why quantum gravity-an introductory survey.pdf
TRANSCRIPT
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)
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Why Quantum Gravity?
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Why Quantum Gravity?
General Relativity and Quantum Theory: not only verydifferent domains of validity, but very different theories:
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Why Quantum Gravity?
General Relativity and Quantum Theory: not only verydifferent domains of validity, but very different theories:
General Relativity: smoothness and geometry
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Why Quantum Gravity?
General Relativity and Quantum Theory: not only verydifferent domains of validity, but very different theories:
General Relativity: smoothness and geometryQuantum Mechanics: probability and uncertainty
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Why Quantum Gravity?
General Relativity and Quantum Theory: not only verydifferent domains of validity, but very different theories:
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 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
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
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
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
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
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
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
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
Einstein’s equations once more:
Rµν
− 1
2
gµν R Marble?
= κT µν Timber?
Question: can we understand the r.h.s. geometrically?
. – p.4/25
G d M
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Geometry and Matter
Einstein’s equations once more:
Rµν
− 1
2
gµν R Marble?
= κT µν Timber?
Question: can we understand the r.h.s. geometrically?
Or: what is the relation between gravitation and theother fundamental (strong and electroweak) forces?
. – p.4/25
G d M
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Geometry and Matter
Einstein’s equations once more:
Rµν
− 1
2
gµν R Marble?
= κT µν Timber?
Question: can we understand the r.h.s. geometrically?
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
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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]
Two possible conclusions:
. – p.5/25
A B i P bl
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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]
Two possible conclusions:
Consistent quantization of gravity requires inclusion of(supersymmetric) matter to cancel UV divergences; or
. – p.5/25
A B i P bl
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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]
Two possible conclusions:
Consistent quantization of gravity requires inclusion of(supersymmetric) matter to cancel UV divergences; or
UV divergences are artefacts of perturbative treatment⇒ disappear upon non-perturbative quantization?
. – p.5/25
A B i P bl
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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]
Two possible conclusions:
Consistent quantization of gravity requires inclusion of(supersymmetric) matter to cancel UV divergences; or
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)
. – p.5/25
Approaches to Quantum Gravity
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Approaches to Quantum Gravity
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Approaches to Quantum Gravity
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Approaches to Quantum Gravity
Supergravity, Superstring and M Theory... preferred by particle theorists
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Approaches to Quantum Gravity
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Approaches to Quantum Gravity
Supergravity, Superstring and M Theory... preferred by particle theorists
Canonical quantum gravity: geometrodynamics,loop and spin foam quantum gravity (LQG)... preferred by general relativists
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Approaches to Quantum Gravity
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Approaches to Quantum Gravity
Supergravity, Superstring and M Theory... preferred by particle theorists
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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Canonical Quantization
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Canonical Quantization
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Canonical Quantization
Non-perturbative and background independent approach:quantum metric fluctuations and quantum geometry
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Canonical Quantization
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Canonical Quantization
Non-perturbative and background independent approach:quantum metric fluctuations and quantum geometry
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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Canonical Quantization
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Canonical Quantization
Non-perturbative and background independent approach:quantum metric fluctuations and quantum geometry
Hamiltonian approach: manifest space-time covarianceis lost through split (‘foliation’) of space-time accordingto M = Σ× R with spatial manifold Σ and time R
→ Space-time geometry as the evolution of spatial geometry in time according to Einstein’s equations
. – p.7/25
Canonical Quantization
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Canonical Quantization
Non-perturbative and background independent approach:quantum metric fluctuations and quantum geometry
Hamiltonian approach: manifest space-time covarianceis lost through split (‘foliation’) of space-time accordingto M = Σ× R with spatial manifold Σ and time R
→ Space-time geometry as the evolution of spatial geometry in time according to Einstein’s equations
Textbook method: replace { p, q } = 1 → [ p, q ] = i
. – p.7/25
Canonical Quantization
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Canonical Quantization
Non-perturbative and background independent approach:quantum metric fluctuations and quantum geometry
Hamiltonian approach: manifest space-time covarianceis lost through split (‘foliation’) of space-time accordingto M = Σ× R with spatial manifold Σ and time R
→ Space-time geometry as the evolution of spatial geometry in time according to Einstein’s equations
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)
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Geometrodynamics(I)
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Geometrodynamics(I)
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Geometrodynamics(I)
= (formal) canonical quantization with metric variables
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Geometrodynamics(I)
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Geometrodynamics(I)
= (formal) canonical quantization with metric variables
4-Metric gµν decomposes into dynamical variables
gij (spatial components of 4-metric)
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Geometrodynamics(I)
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G y a ( )
= (formal) canonical quantization with metric variables
4-Metric gµν decomposes into dynamical variables
gij (spatial components of 4-metric)
and Lagrange multipliers (→ constraints)
N ∼ g00 (Lapse) , N i ∼ g0i (Shift)
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Geometrodynamics(I)
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y ( )
= (formal) canonical quantization with metric variables
4-Metric gµν decomposes into dynamical variables
gij (spatial components of 4-metric)
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)
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y ( )
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Geometrodynamics (II)
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y ( )
Quantization in Schrödinger picture
Πij(x)
−→
i
δ
δgij(x
)
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Geometrodynamics (II)
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y ( )
Quantization in Schrödinger picture
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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Geometrodynamics (II)
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y
Quantization in Schrödinger picture
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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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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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?
