loop quantum gravity (lqg) · naive: quantum gravity cut – off at k‘ pˇ 1 where ‘2 = ~g ? =...
TRANSCRIPT
![Page 1: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/1.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Loop Quantum Gravity (LQG)From secured land to unknown territory
Thomas Thiemann1,2
1 Albert Einstein Institut, 2 Perimeter Institute for Theoretical Physics
Loops’07, Morelia
h G
c
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 2: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/2.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Contents
Motivation
The Challenge of Quantum Gravity
Secured Land
Unknown Territory, first Steps and open Problems
Open Questions and Summary
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 3: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/3.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Contents
Motivation
The Challenge of Quantum Gravity
Secured Land
Unknown Territory, first Steps and open Problems
Open Questions and Summary
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 4: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/4.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Contents
Motivation
The Challenge of Quantum Gravity
Secured Land
Unknown Territory, first Steps and open Problems
Open Questions and Summary
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 5: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/5.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Contents
Motivation
The Challenge of Quantum Gravity
Secured Land
Unknown Territory, first Steps and open Problems
Open Questions and Summary
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 6: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/6.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Contents
Motivation
The Challenge of Quantum Gravity
Secured Land
Unknown Territory, first Steps and open Problems
Open Questions and Summary
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 7: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/7.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Cosmological Puzzles A: Isotropy of the CMB
New era: high precision cosmology (COBE, WMAP, PLANCK)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 8: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/8.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Classical GR: Time had a beginning
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 9: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/9.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Usual matter: Horizon Problem
η
η
η
η
rec
0
(0)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 10: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/10.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Exotic matter (inflaton): solves horizon problem
η
η
η0
(0)
ηrec
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 11: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/11.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Cosmological standard model
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 12: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/12.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Inflation also explains temperature fluctuations, but:
Is GR really valid until tP?
Maybe quantum mech. absence of singularity
Inflaton would no longer be necessary! Pre – big bang?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 13: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/13.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Inflation also explains temperature fluctuations, but:
Is GR really valid until tP?
Maybe quantum mech. absence of singularity
Inflaton would no longer be necessary! Pre – big bang?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 14: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/14.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Inflation also explains temperature fluctuations, but:
Is GR really valid until tP?
Maybe quantum mech. absence of singularity
Inflaton would no longer be necessary! Pre – big bang?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 15: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/15.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Cosmological Puzzle B: Dark Energy
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 16: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/16.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Problem of cosmological constant:
Dark energy = vacuum fluct.? E.g. zero point energy
< bH >scalar − <: bH :>scalar=~
2
Z
R3d3x
Z
R3d3k |k|
Naive: Quantum gravity Cut – off at k`P ≈ 1 where `2P = ~G ?
< bH > − <: bH :>=~
2
Z
R3d3x `−4
P
Comparison with cosmological term
Hcosmo =Λ
G
Z
R3d3x
p| det(g)| ⇒ Λ`2
P ≈ 1
Worst prediction in history of physics:Experimentally: Λ`2
P ≈ 10−120.
What is dark energy (matter), why Λ so unnaturally tiny?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 17: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/17.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Problem of cosmological constant:
Dark energy = vacuum fluct.? E.g. zero point energy
< bH >scalar − <: bH :>scalar=~
2
Z
R3d3x
Z
R3d3k |k|
Naive: Quantum gravity Cut – off at k`P ≈ 1 where `2P = ~G ?
< bH > − <: bH :>=~
2
Z
R3d3x `−4
P
Comparison with cosmological term
Hcosmo =Λ
G
Z
R3d3x
p| det(g)| ⇒ Λ`2
P ≈ 1
Worst prediction in history of physics:Experimentally: Λ`2
P ≈ 10−120.
What is dark energy (matter), why Λ so unnaturally tiny?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 18: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/18.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Problem of cosmological constant:
Dark energy = vacuum fluct.? E.g. zero point energy
< bH >scalar − <: bH :>scalar=~
2
Z
R3d3x
Z
R3d3k |k|
Naive: Quantum gravity Cut – off at k`P ≈ 1 where `2P = ~G ?
< bH > − <: bH :>=~
2
Z
R3d3x `−4
P
Comparison with cosmological term
Hcosmo =Λ
G
Z
R3d3x
p| det(g)| ⇒ Λ`2
P ≈ 1
Worst prediction in history of physics:Experimentally: Λ`2
P ≈ 10−120.
What is dark energy (matter), why Λ so unnaturally tiny?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 19: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/19.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Problem of cosmological constant:
Dark energy = vacuum fluct.? E.g. zero point energy
< bH >scalar − <: bH :>scalar=~
2
Z
R3d3x
Z
R3d3k |k|
Naive: Quantum gravity Cut – off at k`P ≈ 1 where `2P = ~G ?
< bH > − <: bH :>=~
2
Z
R3d3x `−4
P
Comparison with cosmological term
Hcosmo =Λ
G
Z
R3d3x
p| det(g)| ⇒ Λ`2
P ≈ 1
Worst prediction in history of physics:Experimentally: Λ`2
P ≈ 10−120.
What is dark energy (matter), why Λ so unnaturally tiny?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 20: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/20.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Problem of cosmological constant:
Dark energy = vacuum fluct.? E.g. zero point energy
< bH >scalar − <: bH :>scalar=~
2
Z
R3d3x
Z
R3d3k |k|
Naive: Quantum gravity Cut – off at k`P ≈ 1 where `2P = ~G ?
< bH > − <: bH :>=~
2
Z
R3d3x `−4
P
Comparison with cosmological term
Hcosmo =Λ
G
Z
R3d3x
p| det(g)| ⇒ Λ`2
P ≈ 1
Worst prediction in history of physics:Experimentally: Λ`2
P ≈ 10−120.
What is dark energy (matter), why Λ so unnaturally tiny?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 21: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/21.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Astrophysical Puzzles: Black Holes
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 22: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/22.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Entropy of Black Holes
Class. GR (Penrose & Hawking): δAr(H) ≥ 0 ⇒ S ∝ Ar(H)(cf. 2nd law).
QFT on CST (Hawking – effect): Black holes = blackradiators kT = ~ω ≈ ~c/r.
Entropy (Bekenstein): For Schwarzschild solutionr = 2Gm/c2 with S = E/(kT) = mc2/(kT)
SBH =Ar(H)
4`2p
What is the microscopic origin of this entropy?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 23: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/23.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Entropy of Black Holes
Class. GR (Penrose & Hawking): δAr(H) ≥ 0 ⇒ S ∝ Ar(H)(cf. 2nd law).
QFT on CST (Hawking – effect): Black holes = blackradiators kT = ~ω ≈ ~c/r.
Entropy (Bekenstein): For Schwarzschild solutionr = 2Gm/c2 with S = E/(kT) = mc2/(kT)
SBH =Ar(H)
4`2p
What is the microscopic origin of this entropy?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 24: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/24.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Entropy of Black Holes
Class. GR (Penrose & Hawking): δAr(H) ≥ 0 ⇒ S ∝ Ar(H)(cf. 2nd law).
QFT on CST (Hawking – effect): Black holes = blackradiators kT = ~ω ≈ ~c/r.
Entropy (Bekenstein): For Schwarzschild solutionr = 2Gm/c2 with S = E/(kT) = mc2/(kT)
SBH =Ar(H)
4`2p
What is the microscopic origin of this entropy?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 25: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/25.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Entropy of Black Holes
Class. GR (Penrose & Hawking): δAr(H) ≥ 0 ⇒ S ∝ Ar(H)(cf. 2nd law).
QFT on CST (Hawking – effect): Black holes = blackradiators kT = ~ω ≈ ~c/r.
Entropy (Bekenstein): For Schwarzschild solutionr = 2Gm/c2 with S = E/(kT) = mc2/(kT)
SBH =Ar(H)
4`2p
What is the microscopic origin of this entropy?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 26: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/26.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Cosmological Puzzles A: Isotropy of the CMBCosmological Puzzles B: Dark EnergyAstrophysical Puzzles: Black Holes
Holographic Explanation of BH – Entropy? [’t Hooft, Susskind 92]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 27: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/27.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Principle of Background Independence
It is widely believed that only a full fledged quantum theoryof gravity can answer these fundamental questions.
For more than 70 years physicists are looking for a unifiedtheory of general relativity and quantum mechanics – sofar w/o success.
Why?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 28: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/28.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Principle of Background Independence
It is widely believed that only a full fledged quantum theoryof gravity can answer these fundamental questions.
For more than 70 years physicists are looking for a unifiedtheory of general relativity and quantum mechanics – sofar w/o success.
Why?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 29: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/29.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Principle of Background Independence
It is widely believed that only a full fledged quantum theoryof gravity can answer these fundamental questions.
For more than 70 years physicists are looking for a unifiedtheory of general relativity and quantum mechanics – sofar w/o success.
