on the nature of quantum space-time200.145.112.249/webcast/files/06-15-16-freidel.pdf · •dirac:...
Post on 22-Jul-2021
3 Views
Preview:
TRANSCRIPT
On the Nature of Quantum Space-time
Laurent Freidel Perimeter Institute.Sao-Paulo June 2016
based on 1307.7080, 1405.3949, 1502.08005 and 1606.01829…with R.G. Leigh and D. Minic
also based on 1101.0931, 1103.5626, 1104.2019with G. Amelino-Camelia, L. Smolin and J. Kowalski-Glikman
Notion of spaces
Our concepts of space and time have radically evolved over history•Euclid: Notion of absolute space•Galileo: Relativity of observers in space : velocity is relative•Newton: Motion generated by forces : action at a distance•Einstein: Time is relative •Einstein: Relativity of general observers
Quantum mechanic hasn’t affected our fundamental picture of space and time yet
Notion of MatterThe history of the concept of matter is more intertwined, it is a constant dialogue between the idea of individual objects versus continuum fields•Democritus: Atomicity• Aristotle: Continuum hypothesis : `There is no void’ Descartes •Newton: there are individual macroscopical objects acting on each other, due to their charges, mass.
•Faraday : Fields are real•Maxwell: Fields in space are dynamical •Heisenberg-Born-Jordan: Discovery of Quantum mechanics: Atoms are stable after all, fundamental discreteness .
•Dirac: Quantum Fields are relativistic: Anti-particles•Kramers-Heisenberg-Mandelstam-Chew: S-matrix: scattering of asymptotic states are the only observables (fields are not real)
• Veneziano, Nambu,…: String theory, probes are non local
Classical space still appears as wave function, fields or string labels.Quantum matter on classical space-time
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
Singular limits
In any breakthrough, invisible phenomena become visible:The lower order description is a singular limit of the higher one (M-Berry). That is, a mathematically consistent description which cannot reveal certain observables.• Eulerian fluid is a singular limit of the viscous fluid: planes can’t fly• geometrical optic-wave optics: No central bright spot• Classical-Quantum: Aharonov-Bohm phases are invisible• Non relativivistic-relativistic Quantum Field Theory: No anti-particle• Newton-GR: No gravity waves
UnificationIn any breakthrough, invisible phenomena become visible, but also a fundamental form of unification takes place. Seemingly opposite concepts in the original picture are unified in the more advanced one. Each time a fundamental constant is understood as a conversion factor. • h: Unification of wave and particle• c: Unification of space and time• G: Unification of Inertial and gravitational mass• k: Unification of Energy and information• h,c: Unification of quanta and fields • G,c: Unification of matter and geometry
What’s next?
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
HN⇤ ' H⌦⇤ (17)
Note G�1N is a tension Gab = 8⇡GTab
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (18)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(19)
2
Quantum and gravity
h is a composite: a product of length scale and energy scaleG is a composite: a ratio of length scale and energy scale (in 4d)c is a conversion factor between space and timeWe want a theory based on h and G and c: the remaining unification
What are the essential features of quantum theory and gravity?
Both h and G inversely relate space-time concept to an energy-momentum concept
in QM higher energy probes shorter distanceIn GR higher mass leads to bigger BH scaleIn the gravitational memory effect higher energy leads to a larger displacement
Unifying space-time with energy-momentum?What does that mean/implies?
QFT and gravity Miracles
Gauge invariance mediated by massless particles, no fundamental particle with spin higher than 2, anomaly cancellation, etc…, can be deduced QFT as an effective theory (EFT) is unique.
With current knowledge it is possible to build QFT and its main characteristics from first principles•Unitarity•Causality•Locality•Chiral symmetry
With current knowledge it is possible to build Gravity and its main characteristics from first principles•Causality•Locality•Background independence/ General covariance
How can we go beyond ? what principle should we relax?
Quantum + Gravity
• Quantising geometry ? : Background independence and non local observables, space is fundamentally discrete, built-in Hilbert space bases. But the challenge is reconciliation with the General relativity principle outside the classical limit.
• String Theory ?: The probe is fundamental, delocalising the probe, consistent with relativity, but it hasn’t changed yet our understanding of space and time at the fundamental level.
• Emergent models ?: CDT, Causal sets, Horava Gravity, CMT inspired or Holography AdS/CFT
• Quantising gravity? : Doesn’t work — non-renormalisable, Asymptotic Safety
• What have we learned? what are we missing? what haven’t we tried?
Yes: We expect radically new phenomena to become visible, not just small corrections to known phenomena, more than EFT.
• Should we care about putting together gravity and the quantum?
Lessons• We have many different approaches to the problem. What have we
learned so far? What do they all have in common?
• The only common theme between all of them is non-locality• non-local observables in background independent approach• non-local probes• non-locality of holography • discreteness• non-local fixed point, etc…
The Challenges of non-Locality
• Locality is built in Field Theory and General Relativity:
locality of asymptotic states, locality of interactions,locality of RG =separation of scales.
•We expect that any theory of quantum gravity will involve some non-locality. How do we deal with non-locality without opening Pandora’s Box?
These are the foundations of modern physics.We need to specify what type of non-locality is viable, we need a new principle to tame non-locality.
