Quantum Gravity

Prerequisites: General Relativity, Canonical Quantum Field Theory. Full path (30 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• Overview
• FAQ
• History
• Criticism

Intuitive

Quantum gravity is the generic term for theories that try to combine the principles of quantum theories with general relativity.

So far, no experimentally verified theory of quantum gravity exists.

• A great book on quantum gravity for laypersons is "Three Roads to Quantum Gravity" by Lee Smolin

Concrete

Recommended Textbooks:

• Quantum Gravity by Rovelli which focuses on the Loop Quantum Gravity perspective.
• "Approaches to Quantum Gravity" by Daniele Oriti
• "Quantum Gravity" by K. Kiefer.

Recommended Essays:

• Quantum Gravity: The View From Particle Physics by Hermann Nicolai
• Introduction to the Effective Field Theory Description of Gravity by John F. Donoghue
• Gauge Gravity: a forward-looking introduction by Andrew Randono

From a brief analysis of the standard Einstein-Hilbert action, we see that fluctuations of the metric at the Planck scale should become very violent, leading to potential changes in the topology of the spacetime [103, 104]. https://arxiv.org/abs/1711.00864

Abstract

The problem can be stated precisely in the language of quantum field theory. General relativity is described by the Einstein-Hilbert action (with $c = 1$)

$$S_{\text{EH}} = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R .$$

Expanding around flat space, $g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}$ with $\kappa^2 = 32\pi G$, gives a theory of a massless spin-2 field $h_{\mu\nu}$ with infinitely many interaction vertices, all proportional to powers of $\kappa$. In units $\hbar = c = 1$ Newton's constant has mass dimension $-2$, $G = 1/M_{\text{Pl}}^2$, so by power counting the theory is not renormalizable: each loop order requires new counterterms of higher order in the curvature. For pure gravity the one-loop divergences vanish on shell ('t Hooft and Veltman), but at two loops a counterterm cubic in the Riemann tensor is required (Goroff and Sagnotti), and with matter fields divergences appear already at one loop.

As an effective field theory the quantized Einstein-Hilbert action is consistent at energies $E \ll M_{\text{Pl}} \approx 1.2\times 10^{19}$ GeV. The unknown higher-curvature terms are suppressed by powers of $E^2/M_{\text{Pl}}^2$, and the leading quantum corrections, for example to the Newtonian potential at long distances, are computable and unambiguous. Quantum gravity in the strict sense is the question of what replaces this effective theory at and above the Planck scale.

Beyond the ultraviolet problem, two structural issues appear in any approach. First, diffeomorphism invariance means there is no background time. In the canonical (ADM) formulation of a spatially closed universe the Hamiltonian is a sum of constraints, and the Wheeler-DeWitt equation $\hat{\mathcal{H}}\,\Psi[h_{ij}] = 0$ for wave functionals of the spatial metric contains no time parameter (problem of time). Second, there are no local gauge-invariant observables in a diffeomorphism-invariant theory; observables must be relational or defined at an asymptotic boundary.

The main approaches answer these problems in different ways:

• String theory replaces point particles by strings. The massless spin-2 state of the closed string gives the graviton, and perturbative scattering amplitudes are expected to be ultraviolet finite. Through AdS/CFT, quantum gravity in asymptotically anti-de Sitter space is defined non-perturbatively by a conformal field theory on the boundary.
• Loop quantum gravity quantizes the Ashtekar-Barbero formulation, in which the canonical variables are an $SU(2)$ connection and a densitized triad. Kinematical states are spin networks, and geometric operators have discrete spectra; the area of a surface crossed by edges with spins $j_i$ is $A = 8\pi\gamma\,\ell_{\text{P}}^2\sum_i\sqrt{j_i(j_i+1)}$, with $\ell_{\text{P}}^2 = \hbar G/c^3$ and the Barbero-Immirzi parameter $\gamma$.
• Asymptotic safety (proposed by Weinberg) keeps the metric as fundamental field and conjectures a non-Gaussian ultraviolet fixed point of the renormalization group flow with finitely many relevant directions.
• Lattice approaches such as causal dynamical triangulations define the gravitational path integral as a sum over piecewise flat geometries and look for a continuum limit.

A consistency check that every candidate is expected to pass is the Bekenstein-Hawking entropy of black holes, $S = \frac{k_B c^3 A}{4G\hbar}$, which should arise from counting microscopic states. String theory reproduces it, including the factor $\frac14$, for certain supersymmetric extremal black holes (Strominger and Vafa).