Why?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 30: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/30.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Gravity=Geometry, Curvature=Matter Energy Density
Einstein’s equations
Rµν [g] −12
R[g] gµν = 8πG Tµν [g]
Background independence: Geometry g not prescribed butdynamically determined by matter energy density T.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 31: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/31.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Gravity=Geometry, Curvature=Matter Energy Density
Einstein’s equations
Rµν [g] −12
R[g] gµν = 8πG Tµν [g]
Background independence: Geometry g not prescribed butdynamically determined by matter energy density T.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 32: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/32.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Gravity=Geometry, Curvature=Matter Energy Density
Einstein’s equations
Rµν [g] −12
R[g] gµν = 8πG Tµν [g]
Background independence: Geometry g not prescribed butdynamically determined by matter energy density T.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 33: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/33.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Background independence and backreaction (gravitationalwaves)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 34: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/34.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Usual QFT (on CST) background dependent
g0
PSfrag replacementssupp(f)
supp(f’)
E.g. via causality axiom:If supp(f), supp(f’) spacelike separated wrt g0 then
[φ(f), φ(f′)] = 0, φ(f):=∫
Md4x f(x) φ(x)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 35: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/35.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Usual QFT (on CST) background dependent
g0
PSfrag replacementssupp(f)
supp(f’)
E.g. via causality axiom:If supp(f), supp(f’) spacelike separated wrt g0 then
[φ(f), φ(f′)] = 0, φ(f):=∫
Md4x f(x) φ(x)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 36: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/36.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
The structure crucial for ordinary QFT
g0 ⇒ (x − y)2 < 0 ⇒ A
Background Light Cone Algebra
collapses when g0 not available.ignores gravitational backreaction.invalid approx. in extreme cosm. & astrophys. situat.Perturbative approach
g = g0 + h↑ ↑ ↑
Total Metric Background Perturbation (Graviton)
violates BI, inacceptable due to non – renormalisability,merely effect. graviton QFT over g0.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 37: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/37.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
The structure crucial for ordinary QFT
g0 ⇒ (x − y)2 < 0 ⇒ A
Background Light Cone Algebra
collapses when g0 not available.ignores gravitational backreaction.invalid approx. in extreme cosm. & astrophys. situat.Perturbative approach
g = g0 + h↑ ↑ ↑
Total Metric Background Perturbation (Graviton)
violates BI, inacceptable due to non – renormalisability,merely effect. graviton QFT over g0.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 38: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/38.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
The structure crucial for ordinary QFT
g0 ⇒ (x − y)2 < 0 ⇒ A
Background Light Cone Algebra
collapses when g0 not available.ignores gravitational backreaction.invalid approx. in extreme cosm. & astrophys. situat.Perturbative approach
g = g0 + h↑ ↑ ↑
Total Metric Background Perturbation (Graviton)
violates BI, inacceptable due to non – renormalisability,merely effect. graviton QFT over g0.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 39: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/39.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
The structure crucial for ordinary QFT
g0 ⇒ (x − y)2 < 0 ⇒ A
Background Light Cone Algebra
collapses when g0 not available.ignores gravitational backreaction.invalid approx. in extreme cosm. & astrophys. situat.Perturbative approach
g = g0 + h↑ ↑ ↑
Total Metric Background Perturbation (Graviton)
violates BI, inacceptable due to non – renormalisability,merely effect. graviton QFT over g0.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 40: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/40.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
The structure crucial for ordinary QFT
g0 ⇒ (x − y)2 < 0 ⇒ A
Background Light Cone Algebra
collapses when g0 not available.ignores gravitational backreaction.invalid approx. in extreme cosm. & astrophys. situat.Perturbative approach
g = g0 + h↑ ↑ ↑
Total Metric Background Perturbation (Graviton)
violates BI, inacceptable due to non – renormalisability,merely effect. graviton QFT over g0.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 41: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/41.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
We need the
Rµν −
12 R gµν = 8πG Tµν(g)
Quantum - Einstein - Equations
Radical generalisation of principles of QFT on CST.
QFT on diff. mfd. M rather than QFT on spacetime (M,g0).
No Fock spaces but new BI Hilbert spaces.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 42: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/42.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
We need the
Rµν −
12 R gµν = 8πG Tµν(g)
Quantum - Einstein - Equations
Radical generalisation of principles of QFT on CST.
QFT on diff. mfd. M rather than QFT on spacetime (M,g0).
No Fock spaces but new BI Hilbert spaces.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 43: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/43.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
We need the
Rµν −
12 R gµν = 8πG Tµν(g)
Quantum - Einstein - Equations
Radical generalisation of principles of QFT on CST.
QFT on diff. mfd. M rather than QFT on spacetime (M,g0).
No Fock spaces but new BI Hilbert spaces.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 44: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/44.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
We need the
Rµν −
12 R gµν = 8πG Tµν(g)
Quantum - Einstein - Equations
Radical generalisation of principles of QFT on CST.
QFT on diff. mfd. M rather than QFT on spacetime (M,g0).
No Fock spaces but new BI Hilbert spaces.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 45: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/45.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Ordinary QFT: Matter propagates on rigid spacetime
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 46: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/46.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
New QFT: Matter can exist only where geometry is excited
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 47: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/47.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Principle of Background Independence
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 48: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/48.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Classical Formulation
Canonical formulation: M ∼= R×σ
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 49: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/49.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Palatini action with (on shell) topological term
S =1
8πG
Z
MΩIJ ∧ [εIJKLeK ∧ eL +
1β
eI ∧ eJ]
Legendre transformation reveals canonical pair
(ωIJa ,E
aIJ =
12ε
abc[εIJKLeKb eL
c +1β
ebIecJ])
plus constraints, in particular simplicity constraints,
Sab = εIJKLEa
IJEbKL = 0
spatial diffeo, Hamiltonian and Spin(1,3) Gauß constraints
GIJ = ∂aEaIJ − 2ωa[I
KEaJ]K = 0
2nd class system! Solving 2nd class constr. yields canonical pair
(Ajka =Γjk
a + βKabeblεjkl, Ea
jk = | det(e)|εabcebjeck)
where Γ = spin connection of triad and K= extrinsic curvatureassociated to qab = ej
aejb
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 50: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/50.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Palatini action with (on shell) topological term
S =1
8πG
Z
MΩIJ ∧ [εIJKLeK ∧ eL +
1β
eI ∧ eJ]
Legendre transformation reveals canonical pair
(ωIJa ,E
aIJ =
12ε
abc[εIJKLeKb eL
c +1β
ebIecJ])
plus constraints, in particular simplicity constraints,
Sab = εIJKLEa
IJEbKL = 0
spatial diffeo, Hamiltonian and Spin(1,3) Gauß constraints
GIJ = ∂aEaIJ − 2ωa[I
KEaJ]K = 0
2nd class system! Solving 2nd class constr. yields canonical pair
(Ajka =Γjk
a + βKabeblεjkl, Ea
jk = | det(e)|εabcebjeck)
where Γ = spin connection of triad and K= extrinsic curvatureassociated to qab = ej
aejb
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 51: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/51.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Palatini action with (on shell) topological term
S =1
8πG
Z
MΩIJ ∧ [εIJKLeK ∧ eL +
1β
eI ∧ eJ]
Legendre transformation reveals canonical pair
(ωIJa ,E
aIJ =
12ε
abc[εIJKLeKb eL
c +1β
ebIecJ])
plus constraints, in particular simplicity constraints,
Sab = εIJKLEa
IJEbKL = 0
spatial diffeo, Hamiltonian and Spin(1,3) Gauß constraints
GIJ = ∂aEaIJ − 2ωa[I
KEaJ]K = 0
2nd class system! Solving 2nd class constr. yields canonical pair
(Ajka =Γjk
a + βKabeblεjkl, Ea
jk = | det(e)|εabcebjeck)
where Γ = spin connection of triad and K= extrinsic curvatureassociated to qab = ej
aejb
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 52: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/52.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Palatini action with (on shell) topological term
S =1
8πG
Z
MΩIJ ∧ [εIJKLeK ∧ eL +
1β
eI ∧ eJ]
Legendre transformation reveals canonical pair
(ωIJa ,E
aIJ =
12ε
abc[εIJKLeKb eL
c +1β
ebIecJ])
plus constraints, in particular simplicity constraints,
Sab = εIJKLEa
IJEbKL = 0
spatial diffeo, Hamiltonian and Spin(1,3) Gauß constraints
GIJ = ∂aEaIJ − 2ωa[I
KEaJ]K = 0
2nd class system! Solving 2nd class constr. yields canonical pair
(Ajka =Γjk
a + βKabeblεjkl, Ea
jk = | det(e)|εabcebjeck)
where Γ = spin connection of triad and K= extrinsic curvatureassociated to qab = ej
aejb
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 53: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/53.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Palatini action with (on shell) topological term
S =1
8πG
Z
MΩIJ ∧ [εIJKLeK ∧ eL +
1β
eI ∧ eJ]
Legendre transformation reveals canonical pair
(ωIJa ,E
aIJ =
12ε
abc[εIJKLeKb eL
c +1β
ebIecJ])
plus constraints, in particular simplicity constraints,
Sab = εIJKLEa
IJEbKL = 0
spatial diffeo, Hamiltonian and Spin(1,3) Gauß constraints
GIJ = ∂aEaIJ − 2ωa[I
KEaJ]K = 0
2nd class system! Solving 2nd class constr. yields canonical pair
(Ajka =Γjk
a + βKabeblεjkl, Ea
jk = | det(e)|εabcebjeck)
where Γ = spin connection of triad and K= extrinsic curvatureassociated to qab = ej
aejb
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 54: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/54.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Palatini action with (on shell) topological term
S =1
8πG
Z
MΩIJ ∧ [εIJKLeK ∧ eL +
1β
eI ∧ eJ]
Legendre transformation reveals canonical pair
(ωIJa ,E
aIJ =
12ε
abc[εIJKLeKb eL
c +1β
ebIecJ])
plus constraints, in particular simplicity constraints,
Sab = εIJKLEa
IJEbKL = 0
spatial diffeo, Hamiltonian and Spin(1,3) Gauß constraints
GIJ = ∂aEaIJ − 2ωa[I
KEaJ]K = 0
2nd class system! Solving 2nd class constr. yields canonical pair
(Ajka =Γjk
a + βKabeblεjkl, Ea
jk = | det(e)|εabcebjeck)
where Γ = spin connection of triad and K= extrinsic curvatureassociated to qab = ej
aejb
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 55: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/55.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Full set of first class constraints
SU(2) Gauß constraint
Cj = ∂aEaj + Aaj
kEak = 0
spatial diffeomorphism constraint
Ca = Tr(
FabEb)
= 0
Hamiltonian constraint
C =Tr
(FabEaEb
)√|det(E)|
+ ...
Convenient to summarise them in Master Constraint
M =
∫
σ
d3x
[C2 + qabCaCb + δjkCjCk
]√|det(E)|
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 56: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/56.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Full set of first class constraints
SU(2) Gauß constraint
Cj = ∂aEaj + Aaj
kEak = 0
spatial diffeomorphism constraint
Ca = Tr(
FabEb)
= 0
Hamiltonian constraint
C =Tr
(FabEaEb
)√|det(E)|
+ ...
Convenient to summarise them in Master Constraint
M =
∫
σ
d3x
[C2 + qabCaCb + δjkCjCk
]√|det(E)|
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 57: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/57.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Full set of first class constraints
SU(2) Gauß constraint
Cj = ∂aEaj + Aaj
kEak = 0
spatial diffeomorphism constraint
Ca = Tr(
FabEb)
= 0
Hamiltonian constraint
C =Tr
(FabEaEb
)√|det(E)|
+ ...
Convenient to summarise them in Master Constraint
M =
∫
σ
d3x
[C2 + qabCaCb + δjkCjCk
]√|det(E)|
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 58: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/58.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Full set of first class constraints
SU(2) Gauß constraint
Cj = ∂aEaj + Aaj
kEak = 0
spatial diffeomorphism constraint
Ca = Tr(
FabEb)
= 0
Hamiltonian constraint
C =Tr
(FabEaEb
)√|det(E)|
+ ...
