# What is Loop Quantum Gravity?

15 Mar

Loop Quantum Gravity (also known as Canonical Quantum General Relativity) is a quantization of General Relativity (GR) including its conventional matter coupling. It merges General Relativity and Quantum Mechanics without extra speculative assumptions (e.g., no extra-dimensions, just 4 dimensions; no strings; not assuming that space is formed by individual discrete points). LQG has no ambition to do unification of forces or to add more than 4 spacetime dimension, nor supersymmetry [1]. In this sense, LQG has a less ambitious research program than String Theory and is its biggest competitor.

General Relativity envisages spacetime and the gravitational field as the same entity, “spacetime” itself, that, in many ways, can be seen as a physical object analog to the electromagnetic field. Quantum Mechanics (QM) was formulated by means of an external time variable t like it appears in Schrödinger equation

$i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$

Fig. 1 below shows the solutions of SE when applied to the simplest atom in Nature, the hydrogen atom, where the potential is $V=\frac{1}{r}$. Electrons that rotate around the positive nucleus have the energy is quantified according to the law $\latex E_n=\frac{13.6}{n^2}$ and the wave function $\latex \Psi$ are given by the mathematical functions given in Figure 1.

However, in General Relativity (GR) this external time (represented above by the letter t) is incompatible because the role of time becomes dynamical in the framework of Minkowski spacetime. Time is no longer absolute (as Sir Isaac Newton once stated) but is relative to a frame of measurement. In addition, GR was formulated formerly by Albert Einstein in the framework of Riemannian geometry, where it is assumed that the metric is a smooth and deterministic dynamical field (Fig.2).

Fig.2 – Example of Riemann surface. Image courtesy: http://virtualmathmuseum.org (See for details about this specific surface here: http://virtualmathmuseum.org/Surface/riemann/riemann.html

This raises an immediate problem, since QM requires that any dynamical field be quantized, that is, be made of discrete quanta that follows probabilistic laws… This would mean that we should treat quanta of space and quanta of time…

All the known forces in the universe have been quantized, except gravity. The first approach to quantization of gravity consists of writing the gravitational field as composed of the sum of two terms, a background field $g_{background}$ and a perturbation $h(x)$. So, its full metric $g_{\mu \nu}$:

$g(x)=\eta_{background} (x)+ h(x)$

where $\eta_{\mu \nu}$ represents the background spacetime, normally Minkowski) and $h_{\mu \nu}$ a perturbation of the field (representing the graviton). The Minkowski space united space and time as a single entity introducing the new concept of space-time manifold where two points are distant by

$ds^2= c^2 dt^2-d \bf{x}^2$.

The problem resides in the intrinsic difficulty that this approach face when describing extreme astrophysical (near a black hole) or cosmological scenarios (Big Bang singularity). The inconsistency between GR and QM becomes more clear when looking at Einstein equation of GR:

$R_{\mu \nu} - \frac{1}{2}gR=\kappa T_{\mu \nu}(g)$

$R_{\mu \nu}$ is the Ricci curvature tensor, $R$ is the curvature, and $T_{\mu \nu}$ is the energy-momentum tensor. $\kappa \equiv \frac{8 \pi G}{c^4}$. While the left-hand side is described by a classical theory of fields, the right-hand side is described by the quantum theory of fields…

LQG avoids any background metric structure (described by the metric g), choosing a background independent approach, along the suggestion of Roger Penrose on the spin-networks where a system is supposed to be built of discrete “units” (anything from the system can be known on purely combinatorial principles) and all is purely relational (avoiding the use of space and time…) In GR spacetime is represented as a well-defined grid of lines, even if curved in the presence of a massive body, such as

In LQG, spacetime is represented rather as background-independent, the geometry is not fixed, is a spin network of points defined by field quantities and angular momentum, more like a mesh of polygons; spacetime is more a derived concept rather than a pre-structure, pre-concept on which events take place, as shown here

Image credit: http://www.timeone.ca/glossary/spin-network/

This new representation of fields has the advantage of representing both their intrinsic attributes but also their induction attributes. That is, the field quantities depend not only of the point where it exist but also on the neighboring points connected by a line. That’s why the mathematical idea that best express this representation is the holonomy of the gauge potential $A$ along the loop (line) $\alpha$, $U(A,\alpha)$, which is given by the integral

$U(A,\alpha)=exp{\int_0^{2\pi} ds A_a(\alpha(s))\frac{d\alpha^a(s)}{ds}}.$

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