• For a nice overview, see Towards the map of quantum gravity by Jakub Mielczarek, Tomasz Trześniewski and alos
• Strings, loops and others: a critical survey of the present approaches to quantum gravity by Carlo Rovelli
• Conceptual Problems in Quantum Gravity and Quantum Cosmology by Claus Kiefer
• What Quantum Gravity Needs Is More Experiments - Math won’t solve quantum gravity. Experimentation will. by Sabine Hossenfelder

Why is it interesting?

One of the big problems in physics — perhaps the biggest! — is figuring out how our two current best theories fit together. On the one hand we have the Standard Model, which tries to explain all the forces except gravity, and takes quantum mechanics into account. On the other hand we have General Relativity, which tries to explain gravity, and does not take quantum mechanics into account. Both theories seem to be more or less on the right track — but until we somehow fit them together, or completely discard one or both, our picture of the world will be deeply schizophrenic.

It seems plausible that as a step in the right direction we should figure out a theory of gravity that takes quantum mechanics into account, but reduces to General Relativity when we ignore quantum effects (which should be small in many situations). This is what people mean by "quantum gravity" — the quest for such a theory.

John Baez

At the beginning of the XX century,General Relativity (GR) and Quantum Mechanics (QM) once again began reshaping our basic understanding of space and time and, respectively, matter, energy and causality —arguably to a no lesser extent. But we have not been able to combine these new insights into a novel coherent synthesis, yet. The XX century scientific revolution opened by GR and QM is therefore still wide open. We are in the middle of an unfinished scientific revolution. Quantum Gravity is the tentative name we give to the “synthesis to be found”."Unfinished Revolution" by C. Rovelli

See also

• Why Quantize Gravity? By Sabine Hossenfelder

Overview

Here's an incomplete list of popular approaches to formulate a theory of quantum gravity:

Traditional

• Discrete methods
  • Dynamical triangulations
  • Regge calculus
  • Simplicial models
• Approximate theories
  • Euclidean quantum gravity
  • Perturbative quantum gravity
  • QFT on curved space–times
• Unorthodox approaches
  • Sorkin’s Posets
  • Finkelstein
  • Twistors

New

• Non-commutative geometry
• Null surfaces
• Spin foam models

Most popular

• String Theory
• Loop Quantum Gravity
• Asymptotic Safety
• Supergravity

List adapted from Strings, loops and others: a critical survey of the present approaches to quantum gravity by Carlo Rovelli

FAQ

History

• See also: Unfinished revolution by Rovelli

Criticism

“Pursuing different and independent ideas is certainly a good strategy as long as we do not know the final answer, but it is a bit worrisome (at least to me) that the proponents of the different approaches not only base their approaches on very different assumptions, but continue to speak languages that are foreign to one another. Surely, when zeroing in on the ‘correct’ theory there should be a convergence of ideas and concepts: when Schrödinger proposed wave mechanics and Heisenberg formulated matrix mechanics, these were initially regarded as very different, but it did not take long before it became clear that they were just equivalent descriptions of the same theory. Unfortunately, at this time there is no such convergence in existing approaches to quantum gravity – a sign that we are probably still very far from the correct answer! […] In one form or another the existing approaches to quantum gravity suffer from a very large number of ambiguities, so far preventing any kind of prediction with which the theory will stand or fall. Even at the risk of sounding polemical, I would put this ambiguity at 10^500 (or even more) – in any case a number too large to cut down for any conceivable kind of experimental or observational advance. […] Being exposed to many talks from the different ‘quantum gravity camps’ I am invariably struck by the success stories I keep hearing, and the implicit or explicit claims that ‘we are almost there’. I, for one, would much prefer to hear once in a while that something does not work, and to see some indications of inconsistencies that might enable us to discriminate between a rapidly growing number of diverging ideas on quantum gravity [27, 28]. If, however, the plethora of theory ambiguities were to stay with us I would conclude that our search for an ultimate explanation, and with it the search for quantum gravity, may come to an ignominious end” from Quantum Gravity: The View From Particle Physics by Hermann Nicolai

Recommended Resources:

• Should I be thinking about quantum gravity? by John Baez

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
7. Legendre Transformation
8. Manifold
9. Newton's Second Law
10. Vector Calculus
11. Curvature
12. Fiber Bundles
13. Newton's law of Gravity
14. Newtonian Mechanics
15. Parallel Transport
16. Variational Calculus
17. Configuration Space
18. Connections
19. Euler-Lagrange Equations
20. Special Relativity
21. Lagrangian Formalism
22. Levi-Civita Connection
23. Phase Space
24. Classical Field Theory
25. Einstein Equation
26. Hamiltonian Formalism
27. General Relativity
28. Schrödinger Equation
29. Canonical Quantum Mechanics
30. Canonical Quantum Field Theory