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
ESFQG – SISSA, SEPTEMBER 2014
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* Based on the (partially inpublished) work done with T
The assumpti mption n that t the angle sum m is less than n 180 80 leads s to a geome metr try y quite e different from Euclid’s. . It depe pends nds on
constant, tant, which ch is not given n a pr priori.
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
length h a pr priori
. (Gauss, s, 182 827) 7) 3 If there re is an a pr priori ri scale, , you expe pect nontri rivial al geome metr try. y.
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:
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Spacetime Momentum space
The phase space is a a cotan tange gent nt bun undle over flat t momentum m space
No scale required
For deformed d relativistic istic particle icle we have flat spaceti time me and curved d momentum um sp space:
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Spacetime Curved momentum space (the curvature scale needed!)
The phase space is a a cotan tange gent nt bundle over curved d momentum m space
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 with the no nontrivial ial momentum m space frame e field.
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a a a a a
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 pμ(P): We need metric c to d define the m mass-sh shell ll relation tion ! If f the metric is n s nonlinear ar we need the mass ss sc scale to defi fine it.
2 2
2
( )
p geodesic
D p ds g p p
In order r to add two momenta ta the notion of connecti tion
lel transpor port) ) is ne needed. d. If the c composition sition pq is nonlinear ar we need a mass scale to define it.
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p q pq
In theories with curved d momentum m space, , localit lity y might be relative: e: 1. 1. The transl slati ation
z transf sforma
ion) ) of f a particle icle wordline dline depends s on the momentum um that the particl icle e carrie ries; 2. 2. Therefore
dlines of the particle cles with differe rent t momenta ta transform
tly; 3. 3. As a result, locali ality of events (defined d by worldl ldline ines intersecti tion
) is not absolut lute, e, and become
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. 9
iple e of relati tive e localit
Relati tive e localit ity: y: A deepenin ing g of the relati tivit ity y principl
43 2547arXiv:1106.0313 [hep-th]
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 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)
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4 Pl Pl
In 2+1 dimensions the Newton’s constant G3 has the dimension ion of inverse mass G3=1/ 1/κ. 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.
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The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is
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2 1 2 1 2
ij i j ij ij
The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is
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2 1 2 1 2
ij i j ij ij
Kinetic term
The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is
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2 1 2 1 2
ij i j ij ij
Kinetic term Particle term
is ' , ' translation C m Lorentz J sP h
The Lagrangian angian of 2+1 grav avit ity with one (massive) ) particle icle at at the origin in is is
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2 1 2 1 2
ij i j ij ij
Kinetic term Particle term Constraint: curvature is nonzero only at the position of the particle
Poincare group C mJ P h s
The idea is is to solve the constrai aint and plug the solution
angian. . (This can be done explicit citly ly!) !)
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2 1
ij i j
1 2
1 ( )
ij ij
F hCh x
A.Y. Alekseev and A.Z. Malkin, Commun. Math. Phys. 169, 99 (1995) [arXiv:hep-th/9312004; C. Meusburger and B. J. Schroers,
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1 2
1 ( )
ij ij
F hCh x
1
(0) h
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The connection ion must be continuou
dary of the disk. Decompo mposing ing the P Poincare are group elements into Lorentz and translat lational
ity condition tion reads ds
, ql q l
exp 2 m J l(t, ) n(t) l(t, )
Lorentz group element!
Since the connection ion AH is is gauge ge trivial ial, , its holonomy
dary is is given by by Here is is the group valued momentum charact acteriz rizing ng motion
icle. . In terms П of f the lagrangi angian an of f the particle cle has the form
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1 1
2 Hol ( ) (0) (2 ) (0)exp (0)
H J
A C k
l l l l
1 2
(spin part)=- ( ( ) ( ) is the momentum space frame fiel ) d
a a a
E L P x P E N P m P P
x
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
. 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
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2 2 2 2 3 1 2 2
1 1 4 P P P P
By dualit ity (Born reciprocit
, Majid id co co-gravit ity) ) the position
is non- commu mutat tative
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1 ,
a b ab c c
x x x
In the case of two (or
) particle icles the procedur dure is is very similar lar:
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In the case of two (or
) particle icles the procedur dure is is very similar lar:
D1 D2 H
П1 П2
The deformed lagrangian angian has the form (no spin) and there is is a a nontrivial ial coup uplin ing between particle icles. . This is is this coup upling ng that makes the total momentum um of the system equal to the deformed composit ition ion
dual parti ticle cles momenta ta, , given by group product ct. T
So So that the lagrangi gian an is is invaria iant under rigid id translat lation ions which do not change ge the partic icle les relative position
, with the associa iated total momentum um being П.
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1 1 1 1 1 1 1 1 2 2 2 2 1 1 2 1 1 2
L
x x x
1 2 1 2
1 1 , 2 2 x x x d x x
1 2 1
terms, L
x d
No relati tive e localit ity! y! In this s mo model there e is no sign of r relati tive e lo localit lity: y: the rela lati tive e po positio tion n of
the pa partic ticle les s is invaria ariant nt under r rigi gid d translation slations. s.
perts) s) this ma makes the construc uction tion of t the vertex quite ite natural and unique ue (no sudoku needed!) d!). This s effecti tive e pa partic icle le mo model works s only in 2+1D and cannot t be extende nded d to higher her dime mensio sions. . There e is a mo model with h (kind nd of) the κ-Poinc ncar are struct cture, ure, which h can be extende nded d to higher er dime mensio sion n (talk k in Rome me). 25 25
There are only circumstan mstantial tial evidences: 1. 1. Grav avit ity in 3+ 3+1D D can be defi fined as s a const straine ained topologic
al fi field theory, , when the constrain aint is is forced to vanish one has a TFT; 2. 2. One One can couple this theory to parti ticle cle(s) (s) and even argue that such a a system can be describe bed by CS theory in 2 2+1D, like 2+1 gravit ity, , but with more complic licate ated gauge ge group;
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1. 1. In 2+1D we c can honesty derive the deforme med d single- and multi-pa parti ticle cle action ions s with momentum um sp space being a group up manifold;
ta of f the particle( icle(s) ) represented d by group p elements; 2. 2. The total momentum m of the multi-par parti ticle cle system is gi given by the group p produc duct t of the group p elements representing ing momenta ta of the particle icles; 3. 3. The multipar particle icle lagrangi gian contain ins the „topological interaction terms” and the form of these t terms is such that local alit ity turns out to be a absolu
e (not relative). e). 4. 4. It is un unclear ar if this results can be applie ied d beyond the 2+1D setup.
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