From Osher Doctorow

Harold Steinacker of U. Wien (Vienna) Austria, who has 20 papers in
arXiv including 3 in 2007, including many on fuzzy spheres and one on
fuzzy instantons, in "Emergent gravity from noncommutative gauge
theory," arXiv: 0708.2426 v1 [hep-th] 20 Aug 2007, 37 pages, provides
much of an outline of a theory of Noncommutative-based Quantum Gravity
via traces or integrals of traces in matrix models.

I have been discussing traces in the previous Section of this thread,
pointing out their fundamental importance in Probable Causation/
Influence (PI).

To rapidly peruse Steinacker's paper, look first at his two key
(boxed) equations 47 and 51 for (scalar) action, then his equation
(1), then his equation (12), then his equation (15). Read the
Introduction and Abstract to his paper, the beginning of Section 3 of
his paper (Effective Metric), and the Conclusion of his paper. Both
equation 47 and 51 are based on integrals of traces, (1) is based on
the trace, so is (12) and so is (15).

He shows how the matrix model formulation of Noncommutative gauge
theory contains gravitation via an su(n) gauge theory coupled to
gravitation with u(1) components of covariant coordinates X^a
determining dynamical geometry. Until this, the problem of how to
define Noncommutative su(n) gauge theory was unsolved because the u(1)
sector of this theory can't be apparently disentangled from the su(n)
sector, but here he explains this by the coupling of su(n) gauge
fields to gravity. The dynamical Poisson tensor is involved.

His basic message is that gravitation is already contained in the
simplest matrix models of Noncommutative gauge theory and that new
ideas in additional to those aren't required for Quantum Gravity.

SInce Noncommutativity is required in an essential way in his theory
and there is no Commutative analog, doesn't this affect Probable
Causation/Influence (PI)? No, because PI is one-sided via P(A-->B) =
1 + P(AB) - P(A) and P ' (A-->B) = 1 + P(B) - P(A) with P(B) < = P(A)
in the latter, which are asymmetric and ordinarily Noncommutative in
the --> or P( --> ) operation. In special cases, Commutativity can
arise in PI via P(A<-->B) = P(AB) + P(A ' B ' ), where (A<-->B) is the
intersection of (A-->B) = (AB ' ) ' = A ' U B and (B-->A) = (BA ' ) '
= B ' U A, which are equivalent to Probable Correlation, but these are
not the usual situation.

Osher Doctorow