Convenient to summarise them in Master Constraint
M =
∫
σ
d3x
[C2 + qabCaCb + δjkCjCk
]√|det(E)|
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 59: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/59.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Problem of Time
Master Constraint M = 0 (Constraint Hypersurface)
Gauge invariant (Dirac) observables
O, O,MM=0 = 0
Problem of Time:1. No canonical Hamiltonian in generally covariant theories2. No time evolution of observables, frozen picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 60: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/60.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Problem of Time
Master Constraint M = 0 (Constraint Hypersurface)
Gauge invariant (Dirac) observables
O, O,MM=0 = 0
Problem of Time:1. No canonical Hamiltonian in generally covariant theories2. No time evolution of observables, frozen picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 61: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/61.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Problem of Time
Master Constraint M = 0 (Constraint Hypersurface)
Gauge invariant (Dirac) observables
O, O,MM=0 = 0
Problem of Time:1. No canonical Hamiltonian in generally covariant theories2. No time evolution of observables, frozen picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 62: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/62.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Problem of Time
Master Constraint M = 0 (Constraint Hypersurface)
Gauge invariant (Dirac) observables
O, O,MM=0 = 0
Problem of Time:1. No canonical Hamiltonian in generally covariant theories2. No time evolution of observables, frozen picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 63: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/63.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Observables & Physical Hamiltonian
Relational Ansatz [Rovelli 92 – ], [Dittrich 04 – ]
Suitable scalar matter φ: Brown – Kuchar deparam. [Brown,Kuchar 92],[T.T. 06]
M = 0 ⇔ π(x) + H(x) = 0
Physical Hamiltonian (dep. only on non scalar d.o.f.)
H =
Z
σ
d3x H(x)
Dirac observables (f: spat. diff. inv.)
Of(τ ) :=∞X
n=0
1n!
Hτ , f(n), Hτ =
Zd3x (τ − φ)H
Physical time evolution
ddτ
Of(τ ) = H,Of(τ )
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 64: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/64.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Observables & Physical Hamiltonian
Relational Ansatz [Rovelli 92 – ], [Dittrich 04 – ]
Suitable scalar matter φ: Brown – Kuchar deparam. [Brown,Kuchar 92],[T.T. 06]
M = 0 ⇔ π(x) + H(x) = 0
Physical Hamiltonian (dep. only on non scalar d.o.f.)
H =
Z
σ
d3x H(x)
Dirac observables (f: spat. diff. inv.)
Of(τ ) :=∞X
n=0
1n!
Hτ , f(n), Hτ =
Zd3x (τ − φ)H
Physical time evolution
ddτ
Of(τ ) = H,Of(τ )
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 65: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/65.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Observables & Physical Hamiltonian
Relational Ansatz [Rovelli 92 – ], [Dittrich 04 – ]
Suitable scalar matter φ: Brown – Kuchar deparam. [Brown,Kuchar 92],[T.T. 06]
M = 0 ⇔ π(x) + H(x) = 0
Physical Hamiltonian (dep. only on non scalar d.o.f.)
H =
Z
σ
d3x H(x)
Dirac observables (f: spat. diff. inv.)
Of(τ ) :=∞X
n=0
1n!
Hτ , f(n), Hτ =
Zd3x (τ − φ)H
Physical time evolution
ddτ
Of(τ ) = H,Of(τ )
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 66: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/66.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Observables & Physical Hamiltonian
Relational Ansatz [Rovelli 92 – ], [Dittrich 04 – ]
Suitable scalar matter φ: Brown – Kuchar deparam. [Brown,Kuchar 92],[T.T. 06]
M = 0 ⇔ π(x) + H(x) = 0
Physical Hamiltonian (dep. only on non scalar d.o.f.)
H =
Z
σ
d3x H(x)
Dirac observables (f: spat. diff. inv.)
Of(τ ) :=∞X
n=0
1n!
Hτ , f(n), Hτ =
Zd3x (τ − φ)H
Physical time evolution
ddτ
Of(τ ) = H,Of(τ )
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 67: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/67.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Quantum Kinematics
Lattice – inspired canon. q’ion [Gambini,Trias et al 81], [Jacobson,Rovelli,Smolin 88]
Magnet. dof.: Holonomy (Wilson – Loop)
A(e) = P exp(
∫
eA)
Electr. dof: flux
Ej(S) =
∫
S
12
Eakl εjkl εabc dxb ∧ dxc
Poisson – brackets:
Ej(S),A(e) = G A(e1) τj A(e2); e = e1 e2, e1 ∩ e2 = e ∩ S
Reality conditions:
A(e) = [A(e−1)]T, Ej(S) = Ej(S)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 68: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/68.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Quantum Kinematics
Lattice – inspired canon. q’ion [Gambini,Trias et al 81], [Jacobson,Rovelli,Smolin 88]
Magnet. dof.: Holonomy (Wilson – Loop)
A(e) = P exp(
∫
eA)
Electr. dof: flux
Ej(S) =
∫
S
12
Eakl εjkl εabc dxb ∧ dxc
Poisson – brackets:
Ej(S),A(e) = G A(e1) τj A(e2); e = e1 e2, e1 ∩ e2 = e ∩ S
Reality conditions:
A(e) = [A(e−1)]T, Ej(S) = Ej(S)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 69: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/69.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Quantum Kinematics
Lattice – inspired canon. q’ion [Gambini,Trias et al 81], [Jacobson,Rovelli,Smolin 88]
Magnet. dof.: Holonomy (Wilson – Loop)
A(e) = P exp(
∫
eA)
Electr. dof: flux
Ej(S) =
∫
S
12
Eakl εjkl εabc dxb ∧ dxc
Poisson – brackets:
Ej(S),A(e) = G A(e1) τj A(e2); e = e1 e2, e1 ∩ e2 = e ∩ S
Reality conditions:
A(e) = [A(e−1)]T, Ej(S) = Ej(S)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 70: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/70.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Quantum Kinematics
Lattice – inspired canon. q’ion [Gambini,Trias et al 81], [Jacobson,Rovelli,Smolin 88]
Magnet. dof.: Holonomy (Wilson – Loop)
A(e) = P exp(
∫
eA)
Electr. dof: flux
Ej(S) =
∫
S
12
Eakl εjkl εabc dxb ∧ dxc
Poisson – brackets:
Ej(S),A(e) = G A(e1) τj A(e2); e = e1 e2, e1 ∩ e2 = e ∩ S
Reality conditions:
A(e) = [A(e−1)]T, Ej(S) = Ej(S)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 71: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/71.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
e
S
e
+
1
2
e S
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 72: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/72.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
HS Reps.: In QFT no Stone – von Neumann Theorem!!!
Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]
Diff(σ) inv. states on hol. – flux algebra A unique.
wave functions
ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C
Holonomy = multiplication – operator
[A(e) ψ](A) := A(e) ψ(A)
Flux = derivative – operator
[Ej(S) ψ](A) := i~ Ej(S), ψ(A)
Scalar product
< ψ,ψ′>:=
Z
SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′
γ(h1, .., hN)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 73: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/73.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
HS Reps.: In QFT no Stone – von Neumann Theorem!!!
Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]
Diff(σ) inv. states on hol. – flux algebra A unique.
wave functions
ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C
Holonomy = multiplication – operator
[A(e) ψ](A) := A(e) ψ(A)
Flux = derivative – operator
[Ej(S) ψ](A) := i~ Ej(S), ψ(A)
Scalar product
< ψ,ψ′>:=
Z
SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′
γ(h1, .., hN)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 74: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/74.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
HS Reps.: In QFT no Stone – von Neumann Theorem!!!
Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]
Diff(σ) inv. states on hol. – flux algebra A unique.
wave functions
ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C
Holonomy = multiplication – operator
[A(e) ψ](A) := A(e) ψ(A)
Flux = derivative – operator
[Ej(S) ψ](A) := i~ Ej(S), ψ(A)
Scalar product
< ψ,ψ′>:=
Z
SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′
γ(h1, .., hN)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 75: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/75.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
HS Reps.: In QFT no Stone – von Neumann Theorem!!!
Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]
Diff(σ) inv. states on hol. – flux algebra A unique.
wave functions
ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C
Holonomy = multiplication – operator
[A(e) ψ](A) := A(e) ψ(A)
Flux = derivative – operator
[Ej(S) ψ](A) := i~ Ej(S), ψ(A)
Scalar product
< ψ,ψ′>:=
Z
SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′
γ(h1, .., hN)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 76: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/76.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
HS Reps.: In QFT no Stone – von Neumann Theorem!!!
Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]
Diff(σ) inv. states on hol. – flux algebra A unique.
wave functions
ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C
Holonomy = multiplication – operator
[A(e) ψ](A) := A(e) ψ(A)
Flux = derivative – operator
[Ej(S) ψ](A) := i~ Ej(S), ψ(A)
Scalar product
< ψ,ψ′>:=
Z
SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′
γ(h1, .., hN)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 77: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/77.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
HS Reps.: In QFT no Stone – von Neumann Theorem!!!
Theorem [Ashtekar,Isham,Lewandowski 92-93], [Sahlmann 02], [L., Okolow,S.,T.T. 03-05], [Fleischhack 04]
Diff(σ) inv. states on hol. – flux algebra A unique.
wave functions
ψ(A) = ψγ(A(e1), ..,A(eN)), ψγ : SU(2)N → C
Holonomy = multiplication – operator
[A(e) ψ](A) := A(e) ψ(A)
Flux = derivative – operator
[Ej(S) ψ](A) := i~ Ej(S), ψ(A)
Scalar product
< ψ,ψ′>:=
Z
SU(2)NdµH(h1) .. dµH(hN) ψγ(h1, .., hN) ψ′
γ(h1, .., hN)
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 78: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/78.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Spin Network Basis Tγ,j,I ∼ Hj Hermite Polynomials
PSfrag replacements
v1,I1
v2,I2
v3,I3
v4,I4 v5,I5
v6,I6
v7,I7
v8,I8
v9,I9
v10,I10
vm,Im
e1,j1
e2,j2 e3,j3
e4,j4
e5,j5
e6,j6 e7,j7
e8,j8
e9,j9
e10,j10
e11,j11
e12,j12 e13,j13
e14,j14
e15,j15
e16,j16
e17,j17
e18,j18
e19,j19
e20,j20
e21,j21
e22,j22
en,jn
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 79: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/79.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Colour Coding of Spin Quantum Numbers on graph edges ...
