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1 A A.1 G = A,B,C, A,B, (1) A,B AB (2) (AB)C = A(BC) (3) 1 A 1A = A1 = A (4) A A 1 A 1 A = AA 1 = 1 AB = BA ( ) AB BA ( ) 3 SU(N),N 2 (Lie) A(θ 1,θ 2,θ n ) = exp(i n i=1 θ i F i ) (A.1) F i 2 0 θ 2π 1 < v < 1 < η < U(A) U(A)U(B) = U(AB) U(A 1 ) = U 1 (A) U(1) = 1( ) G (n) SO(N) x i (i = 1 N) N L = N i=1 x2 i O(N) n(n 1) SO(N) 2 SU(N) Special Unitary Group u i (i = 1 N) N L = N i=1 u i 2 1 n(n 1) SU(N) N N N 2 1 θ i F i ( U = exp i N i=1 θ i F i ) TrF i = 0, F i, F j = i f i jk (A.2)

A 2 * 1) F i SU(N) f i jk SU(2) F i = τ i 2 SU(3) F i = λ i 2 A.2 x x r y = R i j y = Rr z r = x 2 + y 2 + z 2 z r 2 = r T r = r T R T Rr = r 2 = r T r (A.3) (A.4a) R T R = 1 (A.4b) R 3X3 V = (V x,v y,v z ) z θ cosθ sinθ 0 = sinθ cosθ 0 0 0 1 V x V y V z V x V y V z V = R z (θ)v (A.5) θ θ δθ R z (δθ) = 1 + is z δθ S z = 1 dr 0 i 0 z = i 0 0 (A.6) i dθ θ=0 0 0 0 R z (θ) = lim n δθ=θ/n (1 + ij z δθ) n = expis z θ (A.7) S z z (generator) x 0 0 0 0 0 i 0 i 0 S x = 0 0 i S y = 0 0 0 S z = i 0 0 0 i 0 i 0 0 0 0 0 * 1) expiλ n( θ 2 ) = expiq U = exp iq TrU = TrS 1 US S U η i detu = i η i = exp( i logη i ) = e Tr(logU) = e itrq TrQ = 0 detu = 1 (A.8)

A 3 +1 i jk 123 S i, S j = iε i jk S k ε i jk = 1 i jk 123 (A.9) ε i jk O(3) L = r p = i(x i j x j i ) x i = (x,y,z) (A.10) S = (S x,s y,s z ) S J = L + S (A.11) n θ ( ) R (A.4b) O(3) ( ) R(n,θ) = exp ij nθ (A.12) A.3 SU(2) U(2) 2 ( ) ψ = (A.13) 2 2 SU(2) (detu = 1) ψ ψ = Uψ ξ 1 ξ 2 = a c b d (A.14) U 1 = U detu = 1 (A.15) a U = b b, a 2 + b 2 = 1 (A.16) a

A 4 A.1 (A.16) ξ 2 0 1 ξ = 1 1 0 ψ = ξ 1 ξ 2 ******************************************: = iσ 2 ψ (A.17) N N 2 1 λ (λ i ; i = 1 N 2 1) θ n (n 2 = 1) U = exp iλ n ( θ2 ) (A.18) ******************************************: N = 2 λ i = σ i s i = σ i /2 SU(2) U = exp i σ 2 θ = cos θ 2 + sin θ 2 iσ n (A.19) s i, s j = iε i jk s k (A.20) SU(2) O(3) O(3) H i j = ξ i ξ j (i, j = 1 2) H i j 2 2 4 H = a1 + b σ = a1 + b z b x + ib y b x ib y = A + B (A.21) b z SU(2) H H = UHU = A + B = a 1 + b σ (A.22) A = a 1 = UAU = Ua1U = a1 a = a (A.23) a SU(2) a (A.21) ( ) 2a = TrH = ξ 1 + ξ 2 = 2 + 2 = ψ ψ SU(2) B (A.24) detb = detubu = detb b 2 x + b 2 y + b 2 z = b 2 x + b 2 y + b 2 z (A.25)

A 5 (A.21) b = (b x,b y,b z ) SU(2) O(3) b V (A.5) b x (A.21) σ x Trσ 2 x = 2, Trσ x = Trσ x σ y = Trσ x σ z = 0 b x = 1 2 Trσ xh = 1 2 (σ x,i j)ξ j ξ i = ξ i (σ x,i j)ξ j = ψ σ x ψ b y = ψ σ y ψ, b z = ψ σ z ψ (A.26) SU(2) z (A.19) n = (0,0,1) b x b x = ξ σ x ξ = ξ U σ x Uξ (A.27) = ξ (cos θ 2 sin θ ( 2 iσ z )σ x cos θ 2 + sin θ ) 2 iσ z ξ (A.28) σ z σ x σ z = σ x, σ z σ x = iσ y b x = cosθ b x + sinθ b y (A.29a) b y = sinθ b x + cosθ b y (A.29b) b z = b z (A.29c) b = ψ σψ O(3) A.2 ξ = ξ = η 1 η 2 1/2 v s = η 2 η x = η 2 2 η 1 1 v : v y = η 1 2i + η 2 2 v z = η 2 2 + η 1 (A.30) 0 1 x SU(2) y O(3) z U = e iσ θ/2 = cos θ 2 + sin θ 2 iσ n R = e ij θ (A.31) SU(2) 4π O(3) 2π 2π 4π U U O(3) SU(2) O(3) 1 : 2 O(3) SU(2) 1/2 1/2 4π SU(2) A.1

付録A 回転群 6 図 A.1: SU(2) トポロジーの図示 4π 回転後紐はステップ 3-8 を実行することによりほぐすことができ る 2π 回転ではほぐすことがでない C.W. Misner, K.S.Thorne and J.A Wheeler, Gravitation (W.H Freeman 1973)

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