why quantum gravity-an introductory survey.pdf

8/19/2019 Why Quantum Gravity-An introductory survey.pdf

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Quantum Gravity

An introductory survey

Hermann Nicolai

Max-Planck-Institut f ¨ ur Gravitationsphysik

(Albert–Einstein–Institut, Potsdam)

. – p.1/25



Why Quantum Gravity?

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General Relativity and Quantum Theory: not only verydifferent domains of validity, but very different theories:

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General Relativity: smoothness and geometry

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General Relativity: smoothness and geometryQuantum Mechanics: probability and uncertainty

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General Relativity: smoothness and geometryQuantum Mechanics: probability and uncertainty

Einstein equations tie together matter and geometry :

Rµν − 1

2gµν R

classical?

= κT µν

quantum?

→ mathematical and conceptual inconsistencies?

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Unsolved Questions

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Unsolved Questions

Singularities in General Relativity (GR) – Black holes: gravitational collapse generically unavoidable

– Cosmological ("big bang") singularity: what ‘happened’ at t = 0?

– Singularity theorems: space and time ‘end’ at the singularity

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Unsolved Questions




Singularities in Quantum Field Theory (QFT) – Perturbation theory: UV divergences in Feynman diagrams

– Can be removed by infinite renormalizations order by order

– But: Standard Model (or its extensions) unlikely to exist as rigorous QFTs

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Unsolved Questions







Difficulties have similar origin: – Spacetime as a continuum ( = differentiable manifold) in GR and QFT

– Elementary particles as pointlike excitations in GR and QFT

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Unsolved Questions







Difficulties have similar origin: – Spacetime as a continuum ( = differentiable manifold) in GR and QFT

– Elementary particles as pointlike excitations in GR and QFT

Structure of space-time at the smallest distances?‘Smallest distance’: Planck scales P ∼ 10−33cm and tP = 10−43sec

[Planck mass: M P = 1019 GeV ∼ 10−5g ]

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Geometry and Matter

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Geometry and Matter

Einstein’s equations once more:

Rµν

− 1

2

gµν R Marble?

= κT µν Timber?

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Geometry and Matter


Rµν

− 1

2

gµν R Marble?

= κT µν Timber?

Question: can we understand the r.h.s. geometrically?

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G d M



Geometry and Matter


Rµν

− 1

2

gµν R Marble?

= κT µν Timber?


Or: what is the relation between gravitation and theother fundamental (strong and electroweak) forces?

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G d M



Geometry and Matter


Rµν

− 1

2

gµν R Marble?

= κT µν Timber?


Or: what is the relation between gravitation and theother fundamental (strong and electroweak) forces?

And: is the ‘geometrization’ of matter a necessary prerequi-

site for consistent quantization of gravity?

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A Basic Problem

Perturbative quantum gravity is non-renormalizable

Γ(2)div =

1

ε

209

2880

1

(16π2

)2 dV C µνρσC ρσλτ C λτ

µν

[Goroff, Sagnotti; van de Ven]

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A B i P bl



A Basic Problem


Γ(2)div =

1

ε

209

2880

1

(16π2


µν


Two possible conclusions:

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A B i P bl



A Basic Problem


Γ(2)div =

1

ε

209

2880

1

(16π2


µν



Consistent quantization of gravity requires inclusion of(supersymmetric) matter to cancel UV divergences; or

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A B i P bl



A Basic Problem


Γ(2)div =

1

ε

209

2880

1

(16π2


µν




UV divergences are artefacts of perturbative treatment⇒ disappear upon non-perturbative quantization?

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A B i P bl



A Basic Problem


Γ(2)div =

1

ε

209

2880

1

(16π2


µν




UV divergences are artefacts of perturbative treatment⇒ disappear upon non-perturbative quantization?

The real problem: infinite number of ambiguities!

(NB: This is also a problem for existing non-perturbative approaches)

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Approaches to Quantum Gravity




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Supergravity, Superstring and M Theory... preferred by particle theorists

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Canonical quantum gravity: geometrodynamics,loop and spin foam quantum gravity (LQG)... preferred by general relativists

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Canonical quantum gravity: geometrodynamics,loop and spin foam quantum gravity (LQG)... preferred by general relativists

Other: – (Euclidean) Path integrals

– Discrete quantum gravity: Regge calculus and dynamical triangulations

– Non-commutative geometry and non-commutative space-time

– Asymptotic safety and RG fixed points – Causal histories

– Cellular automata

– . . . . . .

