I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2

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1 III Jan 30th, 2006 I : II : I : [ I ] (Landau and Lifshitz, Quantum Mechanics chapter 12, 13, 9: Pergamon Pr.) [ ] ( ) (H. Georgi, Lie algebra in particle physics, Perseus Books) [ ] II : H. (H. Flanders, Differential Forms With Applications to the Physical Sciences, Dover) I (M. Nakahara, Geometry, Topology and Physics, Inst of Physics Pub Inc, chap 5,6) M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (Perseus Books) 1 matsuo( )phys.s.u-tokyo.ac.jp 1

2 I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2

3 Maxwell Lorentz Yang-Mills 2 G 1. a, b G a b G 2. a, b, c G (a b) c = a (b c) 3. e G a G a e = e a = a 4. a G a 1 G a a 1 = a 1 a = e 3

4 G (G ) ( ) ( ) a, b G a b = b a ( (Nonabelian group)) ( ) ( ) ( SO(3), (R)) (Z 2 ) x x G = {e, σ} e : x x σ : x x e e = σ σ = e, e σ = σ e = σ (Z n ) z = x + iy σ : z ωz (ω = e 2πi/n ) 360/n G = {e, σ, σ 2,, σ n 1 } σ r σ s = σ r+s r + s mod n Z n = n (S n ) (n ) ( 1 2 n a 1 a 2 a n i a i S n = n! ) GL(n, R) (GL(n, C)): ( ) n n ( ) 4

5 O(n): {a GL(n, R), a t a = E} (E : ) n U(n): { a GL(n, C), a a = E } (a = (a ) t ) n SO(n) (SU(n)) O(n) (U(n)) 1 O(n, m) (U(n, m)) GL(n+m, R) (GL(n+m, C)) a t Ja = J (a Ja = J) J = diag(1,, 1, 1,, 1) (1 n -1 m ) O(3) O(3, 1) ( SU(n)) b.. a a b. Z 2. e σ e e σ σ σ e 3 (group homomorphism) 1. f : G 1 G 2 5

6 2. G 1 2 a 1, a 2 f(a 1 ) f(a 2 ) = f(a 1 a 2 ) Ker f = {g G 1 f(g) = e} G 1, Im f = {f(g) g G 1 } G 2 (isomorphism) G 1 G 2 f (bijection) (subgroup) H G 1. H G H G 2. H G a, b H a b H (coset) H G H Q G g H H g = {h g h H} G g H = {g h h H} G G H H 2 H G g 1 H g 1 H H g 1 = φ h 1 H H g 1 h 2 H h 1 = h 2 g 1 g 1 = h 1 2 h 1 H 3 4 G G = H (H g 1 ) (H g n 1 ) (right coset) H\G H\G {H, H g 1,, H g n 1 } (left coset) G/H {H, g 1 H,, g n 1 H} n = G / H (index) (G : H) 6

7 ( ) (invariant subgroup, normal subgroup) G H G g g H = H g H G H G/H = H\G G/H (coset group), (factor group) (g 1 H) (g 2 H) = (g 1 g 2 ) H H [H H = H H ] (g H) (g 1 H) [ ] (g H) (g 1 H) = g g 1 H H = H 2 a, b (conjugate) G g b = g a g 1 a b a b a b a = b (transitive law) a b b c a c [ ] a = g 1 b g1 1, b = g 2 c g2 1 a = (g 1 ) (g 2 c g 1 2 ) g 1 1 = (g 1 g 2 ) c (g 1 g 2 ) 1 (conjugacy class) G G = C 1 C 2 C n C i G a, b C i a b i j C i C j = φ [ ] e g G g e g 1 = e e e (group ring) G g e g R G = a g e g a g R g G 7

8 : R G R G R G e g1 e g2 = e g1 g 2 a g e g b g e g = a g b g e g g g G g G g,g G R G G C i R G Ĉ i Ĉ i = e a a C i g G e g Ĉi = Ĉi e g [ ] h C i g h g 1 C i g C i = C i g ( ) Ĉ i Ĉj = Ĉj Ĉi Ĉ i Ĉj = k N ij k Ĉ k N k ij S 3 3 3! = 6 ( ) ( ) ( ) e = σ 1 = σ 2 = ( ) ( ) ( ) σ 3 = ω = ω 2 = σ 1 σ 2 = ( ) ( ) ( ) ( = = ( ) = ω )

9 e ω ω 2 σ 1 σ 2 σ 3 e e ω ω 2 σ 1 σ 2 σ 3 ω ω ω 2 e σ 3 σ 1 σ 2 ω 2 ω 2 e ω σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 e ω ω 2 σ 2 σ 2 σ 3 σ 1 ω 2 e ω σ 3 σ 3 σ 1 σ 2 ω ω 2 e S 3 {e}, {e, σ 1 }, {e, σ 2 }, {e, σ 3 }, { e, ω, ω 2 }, S 3 S 3 {e}, { e, ω, ω 2 }, S 3 S 3 S 3 / {e} = S 3, S 3 /S 3 = {e}, S 3 / { e, ω, ω 2} = {{ e, ω, ω 2}, {σ 1, σ 2, σ 3 } } = Z 2 S 3 C 1 = {e}, C 2 = { ω, ω 2}, C 3 = {σ 1, σ 2, σ 3 } Ĉ 1 Ĉ1 = Ĉ1, Ĉ 1 Ĉ2 = Ĉ2, Ĉ 1 Ĉ3 = Ĉ3, Ĉ 2 Ĉ2 = 2Ĉ1 + Ĉ2, Ĉ 2 Ĉ3 = 2Ĉ3, Ĉ 3 Ĉ3 = 3Ĉ1 + 3Ĉ2 4 (representation) G GL(n, C) ρ : G GL(n, C) ρ G n ρ g 1 g 2 = g 3 ρ(g 1 ) ρ(g 2 ) = ρ(g 3 ) 9

10 ρ G GL(n, C) ρ(e) = E ( ), ρ(g 1 ) = (ρ(g)) 1 ρ n (representation space) : (trivial ) g G ρ(g) = 1 GL(1, C) (1 ) : g 1 g 2 ρ(g 1 ) ρ(g 2 ) : (unitary ) g G ρ(g) U(n) ρ(g 1 ) = (ρ(g)) 1 = ρ(g) : (direct sum ) ρ 1, ρ 2 n 1, n 2 ρ 1 ρ 2 ρ 1 ρ 2 n 1 + n 2 ρ 1 ρ 2 : g ( ρ1 (g) 0 0 ρ 2 (g) : (direct product ) ρ 1, ρ 2 n 1, n 2 ρ 1 ρ 2 ρ 1 ρ 2 n 1 n 2 ) ρ 1 ρ 2 : g ρ(g) ik,jl = ρ 1 (g) ij ρ 2 (g) kl ρ(g) ik,jl ik jl :(equivalent ) ρ 1, ρ 2 GL(n, C) g G T GL(n, C) ρ 1 (g) = T ρ 2 (g)t 1 ρ 1 ρ 2 (invariant subspace): ρ V ρ(g) ρ(g)v V (irreducible ): : (decomposition into irreducible representations) ρ = ρ 1 ρ 2 ρ n 10

