01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1
e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,, A, B, C (, 1, ) B, A, C 1,, A, B, C (, 1, ) B, C, A A, B, C (, 1, ) (, 1, ) A, C, B (1,, ) (, 1, ) (, 1, ) = (1,, ). (1.1) σ τ τ σ. n S n? ( : n! ) S n σ = (σ 1, σ,..., σ n ) i < j σ i > σ j (i, j) σ σ. S n σ τ σ τ
(x i x j ) (1.) i<j σ i<j(x σi x σj ) = (±1) i<j (x i x j ) (1.) ± +. S n n A n. A n ( : n!/ n = 1,,, ). G 1, G G 1 a G f(a) f(a b) = f(a) f(b) : 5. A S 5 A 5 x 5 1 5 5! = 10 5!/ = 60 5 60/5 = 1 1 1 xy θ R(θ) R(θ)R(ϕ) = R(θ + ϕ) (.1) θ R(θ) = R(θ + π). (.) xy (x, y) R(θ)
y w' θ φ w x r Figure 1: (x, y) w = x + iy ( z z w ) w = x iy w = w w = x + y r w w = r (.) r e iθ e iθ = cos θ + i sin θ (.) 6. e iθ e iϕ = e i(θ+ϕ) ( : ) 1 (x, y) w = x + iy w r x ϕ w = re iϕ (.5) θ x θ + ϕ w = re i(ϕ+θ) = e iθ re iϕ = e iθ w (.6) θ R(θ) e iθ 7. w = x + iy w = x + iy (x, y ) x, y, cos θ, sin θ (I) (x, y, z) θ ϕ ψ
z y w x Figure : : x + y + z = 1. (.1) ( ) (x, y, z) (.1) (0, 0, 1) (x, y, z) xy (u, v, 0) xy w = u + iv (.) w w = 0 (0, 0, 1) w = (0, 0, 1) 8. w (x, y, z) ( : w = (x + iy)/(1 z) ) 9. (x, y, z) w ( : w Re w, Im w x = Re w/(1+ w ), y = Im w/(1 + w ), z = ( w 1)( w + 1). ) z θ w w e iθ w w : w f[ a b aw + b ](w) := c d cw + d. (.1) 5
a, b, c, d a : b c : d ( a : b = c : d a = kc, b = kd = k ) e iθ f[ eiθ 0 ] x t 0 1 w w + t f[ 1 t 0 1 ] 10. f[ a b p q ] f[ c d r s ] f[ p q r s ](f[ a b ](w)) (.) c d A, B,C, D f[ A B ](w) C D ( : A = pa + qc, B = pb + qd, C = ra + sc, D = rb + sd ) 11. f[ a b A B ] f[ ] A, B, C, D ( : A = d, B = b, c d C D C = c, D = a ) k f[ a b ka kb ] f[ c d kc kd ] (a, b, c, d) k 5 (II) z θ R z,θ e iθ f[ eiθ 0 0 1 ] x 90 y z R x,90 (x, y, z) (x, y, z) (x, z, y) 1. (x, y, z) w (x, z, y) w w = w + i iw + 1 = f[ 1 i ](w) (5.1) i 1 ( : ) R x,90 6
1. R x, 90 y θ R y,θ : x 90 : R x,90 z θ : R z,θ x 90 R x, 90 1. R y,θ = R x, 90 R z,θ R x,90 θ, ϕ ψ : 1. θ, ϕ (0, 0, 1) (a) z ϕ R z, ϕ 0, xz (b) y θ z R y, θ. ψ R z,ψ. (b ) y θ R y,θ (a ) z ϕ R z,ϕ R z,ϕ R y,θ R z,ψ R y, θ R z, ϕ (5.) (x, y, z) (x, y, z) (F. Klein ) θ, ϕ, ψ f[ a b ka kb ] f[ ] k c d kc kd 7
