Table of Contents

1. Introduction
2. Classical Black Holes and the Laws of Mechanics
3. The Zeroth Law
4. The First Law
5. The Second Law
6. The Third Law
7. Bekenstein’s Entropy
8. Hawking Radiation
9. Temperature of a Black Hole
10. Black Hole Entropy Formula
11. Area Law for Entropy
12. Generalized Second Law
13. Statistical Interpretation of Entropy
14. Entropy from Quantum Gravity
15. Loop Quantum Gravity and Entropy
16. String Theory Microstates
17. Black Hole Evaporation
18. Information Paradox
19. Firewall Hypothesis
20. Black Hole Complementarity
21. Page Curve and Entanglement Entropy
22. Holographic Principle
23. AdS/CFT and Black Hole Thermodynamics
24. Observational Implications
25. Conclusion

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1. Introduction

Black hole thermodynamics is the study of black holes using the laws of thermodynamics. Surprisingly, black holes exhibit temperature, entropy, and obey laws that closely mirror classical thermodynamic laws. This profound connection provides a bridge between general relativity, quantum mechanics, and statistical physics.

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2. Classical Black Holes and the Laws of Mechanics

The laws of black hole mechanics, discovered in the 1970s, mirror the laws of thermodynamics:

• Surface gravity \( \kappa \): analogous to temperature
• Horizon area \( A \): analogous to entropy
• Mass \( M \): analogous to energy

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3. The Zeroth Law

Zeroth Law of Black Hole Mechanics:

The surface gravity \( \kappa \) is constant over the event horizon of a stationary black hole.

Analogous to the thermodynamic zeroth law: temperature is uniform at equilibrium.

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4. The First Law

\[
dM = \frac{\kappa}{8\pi} dA + \Omega dJ + \Phi dQ
\]

• \( M \): mass
• \( A \): horizon area
• \( J \): angular momentum
• \( Q \): charge
• \( \Omega \), \( \Phi \): horizon angular velocity, electric potential

Matches the first law of thermodynamics: \( dE = TdS + \dots \)

---

5. The Second Law

Hawking’s Area Theorem:

The area of a classical black hole horizon never decreases:
\[
\frac{dA}{dt} \geq 0
\]

Corresponds to the second law: entropy increases.

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6. The Third Law

It is impossible to reduce the surface gravity \( \kappa \) to zero in a finite number of steps.

Analogous to: \( T \to 0 \) cannot be reached by finite processes.

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7. Bekenstein’s Entropy

Jacob Bekenstein proposed that black holes carry entropy proportional to their area:

\[
S \propto A
\]

This made the analogy between thermodynamics and black holes more than formal.

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8. Hawking Radiation

Stephen Hawking showed quantum effects lead black holes to emit thermal radiation:

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

This gives black holes a real temperature and confirms Bekenstein’s idea.

---

9. Temperature of a Black Hole

For Schwarzschild black holes:

\[
T_H = \frac{\hbar c^3}{8\pi G M k_B}
\]

As mass decreases, temperature increases — black holes get hotter as they evaporate.

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10. Black Hole Entropy Formula

\[
S = \frac{k_B A}{4 \ell_P^2}
\]

• \( A \): event horizon area
• \( \ell_P = \sqrt{\frac{\hbar G}{c^3}} \): Planck length

This is known as the Bekenstein–Hawking entropy.

---

11. Area Law for Entropy

Entropy is proportional to area, not volume — a deep insight suggesting that the information content of a region resides on its boundary.

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12. Generalized Second Law

The total entropy of matter + black hole does not decrease:

\[
\frac{d}{dt} (S_{\text{matter}} + S_{\text{BH}}) \geq 0
\]

This generalizes the second law to include gravitational entropy.

---

13. Statistical Interpretation of Entropy

The formula \( S = k_B \log \Omega \) begs the question: what are the microstates \( \Omega \) of a black hole?

Various approaches attempt to answer this using quantum gravity.

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14. Entropy from Quantum Gravity

A quantum theory of gravity should explain black hole entropy microscopically. Two major frameworks:

• Loop Quantum Gravity
• String Theory

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15. Loop Quantum Gravity and Entropy

In LQG, black holes are modeled using isolated horizons. Counting spin network punctures on the horizon yields:

\[
S = \frac{k_B A}{4 \ell_P^2}
\]

after fixing the Immirzi parameter.

