Canonical Quantum Gravity

Table of Contents

1. Introduction
2. Motivation for Canonical Quantization
3. 3+1 Decomposition of Spacetime
4. Hamiltonian Formulation of General Relativity
5. The ADM Formalism
6. Canonical Variables and Constraints
7. Dirac’s Procedure for Constrained Systems
8. Wheeler–DeWitt Equation
9. Problem of Time
10. Quantization of the Constraint Algebra
11. Wavefunction of the Universe
12. Minisuperspace Models
13. Midisuperspace and More General Configurations
14. Loop Quantum Gravity: Canonical Roots
15. Ashtekar Variables
16. Holonomies and Fluxes
17. Spin Networks and Hilbert Space
18. Discrete Geometry in Canonical Framework
19. Semiclassical Limit and Recovering GR
20. Observables and Physical States
21. Gauge Invariance and Diffeomorphism Constraint
22. Boundary Conditions and Black Holes
23. Canonical Quantum Cosmology
24. Comparison with Path Integral Methods
25. Conclusion

1. Introduction

Canonical quantum gravity is a non-perturbative approach to quantizing general relativity. It is based on the Hamiltonian (canonical) formulation of gravity, aiming to promote classical variables to quantum operators and formulate a consistent quantum dynamics for spacetime geometry.

2. Motivation for Canonical Quantization

Canonical quantization mirrors the standard approach in quantum mechanics: promote classical phase space variables to operators and impose commutation relations. In quantum gravity, this is applied to the metric of space and its conjugate momentum.

3. 3+1 Decomposition of Spacetime

Spacetime is foliated into spatial hypersurfaces \( \Sigma_t \), each labeled by a time parameter \( t \). The full metric is written as:

\[
ds^2 = -N^2 dt^2 + q_{ij} (dx^i + N^i dt)(dx^j + N^j dt)
\]

• \( N \): lapse function
• \( N^i \): shift vector
• \( q_{ij} \): 3-metric on \( \Sigma_t \)

4. Hamiltonian Formulation of General Relativity

In this decomposition, GR is rewritten in terms of canonical variables:

• Configuration: 3-metric \( q_{ij} \)
• Momentum: \( \pi^{ij} \) (related to extrinsic curvature)

The Einstein–Hilbert action becomes:

\[
S = \int dt \int_\Sigma d^3x \left( \pi^{ij} \dot{q}{ij} – \mathcal{H}{\text{total}} \right)
\]

5. The ADM Formalism

The Arnowitt–Deser–Misner (ADM) formalism provides the canonical structure:

• Hamiltonian constraint \( \mathcal{H} \approx 0 \)
• Momentum (diffeomorphism) constraint \( \mathcal{H}_i \approx 0 \)

These constraints must be satisfied by physical states.

6. Canonical Variables and Constraints

Poisson brackets:

\[
\{ q_{ij}(x), \pi^{kl}(y) \} = \delta^{(k}_i \delta^{l)}_j \delta^3(x – y)
\]

Constraints generate diffeomorphisms and time reparametrization — key to background independence.

7. Dirac’s Procedure for Constrained Systems

General relativity is a constrained system, requiring:

• Identification of first-class constraints
• Promotion to quantum operator equations
• Imposition on physical states

8. Wheeler–DeWitt Equation

Quantizing the Hamiltonian constraint gives the Wheeler–DeWitt equation:

\[
\hat{\mathcal{H}} \Psi[q_{ij}] = 0
\]

This equation governs the quantum state of the entire universe — a central concept in canonical quantum gravity.

9. Problem of Time

In canonical quantum gravity:

• Time evolution is generated by constraints
• The Wheeler–DeWitt equation lacks an explicit time variable
• Raises conceptual questions about the nature of time and change

10. Quantization of the Constraint Algebra

The constraint algebra includes the Dirac algebra, reflecting the hypersurface deformation algebra. Ensuring this algebra is preserved at the quantum level is essential but challenging.

11. Wavefunction of the Universe

The solution \( \Psi[q_{ij}] \) to the Wheeler–DeWitt equation represents the wavefunction of the universe, encoding all possible spatial geometries and their quantum superpositions.

12. Minisuperspace Models

Simplify the theory by restricting to highly symmetric geometries (e.g., homogeneous and isotropic). This reduces infinite degrees of freedom to a finite-dimensional system, allowing analytical solutions.

13. Midisuperspace and More General Configurations

Involves more complex models with some local degrees of freedom. Examples include:

• Spherically symmetric models
• Inhomogeneous cosmologies
• Black hole interiors

14. Loop Quantum Gravity: Canonical Roots

Loop quantum gravity arises from reformulating canonical quantum gravity using Ashtekar variables, leading to a background-independent quantization of geometry.

15. Ashtekar Variables

Complex connection \( A^i_a \) and densitized triad \( E^a_i \) replace the metric and its momentum. The Poisson bracket becomes:

\[
\{ A^i_a(x), E^b_j(y) \} = \delta^i_j \delta^b_a \delta^3(x – y)
\]

16. Holonomies and Fluxes

The configuration variable is replaced by holonomies (Wilson lines):

\[
h_e = \mathcal{P} \exp \left( \int_e A \right)
\]

Fluxes are integrals of \( E \) over surfaces. These become the fundamental operators in LQG.

17. Spin Networks and Hilbert Space

Spin networks are graphs with SU(2) labels on edges and nodes. They provide an orthonormal basis of the Hilbert space in loop quantum gravity and encode discrete geometrical data.

18. Discrete Geometry in Canonical Framework

Operators corresponding to area and volume have discrete spectra:

\[
\hat{A} \Psi = \sum_i \sqrt{j_i(j_i + 1)} \, \Psi
\]

This implies a fundamentally quantum (granular) structure of space.

19. Semiclassical Limit and Recovering GR

A major challenge is recovering classical general relativity in the low-energy limit. Semiclassical states (coherent states) are constructed to peak around classical geometries.

20. Observables and Physical States

Physical observables must commute with all constraints. Finding such Dirac observables is difficult but essential for extracting predictions.

21. Gauge Invariance and Diffeomorphism Constraint

States must be invariant under spatial diffeomorphisms. Implemented by considering equivalence classes of spin networks under smooth deformations.

22. Boundary Conditions and Black Holes

Canonical methods are used to describe:

• Isolated horizons
• Black hole entropy via boundary degrees of freedom
• Punctures on spin networks contributing to entropy

23. Canonical Quantum Cosmology

Applies the formalism to cosmological models:

• Produces discrete evolution equations
• Resolves classical singularities
• Predicts bouncing cosmologies

24. Comparison with Path Integral Methods

• Canonical: emphasizes states and operators
• Path integral: sums over geometries
• Spin foam models attempt to unify both views

25. Conclusion

Canonical quantum gravity provides a rigorous, background-independent route to quantizing general relativity. Despite technical and conceptual challenges, it has laid the groundwork for loop quantum gravity, quantum cosmology, and the exploration of discrete quantum spacetime. As research advances, its interplay with other approaches like path integrals and string theory continues to enrich our understanding of quantum spacetime.