Non-Locality of QM•Both QM and GR exhibit non-locality:• QM:
•Heisenberg non-locality •Entanglement non-locality : “Spooky action”. A form of
kinematical non-locality•Aharanov-Bohm non-locality = non-locality of
interferences. A form of dynamical non-locality
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
September 5, 2014
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = �✏, GN =�
✏. (3)
What operator can measure the interference?These are the modular operators: A new type of observable with no classical analogue. Aharonov
Modular operators 1(x), 2(x)
hx |↵i = 1(x) + e
i↵ 2(x)
x , p ↵
L ⇠ x1 � x2
Suppose we have two waves of non overlapping supportConsider a wave
No polynomial in can detect
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
HN⇤ ' H⌦⇤ (17)
Note G�1N is a tension Gab = 8⇡GTab
@↵h↵|xnpm|↵i = 0 (18)
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (19)
2
But if
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
HN⇤ ' H⌦⇤ (17)
Note G�1N is a tension Gab = 8⇡GTab
@↵h↵|xnpm|↵i = 0 (18)
@↵h↵|eiLp/~|↵i 6= 0 (19)
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (20)
2
happens because wavefunctions can be non-analytic but for any func with a Fourier transform
e
iLp/~f (x) = f (x + L)
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
HN⇤ ' H⌦⇤ (17)
Note G�1N is a tension Gab = 8⇡GTab
@↵h↵|xnpm|↵i = 0 (18)
@↵h↵|eiLp/~|↵i 6= 0 (19)
i@teiLp/~ = (V (q)� V (q + L))eiLp/~ (20)
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (21)
2
modular operator such as •have no classical analog•satisfy non local equations
of motions
dynamical non locality
Boundary dof in gravity, gauge and symmetry
June 15, 2016
1 Local subsystem
The question that interest is wether we can define the notion of local subsytem in Gravity.That is given a Hilbert space H can we decompose it in terms of Hilbert spaces associated withsubregions. For a scalar field theory or spin system we have that
H = H⌃ ⌦H⌃ (1)
H 6= H⌃ ⌦H⌃ (2)
⌃ ⌃ (3)
From which we derive our notion of locality. The hilbert space splits as a tensor product. Thisis no longer true for gauge and gravity in fact
H⌃[⌃ � H⌃ ⌦H⌃, @⌃ = S. (4)
H⌃[⌃ ⇢ H⌃ ⌦ H⌃, @⌃ = S. (5)
[O⌃, O⌃] = 0 (6)
eipL/~ (7)
The split HS is too small :( This follows from the fact that certain gauge invariant observablesare no longer gauge invariant for the subsytems. A typical observable in gravity is the holonomyof the gravitational connection along a closed curve. If that curve cross S = @⌃ gauge invariantis broken for the subsystem.
Present the geometrical setting and intrduce the concept of entangling spheres.How do we solve this ? how can we split our Hilbert space our our set of gauge invariantoperators. This is a key definition in order to understand locality
The key idea is to design an extended phase space
H⌃ = H⌃ ⌦HS
� H⌃ ⌃ S (8)
that contains new and physical degrees of freedom associated with the presence of the boundary.What are these? How do we find them ? How do we know they are physical degrees of freedom?• They represent observer degrees of freedom, which are necessary to even define the problem.Extension analogous to stuckelber mechanism. Excpet that in the presence of a broken gaugewe get analogs of Goldstone modes These are the physical modes we are looking for.•We find them by demanding that they restore gauge invariance in the presence of the boundary.
1
Non Locality of GR
Both gravity and quantum mechanics lead to new class of non local observables
•Diffeomorphism invariance gravity observables are non local: Generalised Gauss Law = Holography.
•Due to causality there is no-screening, the gravity charges is the energy E: it has to be positive.
Non local memory effect:
ei↵xei�x = e2i⇡↵�
eixR eiRx
= e2i⇡eiRxeixR
[[q], [p]] = 0
NeuroLinguisticProgramming
NameofthedemonsofPandorabox : confusion, ambiguities, Lorentzsymmetrybreaking, non�unitarity, strong�coupling, causalityviolation, UV�IRmixing, non�renormalisability, abscenceofinterpretations, arbitrariness, ambiguities, lackofhierarchy.
E = 0
F (A) = 0
� �
⇤
P =
ZdTeiTH
gµ⌫(x, p) ⌘ H|L|(a, b) + (a, b0) + (a0, b)� (a0, b0)| < X
@�X $ @⌧X
�(q, p)
2H� = 0 2⌘� = 0
@p� = 0
2⌘� = ⌘AB@A@B = @pa@qa
P = ⌘�1H(�X)
(q, p) ! (⇤q,⇤�1p)
(q, p) ! (↵q + �p,↵p+ �q)
�T= ��
↵2 � �2
= 1
�z ⇠ Gp�x
x
2
t’Hooft: One should introduce a generalisation of S-matrix in the gravitational context to account for these effects
(p·x) > 0(p·x) < 0
CB AFigure
4. Snapshot of the gravitational shock wave caused by a highly relativistic particle. If we have arectangular grid and synchronized clocks before the particle passed by (region A), then, behindthe particle (region B), the grid will be deformed, and the clocks desynchronized. The shift isproportional to the logarithm of the transverse distance.
begin to move after the shock wave passed. The artist stopped them by giving them a
little kick to keep them in a fixed position afterwards.
In terms of the light cone coordinates of Eq. (4.2) in section 4, we have the connection
formulax+
(+) − x+(−) = 4Gp+ log |x| = 4
√2 Gp log |x| ;
x−(+) − x−
(−) = 0 .(10.12)
Here, x is the transverse distance from the source particle, which is moving (highly rel-
ativistically) along the line x− = x = 0 . The r.h.s. of Eq. (10.12) is a Green function,
−√
2f(x)p , satisfying the equation
∂ 2f(x) = −8πG δ2(x) , (10.13)
where the sign is chosen such that f is large for small values of x .
Next, we generalize this result for a particle moving into a black hole. For the
Schwarzschild observer, its energy is taken to be so small that its gravitational field appears
to be negligible. However, in Kruskal space, Section 2, the energy is seen to grow exponen-
tially as Schwarzschild time t progresses. Let us therefore choose the Kruskal coordinate
frame such that the particle came in at large negative t . This means (Eq. (2.3)), that
x ≈ 0 , or, the particle moves in along the past horizon. In view of the result derived above
one can guess in which way the ingoing particle will deform the metric: we cut Kruskal
space in halves across the x -axis, and glue the pieces together again after a shift, defined
by
37
Holographic observables
What kind of non locality?
One of the fundamental challenges is to reconcile having a fundamental scale with Lorentz invariance
•Relative locality
A new take on quantum gravity: It should emerge from a theory which is quantum has a fundamental delocalisation scale and satisfies
the relativity principle. Non-locality cannot be arbitrary.
•Born Duality
Relative Locality is taken as the organizational featureallowing us to tame non locality.
ability to change polarization
Instead of assuming the existence of an absolute space time one define it through localization/interaction of fundamental probes. Each probes might define its own notion of space-time: Locality is relative
What is Relative locality?•Absolute locality is the hypothesis that the concept of spacetime is independent of the nature of probe used. That it is a universal notion.
