## Canonical quantum gravity

*Notes on canonical classical and quantum gravity.*

### 1. Hamiltonian General Relativity

Lagrangian formulations General Relativity can be cast to the Hamiltonian form. The Hamilton's function of gravity appears to be a linear combination of constraints. These constraints come from diffeomorphism (and Lorentz, in tetrad formulation) redundance. Hamilton's function vanishes on the constrained surface, leading to the problem of time.

##### 1.1. ADM decomposition

##### 1.2. Derivation of the Wheeler-DeWitt equation

##### 1.3. Self-dual connection

### 2. Loop Quantum Gravity

After reformulation in terms of Ashtekar variables, General Relativity admits Hamiltonian description in which the (Ashtekar) SU(2) connection and "electric" field are the canonical conjugate pair. Passing to the discretized integral variables, we arrive at the description in terms of holonomies of links of the graph. This description admits rigorous quantization and all of the constraints of General Relativity can be appropriately represented as quantum operators in this new loop representation.

##### 2.1. Variation of the Holonomy

##### 2.2. Group of loops

##### 2.3. Spin networks in LQG

##### 2.4. FAQ on Canonical LQG

### 3. Quantum spacetime: spinfoams

Discretization of General Relativity on 2-complexes (higher-dimensional topological structures, generalizing the notion of graphs) is considered. We arrive at the background-independent concept called "spinfoam", which is analogous to the Feynman history of the quantized spacetime. Boundaries of spinfoams are spin networks. The sum over spinfoams is hypothesized to describe the quantization of the Hamiltonian constraint of LQG.

##### 3.1. Ponzano-Regge spinfoams

##### 3.2. 3D Loop Quantum Gravity amplitudes

##### 3.3. Turaev-Ooguri-Crane-Yetter spinfoams

##### 3.4. Recoupling theory and triangulation-independence of 2D TOCY model

##### 3.5. Barrett-Crane spinfoams

##### 3.6. EPRL spinfoams: Quantum General Relativity

### 4. Applications

Applications of LQG and spinfoams to cosmological and other problems where quantum gravity plays the leading role.