Pregeometry of κ-Minkowski/κ-Poincar´espacetime and relative locality for fuzzy worldlines

Giacomo Rosati

Institute for Theoretical Physics, University of Wroclaw

July 3, 2019

Giovanni Amelino-Camelia, Valerio Astuti, G. R., arXiv:1206.3805, Eur.Phys.J. C73 (2013) 2521; arXiv:1304.7630, Phys.Rev. D87 (2013) 084023 One way of characterizing such a “quantum spacetime” is through a sort of “Heisenberg uncertainty principle for spacetime”, where, similarly to position and momentum in quantum mechanics ([ˆp, xˆ] = i~ ⇒ δpδˆ xˆ & ~), spacetime coordinates µ ν µν do not commute among themeselves: [ˆx , xˆ ] = iΘ (ˆx) ∝ `P (Doplicher+Fredenaghen+Roberts,Phys.Lett.B(1994)331) µ ν µν ρ µν Consider [ˆx , xˆ ] = iλρ xˆ , with λρ = O(`P ). Is this a preferred-frame picture or still a fully relativistic picture?

If λµν is the same for all inertial observers, one needs to modify (“deform”) the relativistic laws of transformations between their coordinates.

κ-Minkowski (c = ~ = 1) £ κ ∼ Mp = ~ = `P c ¢ ¡ q i ~c ∼ 1019GeV/c2 [xj , x0] = xj [xj , x ] = 0 G κ k

κ-Poincar´e is the deformed (Hopf-)algebra of symmetries £ (See the talks of Gubitosi, (Lukierski,Ruegg,Majid,...’90) ¢ ¡ Amelino-Camelia,...)

Several arguments suggest that, when both gravitational and quantum effects are considered, localization measurements in spacetime can be performed down to a minimum length scale, usually thought to be close to the Planck scale q ` = ~G ∼ 10−35 m. P c3 µ ν µν ρ µν Consider [ˆx , xˆ ] = iλρ xˆ , with λρ = O(`P ). Is this a preferred-frame picture or still a fully relativistic picture?

If λµν is the same for all inertial observers, one needs to modify (“deform”) the relativistic laws of transformations between their coordinates.

κ-Minkowski (c = ~ = 1) £ κ ∼ Mp = ~ = `P c ¢ ¡ q i ~c ∼ 1019GeV/c2 [xj , x0] = xj [xj , x ] = 0 G κ k

κ-Poincar´e is the deformed (Hopf-)algebra of symmetries £ (See the talks of Gubitosi, (Lukierski,Ruegg,Majid,...’90) ¢ ¡ Amelino-Camelia,...)

Several arguments suggest that, when both gravitational and quantum effects are considered, localization measurements in spacetime can be performed down to a minimum length scale, usually thought to be close to the Planck scale q ` = ~G ∼ 10−35 m. P c3 One way of characterizing such a “quantum spacetime” is through a sort of “Heisenberg uncertainty principle for spacetime”, where, similarly to position and momentum in quantum mechanics ([ˆp, xˆ] = i~ ⇒ δpδˆ xˆ & ~), spacetime coordinates µ ν µν do not commute among themeselves: [ˆx , xˆ ] = iΘ (ˆx) ∝ `P (Doplicher+Fredenaghen+Roberts,Phys.Lett.B(1994)331) κ-Minkowski (c = ~ = 1) £ κ ∼ Mp = ~ = `P c ¢ ¡ q i ~c ∼ 1019GeV/c2 [xj , x0] = xj [xj , x ] = 0 G κ k

κ-Poincar´e is the deformed (Hopf-)algebra of symmetries £ (See the talks of Gubitosi, (Lukierski,Ruegg,Majid,...’90) ¢ ¡ Amelino-Camelia,...)

Several arguments suggest that, when both gravitational and quantum effects are considered, localization measurements in spacetime can be performed down to a minimum length scale, usually thought to be close to the Planck scale q ` = ~G ∼ 10−35 m. P c3 One way of characterizing such a “quantum spacetime” is through a sort of “Heisenberg uncertainty principle for spacetime”, where, similarly to position and momentum in quantum mechanics ([ˆp, xˆ] = i~ ⇒ δpδˆ xˆ & ~), spacetime coordinates µ ν µν do not commute among themeselves: [ˆx , xˆ ] = iΘ (ˆx) ∝ `P (Doplicher+Fredenaghen+Roberts,Phys.Lett.B(1994)331) µ ν µν ρ µν Consider [ˆx , xˆ ] = iλρ xˆ , with λρ = O(`P ). Is this a preferred-frame picture or still a fully relativistic picture?