11 (ρ i ) ρ (α) n (α) ρ = α n (α) ρ (α) : (regular ) e g e a = ρ (reg) (a) gg e g g G ρ (reg) 0 1 ρ (reg) (e) (e g e a ) e b = ρ (reg) (a) gg e g e b = g G = e g (e a e b ) = g G ρ (reg) (a b) gg e g. g,g G ρ (reg) (a) gg ρ (reg) (b) g g = ρ(reg) (a b) gg g G ρ (reg) (a) gg ρ (reg) (b) g g e g Shur (Schur s Lemma) 1. ρ i (i = 1, 2) G n i V i M V 1 V 2 g G Mρ 1 (g) = ρ 2 (g)m M ( n 1 = n 2 M 1 ) M = 0 2. ρ, V M V V g G ρ(g)m = Mρ(g) M 11

12 1. KerM = {v V 1 Mv = 0}, ImM = {Mv V 2 v V 1 } V 1, V 2 v Ker M Mv = 0 M Mρ 1 (g)v = ρ 2 (g)mv = 0 ρ 1 (g)v Ker M Ker M ρ 1 0 Ker M = 0 Ker M = V 1 M = 0 Im M 0 V 2 Ker M = 0 Im M = V 2 M 2. M v V 1 Mv = λv (λ C ) g G (M λe)ρ(g) = ρ(g)(m λe) M λe M λe v M λe = 0 { } ρ (α) (α = 1,, #( )) ρ (α) ji (g 1 )ρ (β) kl (g) = G δ ik δ jl δ αβ d α g G ρ (α), ρ (β) V (α), V (β) B V (β) V (α) M = g G ρ (α) (g 1 )Bρ (β) (g) M V (β) V (α) g G ρ (α) (g)m = Mρ (β) (g) ρ (α) (g)m = ρ (α) (g) ρ (α) (g 1 )Bρ (β) (g ) g G = ρ (α) (gg 1 )Bρ (β) (g ) = ρ (α) (g 1 )Bρ (β) (g g) g G g = g G ρ (α) (g 1 )Bρ (β) (g ) ρ (β) (g) = Mρ (β) (g) Schur 1 α β 12

13 2 M = 0 B rs = δ ri δ sk M jl = g r,s ρ (α) jr (g 1 )δ ri δ sk ρ (β) sl (g) = g ρ (α) ji (g 1 )ρ (β) kl (g) = 0 α = β Shur 2 M = ce B rs = δ ri δ sk ρ (α) ji (g 1 )ρ (α) kl (g) = c ik δ jl g c ik j, l d α c ik = ρ (α) ji (g 1 )ρ (α) kj (g) = ρ (α) kj (g)ρ(α) ji (g 1 ) = ρ (α) ki (g g 1 ) = G δ ki g g g j j (d α ρ (α) G G ρ (α) (e) ki = δ ki ) c ik = G d α δ ik ρ g G ρ (α) ij (g)ρ (β) kl (g) = G δ ik δ jl δ αβ d α (character) ρ χ(g) = Tr(ρ(g)) ρ, ρ ρ (g) = T ρ(g)t 1 χ (g) = χ(g) g, g χ(g) = χ(g ) [ ] g g g = k 1 gk (k G) χ(g ) = Trρ(k 1 gk) = Trρ(k 1 )ρ(g)ρ(k) = χ(g) ρ (α) ρ (β) χ(ρ (α) ρ (β) ) = χ (α) + χ (β) χ(ρ (α) ρ (β) ) = χ (α) χ (β) 2 α β d α = d β M 0 Mρ 1 (g) = ρ 2 (g)m ρ 2 (g) = Mρ 1 (g)m 1 ρ 1 ρ 2 M = 0 13

14 (I) ρ (α), ρ (β) χ α, χ (β) χ (α) (g)χ (β) (g) = G δ αβ g G [ ] i = j, k = l χ {C i } (i I) G χ (α) = χ (α) (g) g Ci i I C i χ (α) i χ (β) i = G δ αβ i (I) ρ(g) ρ(g) = α q α ρ (α) (g) ( α ) q α χ(g) = α q α χ α (g) χ α(g) g χ α(g)χ(g) = g β q α = 1 G g q β χ α(g)χ β (g) = g β χ (α ) (g)χ (α) (g) = 1 G i I q β G δ αβ = q α G C i χ (α) i χ (α) i ρ (reg) (g) g = e χ (reg) (e) = E χ (reg) (g) = { G (for g = e) 0 (otherwize) q α = 1 G g χ (reg) (g)χ (α) (g) = 1 G χ(reg) (e)χ (α) (e) = d α 14

15 χ (reg) (e) = G, χ (α) (e) = d α ρ (reg) (g) = α d α ρ (α) (g) g = e G = α (d α ) 2 (II) n n α=1 χ (α) i χ (α) j = G C i δ ij [ ] Ĉ i = g C i e g ρ (α) (Ĉi) = [ ] g C i ρ (α) (g) ( ) eg, Ĉi = 0 g G [ ρ (α) (g), ρ (α) (Ĉi) ] = 0 Schur ρ (α) (Ĉi) = λe trace d α ρ (α) λ C i χ (α) i = λd α λ = C i χ (α) i d α ρ (α) (Ĉi)ρ (α) (Ĉj) = k N ij k ρ (α) (Ĉk) ρ (α) (Ĉi) C i C j d 2 α χ (α) i χ (α) j = k C i C j χ (α) i χ (α) j = k k N C k χ (α) k ij d α N ij k d α C k χ (α) k C i C j α χ (α) i χ (α) j = C k N k ij d α χ (α) k = α,k k C k N ij k χ (reg) k = N ij 1 χ (reg) 1 3 d α χ (α) k 15

16 C 1 N 1 ij = δîj C j, χ (reg) 3 1 = G α χ (α) i χ (β) G j = δ ij C j U (α) = C i #class i U (α) i U (β) i = δ αβ, #irreps α U (α) i U (α) j = δ ij. i G χ(α) i U Trρ (α) (g) ρ (β) (g) = Trρ (α) (g)trρ (β) (g) ρ (α) ρ (β) = irreps γ C αβ γ ρ (γ) (g) C αβ γ = 1 G χ (γ) (g)χ (α) (g)χ (β) (g) = g i C i G χ(γ) i χ (α) i χ (β) i (II) C i C j χ (α) i χ (α) j = k N ij k d α C k χ (α) k d α l α k α χ (α) N k ij C k χ (α) k χ(α) l = N k ij C k G C k δ kl = N k ij G 3 î (complex conjugate class) C i g g 1 ρ(g 1 ) = ρ(g) χî = χ i 16

17 N ij k = α C i C j χ (α) i d α G χ (α) j χ (α) l 5 (point group and its representation) 5.1 (point group) (symmetry transformation of point group) (rotation) 2π/n C n (C n ) n = e (reflection) σ σ 2 = e σ h σ v, rotation-reflection S n S n = C n σ h (inversion) x x I I = S 2 = σ h C 2 (classification of point group) 5 C n : e, C n, (C n ) 2,, (C n ) n 1 n Z n 2 D n : n 2 n 2 n 2 n 2n (C n n C 2 n ) D 3 = V 17

18 4 T: T = 12 8 O: O = I: C 5 6 C 3 10 C 2 15 I = 60 ( ) S 2n : ( S n ) 2n S 2n n = 2p + 1 (S 4p+2 ) 2p+1 = I S 4p+2 = C 2p+1 C i C i {e, I} C nh : n (C n ) p, (C n ) p σ h (p = 0,, n 1) 2n C nv : n n C nv = 2n D nh : D n n 2 D nd : n 2 2 T d : T T h : T T h = T C i O h : O O h = O C i I h : I I h = I C i H 2 O : C 2v NH 3 : C 3v CH 3 Cl : C 3v CH 4 : T d 18

