When the dynamical variables do no commute, the underlying coordinate geometry must also be non-commutative.

The Einstein hole argument shows that for the points of spacetime to be distinguishable from each other, the spacetime has to be overlaid by a metric. The metric at a point then acts as a flag, a marker so to say, labelling the point.

In quantum theory as currently formulated, there is assumed to exist a background spacetime and a universe dominated by classical bodies and fields. This permits a metric which serves as marker.

However, when there are no classical objects around, the metric undergoes quantum fluctuations [because it is being produced by quantum sources] and hence can no longer serve as a marker. The distinguishability of spacetime points is lost.

We see that in conventional quantisation, we necessarily keep the universe dominantly classical, and quantise a negligible fraction of it. What should we do if we want to quantise everything that is classical, in one go?

Consider Newtonian mechanics, or special relativity as a starting point. Raise every dynamical degree of freedom (configuration variables and their corresponding canonical momenta) to the status of matrices. Do not impose Heisenberg quantum commutation relations [q,p]=i\hbar by hand (they will emerge). Replace the four dimensional spacetime coordinate geometry by a non-commuting coordinate system. The Einstein hole argument no longer applies.

We have a matrix-valued polynomial Lagrangian. It's trace defines the trace Lagrangian to be used in the action principle. Use a new absolute time to describe evolution. The Lagrangian dynamics so defined is a pre-quantum theory, from which quantum theory emerges. In a sense this pre-quantum theory can be called true quantisation, a no-holds barred purist quantisation wherein all classical elements are removed. What we call quantum theory is a stop-gap, a half-way home, which serves very well phenomenologically, but leaves a lot unexplained as well.

The non-commutative geometry dictates what we mean by elementary particles, and what properties they have. It also dictates what form the Lagrangian takes. In Newton's mechanics the simplest description of the universe is as a collection of colliding point particles, and the Lagrangian for each one of them is simply it's kinetic energy. So we raise the position variable of each point particle to a (Grassmann number valued) matrix, and define its velocity as the time rate of change of its matrix-valued position vector in the non-commutative space. Then we can define the kinetic energy as the trace Lagrangian - in place of the mass of the particle there is a length scale, which is the only parameter associated with the particle. The universe consists of such colliding matrix-particles.

Because the entries in the matrices are Grassmann numbers, there is a natural place for bosonic and fermionic degrees of freedom, and for particles and forces. And there seems to be a possibility that we can derive our observed universe from the above pre-quantum, pre-spacetime dynamics, because the observed symmetries possibly coincide with those of the underlying non-commutative geometry.