If λµν is the same for all inertial observers, one needs to modify (“deform”) the relativistic laws of transformations between their coordinates. Several arguments suggest that, when both gravitational and quantum effects are considered, localization measurements in spacetime can be performed down to a minimum length scale, usually thought to be close to the Planck scale q ` = ~G ∼ 10−35 m. P c3 One way of characterizing such a “quantum spacetime” is through a sort of “Heisenberg uncertainty principle for spacetime”, where, similarly to position and momentum in quantum mechanics ([ˆp, xˆ] = i~ ⇒ δpδˆ xˆ & ~), spacetime coordinates µ ν µν do not commute among themeselves: [ˆx , xˆ ] = iΘ (ˆx) ∝ `P (Doplicher+Fredenaghen+Roberts,Phys.Lett.B(1994)331) µ ν µν ρ µν Consider [ˆx , xˆ ] = iλρ xˆ , with λρ = O(`P ). Is this a preferred-frame picture or still a fully relativistic picture?

If λµν is the same for all inertial observers, one needs to modify (“deform”) the relativistic laws of transformations between their coordinates.

κ-Minkowski (c = ~ = 1) £ κ ∼ Mp = ~ = `P c ¢ ¡ q i ~c ∼ 1019GeV/c2 [xj , x0] = xj [xj , x ] = 0 G κ k

κ-Poincar´e is the deformed (Hopf-)algebra of symmetries £ (See the talks of Gubitosi, (Lukierski,Ruegg,Majid,...’90) ¢ ¡ Amelino-Camelia,...) It turns out that in theories where relativistic symmetries are deformed in terms of a (inverse-) momentum invariant scale 1/κ (DSR theories (Amelino-Camelia,Kowalski-Glikman,Smolin,Magueijo ∼ 2000)) the notion of absolute spacetime locality is relaxed in favor of a relativity of locality (Amelino-Camelia+Matassa+Mercati+G.R.(2010)Phys.Rev.Lett.106), similarly to how, in the transition from Galilean to special relativity, the deformation of relativistic symmetries in terms of an invariant speed c, enforces relativity of simultaneity.

1 xB

detector ℓb ∆p Bob | | 0 xB

1 xA p s p h

emitter 0 Alice xA

(Amelino-Camelia+Loret+G.R.,(2011)Phys.Lett.B700; Amelino-Camelia+Arzano+Kowalski-Glikman+G.R.+Trevisan(2011)Class.Quant.Grav.29; Amelino-Camelia+Freidel+Kowalski-Glikman+Smolin(2011)Phys.Rev.D84) What is the description of spacetime points in κ-Minkowski? One expect some kind of fuzziness. How can it be characterized? Is it possible to provide a quantum mechanical description of κ-Minkowski/κ-Poincar´e?

In a “pregeometrical” interpretation, κ-Minkowski emerges from quantum gravity at some level of effective description, and non-commutativity is represented in terms of standard Heisenberg quantum mechanics introduced at some deeper level.

Some previous attempts (Lukierski+Ruegg+Zakrzewski,Ann.Phys.243(1995)90) (Livine+Oriti,J.HighEnergyPhys.0406(2004)050) (Agostini,J.Math.Phys.48(2007)052305) (D’andrea,J.Math.Phys.47(2006)062105) (Ghosh+Pal,Phys.Rev.D75(2007)105021) (Dabrowski+Piacitelli,ArXiv:1004.5091) (Mignemi,Phys.Rev.D84(2011)025021) (Kovacevic+Meljanac,J.Phys.A45(2012)135208) κ-Poincar´ealgebra and κ-Minkowski spacetime (1+1D) Lukierski,Ruegg,Majid,...’90

κ-Minkowski Xκ is the homogeneous space of the κ-Poincar´eHopf algebra Pκ i [x1, x0] = x1 κ [P1,P0] = 0 [N,P0] = iP1 1 1 κ  2  i ∆ (xµ) = xµ ⊗ + ⊗ xµ − P0 2 [N,P1] = i 1 − e κ − P 2 2κ 1 S (xµ) = −xµ  (xµ) = 0