•Relative locality is on the contrary, exploring the idea that spacetime is a notion which depends on the quantum nature of probe used i-e energy and quantum numbers.The usual spacetime notion is adapted to probes which are point-like and classical.What is the proper notion of quantum spacetime adapted to quantum and non-local probes ?
Why? How to implement it? What are the elements?
Spacetime is relative to energy-momentum in phase space.
Relative Locality: Illustration full sky survey:
Wise infrared
Rosat X-ray
Planck microwave
Fermi Gamma ray
Visualization of wave function
In the same region of space we can have different eigenstates of different energy. It is disordered at a given energy and ordered at another.
High E Low E
The classical question: is it ordered or disordered? is ill-defined in QM. It depends on the observation not just the system!
Analogy: Quantum crystal = spacetime electrons= probes.
Quasi-Particle interferences: Friedel oscillations
ordereddisordered translation invariant
localization is in beholder’s eye
Locality is relative but special relativity is missing
S. Davis
key element: lattice scale
G as a conversion factorIn special relativity there is a universal speed that is also understood as a universal conversion factor between space and time. This leads to observer dependent space and time and their unification in a geometrical structure the Minkowski space-time. In General Relativity due to equivalence principle GN can be understood for matter-like probes, as a universal inverse tension, a conversion factor between space-time and energy-momentum. This leads to observer dependent space-times and energy-momentum and their unification in a geometrical structure the relativistic phase space of individual probes.
G as a conversion factorIn General Relativity due to equivalence principle GN can be understood as a universal inverse tension for matter probes.
The running of a dimensional coupling is always fixed in terms of another dimensionfull parameter. We choose Planck units as units of energy and time where GN doesn’t run. Thus we are extending the equivalence principle to quantum probesWe emphasize that we are talking the relativistic phase space of quantum probes as a realm for the mathematical expression of relative locality. What kind of geometry on phase space expresses the relative locality principle?
For gravitons probes GN is usually seen as a running coupling constant.
Relative locality and Phase space •The geometry of spacetime is encoded in a Lorentzian metric which encodes in its causal structure the difference between space like and time like.
• Similarly the geometry of phase space is encoded into a geometric structure which encodes the difference between space-time like and energy-momentum like. We call this geometry Born geometry of phase space.To unify space-time with energy and momenta we need, a fundamental length scale and energy scale
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
(e, n, p, s, ua) (2.12)
T = T|{z}Tab= T ab
e
+ T abv
, S = Se
+ Sv
(2.13)
T abv
= ✏uaub+ qaub
+ qbub+ ⇡qab + ⇧
ab(2.14)
f(x, p) = fe
+ fv
(2.15)
T abv
ub = 0 (2.16)
1
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
(e, n, p, s, ua) (2.12)
T = T|{z}Tab= T ab
e
+ T abv
, S = Se
+ Sv
(2.13)
T abv
= ✏uaub+ qaub
+ qbub+ ⇡qab + ⇧
ab(2.14)
1
with
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = p
rG
2⇡~ 2 P
{x, x} =
1
2⇡[x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
(2, 2(d� 1)) (d, d)
ds2⌘ = ⌘ABdXAdXB
=
2
~dpdq
T =
c2
G⇠ 10
17
kg
˚A
2H�H = 0 2H�⌘ = 0
2⌘�⌘AB = 4(�⌘++
AB � �⌘��AB )
2⌘�HAB = 4(�⌘++
AB + �⌘��AB )
/ @qµ@pµ
�⌘ = 0
[q] = [x] ⌘ xmod(2⇡R)
[x] ⌘ xmod(2⇡/R)
1
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
June 12, 2016
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = 2⇡�✏, GN =�
✏. (3)
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
June 12, 2016
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = 2⇡�✏, GN =�
✏. x =
q
�x =
p
✏(3)
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
June 12, 2016
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = 2⇡�✏, GN =�
✏. x =
q
�x =
p
✏(3)
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
June 12, 2016
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = 2⇡�✏, GN =�
✏. x =
q
�x =
p
✏x = q/� (3)
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
June 12, 2016
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = 2⇡�✏, GN =�
✏. x =
q
�x =
p
✏x = q/� x = p/✏ (3)
New perspectives on Gravity and the Quantum
Laurent Freidela
aPerimeter Institute for Theoretical Physics,
31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
June 12, 2016
Abstract
Talk given at the La Sapienza Roma.
1 Intro
Non-locality Relative Locality Infra-red UV mixing Geometry of phase space Born reciprocityGravitisation of quantum mechanics.
2 Non-Locality
First and Foremost. We expect that any theory of quantum gravity will involve some non-locality. The question is then what type and how do deal with it.
Why:? Both Quantum mechanics and GR involves non locality. Naively Quantum mechanicspossess two type of non locality:• Heisenberg non locality
�x�p � ~/2 (1)
Any finite amount of energy resolve cannot resolve x fully. We need an infinite amount of energyto do so.Entanglement no locality: Bell’s inequality maintain relativistic locality.
|(a, b) + (a, b0) + (a0, b0)� (a0, b0)| < X, Xcl = 2, XQ = 2p2. (2)
A quantum system can be entangled acrossGeneral Relativity- Classical- Non locality:
Due to di↵eomorphism invariance there are no local observables.Gravity Non locality in the process of localization QM non locality of correlated measure-
ments , correlation.Non locality due Quantum and Gravity put two together. First and foremost any theory
of quantum gravity incorporate both ~ and GN . Putting c = 1 This implies that there exist afinite length scale � and a finite energy scale ✏ such that
~ = 2⇡�✏, GN =�
✏. x =
q
�x =
p
✏x = q/� x = p/✏ (3)
Born Geometry I Phase space naturally possesses 3 natural structures:A symplectic structure and 2 metrics. The Quantum metric and the locality metric
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
At the quantum level
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
(e, n, p, s, ua) (2.12)
T = T|{z}Tab= T ab
e
+ T abv
, S = Se
+ Sv
(2.13)
1
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
! (P ,!, H, ⌘)
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
(e, n, p, s, ua) (2.12)
1
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
! (P ,!, H, ⌘)
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
1
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
1
Born Geometry:
The symplectic structure is encoded in the commutatorIn ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �
✏
[xa, xb] =i
2⇡�ab (4)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (5)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(6)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
As physicist we usually believe both depending on the context. If one think that Geometryis primordial then the challenge is to ”quantize” it, And then define fundamental objects interms of it. (Locality, symmetry principle,...) (E.g particle defined as rep of Poincare group)—¿ LQG.