ad bc = 1 a : b c : d ad bc 0 a, b, c, d ad bc = 1 1 R z,θ R x,90 R z,θ = f[ eiθ 0 0 1 ], R x,90 = f[ 1 i i 1 ] (5.) ad bc = 1 R z,θ = f[ eiθ/ 1 0 0 e iθ/ ], R x,90 = f[ i i 1 ] (5.) f[ α β β ᾱ ] α, β α + β = 1 15. α + β = 1 f[ α β β ᾱ ] ( : α, β α, β f[ α β β ᾱ ] f[ α β α, β α + β = 1 ) β ᾱ ] = f[ α β β ᾱ ] α β α + β = 1 α = + i, β = + i α + β = 1 + + + = 1 (5.5) 6 (x, y, z) t x, y z, t 1 f[ a b a b a b ] ad bc = 1 f[ ] = f[ ] ( a)( d) ( b)( c) = 1 c d c d c d ad bc = 1 (,,, ) (,,, ) 8
t z=vt z=wt z=ct y z=0 y=bx z φ θ y=0 y=ax x Figure : : z-t : x-y zt (?) z = 0 v z = vt c z = ct c = 1 z = t v z = vt w z = wt v w w v y = 0 y = ax tan θ = a y = ax y = bx tan ϕ = b a x + y θ w = x + iy e iθ x = x cos θ y sin θ, (6.1) y = x sin θ + y cos θ (6.) w x + y = (x + iy)(x iy) w = e iθ w, (6.) w = e iθ w (6.) z = ±t z t 9
z t = (z t)(z + t) z + = z + t, z = z t η z + = z + t, z = z t 16. z t = z t sinh, cosh z + = e η z +, (6.5) z = e η z (6.6) z = z cosh η t sinh η, (6.7) t = z sin η + t cosh η (6.8) e η = cosh η + sinh η, e η = cosh η sinh η (6.9) sinh cosh 17. cosh η, sinh η e η, e η 18. cosh(ξ + η) = cosh ξ cosh η + sinh ξ sinh η, (6.10) sinh(ξ + η) = sinh ξ cosh η + cosh ξ sinh η (6.11) z = vt z = wt z = vt z = 0 (6.7) v cosh η = sinh η (6.1) tanh η := sinh η/ cosh η v = tanh η 19. v = tanh η η v, log ( : η = log( (1 + v)/(1 v)) w = tanh ξ z = wt (z, t) (z, t) = (k sinh ξ, k cosh ξ) (6.7), (6.8) z = k sinh(ξ η), t = k cosh(ξ η) (6.1) z = t tanh(ξ η) z = vt z = wt v = tanh η, w = tanh ξ z = ut u = tanh(ξ η) 0. u = (v w)/(1 vw) 10
y = ax y = bx y = ax y = 0 (6.) a = tan θ θ y = bx b = tan ψ y = bx (x, y) = (k cos ψ, k sin ψ) (6.1), (6.) (x, y ) = (k cos(ψ θ), k sin(ψ θ)) y = tan(ψ θ)x y = ax y = bx a = tan θ, b = tan ψ y = cx c = tan(ψ θ) 1. c = (a b)/(1 + ab) θ, ϕ a = tan θ, b = tan ϕ y = ax y = bx a, b z = vt z = wt v, w v = tanh η, w = tanh ξ η ξ η, ξ 7 (x, y, z, t) = (0, 0, 0, 0) x + y + z t = 0 (7.1) : (x, y, z) = ( x t, y t, z ). (7.) t. x + y + z = 1 w. w ( : w = (x + iy)/(t z)) z z-t (6.5), (6.6), (6.7), (6.8) η w e η w (7.). 11
Figure : : v = 0 : v = 0.5c : v = 0.9c : v = 0.99c 0, 0.5c, 0.9c, 0.99c http://demonstrations.wolfram.com/thecelestialsphere/ w e η w 1
1 5 5 5 1 1 1 5 1 1 1 1