Noncommutative spacetime

As I explained in a previous post, it is only because one drops commutativity that, in the calculus, variables with continuous range can coexist with variables with countable range. In the classical formulation of variables, as maps from a set X to the real numbers, we saw above that discrete variables cannot coexist with continuous variables.
The uniqueness of the separable infinite dimensional Hilbert space cures that problem, and variables with continuous range coexist happily with variables with countable range, such as the infinitesimal ones. The only new fact is that they do not commute.

One way to understand the transition from the commutative to the noncommutative is that in the latter case one needs to care about the ordering of the letters when one is writing.
As an example, use the "commutative rule" to simplify the following cryptic message I received from a friend :"Je suis alençonnais, et non alsacien. Si t'as besoin d'un conseil nana, je t'attends au coin annales. Qui suis-je?"
It is Heisenberg who discovered that such care was needed when dealing with the coordinates on the phase space of microscopic systems.
At the philosophical level there is something quite satisfactory in the variability of the quantum mechanical observables. Usually when pressed to explain what is the cause of the variability in the external world, the answer that comes naturally to the mind is just: the passing of time. But precisely the quantum world provides a more subtle answer since the reduction of the wave packet which happens in any quantum measurement is nothing else but the replacement of a "q-number" by an actual number which is chosen among the elements in its spectrum. Thus there is an intrinsic variability in the quantum world which is so far not reducible to anything classical. The results of observations are intrinsically variable quantities, and this to the point that their values cannot be reproduced from one experiment to the next, but which, when taken altogether, form a q-number.

Heisenberg's discovery shows that the phase-space of microscopic systems is noncommutative inasmuch as the coordinates on that space no longer satisfy the commutative rule of ordinary algebra. This example of the phase space can be regarded as the historic origin of noncommutative geometry. But what about spacetime itself ? We now show why it is a natural step to pass from a commutative spacetime to a noncommutative one.
The full action of gravity coupled with matter admits a huge natural group of symmetries. The group of invariance for the Einstein-Hilbert action is the group of diffeomorphisms of the manifold and the invariance of the action is simply the manifestation of its geometric nature. A diffeomorphism acts by permutations of the points so that points have no absolute meaning.
The full group of invariance of the action of gravity coupled with matter is however richer than the group of diffeomorphisms of the manifold since one needs to include something called ``the group of gauge transformations" which physicists have identified as the symmetry of the matter part.
This is defined as the group of maps from the manifold to some fixed other group, G, called the `gauge group', which as far as we known is: G=U(1).SU(2).SU(3). The group of diffeomorphisms acts on the group of gauge transformations by permutations of the points of the manifold and the full group of symmetries of the action is the semi-direct product of the two groups (in the same way, the Poincaré group which is the invariance group of special relativity, is the semi-direct product of the group of translations by the group of Lorentz transformations). In particular it is not a simple group (a simple group is one which cannot be decomposed into smaller pieces, a bit like a prime number cannot be factorized into a product of smaller numbers) but is a ``composite" and contains a huge normal subgroup.
Now that we know the invariance group of the action, it is natural to try and find a space X whose group of diffeomorphisms is simply that group, so that we could hope to interpret the full action as pure gravity on X. This is the old Kaluza-Klein idea. Unfortunately this search is bound to fail if one looks for an ordinary manifold since by a mathematical result, the connected component of the identity in the group of diffeomorphisms is always a simple group, excluding a semi-direct product structure as that of the above invariance group of the full action of gravity coupled with matter.
But noncommutative spaces of the simplest kind readily give the answer, modulo a few subtle points. To understand what happens note that for ordinary manifolds the algebraic object corresponding to a diffeomorphism is just an automorphism of the algebra of coordinates i.e. a transformation of the coordinates that does not destroy their algebraic relations. When an involutive algebra A is not commutative there is an easy way to construct automorphisms.

One takes a unitary element u of the algebra i.e. such that u u*=u*u=1. Using u one obtains an automorphism called inner, by the formula x -> uxu*.
Note that in the commutative case this formula just gives the identity automorphism (since one could then permute x and u*). Thus this construction is interesting only in the noncommutative case. Moreover the inner automorphisms form a subgroup denoted Int(A) which is always a normal subgroup of the group of automorphisms of A.
In the simplest example, where we take for A the algebra of smooth maps from a manifold M to the algebra of matrices of complex numbers, one shows that the group Int(A) in that case is (locally) isomorphic to the group of gauge transformations i.e. of smooth maps from M to the gauge group G= PSU(n) (quotient of SU(n) by its center). Moreover the relation between inner automorphisms and all automorphisms becomes identical to the exact sequence governing the structure of the above invariance group of the full action of gravity coupled with matter.

It is quite striking that the terminology coming from physics: internal symmetries agrees so well with the mathematical one of inner automorphisms. In the general case only automorphisms that are unitarily implemented in Hilbert space will be relevant but modulo this subtlety one can see at once from the above example the advantage of treating noncommutative spaces on the same footing as the ordinary ones. The next step is to properly define the notion of metric for such spaces and we shall indulge, in the next post, in a short historical description of the evolution of the definition of the ``unit of length" in physics. This will prepare the ground for the introduction to the spectral paradigm of noncommutative geometry.