Η Ομάδα SL(2,C) και οι αναπαραστάσεις της

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Καλλιόπη Καραμήτσος
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1 SL(2, C) SO(3, 1) D : Λ D(Λ) SO(3, 1) 2 1 D : ±A D(π(±A)) SL(2, C) SL(2, C) SO(3, 1) SL(2, C) SO(3, 1) ξ i (, ) K i x µ p µ J µν T µν A µ ψ α

2 J i = J i, () K i = K i, ( ) K i M 0i = (iξ i K i ) A i = 1 2 (J i + ik i ) B i = 1 2 (J i ik i ), [A i, A j ] = iε ijk A k [B i, B j ] = iε ijk B k [A i, B j ] = 0. A, B SL(2, C) J i = A i + B i K i = i(b i A i ) SL(2, C) SU(2) A SU(2) B SL(2, C) = SU(2) SU(2). SL(2, C) (j A, j B ) j A j B (j A, j B ) j = j A + j B, j A + j B 1,..., j A j B SL(2, C)

3 SL(2, C) (1/2, 0) SU(2) C 2 j A = 1/2, j B = 0 A i = 1 2 σ i, B i = 0, J i = 1 2 σ i, K i = i 2 σ i. SL(2, C) [ i ( ) ] [ D(A) = A R = θ σ iξ σ = i σ ] 2 2 ( θ iξ), A SL(2, C) ξ a C 2 ξ α ξ α = (A R ) α βξ β. ξ a D(A) = (A 1 R )T ξ α ξ α = ξ β (A 1 R )β α = [(A 1 R )T ] β α ξ β (A R ) α β(δω) = δ α β + i 2 ω µν(s µν R )α β. ( 0 1 ε αβ = 1 0 ) ε αβ = ( ) (A R ) α γ(a R ) β δ εγδ = ε αβ ε γδ (A 1 R )γ α(a 1 R )δ β = ε αβ

4 C 2 ξ α = ε αβ ξ β, ξ α = ε αβ ξ β ξ α Ξ α = ξ 1 Ξ 2 ξ 2 Ξ 1, j = 0 j 1 = j 2 = 1 2 (0, 1/2) SU(2) j A = 0, j B = 1/2 A i = 0, B i = 1 2 σ i. J i = 1 2 σ i, K i = i 2 σ i. SL(2, C) [ i ( ) ] [ A A L = θ σ + iξ σ = i σ ] 2 2 ( θ + iξ), η α η α η α = (A L ) β. α βη σ 2 σ σ 2 = σ A L = ζa R ζ 1, ζ = iσ 2 A L A R η α ξ α η α η α = (A β. R) α βη PJ i P 1 = J i, PK i P 1 = K i

5 PA i P 1 = B i, PB i P 1 = A i Pξ = η, Pη = ξ P (1/2, 0) (0, 1/2) (n/2, m/2) SL(2, C) (1/2, 0), (0, 1/2) ζ α 1...α n ; β 1... β m ζ α 1...α n ; β 1... β m = (A R ) α 1 γ1... (A R ) α n γn (A R) β 1 δ1... (A R) β ṁ δ m ζ γ 1...γ n ; δ 1... δ m ε αβ SL(2, C) ζ (α 1α 2...α n);( β 1 β2... β m), (n/2, m/2) (n + 1)(m + 1) v µ j = 0, 1 j = 0 v 0 j = 1 v i x µ, p µ, A µ 4D (M µν ) λ ρ = i(δ λ µη νρ δ λ νη µρ )

6 (1, 0) (0, 1) t µν A t µν (1, 0) (0, 1) (1, 0) t µν A,c = 1 2 εµν λρ tλρ A,c (0, 1) t µν A,a = 1 2 εµν λρ tλρ A,a F µν (0, 0) (1, 1) t µν S t µν (0, 0) (1, 1) T µν SL(2, C) Dirac (1/2, 0) (0, 1/2) ( ) ψr Ψ =, ψ L ψ R ξ, ψ L η SL(2, C) ( ) Ψ Ψ AR 0 = Ψ 0 A L P : Ψ ( ψl ψ R ) = ( ) Ψ. θ = 0 ( ) [ ( ) ( )] σ ψ R 2 ξ ξ ξ ψ R = + σ ˆn ψ R 2 2 ( ψ L σ ) [ ( ) ( )] 2 ξ ξ ξ ψ L = σ ˆn ψ L 2 2 ψ R (0) = ψ L (0) ξ γ β

7 ψ R ( p) = ψ L ( p) = ( ) E + m + σ p ψ [2m(E + m)] 1/2 R (0) ( ) E + m σ p ψ [2m(E + m)] 1/2 L (0) E + σ p ψ R ( p) = m ψ L( p) E σ p ψ L ( p) = m ψ R( p) ( m p 0 σ p p 0 + σ p m ) ( ψr ( p) ψ L ( p) ) = 0 (γ µ p µ m)ψ(p) = 0, γ µ ( 0 σµ σ µ 0 ).