If one think that fundamental quantum objects are fundamental. The challenge is to de-scribe how one can obtain from their interaction the space-time structure. And then how thisinteraction structure can be deformed ”gravitised” in order to account for dynamical spacetime.–¿ String.
In summary we can either ”Quantise gravity” or ”Gravitise the quantum”.
2
Born Geometry II Quantum metric H
For weakly gravitating objects
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
! (P ,!, H, ⌘)
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
1
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
1
This metric reduces to the usual spacetime metric where spacetime is viewed as a slice of phase space
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
1
signature
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
(2, 2(d� 1))
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
1
gravitational tension is huge
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
(2, 2(d� 1)) (d, d)
ds2⌘ = ⌘ABdXAdXB
=
2
~dpdq
T =
c2
G⇠ 10
17
kg
˚A
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
1
In relative locality the spacetime metric is the leftover of the quantum metric
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
(2, 2(d� 1)) (d, d)
ds2⌘ = ⌘ABdXAdXB
=
2
~dpdq
T =
c2
G⇠ 10
17
kg
˚A
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
1
Born geometry III The locality metric It encodes the distinction between spacetime like and energy-momentum like.
Vector tangents to spacetime are null with respect to
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
! (P ,!, H, ⌘)
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
1
signature
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
(2, 2(d� 1))
ds2⌘ = ⌘ABdXAdXB
=
2
~dpdq
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
1
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡ [x, x] = 2i⇡
! (P ,!, H, ⌘)
! = !ABdXAdXB
=
1
~dpa ^ dqa
ds2H = HABdXAdXB
=
1
~
✓dq2
G+Gdp2
◆
G�E << �L
ds2H / dq2
(2, 2(d� 1)) (d, d)
ds2⌘ = ⌘ABdXAdXB
=
2
~dpdq
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
1
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
Vector tangents to momentum space are also null wrt
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
This new metric captures the essence of relative locality.Curving : Gravitising the quantum
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
Geometry of Phase space
determines which direction in phase space is a spacetime-like direction
A subset of max dim of P such that
In relative locality space-time is not an absolute notion it is a Lagrangian manifold.
Phase Space Backgrounds
Phase Space Backgrounds
conjecture: there are consistent string theories (CFTs) for which⌘ = ⌘(X) and H = H(X) (etc.)however, the phase space geometry is not arbitrary: we haveseen the importance of J2 = 1 and presumably this is to be kept inthe curved caseas well we have two notions of ‘metric’, ⌘ and HM2 ⌘ p2 + �2 = 2(N + N � 2) (1, 1) ! H (2, 0) ! (H + ⌘)(0, 2) ! (H � ⌘). J2 = 1 TP = L � L? !L = !L? = 0. L L?
in such a background, the equations of motion become
r�SA = �12(rAHBC)@�XB@�XC
where r has been assumed to be ⌘-compatiblethis should be supplemented by the constraintswhat are solutions – what is the geometry of phase space?
Laurent Freidel (Perimeter institute) Born Reciprocity SPOCK: October 2013 22 / 27
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘ (P,!) !T = �!
I2 = �1 IT!I = ! H ⌘ !I
HT = IT!T =|{z}symplectic
�IT! =|{z}complex
IT!I2 =|{z}compatibility
!I = H
ds2H =1
✏2dp2 +
1
�2dx2
dx ⇠ �x dp ⇠ �p
dx
�>>>
dp
✏
�x/� ' 1031 �p/✏ ' 10�31
.ds2H / dx2
!|L = 0
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
3
Phase Space Backgrounds
Phase Space Backgrounds
conjecture: there are consistent string theories (CFTs) for which⌘ = ⌘(X) and H = H(X) (etc.)however, the phase space geometry is not arbitrary: we haveseen the importance of J2 = 1 and presumably this is to be kept inthe curved caseas well we have two notions of ‘metric’, ⌘ and HM2 ⌘ p2 + �2 = 2(N + N � 2) (1, 1) ! H (2, 0) ! (H + ⌘)(0, 2) ! (H � ⌘). J2 = 1 TP = L � L? !L = !L? = 0. L L?
in such a background, the equations of motion become
r�SA = �12(rAHBC)@�XB@�XC
where r has been assumed to be ⌘-compatiblethis should be supplemented by the constraintswhat are solutions – what is the geometry of phase space?
Laurent Freidel (Perimeter institute) Born Reciprocity SPOCK: October 2013 22 / 27
Phase Space Backgrounds
Phase Space Backgrounds
conjecture: there are consistent string theories (CFTs) for which⌘ = ⌘(X) and H = H(X) (etc.)however, the phase space geometry is not arbitrary: we haveseen the importance of J2 = 1 and presumably this is to be kept inthe curved caseas well we have two notions of ‘metric’, ⌘ and HM2 ⌘ p2 + �2 = 2(N + N � 2) (1, 1) ! H (2, 0) ! (H + ⌘)(0, 2) ! (H � ⌘). J2 = 1 TP = L � L? !L = !L? = 0. L L?
in such a background, the equations of motion become
r�SA = �12(rAHBC)@�XB@�XC
where r has been assumed to be ⌘-compatiblethis should be supplemented by the constraintswhat are solutions – what is the geometry of phase space?
Laurent Freidel (Perimeter institute) Born Reciprocity SPOCK: October 2013 22 / 27
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
Which one?
p
q
q
p
By the same token it also determines which direction in phase space is energy-momentum like
What about quantum?So far we have presented the classical side of relative locality.At the quantum level phase is promoted to a non-commutative Heisenberg algebra.In QM Euclidean space appears simply as a choice of polarization: That is in the argument of the wave function. This is the quantum analog of a choice of Lagrangian
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (5)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (6)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(7)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
As physicist we usually believe both depending on the context. If one think that Geometryis primordial then the challenge is to ”quantize” it, And then define fundamental objects interms of it. (Locality, symmetry principle,...) (E.g particle defined as rep of Poincare group)—¿ LQG.
