Hyperbolic Geometry and the Lorentz Group

Transcription

1 Hyperbolic Geometry and the Lorentz Group Motivation Norbert Dragon Bad Honnef 18. März 2015 Fermi-Walker transport Wigner Rotation, Unitary Representations of the Poincaré Group Hyperbolic Geometry Poincaré Disk Wigner Angle

2 Motivation Just for fun, Curiosity Not in the books, professional training Uniformly accelerated worldlines (alternative mechanics, space flight) Rotationfree transport Thomas Precession Quantum Mechanics Nonassociativity (Gyro group, Ungar, Karzel)

3 Fermi-Walker Transport Observer, vierbein, infinitesimal rigid body: Γ : τ (x(τ),e 1 (τ),e 2 (τ),e 3 (τ)), e 0 = dx dτ, e a e b = η ab Λ = ( e 0, e 1, e 2, e 3 ) Lorentzgroup SO(1,3) = H 3 SO(3), Λ = L(Λ)O(Λ), L = L T,O 1 = O T { e0 } = H 3 = { (q 0,q 1,q 2,q 3 ) : (q 0 ) 2 + (q 1 ) 2 + (q 2 ) 2 + (q 3 ) 2 = 1 } e 1,e 2,e 3 span tangent space at e 0 de a dτ = e cω c a, ω ab = ω ba ( ) 0 b T rotation free ω ij = 0, ω = b 0, de 0 dτ = b1 e 1 + b 2 e 2 + b 3 e 3, de 1 dτ = b1 e 0, de 2 dτ = b2 e 0, de 3 dτ = b3 e 0

4 Rotationfree (Fermi-Walker) Transport Reflected light returns from the direction into which it was emitted τ v Γ τ u Directions of light from distant stars changed by aberration

5 Constant, rotation free acceleration Γ : a x(a) = ( ) sh a (ch a 1)n + x(0), ˆΓ : a dx da = ( ) ch a n sh a Rapidity a: lenght of Γ and ˆΓ Rotation free, linearly accelerated vierbein, initially at rest ( ) ( ) ch a, n ˆΓ : a L a,n = T sh a 0 n T n sh a, 1 + (cha 1)nn T = expa n 0 with initial velocity v = n b thb, (read a to denote (a, n a ) as appropriate) L bˆγ : a Lb L a L a multiplies from the right, acceleration is constant in the observer frame

6 Relativistic Quantum Mechanics Each unitary, irreducible representation of the Poincaré group is unitarily equivalent to the representation, which is induced by an irreducible, unitary representation R of the little group of a vector p on its mass shell. For m 2 > 0 and p 0 > 0 and with a Lorentz invariant integration measure it is given by (U Λ Ψ)(Λp) = R(W(Λ,p))Ψ(p) Wigner Rotation: W(Λ,p) = L Λp 1 ΛL p, L p p = p L p = ( p 0 m p T m p m 1 + p pt m(p 0 +m) W(O,p) = O, L Lq p W(L q,p) = L q L p ) ( ) p cha, m =, L n sha p = L a, independent of m

7 Infinitesimal Transformations infinitesimal transformation (antihermitean) l ij = (p i p j pj p i) + Γ ij, Γ ij = Γ ji = (Γ ij ) [Γ ij,γ kl ] = δ ik Γ jl δ il Γ jk δ jk Γ il + δ jk Γ il p k l 0i = p 0 p + Γ i ik p 0 + m independent of mass

8 Hyperbolic Geometry L 1 c W(b,a) = L b L a ( ) chc = ( )( ) chc W(b, a) ( chb sh bn T = b ) ( ) ch a shan a ch c = ch a chb shash b cosγ,γ subtends c chc = ( chb sh bn b ) ( cha shan a ) infinitesimally Euclidean c 2 = a 2 + b 2 2ab cosγ (γ subtends c) Circumference and area of circle of radius r: U = 2π sh r, A = 2π(chr 1) Thomas precession: deficit angle δ = 2π(ch r 1)

