Question: How is space represented with quantum gravity?
About the article "General relativity", section "Quantum gravity".

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Main contributions.
Loop quantum gravity.
In 1988 Carlo Rovelli, Lee Smolin, and Abhay Ashtekar introduced a theory of quantum gravity called loop quantum gravity.
In 1995 Rovelli and Smolin obtained a basis of states of quantum gravity, labelled by Penrose's spin networks, and using this basis they were able to show that the theory predicts that area and volume are quantized.
This result indicates the existence of a discrete structure of space at very small scale.
In 1997 Rovelli and Michael Reisenberger introduced a "sum over surfaces" formulation of theory, which has since evolved into the currently covariant "spinfoam" version of loop quantum gravity.
In 2008, in collaboration with Jonathan Engle and Roberto Pereira, he has introduced the spin foam vertex amplitude which is the basis of the current definition of the loop quantum gravity covariant dynamics.
The loop theory is today considered a candidate for a quantum theory of gravity.
It finds applications in quantum cosmology, spinfoam cosmology, and quantum black hole physics. 
The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.
Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.
At low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.
At very high energies, however, the result are models devoid of all predictive power ("non-renormalizability").
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.
The theory promises to be a unified description of all particles and interactions, including gravity; the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.
In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.
Another approach starts with the canonical quantization procedures of quantum theory.
Using the initial-value-formulation of general relativity (cf.
evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined.
However, with the introduction of what are now known as Ashtekar variables, this leads to a promising model known as loop quantum gravity.
Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being dynamical triangulations, causal sets, twistor models or the path-integral based models of quantum cosmology.
All candidate theories still have major formal and conceptual problems to overcome.
They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.
Non-associative Ashketar gravity.
There is a paper on Non-associative Ashketar gravity or Loop quantum gravity in seven dimensions. 
Much of the difficulty in meshing these theories at all energy scales comes from the different assumptions that these theories make on how the universe works.
Quantum field theory depends on particle fields embedded in the flat space-time of special relativity.
General relativity models gravity as a curvature within space-time that changes as a gravitational mass moves.
Historically, the most obvious way of combining the two (such as treating gravity as simply another particle field) ran quickly into what is known as the renormalization problem.
In the old-fashioned understanding of renormalization, gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results.
This is in contrast with quantum electrodynamics where, while the series still do not converge, the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization. 
Loop quantum gravity is formulated in a background-independent language.
No spacetime is assumed a priori, but rather it is built up by the states of theory themselves - however scattering amplitudes are derived from formula_324-point functions and these, formulated in conventional quantum field theory, are functions of points of a background space-time.
The relation between the background-independent formalism and the conventional formalism of quantum field theory on a given spacetime is far from obvious, and it is far from obvious how to recover low-energy quantities from the full background-independent theory.
One would like to derive the formula_324-point functions of the theory from the background-independent formalism, in order to compare them with the standard perturbative expansion of quantum general relativity and therefore check that loop quantum gravity yields the correct low-energy limit.
A strategy for addressing this problem has been suggested; the idea is to study the boundary amplitude, namely a path integral over a finite space-time region, seen as a function of the boundary value of the field.
In conventional quantum field theory, this boundary amplitude is well–defined and codes the physical information of the theory; it does so in quantum gravity as well, but in a fully background–independent manner.
A generally covariant definition of formula_324-point functions can then be based on the idea that the distance between physical points –arguments of the formula_324-point function is determined by the state of the gravitational field on the boundary of the spacetime region considered.
Progress has been made in calculating background independent scattering amplitudes this way with the use of spin foams.
This is a way to extract physical information from the theory.
Claims to have reproduced the correct behaviour for graviton scattering amplitudes and to have recovered classical gravity have been made.
"We have calculated Newton's law starting from a world with no space and no time."
- Carlo Rovelli.