19 OsF 8 : O h UF 6 : O h C 2 H 6 : D 3d C 2 H 4 : D 2h 5.2 (Representation of point group) C n (=Z n ) g 1, g 2 G [g 1, g 2 ] = 0 [ρ(g 1 ), ρ(g 2 )] = 0 ρ(g) (g G) v ρ(g) v = λ(g) v, λ(g) C v 1 λ(g) 1 C n = Z n {e, C n, (C n ) 2,, (C n ) n 1 } (C n ) n = e ρ(c n ) = λ C 1 (C n ) n = e λ n = 1 n 1 ρ (α) ((C n ) p ) = e 2πiαp/n p = 0, 1,, n 1, α = 0, 1,, n 1 n χ (α) ((C n ) p ) = e 2πiαp/n ( ρ (0) ) discrete Fourier n 1 p=0 (χ (α) ((C n ) p )) χ (β) ((C n ) p ) = n 1 p=0 e 2πipα/n e 2πipβ/n = nδ α,β 19

20 α irrep. (χ (α) ((C n ) p ) χ (α) ((C n ) q ) = n 1 α=0 e 2πipα/n e 2πiqα/n = nδ p,q x p χ (α) (g) C nh C 3v (= S 3 3 ) {e}, {ω, ω 2 }, {σ 1, σ 2, σ 3 } ρ (1) ρ (2), ρ (1) (g) = 1 ρ (2) (e) = ρ (2) (ω) = ρ (2) (ω 2 ) = 1, g C 3v ρ (2) (σ 1 ) = ρ (2) (σ 2 ) = ρ (2) (σ 3 ) = 1 3 d d 2 = 6 d = 2 ( ) ( ) ( ) 1 0 c s c s ρ (3) (e) =, ρ (3) (ω) =, ρ (3) (ω 2 ) = 0 1 s c s c ( ) ( ) ( ) 1 0 c s c s ρ (3) (σ 1 ) =, ρ (3) (σ 2 ) =, ρ (3) (σ 3 ) =. 0 1 s c s c c = cos(2π/3) = 1/2, s = sin(2π/3) = 3/2 2 ρ (α) (g) = χ (α) (g) ρ (3) χ (3) (e) = 2, χ (3) (ω) = χ (3) (ω 2 ) = 1, χ (3) (σ 1 ) = χ (3) (σ 2 ) = χ (3) (σ 3 ) = ρ(e) = 0 1 0, ρ(ω) = ρ(σ 1 ) = , ρ(σ 2) = , ρ(ω2 ) =, ρ(σ 3) =

21 D 2n a (2π/n ) b (2π/2 ) a n = b 2 = e, b 1 a b = a 1 D 2n = {e, a,, a n 1, b, ba,, ba n 1 } a p a l a p = a l, a r (a l b)a r = a l+2r b, a l a n l, (a p b)a l (a p b) 1 = a n l (a r b)(a l b)(a r b) 1 = a 2r l b a l b a l+2 b a n l b n: : {e}, { a i, a n i} { (1 i n/2), } { ba 2i, } ba 2i 1 (1 i n/2) n: : {e}, { a i, a n i} { (1 i (n 1)/2), } ba i (1 i n) n n/2 + 3, (n 1)/ G 1 (G : G ) G G 4 D2n a 2 n 1 4 n 2 n ρ (1) (a) = ρ (1) (b) = 1 ρ (2) (a) = 1, ρ (2) (b) = 1 ρ (3) (a) = 1, ρ (2) (a) = 1, ρ (2) (b) = 1 ρ (1) (a) = ρ (1) (b) = 1 ρ (2) (a) = 1, [ ] 2 n 2 ( ρ (k) ω k 0 2 (a) = 0 ω k ) ρ (2) (b) = 1 ρ (2) (b) = 1 (, ρ (k) (b) = G 2 g 1, g 2 g 1 g 2 g1 1 g )

22 (k = 1, 2,, [ ] n 2 ) ω = e 2πi/n : x i (i = 1,, N) H = 1 M ij ẋ i ẋ j + 1 K ij x i x j 2 i,j 2 i,j x i (x q = Rx, R t R = M) H = 1 2 i ( ) 2 dqi + 1 dt 2 L ij q i q j i,j q (q Q = Sq, S t LS = diag(ω 2 i )) ( S O(N) ) H = 1 2 i ( ) 2 dqi + 1 dt 2 Ω 2 i Q 2 i i G Ω G q G q i = ρ ij (g)q i H H(ρ(g)q) = H(q) ρ ρ ρ = α n α ρ (α) 22

23 ρ 1 (g)... ρ(g) = S ρ2(g) S t Sρ (diag) (g)s t S q Q = Sq L Ω = S t LS Hamiltonian q ρ(g)q ρ t (g)lρ(g) = L S t LSρ (diag) (g) = ρ (diag) S t LS S t LS g G ρ (diag) (g) Schur Schur I Ω Schur II Ω Hamiltonian H = 1 2 n α d α α i=1 s=1 ( ( Q (α,i) s ) 2 + (Ω (α) i ) 2 (Q (α,i) s ) 2) i s Ω (α) (NH 3 ) C 3v = S 3 N x 1, H x 2,3,

24 =6 6 C 3v 3 {e}, {ω, ω 2 }, σ 1, σ 2, σ 3 e, ω, σ 3 e χ(e) = 6 ω x 1 R x 1 x R 0 x 2 = x R x 3 x 4 0 R 0 0 x 4 R = c s 0 s c (c = cos 2π 3 = 1 2, s = sin 2π 3 3 = 2 ) χ(ω) = 0 σ 3 x 1 Σ x 1 x 2 0 Σ 0 0 x 2 = x Σ x 3 x Σ 0 x 4 Σ = χ(σ 3 ) = 2trΣ = 2 C 3v C 1 = {e}, C 2 = {ω, ω 2 }, C 3 = {σ 1, σ 2, σ 3 } C 1 C 2 C 3 ρ (1) ρ (2) ρ (3) n α = 1 6 g G χ (α) (g) χ(g) n 1 = 2, n 2 = 0, n 3 = 2 ρ (1) 2 ρ (2) 0 ρ (3) = 6 24

25 6 ( ) (Symmetry (Permutation) group) Young 6.1 S n 1,, n σ ( 1 n ) σ(1) σ(n) S n σ σ τ = ( ) ( ) 1 n 1 n σ(1) σ(n) τ(1) τ(n) = ( ) ( ) τ(1) τ(n) 1 n σ τ(1) σ τ(n) τ(1) τ(n) = ( 1 n ) σ τ(1) σ τ(n) σ, τ ( ) ( ) σ(1) σ(n) σ 1 1 n = = 1 n σ 1 (1) σ 1 (n) (Transposition) ( 1 i j ) n 1 j i n (ij) ( ) :

26 2 (ij) (ij) = e p i σ p n p 1 σ 1 = p 1 p n σ ( ) Van der Monde (x 1,, x n ) = 1 1 x 1 x n = (x.. i x j ) i<j x n 1 1 xn n 1 S n σ (x 1,, x n ) = (x σ(1),, x σ(n) ) σ = ± σ σ = σ = ± ( +, ) (cycle) l ( ( ) a1 a 2 a l 1 a l (a 1,, a l ) a 2 a 3 a l a 1 (cycle) S n ) = (134)(25)(6) 3, 2, 1 S n n = λ λ n λ i 0, λ i λ i+1, n i=1 λ i = n [λ 1,, λ n ] n (partition) n p(n) n p(n) 1 1 [1] 2 2 [2], [1, 1] 3 3 [3], [2, 1], [1, 1, 1] 4 5 [4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1] 26