Mathematical Challenges – Singular functional differential equation
– Just the old UV divergences in a new guise? – No Hilbert space of wave functionals?
. – p.10/25
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?
Mathematical Challenges – Singular functional differential equation
– Just the old UV divergences in a new guise? – No Hilbert space of wave functionals?
Very little progress in more than 40 years:
“... that damned equation!” (Bryce DeWitt)
. – p.10/25
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?
Mathematical Challenges – Singular functional differential equation
– Just the old UV divergences in a new guise? – No Hilbert space of wave functionals?
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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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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The ‘Superworld’
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Basic strategy: render gravity perturbatively consistent(i.e. finite) via inclusion of suitable matter couplings
Supersymmetry: matter (fermions) ↔ forces (bosons)
→ cancellation of UV divergences in perturbation theory
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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
Supersymmetry: matter (fermions) ↔ forces (bosons)
→ cancellation of UV divergences in perturbation theoryMaximally symmetric point field theories
D = 4, N = 8 Supergravity
D = 11 Supergravity
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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
Supersymmetry: matter (fermions) ↔ forces (bosons)
→ 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?
. – p.12/25
Successes....
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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....
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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
Gravity is ‘predicted’ by string theory
. – p.13/25
Successes....
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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
Gravity is ‘predicted’ by string theory
Semi-realistic spectrum of low energy physics?
. – p.13/25
Successes....
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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
Gravity is ‘predicted’ by string theory
Semi-realistic spectrum of low energy physics?
Microscopic BH Entropy: S = 14A
. – p.13/25
Successes....
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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
Gravity is ‘predicted’ by string theory
Semi-realistic spectrum of low energy physics?
Microscopic BH Entropy: S = 14A
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’
. – p.13/25
Successes....
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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
Gravity is ‘predicted’ by string theory
Semi-realistic spectrum of low energy physics?
Microscopic BH Entropy: S = 14A
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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An important development
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Very recent work has shown that N = 8 supergravity
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An important development
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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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An important development
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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]
... and might thus be finite to all orders for D = 4!
. – p.14/25
An important development
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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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]
... and might thus be finite to all orders for D = 4!
String inspired techniques are crucial for derivation.
. – p.14/25
An important development
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 63/122
Very recent work has shown that N = 8 supergravity... is super-finite up to four loops[Bern et al., hep-th/0905.2326]
... and might thus be finite to all orders for D = 4!
String inspired techniques are crucial for derivation.
New feature: cancellations beyond supersymmetry .
. – p.14/25
An important development
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 64/122
Very recent work has shown that N = 8 supergravity... is super-finite up to four loops[Bern et al., hep-th/0905.2326]
... and might thus be finite to all orders for D = 4!
String inspired techniques are crucial for derivation.
New feature: cancellations beyond supersymmetry .
If true, this could change the picture completely:
. – p.14/25
An important development
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 65/122
Very recent work has shown that N = 8 supergravity... is super-finite up to four loops[Bern et al., hep-th/0905.2326]
... and might thus be finite to all orders for D = 4!
String inspired techniques are crucial for derivation.
New feature: cancellations beyond supersymmetry .
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?
. – p.14/25
The esoteric road?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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[with T. Damour, M. Henneaux, A. Kleinschmidt,...]
. – p.15/25
The esoteric road?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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[with T. Damour, M. Henneaux, A. Kleinschmidt,...]
Symmetry: the most successful principle of physics
. – p.15/25
The esoteric road?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 68/122
[with T. Damour, M. Henneaux, A. Kleinschmidt,...]
