Introduction to Loop Quantum Gravity

Home , Loop quantum cosmology, Unification (physics)

Abhay Ashtekar Institute for Gravitation and the Cosmos, Penn State

A broad perspective on the challenges, structure and successes of loop quantum gravity. Focus on conceptual issues, coherence of physical principles and viability of predictions

CUP Monographs by Carlo Rovelli (2004) and Thomas Thiemann (2008); Introduction: Gravity and the Quantum by AA, New J.Phys. 7 (2005) 198

Overview Lecture for PHY565

– p. Organization

1. Historical and Conceptual Setting 2. Loop Quantum Gravity: Quantum Geometry 3. Black Holes: Zooming in on Quantum Geometry 4. Big Bang: Loop Quantum Cosmology 5. Summary

– p. 1. Historical and Conceptual Setting

Einstein’s resistance to accept quantum mechanics as a fundamental theory is well known. However, he had a deep respect for quantum mechanics and was the ﬁrst to raise the problem of unifying general relativity with quantum theory.

“Nevertheless, due to the inner-atomic movement of electrons, atoms would have to radiate not only electro-magnetic but also gravitational energy, if only in tiny amounts. As this is hardly true in Nature, it appears that quantum theory would have to modify not only Maxwellian electrodynamics, but also the new theory of gravitation.”

(Albert Einstein, Preussische Akademie Sitzungsberichte, 1916)

– p. Physics has advanced tremendously in the last 90 years but the the problem• of uniﬁcation of general relativity and quantum physics still open. Why? ⋆ No experimental data with direct ramiﬁcations on the quantum nature of Gravity.

In general relativity, gravity is coded in space-time geometry. Most •spectacular predictions —e.g., the Big-Bang, Black Holes & Gravitational Waves— emerge from this encoding. Suggests: Geometry itself must become quantum mechanical. How do you do physics without a space-time continuum in the background?

Several approaches: Causal sets, twistors, AdS/CFT • conjecture of string theory. Loop Quantum Gravity grew out of the Hamiltonian approach pioneered by Bergmann, Dirac, and developed by Wheeler, DeWitt and others.

– p. Contrasting LQG with String theory

Because there are no direct experimental checks, approaches are driven by intellectual prejudices about what the core issues are and what will “take care of itself” once the core issues are resolved.

Particle Physics: ‘Uniﬁcation’ Central: Extend Perturbative, ﬂat space QFTs; Gravity just another force. Higher derivative theories; Supergravity • String theory incarnations: • •⋆ Perturbative strings; ⋆ M theory & F theory; ⋆ Matrix Models; ⋆ AdS/CFT Correspondence.

– p. Contrasting LQG with String theory

Particle Physics: ‘Uniﬁcation’ Central: Extend Perturbative, ﬂat space •QFTs; Gravity just another force. Higher derivative theories; Supergravity • String theory incarnations: • •⋆ Perturbative strings; ⋆ M theory; ⋆ Matrix Models; ⋆ AdS/CFT Correspondence.

General Relativity: ‘Background independence’ Central: LQG ⋆• Quantum Geometry (Hamiltonian Theory used for cosmology & BHs), ⋆ Spin-foams (Path integrals used to bridge low energy physics.) Issues: Uniﬁcation: Ideas proposed in LQG but strong limitations; Recall• however, QCD versus Grand Uniﬁed Theories. Background Independence: Progress through AdS/CFT; but a ‘small corner’• of QG; Physics beyond singularities and S-matrices?

A. Ashtekar: LQG: Four Recent Advances and a dozen FAQs; arXiv:0705.2222

– p. Organization

1. Historical and Conceptual Setting 2. Loop Quantum Gravity: Quantum Geometry 3. Black Holes: Zooming in on Quantum Geometry 4. Big Bang: Loop Quantum Cosmology 5. Summary

– p. 2. Loop Quantum Gravity: Quantum Geometry

Geometry: Physical entity, as real as tables and chairs. •Riemann 1854: Göttingen Address; Einstein 1915: General Relativity Matter has constituents. GEOMETRY?? ‘Atoms• of Geometry’? Why then does the continuum picture work so well? Are there physical processes which convert Quanta of Geometry to Quanta of Matter and vice versa?