27 (p(0) = 1 ) p(n)q n 1 = n=0 n=1 1 q n : σ 1 σ 2 ( S n ) σ 1 σ 2 [λ 1,, λ n ] σ = (σ(1) σ(λ 1 )) (σ(λ 1 + 1) σ(λ 1 + λ 2 )) (σ(λ λ n 1 + 1),, σ(λ λ n )) τ = (τ(1) τ(λ 1 )) (τ(λ 1 + 1) τ(λ 1 + λ 2 )) (τ(λ λ n 1 + 1),, τ(λ λ n )) µ µ = ( σ(1) σ(n) τ(1) τ(n) σ = µ 1 τµ ) Young diagram (Young ) [λ 1, λ 2,, λ n ] λ 2 λ 1 λ 3 1: Young diagram Young Young 27

28 ( ) S 3 ρ 1 [3], ρ 2 [1, 1, 1], ρ 3 [2, 1] 6.2 S n S n n Young Young λ = [λ 1,, λ n ] d λ = f! s 1 s 2 s f f = i λ i ( ) s i i (hook length)= + +1 [ ] S 5 5! = λ d 2 λ 1,4,5,6,5,4,1 [ ] d [n] = d [1,1,,1] = 1 ρ(σ) = 1 ρ(σ) = ( 1) σ Young ( n ) 1,, n (board) S 5 Young [2, 2, 1] 2: (board) B H B ( (2, 4), (1, 5) ) S n 28

29 R B ( (2, 1, 3), (4, 5) ) a B = b B = 1 H B 1 R B e σ σ H B σ R B ( 1) σ e σ, a B a B = a B, b B b B = b B e B = r n! a B b B (r = d λ, n = λ ) e B e B = e B tre B = r e B V B = e B C S n C Sn r S n λ ( 1, 2,, n) ψ(1, 2,, n) (1,, n ) S n σ ψ(1, 2,, n) = ψ(σ 1,, σ n ) S n n! S n Young e B [n] ψ [n] (1, 2,, n) = σ S n ψ(σ 1,, σ n ) 1 [1, 1,, 1] ψ [1,1,,1] (1, 2,, n) = σ S n ( 1) σ ψ(σ 1,, σ n ) 29

30 1 Young λ B n! σ ψ Young e B d λ (= ) ρ (reg) (g) = i d α ρ (α) (g) d λ ρ ρ (α) π α = d α χ G α(g)ρ(g) g G ρ (α) π α π β = d αd β G 2 g,g G g g G χ α(g)χ α(g )ρ(g g ) = d αd β χ (α) (g g 1 )χ (β) (g ) ρ(g) G 2 = δ αβ d α G g G χ α(g)ρ(g) = δ αβ π β 2 3 g G χ (α) (g 1 )χ (β) (g g) = δ αβ G χ (α) (g) d α ρ(g) = α n (α) ρ (α) (g) Tr (π α ) = d α χ G α(g)trρ(g) = n α d α g G π α n α d α ρ (α) d α n α 30

31 6.3 SU(n) GL(n, C) n ( V = C n ) M GL(n, C) (i, j = 1,, n) M j i m V V v 1 v m M M v 1 M v m GL(n, C) S m σ ( v 1 v m ) = v σ1 v σm S m GL(n, C) σ (M( v 1 v m )) = M(σ( v 1 v m )) = M v σ1 M v σm S m (Young ) e B e B (V V ) GL(n, C) GL(n, C) GL(n, C) Young V e i (i = 1,, n) 1 m = 2 λ = [2] 2 ( e 2 i e j + e j e i ) n(n + 1)/2 1 λ = [1, 1] 2 ( e 2 i e j e j e i ) n(n 1)/2 m = 3 : [3] 3 n(n+1)(n+2)/6 [1, 1, 1] 3 n(n 1)(n 2)/6 λ = [2, 1] Young n(n 2 1)/3 2 n n + 1 Young n 31

32 SU(n) SU(n) n GL(n, C) M SU(n) det(m) = 1 n e 1 e n det(m) e 1 e n = e 1 e n e 1 e n = 1 ( 1) σ e σ1 e σn n! σ S n 5 SU(n) Young 1. n 2. n Young n Young [λ 1,, λ n 1, λ n ] n 1 Young [λ 1 λ n,, λ n 1 λ n ] ( n 1 ) SU(2) SU(2) Young 2 1 = 1 λ Young [λ] σ S λ s σ1 s σλ s 2 Up Down) λ + 1 λ λ/2 SU(3) Young 2 λ 1 λ 2 SU(3) Young [1], 3; [1, 1], 3; [2], 6; 5 32

33 [2, 2], 6; [2, 1], 8; [3], 10 [1] (3 ) u, d, s [3], [2, 1] (baryon) [2, 1] (meson) (color) SU(3) [1] (3 ) [1 2 ] (3 ) [2, 1] (8 ) SU(m) Young λ ( n ) F/H, F = f 1 f n, H = s 1 s n s i i hook length (S n ) (factor)f i i f = m +1, 1 f 3 λ = [2, 2, 1] hook length 3: hook length m 2 (m + 1)(m 1)(m 2) = m2 (m 2 1)(m 2) 24 SU(2) SU(3) 33

34 7 (Representation theory of Lie group and Lie algebra) SU(n) 7.1 GL(n, C), GL(n, R): (general linear group) (det g 0) SL(n, C), SL(n, R): (special linear group) (det g = 1) U(n): (unitary group) g GL(n, C), g g = E O(n): (orthogonal group) g GL(n, R), g t g = E. SU(n) = U(n) SL(n, C) SO(n) = O(n) SL(n, R) Sp(n, K): (symplectic group) (K = R, C). ω = n i=1 (ξ i η i+n η i ξ i+n ) g GL(2n, K) ( 0 g t En Jg = J J = E n 0 ) Sp(n, C) U(2n) Sp(n) (classical Lie group) U(n), SU(n), O(n), SO(n), Sp(n) G 2, F 4, E 6, E 7, E 8 (exceptional Lie group) 34

35 ( ) (Lie algebra, ring) g 1. (linearity) X, Y g ax + by g (a, b C) 2. (commutator) X, Y g [X, Y ] g ( [X, Y ] = XY Y X) 1. [X, ay + bz] = a [X, Y ] + b [X, Z] [ax + by, Z] = a [X, Y ] + b [Y, Z] 2. [X, Y ] = [Y, X] 3. Jacobi [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 g T 1,, T d X g X = d i=1 a i T i d [T A, T B ] = i f C AB T C C=1 f C AB f AB C = f BA C ( ) f AB D f CD E + f BC D f AD E + f CA D f BD E = 0 (Jacobi ) ɛ g = e + iɛx ɛ2 2 X2 + O(ɛ 3 ) = exp(iɛx), X g. 35

36 1. U(n) u(n) E = g g = E + iɛ(x X ) + O(ɛ 2 ) T = T Hermite 2. SU(n) su(n): det(g) = 1 + iɛtr(x) + O(ɛ 2 ) = 1 trx = 0. Hermite 3. O(n) o(n): X t = X 4. SO(n) so(n): X t = X tr(x) = 0 X tr(x) = 0 O(n) SO(n) o(n) = so(n) 5. Sp(n, C) sp(n, C): ( E + iɛa iɛb g = iɛc E + iɛd ) + O(ɛ 2 ) d = a t, b t = b, c t = c 6. Sp(n) = Sp(n, C) U(2n) sp(n): a = a, b = c Campbell-Hausdorff g 1 = e X 1, g 2 = e X 2 g 1 g 2 e X 3 Campbell-Haussdorff X 3 = X 1 + X [X 1, X 2 ] [X 1 X 2, [X 1, X 2 ]] + = 1 m=1 m {Z m(x 1, X 2 ) + ( 1) m Z m (X 2, X 1 )} Z m (X, Y ) = ( 1) n+1 [ad X] p 1 [ad Y ] q1 [ad X] p n 1 [ad Y ] q n 1 X n=1 n (p i,q i p ) 1!q 1! p n 1!q n 1! n 1 (p i + q i ) = m 1, p i + q i > 0 i=1 36

