abc a, b, c 12.10 SU(3) SU(3) T a = λ a /2. 0 1 0 0 i 0 1 0 0 λ 1 = 1 0 0, λ 2 = i 0 0, λ 3 = 0 1 0, 0 0 0 0 0 0 0 0 0 0 0 1 0 0 i λ 4 = 0 0 0, λ 5 = 0 0 0, 1 0 0 i 0 0 0 0 0 0 0 0 λ 6 = 0 0 1, λ 7 = 0 0 i, λ 8 = 1 1 0 0 0 1 0 3 0 1 0 0 i 0 0 0 2 T a 0 tr(t a T b ) = (1/2)δ ab SU(3) SU(3) = {e iθ aλ a /2 θ a R, a = 1, 2,, 8 } 13

<SU(3)> f abc tr ([λ a, λ b ] λ c ) = 4if abc f abc 8C 3 = 56 a < b < c f 123 = 1, f 147 = 1 2, f 156 = 1 2, f 246 = 1 2, f 257 = 1 2, f 345 = 1 2, f 367 = 1 2, f 458 = 3 2, f 678 = 3 2 0 ( ) SU(n) SU(n) SO(3) (* ) Excel VBA / for VBA URL: http://amonphys.web.fc2.com/ 14

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