2
Similarly Lorentzian space appears simply as a field label. Classical locality is built in the field definition.
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (6)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(7)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
As physicist we usually believe both depending on the context. If one think that Geometryis primordial then the challenge is to ”quantize” it, And then define fundamental objects interms of it. (Locality, symmetry principle,...) (E.g particle defined as rep of Poincare group)—¿ LQG.
2
Can we define a notion of quantum space? quantum space-time?
Quantum spaces are?Quantum space are simply choices of polarizations of the Heisenberg algebra= quantum Born geometry
The Schrodinger representation is the representation diagonalising the maximally commuting sub-algebra
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i (6)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (7)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(8)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
As physicist we usually believe both depending on the context. If one think that Geometryis primordial then the challenge is to ”quantize” it, And then define fundamental objects interms of it. (Locality, symmetry principle,...) (E.g particle defined as rep of Poincare group)—¿ LQG.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (5)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(6)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
As physicist we usually believe both depending on the context. If one think that Geometryis primordial then the challenge is to ”quantize” it, And then define fundamental objects interms of it. (Locality, symmetry principle,...) (E.g particle defined as rep of Poincare group)—¿ LQG.
If one think that fundamental quantum objects are fundamental. The challenge is to de-scribe how one can obtain from their interaction the space-time structure. And then how thisinteraction structure can be deformed ”gravitised” in order to account for dynamical spacetime.–¿ String.
In summary we can either ”Quantise gravity” or ”Gravitise the quantum”.
2
In general a Schrodinger like polarization is associated with classical Lagrangian sub-manifold of phase space.
By definition a quantum space is a commutative *-subalgebra of the Heisenberg algebra
Is there more than Lagrangian?
Flat Modular spaceFlat Modular space are quantum space associated with abelian subgroup of the Heisenberg group.Such groups are generated by modular observables
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (5)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(6)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
As physicist we usually believe both depending on the context. If one think that Geometryis primordial then the challenge is to ”quantize” it, And then define fundamental objects interms of it. (Locality, symmetry principle,...) (E.g particle defined as rep of Poincare group)—¿ LQG.
If one think that fundamental quantum objects are fundamental. The challenge is to de-scribe how one can obtain from their interaction the space-time structure. And then how thisinteraction structure can be deformed ”gravitised” in order to account for dynamical spacetime.–¿ String.
In summary we can either ”Quantise gravity” or ”Gravitise the quantum”.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 (7)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (8)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(9)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
2
A quantum algebra possesses more commutative directions than a classical Poisson algebra
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (8)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(9)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
2
Modular uncertainty: can specify within a cell but no knowledge of which cell
classical Lagrangian
quantum Lagrangian
p
x
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (9)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(10)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
2
Schrodinger rep is a singular limit.
Modular space
Usually space determines what set of commuting measurement can be performed. We reverse the logic and define space as the maximal set of commutative operations. In this way modular spaces are fundamentally quantum. Also Modular space has a built-in length and energy scales.
Is there a physical system where modular spacetime is realized ?Yes there is: In string theory
We can show that the target space of closed string is a modular space. It is not doubled. It is not compactified.
Modularity is the target space realization of T-duality.
Modular space
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 (7)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (8)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(9)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
2
The quasi-periods correspond to the tails of an Aharonov-Bohm potential attached to a unit flux
A generic commutative subalgebra is associated with a lattice in phase space
The Hilbert space corresponds to sections of a U(1) bundle
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (8)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (9)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(10)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
This dichotomy is the dichotomy between GR and QM. Fundamental object of study isquantum objects or space-time.
2
A modular wave function is quasi-periodic
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (10)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(11)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (10)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(11)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
In the tradition of the Greek school, Leibniz, particle physics today, The fundamental objectsare the microscopical constituent (matter) and their relation defines what space and time is.
2
•
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
1d Euclidean space is non compact, simply connected 1d modular space is 2d, compact and not simply-connected
It carries flux and the doubling is a choice of polarisation.
Lorentz covariance of modular spaceIn order to construct we need to choose a lift
There are no translational invariant vacua, since translations do not commute, space corresponds to a broken phase of the Heisenberg group viewed as a translational group.
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
A lift determines a polarisation metricA vacuum determines a quantum metric
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
I. LOOPS 15
XA=
0
@ xµ
xµ
1
A
x =
1p~
qpG
x =
2⇡p~pGp
x =
qp~G
x = 2⇡p
rG
~ 2 P
{x, x} = 2⇡
! (P ,!, H, ⌘)
II. PI-DAY
1 � R (2.1)
Kn =
�
R<< 1 (2.2)
ua Na Sa T ab(2.3)
DaNa= 0, DaS
a � 0 DaTab= 0 (2.4)
Nae
= nua Sae
= sua T abe
= euaub+ pqab qab = hab
+ uaub. (2.5)
s(e, n) ds = �de� �µdn, p+ ⇢ = Ts+ µn. (2.6)
p � = 1/T µ e (2.7)
n = �✓n (2.8)
e = �✓(e+ p) (2.9)
(e+ p)ua = �dap (2.10)
uadaf = 0 Daf = �ua
˙f + daf (2.11)
1
The unbroken symmetry group is the lattice translation.Space corresponds to a Cartan subgroup.
This is why we can reconcile for the first time fundamental discreetness and translational and Lorentz symmetries.
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (12)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(13)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (14)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(15)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: There
2
Lorentz covariance of modular spaceBesides the translations the Heisenberg group is invariant under
Lorentz
Under a boost but
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(2, (d� 1)) (12)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (13)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(14)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
2
The choice of modular polarisation break it down to
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(2, (d� 1)) (12)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (13)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(14)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
2
The choice of vacua break it further to
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (13)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(14)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
2
a frame
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (14)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(15)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravity
2
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (15)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(16)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.
2
A boost is a change of polarization
inthe
Schrod
inger
repres
entation
ought
tobe
realize
d inany
other
(unitarily
equiva
lent)
quanti
zation
.
Befor
e presen
ting our
main
line of
argum
ent, le
t us c
omme
nttha
t reso
lving (a
close
relative
of)thi
s issue
is of fu
ndam
entalimpor
tance
toqua
ntum
gravity.