37 M. Reinsch: arxiv:mathphys/ (Homotopy ) 7.2 (Global structure of Lie group) (π 0 (G)) G G 2 g 1, g 2 G g(t) G, t = [0, 1] g(0) = g 1, g(1) = g 2 O(3) = {g GL(3, R) g g t = E} g t g = E det(g) 2 = 1 det(g) = ±1 det(g 1 ) = 1 ( g 1 = E) det(g 2 ) = 1 ( g 2 = E) O(3) det(g) +1 1 G 0 G G G 0 G g G g G 0 g 1 = G 0 G/G 0 G = O(3) G/G 0 = Z 2 π 0 (G) (homotopy group) (π 1 (G)) e e G 0 G e e g(t) G (t [0, 1]) g(0) = g(1) = e g 0 (t) = e (t [0, 1]) g 1 g 2 (t) g 12 (t, s), t, s [0, 1] g 12 (t, 0) = g 1 (t), g 12 (t, 1) = g 2 (t) g 1 g 2 e e g 1 g 2 g 1 g 2 (t) = { g2 (2t) t [0, 1] 2 g 1 (2t 1) t [ 1, 1] 2 37

38 g g 1 (t) = g(1 t) t [0, 1] e e G (π 1 (G 0 )) G π 1 (G 0 ) 1. U(1) = {a C a 2 = 1}: n g n (t) = e 2πint (t [0, 1]) g n g m g n+m π 1 (U(1)) = Z 2. SO(3): SU(2) SO(3) g SU(2) g SO(3)) : 3 g σ i g = σ j g ji j=1 σ i Pauli SU(2) ( ) e πit 0 g(t) =, t [0, 1] 0 e πit SU(2) E E SO(3) g(t) E E SU(2) 2 g g SU(2) SO(3) g g π 1 (SO(3)) = Z 2 SU(2) S 3 (3 ) SO(3) SU(2)/ {E, E} = RP 3 (3 ) 38

39 Lie Lie g Lie G Lie Lie (universal covering group) UG Lie Lie UG/D D G g UG, d D g 1 dg D Lie Schur D {λe} ( λ C) SU(n) Lie D g g = E λ 2 = 1, det(g) = 1 λ n = 1 D = { ω l E }, ω = e 2πi/n, l = 0, 1,, n 1 SU(n) SU(n)/Z n Lie n = 2 SU(2)/Z 2 = SO(3) Lie Lie Lie g ρ : g GL(n, C) X 1, X 2 g [ρ(x 1 ), ρ(x 2 )] = ρ([x 1, X 2 ]) ρ e X Lie Lie ρ(e X ) = e ρ(x) Lie Lie UG/D D d ρ(d) = E Lie Lie UG/D 39

40 SU(2) j (j = 0, 1, 1, 3, ) 2j SO(3) Lie j j ρ( E) = E SO(3) 7.3 su(2) su(3) su(2) Lie su(2) [J i, J j ] = i k ɛ ijk J k J ± = J 1 ± ij 2 [J 3, J ± ] = ±J ±, [J +, J ] = 2J 3 1. J 3 J 2 = i Ji 2 2. J + j, j = 0, J 3 j, j = j j, j 2 j, j J 2 j(j + 1) 3. j, j J J 3 J 3 (J ) p j, j = (j p)(j ) p j, j J 2 J J 2 (J ) p j, j j, j p 4. j, j j, m j, m j, m = 1 J j, m = N j,m j, m 1 N j,m 2 = j, m J + J j, m = j, m (J 2 + J J 3 ) j, m = (j m + 1)(j + m) 40

41 j + m > 0 N j,m = (j m + 1)(j + m) 5. j (J ) p j, j (p = 0, 1, 2, ) j +m = j +(j p) < 0 p > 2j J j, p j = 0 j 1/2 l/2 2j + 1 j, j, j, j 1,, j, j + 1, j, j su(3) Lie su(3) su(2) su(3) 3 3 X X = X, tr(x) = 0. 8 Pauli Gellmann i λ 1 = 1 0 0, λ 2 = i 0 0, λ 3 = 0 1 0, λ 4 = λ 7 = i 0 i , λ 5 =, λ 8 = i i 0 0, λ 6 = T a λ a , tr(t a T b ) = 1 2 δ ab 41

42 su(2) J 3 T 3 T 8 2 H 1 = T 3, H 2 = T 8 Cartan (Cartan subalgebra) su(2) J ± 1 2 (T 1 ± it 2 ) = E ± α1, 1 2 (T 4 ± it 5 ) = E ± α2, 1 2 (T 6 it 7 ) = E ± α3 α i R 2 (i = 1, 2, 3) ( ) ( ) ( 1 1/2 1/2 α 1 =, α 2 =, α 3 = 0 3/2 3/2 H i (i = 1, 2) [ ] Hi, E αj = ( αj ) i E α ) ( α j ) i α j i Cartan 2 (2 ) 2 su(3) 4 2 H 1,2 Cartan (root system) ± α i (root vector) SU(2) Cartan 1 (±1), (0) ( J ±, J 3 ) SU(3) 2 su(3) (Young [1]) e 1 = 0, e 2 = 1, e 3 = Cartan H 1,2 H 1 e 1 = 1 2 e 1, H 1 e 2 = 1 2 e 2, H 1 e 3 = 0, H 2 e 1 = e 1, H 2 e 2 = e 2, H 2 e 3 = 1 3 e 3 42

43 4: SU(3) H i e j = ( ω j ) i e j ( ω j ) i 2 ω j i 3 ω j (weight vector) H i (fundamental weight) SU(3) Young e j1 e jl Cartan H i H i e j1 e jl = ( ω j1 + + ω jl ) i e j1 e jl Young 2 [2] (6 ) [1, 1](3 ) [2] e i e i, (i = 1, 2, 3), 1 2 ( e i e j + e j e i ), (i < j) 6 [1, 1] 1 2 ( e i e j e j e i ), (i < j)

44 Η 2 Η 2 Η 2 2ω 2 2ω 1 ω 2 ω 1 ω 3 Η 1 Η 1 Η 1 2ω 3 5: SU(3) [ ] 3 [3], [2, 1], [1 3 ] [2, 1] ( 4) ω ω E α ω ω + α : H i E α ω = [H i, E α ] ω + E α H i ω = (α i + ω i ) ω SU(3) ω 1 ω 2 = α 1, ω 1 ω 3 = α 2, ω 3 ω 2 = α 3 α i SU(2) j, m j, m ± 1 J ± 7.4 SU(3) quark SU(3) quark quark 2 quark, lepton, Higgs (gauge ) ( U(1)) photon γ, (SU(2)) weak boson Z, W ±, (SU(3)) gluon 44