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Canonical Quantization




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Non-perturbative and background independent approach:quantum metric fluctuations and quantum geometry

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Hamiltonian approach: manifest space-time covarianceis lost through split (‘foliation’) of space-time accordingto M = Σ× R with spatial manifold Σ and time R

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→ Space-time geometry as the evolution of spatial geometry in time according to Einstein’s equations

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Textbook method: replace { p, q } = 1 → [ p, q ] = i

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Textbook method: replace { p, q } = 1 → [ p, q ] = i

Matter interactions are to be ignored in a first step:unification is not the primary goal of this approach!

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Geometrodynamics(I)



Geometrodynamics(I)

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Geometrodynamics(I)



Geometrodynamics(I)

= (formal) canonical quantization with metric variables

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Geometrodynamics(I)



Geometrodynamics(I)


4-Metric gµν decomposes into dynamical variables

gij (spatial components of 4-metric)

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Geometrodynamics(I)



G y a ( )




and Lagrange multipliers (→ constraints)

N ∼ g00 (Lapse) , N i ∼ g0i (Shift)

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Geometrodynamics(I)



y ( )




and Lagrange multipliers (→ constraints)

N ∼ g00 (Lapse) , N i ∼ g0i (Shift)

thus: canonical dynamical degrees of freedom

gij(t,x) , Πij(t,x) = δ LEinsteinδ gij(t,x)

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Geometrodynamics (II)



y ( )

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y ( )

Quantization in Schrödinger picture

Πij(x)

−→

i

δ

δgij(x

)

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y ( )


Πij(x)

−→

i

δ

δgij(x

)

leads to Wheeler-DeWitt Equation (1962)

− 2Gijkl δ 2

Ψ[g]δgik(x)δg jl(x)

−√ gR(3)(x)Ψ[g] = 0

for Wave Function(al) of the Universe Ψ[g].

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y


Πij(x)

−→

i

δ

δgij(x

)leads to Wheeler-DeWitt Equation (1962)

− 2Gijkl δ 2

Ψ[g]δgik(x)δg jl(x)

−√ gR(3)(x)Ψ[g] = 0

for Wave Function(al) of the Universe Ψ[g].

Schrödinger equation of quantum gravity!

‘Timeless’→

contains information ‘from beginning to end’

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Numerous Questions...



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Conceptual and interpretational problems – Physical interpretation of ‘wave function of the universe’ Ψ[g]?

– Quantum theory in the cosmological context: Copenhagen vs. many worlds?

– Decoherence and the emergence of a classical (FRW or de Sitter) universe?

– Origin of time and time arrow?

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Mathematical Challenges – Singular functional differential equation

– Just the old UV divergences in a new guise? – No Hilbert space of wave functionals?

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Very little progress in more than 40 years:

“... that damned equation!” (Bryce DeWitt)

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Very little progress in more than 40 years:

“... that damned equation!” (Bryce DeWitt)

Recent advances based on ‘change of variables’: – Replace metric variables gij by (Ashtekar) connection Ai

a

– Loop Quantum Gravity: holonomies and fluxes as basic variables

– Space (and time) at the Planck scale: spin network or spin foam?

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Spin networks and quantum geometry?



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The ‘Superworld’



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Basic strategy: render gravity perturbatively consistent(i.e. finite) via inclusion of suitable matter couplings

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Supersymmetry: matter (fermions) ↔ forces (bosons)

→ cancellation of UV divergences in perturbation theory

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→ cancellation of UV divergences in perturbation theoryMaximally symmetric point field theories

D = 4, N = 8 Supergravity

D = 11 Supergravity

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→ cancellation of UV divergences in perturbation theoryMaximally symmetric point field theories

D = 4, N = 8 Supergravity

D = 11 Supergravity

Supersymmetric extended objects

IIA/IIB und heterotic superstrings (D = 10)

Supermembranes and M(atrix)-Theory (D = 11)

No point-like interactions→

no UV singularities?

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Successes....



No UV Divergences (so far...) – superstrings: up to and including two loops

– N=8 supergravity: see below!

– no results for supermembranes or M(atrix) theory

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Successes....






Gravity is ‘predicted’ by string theory

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Successes....







Semi-realistic spectrum of low energy physics?

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Successes....








Microscopic BH Entropy: S = 14A

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Successes....









New ideas for physics beyond the Standard Model: – Low energy supersymmetry and the MSSM

– Large extra dimensions and brane worlds – Multi- (or mega-)verses and the ‘string landscape’

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Successes....









New ideas for physics beyond the Standard Model: – Low energy supersymmetry and the MSSM

– Large extra dimensions and brane worlds – Multi- (or mega-)verses and the ‘string landscape’

If true, a new El Dorado for experimentalists:new particles, new forces, new dimensions,...

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An important development



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Very recent work has shown that N = 8 supergravity

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Very recent work has shown that N = 8 supergravity... is super-finite up to four loops[Bern et al., hep-th/0905.2326]

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... and might thus be finite to all orders for D = 4!