Indeed
, the m
ain
conundru
mfor
anythe
oryof
quantu
mgra
vity is t
o beabl
e to fin
d a way to
reconc
ilethe
presen
ceofa fundam
ental l
ength
andene
rgysca
le with
thepri
nciple
ofrelativity.Th
e basic
issue is
that the
Loren
tzcon
traction
oflen
gthand
thedilation
oftim
e for rela
tive o
bserve
rs
render
s seem
inglyimpos
sible t
heabi
lityto
have a
scaleon
which
allobs
ervers
agree.
This
fundam
entalissu
e has
attrac
teda lot
ofatt
ention
inrec
entyea
rsand
is still
atthe
core o
f
anyatt
empt
topro
duce
a viab
le theo
ryofqua
ntumgra
vity.
Ause
fulana
logy in qua
ntumme
chanic
s ispro
vided
byang
ular m
oment
um. C
lassically
,
theang
ular m
oment
umdes
cribes
a point
ona sph
ere, a
ndits
canoni
calact
ions a
regiv
en
byrot
ations
which
aresym
metrie
s of the
sphere
. Quant
umme
chanic
ally, repr
esenta
tions
aredis
crete;
theeig
envalu
e ofthe
angula
r mom
entum
iseve
nlyspa
cedand
this c
anbe
interp
reted
asa d
iscret
ization o
f the sphere
, inwh
ichit i
s repla
cedby
a discret
e setofcircles
equally
spaced
along
thez-a
xis, sa
y, ageo
metric
alpic
ture rigo
rously
realize
d in t
heKir
illov
orbit m
ethod
[33]. T
hese o
rbitscor
respon
d tothe
weigh
t lattic
e of th
e repr
esenta
tion.
The
lattice app
arentl
y destro
ysthe
rotationa
l sym
metry
ofthe
classic
aldes
cripti
on.Bu
t we
knowtha
t isnot
true:
quantu
mthe
oryres
tores
therot
ationa
l symm
etry by
arrang
ingfor
a superp
osition of
spin sta
teswh
ichin
turn par
ametr
izea sph
ere’s-w
orth of
states
. The
spin s
tatesare
merely a
basis f
orthe
entire
statespa
ce,and
thebas
is is n
otinv
ariant
under
rotations
.
Figure
1:Dis
cretization
ofsph
ere.
Sohow
does t
hisana
logy wo
rkin
thecon
text o
f modu
larspa
ce?In
theSch
roding
er
repres
entation,
rotations
actwit
hinthe
setofsta
tes|xi,
giving
back a
linear
combin
ation
of
these
states
. In the
modular
polarizat
ion, clea
rly, w
e shou
ldreg
ardthe
rotations
asact
ing
onthe
basis,
result
ingina n
ewcho
iceofmo
dular
cell, i
.e.,a n
ew’qu
antiza
tion axi
s’.Such
a tran
sform
ation
corres
ponds
toa c
anonic
altra
nsform
ation.
Letus
startaga
inwit
h thepuzzle w
e are
facing
. As w
e have
seen,
once w
e intro
duce
a
fundam
ental s
cale, t
henat
ural pola
rization
that re
spects
thepre
senceof t
hissca
le intr
oduces
a bilag
rangia
n lattice ⇤
=` � ˜ em
bedded
inphase
space
P , and
equipp
edwit
h a neut
ral
metric
⌘.More
over, w
e have
seen tha
t the
Hilber
t spac
e isgiv
enby
a spac
e of se
ctions
ofa
line b
undle
L⇤
over the
torus
T⇤
= P/⇤and
that the
embed
ding T
⇤
,! L⇤
is char
acterized
bya lift,
↵⌘2 U(
1).Ro
tation
s acton
phase
space
assym
plectic tran
sform
ations
X = (x,x)
7! O · X = (Ox,O
T x),O 2 O(
d).
(69)
21
Analog to rotation of the frame of a spin
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (16)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(17)
Both radia are physically indistinguishable.
2
inthe Schrodinger representation
ought tobe realized
inany
other (unitarilyequivalent)
quantization.Before presenting
our mainline of argument, let us comment that resolving
(aclose
relative of) this issue is of fundamental importance to quantumgravity. Indeed, the main
conundrumfor any theory of quantum
gravity is to be able to finda way to reconcile the
presence of a fundamental length and energy scale with the principle of relativity. The basic
issue is that the Lorentz contraction of length and the dilation of time for relative observers
renders seemingly impossible the ability to have a scale on which all observers agree. This
fundamental issue has attracted a lot of attention in recent years and is still at the core of
any attempt to produce a viable theory of quantumgravity.
Auseful analogy in quantum
mechanics is provided by angular momentum. Classically,
the angular momentumdescribes a point on
a sphere, andits canonical actions are given
byrotations which
are symmetries of the sphere.Quantum
mechanically, representations
are discrete; the eigenvalue of the angular momentumis evenly
spacedand
this canbe
interpreted as a discretization of the sphere, in which it is replaced by a discrete set of circles
equally spaced along the z-axis, say, a geometrical picture rigorously realized in the Kirillov
orbit method [33]. These orbits correspond to the weight lattice of the representation. The
lattice apparentlydestroys the rotational symmetry
of the classical description.But we
knowthat is not true: quantum
theory restores the rotational symmetry by arranging for
asuperposition
of spinstates which
inturn
parametrize asphere’s-worth
of states.The
spin states are merely a basis for the entire state space, and the basis is not invariant under
rotations.
Figure 1: Discretization of sphere.
Sohow
does this analogywork
inthe context of modular space?
Inthe Schrodinger
representation, rotations act within the set of states |xi, giving back a linear combination of
these states. In the modular polarization, clearly, we should regard the rotations as acting
on the basis, resulting in a newchoice of modular cell, i.e., a new
’quantization axis’. Such
a transformation corresponds to a canonical transformation.
Let us start again with the puzzle we are facing. As we have seen, once we introduce a
fundamental scale, the natural polarization that respects the presence of this scale introduces
a bilagrangianlattice ⇤
=` � ˜
embeddedinphase space P , and
equippedwith
a neutral
metric ⌘. Moreover, we have seen that the Hilbert space is given by a space of sections of a
line bundle L⇤ over the torus T
⇤ = P/⇤and that the embedding T
⇤ ,!L⇤ is characterized
by a lift, ↵⌘ 2
U(1). Rotations act on phase space as symplectic transformations
X=(x, x) 7!