9 Poincaré Disk u 2 < 1 W(b,a) rotates the plane, spanned by a and b (SO(1,2) SL(2,R 2 )), R 1,2 Stereographic projection u of H 2 from the south pole ( 1,0,0) u i = qi 1 + q 0, inverted: q0 = 1 + u2 1 u 2, qi = 2ui 1 u 2 bijective, conformal: circles to circles, metric g(u,du) = 4 (1 u 2 ) 2 du i du i a u(l v L a ) circle in Poincaré disk u 2 < 1, which intersects u 2 = 1 orthogonally, straight lines

10 Wigner Rotation W(b,a) rotates the plane, spanned by a and b (SO(1,2) SL(2,R 2 )) 1 W(b, a) = cosδ sinδ opposite to the rotation from a to b, sin δ cosδ W(a,b) = W(b,a) 1 Its angle δ is the hyperbolic area of the triangle a,b,c δ in terms of the lengths of the sides: Heron s formula (some calculation) cos δ 2 = 1 + ch a + ch b + chc 4 ch a 2 ch b 2 ch c 2 In terms a, b and the angle γ: chc = chach b sh a sh b cosγ cosδ = ( m 2 + mp 0 + mq 0 + p 0 q 0 + p q ) 2 (m + p 0 )(m + q 0 )(m 2 + p 0 q 0 + p q) 1

11 Massless Representations p 0 = W(Λ,p) = L 1 Λp ΛL p,l p p = p (p z ) 2 + p 2, p = (1,1,0,...0) little group ISO(D 2) L p = D p B p 1 2 (p0 + 1 ) 1 p 0 2 (p0 1 ) p 0 1 B p = 2 (p0 1 ) 1 p 0 2 (p0 + 1 ) p 0 D p = ( 1 ˆDp), ˆDp = D p shortest rotation from (p 0,p 0,0...) to (p 0,p z, p), not defined for (p 0,p z, p) = (p 0, p z,0) p z p 0 1 (D 2) (D 2) pt p 0 p p 0 1 p pt p 0 (p 0 +p z )

12 Infinitesimal Transformations (antihermitean) l ij = (p i p j pj p i) + γ ij, i,j {2,...D 1} [γ ij,γ kl ] = δ ik γ jl δ il γ jk δ jk γ il + δ jl γ ik l 0i = p p + γ p k i ik, p = p p + p 2 z + p i p i z l 0z = p p z Singular on the negative z-axis. l zi = (p z p i pi ) + γ ik p z Spin bundle with northern and southern hemisphere p k p + p z

13 Southern Chart L S p = D S pb S pd zx D zx rotation by π in zx-plane, D zx (1,1,0...) T = (1, 1,0...) T B S p = 1 2 (p + 1 p ) 1 2 (p 1 p ) 1 2 (p 1 p ) 1 D S p = ( 1 ˆD S p ), ˆDS p = p z p p p 2 (p + 1 p ) 1 (D 2) (D 2) p T p 1 p pt p(p p z ) D S p shortest rotation from (p, p,0...) to (p,p z, p), not defined for (p,p z, p) = (p, p z,0).

14 Southern Infinitesimal Transformations l ij = (p i p j pj p i) + γ ij, i,j {2,...D 1} l 0i = p p + γ p k i ik p p z l zi = (p z p p k i pi ) γ ik p z p p z l 0z = p p z Singular on the positive z-axis.

15 Transition Function 1 T p =(L S p) 1 L N p = p2 x p 2 2 p x p p 2 2 p x p T p p pt p 2 1 = 1 C S n T S n 1 + (C 1) n n T p 2 = p 2 x + p 2 (no p 2 z), C = cos2ϕ, S = sin 2ϕ, p (p 0, p z,0,...,0) Transition well defined also for half-integer Spin(D-2).

16 Spherical Geometry cosc = cosacos b + sin a sin b cosγ ( ) ( ) cosb cosa cosc = sin bn b sin an a infinitesimally Euclidean c 2 = a 2 + b 2 2ab cosγ