---

16. String Theory Microstates

Strominger and Vafa showed that counting D-brane configurations in string theory matches the Bekenstein–Hawking entropy for extremal black holes.

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17. Black Hole Evaporation

Due to Hawking radiation, black holes lose mass and eventually evaporate completely, unless new physics intervenes.

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18. Information Paradox

Evaporation produces thermal radiation — seemingly uncorrelated with the initial state. Does information disappear?

This violates unitarity in quantum mechanics, leading to the black hole information paradox.

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19. Firewall Hypothesis

One radical proposal is the firewall: a high-energy surface at the event horizon that destroys infalling information. Controversial and debated.

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20. Black Hole Complementarity

Suggests:

• No single observer sees a paradox
• Information is both reflected and absorbed
• No contradiction due to causal limitations

An attempt to resolve the information loss paradox.

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21. Page Curve and Entanglement Entropy

Don Page proposed the Page curve: entropy of Hawking radiation rises and then falls, consistent with unitary evolution. Recent work using replica wormholes supports this view.

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22. Holographic Principle

The area law inspires the holographic principle: all information in a volume can be encoded on its boundary. Central to modern theories like AdS/CFT.

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23. AdS/CFT and Black Hole Thermodynamics

Black holes in AdS space are dual to thermal states in CFT. This duality provides a unitary description of black hole dynamics, including entropy and evaporation.

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24. Observational Implications

While Hawking radiation is too weak to observe for astrophysical black holes, analog systems (e.g., sonic black holes) may provide indirect evidence.

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25. Conclusion

Black hole thermodynamics reveals deep connections between gravity, thermodynamics, and quantum mechanics. The laws governing black holes echo those of entropy and temperature, while quantum effects reveal a rich structure behind the classical event horizon. As a testing ground for quantum gravity, black holes remain central to our quest for a unified theory of the fundamental forces.

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Table of Contents

1. Introduction
2. Motivation for Loop Quantum Gravity
3. Background Independence
4. Reformulating General Relativity
5. Ashtekar Variables
6. Holonomies and Wilson Loops
7. The Loop Representation
8. Quantum Configuration Space
9. Spin Networks
10. Kinematical Hilbert Space
11. Quantum Geometry: Discrete Spectra
12. Area and Volume Operators
13. Diffeomorphism Invariance
14. Physical Hilbert Space
15. Dynamics and the Hamiltonian Constraint
16. Thiemann’s Construction
17. Semiclassical Limit and Coherent States
18. Loop Quantum Cosmology
19. Resolution of Singularities
20. Black Hole Entropy in LQG
21. Immirzi Parameter
22. Spinfoam Models and Path Integral Formulation
23. Covariant LQG and EPRL Model
24. Relation to Other Approaches
25. Conclusion

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1. Introduction

Loop Quantum Gravity (LQG) is a background-independent, non-perturbative approach to quantizing general relativity. It predicts a fundamentally discrete structure of space at the Planck scale and is one of the major contenders for a quantum theory of gravity.

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2. Motivation for Loop Quantum Gravity

LQG aims to:

• Respect the background independence of general relativity
• Provide a consistent canonical quantization
• Predict quantum geometry without introducing extra dimensions or strings

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3. Background Independence

Unlike many quantum field theories, LQG does not assume a fixed background spacetime. Instead, spacetime geometry emerges from quantum states themselves.

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4. Reformulating General Relativity

General relativity is reformulated using variables more amenable to quantization:

• Triads (instead of metric)
• SU(2) gauge connections

This leads to a gauge-theoretic formulation of gravity.

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5. Ashtekar Variables

Ashtekar variables recast GR into a form similar to Yang–Mills theory:

• Connection: \( A^i_a \)
• Conjugate momentum: densitized triad \( E^a_i \)

Poisson bracket:

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

---

6. Holonomies and Wilson Loops

Instead of local fields, LQG uses holonomies of the connection:

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

These define Wilson loops, gauge-invariant observables.

---

7. The Loop Representation

States are functionals of loops (closed holonomies). This leads to a loop representation of quantum states, where loops encode quantum geometry.

