Quantum Gravity & Quantum Cosmology

Introduction

Quantum gravity seeks to reconcile General Relativity with quantum mechanics. At the Planck scale ($\ell_P \sim 10^{-35}$ m, $t_P \sim 10^{-43}$ s), quantum fluctuations of spacetime itself become important.

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}}, \quad t_P = \sqrt{\frac{\hbar G}{c^5}}, \quad M_P = \sqrt{\frac{\hbar c}{G}} \approx 10^{19}\text{ GeV/c}^2$$

1. Canonical Quantization of Gravity

The ADM (Arnowitt-Deser-Misner) formulation splits spacetime into space + time:

$$ds^2 = -(N^2 - N_iN^i)dt^2 + 2N_idx^idt + h_{ij}dx^idx^j$$

where $N$ is the lapse function, $N^i$ is the shift vector, and $h_{ij}$ is the spatial metric.

Canonical Momenta:

$$\pi^{ij} = \frac{\sqrt{h}}{16\pi G}(K^{ij} - Kh^{ij})$$

where $K_{ij} = \frac{1}{2N}(\dot{h}_{ij} - \nabla_iN_j - \nabla_jN_i)$ is the extrinsic curvature.

2. Wheeler-DeWitt Equation

The quantum constraint equation for the wavefunction of the universe:

$$\hat{H}\Psi[h_{ij}] = 0$$

Explicitly:

$$\left[G_{ijkl}\frac{\delta^2}{\delta h_{ij}\delta h_{kl}} + \sqrt{h}R\right]\Psi[h_{ij}] = 0$$

where $G_{ijkl} = \frac{1}{2\sqrt{h}}(h_{ik}h_{jl} + h_{il}h_{jk} - h_{ij}h_{kl})$ is the DeWitt supermetric.

This equation has no explicit time dependence - the "problem of time" in quantum gravity.

3. Minisuperspace Quantum Cosmology

For a closed FLRW universe with scale factor $a(t)$:

$$\left[-\frac{\hbar^2}{2}\frac{\partial^2}{\partial a^2} + V(a)\right]\Psi(a) = 0$$

where the potential is:

$$V(a) = -\frac{3c^2a^2}{8\pi G} + \frac{\Lambda a^4}{8\pi G} + \frac{3c^2a^4}{8\pi G}\rho(a)$$

4. Loop Quantum Gravity (LQG) - In Depth

Loop Quantum Gravity is a non-perturbative, background-independent approach to quantum gravity that quantizes spacetime geometry itself. Unlike string theory, it requires no extra dimensions and makes no assumption about a background spacetime metric.

4.1 Ashtekar Variables

The key innovation of LQG is reformulating GR using new variables introduced by Abhay Ashtekar in 1986. Instead of the metric $g_{ij}$, we use:

$$A_i^a(x) \quad \text{(SU(2) connection)}$$$$E_a^i(x) \quad \text{(densitized triad)}$$

These are conjugate variables with Poisson bracket:

$$\{A_i^a(x), E_b^j(y)\} = 8\pi\gamma G\delta_b^a\delta_i^j\delta^3(x,y)$$

where $\gamma$ is the Barbero-Immirzi parameter, a dimensionless free parameter that must be fixed by comparison with the semiclassical limit. The relation to the metric is:

$$q_{ij} = E_a^i E_b^j \delta^{ab}$$$$\sqrt{q} = \sqrt{\det(E_a^i E_b^j)}$$

4.2 Constraint Equations

The Hamiltonian formulation of GR contains three constraints that quantum states must satisfy:

Gauss Constraint (gauge invariance):

$$\mathcal{G}_a = D_iE_a^i = \partial_i E_a^i + \epsilon_{abc}A_i^b E_c^i = 0$$

This generates SU(2) gauge transformations - physical states must be gauge invariant.

Diffeomorphism Constraint (spatial covariance):

$$\mathcal{D}_i = F_{ij}^a E_a^j - (1+\gamma^2)K_{[i}^a E_{j]}^a = 0$$

where $F_{ij}^a = 2\partial_{[i}A_{j]}^a + \epsilon^{abc}A_i^b A_j^c$ is the curvature of the connection. This generates spatial diffeomorphisms.

Hamiltonian Constraint (time evolution):

$$\mathcal{H} = \frac{1}{\sqrt{\det E}}\epsilon^{ijk}E_a^i E_b^j\left(F_{kl}^c\epsilon^{abc} - 2(1+\gamma^2)K_{[k}^a K_{l]}^b\right) = 0$$

This is the most difficult constraint to implement quantum mechanically and embodies the dynamics of GR.