45 quark 6 u (up), d(down), s(strange), c(charm), b(bottom), t(top) quark SU(3) u, d, s 3 3 quark 2 SU(3) SU(3) 3 quark u i, d i, s i (i = 1, 2, 3) SU(3) u d s (flavor) SU(3) 2 SU(3) quark quark baryon meson color color color SU(3) singlet (confinement) (QCD) SU(3) singlet 3 q i quark ɛ ijk q i q j q k quark (baryon) [1, 1] (3 ) 3 ([1]) 3 ([1, 1]) = 8 ([2, 1]) 1 ([1, 1, 1]) quark ( q) [1, 1] q q (meson) quark H i (i = 1, 2) quark (Q), (B), strangeness (S), (Y), (T 3 ) Y = B + S, Q = T 3 + Y 2 u,d,s 45

46 Q B S Y T 3 u 2/3 1/3 0 1/3 1/2 d 1/3 1/3 0 1/3 1/2 s 1/3 1/3 1 2/3 0 T 3 Y Y d u s Τ3 6: quark T 3 H 1 3Y/2 H 2 (u ω 1, d ω 2, s ω 3 ) SU(3) quark quark fermion baryon flavor SU(3) flavor SU(3) Young = 10 ([3]) 2 8 ([2, 1]) 1 ([1, 1, 1]) SU(2) = 4 ([3]) 2 2 ([2, 1]) Young (Young Young 46

47 ) SU(2) SU(3) 1 SU(2) 4 3/2, 2 1/2 10 3/2 8 1/2 baryon - 0 Y + ++ N Y P Σ Ξ *- *- Σ Ω *0 - Ξ Σ *0 *+ T Σ - Ξ - 0 Σ Λ Ξ 0 Σ + T 7: Baryon [ ] 7.5 Lie Cartan Cartan H i (i = 1,, m) r m E α r m (rank) α m root [H i, H j ] = 0, [H i, E α ] = α i E α weight m ω H i ω = ω i ω H i E α ω = (α i + ω i )E α ω weight ω root E α ω ω + α 47

48 λ Tr (T a T b ) = λδ ab E α E β := λ 1 Tr ( E α E β ) = δ α, β, H i H j := λ 1 Tr (H i H j ) = δ i,j λ su(3) 1/2 m [E α, E α ] = α i H i, i=1 E α E α weight E α E α = mi=1 β i H i β i β i = H i E α E α = λ 1 Tr (H i [E α, E α ]) = λ 1 Tr (E α [H i, E α ]) = α i λ Tr (E αe α ) = α i root su(2) J + j, j = 0 j, j J j, j, j, j 1,, j, j E α root α root α α i i = 1 m α i α α E α su(2) E α ω = 0 ( α > 0) su(3) (1, 0), (1/2, ± 3/2) su(3) (1/2, ± 3/2) (1, 0) = (1/2, 3/2) + (1/2, 3/2) (1, 0) (Cartan ) weight 48

49 E α ω = N α, ω ω + α, N α, ω 1. N α, ω α 2 N α, ω 2 = α ω ( ) ( )[E α, E α ] = m i=1 α i H i ω [E α, E α ] ω = m i=1 α i ω H i ω = mi=1 α i ω i ω E α E α ω ω E α E α ω = N α, ω 2 N α, ω 2, N α, ω = ω α E α ω = ω α E α ω = (N α, ω α) 2. ω p, q E α ω+p α = E α ω q α = 0 α ω α = 1 (p q). ( ) 2 2 ( ) (*) ω N α, ω+(p 1) α 2 0 = α ( ω + p α) N α, ω+(p 2) α 2 N α, ω+(p 1) α 2 = α ( ω + (p 1) α) 0 N α, ω q α 2 = α ( ω q α). ( ) p(p + 1) (p + q + 1) α ω + α 2 q(q + 1) 2 2 { = (p + q + 1) α ω + 1 } 2 α 2 (p q) p + q root (**) ω root α β (m ) α β β = q p := m β α = q p := m α α β 2 α 2 β = mm 2 4 = cos 2 θ. 49

50 θ root mm 4 mm = 0 (θ = π/2), mm = 1 (θ = π/3, 2π/3), mm = 2 (θ = π/4, 3π/4), mm = 3 (θ = π/6, 5π/6) Dynkin α, β 6 E α E β = 0 q = q = 0 = p/2 0, α 2 p /2 0 α β π/2 θ < π θ = π 2, 2π 3, 3π 4, 5π 6. β α β 2 = θ = π (p, p ) cos θ = 1 2 pp, β 2 / α 2 = p/p θ = 2π/3 p = p = 1 θ = 3π/4, 5π/6 (p, p ) = (1, 2), (1, 3) 2 3 SU(3) (1/2, ± 3/2) 2π/3 Dynkin su(n + 1) ( A n so(2n + 1) (B n ), sp(n) (C n ), so(2n) (D n ), G 2, F 4, E 6, E 7, E 8 Dynkin A 3 = D 3, B 2 = C 2, D 2 = A 1 A 1 6 [E α, E β ] = 0 β α root β α = γ γ root β = α + γ β γ root γ root 50

51 θ=5π/6 θ=2π/3 θ=3π/4 θ=π/2 8: (Dynkin ) su(4) = so(6), so(5) = sp(2), so(4) = su(2) su(2) A D E simply laced Lie algebra A B C D G F E 9: Dynkin (fundamental weight) E α ω = 0 (**) 2 α i ω α 2 = q i 0, (i = 1, 2,, m) m q i ω m ω = q i ω i i=1 51

52 ω i (fundamental weight) 2 α i ω j α i 2 = δ ij q i su(2) j 52

53 II 8 wedge V n (R n ) wedge ( ) α 1 α p = 1 ( 1) σ α σ(1) α σ(p), p! σ S p α i V p p V p wedge p- dim( p V ) = ( n p ) = n(n 1) (n p + 1) p! wedge 1. α 1 α i α j α p = α 1 α j α i α p 2. α 1 (a 1 α i + a 2 α i) α p = a 1 α 1 α i α p + a 2 α 1 α i α p a 1, a 2 R, α i V V = R 3 v = 3 i=1 v i e i u = 3 i=1 u i e i v u = i<j(v i u j u i v j ) e i e j = w 1 e 2 e 3 + w 2 e 3 e 1 + w 3 e 1 e 2 w i v u v u i 53

54 p V 1. V e i (i = 1,, n) p V 1 i 1 < i 2 < < i p n e i1 e ip 2. p = n dim( n V ) = 1 e 1 e n v i = nj=1 R ij e j v 1 v n = det(r ij ) e 1 e n 3. p > n dim( p V ) = 0 n n p V p V λ, µ λ = v 1 v p, µ = u 1 u p V (λ, µ) = det i,j=1,,p ( v i, u j ) p V λ = λ i1 i p e i1 e ip, µ = µ i1 i p e i1 e ip i 1 < <i p i 1 < <i p V ( e i, e j ) = G ij (λ, µ) = i 1 < <i p G i1 j 1 G ip j 1 λ i1 i p µ j1 j p.. j 1 < <j p G i1 j p G ip j p e i V e i ( e i, e j ) = δ j i V p V (p = 0, 1,, n) 2 1. ψi : λ p V ψ i (λ) = e i λ p+1 V 54

55 2. ψ i : λ p V ψ i (λ) = i( e i )λ p 1 V i( v) λ = v 1 v p p i( u)λ = ( 1) j 1 ( u, v j ) v 1 v j 1 v j+1 v p j=1 { ψ i, ψ j } = δ i j, { ψ i, ψ j} = 0, { ψi, ψ j } = 0. V e i e 1 e p = ψ 1 ψ p (1) (1) p Hodge dim( p V ) = dim( n p V ) Hodge e i V e i = e i λ = λ i1 i p e i1 e ip = λ i1 i p ψi1 ψ ip (1) p V i 1 < <i p i 1 < <i p Hodge λ n p V λ = λ i1 i p ψ ip ψ i1 σ i 1 < <i p σ = e 1 e n Hodge λ p V λ = ( 1) p(n p) λ : V = R 3 v u = ( v u) 55