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8. Quantum Configuration Space

The space of generalized connections modulo gauge transformations provides the quantum configuration space. This is a non-separable, compact space suitable for defining a diffeomorphism-invariant measure.

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9. Spin Networks

Spin networks are graphs labeled by SU(2) representations:

• Edges: labeled by spins \( j \)
• Vertices: intertwining operators

They form an orthonormal basis of the Hilbert space and encode geometric information.

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10. Kinematical Hilbert Space

The Hilbert space of LQG is built from cylindrical functions of connections, equipped with the Ashtekar–Lewandowski measure. It admits a basis of spin network states.

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11. Quantum Geometry: Discrete Spectra

Operators measuring geometric quantities, like area and volume, have discrete spectra:

\[
\hat{A}_S \Psi = \sum_i 8\pi \gamma \ell_P^2 \sqrt{j_i(j_i + 1)} \Psi
\]

\[
\hat{V}R \Psi = \sum{v \in R} V_v \Psi
\]

---

12. Area and Volume Operators

Defined in terms of fluxes of the densitized triad:

• Area operator acts on surfaces intersected by spin network edges
• Volume operator acts on nodes of spin networks

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13. Diffeomorphism Invariance

States invariant under spatial diffeomorphisms are constructed by averaging spin network states over all diffeomorphisms. These encode the relational nature of quantum geometry.

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14. Physical Hilbert Space

Defined by solving all constraints:

• Gauss constraint: gauge invariance
• Diffeomorphism constraint: spatial diffeomorphism invariance
• Hamiltonian constraint: dynamics

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15. Dynamics and the Hamiltonian Constraint

Defining the Hamiltonian constraint operator is challenging. Thiemann constructed a version that is:

• Finite
• Background-independent
• Anomaly-free (in some versions)

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16. Thiemann’s Construction

Thiemann defined the Hamiltonian constraint using holonomies and volume operators, regularizing the expressions to avoid infinities.

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17. Semiclassical Limit and Coherent States

Efforts are made to construct coherent states that approximate classical geometries. These are crucial for recovering general relativity in the semiclassical limit.

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18. Loop Quantum Cosmology

A symmetry-reduced version of LQG applied to cosmology:

• Predicts a Big Bounce replacing the Big Bang
• Provides a discrete evolution equation for the universe

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19. Resolution of Singularities

LQC shows that classical singularities (e.g., in the early universe or black holes) are resolved by quantum geometry effects.

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20. Black Hole Entropy in LQG

Black hole horizons are modeled by punctured spin networks. Entropy arises from counting microstates:

\[
S = \frac{A}{4 \ell_P^2} + \text{corrections}
\]

The Immirzi parameter is fixed to match the Bekenstein–Hawking result.

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21. Immirzi Parameter

A free parameter in LQG affecting spectra of geometric operators. Its value is fixed phenomenologically (e.g., via black hole entropy).

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22. Spinfoam Models and Path Integral Formulation

A spinfoam is a sum-over-histories of spin networks. It provides a covariant formulation of LQG, encoding quantum dynamics through 2-complexes.

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23. Covariant LQG and EPRL Model

The Engle–Pereira–Rovelli–Livine (EPRL) model is a well-studied spinfoam model satisfying:

• Correct classical limit
• Matching with canonical theory
• Convergence properties

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24. Relation to Other Approaches

• LQG shares background independence with causal dynamical triangulations
• Differs from string theory in assumptions, mathematical tools, and phenomenology

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25. Conclusion

Loop Quantum Gravity provides a background-independent, mathematically rigorous framework for quantizing spacetime geometry. With discrete spectra for geometric operators, a rich structure of spin networks and spinfoams, and applications to cosmology and black holes, LQG is a promising contender in the quest for a quantum theory of gravity. Continued research is deepening its foundations and exploring potential observational consequences.

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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

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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.

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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.

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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 \( \Sigma_t \)

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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:

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

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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.

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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.

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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

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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20. Observables and Physical States

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

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21. Gauge Invariance and Diffeomorphism Constraint

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

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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

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23. Canonical Quantum Cosmology

Applies the formalism to cosmological models:

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

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24. Comparison with Path Integral Methods

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

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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.

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