∆P0 = P0 ⊗ 1 + ⊗P0 P − 0 ∆P1 = P1 ⊗ 1 + e κ ⊗ P1 P − 0 ∆N = N ⊗ 1 + e κ ⊗ N

P0 P0 S (P0) = −P0 S (P1) = −e κ P1 S(N) = −e κ N1

  P 2 2 P0 0 2 2κ = (2κ) sinh − e κ P~ 2κ Translations and differential calculus

Coproduct ∆ =⇒ action . : Pκ ⊗ Xκ → Xκ £ ¢ ¡  P  − 0 P1 . f(x)g(x) = (P1 . f(x)) g(x) + e κ . f(x) (P1 . g(x)) Leibniz

i P1 . [x1, x0] = P1 . x1 κ

0  0 0  i 0 Translations: xµ = xµ − aµ ⇐⇒ x1, x0 = κ x1

 κ  i κ  κ   κ  a , x0 = a a , x1 = 0 a , x0 = 0 § 1 κ 1 µ 0 ¤

¦ 1 κ µ ¥ T = + d d = −iaµP

differential calculus df(x)g(x) = (df(x)) g(x) + f(x)dg(x) Woronowicz,Majid,Oeckl,...’90 Pregeometry of κ-Minkowski/κ-Poincar´e Covariant quantum mechanics

Pregeometry spacetime noncommutativity arises from a more fundamental theory £ ¢ Ordinary Heisenberg¡ algebra ⇐⇒ κ-Minkowski/κ-Poincar´e pregeometric representation

But what framework?

κ-Minkowski Quantum Mechanics time coordinate = noncommutative time coordinate = evolution parameter observable ⇓

Covariant formulation of ordinary quantum mechanics Reisenberger,Rovelli Phys.Rev.D65(2002) The spatial coordinates£ and the time coordinate play the same type of role. Both operators on¢ a “kinematical Hilbert space” ¡ (ordinary Hilbert space of normalizable wave functions)

(See also the talks of Hoehn, Giacomini) Covariant quantum mechanics Reisenberger,Rovelli,Phys.Rev.D65(2002) Kinematical Hilbert space

2 2 Space of (1+1D) “spacetime regions” L (R , dq0, dq1)

Standard multiplicative operators qˆ0, qˆ1 and derivative operators πˆ ≡ i ∂ , πˆ ≡ i ∂ 0 ∂q0 1 ∂q1

Standard commutator algebra

[ˆπ0, qˆ0] = i [ˆπ0, qˆ1] = 0

[ˆπ1, qˆ0] = 0 [ˆπ1, qˆ1] = −i

“Regions of spacetime” hqˆ0i , hqˆ1i, δq0, δq1 Covariant quantum mechanics Reisenberger,Rovelli,Phys.Rev.D65(2002) Dynamics

There is no “evolution” £ ¢ ¡ Kinematical Hilbert space ⇐⇒ Physical Hilbert space Hamiltonian constraint H = 0 relationships between the properties of the (partial) observables for ⇒ dynamics = £ spatial coordinates and the properties of the time (partial) observable ¢ ¡ −→ “fuzzy worldlines”

”Geometrical interpretation“

(Classical) Special Relativity: Covariant Quantum Mechanics arena = Minkowski spacetime arena = kinematical Hilbert space H = 0 H = 0 ⇓ ⇓ dynamics of relativistic classical dynamics of relativistic quantum particles unfolds particles unfolds Geometrical interpretation

Minkowski spacetime classical particles “spacetime observables” physically observable experiment property =⇒

properties

kinematical Hilbert space quantum particles “spacetime observables” physically observable experiment property =⇒ (physical Hilbert space) Pregeometry of κ-Minkowski/κ-Poincar´e

