RELA TIVE LOCALITY IN 2+1 DIMENSIONS ESFQG SISSA, SEPTEMBER 2014 - PowerPoint PPT Presentation

„ RELA TIVE LOCALITY ” IN 2+1 DIMENSIONS ESFQG – SISSA, SEPTEMBER 2014 1

THE PLAN 1. 1. Curved momentum spac ace an and relat ative local ality; 2. 2. Alekseev-Mal alkin kin construct ction on of f effect ctive, , deformed an in 2+1 dimensions * ; par articl cle lag agran angian 3. 3. Comments ts. omasz Trze ś niewski. * Based on the (partially inpublished) work done with T 2

SCALE AND GEOMETRY The assumpti mption n that t the angle sum m is less 80  leads than n 180 s to a geome metr try y quite e different from Euclid’s . . It depe pends nds on on a constant, tant, which ch is not given n a pr priori. ori. As As a a joke I e even wishe hed d Euclide idean an geome metr try was not true, , for then we would d have an absolute te me measure ure of of le length h a pr priori ori. . (Gauss, s, 182 827) 7) If there re is an a pr priori ri scale, , you expe pect nontri rivial al geome metr try. y. Ever erything ything is curved, d, unless s it cannot b t be 3

FLA T/CURVED MOMENTUM SPACE In the case of a standar ard d relati tivisti istic c particle cle we have flat spacetime ime and flat momentum m sp space: Spacetime No scale required The phase space is a a Momentum space cotan tange gent nt bun undle over flat t momentum m space 4

FLA T/CURVED MOMENTUM SPACE For deformed d relativistic istic particle icle we have flat spaceti time me and curved d momentum um sp space: Spacetime The phase space is a a cotan tange gent nt bundle over Curved momentum space curved d momentum m space (the curvature scale needed!) 5

RELA TIVE LOCALITY In the RL framework work the no-trivia ial geometr try y of momentum m space exhibits its itse self f in a num umber of f ways ys: 1. 1. The kinetic term for a particle icle has the form L ~ p E ( p x ) , a     a  1 E ( p ) f p a a a            with the no nontrivial ial momentum m space frame e field. 6

RELA TIVE LOCALITY The mass-shell ll relation ion is defined d as a square of the distan ance from zero to the po point P, with coordin dinat ates p μ (P): C ( p ) D ( p ) m 2 2   We need metric c to d define the m mass-sh shell ll relation tion ! p   D ( p ) ds g p p 2    0 geodesic If f the metric is n s nonlinear ar we need the mass ss sc scale to defi fine it.

MOMENTUM ADDITION In order r to add two momenta ta the notion of connecti tion on (paralle lel transpor port) ) is ne needed. d. p  q q p 0 sition p  q is nonlinear If the c composition ar we need a mass scale to define it. 1 ( p q ) p q p q                8

RELA TIVE LOCALITY In theories with curved d momentum m space, , localit lity y might be relative: e: 1. 1. The transl slati ation on (and/or or Lorentz z transf sforma ormation ion) ) of f a particle icle wordline dline depends s on the momentum um that the particl icle e carrie ries; 2. 2. Therefore ore the worldline dlines of the particle cles with differe rent t momenta ta transform orm differently tly; 3. 3. As a result, locali ality of events (defined d by worldl ldline ines intersecti tion ons) ) is not absolut lute, e, and become omes s relative. It seems ms that relati ative localit lity y is neither her logical ally ly inconsist sistent nt nor does it contrad adict t any observat atio ional nal data. ity . Phys.Rev. D84 084010, arXiv:1101.0931 [ hep-th ] ; G. Amelino-Camelia, L. Freidel, JKG, L. Smolin The principl iple e of relati tive e localit 43 2547arXiv:1106.0313 [ hep-th ] Relati tive e localit ity: y: A deepenin ing g of the relati tivit ity y principl iple . Gen.Rel.Grav. 43 9

FUNDAMENTAL OR EMERGENT? Is the curved d momentum m space fundame mental tal or emergent? 1. 1. If f it is s fun undame mental, tal, what is s the ass ssocia iated d dyn ynamic ics? s? 2. 2. If it is is emergent nt, , how does it arise? In the emergent t case the master r theory must provide ide a momentum m scale. In 3+1 3+1D D G l M  4 Pl Pl One can imagi gine the re regime me, , in wh which Planck length is small, l, while Planck k mass stays finite (relat lative to the charact acterist eristic ic scales of the pr proble blem) m) 10 10