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 80: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/80.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
... or on Faces of Dual Cell Complex (Triangulation)
Animation:
http://www.einstein-online.info/de/vertiefung/Spinnetzwerke/index.html.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 81: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/81.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical spatial geometry operators
Area – Operator [Rovelli & Smolin 94], [Ashtekar & Lewandowski 95]
Class. Area functional for 2 – mfd. S
Ar(S) =
∫
S
√Tr
([Eaεabcdxb ∧ dxc
]2)
Operator Ar(S) exists on Hgeo! Not possible on Fockspace.
Spectrum explicitly known, purely discrete. Eigenstates =spin – network – states Tγ,j,I
In LQG: spacetime distances & `P , discrete, discont.Planck – scale structure.
Similar results for length and volume operators.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 82: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/82.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical spatial geometry operators
Area – Operator [Rovelli & Smolin 94], [Ashtekar & Lewandowski 95]
Class. Area functional for 2 – mfd. S
Ar(S) =
∫
S
√Tr
([Eaεabcdxb ∧ dxc
]2)
Operator Ar(S) exists on Hgeo! Not possible on Fockspace.
Spectrum explicitly known, purely discrete. Eigenstates =spin – network – states Tγ,j,I
In LQG: spacetime distances & `P , discrete, discont.Planck – scale structure.
Similar results for length and volume operators.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 83: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/83.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical spatial geometry operators
Area – Operator [Rovelli & Smolin 94], [Ashtekar & Lewandowski 95]
Class. Area functional for 2 – mfd. S
Ar(S) =
∫
S
√Tr
([Eaεabcdxb ∧ dxc
]2)
Operator Ar(S) exists on Hgeo! Not possible on Fockspace.
Spectrum explicitly known, purely discrete. Eigenstates =spin – network – states Tγ,j,I
In LQG: spacetime distances & `P , discrete, discont.Planck – scale structure.
Similar results for length and volume operators.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 84: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/84.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical spatial geometry operators
Area – Operator [Rovelli & Smolin 94], [Ashtekar & Lewandowski 95]
Class. Area functional for 2 – mfd. S
Ar(S) =
∫
S
√Tr
([Eaεabcdxb ∧ dxc
]2)
Operator Ar(S) exists on Hgeo! Not possible on Fockspace.
Spectrum explicitly known, purely discrete. Eigenstates =spin – network – states Tγ,j,I
In LQG: spacetime distances & `P , discrete, discont.Planck – scale structure.
Similar results for length and volume operators.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 85: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/85.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical spatial geometry operators
Area – Operator [Rovelli & Smolin 94], [Ashtekar & Lewandowski 95]
Class. Area functional for 2 – mfd. S
Ar(S) =
∫
S
√Tr
([Eaεabcdxb ∧ dxc
]2)
Operator Ar(S) exists on Hgeo! Not possible on Fockspace.
Spectrum explicitly known, purely discrete. Eigenstates =spin – network – states Tγ,j,I
In LQG: spacetime distances & `P , discrete, discont.Planck – scale structure.
Similar results for length and volume operators.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 86: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/86.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
.25
.50
.75
1.00
1.25
Area Gap
A/lp2
A/lp5
7
9
11
10
8
6
1.4 1.6 1.8 2.0 2.2 2.4
ln(E(A))
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 87: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/87.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical Coherent States
Non – pert. approach, no pert. theory, rather different tool:[T.T. 00], [T.T., Winkler 00 – 02], [Varadarajan 02], [Ashtekar & Lewandowski 02]
∃ minimal uncertainty states f. A, i.e.1. F. each point (A0,E0) obtain ψ(A0,E0) s.t.
< ψA0,E0 ,A(e)ψA0,E0 >= A0(e), < ψA0,E0 ,Ej(S)ψA0,E0 >= Ej0(S)
2. and
<∆A(e)> <∆Ej(S))>=12| < [A(e), Ej(S))]> |
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 88: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/88.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical Coherent States
Non – pert. approach, no pert. theory, rather different tool:[T.T. 00], [T.T., Winkler 00 – 02], [Varadarajan 02], [Ashtekar & Lewandowski 02]
∃ minimal uncertainty states f. A, i.e.1. F. each point (A0,E0) obtain ψ(A0,E0) s.t.
< ψA0,E0 ,A(e)ψA0,E0 >= A0(e), < ψA0,E0 ,Ej(S)ψA0,E0 >= Ej0(S)
2. and
<∆A(e)> <∆Ej(S))>=12| < [A(e), Ej(S))]> |
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 89: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/89.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Kinematical Coherent States
Non – pert. approach, no pert. theory, rather different tool:[T.T. 00], [T.T., Winkler 00 – 02], [Varadarajan 02], [Ashtekar & Lewandowski 02]
∃ minimal uncertainty states f. A, i.e.1. F. each point (A0,E0) obtain ψ(A0,E0) s.t.
< ψA0,E0 ,A(e)ψA0,E0 >= A0(e), < ψA0,E0 ,Ej(S)ψA0,E0 >= Ej0(S)
2. and
<∆A(e)> <∆Ej(S))>=12| < [A(e), Ej(S))]> |
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 90: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/90.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Classical FormulationProblem of TimeQuantum KinematicsKinematical spatial geometry operatorsKinematical Coherent States
Coherent States
-0.2
-0.1
0
0.1
0.2-0.1
-0.05
0
0.05
0.1
0
0.25
0.5
0.75
1
-0.2
-0.1
0
0.1
0.2
PSfrag replacements
arcsin(=(A(e)))
E(S)/L2E(S)/L2
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 91: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/91.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Definition
For simplicity: cubic graph and dual cell complex
e
PSfrag replacements
~nSeSe
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 92: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/92.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Difference: BD & BI theories
Yang – Mills on (R4, η) [Kogut & Susskind 74]
H =~
2 g2 ε
X
v∈V(γ)
3X
a=1
Tr“
E(Sav)
2 + [A(αav) − A(αa
v)−1]2
”
Gravity on R × σ [T.T. 96 – 05, Giesel & T.T. 06]
M =~
`4P
X
v∈V(γ)
3X
j=0
˛˛˛
3X
a=1
Tr“τj A(αa
v) A(eav) [A(ea
v)−1,V1/2
v ]”˛
˛˛
2
Volume operator
Vv =q
|εabcTr`E(Sa
v) E(Sbv) E(Sc
v)´|
Lattice spacing ε disappears, replaced by `P
No continuum limit, automat. UV finite.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 93: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/93.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Difference: BD & BI theories
Yang – Mills on (R4, η) [Kogut & Susskind 74]
H =~
2 g2 ε
X
v∈V(γ)
3X
a=1
Tr“
E(Sav)
2 + [A(αav) − A(αa
v)−1]2
”
Gravity on R × σ [T.T. 96 – 05, Giesel & T.T. 06]
M =~
`4P
X
v∈V(γ)
3X
j=0
˛˛˛
3X
a=1
Tr“τj A(αa
v) A(eav) [A(ea
v)−1,V1/2
v ]”˛
˛˛
2
Volume operator
Vv =q
|εabcTr`E(Sa
v) E(Sbv) E(Sc
v)´|
Lattice spacing ε disappears, replaced by `P
No continuum limit, automat. UV finite.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 94: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/94.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Difference: BD & BI theories
Yang – Mills on (R4, η) [Kogut & Susskind 74]
H =~
2 g2 ε
X
v∈V(γ)
3X
a=1
Tr“
E(Sav)
2 + [A(αav) − A(αa
v)−1]2
”
Gravity on R × σ [T.T. 96 – 05, Giesel & T.T. 06]
M =~
`4P
X
v∈V(γ)
3X
j=0
˛˛˛
3X
a=1
Tr“τj A(αa
v) A(eav) [A(ea
v)−1,V1/2
v ]”˛
˛˛
2
Volume operator
Vv =q
|εabcTr`E(Sa
v) E(Sbv) E(Sc
v)´|
Lattice spacing ε disappears, replaced by `P
No continuum limit, automat. UV finite.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 95: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/95.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Difference: BD & BI theories
Yang – Mills on (R4, η) [Kogut & Susskind 74]
H =~
2 g2 ε
X
v∈V(γ)
3X
a=1
Tr“
E(Sav)
2 + [A(αav) − A(αa
v)−1]2
”
Gravity on R × σ [T.T. 96 – 05, Giesel & T.T. 06]
M =~
`4P
X
v∈V(γ)
3X
j=0
˛˛˛
3X
a=1
Tr“τj A(αa
v) A(eav) [A(ea
v)−1,V1/2
v ]”˛
˛˛
2
Volume operator
Vv =q
|εabcTr`E(Sa
v) E(Sbv) E(Sc
v)´|
Lattice spacing ε disappears, replaced by `P
No continuum limit, automat. UV finite.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 96: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/96.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Difference: BD & BI theories
Yang – Mills on (R4, η) [Kogut & Susskind 74]
H =~
2 g2 ε
X
v∈V(γ)
3X
a=1
Tr“
E(Sav)
2 + [A(αav) − A(αa
v)−1]2
”
Gravity on R × σ [T.T. 96 – 05, Giesel & T.T. 06]
M =~
`4P
X
v∈V(γ)
3X
j=0
˛˛˛
3X
a=1
Tr“τj A(αa
v) A(eav) [A(ea
v)−1,V1/2
v ]”˛
˛˛
2
Volume operator
Vv =q
|εabcTr`E(Sa
v) E(Sbv) E(Sc
v)´|
Lattice spacing ε disappears, replaced by `P
No continuum limit, automat. UV finite.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 97: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/97.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Solution
Physical Hilbert Space
Dirac’s basic idea MΨ = 0
General solution Ψ = δ(M)ψ, ψ ∈ Hkin in general ill defined
Must equip solution space with new inner product. Heuristically:
< Ψ,Ψ′> phys :=
< δ(M) ψ, δ(M) ψ′ >
< δ(M) ψ0, δ(M) ψ0 >=δ(0)
δ(0)
< ψ, δ(M) ψ′ >
< ψ0, δ(M) ψ0 >
< Ψ,Ψ′> phys = c < ψ, δ(M) ψ′
>
Rigorously: Spectral resolution of s.a. M [Dittrich, T.T. 04]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 98: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/98.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Solution
Physical Hilbert Space
Dirac’s basic idea MΨ = 0
General solution Ψ = δ(M)ψ, ψ ∈ Hkin in general ill defined