– p. 10 2. Loop Quantum Gravity: Quantum Geometry

A Paradigm shift to address these issues • “The major question for anyone doing research in this ﬁeld is: Of which mathematical type are the variables ...which permit the expression of physical properties of space... Only after that, which equations are satisﬁed by these variables?” Albert Einstein (1946); Autobiographical Notes.

Choice in General Relativity: Metric, gµν . Directly determines • Riemannian geometry; Geometrodynamics. In all other interactions, by contrast, the basic variable isa Connection, i i.e., a matrix valued vector potential Aa; Gauge theories: Connection-dynamics

Key new idea: Cast GR also as a theory of connections. Import into GR •techniques from gauge theories. ‘Kinematic uniﬁcation’ (AA, 1986)

– p. 11 Holonomies/Wilson Lines and Frames/Triads

Connections: Vehicles for parallel transport. •In QED: parallel transport of the state of an electron Recall: ∂ψ (∂ ieA)ψ In QCD: parallel transport→ of the− state of a quark In gravity: parallel transport a chiral spinor q p > q ψ(q)=[ exp R A dS] ψ(p) •−−− −−−• P p ·

a In Gravity: the (canonically conjugate) non-Abelian electric ﬁelds Ei •interpreted as orthonormal frames/triads. They determine the physical, curved geometry. (Gauge group: Rotations of triads.)

– p. 12 Holonomies/Wilson Lines and Frames/Triads

Connections: Vehicles for parallel transport. •In QED: parallel transport of the state of an electron Recall: ∂ψ (∂ ieA)ψ In QCD: parallel transport→ of the− state of a quark In gravity: parallel transport a spinor q p > q ψ(q)=[ exp R A dS] ψ(p) •−−− −−−• P p ·

Surprising uniqueness result: The quantum algebra of holonomies and triad-ﬂuxes• admits a unique diff invariant representation Thanks to background independence, quantum kinematics is unique⇒in LQG! (Lewandowski, Okolow, Sahlmann, & Thiemann (2006); Fleischhack(2009))

– p. 13 Polymer Geometry

This unique kinematics was ﬁrst constructed explicitly in the early 1990s.• High mathematical precision. Provides a Quantum Geometry. Replaces the Riemannian geometry used in classical gravity theories. AA, Baez, Corichi, Lewandowski, Marolf, Mourão, Rovelli, Smolin, Thiemann, Zapata,...

2 Quantum States: Ψ = L ( ¯,dµo) • ∈ H A µo a diffeomorphism invariant, regular measure on the space ¯ of (generalized) connections. A

Fundamental excitations of geometry 1-dimensional. Polymer geometry at• the Planck scale. Continuum arises only in the coarse rained approximation.

– p. 14 Flux lines of area. Background independence!

Examples of Novel features: • ⋆ All eigenvalues of geometric operators discrete. Area gap. Eigenvalues not just equally-spaced but crowd in a rather sophisticated way. Geometry quantized in a very speciﬁc way. (Recall Hydrogen atom.) ⋆ Inherent non-commutativity: Areas of intersecting surfaces don’t commute. Inequivalent to the Wheeler-DeWitt theory (quantum geometrodynamics).

Summary: AA & Lewandowski, Encyclopedia of Mathematical Physics; available at http://www.gravity.psu.edu/research/poparticle.shtml

– p. 15 Organization

1. Historical and Conceptual Setting 2. Loop Quantum Gravity: Quantum Geometry 3. Black Holes: Zooming in on Quantum Geometry 4. Big Bang: Loop Quantum Cosmology 5. Summary

– p. 16 3. Black Holes: Zooming in on Quantum Geometry

First law of BH Mechanics + Hawking’s discovery that TBH = κ~/2π • 2 ⇒ for large BHs, SBH = ahor/4ℓpl (Bekenstein 1973)

Entropy: Why is the entropy proportional to area? For a M⊙ black hole, • we must have exp 1076 micro-states, a HUGE number even by standards of statistical mechanics. Where do these micro-states come from? For gas in box, the microstates come from molecules; for a ferromagnet, from Heisenberg spins; Black hole ? Cannot be gravitons: gravitational ﬁelds stationary.

To answer these questions, must go beyond the classical space-time approximation• used in the Hawking effect. Must take into account the quantum nature of gravity.

Distinct approaches. In Loop Quantum Gravity, this entropy arises from •the huge number of microstates of the quantum horizon geometry. ‘Atoms’ of geometry itself!