Symmetry: the most successful principle of physics
Thus: search for a fundamental symmetry of space,time and matter (and quantum gravity)
. – p.15/25
The esoteric road?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 69/122
[with T. Damour, M. Henneaux, A. Kleinschmidt,...]
Symmetry: the most successful principle of physics
Thus: search for a fundamental symmetry of space,time and matter (and quantum gravity)
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
. – p.15/25
The esoteric road?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 70/122
[with T. Damour, M. Henneaux, A. Kleinschmidt,...]
Symmetry: the most successful principle of physics
Thus: search for a fundamental symmetry of space,time and matter (and quantum gravity)
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?
. – p.15/25
Quantum gravity and experiment?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.16/25
Quantum gravity and experiment?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 72/122
Any direct check must contend with smallness ofPlanck scale → seems completely hopeless....
. – p.16/25
Quantum gravity and experiment?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 73/122
Any direct check must contend with smallness ofPlanck scale → seems completely hopeless....
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?
. – p.16/25
Quantum gravity and experiment?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 74/122
Any direct check must contend with smallness ofPlanck scale → seems completely hopeless....
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?
Or do there exist cumulative effects, such as e.g.– Modifi ed dispersion laws for cosmic rays?
– Unexplained noise in GEO600 detector?
. – p.16/25
Quantum gravity and experiment?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 75/122
Any direct check must contend with smallness ofPlanck scale → seems completely hopeless....
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?
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.
. – p.16/25
Hints from the sky?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.17/25
Hints from the sky?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.17/25
Or from accelerator experiments?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.18/25
Or from accelerator experiments?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.18/25
What LHC can tell us
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 81/122
g g /... and (most probably) not a mini-black-hole factory either!
. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 82/122
... and (most probably) not a mini-black-hole factory either!
Coupling gravity to matter: how mass is generated
. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 83/122
... and (most probably) not a mini-black-hole factory either!
Coupling gravity to matter: how mass is generated
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)
. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 84/122
... and (most probably) not a mini-black-hole factory either!
Coupling gravity to matter: how mass is generated
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)
Low energy supersymmetry — to be or not to be?
. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 85/122
... and (most probably) not a mini-black-hole factory either!
Coupling gravity to matter: how mass is generated
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)
Low energy supersymmetry — to be or not to be?
Crucial question: are there new scales between theweak scale and the Planck scale associated with..., substructure (e.g. technicolor), GUTs, supersymmetry, brane worlds,...?
. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 86/122
... and (most probably) not a mini-black-hole factory either!
Coupling gravity to matter: how mass is generated
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)
Low energy supersymmetry — to be or not to be?
Crucial question: are there new scales between theweak scale and the Planck scale associated with..., substructure (e.g. technicolor), GUTs, supersymmetry, brane worlds,...?
Or is there a ‘grand desert’: can the standard modelsurvive all the way up to the Planck scale?
. – p.19/25
What LHC can tell us
Not a "big bang machine": E LHC/M P = 10−15 !
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 87/122
... and (most probably) not a mini-black-hole factory either!
Coupling gravity to matter: how mass is generated
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)
Low energy supersymmetry — to be or not to be?
Crucial question: are there new scales between theweak scale and the Planck scale associated with..., substructure (e.g. technicolor), GUTs, supersymmetry, brane worlds,...?
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?
. – p.19/25
Prospects (I)
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. – p.20/25
Prospects (I)
E i ti h t diff t t
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Existing approaches stress different aspects:
. – p.20/25
Prospects (I)
E i ti h t diff t t
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 90/122
Existing approaches stress different aspects:
Background independence and quantum geometry
. – p.20/25
Prospects (I)
Existing approaches stress different aspects
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 91/122
Existing approaches stress different aspects:
Background independence and quantum geometry
UV finiteness and perturbative consistency
. – p.20/25
Prospects (I)
Existing approaches stress different aspects:
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 92/122
Existing approaches stress different aspects:
Background independence and quantum geometry
UV finiteness and perturbative consistencyUnification of matter and gravitation
. – p.20/25
Prospects (I)
Existing approaches stress different aspects:
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 93/122
Existing approaches stress different aspects:
Background independence and quantum geometry
UV finiteness and perturbative consistencyUnification of matter and gravitation
However ...