Must equip solution space with new inner product. Heuristically:
< Ψ,Ψ′> phys :=
< δ(M) ψ, δ(M) ψ′ >
< δ(M) ψ0, δ(M) ψ0 >=δ(0)
δ(0)
< ψ, δ(M) ψ′ >
< ψ0, δ(M) ψ0 >
< Ψ,Ψ′> phys = c < ψ, δ(M) ψ′
>
Rigorously: Spectral resolution of s.a. M [Dittrich, T.T. 04]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 99: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/99.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Solution
Physical Hilbert Space
Dirac’s basic idea MΨ = 0
General solution Ψ = δ(M)ψ, ψ ∈ Hkin in general ill defined
Must equip solution space with new inner product. Heuristically:
< Ψ,Ψ′> phys :=
< δ(M) ψ, δ(M) ψ′ >
< δ(M) ψ0, δ(M) ψ0 >=δ(0)
δ(0)
< ψ, δ(M) ψ′ >
< ψ0, δ(M) ψ0 >
< Ψ,Ψ′> phys = c < ψ, δ(M) ψ′
>
Rigorously: Spectral resolution of s.a. M [Dittrich, T.T. 04]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 100: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/100.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Solution
Physical Hilbert Space
Dirac’s basic idea MΨ = 0
General solution Ψ = δ(M)ψ, ψ ∈ Hkin in general ill defined
Must equip solution space with new inner product. Heuristically:
< Ψ,Ψ′> phys :=
< δ(M) ψ, δ(M) ψ′ >
< δ(M) ψ0, δ(M) ψ0 >=δ(0)
δ(0)
< ψ, δ(M) ψ′ >
< ψ0, δ(M) ψ0 >
< Ψ,Ψ′> phys = c < ψ, δ(M) ψ′
>
Rigorously: Spectral resolution of s.a. M [Dittrich, T.T. 04]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 101: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/101.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Semiclassical Limit
Semiclass. Limit of Q. Dynamics testable only with kinemat. coh. states
Theorem [Giesel & T.T. 06] For any (A0,E0)
1. Exp. Value < ψA0,E0 ,bMψA0,E0 >= M(A0,E0) + O(~)
2. Fluctuation< ψA0,E0 ,bM2ψA0,E0 > − < ψA0,E0 ,
bMψA0,E0 >2= O(~)
Corollary
i. Quantum Master Constraint correctly implemented
ii. If M(A0,E0) = 0 obtain approx. phys. states
< ψA0,E0 ,bM ψA0,E0 >≈< ψA0,E0 ,
bM2ψA0,E0 >≈ 0
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 102: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/102.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Semiclassical Limit
Semiclass. Limit of Q. Dynamics testable only with kinemat. coh. states
Theorem [Giesel & T.T. 06] For any (A0,E0)
1. Exp. Value < ψA0,E0 ,bMψA0,E0 >= M(A0,E0) + O(~)
2. Fluctuation< ψA0,E0 ,bM2ψA0,E0 > − < ψA0,E0 ,
bMψA0,E0 >2= O(~)
Corollary
i. Quantum Master Constraint correctly implemented
ii. If M(A0,E0) = 0 obtain approx. phys. states
< ψA0,E0 ,bM ψA0,E0 >≈< ψA0,E0 ,
bM2ψA0,E0 >≈ 0
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 103: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/103.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Semiclassical Limit
Semiclass. Limit of Q. Dynamics testable only with kinemat. coh. states
Theorem [Giesel & T.T. 06] For any (A0,E0)
1. Exp. Value < ψA0,E0 ,bMψA0,E0 >= M(A0,E0) + O(~)
2. Fluctuation< ψA0,E0 ,bM2ψA0,E0 > − < ψA0,E0 ,
bMψA0,E0 >2= O(~)
Corollary
i. Quantum Master Constraint correctly implemented
ii. If M(A0,E0) = 0 obtain approx. phys. states
< ψA0,E0 ,bM ψA0,E0 >≈< ψA0,E0 ,
bM2ψA0,E0 >≈ 0
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 104: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/104.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Semiclassical Limit
Semiclass. Limit of Q. Dynamics testable only with kinemat. coh. states
Theorem [Giesel & T.T. 06] For any (A0,E0)
1. Exp. Value < ψA0,E0 ,bMψA0,E0 >= M(A0,E0) + O(~)
2. Fluctuation< ψA0,E0 ,bM2ψA0,E0 > − < ψA0,E0 ,
bMψA0,E0 >2= O(~)
Corollary
i. Quantum Master Constraint correctly implemented
ii. If M(A0,E0) = 0 obtain approx. phys. states
< ψA0,E0 ,bM ψA0,E0 >≈< ψA0,E0 ,
bM2ψA0,E0 >≈ 0
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 105: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/105.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Master Constraint: Semiclassical Limit
Semiclass. Limit of Q. Dynamics testable only with kinemat. coh. states
Theorem [Giesel & T.T. 06] For any (A0,E0)
1. Exp. Value < ψA0,E0 ,bMψA0,E0 >= M(A0,E0) + O(~)
2. Fluctuation< ψA0,E0 ,bM2ψA0,E0 > − < ψA0,E0 ,
bMψA0,E0 >2= O(~)
Corollary
i. Quantum Master Constraint correctly implemented
ii. If M(A0,E0) = 0 obtain approx. phys. states
< ψA0,E0 ,bM ψA0,E0 >≈< ψA0,E0 ,
bM2ψA0,E0 >≈ 0
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 106: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/106.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 107: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/107.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 108: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/108.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 109: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/109.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 110: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/110.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 111: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/111.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 112: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/112.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 113: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/113.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
What if 0 6∈ spec(M)? Anomalies in Ham. Constr.?
How to fix q’ion ambiguities in definition of M?
So far only existence of Hphys if 0 ∈ spec(M). Can we get more controlfor explicit calculations or a systematic approximation scheme?
Connection BI QFT with QFT on CST [Brunetti, Fredenhagen, Verch 02]?
Gravitons, Feynman graphs [Varadarajan 01, Rovelli 06]
Renormalisation group [Markopoulou 02], [Oeckl 04], [Manrique, Oeckl, Weber, Zapata 07]
and effective theories [Niedermaier, Reuter et al 03-]?
So far only classical Dirac observables. Q’ion on Hphys [Sahlmann 06], [T.T. 04]?What are good clocks [Kuchar 90’s-]?
Infinitesimal quantum dynamics wrt M sufficient but how about physicalHamiltonian? Physical coherent states (preserved by H evolution)[Bojowald, Skirzewski 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 114: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/114.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Spin Foam Models
Heuristic starting point [Reisenberger & Rovelli 96]
Instead of using single M use ∞ number of Ham. Constraints C(x)
Formal solutions Ψ= δ[C] ψ :=Q
x δ(C(x)) ψ
Formal physical inner product (functional integral)
< Ψ,Ψ′> phys =< ψ, δ[C] ψ′
>=
Z[dN] < ψ, exp(iC(N)) ψ′
>kin
Explicit action of C(N) [T.T. 96-98] suggests spin foam picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 115: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/115.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Spin Foam Models
Heuristic starting point [Reisenberger & Rovelli 96]
Instead of using single M use ∞ number of Ham. Constraints C(x)
Formal solutions Ψ= δ[C] ψ :=Q
x δ(C(x)) ψ
Formal physical inner product (functional integral)
< Ψ,Ψ′> phys =< ψ, δ[C] ψ′
>=
Z[dN] < ψ, exp(iC(N)) ψ′
>kin
Explicit action of C(N) [T.T. 96-98] suggests spin foam picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 116: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/116.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Spin Foam Models
Heuristic starting point [Reisenberger & Rovelli 96]
Instead of using single M use ∞ number of Ham. Constraints C(x)
Formal solutions Ψ= δ[C] ψ :=Q
x δ(C(x)) ψ
Formal physical inner product (functional integral)
< Ψ,Ψ′> phys =< ψ, δ[C] ψ′
>=
Z[dN] < ψ, exp(iC(N)) ψ′
>kin
Explicit action of C(N) [T.T. 96-98] suggests spin foam picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 117: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/117.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Spin Foam Models
Heuristic starting point [Reisenberger & Rovelli 96]
Instead of using single M use ∞ number of Ham. Constraints C(x)
Formal solutions Ψ= δ[C] ψ :=Q
x δ(C(x)) ψ
Formal physical inner product (functional integral)
< Ψ,Ψ′> phys =< ψ, δ[C] ψ′
>=
Z[dN] < ψ, exp(iC(N)) ψ′
>kin
Explicit action of C(N) [T.T. 96-98] suggests spin foam picture
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 118: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/118.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
PSfrag replacementse1
e1
e2
e2
e3
e3
Σt
Σt+δt
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 119: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/119.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Spin Foam Models
Heuristic starting point [Reisenberger & Rovelli 96]
Instead of using single M use ∞ number of Ham. Constraints C(x)
Formal solutions Ψ= δ[C] ψ :=Q
x δ(C(x)) ψ
Formal physical inner product (functional integral)
< Ψ,Ψ′> phys =< ψ, δ[C] ψ′
>=
Z[dN] < ψ, exp(iC(N)) ψ′
>kin
Explicit action of C(N) [T.T. 96-98] suggests spin foam picture
Problem: The C(x) are mutually not commuting, henceC(x0)
Qx δ(C(x)) 6= 0. Moreover, C(x) not s.a.
Ways out?: MCP and completely independent approach to spin foams
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 120: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/120.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Spin Foam Models
Heuristic starting point [Reisenberger & Rovelli 96]
Instead of using single M use ∞ number of Ham. Constraints C(x)
Formal solutions Ψ= δ[C] ψ :=Q
x δ(C(x)) ψ
Formal physical inner product (functional integral)
< Ψ,Ψ′> phys =< ψ, δ[C] ψ′
>=
Z[dN] < ψ, exp(iC(N)) ψ′
>kin
Explicit action of C(N) [T.T. 96-98] suggests spin foam picture
Problem: The C(x) are mutually not commuting, henceC(x0)
Qx δ(C(x)) 6= 0. Moreover, C(x) not s.a.