– p. 17 Quantum Horizon Geometry & Entropy

Heuristics: Wheeler’s It from Bit • 2 Divide the horizon into elementary cells, each carrying area ℓpl . Assign to each cell a ‘Bit’ i.e. 2 states. 2 n Then, # of cells n ao/ℓpl ; No of states 2 ; ∼ 2 N ∼ 2 Shor ln n ln 2 ao/ℓpl . Thus, Shor ao/ℓpl . ∼ N ∼ ∼ ∝

Argument made rigorous in quantum geometry. Many inaccuracies of • 2 the heuristic argument have to be overcome: Quanta of area not ℓpl but 2 4πγ pj(j+) ℓpl ; Calculation has to know that the surface is black hole horizon; What is a quantum horizon?

Interesting mathematical structures U(1) Chern-Simons theory; • non-commutative torus, quantum U(1), mapping class group, ... (AA, Baez, Krasnov)

– p. 18 Quantum Horizon

Polymer excitations of geometry in the bulk puncture the horizon. Quantum horizon geometry described by the U(1) Chern-Simons theory.

– p. 19 Quantum Horizon Geometry and Entropy

Horizon geometry ﬂat everywhere except at punctures. At punctures •the bulk polymer excitations cause a ‘tug’ giving rise to quantized deﬁcit angles. They add up to 4π providing a 2-sphere quantum geometry. ‘Quantum Gauss-Bonnet Theorem’.

As in Statistical mechanics, • have to construct a suitable ensemble ρ by specifying macroscopic parameters (multipoles) characterizing the classical horizon geometry. Shor = Trρ ln ρ gives the log of the number of quantum horizon geometry states compatible with the classical geometry.

2 2 2 Shor = ahor/4ℓpl (1/2)ln(ahor/ℓpl )+ o(ahor/ℓpl ) pi • − for a speciﬁc value of the parameter γ 0.24. Procedure incorporates all physically interesting≈ BHs and Cosmological horizons in one swoop. gamma

(AA, Baez, Krasnov; Domagala, Lewandowski; Meissner; AA, Engle, Van Den Broeck)

– p. 20 Organization

1. Historical and Conceptual Setting 2. Loop Quantum Gravity: Quantum Geometry 3. Black Holes: Zooming in on Quantum Geometry 4. Big Bang: Loop Quantum Cosmology 5. Summary

– p. 21 4. Big-Bang: Loop Quantum Cosmology

Issue: In classical General Relativity, assumption of spatial homogeneity• & isotropy implies that the metric has the FLRW form; a(t): Scale Factor Volume [a(t)]3 Curvature [a(t)]−n ∼ ∼ Einstein Equations volume 0 and Curvature : BIG BANG!! Classical SPACE-TIME⇒ ENDS;→ PHYSICS STOPS!!→ ∞

– p. 22 The Big Bang in classical GR

In classical general relativity the fabric of space-time is violently torn apart at the Big Bang singularity

– p. 23 Expectation: Just an indication that the theory is pushed beyond its domain• of validity. Example: H-atom. Energy unbounded below in the classical theory; 4 me instability. Quantum theory: Eo = 2 − 2~

Is this the case? If so, what is the true physics near the Big Bang? Need• a theory which can handle both strong gravity/curvature and quantum physics, i.e., Quantum Gravity. 3 1 −33 Scale for new phenomena: ℓPl =(G~/c ) 2 10 cm ∼

– p. 24 Some Long-Standing Questions expected to be answered by Quantum •Gravity from ﬁrst principles: ⋆ Is the Big-Bang singularity naturally resolved by quantum gravity? Or, Is a new principle/ boundary condition at the Big Bang essential? ⋆ Is the quantum evolution across the ‘singularity’ deterministic? (answer ‘No’ e.g. in the Pre-Big-Bang and Ekpyrotic scenarios) ⋆ What is on the other side? A quantum foam? Another large, classical universe? ...

– p. 25 Some Long-Standing Questions expected to be answered by Quantum •Gravity from ﬁrst principles: ⋆ Is the Big-Bang singularity naturally resolved by quantum gravity? Or, Is a new principle/ boundary condition at the Big Bang essential? ⋆ Is the quantum evolution across the ‘singularity’ deterministic? (answer ‘No’ e.g. in the Pre-Big-Bang and Ekpyrotic scenarios) ⋆ What is on the other side? A quantum foam? Another large, classical universe? ...