IN 2+1 D THERE IS A MASS SCALE! In 2+1 dimensions the Newton’s constant G 3 has the dimension ion of inverse 1/ κ. mass G 3 =1/ This suggests s that the momentum um space might t be curved, d, and I wi will show that it indeed d is. s. Of course, , 2+1D gravity is a a toy model, l, and we do not have to believe that it tells you anything ing relevant ant for the real world. d. However, r, it is in interesting ng becau ause: 1. 1. This s is s the only y example mple that we have; 2. 2. There might be real physical al systems, which are effectively ly 2+1 dimensional ional. 11 11

GRAVITY IN 2+1 D WITH PARTICLES The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is 1 L d x A A d h hC ij 2 1       2 i j  k   d x A F hCh ( x ) ij 2 1 2        2 0 ij    12 12

GRAVITY IN 2+1 D WITH PARTICLES The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is Kinetic term 1 L d x A A d h hC ij 2 1       2 i j  k   d x A F hCh ( x ) ij 2 1 2        2 0 ij    13 13

GRAVITY IN 2+1 D WITH PARTICLES The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is Particle Kinetic term term 1 L d x A A d h hC ij 2 1       2 i j  k   d x A F hCh ( x ) ij 2 1 2        2 0 ij    C m J sP ,   0 0 h is ' translation Lorentz '  14 14

GRAVITY IN 2+1 D WITH PARTICLES The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is Particle Kinetic term term 1 L d x A A h hC ij 2 1     2 i j  1   d x A F hCh ( x ) ij 2 1 2        0 ij    Constraint: curvature is h Poincare group  nonzero only at the position of the particle C mJ s P   15 0 0 15

GRAVITY IN 2+1 D WITH PARTICLES 1 L d x A A h hC 2 ij 1     2 i j  The idea is is to solve the constrai aint 1 F hCh ( x ) ij 1 2    ij  and plug the solution on back to the lagrangian angian. . (This can be done explicit citly ly!) !) A.Y. Alekseev and A.Z. Malkin, Commun. Math. Phys. 169, 99 (1995) [ arXiv:hep-th/9312004; C. Meusburger and B. J. Schroers, 16 Class. Quant. Grav. 20 (2003) 2193 [ arXiv:gr-qc/0301108. 16

1 F hCh ( x ) ij 1 2    SOL VING THE CONSTRAINT ij  (0) h 1    17 17

CONTINUITY CONDITION The connection ion must be continuou ous across the boundar dary of the disk. Decompo mposing ing the P Poincare are group elements into Lorentz and translat lational onal parts ,     ql q l one finds that the Lorentz part of the continuit ity condition tion reads ds m J    exp      l(t, ) n(t) l(t, ) 2 0     Lorentz group element! 18 18

HOLONOMY AND MOMENTUM ion A H is Since the connection is gauge ge trivial ial, , its holonomy omy along the boundar dary is is given by by 2   1   Hol ( A ) (0) (2 ) (0)exp C (0) H 1         l l l l k J    Here is is the group valued momentum charact acteriz rizing ng motion on of the particle icle. . In terms П of f the lagrangi angian an of f the particle cle has the form L (spin part)=- P E ( P ) x N ( P P m ) 1 a 2           x a    E ( P ) is the momentum space frame fiel d  a 19 19

THE MOMENTUM SPACE The momentum of П the particle icle is is defined by the group element and thus the momentum sp space is is a a group up manifold old. . In fact, the 2+1 +1D D Lorentz group up, , to which П belongs gs is is the 2 2+1D Anti de Sitter space 1   P P P P 1 2 2 2 2     3 4 0 1 2 2  20 20

NONCOMMUTA TIVE POSITIONS By dualit ity (Born reciprocit ocity, , Majid id co co-gravit ity) ) the position on space is is non- commu mutat tative 1   x , x x a b ab c  c  21 21

TWO PARTICLES In the case of two (or or many) ) particle icles the procedur dure is is very similar lar: 22 22

TWO PARTICLES In the case of two (or or many) ) particle icles the procedur dure is is very similar lar: D 2 H П 2 D 1 П 1 23 23