Ways out?: MCP and completely independent approach to spin foams
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 121: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/121.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 122: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/122.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 123: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/123.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 124: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/124.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 125: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/125.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 126: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/126.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 127: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/127.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 128: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/128.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea: [Baez, Barrett, Bouffenoir, Crane, Freidel, Girelli, Henneaux, Krasnov, Livine, Noui, Oeckl, Oriti, Perez,
Pfeiffer, Reisenberger, Roche, Rovelli, Starodubtsev, Speziale,.., Zapata 96 – ]
Palatini:R
F ∧ ∗(e ∧ e) = BF:R
B ∧ F
plus simplicity constr.: C(B) = 0 ⇒ B = ±e ∧ e, ± ∗ (e ∧ e)
Palatini PI = perturbation theory of BF – TQFT
Z =
Z[dω de] eiSPalatini =
Z[dω dB dΦ] ei[SBF+
R
Φ C(B)]
Regularisation of PI: Triangulation τ 7→ Zτ
Triangles ∆ ∈ τ ↔ B∆, Edges e ∈ τ∗ ↔ g(e)
Integrate out Φ, B; use Peter & WeylR
G dµH(g) ↔P
π
Z :=Στ wτ Zτ : Group Field Theory [Boulatov,Ooguri 90’s]
Z[dµH(g)] ei[S0[g]+λS1 [g]] =
∞X
n=0
λn
X
n=|π1(τ)|
Zτ
Proposal [Freidel 05]: Physical inner product: n=0 contribution
All of this works beautifully in 3D, since TQFT
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 129: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/129.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 130: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/130.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 131: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/131.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 132: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/132.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 133: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/133.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 134: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/134.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 135: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/135.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS in 4D:
Current (Barrett – Crane) models peaked on degenerate histories [Baez,
Christensen, Willis 03-07]
2nd class simplicity constraints not correctly implemented into measure[Bouffeneoir, Henneaux, Noui, Roche 04]
quantum version of simplicity constraints anomalous[Engle, Pereira, Rovelli 07], [Livine, Speziale 07]
physical significance of the four sectors?
Convergence of GFT perturbation series?
Posivity of GFT scalar product (tree diagrammes) [Pfeiffer 06]?
what is the precise connection between canonical (MCP) and covariant(spin foam) formulation of LQG anyway?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 136: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/136.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Quantum Black Hole Physics
Careful analysis: isolated horizons[Ashtekar,Baez,Corichi,Dreyer,Engle,Fairhurst,Krasnov, Krishnan,Lewandowski,Pawlowski,Willis 97–]
Isolated horizon = local generalisation of event horizon
Precise definition leads to σ with interior H∼= S2 boundary andmodifications of classical canonical analysis
Most important consequences for us:
i. Interior bdry conditions make Ar(H) a Dirac observableii. The gauge invariant surface information consists of ordered
N – tuples of intersection points (p1, .., pN), N = 0, 1, 2, ....up to diffeomorphisms together with the degeneracies ofcorresponding (total) spin configurations (j1, .., jN) onintersecting edges
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 137: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/137.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Quantum Black Hole Physics
Careful analysis: isolated horizons[Ashtekar,Baez,Corichi,Dreyer,Engle,Fairhurst,Krasnov, Krishnan,Lewandowski,Pawlowski,Willis 97–]
Isolated horizon = local generalisation of event horizon
Precise definition leads to σ with interior H∼= S2 boundary andmodifications of classical canonical analysis
Most important consequences for us:
i. Interior bdry conditions make Ar(H) a Dirac observableii. The gauge invariant surface information consists of ordered
N – tuples of intersection points (p1, .., pN), N = 0, 1, 2, ....up to diffeomorphisms together with the degeneracies ofcorresponding (total) spin configurations (j1, .., jN) onintersecting edges
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 138: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/138.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Quantum Black Hole Physics
Careful analysis: isolated horizons[Ashtekar,Baez,Corichi,Dreyer,Engle,Fairhurst,Krasnov, Krishnan,Lewandowski,Pawlowski,Willis 97–]
Isolated horizon = local generalisation of event horizon
Precise definition leads to σ with interior H∼= S2 boundary andmodifications of classical canonical analysis
Most important consequences for us:
i. Interior bdry conditions make Ar(H) a Dirac observableii. The gauge invariant surface information consists of ordered
N – tuples of intersection points (p1, .., pN), N = 0, 1, 2, ....up to diffeomorphisms together with the degeneracies ofcorresponding (total) spin configurations (j1, .., jN) onintersecting edges
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 139: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/139.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Quantum Black Hole Physics
Careful analysis: isolated horizons[Ashtekar,Baez,Corichi,Dreyer,Engle,Fairhurst,Krasnov, Krishnan,Lewandowski,Pawlowski,Willis 97–]
Isolated horizon = local generalisation of event horizon
Precise definition leads to σ with interior H∼= S2 boundary andmodifications of classical canonical analysis
Most important consequences for us:
i. Interior bdry conditions make Ar(H) a Dirac observableii. The gauge invariant surface information consists of ordered
N – tuples of intersection points (p1, .., pN), N = 0, 1, 2, ....up to diffeomorphisms together with the degeneracies ofcorresponding (total) spin configurations (j1, .., jN) onintersecting edges
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 140: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/140.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Quantum Black Hole Physics
Careful analysis: isolated horizons[Ashtekar,Baez,Corichi,Dreyer,Engle,Fairhurst,Krasnov, Krishnan,Lewandowski,Pawlowski,Willis 97–]
Isolated horizon = local generalisation of event horizon
Precise definition leads to σ with interior H∼= S2 boundary andmodifications of classical canonical analysis
Most important consequences for us:
i. Interior bdry conditions make Ar(H) a Dirac observableii. The gauge invariant surface information consists of ordered
N – tuples of intersection points (p1, .., pN), N = 0, 1, 2, ....up to diffeomorphisms together with the degeneracies ofcorresponding (total) spin configurations (j1, .., jN) onintersecting edges
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 141: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/141.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
PSfrag replacementsp1
p2
pN
j1
j2
jN
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 142: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/142.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Entropy – Calculation[Krasnov 95], [Ashtekar,Baez,Krasnov 97], [Meissner 04], [Domagala,Lewandowski 04], [Ghosh,Mitra 06]
Given Ar0, count number N(Ar(H)) of SNW Eigenstates Tλ of Ar(H)with EV λ ∈ [Ar0 − `2
P,Ar0 + `2P].
spec( bAr(H)) = β`2P
PNk=1
pjk(jk + 1)
In microcanon. ensemble we find
S(Ar0) = ln(N(Ar0)) =Ar0
4~G+ O(ln(
Ar0
`2P
))
provided that Immirzi β = β0
Microscopical explanation: polymer – like excitations of geometry definesurface.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 143: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/143.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Entropy – Calculation[Krasnov 95], [Ashtekar,Baez,Krasnov 97], [Meissner 04], [Domagala,Lewandowski 04], [Ghosh,Mitra 06]
Given Ar0, count number N(Ar(H)) of SNW Eigenstates Tλ of Ar(H)with EV λ ∈ [Ar0 − `2
P,Ar0 + `2P].
spec( bAr(H)) = β`2P
PNk=1
pjk(jk + 1)
In microcanon. ensemble we find
S(Ar0) = ln(N(Ar0)) =Ar0
4~G+ O(ln(
Ar0
`2P
))
provided that Immirzi β = β0
Microscopical explanation: polymer – like excitations of geometry definesurface.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 144: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/144.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Entropy – Calculation[Krasnov 95], [Ashtekar,Baez,Krasnov 97], [Meissner 04], [Domagala,Lewandowski 04], [Ghosh,Mitra 06]
Given Ar0, count number N(Ar(H)) of SNW Eigenstates Tλ of Ar(H)with EV λ ∈ [Ar0 − `2
P,Ar0 + `2P].
spec( bAr(H)) = β`2P
PNk=1
pjk(jk + 1)
In microcanon. ensemble we find
S(Ar0) = ln(N(Ar0)) =Ar0
4~G+ O(ln(
Ar0
`2P
))
provided that Immirzi β = β0
Microscopical explanation: polymer – like excitations of geometry definesurface.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 145: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/145.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Entropy – Calculation[Krasnov 95], [Ashtekar,Baez,Krasnov 97], [Meissner 04], [Domagala,Lewandowski 04], [Ghosh,Mitra 06]
Given Ar0, count number N(Ar(H)) of SNW Eigenstates Tλ of Ar(H)with EV λ ∈ [Ar0 − `2
P,Ar0 + `2P].
spec( bAr(H)) = β`2P
PNk=1
pjk(jk + 1)
In microcanon. ensemble we find
S(Ar0) = ln(N(Ar0)) =Ar0
4~G+ O(ln(
Ar0
`2P
))
provided that Immirzi β = β0
Microscopical explanation: polymer – like excitations of geometry definesurface.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 146: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/146.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Entropy – Calculation[Krasnov 95], [Ashtekar,Baez,Krasnov 97], [Meissner 04], [Domagala,Lewandowski 04], [Ghosh,Mitra 06]
Given Ar0, count number N(Ar(H)) of SNW Eigenstates Tλ of Ar(H)with EV λ ∈ [Ar0 − `2
P,Ar0 + `2P].
spec( bAr(H)) = β`2P
PNk=1
pjk(jk + 1)
In microcanon. ensemble we find
S(Ar0) = ln(N(Ar0)) =Ar0
4~G+ O(ln(
Ar0
`2P
))
provided that Immirzi β = β0
Microscopical explanation: polymer – like excitations of geometry definesurface.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 147: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/147.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Warning: All spins contribute, no bit picture!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 148: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/148.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Secret assumption: For each boundary configuration ∃ at leastone matching bulk solution of M=0. Otherwise S decreases andchanges β0! Can one check this?
Analysis has significant classical input (semiclassical quantumblack hole). Can one define purely quantum trapping conditions[Dasgupta 05], [Husain,Winkler 05]?
Hawking effect from first principles? Geometry transmutationsgenerate matter QFT modes [Krasnov 00]?
Is there renormalisation of G in LQG and can this be absorbedinto β [Jacobson 04]?
Quasinormal mode puzzle [Dreyer 02]
Entropy range q’ion for specific [Ar0 − δ`2P,Ar0 + δ`2
P] [Corichi et al 06]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 149: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/149.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Secret assumption: For each boundary configuration ∃ at leastone matching bulk solution of M=0. Otherwise S decreases andchanges β0! Can one check this?
Analysis has significant classical input (semiclassical quantumblack hole). Can one define purely quantum trapping conditions[Dasgupta 05], [Husain,Winkler 05]?
Hawking effect from first principles? Geometry transmutationsgenerate matter QFT modes [Krasnov 00]?
Is there renormalisation of G in LQG and can this be absorbedinto β [Jacobson 04]?
Quasinormal mode puzzle [Dreyer 02]
Entropy range q’ion for specific [Ar0 − δ`2P,Ar0 + δ`2
P] [Corichi et al 06]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 150: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/150.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Secret assumption: For each boundary configuration ∃ at leastone matching bulk solution of M=0. Otherwise S decreases andchanges β0! Can one check this?