Emerging Scenario: vast classical regions bridged deterministically by •quantum geometry. No new principle needed. (AA, Bojowald, Chiou, Corichi, Pawlowski, Singh, Vandersloot,... )

In the classical theory, don’t need full Einstein equations in all their complexity.• Almost all work in physical cosmology based on homogeneous isotropic models and perturbations thereon. At least in a ﬁrst step, can use the same strategy in the quantum theory: mini and midi-superspaces.

– p. 26 A Simple Model A FRW Model: Gravity coupled to a massless scalar ﬁeld φ. Instructive because every classical solution is singular. Provides a foundation for more complicated models.

-0.2

-0.4

-0.6

-0.8

-1

-1.2

4 4 4 4 4 0 1*10 2*10 3*10 4*10 5*10 v Classical trajectories

– p. 27 Older Quantum Cosmology (DeWitt, Misner, . . . 70's)

Since only finite number of DOF a(t),φ(t), field theoretical difficulties bypassed;• analysis reduced to standard quantum mechanics. Quantum States: Ψ(a, φ); aˆΨ(a, φ)= aΨ(a, φ) etc. • Quantum evolution governed by the Wheeler-DeWitt equation on Ψ(a, φ) Without additional assumptions, singularity is not resolved by the equation.

General belief since late seventies: This situation can not be remedied because• of von-Neumann’s uniqueness theorem which lies at the heart of QM.

New Development: A key assumption of the theorem fails in LQG • von-Neumann’s uniqueness result naturally bypassed. ⇒ (AA, Bojowald, Lewandowski). 2 2 New Quantum Mechanics! ( = L (R) but rather = L (R¯ Bohr).) Novel features precisely in theH 6 deep Planck regime.H

– p. 28 0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

4 4 4 4 4 0 1*10 2*10 3*10 4*10 5*10

v Classical trajectories

– p. 29 0 LQC classical

-0.2

-0.4

-0.6 φ

-0.8

-1

-1.2

0 1*104 2*104 3*104 4*104 5*104 v

Expectations values and dispersions of Vˆ φ & classical trajectories. |

– p. 30 1.6 |Ψ(v,φ)| 1.4 1.2 1.5 1 0.8 1 0.6 0.5 0.4 0.2 0 0

5*103 1.0*104 -1.2 1.5*104 -1 4 2.0*10 -0.8 4 v 2.5*10 -0.6 3.0*104 -0.4 φ 4 3.5*10 -0.2 4.0*104

Absolute value of the physical state Ψ(v,φ)

– p. 31 Results

Assume that the quantum state is semi-classical at late times (e.g., NOW) and evolve backwards. Then:

The state remains semi-classical till very early times! Till ρ ρPl. •Space-time can be taken to be classical during the inﬂationary∼ era. ⇒ In the deep Planck regime, semi-classicality fails. But quantum evolution• is well-deﬁned through the Planck regime, and remains deterministic. No new principle needed.

– p. 32 Results Assume that the quantum state is semi-classical at late times (e.g., NOW) and evolve backwards. Then:

The state remains semi-classical till very early times! Till ρ ρPl. •Space-time can be taken to be classical during the inﬂationary∼ era. ⇒

In the deep Planck regime, semi-classicality fails. But quantum evolution• is well-deﬁned through the Planck regime, and remains deterministic. No new principle needed.

Big bang replaced by a quantum bounce. Matter satisfies standard energy• conditions. Exact quantum analysis using semi-classical states not WKB. Quantum geometry ‘bridges’ an infinite expanding branch with an infinite contracting branch.

A new ‘repulsive force’ due to quantum geometry. Unlike in other approaches• with bounces, unambiguous evolution across the ‘bridge’, provided by the quantum Einstein equation. Effective Friedmann Eq: 2 (˙a/a) = (8πGρ/3) [1 ρ/ρcrit] with ρcrit 0.4ρPlanck − ≈ Results robust for all simple models (open, closed, Λ = 0, Λ > 0, Λ < 0, •anisotropies). – p. 33 The Big Bang in classical GR

In classical general relativity the fabric of space-time is violently torn apart at the Big Bang singularity

– p. 34 The Big Bang in LQC

In loop quantum cosmology our post-big-bang branch of the universe is joined to a pre-big-bang branch by a quantum bridge.

