23th February 2004

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

- 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.

- String Theory; &
- Loop Quantum Gravity.

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

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

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

**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**!

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

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

- Hamiltonian constraint:

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

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

- 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)!

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

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

- 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).

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

- "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".

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

- 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.

- 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.

- 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.

- 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!