Pregeometric representations [ˆπ0, qˆ0] = i [ˆπ0, qˆ1] = 0

Amelino-Camelia,Astuti,G.R.,arXiv:1206.3805,Eur.Phys.J.C73(2013) [ˆπ1, qˆ0] = 0 [ˆπ1, qˆ1] = −i

xˆ0 =q ˆ0 i πˆ0 ⇒ [ˆx1, xˆ0] = xˆ1 xˆ1 =q ˆ1e κ κ

πˆ0  P  P0 . f(ˆx0, xˆ1) ←→ [ˆπ0, f(ˆq0, qˆ1e κ )] − 0 ⇒ P1.fg = (P1 .f) g+ e κ .f (P1 .g) πˆ πˆ − 0 0 P1 . f(ˆx0, xˆ1) ←→ e κ [ˆπ1, f(ˆq0, qˆ1e κ )]

κ aˆ0 = a0 κ  κ  ⇒ [ˆa0 , xˆ0] = 0 aˆµ, xˆ1 = 0 πˆ κ 0 aˆ1 = a1e κ κ i κ [ˆa , xˆ0] = aˆ 1 κ 1

Translations ⇒ 1 κ µ µ T = + dP dP = −iaµP = −ia [ˆπµ, ·] dP fg = (dP f) g + fdP g Pregeometry of κ-Minkowski/κ-Poincar´e P 1 − 0 Boosts ∆P1 = P1 ⊗ + e κ ⊗ P1 P − 0 Infinitesimal boost transformation ∆N = N ⊗ 1 + e κ ⊗ N κ κ  2  i B = 1 + d d = iξˆ N − P0 2 N N [N,P1] = i 1 − e κ − P 2 2κ 1 dN fg = (dN f) g + fdN g

⇑

h i i h i πˆ ˆκ ˆκ ˆκ κ 0 ξ , xˆ0 = ξ ξ , xˆ1 = 0 ⇐ ξˆ = ξe κ κ

 πˆ0  πˆ 2 2 − 0 e κ − 1 πˆ1 N . f(ˆx) ≡ e κ [ˆη, f(ˆx)]η ˆ ≡ κ +  qˆ1 − πˆ1qˆ0 2 2κ

B = 1 + iξ [ˆη, ·] ˆ ˆ† ˆ ˆ iξηˆ† ˆ −iξηˆ Finite boost −→ Bfinite . O → B OB = e Oe

  πˆ on-shell condition 2 2 πˆ0 − 0 2 Hˆ = κ = (2κ) sinh − e κ πˆ l  2κ 1 Hamiltonian constraint Pregeometry of κ-Minkowski/κ-Poincar´e Deformed measure and hermiticity

“Pregeometric Momentum-space representation” ψ (π ) µ ←→ Measure of integration of states: square-integrable functions of πˆ0 and πˆ1

Z ? Scalar product hOˆi = hψ|Oˆ|ψi = D(πµ)ψ (πµ)O(πµ)ψ(πµ)

0 π = π0 − ξπ1 0 π  2 0 2  B : 0 e κ −1 π1 π1 = π1 − ξ κ 2 + 2κ

π − 0 † ⇒ Invariant measure D(πµ) = dπ0dπ1e κ D(πµ) :η ˆ =η ˆ

Bˆ : hψ0|ψ0i = hψ|eiξηˆe−iξηˆ|ψi = hψ|ψi scalar product is invariant Pregeometry of κ-Minkowski/κ-Poincar´e Time

π0 π0 † − κ − κ qˆ1 ≡ i∂π1 ⇒ qˆ1e = e qˆ1 ⇒ xˆ1 =x ˆ1

π π   − 0 − 0 i † i qˆ0 ≡ −i∂π ⇒ qˆ0e κ = e κ qˆ0 + ⇒ xˆ =x ˆ0 + (ˆx0 =q ˆ0) 0 κ 0 κ

∗ i † xˆ0 ≡ qˆ0 − 2κ xˆ0 =x ˆ0

h i Anyway: xˆ0 not observable: Hˆ, xˆ0 6= 0

as a “partial” observable we are interested in “differences in time” No state in the pregeometric Hilbert space gives absolutely sharp values to xˆ0 and xˆ1:

`πˆ xˆ0 =q ˆ0, xˆ1 =q ˆ1e 0 a sharply specified xˆ0 requires an eigenstate of qˆ0 but on such eigenstates of qˆ0 one finds `πˆ that πˆ0 is infinitely fuzzy δπˆ0 = ∞ which in turn implies that xˆ1 =q ˆ1e 0 cannot be sharp

