Table of Contents
- Introduction
- Motivation for Loop Quantum Gravity
- Background Independence
- Reformulating General Relativity
- Ashtekar Variables
- Holonomies and Wilson Loops
- The Loop Representation
- Quantum Configuration Space
- Spin Networks
- Kinematical Hilbert Space
- Quantum Geometry: Discrete Spectra
- Area and Volume Operators
- Diffeomorphism Invariance
- Physical Hilbert Space
- Dynamics and the Hamiltonian Constraint
- Thiemann’s Construction
- Semiclassical Limit and Coherent States
- Loop Quantum Cosmology
- Resolution of Singularities
- Black Hole Entropy in LQG
- Immirzi Parameter
- Spinfoam Models and Path Integral Formulation
- Covariant LQG and EPRL Model
- Relation to Other Approaches
- Conclusion
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.
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
3. Background Independence
Unlike many quantum field theories, LQG does not assume a fixed background spacetime. Instead, spacetime geometry emerges from quantum states themselves.
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.
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.
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.
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.
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.
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
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.
14. Physical Hilbert Space
Defined by solving all constraints:
- Gauss constraint: gauge invariance
- Diffeomorphism constraint: spatial diffeomorphism invariance
- Hamiltonian constraint: dynamics
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)
16. Thiemann’s Construction
Thiemann defined the Hamiltonian constraint using holonomies and volume operators, regularizing the expressions to avoid infinities.
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.
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
19. Resolution of Singularities
LQC shows that classical singularities (e.g., in the early universe or black holes) are resolved by quantum geometry effects.
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.
21. Immirzi Parameter
A free parameter in LQG affecting spectra of geometric operators. Its value is fixed phenomenologically (e.g., via black hole entropy).
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.
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
24. Relation to Other Approaches
- LQG shares background independence with causal dynamical triangulations
- Differs from string theory in assumptions, mathematical tools, and phenomenology
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.