4.3 Holonomies and Fluxes

The connection $A_i^a$ is not well-defined as an operator on the quantum Hilbert space. Instead, LQG uses holonomies (parallel transport) along edges $e$:

$$h_e[A] = \mathcal{P}\exp\left(\int_e A_i^a\tau_a dx^i\right) \in SU(2)$$

where $\tau_a = -i\sigma_a/2$ are SU(2) generators and $\mathcal{P}$ denotes path ordering. The conjugate variables are fluxes through 2-surfaces $S$:

$$E_S^a = \int_S E_a^i n_i d^2x$$

The fundamental quantum commutation relation becomes:

$$[\hat{h}_e, \hat{E}_S^a] = 8\pi\gamma\ell_P^2 \hat{h}_e \tau^a \delta(e,S)$$

where $\delta(e,S)$ is non-zero only if edge $e$ pierces surface $S$.

4.4 Kinematical Hilbert Space

Quantum states are functionals $\Psi[A]$ of the connection. The kinematical Hilbert space$\mathcal{H}_{\text{kin}}$ is constructed using spin networks as basis states:

$$|\Gamma, j_e, i_n\rangle$$

where:

  • • $\Gamma$ is an embedded graph (nodes and edges in 3-space)
  • • $j_e \in \{0, 1/2, 1, 3/2, \ldots\}$ is the spin label on edge $e$
  • • $i_n$ is an intertwiner at node $n$, coupling the spins of adjacent edges

The wave function is evaluated on a holonomy configuration:

$$\Psi_\Gamma[A] = \text{Tr}_{j_1,\ldots,j_E}\left[i_{n_1} h_{e_1}[A] i_{n_2} h_{e_2}[A] \cdots\right]$$

4.5 Geometric Operators - Area

The area operator $\hat{A}_S$ for a surface $S$ acts on spin network states:

$$\hat{A}_S = \sum_{p \in S \cap \Gamma} \sqrt{\hat{E}_p^a \hat{E}_{p\,a}}$$

The eigenvalues are quantized:

$$A_S = 8\pi\gamma\ell_P^2 \sum_{p \in S\cap\Gamma} \sqrt{j_p(j_p+1)}$$

where the sum is over punctures $p$ where edges pierce the surface $S$. This is the discrete spectrum of area - a fundamental prediction of LQG!

For $\gamma \approx 0.2375$ (fixed by black hole entropy), the minimal non-zero area is:

$$A_{\min} = 4\sqrt{3}\pi\gamma\ell_P^2 \approx 5.17 \ell_P^2$$

4.6 Geometric Operators - Volume

The volume operator for a region $R$ is more complex. At a node $n$ where$N$ edges meet, the volume contribution is:

$$\hat{V}_n = \left(\frac{8\pi\gamma\ell_P^2}{6}\right)^{3/2}\sqrt{|\epsilon^{ijk}\hat{E}_i^a\hat{E}_j^b\hat{E}_k^c \epsilon_{abc}|}$$

Volume eigenvalues are also discrete and vanish for graphs without nodes (pure 1D structures have zero volume). For a 4-valent node with spins $(j_1,j_2,j_3,j_4)$, typical volumes scale as:

$$V \sim \ell_P^3\sqrt{j_1j_2j_3}$$

4.7 Physical Interpretation - Atoms of Space

LQG predicts that space is quantized at the Planck scale. A spin network represents:

  • • Edges: Elementary quanta of area (surfaces piercing them have quantized area)
  • • Nodes: Elementary quanta of volume (regions containing them have quantized volume)
  • • Spins $j_e$: Determine how much area/volume each quantum contributes
  • • Intertwiners $i_n$: Encode how areas and volumes connect at nodes

This gives a picture of space as a "quantum fabric" or "spin foam" - a discrete structure at Planck scale that appears continuous at macroscopic scales, analogous to how matter appears continuous but is atomic.

