Table of Contents
- Introduction
- Why Quantum Gravity?
- Classical Gravity: General Relativity
- Quantum Field Theory Overview
- Incompatibility Between GR and QFT
- Conceptual Challenges in Quantizing Gravity
- Approaches to Quantum Gravity
- Perturbative Quantum Gravity
- Graviton and Linearized Gravity
- Ultraviolet Divergences and Non-Renormalizability
- Effective Field Theory Approach
- Canonical Quantization and Wheeler–DeWitt Equation
- Path Integral Quantization
- Covariant vs Canonical Approaches
- Loop Quantum Gravity: Core Ideas
- Ashtekar Variables and Spin Networks
- Loop Quantum Cosmology
- String Theory and Gravity
- Holography and AdS/CFT
- Background Independence
- Black Hole Thermodynamics
- Hawking Radiation and Information Paradox
- Entropy and Microstates
- Discrete Spacetime and Quantum Geometry
- 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.
