Loop Quantum Cosmology

Daniel Doro Ferrante
23th February 2004

Outline

• Quantum Gravity's importance;
• Present Approaches to Quantum Gravity;
• Basics of Loop Quantum Gravity;
• LQG Results;
• Loop Quantum Cosmology; &
• LQC Recent Applications.

Quantum Gravity: Why is it important?!

Relation to other problems

• Quantum Cosmology & Origin of the Universe;
• Quantum Theory ``without time'', Unitarity;
• Structure and Interpretation of Quantum Mechanics, Topos theory;
• Wave Function Collapse;
• Unification of All Interactions, TOEs;
• Final State of a Black Hole; &
• Ultraviolet Divergences.

Present Approaches to Quantum Gravity

Main Directions

• String Theory; &
• Loop Quantum Gravity.

Traditional Approaches

• Discrete Approaches (Regge calculus, dynamical triangulations);
• Old Hopes, Approximate Theories (Euclidean QG, QFT on curved spacetime); &
• ``Unorthodox Approaches'' (causal sets, twistors, Finkelstein's ideas).

New Directions

• Noncommutative Geometry;
• Null Surface Formulations; &
• Spin Foam Models (TFTs).

Loop Quantum Gravity: The Basics

• Quantum Field Theory on a Differentiable Manifold
• background independence;
• pseudo-1-form (densitized triad); &
• SU(2)-connection.

Loop Quantum Gravity: The Basics (cont'd)

• Wilson Loop Operators
• well defined in the Hilbert space of the theory; &
• non-perturbative.
• Physical meaning of diffeomorphism invariance and its implementation in the quantum theory
• all variables are dynamical!

LQG: Results

• Technical
• Solution of the Hamiltonian constraint;
• Time Evolution, Topological Feynman rules;
• Fermions;
• Maxwell & Yang-Mills; &
• Lattice & Simplicial models.

LQG: Results (cont'd)

• Physical
• Planck Scale Discreteness of space (area or volume operators have a discrete spectrum);
• Classical Limit, quantum states for flat spacetimes; &
• Black Hole Entropy.

Loop Quantum Cosmology

• Hamiltonian constraint:
  where , , for a flat or closed model, is the matter Hamiltonian. ( and are part of the isotropic triad and connection, respectively)

Loop Quantum Cosmology (cont'd)

• This is just Friedman's eq, ; &
• LQC: symmetric LQG states.

Loop Quantum Cosmology (cont'd)

• Discrete Evolution
• Indirect quantization: via holonomies, i.e., operator, just a holonomy related to it.
• Quantized Friedman eq: difference eq!
• no cosmological singularities (homogeneous models)!

Loop Quantum Cosmology (cont'd)

• Finite Inverse Scale Factor Operator
• Matter Hamiltonian: need to quantize
• Inverse power of : discrete spectrum with 0
• Alternative way yields 3 phases:
1. 
2. Inflation
3. Transition to the classical

Recent Applications

• Quantum Structure of Classical Singularities
• All homogeneous models: no cosmological singularities!
• Inhomogeneous models: can be treated in the BKL scenario.

Recent Applications (cont'd)

• Phenomenology
• Inflation
• No graceful exit problem (phase transition on the scale factor implies classical regime); &
• Exists for any matter content, even without an inflaton (spacetime property).
• Quantized spacetime: photons with different energies should have slightly different speeds (modified dispersion relations; gamma & cosmic rays).

Recent Applications (cont'd)

• Perturbative Corrections
• Leading order: Friedman eq (as seen); &
• Two types of corrections:
1. Higher curvature corrections (i.e., effective action); &
2. Other terms: reflect the fact that there is no unique coherent state such as the Minkowski vacuum for an effective action.

Bibliography

• "Throwing Einstein for a Loop";
• "Atoms of Space and Time";
• "A Spin on Spin Foam";
• "The Future of String Theory - A Conversation with Brian Greene";
• "How far are we from the quantum theory of gravity?";
• "Strings, loops and others: a critical survey of the present approaches to quantum gravity";
• "Loop Quantum Gravity";
• "Loop quantum gravity and quanta of spacetime: a primer";
• "Lectures on Loop Quantum Gravity";
• "Canonical quantum gravity and consistent discretizations";
• "Absence of Singularity in Loop Quantum Cosmology";
• "Quantum Geometry in Action: Big-Bang and Black Holes";
• "Cosmological applications of loop quantum gravity"; &
• "Loop Quantum Cosmology: Recent Progress".

Just in case...

Oh, you tought I wouldn't be prepared for some questions, huh?! >;-)

Not at all: In what follows there are some slides to entertain the more curious characters...

Appendix: Loop Quantization

• Quantum variables:
• Holonomies:
• Fluxes:
• : Pauli matrices; : tangent vector to the edge ; : co-normal to the surface .
• If all curves and all surfaces are allowed, holonomies and fluxes contain the same information as the original fields.

Appendix: Loop Quantization (cont'd)

• Holonomies and fluxes are better for quantization: Smeared versions of the fields obtained by natural integration along curves and surfaces.
• One- and two-dimensional smearings can be done without introducing a background metric! Background independent quantization! >:-)
• Quantum Theory: defined on a representation of the background independent holonomy-flux Poisson -algebra.

Appendix: Loop Quantization (cont'd)

• The basis for the states is given by spin network states which are associated to graphs in Sigma whose edges are labeled by irreducible SU(2) representations; &
• Diffeomorphism-invariant inner product: Ashtekar-Lewandowski.

Appendix: Loop Quantization (cont'd)

• Characteristics of Loop Quantization:
• Hilbert space before imposing the [diffeomorphism invariance and Hamiltonian] constraints is non-separable: all spin network states in different graphs are orthogonal to each other!
• Holonomies are well defined operators by definition!
• Flux operators have discrete spectra which implies discrete spatial geometry!