4.8 The Hamiltonian Constraint - Dynamics

The most challenging aspect of LQG is implementing the Hamiltonian constraint $\hat{\mathcal{H}}\Psi = 0$. The Thiemann operator is:

$$\hat{\mathcal{H}} = -\frac{1}{16\pi\gamma^3 G^2\hbar}\sum_n \text{Tr}\left[\hat{h}_{\square_n}\hat{V}_n^{-1}[\hat{h}_{\square_n}, \hat{V}_n]\right]$$

where $\square_n$ denotes a small loop around node $n$ and $[\,,\,]$ is a commutator. This operator acts on spin networks by:

  • • Adding new nodes and edges to the graph
  • • Changing spin labels $j_e \to j_e \pm 1$
  • • Modifying intertwiners at nodes

4.9 Semiclassical Limit and Coherent States

For LQG to reproduce classical GR, we need states that peak on classical geometries.Coherent states are constructed to approximate classical metrics:

$$|\Gamma, \{j_e\}_{\text{large}}, g_{\text{cl}}\rangle$$

where $j_e \gg 1$ (large spins) and the graph $\Gamma$ is chosen to approximate a continuous manifold. In the limit $j_e \to \infty$ at fixed $j_e\ell_P^2 = \text{const}$, these states reproduce classical geometry with quantum corrections suppressed by powers of $\ell_P/L$ where $L$ is the macroscopic scale.

4.10 Key Predictions and Observables

1. Discrete Spectra:

All geometric quantities (area, volume, length) have discrete eigenvalues at Planck scale.

2. Singularity Resolution:

Classical singularities (Big Bang, black hole centers) are replaced by quantum bounces or bridges due to Planck-scale repulsive effects.

3. Black Hole Entropy:

$$S_{\text{BH}} = \frac{\gamma}{4}\frac{A_H}{\ell_P^2}\ln(2) + \mathcal{O}(1)$$

Counting spin network states on the horizon reproduces Bekenstein-Hawking entropy for $\gamma = \ln(2)/(2\pi\sqrt{3}) \approx 0.2375$.

4. Modified Dispersion Relations:

$$E^2 = p^2c^2 + m^2c^4 + \alpha\frac{E^3}{\ell_P M_Pc^2} + \ldots$$

Planck-scale corrections could cause energy-dependent speed of light, testable with astrophysical observations (gamma-ray bursts, active galactic nuclei).

5. Cosmological Implications:

Loop Quantum Cosmology (LQC) predicts a "Big Bounce" replacing the Big Bang singularity, potentially observable in CMB anomalies and primordial gravitational waves.

4.11 Open Challenges

  • • Dynamics: Complete understanding of Hamiltonian constraint and time evolution
  • • Semiclassical limit: Rigorous derivation of classical GR from spin foam amplitudes
  • • Matter coupling: Including fermions and gauge fields consistently
  • • Observational signatures: Finding testable predictions distinct from alternatives
  • • Barbero-Immirzi parameter: Deriving $\gamma$ from first principles

5. Spin Networks and Spin Foams

Quantum states of geometry are represented by spin networks - graphs with edges labeled by spins:

$$|\Gamma, j_e, i_n\rangle$$

where $\Gamma$ is a graph, $j_e$ are spin labels on edges, and $i_n$ are intertwiners at nodes.

Spin foams extend this to spacetime: 2-complexes with faces labeled by spins, representing quantum spacetime histories (path integral formulation).

6. String Theory

Fundamental objects are 1-dimensional strings, not point particles. The Nambu-Goto action:

$$S = -\frac{1}{2\pi\alpha'}\int d^2\sigma\sqrt{-\det(\partial_\alpha X^\mu\partial_\beta X^\nu g_{\mu\nu})}$$

where $\alpha' = \ell_s^2$ is the string length parameter. Equivalent Polyakov action:

$$S = -\frac{1}{4\pi\alpha'}\int d^2\sigma\sqrt{-h}h^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu g_{\mu\nu}$$

Virasoro Constraints:

$$L_m|\psi\rangle = 0, \quad \bar{L}_m|\psi\rangle = 0$$

Consistency requires $D = 26$ (bosonic) or $D = 10$ (superstring).

7. Extra Dimensions and Compactification

String theory requires extra spatial dimensions, compactified on small manifolds (typically Calabi-Yau).

Kaluza-Klein Theory:

A simple example: 5D metric with one compact dimension of radius $R$:

$$ds^2 = g_{\mu\nu}dx^\mu dx^\nu + R^2(dy + A_\mu dx^\mu)^2$$

This produces 4D gravity + electromagnetism from pure 5D gravity!

Momentum in extra dimension: $p_y = n/R$ gives tower of Kaluza-Klein modes.

8. D-branes and M-theory

D-branes are extended objects where open strings can end. A Dp-brane has $p$ spatial dimensions.