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String inspired techniques are crucial for derivation.

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New feature: cancellations beyond supersymmetry .

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If true, this could change the picture completely:

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If true, this could change the picture completely:

– A uniquely distinguished UV finite field theory consistently incorporating Einstein gravity?

– Superstring theory (partly) motivated by expectation that N = 8 SUGRA is divergent !

– N = 8 SUGRA has ‘right’ number of fermions (48 = 3 × 16): is this just a curiosity?

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The esoteric road?



[with T. Damour, M. Henneaux, A. Kleinschmidt,...]

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The esoteric road?




Symmetry: the most successful principle of physics

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The esoteric road?





Thus: search for a fundamental symmetry of space,time and matter (and quantum gravity)

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The esoteric road?






A uniquely distinguished symmetry: E 10 → encodes allknown facts about maximally supersymmetric theories

– different ‘slicings’ of E 10 yield D = 11, mIIA, IIB, . . . supermultiplets

– field equations as ‘null geodesic motion’ on E 10/K (E 10)

– expansion in spatial gradients = expansion in heights of E 10 roots

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The esoteric road?






A uniquely distinguished symmetry: E 10 → encodes allknown facts about maximally supersymmetric theories

– different ‘slicings’ of E 10 yield D = 11, mIIA, IIB, . . . supermultiplets

– field equations as ‘null geodesic motion’ on E 10/K (E 10)

– expansion in spatial gradients = expansion in heights of E 10 roots

A Lie algebraic mechanism for the ‘de-emergence’ ofspace(-time) at the cosmological singularity? Or: isthere physics ‘before’ (i.e. ‘without’) space and time?

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Quantum gravity and experiment?



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Any direct check must contend with smallness ofPlanck scale → seems completely hopeless....

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But there may exist indirect checks : is it possible to

link known physics to Planck scale physics, e.g. via– Astrophysics and the early universe?

– Elementary particle physics?

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Or do there exist cumulative effects, such as e.g.– Modifi ed dispersion laws for cosmic rays?

– Unexplained noise in GEO600 detector?

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Or do there exist cumulative effects, such as e.g.– Modifi ed dispersion laws for cosmic rays?

– Unexplained noise in GEO600 detector?

However: without a tight theoretical framework thereare almost unlimited possibilities for such ‘predictions’,and they will not have much discriminatory power.

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Hints from the sky?



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Hints from the sky?



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Or from accelerator experiments?



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Or from accelerator experiments?



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What LHC can tell us



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Not a "big bang machine": E LHC/M P = 10−15 !



g g /... and (most probably) not a mini-black-hole factory either!

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... and (most probably) not a mini-black-hole factory either!

Coupling gravity to matter: how mass is generated

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If Higgs is found and mH measured, many ‘beyondthe standard model’ scenarios will be ruled out! – Example: MSSM type models require mH < O(130 GeV)

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Low energy supersymmetry — to be or not to be?

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Crucial question: are there new scales between theweak scale and the Planck scale associated with..., substructure (e.g. technicolor), GUTs, supersymmetry, brane worlds,...?

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If Higgs is found and mH

measured, many ‘beyondthe standard model’ scenarios will be ruled out!

– Example: MSSM type models require mH < O(130 GeV)



Or is there a ‘grand desert’: can the standard modelsurvive all the way up to the Planck scale?

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If Higgs is found and mH

measured, many ‘beyondthe standard model’ scenarios will be ruled out!

– Example: MSSM type models require mH < O(130 GeV)



Or is there a ‘grand desert’: can the standard modelsurvive all the way up to the Planck scale?

→an unobstructed view of Planck scale physics?

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Prospects (I)



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Prospects (I)

E i ti h t diff t t



Existing approaches stress different aspects:

. – p.20/25

Prospects (I)

E i ti h t diff t t




Background independence and quantum geometry

. – p.20/25

Prospects (I)

Existing approaches stress different aspects





UV finiteness and perturbative consistency

. – p.20/25

Prospects (I)






UV finiteness and perturbative consistencyUnification of matter and gravitation

. – p.20/25

Prospects (I)







However ...

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Prospects (I)







However ...The hypotheses underlying the different approachesmay not be mutually compatible

→

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Prospects (I)








→‘Grand synthesis’ appears unlikely (at least for now)

. – p.20/25

Prospects (I)








→‘Grand synthesis’ appears unlikely (at least for now)

To discriminate between numerous different ansätzeand ideas, need to rely more on Occam’s razor!

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Prospects (II)



. – p.21/25

Prospects (II)

Theoretical search has generated many ideas



Theoretical search has generated many ideas...