O · X=(Ox,O T
x),
O 2O(d).
(69)
21
A boosted state is a superposition of unboosted ones
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (17)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(18)
Both radia are physically indistinguishable.
2
3 Born reciprocity
From the point of view of probes what is the fundamental clash between Quantum mechanicsand Gravity?
I :
⇢P ! QQ ! �P
(6)
Hint 3: The geometrical realm in which one should formulate the theory is phase space andnot
4 Geometry of phase space
P ! H ⌘
5 Relative locality: What does it have to do with non locality
6 Opening Pandora’s Box
What non locality?
7 What are the examples
8 Gravitising the Quantum
9 From non local probes to Gravity equation
The quantum equivalence principle.
10 Meta-String theory
11 What equations for phase space geometry
12 Conclusion
The point of view I want to develop in this talk is the point of view that start from the ideathat microscopical probes are fundamental.
If we think in that language what is the problem of quantum gravity. Born duality principle.expose Relative Locality Is an investigation into the geometry of energy momentum space.
Geometry of momentum space = non localityThe hope is that there is a geometrical principle that allow to put together geometry on
momentum space and on space time consistently.
3
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (12)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(13)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
Many is Large
Under a boost
Space is definitely not compactified: it is modular
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (17)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(18)
Both radia are physically indistinguishable.
2
This is the essence of Relative locality: Different boosted observers experience different space-times
We get from quantum to classical through a many body limit of extensification
are zeroes of order d of ⇥(N)
(⇤,k) labeled by elements of the quotient lattice N ˜/˜. This followsfrom the fact that for these values the exchange na ! �na � 1 changes the summand by asign. We see that a general extensified state is a state with a particular coherency in thestructure of its zeroes. We also see that the zeroes of the extensified vacuum states ⇥(N)
(⇤,k) arealigned along a Lagrangian manifold given by the condition x := (z� z)/2i = k/N ( see Fig.3). The choice of Lagrangian is controlled by the choice of complex structure that entersthe definition of the Hamiltonian. The coherency of these states is manifest in the fact thatthe zeroes are equally spaced with regular spacing �X 2 N�1 ˜ along the dual LagrangianN�1 ˜= (N`)?.
An important fact is that the position of the zeroes determines the vacuum wave-function,up to a constant. This follows from the fact that the ratio of two wave-functions with thesame zeroes is analytic and periodic and hence constant. Using this fact, we can introduceanother basis of H(N)
⇤
whose zeroes are distributed arbitrarily. One first considers the theta
function ⇥⇤
(z) = ⇥(1)
⇤
(z) which is the unique vacuum state of H(1)
⇤
and we define
⇥(N)
⇤
(z; zi) := e�2i⇡Nc·(z�zc
)
NYi=1
⇥⇤
(z � zi), (98)
where we have introduced the ‘center of mass’ position: zc := (PN
i=1
zi)/N = ic � c. Thenormalization is chosen so that it is translation covariant
⇥(N)
⇤
(z + a; zi + a) = e�2i⇡NIm(a)(z�zc
) ⇥(N)
⇤
(z; zi). (99)
x
x~
1
1
x
x~
↪⇣H(1)
�
⌘�NH(N)�
Figure 4: Organizing the zeroes of the vacuum wavefunction in H(N)
⇤
in a coherent fashion
identifies an embedding H(N)
⇤
,!⇣H(1)
⇤
⌘⌦N
.
This means that a change of the center of mass can be reabsorbed, up to an exponentialfactor, by a translation in z. When a 2 ⇤ we know that ⇥(N)
⇤
is mapped onto itself. Thismeans that the translation of the center of mass position by an element of ⇤ is trivial.
This wave-function possesses zeroes at the points z = i2
+ 1
2
+ zi , for i = 1, · · · , N . Thequasi-periodicity condition of this function reads
⇥(N)
⇤
(z + in+ n; zi) = e2i⇡N(nc�nc)e⇡Nn2e�2i⇡nN ·z⇥(N)
⇤
(z; zi). (100)
This shows that it is in H(N)
⇤
provided the center of mass (c, c) belongs to the lattice (N�1⇤).When this is the case the phase factor is equal to the identity for all (n, n) 2 ⇤. Using thetranslation covariance we can therefore label this function in terms of N elements zi suchthat
Pi zi = 0 and an element (c, c) 2 ⇤/(N�1⇤).
29
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
H (11)
Sp(2d) Sp(2d) \O(d, d) = GL(d) Sp(2d) \O(d, d) \O(2, 2(d� 1)) = O(1, (d� 1)) (12)
O⇤ 6= ⇤ (13)
G⇤
[G⇤, GO⇤] 6= 0 (14)
�x ! �z (15)
⇤ ! O⇤ = UO⇤ ⇤ (16)
HN⇤ ' H⌦⇤ (17)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (18)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(19)
2
It leads to a form of unification between matter-like dof and pure geometry dof : flux unit
Conclusion
‘It is only the beginning’
We have reviewed concepts of space and time and proposed a new notion of space which is•fundamentally quantum•fundamentally non-local•respecting the principle of relative locality•reconciling discreetness with relativity
We haven't discuss how it leads to a•generalization of the concept of causality•a generalization of the concepts of fields•or how to curve it
Relative Locality: An IllustrationWhat do we see when we
visualise the electron quantum wave function?
Tunelling current
1 Introduction
2 Measure the electron wave function
In electron microscop techniques ( spectroscopic imaging scanning tunnellingmicroscopy) one position the tip of the microscope, which consist of one atom,at a distance z over a point r of the sample and under a applied potentialdi↵erence V and measure the tunelling currentI(r, z, E) where E = eV isthe electrostatic energy. This current is proportional to the localnumber ofelectronic state n(r, E). This means that the conductance g(r, E) = dI
dVmeasures the local density of states. Explicitely we have
I(r, z, E) = i(r, z)
Z E
0
n(r, E)dE. (1)
and
g(r, E) =ISR ES
0 n(r, E)n(r, E) (2)
where IS, VS is a control current voltage situation which insure that thedistance z is fixed.