Analysis has significant classical input (semiclassical quantumblack hole). Can one define purely quantum trapping conditions[Dasgupta 05], [Husain,Winkler 05]?
Hawking effect from first principles? Geometry transmutationsgenerate matter QFT modes [Krasnov 00]?
Is there renormalisation of G in LQG and can this be absorbedinto β [Jacobson 04]?
Quasinormal mode puzzle [Dreyer 02]
Entropy range q’ion for specific [Ar0 − δ`2P,Ar0 + δ`2
P] [Corichi et al 06]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 151: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/151.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Secret assumption: For each boundary configuration ∃ at leastone matching bulk solution of M=0. Otherwise S decreases andchanges β0! Can one check this?
Analysis has significant classical input (semiclassical quantumblack hole). Can one define purely quantum trapping conditions[Dasgupta 05], [Husain,Winkler 05]?
Hawking effect from first principles? Geometry transmutationsgenerate matter QFT modes [Krasnov 00]?
Is there renormalisation of G in LQG and can this be absorbedinto β [Jacobson 04]?
Quasinormal mode puzzle [Dreyer 02]
Entropy range q’ion for specific [Ar0 − δ`2P,Ar0 + δ`2
P] [Corichi et al 06]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 152: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/152.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Secret assumption: For each boundary configuration ∃ at leastone matching bulk solution of M=0. Otherwise S decreases andchanges β0! Can one check this?
Analysis has significant classical input (semiclassical quantumblack hole). Can one define purely quantum trapping conditions[Dasgupta 05], [Husain,Winkler 05]?
Hawking effect from first principles? Geometry transmutationsgenerate matter QFT modes [Krasnov 00]?
Is there renormalisation of G in LQG and can this be absorbedinto β [Jacobson 04]?
Quasinormal mode puzzle [Dreyer 02]
Entropy range q’ion for specific [Ar0 − δ`2P,Ar0 + δ`2
P] [Corichi et al 06]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 153: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/153.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Secret assumption: For each boundary configuration ∃ at leastone matching bulk solution of M=0. Otherwise S decreases andchanges β0! Can one check this?
Analysis has significant classical input (semiclassical quantumblack hole). Can one define purely quantum trapping conditions[Dasgupta 05], [Husain,Winkler 05]?
Hawking effect from first principles? Geometry transmutationsgenerate matter QFT modes [Krasnov 00]?
Is there renormalisation of G in LQG and can this be absorbedinto β [Jacobson 04]?
Quasinormal mode puzzle [Dreyer 02]
Entropy range q’ion for specific [Ar0 − δ`2P,Ar0 + δ`2
P] [Corichi et al 06]
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 154: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/154.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Singularity avoidance
Singularities: Divergence of Riemann Tensor R(4)µνρσ
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 155: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/155.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea:
Gauss – Codacci∫
Md4X
√| det(g)| R(4) =
∫
Rdt
∫
σ
d3x NTr
(FabEaEb
)√| det(E)|
=:
∫
Rdt R(N)
Consider coh. st. on H concentrated on class. sing. trajectoryt 7→ (A0(t),E0(t)).
Calculate< ψA0(t),E0(t), R(N) ψA0(t),E0(t) >
Result [Brunnemann, T.T. 05]:
1. Curvature operator not bounded2. However: Expectation value finite for bh and cosmological
singularities of order of . `−2P ≈ 1066cm−2.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 156: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/156.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea:
Gauss – Codacci∫
Md4X
√| det(g)| R(4) =
∫
Rdt
∫
σ
d3x NTr
(FabEaEb
)√| det(E)|
=:
∫
Rdt R(N)
Consider coh. st. on H concentrated on class. sing. trajectoryt 7→ (A0(t),E0(t)).
Calculate< ψA0(t),E0(t), R(N) ψA0(t),E0(t) >
Result [Brunnemann, T.T. 05]:
1. Curvature operator not bounded2. However: Expectation value finite for bh and cosmological
singularities of order of . `−2P ≈ 1066cm−2.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 157: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/157.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea:
Gauss – Codacci∫
Md4X
√| det(g)| R(4) =
∫
Rdt
∫
σ
d3x NTr
(FabEaEb
)√| det(E)|
=:
∫
Rdt R(N)
Consider coh. st. on H concentrated on class. sing. trajectoryt 7→ (A0(t),E0(t)).
Calculate< ψA0(t),E0(t), R(N) ψA0(t),E0(t) >
Result [Brunnemann, T.T. 05]:
1. Curvature operator not bounded2. However: Expectation value finite for bh and cosmological
singularities of order of . `−2P ≈ 1066cm−2.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 158: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/158.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea:
Gauss – Codacci∫
Md4X
√| det(g)| R(4) =
∫
Rdt
∫
σ
d3x NTr
(FabEaEb
)√| det(E)|
=:
∫
Rdt R(N)
Consider coh. st. on H concentrated on class. sing. trajectoryt 7→ (A0(t),E0(t)).
Calculate< ψA0(t),E0(t), R(N) ψA0(t),E0(t) >
Result [Brunnemann, T.T. 05]:
1. Curvature operator not bounded2. However: Expectation value finite for bh and cosmological
singularities of order of . `−2P ≈ 1066cm−2.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 159: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/159.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
Idea:
Gauss – Codacci∫
Md4X
√| det(g)| R(4) =
∫
Rdt
∫
σ
d3x NTr
(FabEaEb
)√| det(E)|
=:
∫
Rdt R(N)
Consider coh. st. on H concentrated on class. sing. trajectoryt 7→ (A0(t),E0(t)).
Calculate< ψA0(t),E0(t), R(N) ψA0(t),E0(t) >
Result [Brunnemann, T.T. 05]:
1. Curvature operator not bounded2. However: Expectation value finite for bh and cosmological
singularities of order of . `−2P ≈ 1066cm−2.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 160: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/160.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
However, what one really should be doing:
Compute physical HS and physical coherent states Ψ
Choose physical clock and physical Hamiltonian H
Monitor expectation value of Dirac observables
τ 7→ < Ψ,Of(τ)Ψ > phys = < Ψ,eiτH Of(0) e−iτHΨ > phys
for classically singular spatially diffeo inv. f
All this is possible in certain truncated models of LQG[Bojowald 00 -, Bojowald et al 01-], [Ashtekar, Pawlowski, Singh 06-]
LQC = (cosmological) mini/midisuperspace models q’isedwith LQG methods
Spectacular results, e.g. physical propagation throughsingularity, pre – big bang!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 161: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/161.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
However, what one really should be doing:
Compute physical HS and physical coherent states Ψ
Choose physical clock and physical Hamiltonian H
Monitor expectation value of Dirac observables
τ 7→ < Ψ,Of(τ)Ψ > phys = < Ψ,eiτH Of(0) e−iτHΨ > phys
for classically singular spatially diffeo inv. f
All this is possible in certain truncated models of LQG[Bojowald 00 -, Bojowald et al 01-], [Ashtekar, Pawlowski, Singh 06-]
LQC = (cosmological) mini/midisuperspace models q’isedwith LQG methods
Spectacular results, e.g. physical propagation throughsingularity, pre – big bang!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 162: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/162.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
However, what one really should be doing:
Compute physical HS and physical coherent states Ψ
Choose physical clock and physical Hamiltonian H
Monitor expectation value of Dirac observables
τ 7→ < Ψ,Of(τ)Ψ > phys = < Ψ,eiτH Of(0) e−iτHΨ > phys
for classically singular spatially diffeo inv. f
All this is possible in certain truncated models of LQG[Bojowald 00 -, Bojowald et al 01-], [Ashtekar, Pawlowski, Singh 06-]
LQC = (cosmological) mini/midisuperspace models q’isedwith LQG methods
Spectacular results, e.g. physical propagation throughsingularity, pre – big bang!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 163: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/163.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
However, what one really should be doing:
Compute physical HS and physical coherent states Ψ
Choose physical clock and physical Hamiltonian H
Monitor expectation value of Dirac observables
τ 7→ < Ψ,Of(τ)Ψ > phys = < Ψ,eiτH Of(0) e−iτHΨ > phys
for classically singular spatially diffeo inv. f
All this is possible in certain truncated models of LQG[Bojowald 00 -, Bojowald et al 01-], [Ashtekar, Pawlowski, Singh 06-]
LQC = (cosmological) mini/midisuperspace models q’isedwith LQG methods
Spectacular results, e.g. physical propagation throughsingularity, pre – big bang!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 164: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/164.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
However, what one really should be doing:
Compute physical HS and physical coherent states Ψ
Choose physical clock and physical Hamiltonian H
Monitor expectation value of Dirac observables
τ 7→ < Ψ,Of(τ)Ψ > phys = < Ψ,eiτH Of(0) e−iτHΨ > phys
for classically singular spatially diffeo inv. f
All this is possible in certain truncated models of LQG[Bojowald 00 -, Bojowald et al 01-], [Ashtekar, Pawlowski, Singh 06-]
LQC = (cosmological) mini/midisuperspace models q’isedwith LQG methods
Spectacular results, e.g. physical propagation throughsingularity, pre – big bang!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 165: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/165.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
However, what one really should be doing:
Compute physical HS and physical coherent states Ψ
Choose physical clock and physical Hamiltonian H
Monitor expectation value of Dirac observables
τ 7→ < Ψ,Of(τ)Ψ > phys = < Ψ,eiτH Of(0) e−iτHΨ > phys
for classically singular spatially diffeo inv. f
All this is possible in certain truncated models of LQG[Bojowald 00 -, Bojowald et al 01-], [Ashtekar, Pawlowski, Singh 06-]
LQC = (cosmological) mini/midisuperspace models q’isedwith LQG methods
Spectacular results, e.g. physical propagation throughsingularity, pre – big bang!
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 166: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/166.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Can we repeat this in the full theory?
Are kinematical calculations good approximations to calculationsin the physical HS [Ashtekar,Bombelli,Corichi 06]?
What are appropriate clocks and physical Hamiltonians?[Dittrich, Tambornino 07]
How to obtain gauge invariant (physical) coherent states?[Bahr, T.T. 07]
Can we embed LQC as a sector (invariant subspace) into thephysical LQG Hilbert space [Bojowald 00], [Engle 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 167: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/167.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Can we repeat this in the full theory?
Are kinematical calculations good approximations to calculationsin the physical HS [Ashtekar,Bombelli,Corichi 06]?