. – p.21/25

Prospects (II)

Theoretical search has generated many ideas



Theoretical search has generated many ideas...... but to arrive at testable predictions and to ‘beat’ thesmallness of the Planck scale we must pin down the

correct theory more or less uniquely −→

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Prospects (II)





correct theory more or less uniquely −→The search for the ‘right’ theory is still far from finished!

. – p.21/25

Prospects (II)





correct theory more or less uniquely −→The search for the ‘right’ theory is still far from finished!

Outside Views:

D. Overbye, String theory, at 20, explains it all (or not) (New York Times, 7 Dec. 2004);

H. Nicolai, K. Peeters and M. Zamaklar, Class. Quant. Grav. 22 (2005) R193;

M. Rauner, Aus! (DIE ZEIT, 26 Januar 2006);Tied up with string? (NATURE, November 2006)

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Prospects (II)




eo et ca sea c as ge e ated a y deas... but to arrive at testable predictions and to ‘beat’ thesmallness of the Planck scale we must pin down thecorrect theory more or less uniquely

−→The search for the ‘right’ theory is still far from finished!

Outside Views:

D. Overbye, String theory, at 20, explains it all (or not) (New York Times, 7 Dec. 2004);

H. Nicolai, K. Peeters and M. Zamaklar, Class. Quant. Grav. 22 (2005) R193;

M. Rauner, Aus! (DIE ZEIT, 26 Januar 2006);Tied up with string? (NATURE, November 2006)

Thank you for your attention

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... and open problems



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Semi-classical limit and smooth space-time?



p

ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)

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p

ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)

Status of Wheeler-DeWitt equation?

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p

ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)


Quantum space-time covariance?

. – p.22/25





ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)



Where is the 2-loop divergence?

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ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)



Where is the 2-loop divergence?Matter couplings: anything goes?No divergences, no anomalies,...?

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ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)




Degeneracy: > 10500 consistent Hamiltonians?

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Semi-classical limit and smooth space-time?δ δ



ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)





Idem for spin foam models

. – p.22/25


Semi-classical limit and smooth space-time?δ δ



ggij(x) ∝ δ

δAia(x)

· δ

δA ja(x)





Idem for spin foam models

Is LQG falsifiable?

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Simplifications



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Simplifications

Semi-classical limit

→ 0:i



Ψ[g] ∼ exp

i

S cl[g]

(1 + O( ))

. – p.23/25

Simplifications

Semi-classical limit

→ 0:i



Ψ[g] ∼ exp

i

S cl[g]

(1 + O( ))

‘Mini-superspace’: only finitely many d.o.f.’s

gij(t,x

)dx

i

dx

j

→ a

2

(t) dr2

1− kr2 + r

2

dθ

2

+ r

2

sin

2

θdφ

2⇒ Functional differential equation becomes an‘ordinary’ PDE, that is : Ψ[g]

→ ψ(a)

. – p.23/25

New Variables — New Perspectives?



. – p.24/25


Replace metric gij

by connection [Ashtekar (1986)]

1



Ama = −1

2abcωmbc + γK m

a

Conjugate variable E am ∝ inverse dreibein (triad)

. – p.24/25


Replace metric gij


1



Ama = −1

2abcωmbc + γK m

a


New canonical brackets

{Ama

(x), Anb

(y)} = {E am

(x), E an

(y)} = 0 , {Ama

(x), E bn

(y)} = γδnmδ

ab δ

(3)

(x,y)

. – p.24/25


Replace metric gij


1 b



Ama = −1

2abcωmbc + γK m

a


New canonical brackets

{Ama

(x), Anb

(y)} = {E am

(x), E an

(y)} = 0 , {Ama

(x), E bn

(y)} = γδnmδ

ab δ

(3)

(x,y)

γ = ±i ⇒ equations become polynomial :

E amF mna(A) ≈ 0 , abcE a

mE bnF mnc(A) ≈ 0

with SU (2) (Yang-Mills) field strength

F mna(A) := ∂ mAna − ∂ nAma + abcAm

b

An

c

. – p.24/25

Loop Quantum Gravity (LQG)



. – p.25/25


New canonical variables: holonomy (along link e)

he[A] = P exp

A



he[A] P exp

e

A

. – p.25/25



he[A] = P exp

A



he[A] P exp

e

A

Conjugate variable = flux through area element S

F aS [E ] :=

S

dF a =

S

mnpE amdxn ∧ dx p

. – p.25/25



he[A] = P exp

A



e[ ] p

e

Conjugate variable = flux through area element S

F aS [E ] :=

S

dF a =

S

mnpE amdxn ∧ dx p

Operators act on cylindrical wave functionals

Ψ{Γ,C }[A] = f C he1 [A], . . . , hen [A]which ‘probe’ connection not on all of Σ, but on spin network Γ (= graph consisting of links and vertices).

. – p.25/25

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