The local density of states is defined as a sum over the Eigen-energyEi of the one particle hamiltonian (often called pseudo hamiltonian). Thatis we are in a probe picture, where he probe interacts principally with thebackground but is free from other probes (??). If one denote by i(r) theeigen functions for the Eigenstate i. The Local density of state is defined tobe
n(r, E) =1
N
X
i
| i(r)|2�(E � Ei). (3)
This represent the density of state weighted by the probability to find theelectron at the point r. This can be written as
Z +1
�1dtei(E�Ei)t
X
i
i(r) i(r) (4)
=
Z +1
�1dteiEt
X
i
i(r, t) i(r, 0) (5)
=
Z +1
�1dteiEtG(r, t; r, 0) = G(E; (r, r)). (6)
2
sample position
Local density of state
e Voltage
Tunelling conductivity
1 Introduction
2 Measure the electron wave function
In electron microscop techniques ( spectroscopic imaging scanning tunnellingmicroscopy) one position the tip of the microscope, which consist of one atom,at a distance z over a point r of the sample and under a applied potentialdi↵erence V and measure the tunelling currentI(r, z, E) where E = eV isthe electrostatic energy. This current is proportional to the localnumber ofelectronic state n(r, E). This means that the conductance g(r, E) = dI
dVmeasures the local density of states. Explicitely we have
I(r, z, E) = i(r, z)
Z E
0
n(r, E)dE. (1)
and
g(r, E) =ISR ES
0 n(r, E)n(r, E) (2)
where IS, VS is a control current voltage situation which insure that thedistance z is fixed.
g(r, E) =dI
dV/ n(r, E) (3)
The local density of states is defined as a sum over the Eigen-energyEi of the one particle hamiltonian (often called pseudo hamiltonian). Thatis we are in a probe picture, where he probe interacts principally with thebackground but is free from other probes (??). If one denote by i(r) theeigen functions for the Eigenstate i. The Local density of state is defined tobe
n(r, E) =1
N
X
i
| i(r)|2�(E � Ei). (4)
This represent the density of state weighted by the probability to find theelectron at the point r. This can be written as
Z +1
�1dtei(E�Ei)t
X
i
i(r) i(r) (5)
=
Z +1
�1dteiEt
X
i
i(r, t) i(r, 0) (6)
=
Z +1
�1dteiEtG(r, t; r, 0) = G(E; (r, r)). (7)
2
1 Introduction
2 Measure the electron wave function
In electron microscop techniques ( spectroscopic imaging scanning tunnellingmicroscopy) one position the tip of the microscope, which consist of one atom,at a distance z over a point r of the sample and under a applied potentialdi↵erence V and measure the tunelling currentI(r, z, E) where E = eV isthe electrostatic energy. This current is proportional to the localnumber ofelectronic state n(r, E). This means that the conductance g(r, E) = dI
dVmeasures the local density of states. Explicitely we have
I(r, z, E) = i(r, z)
Z E
0
n(r, E)dE. (1)
and
g(r, E) =ISR ES
0 n(r, E)n(r, E) (2)
where IS, VS is a control current voltage situation which insure that thedistance z is fixed.
g(r, E) =dI
dV/ n(r, E) (3)
The local density of states is defined as a sum over the Eigen-energyEi of the one particle hamiltonian (often called pseudo hamiltonian). Thatis we are in a probe picture, where he probe interacts principally with thebackground but is free from other probes (??). If one denote by i(r) theeigen functions for the Eigenstate i. The Local density of state is defined tobe
n(r, E) =1
N
X
i
| i(r)|2�(E � Ei). (4)
This represent the density of state weighted by the probability to find theelectron at the point r. This can be written as
Z +1
�1dtei(E�Ei)t
X
i
i(r) i(r) (5)
=
Z +1
�1dteiEt
X
i
i(r, t) i(r, 0) (6)
=
Z +1
�1dteiEtG(r, t; r, 0) = G(E; (r, r)). (7)
2
energy dependent measurement of the electron wave function
How do quantum electron localise inside a crystal ?
Why? Quantum theory is our fundamental framework, yet such basic concepts as space and time have not been affected by it. Space and time as they appear in are still treated as classical concepts
In ST the role of GN = g2↵0 is played by the slope parameter ↵0 = �✏
[xa, xb] =i
2⇡�ab (4)
(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)
F (x)| i = F (x)| i F (ˆx) (6)
⇥e2i⇡x, e2i⇡x
⇤= 0 {e2i⇡x, e2i⇡x} 6= 0 (7)
⇤ 2 P (x, x) (8)
(x+ a, x) = e2i⇡ax (x, x), (x, x+ a) = (x, x). (9)
H⇤ = �(L⇤) L⇤ ! T⇤ = P/⇤ T⇤ ! L⇤ (10)
Note G�1N is a tension
� provides a length scale on space-time beyond which we cannot observe ✏ provides a energyscale on momentum space which fundamental object cannot exceed. QG: For Instance in LQGor Dyn Triang: The non locality of QG is endoded in two ways. First it is due to the factthat fundamental building blocks have a fundamental length scale, and also due to the fact thatgeometry fluctuates.
Another way to experience the non-locality is :In order to probe a region of size R we need to localize in it an energy
E � ~RO
. On the other hand such an energy will create a black hole of size RS = GE. So the minimalsize we can observe is
Rmin = Max
✓~E,GE
◆� �. (11)
At some point after we reach the self dual point the size of the region we can observe extendinstead of shrinking. The same phenomena remarkably appear in string model. If we considera theory compactified on a radius R. We can show that both radia
R,2�2
R(12)
Both radia are physically indistinguishable.First Hint: Non locality is due to the presence of a length scale and an energy scale. Second
Hint: There seems to be a operational indistinguishability between R/� and �/R.What Type of Non Locality?
In order to address this issue we need to focus on a more precise definition of non locality: Thereare two fundamentally di↵erent approaches to Quantum gravityBoth approach di↵er about what is the fundamental object of study. Is it the smallest elements?Or is it the Biggest one ( space-time) Heidegger :” What is a thing ? ”
In the tradition of Newton, Galillee, Einstein etc... The fundamental object is space-timeand physical objects are defined by how they move in it.
2
top related