So all points in κ-Minkowski must be fuzzy Fuzzy Points

Gaussian states

2 2 (π0−π0) (π1−π1) − 2 − 2 4σ0 4σ1 iπ0q¯0−iπ1q¯1 Ψπµ,σµ,qµ (πµ)=Ne e

σ2 π¯0 − 0 e κ e 2κ2 N 2 = 2πσ0σ1

πˆ0 i 1 x¯ = hqˆ i x¯ = hqˆ e κ i 0 0 1 1 hxˆ0i = q0 − δxˆ0 = 2κ 2σ0 q 2 2 σ2 δxˆ0 = hqˆ i − x¯  πˆ0  π¯0 0 0 0 − 2 hxˆ1i = hqˆ1i e κ = q1e κ e 2κ s  πˆ 2 1/2 0 2 " σ2 !# δxˆ1 = h qˆ1e κ i − x¯ π¯0 1 − 0 1 κ 2 κ2 δxˆ1 = e 2 + q1 1 − e 4σ1

R ? hOˆi = hψ|Oˆ|ψi = D(πµ)ψ (πµ)O(πµ)ψ(πµ)q ˆµ ≡ −i∂µ Translations and relative locality for fuzzy points

κ µ T. xˆ0 =x ˆ0 − aˆ =q ˆ0 − a0 T = 1 − ia [ˆπµ, ·] 0 πˆ κ 0 T. xˆ1 =x ˆ1 − aˆ1 = e κ (ˆq1 − a1)

xB i 1 hT. xˆ0i = q − a0 − 0 2κ 1 δ (T. xˆ0) = xB 2σ0 0 2 σ L L π0 0 − 2 hT. xˆ1i = (q1 − a1) e κ e 2κ xA 1/2 1 " σ2 !# π0 1 − 0 κ 2 κ2 δ(T. xˆ1)=e 2 +(q1 −a1) 1−e 4σ1 A x0

Alice Bob Physical Hilbert space Amelino-Camelia,Astuti,G.R.,arXiv:1304.7630,Phys.Rev.D87(2013)

π 2 2  π0  − 0 2 Hamiltonian constraint H = (2κ) sinh − e κ π 2κ 1

Physical observables invariant under H −→ new scalar product h·|·iH £projecting all the orbit of the gauge transformation generated by H on the same state

¢ ¡ Z π − 0 ∗ hψ|φiH = hψ|δ (H) Θ(π0)|φi = e κ dπ1dπ0δ (H) Θ(π0)ψ (π)φ(π)

ˆ expectation value hΨq0,q1 |O|Ψq0,q1 iH

2 2 (π0−π0) (π1−π1) − 2 − 2 4σ0 4σ1 iπ0q¯0−iπ1q¯1 Ψq0,q1 (πµ; πµ, σµ)=Ne e

Z π0 −2 − κ 2 N = e dπ1dπ0δ (H) Θ(π0)|Ψq0,q1 (πµ; πµ, σµ)| “Intercept operator”

h i  xˆ1 and xˆ0 are not self-adjoint operators on our physical Hilbert space Hˆ, xˆ0,1 6= 0

πˆ0  1  1 ˆ ˆ ˆ ∂H/∂π h i † A = e κ qˆ1 − Vqˆ0 − [ˆq0, V] V ≡ Hˆ, Aˆ = 0 Aˆ = Aˆ 2 ∂H/∂π0

πˆ0  π0  (κ-deformed Newton-Wigner operator) Xˆ (T ) = Aˆ + e κ VˆT ∂x1/∂x0 ∼ e κ V

→0 ! πˆ0 κ~→∞ Aˆ =x ˆ1 − e κ Vˆxˆ0 −→ x − vt

π0 x1 = q1e κ π − 0 ˆ e κ π1 H=0 1 π0 V ≡ π −→ ' 1− − 0 π1 π κ m=0 1 + κ 0
e 2 π>0 κ κ sinh κ + 2κ π1

Now we can ask what is the observed fuzziness for a particle traveling an arbitrary distance (even cosmological) Translations: Alice and Bob