Dirac-Born-Infeld Action:

$$S = -T_p\int d^{p+1}\xi\sqrt{-\det(g_{ab} + 2\pi\alpha'F_{ab})}$$

M-theory unifies the five consistent superstring theories in 11 dimensions:

$$S = \frac{1}{2\kappa^2}\int d^{11}x\sqrt{-g}(R - \frac{1}{2}|F_4|^2) + S_{\text{CS}}$$

where $F_4$ is the 4-form field strength and $S_{\text{CS}}$ is a Chern-Simons term.

9. AdS/CFT Correspondence

The holographic principle: quantum gravity in $(d+1)$-dimensional Anti-de Sitter space is equivalent to a conformal field theory on the $d$-dimensional boundary.

$$Z_{\text{gravity}}[\phi_0] = Z_{\text{CFT}}[J = \phi_0]$$

AdS$_5$ metric:

$$ds^2 = \frac{R^2}{z^2}(-dt^2 + dx_i^2 + dz^2)$$

The boundary at $z \to 0$ hosts the dual CFT. This provides a non-perturbative definition of quantum gravity!

10. Microscopic Black Hole Entropy

String Theory Calculation:

For certain extremal black holes, count D-brane microstates:

$$S_{\text{micro}} = \ln\Omega(Q_1, Q_5, n) = 2\pi\sqrt{Q_1Q_5n}$$

This matches the Bekenstein-Hawking formula exactly!

LQG Calculation:

Count spin network states on the horizon:

$$S = \frac{\gamma}{4}\frac{A}{\ell_P^2}\ln(j)$$

Reproduces Bekenstein-Hawking for suitable choice of $\gamma$ and $j$.

11. Quantum Cosmology - Wave Function of the Universe

Hartle-Hawking No-Boundary Proposal:

$$\Psi[h_{ij}] = \int_{h_{ij}}\mathcal{D}g_{\mu\nu}\,e^{iS[g_{\mu\nu}]/\hbar}$$

Sum over compact Euclidean geometries with boundary $h_{ij}$. The universe has no initial boundary in time!

Tunneling Proposal (Vilenkin):

The universe tunnels from nothing:

$$\Psi \propto e^{-S_E}$$

where $S_E$ is the Euclidean action. Describes creation of universe from quantum vacuum.

12. Loop Quantum Cosmology - Big Bounce

LQC modifies the Friedmann equation near the Planck density:

$$H^2 = \frac{8\pi G}{3}\rho\left(1 - \frac{\rho}{\rho_{\text{crit}}}\right)$$

where $\rho_{\text{crit}} \sim \rho_{\text{Planck}}$. This prevents the singularity:

  • • As $\rho \to \rho_{\text{crit}}$, $H \to 0$
  • • Universe bounces: contraction → expansion
  • • Singularity replaced by quantum bridge

13. Black Hole Information Paradox

Hawking radiation appears thermal (maximum entropy), but quantum mechanics requires unitary evolution:

$$S_{\text{radiation}} + S_{\text{BH}} = \text{const?}$$

Proposed resolutions:

  • • Information encoded in correlations (subtle)
  • • Firewalls at horizon
  • • ER=EPR: entanglement = wormholes
  • • Holography and Page curve

Page Time:

$$t_{\text{Page}} \sim \frac{M^3G^2}{\hbar c^4}$$

After this time, entanglement entropy of radiation should decrease, restoring unitarity.

14. Emergent Gravity

Gravity might not be fundamental but emergent from quantum entanglement (Verlinde, Jacobson).

Jacobson's Thermodynamic Derivation:

From $\delta Q = TdS$ and horizon thermodynamics, derive:

$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$

Einstein's equations as thermodynamic equations of state!

📺 Video Lectures

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Raphael Bousso & Brian Greene: Is Gravity the Hidden Key to Quantum Physics?

Leading physicist Raphael Bousso joins Brian Greene to explore the almost unreasonable capacity of our theories of gravity to give deep insights into quantum physics. This conversation delves into the profound connections between gravitational physics and quantum information theory, examining how black holes, holography, and spacetime geometry reveal fundamental truths about the quantum nature of reality.

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  • The holographic principle and AdS/CFT correspondence
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John Preskill: Quantum Information and Spacetime

John Preskill, Richard P. Feynman Professor of Theoretical Physics at Caltech, presents "Quantum Information and Spacetime." Aside from enabling revolutionary future technologies, quantum information science is providing powerful new tools for attacking deep problems in fundamental physical science. In particular, the recent convergence of quantum information and quantum gravity is sparking exciting progress on some old and very hard questions.

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