Quantum Gravity Basics

Table of Contents

1. Introduction
2. Why Quantum Gravity?
3. Classical Gravity: General Relativity
4. Quantum Field Theory Overview
5. Incompatibility Between GR and QFT
6. Conceptual Challenges in Quantizing Gravity
7. Approaches to Quantum Gravity
8. Perturbative Quantum Gravity
9. Graviton and Linearized Gravity
10. Ultraviolet Divergences and Non-Renormalizability
11. Effective Field Theory Approach
12. Canonical Quantization and Wheeler–DeWitt Equation
13. Path Integral Quantization
14. Covariant vs Canonical Approaches
15. Loop Quantum Gravity: Core Ideas
16. Ashtekar Variables and Spin Networks
17. Loop Quantum Cosmology
18. String Theory and Gravity
19. Holography and AdS/CFT
20. Background Independence
21. Black Hole Thermodynamics
22. Hawking Radiation and Information Paradox
23. Entropy and Microstates
24. Discrete Spacetime and Quantum Geometry
25. Conclusion

1. Introduction

Quantum gravity is the field of physics that seeks to unify quantum mechanics with general relativity into a consistent theory that describes gravity at the smallest scales. While both frameworks work well in their respective domains, their incompatibility becomes apparent in extreme conditions such as black holes and the early universe.

2. Why Quantum Gravity?

Key motivations include:

• Understanding the Big Bang singularity
• Explaining black hole entropy and information loss
• Achieving a unified theory of all forces
• Exploring spacetime at the Planck scale (\( \sim 10^{-35} \, \text{m} \))

3. Classical Gravity: General Relativity

General Relativity (GR) describes gravity as the curvature of spacetime due to matter and energy:

\[
R_{\mu\nu} – \frac{1}{2} R g_{\mu\nu} = 8\pi G T_{\mu\nu}
\]

• \( R_{\mu\nu} \): Ricci tensor
• \( R \): scalar curvature
• \( T_{\mu\nu} \): energy-momentum tensor

GR is background-independent and nonlinear.

4. Quantum Field Theory Overview

QFT combines quantum mechanics and special relativity to describe particle physics. It treats fields as operator-valued distributions and uses Feynman diagrams to compute interactions.

5. Incompatibility Between GR and QFT

Problems arise when trying to quantize GR:

• Non-renormalizability: UV divergences cannot be absorbed
• Background dependence: QFT requires fixed background, GR does not
• No local gravitational degrees of freedom in traditional QFT framework

6. Conceptual Challenges in Quantizing Gravity

• The gravitational field itself defines spacetime, unlike other fields
• Measurement problem exacerbated by gravitational backreaction
• Time in quantum mechanics is external; in GR, it’s dynamical

7. Approaches to Quantum Gravity

Major approaches include:

• Perturbative QG
• Effective field theories
• Loop Quantum Gravity
• String Theory
• Causal Dynamical Triangulations
• Asymptotic Safety
• Spinfoams

8. Perturbative Quantum Gravity

Expanding the metric around flat spacetime:

\[
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}
\]

Treating \( h_{\mu\nu} \) as a quantum field leads to the graviton, a massless spin-2 particle.

9. Graviton and Linearized Gravity

The graviton mediates gravitational interactions, similar to the photon in electromagnetism. It satisfies a linearized version of Einstein’s equations and is described by symmetric tensor fields.

10. Ultraviolet Divergences and Non-Renormalizability

Graviton loops produce divergences that cannot be canceled by a finite number of counterterms. The perturbative theory is non-renormalizable, requiring new physics at high energies.

11. Effective Field Theory Approach

Quantum gravity can be treated as a low-energy effective theory. Predictions are reliable at energies \( E \ll M_{\text{Planck}} \), with corrections organized in powers of \( E/M_{\text{Planck}} \).

12. Canonical Quantization and Wheeler–DeWitt Equation

Canonical quantization treats the metric as a dynamical variable. The Wheeler–DeWitt equation:

\[
\hat{H} \Psi[g_{ij}] = 0
\]

represents the quantum Hamiltonian constraint for the wavefunction of the universe \( \Psi \).

13. Path Integral Quantization

This approach sums over geometries:

\[
Z = \int \mathcal{D}g_{\mu\nu} \, e^{i S_{\text{GR}}[g]}
\]

Challenges:

• Defining measure over geometries
• Handling non-renormalizable divergences

14. Covariant vs Canonical Approaches

• Covariant: based on path integrals, spacetime treated as a whole
• Canonical: 3+1 decomposition, quantize spatial metric and extrinsic curvature

Both face deep technical and conceptual obstacles.

15. Loop Quantum Gravity: Core Ideas

LQG quantizes geometry using holonomies and fluxes. The key variable is the Ashtekar connection, and states are built from spin networks — graphs with edges labeled by SU(2) representations.

16. Ashtekar Variables and Spin Networks

Ashtekar variables simplify constraints and make the theory more amenable to quantization. The Hilbert space consists of spin network states, providing a discrete geometry of space.

17. Loop Quantum Cosmology

Applies LQG to homogeneous cosmologies. Replaces the Big Bang with a Big Bounce, resolving the classical singularity.

18. String Theory and Gravity

String theory naturally includes gravity via a spin-2 excitation of closed strings. Consistency requires 10 dimensions and supersymmetry, and leads to unification of all fundamental forces.

19. Holography and AdS/CFT

The AdS/CFT correspondence provides a non-perturbative formulation of quantum gravity in certain backgrounds. Gravity in bulk AdS is dual to a conformal field theory on the boundary.

20. Background Independence

Many quantum gravity approaches strive to preserve background independence, a hallmark of general relativity. String theory, however, typically requires a fixed background for quantization.

21. Black Hole Thermodynamics

Black holes exhibit thermodynamic behavior:

• Bekenstein–Hawking entropy:
  \[
  S = \frac{k_B A}{4 \ell_P^2}
  \]
• Laws of black hole mechanics parallel thermodynamics

22. Hawking Radiation and Information Paradox

Hawking showed black holes radiate thermally:

\[
T_H = \frac{\hbar \kappa}{2\pi c k_B}
\]

This leads to the information paradox: does evaporation destroy information?

23. Entropy and Microstates

A key goal is to account for black hole entropy via microstates:

• In string theory: D-brane configurations
• In LQG: counting spin network states intersecting the horizon

24. Discrete Spacetime and Quantum Geometry

Quantum gravity may imply spacetime is not continuous, but built from fundamental quanta. This leads to:

• Minimal length scales
• Modified dispersion relations
• Quantum geometry operators (area, volume)

25. Conclusion

Quantum gravity seeks to unify the smooth spacetime of general relativity with the probabilistic world of quantum mechanics. While no complete theory exists yet, numerous frameworks offer partial insights. The journey involves deep mathematical structures, conceptual innovations, and potentially transformative implications for our understanding of space, time, and reality.