What are appropriate clocks and physical Hamiltonians?[Dittrich, Tambornino 07]
How to obtain gauge invariant (physical) coherent states?[Bahr, T.T. 07]
Can we embed LQC as a sector (invariant subspace) into thephysical LQG Hilbert space [Bojowald 00], [Engle 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 168: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/168.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Can we repeat this in the full theory?
Are kinematical calculations good approximations to calculationsin the physical HS [Ashtekar,Bombelli,Corichi 06]?
What are appropriate clocks and physical Hamiltonians?[Dittrich, Tambornino 07]
How to obtain gauge invariant (physical) coherent states?[Bahr, T.T. 07]
Can we embed LQC as a sector (invariant subspace) into thephysical LQG Hilbert space [Bojowald 00], [Engle 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 169: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/169.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Can we repeat this in the full theory?
Are kinematical calculations good approximations to calculationsin the physical HS [Ashtekar,Bombelli,Corichi 06]?
What are appropriate clocks and physical Hamiltonians?[Dittrich, Tambornino 07]
How to obtain gauge invariant (physical) coherent states?[Bahr, T.T. 07]
Can we embed LQC as a sector (invariant subspace) into thephysical LQG Hilbert space [Bojowald 00], [Engle 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 170: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/170.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Master Constraint ProgrammeSpin Foam ModelsQuantum Black Hole PhysicsSingularity avoidance
OPEN PROBLEMS:
Can we repeat this in the full theory?
Are kinematical calculations good approximations to calculationsin the physical HS [Ashtekar,Bombelli,Corichi 06]?
What are appropriate clocks and physical Hamiltonians?[Dittrich, Tambornino 07]
How to obtain gauge invariant (physical) coherent states?[Bahr, T.T. 07]
Can we embed LQC as a sector (invariant subspace) into thephysical LQG Hilbert space [Bojowald 00], [Engle 06]?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 171: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/171.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Miscellaneous Open Questions
Explanation of the tiny Λ through quantum geometry [Sahlmann, T.T. 02]?(Dynamical) cosmological constant operator?
Connection with experiment? QG Phenomenology [Morales-Tecotl 03-]? E.g.quantum cosmological perturbation theory [Brandenberger, Feldmann, Mukhanov 92]
(important for WMAP, PLANCK) from first principles [Hofmann, Winkler 04],
[Dittrich,Tambornino 06]?DSR type of non comm. matter QFT from integrating out geometry dof[Baratin, Freidel 06] in 3D also in 4D quantum gravity [Hossenfelder 04-]?
Does LQG allow for a background independendent formulation of stringtheory [Baez, Perez 06], [Fairbairn, Perez 07]?
How does LQG relate to other formulations: causal sets [Dowker, Sorkin 90’s -]
or CDT [Ambjorn, Jurkiewicz, Loll 00-] or consistently discretised QG[Gambini, Pullin 03-]?
AIL Rep. highly discontinuous (e.g. infinitesimal diffeos 6 ∃). Are theredifferent A leading to reps. with increased continuity properties, hencedecreased q’ion ambiguities?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 172: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/172.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Miscellaneous Open Questions
Explanation of the tiny Λ through quantum geometry [Sahlmann, T.T. 02]?(Dynamical) cosmological constant operator?
Connection with experiment? QG Phenomenology [Morales-Tecotl 03-]? E.g.quantum cosmological perturbation theory [Brandenberger, Feldmann, Mukhanov 92]
(important for WMAP, PLANCK) from first principles [Hofmann, Winkler 04],
[Dittrich,Tambornino 06]?DSR type of non comm. matter QFT from integrating out geometry dof[Baratin, Freidel 06] in 3D also in 4D quantum gravity [Hossenfelder 04-]?
Does LQG allow for a background independendent formulation of stringtheory [Baez, Perez 06], [Fairbairn, Perez 07]?
How does LQG relate to other formulations: causal sets [Dowker, Sorkin 90’s -]
or CDT [Ambjorn, Jurkiewicz, Loll 00-] or consistently discretised QG[Gambini, Pullin 03-]?
AIL Rep. highly discontinuous (e.g. infinitesimal diffeos 6 ∃). Are theredifferent A leading to reps. with increased continuity properties, hencedecreased q’ion ambiguities?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 173: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/173.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Miscellaneous Open Questions
Explanation of the tiny Λ through quantum geometry [Sahlmann, T.T. 02]?(Dynamical) cosmological constant operator?
Connection with experiment? QG Phenomenology [Morales-Tecotl 03-]? E.g.quantum cosmological perturbation theory [Brandenberger, Feldmann, Mukhanov 92]
(important for WMAP, PLANCK) from first principles [Hofmann, Winkler 04],
[Dittrich,Tambornino 06]?DSR type of non comm. matter QFT from integrating out geometry dof[Baratin, Freidel 06] in 3D also in 4D quantum gravity [Hossenfelder 04-]?
Does LQG allow for a background independendent formulation of stringtheory [Baez, Perez 06], [Fairbairn, Perez 07]?
How does LQG relate to other formulations: causal sets [Dowker, Sorkin 90’s -]
or CDT [Ambjorn, Jurkiewicz, Loll 00-] or consistently discretised QG[Gambini, Pullin 03-]?
AIL Rep. highly discontinuous (e.g. infinitesimal diffeos 6 ∃). Are theredifferent A leading to reps. with increased continuity properties, hencedecreased q’ion ambiguities?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 174: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/174.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Miscellaneous Open Questions
Explanation of the tiny Λ through quantum geometry [Sahlmann, T.T. 02]?(Dynamical) cosmological constant operator?
Connection with experiment? QG Phenomenology [Morales-Tecotl 03-]? E.g.quantum cosmological perturbation theory [Brandenberger, Feldmann, Mukhanov 92]
(important for WMAP, PLANCK) from first principles [Hofmann, Winkler 04],
[Dittrich,Tambornino 06]?DSR type of non comm. matter QFT from integrating out geometry dof[Baratin, Freidel 06] in 3D also in 4D quantum gravity [Hossenfelder 04-]?
Does LQG allow for a background independendent formulation of stringtheory [Baez, Perez 06], [Fairbairn, Perez 07]?
How does LQG relate to other formulations: causal sets [Dowker, Sorkin 90’s -]
or CDT [Ambjorn, Jurkiewicz, Loll 00-] or consistently discretised QG[Gambini, Pullin 03-]?
AIL Rep. highly discontinuous (e.g. infinitesimal diffeos 6 ∃). Are theredifferent A leading to reps. with increased continuity properties, hencedecreased q’ion ambiguities?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 175: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/175.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Miscellaneous Open Questions
Explanation of the tiny Λ through quantum geometry [Sahlmann, T.T. 02]?(Dynamical) cosmological constant operator?
Connection with experiment? QG Phenomenology [Morales-Tecotl 03-]? E.g.quantum cosmological perturbation theory [Brandenberger, Feldmann, Mukhanov 92]
(important for WMAP, PLANCK) from first principles [Hofmann, Winkler 04],
[Dittrich,Tambornino 06]?DSR type of non comm. matter QFT from integrating out geometry dof[Baratin, Freidel 06] in 3D also in 4D quantum gravity [Hossenfelder 04-]?
Does LQG allow for a background independendent formulation of stringtheory [Baez, Perez 06], [Fairbairn, Perez 07]?
How does LQG relate to other formulations: causal sets [Dowker, Sorkin 90’s -]
or CDT [Ambjorn, Jurkiewicz, Loll 00-] or consistently discretised QG[Gambini, Pullin 03-]?
AIL Rep. highly discontinuous (e.g. infinitesimal diffeos 6 ∃). Are theredifferent A leading to reps. with increased continuity properties, hencedecreased q’ion ambiguities?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 176: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/176.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Miscellaneous Open Questions
Explanation of the tiny Λ through quantum geometry [Sahlmann, T.T. 02]?(Dynamical) cosmological constant operator?
Connection with experiment? QG Phenomenology [Morales-Tecotl 03-]? E.g.quantum cosmological perturbation theory [Brandenberger, Feldmann, Mukhanov 92]
(important for WMAP, PLANCK) from first principles [Hofmann, Winkler 04],
[Dittrich,Tambornino 06]?DSR type of non comm. matter QFT from integrating out geometry dof[Baratin, Freidel 06] in 3D also in 4D quantum gravity [Hossenfelder 04-]?
Does LQG allow for a background independendent formulation of stringtheory [Baez, Perez 06], [Fairbairn, Perez 07]?
How does LQG relate to other formulations: causal sets [Dowker, Sorkin 90’s -]
or CDT [Ambjorn, Jurkiewicz, Loll 00-] or consistently discretised QG[Gambini, Pullin 03-]?
AIL Rep. highly discontinuous (e.g. infinitesimal diffeos 6 ∃). Are theredifferent A leading to reps. with increased continuity properties, hencedecreased q’ion ambiguities?
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 177: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/177.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Summary
LQG = candidate f. quantum gravity.
Mathematically rigorous formulation.
Clear, simple conceptual setup, minimalistic.
Promising, but much must and can be done to makecontact with low energy physics.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 178: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/178.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Summary
LQG = candidate f. quantum gravity.
Mathematically rigorous formulation.
Clear, simple conceptual setup, minimalistic.
Promising, but much must and can be done to makecontact with low energy physics.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 179: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/179.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Summary
LQG = candidate f. quantum gravity.
Mathematically rigorous formulation.
Clear, simple conceptual setup, minimalistic.
Promising, but much must and can be done to makecontact with low energy physics.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 180: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/180.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
Summary
LQG = candidate f. quantum gravity.
Mathematically rigorous formulation.
Clear, simple conceptual setup, minimalistic.
Promising, but much must and can be done to makecontact with low energy physics.
Thomas Thiemann Elements of Loop Quantum Gravity
![Page 181: Loop Quantum Gravity (LQG) · Naive: Quantum gravity Cut – off at k‘ Pˇ 1 where ‘2 = ~G ? = ~ 2 Z R3 d3x ‘ 4 P Comparison with cosmological term](https://reader035.vdocuments.mx/reader035/viewer/2022070914/5fb4ff13f498bf78dc3aaabd/html5/thumbnails/181.jpg)
MotivationThe Challenge of Quantum Gravitaty
Secured LandUnknown Territory
Open Questions and Summary
Miscellaneous Open QuestionsSummary
All details can be found in (scheduled Sep. 30th 2007):
Thomas Thiemann Elements of Loop Quantum Gravity