Theory and phenomenology of Planck scale deformed ... · Theory and phenomenology of Planck scale deformed relativistic symmetries Giacomo Rosati Universit a di Roma La Sapienza,

Theory and phenomenology of Planck scale deformed relativisticsymmetries

Giacomo Rosati

Universita di Roma La Sapienza, Dottorato in Fisica XXV Ciclo

Taming non locality in DSR theories, Phys. Rev. Lett. 106, 071301 (2011).

�� ��DSR theories : relativistic theory with invariant lenght scale `,

tipically Planck lenght scale Lp =√

~Gc3 ≈ 10−35m.

It opens the possibility of considering�� ��deformed dispersion relation of the kind

E =√

p2 + m2 −1

2`p2

Consequently the velocity of massless particles is momentum dependent

vγ = c (1− `p)

The covariance of the laws of motions require relativistic symmetry transformations tobe deformed.

Among the many implications: what is the faith of�� ��locality in DSR theories?

Implications for locality had appeared puzzling for several yearsbut only with our work the situation was clarified.

Taming non locality in DSR theories, Phys. Rev. Lett. 106, 071301 (2011).

We estabilished that whatactually happens to locality isthat it remains objective toobservers local to the coincidenceof events, but observers who aredistant in their coordinatizion ofspacetime see those same pairs ofevents as not coincident.

Relative locality

Relative locality in k-Minkowski, Phys. Lett. B 700, 150–156 (2011) .

One of the frameworks where DSR theories have been more extensively studied is the one

of noncommmutative spacetime, and in particularly the so called�� ��k-Minkowski

{x , t} = −`x ,

whose symmetries are described in terms of Hopf algebras as a generalization the Poincare

algebra, the�� ��k-Poincare algebra.

In such a framework we studied the propagation of photons as worldlines at a

classical level (~→ 0) by means of the Hamiltonian approach, where the Hopf

algebraic structures emerging from noncommutativity are implemented through�� ��deformed phase space relations between coordinates and momenta

{E , t} = 1, {E , x} = 0 ,

{p, t} = `p, {p, x} = −1 .

which define a generic translation Tat ,ax acting on the coordinates as

x ′ = x − at {E , x}+ ax {p, x} ,t′ = t − at {E , t}+ ax {p, t} .

Relative locality in k-Minkowski, Phys. Lett. B 700, 150–156 (2011) .

It comes out that the relative locality in this case emerge in a very evident way

Since the worldlines for massless particles appear to not depend on momentum in thecase of k-Minkowski, we see that the distant coincidence in time arrival for the twophotons “seen” by Alice, is a coordinate artifact due to the properties of k-Minkowskispacetime, while Bob, who is local to the event of detection, does actually observe the

expected�� ��delay δt = `Lp .

In the same way, while Bob “sees” the two photons not being emitted together by thedistant emitter, Alice, who is local to the emission event, witnesses the coincidentemission of the two photon.

We thus understand that in k-Minkowski spacetime what appears as a distantlylocal event to an observer may not be local for an observer witness to the event.

Interactions: the PRL approach

A still-open issue: how to describe interactions in this worldline approach to deformedsymmetry?

One difficulty arises from the non linearity in the summation rule for momentaemerging in k-Poincare theories

(p ⊕ q)µ = pµ + qµ − `δjµp0qj

In this direction the new, recently proposed,�� ��PRL1 (Principle of Relative Locality)

approach to the study of deformed symmetry theories comes to aid.

In the PRL approach the momentum space presents curvature, which reflects itself indeformed dispersion relations.

Moreover the deformed summation rule between momenta is implemented through anotion of parallel transport by means of a non metric connection

(p ⊕ q)µ = pµ + qµ − Γαβµ pαpβ

1arXiv:1101.0931.

The leading idea is that the fundamental elements in the observation of events by a givenobserver are momenta and angles between them, while spacetime emerges as aconsequence of coordinatization.

Then vertexes of this kind are theprotagonists of the theory

A proposal for an action (for commutative coordinates) has been made

S =

∫ds(xµJ p

Jµ −NJCJ(p)

)− ξµKµ

Here momenta are the fundamental variables. The interaction at the vertex is implementedthrough the boundary term ξµKµ, where ξµ is a Lagrange multiplier and Kµ is a certain

non linear combination of momenta at the vertex

Kµ =⊕J

pJµ

constraining a deformed conservation law.The translations between different observers then can be defined relying on the symmetriesemerging from the boundary relations

xµ = ξνδKνδkµ

arXiv:1101.0931.

Interactions: the PRL approach1

We then generalized the analysis to the k-Minkowski case exploiting its deformedsymplectic strutcure to build a Lagrangian analogue to the PRL one, and to a systemof a finite number of vertexes.

Thus we are now able to study processes involving a shower of particles like

Our preliminary results are in good agreement with the analysis in “Relativity oflocality in k-Minkowski”, with momentum dependent delays of particles detected bydistant observers, and relativity of locality as coordinate artifacts due tocoordinatization of distant events.

1Work in progress with G. Amelino-Camelia, M. Arzano, J. Kowalski-Glikman and G. Trevisan.

Interactions: the PRL approach1

Our purpose then is to be able to characterize the effects of deformedsymmetries in processes closer to physical models relyable for a QGphenomenology, like the emission of ultraenergetic photons from GammaRay Bursts.

The energy of these photons is high enough to leave traces within thereach of experiments that can be performed in a foreseeable future, likeastronomic observations of delays in arrival times of such photons.

1Work in progress with G. Amelino-Camelia, M. Arzano, J. Kowalski-Glikman and G. Trevisan.

DSR De Sitter1

One step towards this phenomenology is the

generalization of the above results on relative locality to the case of curvedspacetime

A phenomenology of these effects is possible only if the distances involved arecosmological, but for cosmological distances the “flat limit” is evidently inadequate.

To this end we are considering theapplicability of the deformed symmetry

analysis to the case of a�� ��De Sitter like

expanding universe.

With a view to finding results for a more generic FRW expansion.

1Work in progress with G. Amelino-Camelia, A. Marciano, M. Matassa.

DSR De Sitter1

According to our preliminary results, estimates of�� ��upper bounds on ` derived in a DSR

framework differ significantly from the ones derived in a LIV (Lorentz Invariance Violation)

framework, though leaving unchanged the order of magnitude of the effects.

1Work in progress with G. Amelino-Camelia, A. Marciano, M. Matassa.

Noether theorem1

We are also investigating the fate of�� ��Noether theorem in this sort of

theories with Hopf-algebra symmetries.

Results for free theories are already in the literature but once again they

need generalization to�� ��interacting case .

1Work in progress with V. Astuti and G. Amelino-Camelia

G. Amelino-Camelia, M. Matassa, F. Mercati and G. Rosati, “Taming nonlocality intheories with deformed Poincare symmetry”. Phys. Rev. Lett. 106, 071301 (2011).

G. Amelino-Camelia, N. Loret, G. Rosati, “Speed of particles and a relativity oflocality in κ-Minkowski quantum spacetime” Physics Letters B 700 (2011) 150–156.

G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman L. Smolin, “The principle ofrelative locality”. arXiv/hep-th: 1101.0931

A photon as seen by my daughter.