8 L P C 1, C 2 C 1, C 2 /p 2 m, s, p, λ {p, λ : p 2 = m 2, λ = s, s + 1,..., s 1, s} p, λ P k µ L k L µ νk ν = k µ, L L k. k µ U(a, Λ) P P µ k, λ = k, λ k µ. [W µ, P ν ] = 0 W µ P µ P µ W ν k, λ = W ν P µ k, λ = W ν k, λ k µ m, s

9 W µ k µ D s (L) D s (L) k, λ = k, λ D s (L) λ λ, L L k, k, λ H(p) SO(3, 1)/L k k µ p µ = H(p) µ νk ν, L k p, λ = U[H(p)] k, λ. P µ p, λ = P µ U[H(p)] k, λ = U[H(p)]U 1 [H(p)]P µ U[H(p)] k, λ = = U[H(p)]P ν k, λ (H(p)) µ ν = U[H(p)] k, λ H(p) µ νk ν = p, λ p µ. U(0, Λ) p, λ = U(Λ) p, λ = U(Λ)U[H(p)] k, λ = U[H(Λp)]U[H 1 (Λp)ΛH(p)] k, λ = = U[H(Λp)] k, λ D s [L(Λ, p)] λ λ = Λp, λ D s [L(Λ, p)] λ λ, L(Λ, p) = H 1 (Λp)ΛH(p) L k L k P H(p) SO(3, 1)/L k k µ = p µ n = 0 L k = SO(3, 1) Λ µ νp µ n = 0, Λ SO(3, 1)

10 m s (m, s) k µ = p µ t = (m, 0) p µ t W 0 = 0, W i = mj i. W µ p µ t SO(3) L pt = SO(3) R(α, β, γ) SO(3), Rp t = p t SO(3) P SO(3) p t, s, λ = p t, λ, λ = s,..., s, p µ t = (m, 0). P µ p t, λ = p t, λ p µ t J 2 p t, λ = p t, λ s(s + 1) J 3 p t, λ = p t, λ λ. H(p) SO(3, 1)/SO(3) λ p µ t p µ t = H(p) µ νp ν t. p λ p, λ = δ λ λδ 3 ( p p) p, λ = p 0 t p 0 U[H(p)] p t, λ = 1 γ U[H(p)] p t, λ.

11 P R(Λ, p) = H(Λp) 1 ΛH(p) SO(3) T (a) p, λ = p, λ e iaµ p µ U(0, Λ) p, λ = (Λp) 0 Λp, λ D s [R(Λ, p)] λ p λ, 0 D s [R] L pt = SO(3) s k µ = p µ l = (ω 0, 0, 0, ω 0 ) p µ l W 0 = W 3 = ω 0 J 3, W 1 = ω 0 (J 1 + K 2 ), W 2 = ω 0 (J 2 K 1 ). [W 1, W 2 ] = 0, [W 2, J 3 ] = iw 1, [W 1, J 3 ] = iw 2. W µ p µ l E 2 L k = E 2 C 2 C 2 = 0, C 2 0 C 2 = 0 E 2 p l, λ P µ p l, λ = p l, λ p µ l J 3 p l, λ = p l, λ λ W 1 p l, λ = W 2 p l, λ = 0.

12 {p µ = (ω, p) : p = ωˆp} p l, λ H(p) SO(3, 1)/E 2 p, λ = H(p) p l, λ (C 1 = C 2 = 0, λ) T (a) p, λ = p, λ e iaµ p µ U(0, Λ) p, λ = (Λp) 0 Λp, λ e iθ(λ,p), p 0 e iθ(λ,p) = p l, λ D[L(Λ, p)] p l, λ = p l, λ D[H 1 (Λp)ΛH(p)] p l, λ, D E 2 λ λ = ±1, ±2

13 SL(2, C) ( ) D(A) =, A SL(2, C). ε αβ ξ, Ξ (1/2, 0) ξ 1 Ξ 2 ξ 2 Ξ 1, SL(2, C) ξ 1 ξ 1 + ξ 2 ξ 2, E 2 = p 2 + m 2 E = γm, p = γm β γ µ : {γ µ, γ ν } = 2η µν S µν = i 4 [γµ, γ ν ] [S µν, γ λ ] = i(γ µ η νλ γ ν η µλ ) p µ l = (ω 0, 0, 0, ω 0 ) W µ

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