. – p.20/25
Prospects (I)
Existing approaches stress different aspects:
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 94/122
Existing approaches stress different aspects:
Background independence and quantum geometry
UV finiteness and perturbative consistencyUnification of matter and gravitation
However ...The hypotheses underlying the different approachesmay not be mutually compatible
→
. – p.20/25
Prospects (I)
Existing approaches stress different aspects:
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 95/122
Existing approaches stress different aspects:
Background independence and quantum geometry
UV finiteness and perturbative consistencyUnification of matter and gravitation
However ...The hypotheses underlying the different approachesmay not be mutually compatible
→‘Grand synthesis’ appears unlikely (at least for now)
. – p.20/25
Prospects (I)
Existing approaches stress different aspects:
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 96/122
Existing approaches stress different aspects:
Background independence and quantum geometry
UV finiteness and perturbative consistencyUnification of matter and gravitation
However ...The hypotheses underlying the different approachesmay not be mutually compatible
→‘Grand synthesis’ appears unlikely (at least for now)
To discriminate between numerous different ansätzeand ideas, need to rely more on Occam’s razor!
. – p.20/25
Prospects (II)
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http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 97/122
. – p.21/25
Prospects (II)
Theoretical search has generated many ideas
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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Theoretical search has generated many ideas...
. – p.21/25
Prospects (II)
Theoretical search has generated many ideas
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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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 −→
. – p.21/25
Prospects (II)
Theoretical search has generated many ideas...
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 100/122
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 −→The search for the ‘right’ theory is still far from finished!
. – p.21/25
Prospects (II)
Theoretical search has generated many ideas...
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 101/122
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 −→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)
. – p.21/25
Prospects (II)
Theoretical search has generated many ideas...
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 102/122
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
. – p.21/25
... and open problems
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. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?
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p
ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?
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p
ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?
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p
ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
Quantum space-time covariance?
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?
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ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
Quantum space-time covariance?
Where is the 2-loop divergence?
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?
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ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
Quantum space-time covariance?
Where is the 2-loop divergence?Matter couplings: anything goes?No divergences, no anomalies,...?
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?
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ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
Quantum space-time covariance?
Where is the 2-loop divergence?Matter couplings: anything goes?No divergences, no anomalies,...?
Degeneracy: > 10500 consistent Hamiltonians?
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?δ δ
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http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 110/122
ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
Quantum space-time covariance?
Where is the 2-loop divergence?Matter couplings: anything goes?No divergences, no anomalies,...?
Degeneracy: > 10500 consistent Hamiltonians?
Idem for spin foam models
. – p.22/25
... and open problems
Semi-classical limit and smooth space-time?δ δ
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 111/122
ggij(x) ∝ δ
δAia(x)
· δ
δA ja(x)
Status of Wheeler-DeWitt equation?
Quantum space-time covariance?
Where is the 2-loop divergence?Matter couplings: anything goes?No divergences, no anomalies,...?
Degeneracy: > 10500 consistent Hamiltonians?
Idem for spin foam models
Is LQG falsifiable?
. – p.22/25
Simplifications
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. – p.23/25
Simplifications
Semi-classical limit
→ 0:i
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Ψ[g] ∼ exp
i
S cl[g]
(1 + O( ))
. – p.23/25
Simplifications
Semi-classical limit
→ 0:i
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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?
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
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. – p.24/25
New Variables — New Perspectives?
Replace metric gij
by connection [Ashtekar (1986)]
1
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Ama = −1
2abcωmbc + γK m
a
Conjugate variable E am ∝ inverse dreibein (triad)
. – p.24/25
New Variables — New Perspectives?
Replace metric gij
by connection [Ashtekar (1986)]
1
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Ama = −1
2abcωmbc + γK m
a
Conjugate variable E am ∝ inverse dreibein (triad)
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
New Variables — New Perspectives?
Replace metric gij
by connection [Ashtekar (1986)]
1 b
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Ama = −1
2abcωmbc + γK m
a
Conjugate variable E am ∝ inverse dreibein (triad)
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)
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. – p.25/25
Loop Quantum Gravity (LQG)
New canonical variables: holonomy (along link e)
he[A] = P exp
A
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he[A] P exp
e
A
. – p.25/25
Loop Quantum Gravity (LQG)
New canonical variables: holonomy (along link e)
he[A] = P exp
A
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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
Loop Quantum Gravity (LQG)
New canonical variables: holonomy (along link e)
he[A] = P exp
A
8/19/2019 Why Quantum Gravity-An introductory survey.pdf
http://slidepdf.com/reader/full/why-quantum-gravity-an-introductory-surveypdf 122/122
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