– p. 35 Summary and Outlook

In Loop Quantum Gravity, the interplay between geometry and physics •is elevated to quantum level. Just as general relativity is based on Riemannian geometry, LQG is based on a speciﬁc quantum geometry.

Quantum geometry effects negligible under ‘normal conditions’ but dominate• physics at Planck scale. Physics does not end at classical singularities. Many long standing fundamental questions answered both in cosmology and in black hole physics.

Singularity resolution the quantum space-time is vastly larger than •what GR had us believe.⇒ This also provides a novel avenue to resolve the ‘information loss’ issue.

So far focus has been on conceptual challenges. Just beginning to •grapple with the all important phenomenological issues. Interesting challenges and overlap with mathematics, lattice gauge theories, QFT, GR, cosmology, & computational physics.

– p. 36 k=1 Model: WDW Theory

2 WDW classical 1.5

0.5

φ 0

-0.5

-1

-1.5

-2 0 1*104 2*104 3*104 4*104 5*104 6*104 7*104 8*104 9*104 1.0*105 v

Expectations values and dispersions of Vˆ φ. |

– p. 37 k=1 Model: LQC

0 LQC Effective Classical

-1

-2 φ

-3

-4

-5 0 2*104 4*104 6*104 8*104 1.0*105 v

Expectations values and dispersions of Vˆ φ & classical trajectories. (AA, Pawlowski, Singh, Vandersloot)|

– p. 38 classical 0.8 LQC

0.6

0.4

0.2

φ 0

-0.2

-0.4

-0.6

-0.8 0 2*103 4*103 6*103 8*103 1.0*104 1.2*104 1.4*104 v k=0, Positive Cosmological constant: Expectations values and dispersions of Vˆ φ & classical trajectories. (Numerics are currently being improved.) | – p. 39 0 classical LQC

-0.5

-1 φ

-1.5

-2

-2.5

-3 0 2*104 4*104 6*104 8*104 1.0*105 1.2*105 v

– p. 40 k=0, Negative Cosmological constant: Expectations values and Quantum Theory

Finite number of degrees of freedom Quantum Mechanics. ⇒ If x, p is the canonically conjugate pair, [ˆx, pˆ]= i Iˆ. If Uˆ(λ)= eiλxˆ and Vˆ (µ)= eiµpˆ, then Uˆ(λ) Vˆ (µ)= eiλµ Vˆ (µ) Uˆ(λ). von Neumann’s uniqueness theorem: There is a unique IRR of Uˆ(λ), Vˆ (µ) by 1-parameter unitary groups on a Hilbert space satisfying: i) commutation relations; and ii) Weak continuity in λ, µ. This is the standard Schrödinger representation.

– p. 41 Analytical steps in the procedure • ⋆ Equation: ∂2Ψ(µ,φ)= ΘΨ(µ,φ) φ −

⋆ ‘Group averaging procedure’ leads us to consider Positive frequency solutions: i∂φΨ= √ΘΨ − Ψ(µ,φ) = exp[i√Θ(φ φo)]Ψ(µ,φo). ⇒ − ⋆ Inner product:

2 Ψ =<Ψ(µ,φo) Ψ(µφo)>kin || ||

⋆ Dirac observables: Momentum operator pˆφ = i∂φΨ − µ φo operator: µ φo Ψ = exp[i√Θ(φ φo)] µΨ(µ,φo) d| b| − ⋆ Semi-Classical states Coherent states. ∼

– p. 42 Basic Strategy

Analytical Issues: the quantum Hamiltonian constraint takes the form: • ΘΨ(v,φ)= ∂2Ψ(v,φ) (⋆) − φ where Θ is a self-adjoint difference operator independent of φ:

ΘΨ(v,φ)= C+(v) ψ(v + 4, φ)+ Co(v) ψ(v,φ)+ C−(v) ψ(v, φ) suggests φ could be used as ‘emergent time’ also in the quantum theory.

ˆ Strategy: Physical states: solutions to (⋆). Observables: V φ=φo and pˆ φ. | | Inner product: Makes these self-adjoint. Semi-classical states: Generalized coherent states.

Use numerical methods to solve the Quantum Constraint. Several •subtleties had to be carefully handled.

– p. 43