The GEOMETRY of the OCTONIONS

Transcription

1 The GEOMETRY of the OCTONIONS Tevian Dray I: Octonions II: Rotations III: Lorentz Transformations IV: Dirac Equation V: Exceptional Groups VI: Eigenvectors

2 DIVISION ALGEBRAS Real Numbers: R Quaternions: H = C + Cj q = (a + bi) + (c + di)j Complex Numbers: C = R + Ri Octonions: O = H + Hl z = x + yi i 2 = j 2 = l 2 = 1

3 QUATERNIONS k 2 = 1 ij = +k ji = k not commutative j i k

4 VECTORS I v = bi + cj + dk v = b î + cĵ + d ˆk vw v w + v w Dot product exists in any dimension but Cross product exists only in 3 and 7 dimensions

5 kl j i l il k jl

6 kl j i l il k jl

7 kl j i l il k jl

8 kl j l i il k jl

9 kl j l i il k jl

10 kl j l i il k jl

11 OCTONIONS kl each line is quaternionic (ij)l = +kl i(jl) = kl j l i not associative il k jl

12 WHAT STILL WORKS? q Im O = q = q qq = q 2 q 1 = q q 2 (q 0) pq = p q required for supersymmetry [p, p, q] = (pp)q p(pq) = 0 alternativity

13 EXPONENTIAL FORM p = p e s φ (s 2 = 1) = p (cos φ + ssinφ) e kφ i = ie kφ e kφ ie kφ = ie 2kφ

14 ROTATIONS x H x R 4 x x p = 1 = px = x SO(3): θ about k-axis: e kθ x SO(7)? kl e kθ rotates 3 planes perpendicular to k... Need Flips j l i il k jl

15 ROTATIONS x O x R 8 x x p = 1 = pxp = x SO(7): 2θ about k-axis : e kθ xe kθ flips (2θ in ij-plane): (icos θ + j sin θ)(ixi)(icos θ + j sin θ) nesting!

16 ROTATIONS x O x R 8 x x p = 1 = pxp = x SO(8): flips OK 2θ in 1l-plane: e lθ xe lθ triality: pxp, px, xp fails over H!

17 AUTOMORPHISMS pxp 1 = x SO(3): over H: (pxp 1 )(pyp 1 ) = p(xy)p 1 ( p) G 2 SO(7): p = e πs/3 (s 2 = 1) dim G 2 = 14 SU(3) G 2 π/3 phase quarks? Fix 1; Rotate i: 6 choices; Rotate j: 5 choices; k fixed; Rotate l: 3 choices; Done.

18 VECTORS II x = 0 t x y 1 C A t + z x iy X = x + iy t z! z X = X T = X det(x) = t 2 + x 2 + y 2 + z 2 {vectors in (3+1)-dimensional spacetime} {2 2 complex Hermitian matrices} determinant (Lorentzian) inner product X = ti + xσ x + yσ y + zσ z (Pauli matrices)

19 LORENTZ TRANSFORMATIONS Exploit (local) isomorphism: SO(3,1) SL(2, C) x = Λx X = MXM B 0 cos α sin α 0 0 sin α cos α 0 A e iα e iα 2! det(m) = 1 = detx = detx

20 ROTATIONS M zx = cos α 2 sin α 2 sin α 2 cos α 2 M yz = M xy = e iα e iα 2!! cos α 2 i sin α 2 i sin α 2 cos α 2! i j, k,..., l

21 ROTATIONS Still missing: rotations in Im K H: M = e kθ I O: flips! X = M 2 (M 1 XM 1)M 2 M 1 = ii M 2 = (icos θ + j sin θ)i

22 WHICH DIMENSIONS? K = R, C, H, O X = ( p a a m ) (p, m R; a K) dim K + 2 = 3, 4, 6, 10 spacetime dimensions supersymmetry SO(5,1) SL(2, H) SO(9,1) SL(2, O)

23 PENROSE SPINORS v = ( ) c b vv = c 2 cb bc b 2! det(vv ) = 0 (spinor) 2 = null vector Lorentz transformation: v = Mv M(vv )M = (Mv)(Mv) compatibility

24 HOPF FIBRATIONS v K 2 v v = 1 = v S 2k 1 tr (vv ) = v v = 1 = t = const det(vv ) = 0 = vv S k S 2k 1 S k 1 S k phase freedom: «p q pq q q! e sθ (s 2 = 1) OP 1 = {(p, q) (pq 1 χ, χ)} OP 1 = {X 2 = X,trX = 1}

