on the nature of quantum space-time200.145.112.249/webcast/files/06-15-16-freidel.pdf · •dirac:...

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?


[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



E � ~RO


Rmin = Max

✓~E,GE

◆� �. (18)


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


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)

⇤ ! O⇤ = UO⇤ ⇤ (16)

HN⇤ ' H⌦⇤ (17)


@↵h↵|xnpm|↵i = 0 (18)



E � ~RO


Rmin = Max

✓~E,GE

◆� �. (19)

2

But if


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)

⇤ ! O⇤ = UO⇤ ⇤ (16)

HN⇤ ' H⌦⇤ (17)


@↵h↵|xnpm|↵i = 0 (18)

@↵h↵|eiLp/~|↵i 6= 0 (19)



E � ~RO


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)


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)

⇤ ! O⇤ = UO⇤ ⇤ (16)

HN⇤ ' H⌦⇤ (17)


@↵h↵|xnpm|↵i = 0 (18)

@↵h↵|eiLp/~|↵i 6= 0 (19)

i@teiLp/~ = (V (q)� V (q + L))eiLp/~ (20)



E � ~RO


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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙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


Laurent Freidela



June 12, 2016

Abstract


1 Intro


2 Non-Locality



�x�p � ~/2 (1)







~ = 2⇡�✏, GN =�

✏. (3)


Laurent Freidela



June 12, 2016

Abstract


1 Intro


2 Non-Locality



�x�p � ~/2 (1)







~ = 2⇡�✏, GN =�

✏. x =

q

�x =

p

✏(3)


Laurent Freidela



June 12, 2016

Abstract


1 Intro


2 Non-Locality



�x�p � ~/2 (1)







~ = 2⇡�✏, GN =�

✏. x =

q

�x =

p

✏(3)


Laurent Freidela



June 12, 2016

Abstract


1 Intro


2 Non-Locality



�x�p � ~/2 (1)







~ = 2⇡�✏, GN =�

✏. x =

q

�x =

p

✏x = q/� (3)


Laurent Freidela



June 12, 2016

Abstract


1 Intro


2 Non-Locality



�x�p � ~/2 (1)







~ = 2⇡�✏, GN =�

✏. x =

q

�x =

p

✏x = q/� x = p/✏ (3)


Laurent Freidela



June 12, 2016

Abstract


1 Intro


2 Non-Locality



�x�p � ~/2 (1)







~ = 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


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙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


+ uaub. (2.5)


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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (5)


R,2�2

R(6)





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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙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


+ uaub. (2.5)


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


+ uaub. (2.5)


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


+ uaub. (2.5)


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)


=

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)


=

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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙f + daf (2.11)

1

signature

3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





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))


=

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


+ 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)


=

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


+ uaub. (2.5)

1

3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





3

Vector tangents to momentum space are also null wrt

3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





3

This new metric captures the essence of relative locality.Curving : Gravitising the quantum

3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





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


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


I :

⇢P ! QQ ! �P

(6)



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



What non locality?





3















3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





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


[xa, xb] =i

2⇡�ab (4)

(x) ! �(x). (5)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (6)


R,2�2

R(7)








2

Similarly Lorentzian space appears simply as a field label. Classical locality is built in the field definition.


[xa, xb] =i

2⇡�ab (4)

(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (6)


R,2�2

R(7)








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


[xa, xb] =i

2⇡�ab (4)

(x) ! �(x). (i@t �H) = 0 ! ⇤g� = 0 (5)

F (x)| i = F (x)| i (6)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (7)


R,2�2

R(8)








2


[xa, xb] =i

2⇡�ab (4)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (5)


R,2�2

R(6)










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


[xa, xb] =i

2⇡�ab (4)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (5)


R,2�2

R(6)










2


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (8)


R,2�2

R(9)







2

A quantum algebra possesses more commutative directions than a classical Poisson algebra


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (8)


R,2�2

R(9)







2

Modular uncertainty: can specify within a cell but no knowledge of which cell

classical Lagrangian

quantum Lagrangian

p

x


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (9)


R,2�2

R(10)







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


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (8)


R,2�2

R(9)







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


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (9)


R,2�2

R(10)







2

A modular wave function is quasi-periodic


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (10)


R,2�2

R(11)






2


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (10)


R,2�2

R(11)






2

•


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (11)


R,2�2

R(12)





2


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (11)


R,2�2

R(12)





2


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (11)


R,2�2

R(12)





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.


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (11)


R,2�2

R(12)





2


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (11)


R,2�2

R(12)





2

A lift determines a polarisation metricA vacuum determines a quantum metric

3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





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


+ uaub. (2.5)


p � = 1/T µ e (2.7)

n = �✓n (2.8)

e = �✓(e+ p) (2.9)

(e+ p)ua = �dap (2.10)


˙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.


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (12)


R,2�2

R(13)





2


[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)


O⇤ 6= ⇤ (13)

G⇤




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (14)


R,2�2

R(15)



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


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (13)


R,2�2

R(14)




2

The choice of modular polarisation break it down to


[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)





E � ~RO


Rmin = Max

✓~E,GE

◆� �. (13)


R,2�2

R(14)




2

The choice of vacua break it further to


[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)





E � ~RO


Rmin = Max

✓~E,GE

◆� �. (13)


R,2�2

R(14)




2

a frame


[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)


O⇤ 6= ⇤ (13)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (14)


R,2�2

R(15)



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


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (15)


R,2�2

R(16)


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


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (16)


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


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)

⇤ ! O⇤ = UO⇤ ⇤ (16)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (17)


R,2�2

R(18)


2

3 Born reciprocity


I :

⇢P ! QQ ! �P

(6)



P ! H ⌘



What non locality?







12 Conclusion





3


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (12)


R,2�2

R(13)





2

Many is Large

Under a boost

Space is definitely not compactified: it is modular


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)

⇤ ! O⇤ = UO⇤ ⇤ (16)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (17)


R,2�2

R(18)


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


[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)


O⇤ 6= ⇤ (13)

G⇤

[G⇤, GO⇤] 6= 0 (14)

�x ! �z (15)

⇤ ! O⇤ = UO⇤ ⇤ (16)

HN⇤ ' H⌦⇤ (17)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (18)


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




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)


g(r, E) =dI

dV/ n(r, E) (3)


n(r, E) =1

N

X

i

| i(r)|2�(E � Ei). (4)


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




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)


g(r, E) =dI

dV/ n(r, E) (3)


n(r, E) =1

N

X

i

| i(r)|2�(E � Ei). (4)


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


[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)




E � ~RO


Rmin = Max

✓~E,GE

◆� �. (11)


R,2�2

R(12)





2

on the nature of quantum space-time200.145.112.249/webcast/files/06-15-16-freidel.pdf · •dirac:...

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