56 9 n M (R n ) (x 1,, x n ) p- (differential p-form) ω = 0 i 1 < <i p n ω i1 i p (x) dx i 1 dx i p p- Ω p (M) ω = p dx i x i n 7 ( e i ) wedge dx i dx j = dx j dx i ω i1 i p (x) x i ω i1 i r i s i p (x) = ω i1 i s i r i p (x) Ω p (M) ω i1 i ( ) p n p p q ω η = ω = η = 0 i 1 < <i p n 0 j 1 < <j q n 0 i 1 < <i p n 0 j 1 < <j q n ω i1 i p (x) dx i 1 dx ip η j1 j q (x) dx j 1 dx jq ω i1 i p (x)η j1 j q (x) dx i 1 dx ip dx j 1 dx jq ω η = ( 1) pq η ω 7 (cotangent bundle) (fiber) 56

57 M x i y i dx i dx i = x i = φ i (y) n j=1 φ i (y) dy j y j p ω(x) = ω i1 i p (x)dx i 1 dx i p i 1 < i p = ω i1 i p (φ(y)) φi1 i 1 < i p j 1,,j p dy j 1 (φ ω)(y) φip dy j p dyj 1 dy j p y φ:n M M ω N φ ω φ ω(y) M ω (pullback) (exterior derivative) d Ω p (M) Ω p+1 (M) ω = 0 i 1 < <i p n ω i1 i p (x) dx i 1 dx i p dω = 0 i 1 < <i p n j ω i1 i p x j dx j dx i 1 dx i p 1. d(ω + η) = dω + dη 2. d(ω η) = dω η + ( 1) ω ω η 3. d 2 = 0 4. d(φ ω) = φ (dω) 57

58 3 d 2 ω = j,k 2 ω i1 ip i 1 < <i p x j x k dxk dx j dx i 1 dx i p dx j dx k j k 2 ω i1 ip x j x k 4 1 ω = ω i (x)dx i φ ω = ω i (φ(y)) φi y j dyj d(φ ω) = ( y k = ω i (φ(y)) φi y j ) dy k dy j ( ωi x l φ l y k φ i y j + ω i(φ(y)) 2 φ i y k y j = ω i x l φ l y k φ i y j dyk dy j = φ (dω)(y) ) dy k dy j Hodge Hodge dx i x i M ds 2 = ij g ij (x)dx i dx j dx i n e a = Ei a dx i i=1 n ds 2 = e a e a a=1 e a (vierbein) dx i 58

59 Hodge e a ω ω = ω i1 i p (x) dx i 1 dx ip 0 i 1 < <i p n e a = n i=1 Ei a dx i dx i = n a=1 Eae i a ω = ω a1 a p e a 1 e a p a 1 < <a p ω Hodge ω = ω a1 a p ψ ap ψ a 1 σ a 1 < <a p ψ σ σ = e 1 e p Hodge Ω p (M) Ω n p (M) Ω p (M) ω ω = ( 1) p(n p) ω δ Hodge d Ω p (M) Ω p 1 (M) δ δ = ( 1) np n+1 d 8 d, φ,, δ δ d δ 2 = 0 f(x) Ω 0 (M) ( M ) δf(x) = 0 ( 1 ) Laplacian Laplacian Ω p (M) Ω p (M) 2 = (d + δ) 2 = dδ + δd f Ω 0 (M) f(x) = δdf(x) = d df(x) 8 n 1 ( 1) p 59

60 10 M = R 3 x 1, x 2, x 3 ds 2 = (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 dx i : (dx i, dx j ) = δ ij 0 : f(x) 1 : V 1 (x)dx 1 + V 2 (x)dx 2 + V 3 (x)dx 3 2 : A 1 (x)dx 2 dx 3 + A 2 (x)dx 3 dx 1 + A 3 (x)dx 1 dx 2 3 : g(x)dx 1 dx 2 dx 3 x i x i f f, A i A i φ(x), E i (x), B i (x) (1 ) (grad), (div), (rot), Laplacian 1. (gradient) (0 ) (1 ) df = f x 1 dx1 + f x 2 dx2 + f x 3 dx3 0 d 2. (divergence) (1 ) (0 ) 1 ω = i V i (x)dx i δ δω = d (V 1 dx 1 + V 2 dx 2 + V 3 dx 3 ) = d(v 1 dx 2 dx 3 + V 2 dx 3 dx 1 + V 3 dx 1 dx 2 ) = ( x V x V x V 3)dx 1 dx 2 dx 3 = div V 3 δ 60

61 3. (rotation) Hodge (rot) dω = d(v 1 dx 1 + V 2 dx 2 + V 3 dx 3 ) (( V3 = x V ) ) 2 dx 2 dx x 3 = ( ( V ) 1 dx 1 + ( V ) 2 dx 2 + ( V ) 3 dx 3) rot = d 4. Laplacian ( ) (d + δ) 2 ω = ± ( ωi1 i p (x) ) dx i 1 dx i p i 1 < <i p = ( ) 2 ( ) 2 ( ) 2 x + 1 x + 2 x 3 ( ) 1. :(r, θ, ϕ) dx 2 = (dr) 2 + r 2 ( (dθ) 2 + sin 2 θ(dϕ) 2) 2. : (r, θ, z) dx 2 = (dr) 2 + r 2 (dθ) 2 + dz 2 ds 2 = (h 1 (y)) 2 (dy 1 ) 2 + (h 2 (y)) 2 (dy 2 ) 2 + (h 3 (y)) 2 (dy 3 ) 2 (h i (x) > 0 ) e i = h i (y)dy i e i (y) 3 3 ω(y) = V i (y)dy i = (V i /h i )e i i=1 i=1 V i /h i i grad, div, rot 61

62 1. grad: df = i f y i dyi = i (grad f) i = 1 h i f y i 1 f h i y i ei 2. div: Hodge e i dx i δv = d (V 1 e 1 + V 2 e 2 + V 3 e 3 ) 3. rot: = d (V 1 e 2 e 3 + V 2 e 3 e 1 + V 3 e 1 e 2 ) = d ( V 1 h 2 h 3 dx 2 dx 3 + V 2 h 3 h 1 dx 3 dx 1 + V 3 h 1 h 2 dx 1 dx 2) ( = x (V 1h 1 2 h 3 ) + x (V 2h 2 3 h 1 ) + ) x (V 3h 3 1 h 2 ) dx 1 dx 2 dx 3 ( 1 = h 1 h 2 h 3 x (V 1h 1 2 h 3 ) + x (V 2h 2 3 h 1 ) + ) x (V 3h 3 1 h 2 ) divv dv = d (V 1 e 1 + V 2 e 2 + V 3 e 3 ) = d ( h 1 V 1 dx 1 + h 2 V 2 dx 2 + h 3 V 3 dx 3) (( (h3 V 3 ) = (h ) ) 2V 2 ) dy 2 dy 3 + cyclic perm. x 2 x 3 ( ( 1 (h3 V 3 ) = (h ) ) 2V 2 ) e 2 e 3 + cyclic perm. h 2 h 3 x 2 x 3 ( 1 (h3 V 3 ) = (h ) 2V 2 ) e 1 + cyclic perm. h 2 h 3 x 2 x 3 (rot V ) 1 = 1 h 2 h 3 ( (h3 V 3 ) (h ) 2V 2 ), x 2 x 3 4. Laplacian: (d + δ) 2 f = δdf = d df ( f = d y 1 dy1 + f y 2 dy2 + f ) y 3 dy3 ( 1 f = d h 1 y 1 e1 + 1 f h 2 y 2 e2 + 1 ) f h 3 y 3 e3 62