25 WEYL EQUATION Massless, relativistic, spin 1 2 Momentum space Pψ = p µ σ µ ψ = 0 ep = P (tr P ) I Pauli Matrices: «1 0 σ t = 0 1 «0 1 σ x = 1 0 σ z = σ l = 1 0 «0 1 0 «l l 0 σ k = «0 k k 0

26 SOLUTION Weyl equation: (3 of 4 string equations!) One solution: (P, θ complex) Pψ = 0 = det(p) = 0 P = ±θθ f θθ θ = (θθ θ θ)θ = θθ θ θ θθ = 0 General solution: (ξ O) ψ = θξ P, ψ quaternionic

27 P = = P = ( c 2 cb PHASE FREEDOM bc b 2 ) ( b 2 cb bc c 2 = θθ = (θe sφ )(θe sφ ) ) θ = c bc c! = ψ = θξ = ( c bc c ) r e sα Phase freedom is supersymmetry. Solutions are quaternionic (only 2 directions: bc, s).

28 DIRAC EQUATION Gamma matrices: (C Weyl representation) ( ) 0 σµ γ µ = σ µ 0 Dirac equation: (momentum space) (p µ γ µ m) Ψ = 0 Weyl equation: (H Penrose spinors) Ψ = θ! ψ = η + σ k θ η (p µ eσ µ meσ k ) ψ = 0

29 COMPARISON 4 4 complex: 0 = (γ t γ µ p µ m γ t ) Ψ 2 2 quaternionic: 0 = (p t σ t p α σ α m σ k ) ψ = Pψ Isomorphism: (H 2 C 4 )! c kb d + ka 0 a 1 B b c A d

30 DIMENSIONAL REDUCTION SO(3, 1) SL(2, C) SL(2, O) SO(9, 1) Projection: (O C) π(p) = 1 (p + lpl) 2 Determinant: det(p) = 0 = det ( π(p) ) = m 2 Mass Term: P = π(p) + m σ k σ k = 0 k p t + p z p x lp y! km P = p x + lp y + km p t p z k 0!

31 SPIN Finite rotation: R z = ( e l θ e l θ 2 ) Infinitesimal rotation: L z = dr z = 1 dθ 2 θ=0 ( l 0 0 l ) Right self-adjoint operator: ˆL z ψ := (L z ψ)l Right eigenvalue problem: ˆL z ψ = ψλ

32 ANGULAR MOMENTUM REVISITED L x = 1 2 L z = 1 2! 0 l l 0! l 0 0 l! L y = ˆL µ ψ := (L µ ψ)l «1 ψ = e = = k ˆL z ψ = ψ 1 2 ˆL x ψ = ψ k 2 ˆL y ψ = ψ kl 2 Simultaneous eigenvector! (only 1 real eigenvalue)

33 LEPTONS ψ P = ψψ ( ) 1 e = k ( ) k e = 1 ( ) 0 ν z = k ( ) k ν z = 0 ( ) 1 k e e = k 1 ( ) 1 k e e = k 1 ( ) 0 0 ν z ν z = 0 1 ( ) 1 0 ν zν z = 0 0

34 How Many Quaternionic Spaces? Dimensional Reduction: kl = l H Orthogonality: j l i (H 1 H 2 = C) i, j, k il k jl Answer: 3!

35 LEPTONS ( ) 1 e = k ( ) k e = 1 ( ) 0 ν z = k ( ) k ν z = 0 ( ) 0 Ø z = 1 ( ) 1 k e e = k 1 ( ) 1 k e e = k 1 ( ) 0 0 ν z ν z = 0 1 ( ) 1 0 ν zν z = 0 0 ( ) 0 0 Ø z Ø z = 0 1

36 Have: WHAT NEXT? 3 generations of leptons! Neutrinos have just one helicity! Want: What about Ø z? interactions quarks/color (SU(3)!) charge

37 JORDAN ALGEBRAS X = p a c a m b c b n X Y = 1 (X Y + YX) 2 X Y = X Y 1 «X tr (Y) + Y tr(x) «tr (X) tr (Y) tr (X Y) I 2

38 JORDAN ALGEBRAS u, v, w R 3 = 2 uu vv = (u v) (uv + vu ) tr (uu vv ) = (u v) 2 2 uu vv = (u v)(u v) X Y = 1 2 (X Y + YX) X Y = X Y 1 2 X tr (Y) + Y tr (X)! tr (X) tr (Y) tr (X Y)! I