Alice: q0 = q1 = 0 (x0 = x1 = 0) ⇒ hΨ0,0|A|Ψ0,0iH = 0

2 2
hπ0i δA = hΨ0,0|A |Ψ0,0iH ≈ [κ] [κ] 2κσ2

σ1  σ0, π¯1 π1 −→ saddle point approximation 1 1 2 in the π1 integration: 2 = 2 + 2 σ σ1 σ0

(one parameter) family Bob (distant observer reached by the particle): of observers “on the worldline” Ψa0,a1 < A >= hΨa0,a1 |A|Ψa0,a1 iH = 0 1 0   0 1 −1 a = hVia 0 hπ0i a , a : hΨ0,0|T AT |Ψ0,0iH = 0 ⇒ ≈ a 1 − κ

2  2  δA[κ] = hΨa0,hVia0 |A |Ψa0,hVia0 iH [κ] hπ i a2σ2 ≈ 0 + 0 2κσ2 κ2 ~ → 0 Translations: Alice and Bob G.Amelino-Camelia,M.Matassa,F.Mercati,G.R.Phys.Rev.Lett.106(2011) G.Amelino-Camelia,N.Loret,G.R.Phys.Lett.B700(2011)

2
π hΨq ,q |Xˆ (T ) |Ψq ,q iH =x ¯1 − x¯0 + T + O 1/κ 0 0 1 0 1 x1 = q1e κ

B x1 A x1 detector B x0

emitter A x0

Alice Bob

hπ i a2 a σ  hπ i  δA2 ≈ 0 + 0 σ2 ∼ 0 a1 ≈ a0 1 − 0 [κ] 2κσ2 κ2 κ κ

δA|m=0;σ1σ0 ∼ δx1 ~ → 0 Translations: Alice and Bob G.Amelino-Camelia,M.Matassa,F.Mercati,G.R.Phys.Rev.Lett.106(2011) G.Amelino-Camelia,N.Loret,G.R.Phys.Lett.B700(2011)

2
π hΨq ,q |Xˆ (T ) |Ψq ,q iH =x ¯1 − x¯0 + T + O 1/κ 0 0 1 0 1 x1 = q1e κ

x1

x0

emitter

physical picture

hπ i a2 a σ  hπ i  δA2 ≈ 0 + 0 σ2 ∼ 0 a1 ≈ a0 1 − 0 [κ] 2κσ2 κ2 κ κ

δA|m=0;σ1σ0 ∼ δx1 Thanks! Galilean relative rest In Galilean relativity we can say that one has relativity of ”spatial locality“ £ x ¢ ¡ x v (t1, x¯) (t2, x¯) x (t , x¯ v t ) 1 − x 1 v ∆t x| | (t , x¯ v t ) 2 − x 2 t t the rest is relative

We can use this scheme to describe the paradigmatic situation in which Bob is on a boat moving at

velocity vx respect to Alice, who is standing on the dock. Imagine that Alice is bouncing a ball on the dock, and that the two points mark the position and time of two of the ball’s bounces on the ground. While Alice evidently observes the ball bouncing at the same point in space, Bob, who is

moving with velocity vx relative to Alice, observes, in its reference frame, the ball bouncing in two different positions: if Bob is approaching the dock, for example, Bob sees the second bounce closer then the first Galilean relative rest In Galilean relativity we can say that one has relativity of ”spatial locality“ £ x ¢ ¡ x v (t1, x¯) (t2, x¯) x (t , x¯ v t ) 1 − x 1 v ∆t x| | (t , x¯ v t ) 2 − x 2 t t the rest is relative

x x ¯ (t,¯ x2 vxt¯) (t, x2) vx −

(t,¯ x1) (t,¯ x v t¯) 1 − x

t t while time simultaneity is still absolute Thus one has relative space locality and relative (time) simultaneity, but still absolute spacetime locality. There is no observer-independent projection from spacetime to separately space and time. We can say that one ”sees“ spacetime as a whole.

SR: relative simultaneity

In special relativity : £ invariant¢ scale ”c“ ⇒¡ absolute (time) simultaneity → relative (time) simultaneity.

x x (t,¯ x ) 2 v ¯ 1 ¯ x (γ(t 2 v x ),γ(x vt)) − c x 2 2 −

(t,¯ x1) (γ(t¯ 1 v x ),γ(x vt¯)) − c2 x 1 1 −

t γ t c v∆x There is no observer-independent projection from spacetime to separately space and time. We can say that one ”sees“ spacetime as a whole.