63 ( 1 f = d h 1 y 1 e2 e f h 2 y 2 e3 e f h 3 y 3 e1 e 2 ( h2 h 3 f = d = = ( y 1 1 h 1 h 2 h 3 h 1 ( h2 h 3 ( y 1 ) f y 2 dy3 dy 1 + h 1h 2 h 3 y 1 dy2 dy 3 + h 3h 1 h 2 ) ) f + dy 1 dy 2 dy 3 h 1 y 1 ( ) h2 h 3 f + ( h3 h 1 h 1 y 1 y 2 h 2 ) f y 3 dy1 y 2 ) f + ( h1 h 2 y 2 y 3 h 3 )) f y 3 11 Maxwell Maxwell ( c = 1 ) E = ρ, B = 0 B E t = J, E + B t = 0 (t = x 0 ) 4 4 Minkowski ds 2 = (dx 0 ) 2 + (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 e 0 = idx 0, e j = dx j, (j = 1, 2, 3) Hodge 4 2 F = 3 F µν dx µ dx ν = E i (x)dx 0 dx i + 1 µ<ν i=1 2 3 i,j,k=1 ɛ ijk B i dx j dx k F µν 0 E 1 E 2 E 3 E 1 0 B 3 B 2 F = E 2 B 3 0 B 1 E 3 B 2 B

64 1 3 j = ρ(x)dx 0 + J k (x)dx k k=1 Maxwell δf = j, df = 0 df = 1 ( ɛ ijk ( 2 E) k + B ) k dx 0 dx i dx j +( x B)dx 1 dx 2 dx 3 0 i,j,k df = 0 Maxwell 2 4 Hodge F = i j B j dx 0 dx j + i ɛ jkl E j dx j dx l 2 j,k,l i E B Hodge δf = d F δf = j Maxwell 1 3 φ A B = A, E A = φ x 0 F = da 1 A 3 A = φ(x)dx 0 + A j (x)dx j j=1 p ω(x) dω(x) = 0 p 1 µ ω = dµ d 2 = 0 64

65 Maxwell df = 0 F = da 1 F = da A = A + dχ F ( da = da + d(dχ) = F ) χ M p M p 9 p f = f(p ) [P ] p = 1 2 R 2 1 x = x(t), y = y(t) 1 t P 0 P 1 t P 0 = (x(0), y(0)), P 1 = (x(1), y(1)) A = A x dx + A y dy P1 [ 1 A = dt A x (x(t)) dx P 0 0 dt + A y(x(t)) dy ] dt t 1 t t dt dx dt dx dx = d t = d t dt d t dt d t 9 M 65

66 1 A = df (F 0 ) P1 [ 1 df dx df = dt P 0 0 dx dt + df ] dy dy dt 1 df (x(t)) = dt = F (x(1)) F (x(0)) 0 dt = F (P 1 ) F (P 0 ) = F [P 1 ] [P 0 ] Green Green Green ( Vy dxdy D x V ) x = (V x dx + V y dy) y D D R 2 D D D ω = V x dx+v y dy dω = ω D D 12.2 R n+1 n + 1 P 0, P 1,, P n 3 n n n P = t i P i, (t i 0, t i = 1) i=0 i=0 n = 0 P 0, n = 1 2 P 0, P 1 n = 2 3 P 0, P 1, P 2 3 n + 1 P 0, P 1,, P n n (P 0, P 1,, P n ) 66

67 t 1,, t n 0 t i 1, 0 n i=1 t i 1 c = i a i i ( i a i ) (chain) n n 3 (P 1, P 2, P 3 ) 3 3 (P 0, P 1 ) + (P 1, P 2 ) + (P 2, P 0 ) R 2 4 P 1, P 2, P 3, P (2 ) (P 1, P 2, P 3 ) (P 2, P 3, P 4 ) (orientation) 1 (P 0, P 1 ) (P 1, P 0 ) (P 1, P 0 ) = (P 0, P 1 ) (P σ(1),, P σ(n) ) = ( 1) σ (P 1,, P n ) n n 1 n (P 0,, P n ) = ( 1) i (P 0,, P i 1, P i+1,, P n ) i=0 i c = l a l l ( l a l ) c = l a l l (P 0 P 1 ) = (P 1 ) (P 0 ) (P 0 P 1 P 2 ) = (P 1 P 2 ) (P 0 P 2 ) + (P 0 P 1 ) 4 (P 1, P 2, P 3 ) (P 2, P 3, P 4 ) 4 1 (P 1 P 2 ) + (P 2 P 4 ) + (P 4 P 3 ) + (P 3 P 1 ) 67

68 2 2 = 0 ( (P 0 P n )) ( n ) = ( 1) i (P 0 PX i P n ) i=0 n = ( 1) i i 1 ( 1) j (P 0 PX j PX i P n ) + i=0 = 0 j=0 n j=i+1 ( 1) j 1 (P 0 X P i X P j P n ) p ω(x) = 1 ω µ1 µ p! p (x)dx µ 1 dx µp µ 1 µ p p X X p X = X 1 X L X α (α = 1,, L) ϕ α (t) α x µ = ϕ µ α(t), x X α ω X X ω = L α=1 X α ω ω = 1 X α p! µ 1 µ p = (ϕ αω)(t) α t i 0, t i 1 d p t ω µ1 µ p (ϕ α (t)) ϕµ 1 α t 1 ϕµp α t p ϕ αω ω ϕ α α t i (i = 1,, p) 68

69 Stokes Green p p ω (p 1) dω (p 1) = p p ω (p 1) p = dt df(t) dt = f(1) f(0). p = 2 Green X d dω (p 1) = ω (p 1) X X ( )Stokes p M n (ω, µ) = ω µ M (ω, µ Ω p (M)) (ω, ν) = (ν, ω) M ( M = 0) Stokes ω Ω p 1 (M), ν Ω p (M) (dω, ν) = (ω, δν) (dω, ν) = dω ν = d(ω ν) + ( 1) p dω ν M M = ω ν + ( 1) np n+1 ω ( d ν) = ω δν = (ω, δν) M M M Maxwell df = 0, δf = j S[A] = 1 (da, da) (j, A) 2 69

70 F = da df = 0 A (A = A + ɛa 1 ) S[A ] S[A] = ɛ ((da, da 1 ) (j, A 1 )) = ɛ ((δda j, A 1 )) O(ɛ 2 ) A δda j = 0 F A Homology Cohomology M Homology Cohomology d d 2 = 0, 2 = 0 M X X = 0 X = Y Z p (M) = {c c p, c = 0} B p (M) = {c there exists b p 1, c = b} H p (M) = Z p (M)/B p (M) 10 H 0 (T 2 ) = Z, H 1 (T 2 ) = Z Z, H 2 (T 2 ) = Z, H 0 (S 2 ) = Z, H 1 (S 2 ) = 0, H 2 (S 2 ) = Z. p- dω = 0 ω = dµ Z p (M) = {ω Ω p (M) dω = 0}, B p = {ω Ω p (M) there exists µ Ω p 1 (M), ω = dµ} H p (M) = Z p (M)/B p (M) c H p (M), ω H p (M) ω 10 T 2 2 S 2 2 c 70

71 ω = ω + dω = ω, (ω + dµ) = ω + µ = ω c+ b c b c c Stokes dω = 0, c = 0 c c c 71

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