39 JORDAN ALGEBRAS Exceptional quantum mechanics: (Jordan, von Neumann, Wigner) (X Y) X 2 = X `Y X 2 X Y = 1 (X Y + YX) 2 X Y = X Y 1 «X tr (Y) + Y tr(x) «tr (X) tr (Y) tr (X Y) I 2

40 MORE ROTATION GROUPS 3 t = p + m + n 2 3 w = p + m 2n 2z = p m SO(27): tr (X X) = 2( a 2 + b 2 + c 2 + w 2 + z 2 ) + t 2 SO(26, 1): tr (X X) = a 2 + b 2 + c 2 + w 2 + z 2 t 2

41 EXCEPTIONAL GROUPS F 4 : SU(3, O) (MX M ) (MYM ) = M(X Y)M E 6 : SL(3, O) det X = 1 3 tr ((X X) X) SO(3, 1) U(1) SU(2) SU(3) E 6

42 COMPLEX EIGENVALUE PROBLEM Eigenvalue Equation: Hermitian: Av = λv A = A T = A Reality: λv v = (Av) v = v Av = v λv Orthogonality: λ 1 λ 2 = v 1 v 2 = 0 since: λ 1 v 1 v 2 = (Av 1 ) v 2 = v 1 Av 2 = v 1 λ 2v 2 n eigenvalues, all real. basis of orthonormal eigenvectors. Decomposition: A = n m=1 λ m v m v m

43 OCTONIONIC EIGENVALUE PROBLEM Eigenvalue Equation: Av = vλ Hermitian: 1 i «j i 1 l «= A = A «j + il = l k j l «(1 kl) Eigenvalues need not be real! λ(v v) (λv )v = (Av) v = (v A)v v (Av) = v (vλ) (v v)λ

44 3 3 REAL EIGENVALUE PROBLEM Characteristic Equation: A 3 (tr A) A 2 + σ(a) A (det A) I = 0 A 3 = 1 2 (AA2 + A 2 A) But: λ 3 (tr A) λ 2 + σ(a) λ det A = r (r r + )(r r ) = 0 2 sets of 3 real eigenvalues! 4-parameter families of eigenvectors det A = 0 λ = 0!

45 DECOMPOSITIONS I Family: K[v] = rv = A A A(v) «tra A A(v) + σ(a) A(v) det(a) v Theorem: (v, w in same family; λ µ) (vv )w = 0 Proof: 8 hour brute force Mathematica computation! (Analytic proof by Okubo!) Corollary: A = λvv

46 PROJECTIONS Idea: (uu ) z u (u z) Theorem: (z any vector in same family) (uu ) ( (uu )z ) = (uu )z Corollary: (vv ) ( (uu )z ) = 0 Corollary: A ( (uu )z ) = λ ( (uu )z )

47 DECOMPOSITIONS II Into Families: z = K[z] r 2 z r 1 r 2 K[z] r 1 z r 1 r 2 = z 1 + z 2 Within Families: (uu + vv + ww = I) z 1 = (uu )z 1 + (vv )z 1 + (ww )z 1 Theorem: z = (uu )z 1 + (ûû )z 2 This decomposes any vector z into six eigenvectors, one for each eigenvalue of A!

48 JORDAN EIGENVALUE PROBLEM Jordan product: A B = 1 2 (AB + BA) (over R: 2 uu vv = (u v) (uv + vu )) Eigenvalue problem: (eigenmatrices!) A x = λx eigenvalues satisfy characteristic equation (A 3 A A A!) only 3 eigenvalues λ µ = V W = 0 A = P λv

49 OCTONIONIC EIGENVALUE PROBLEM Eigenvalues not necessarily real! New notion of orthogonality: (vv )w = 0 6 eigenvalues in 3 3 case! Decomposition: A = λvv

50 SUPERSPINORS X = M = ( ) X θ θ n ( ) MX M = M ( MXM θ M Mθ n ) Cayley/Moufang plane: OP 2 = {X 2 = X,tr X = 1} = {X X = 0,tr X = 1} = {X = ψψ, ψ ψ = 1} (ψ H 3 )

51 DIRAC EQUATION II ( P θξ ) P = ξθ ξ 2 P P = 0 = Pθ = 0 P = ψψ quaternionic! 3X Furthermore: X = X = X = ψ n ψ n n=1 leptons, mesons, baryons?

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