SR: relative simultaneity

In special relativity : £ invariant¢ scale ”c“ ⇒¡ absolute (time) simultaneity → relative (time) simultaneity.

x x (t,¯ x ) 2 v ¯ 1 ¯ x (γ(t 2 v x ),γ(x vt)) − c x 2 2 −

(t,¯ x1) (γ(t¯ 1 v x ),γ(x vt¯)) − c2 x 1 1 −

t γ t c v∆x

Thus one has relative space locality and relative (time) simultaneity, but still absolute spacetime locality. SR: relative simultaneity

In special relativity : £ invariant¢ scale ”c“ ⇒¡ absolute (time) simultaneity → relative (time) simultaneity.

x x (t,¯ x ) 2 v ¯ 1 ¯ x (γ(t 2 v x ),γ(x vt)) − c x 2 2 −

(t,¯ x1) (γ(t¯ 1 v x ),γ(x vt¯)) − c2 x 1 1 −

t γ t c v∆x

Thus one has relative space locality and relative (time) simultaneity, but still absolute spacetime locality. There is no observer-independent projection from spacetime to separately space and time. We can say that one ”sees“ spacetime as a whole. Loss of simultaneity and synchronization of clocks

Alice and Bob, distant observers in relative motion (with constant speed), have stipulated a procedure of clock synchronization and they have agreed to build emitters of blue photons (blue according to observers at rest with respect to the emitter). They also agreed to then emit such blue photons in a regular sequence, with equal time spacing ∆t∗. Bob’s worldlines are obtained combining a translation and a boost transformation (Bob = B . T . Alice),

A A B  B   A  A  p1 −βp0  B  A  A  x1 x0 = γ x¯ −a1 −β x¯0 −a0 + A A x0 −γ x¯0 −a0 −β x¯ −a1 , p p0 −βp1 We arranged the starting time of each sequence of emissions so that there would be two coincidences between a detection and an emission, which are of course manifest in both coordinatizations, so to obtain a specular description. Relative simultaneity is directly or indirectly responsible for several features that would appear to be paradoxical to a Galilean observer (observer assuming absolute simultaneity). In particular, while they stipulated to build blue-photon emitters they detect red photons, and while the emissions are time-spaced by ∆t∗ the detections are separated by a time greater than ∆t∗.

xA xB

c tB

c tA Think to the Einstein clock x inference L

p p

t

emission detection ℓ x L ℓLp ≃ p p

t

emission detection

Relative locality: an insight

We don’t actually “see” spacetime, but we “see” (detect) time sequences of particles, and then abstract spacetime by inference:

local x

t

inference

We actually “see” (detect) only what we locally witness Relative locality: an insight

Think to the Einstein clock x inference L We don’t actually “see” spacetime, but we p p “see” (detect) time sequences of particles, and then abstract spacetime by inference: t local x emission detection ℓ x t L ℓLp ≃ p p inference

t

emission detection

We actually “see” (detect) only what we locally witness There is no observer-independent projection from a one-particle phase space to a description of the particle separately in spacetime and in momentum space. We thus can say that one ”sees“ phase space as a whole.

Relative locality: DSR theories

DSR theories : £ invariant¢ (inverse-momentum)¡ scale ` ⇒ absolute spacetime locality → relative spacetime locality

(t,¯ x,¯ p ) (t¯+ ℓb p , x¯ b ,p ) 2 x| 2| − x 2 (t,¯ x,¯ p1) x x (t¯+ ℓb p , x¯ b ,p ) x| 1| − x 1 bx

p p

t t ℓb ∆p x| | Relative locality: DSR theories

DSR theories : £ invariant¢ (inverse-momentum)¡ scale ` ⇒ absolute spacetime locality → relative spacetime locality

(t,¯ x,¯ p ) (t¯+ ℓb p , x¯ b ,p ) 2 x| 2| − x 2 (t,¯ x,¯ p1) x x (t¯+ ℓb p , x¯ b ,p ) x| 1| − x 1 bx

p p

t t ℓb ∆p x| | There is no observer-independent projection from a one-particle phase space to a description of the particle separately in spacetime and in momentum space. We thus can say that one ”sees“ phase space as a whole.