Generalizing Einstein's Spacetime

Now that we're in the year 2017, it is a great time to return to Einstein's view of spacetime.  One of his more memorable quotes on the subject is:

"People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."

In our current struggles to formulate quantum gravity, any fruitful approach to the problem must invariably include matter.  Hence, to speak of a purely spacetime-centered formulation of quantum gravity is doomed to failure.

Looking at the problem abstractly, let us return to the mathematics.  General relativity is a theory based on (pseudo) Riemannian geometry, which involves smooth (infinitely differentiable) manifolds with a metric.  We have come so far since the 19th century, in our mathematical view of manifolds.  The 19th century physicists and mathematicians did not have quantum theory in mind while developing their algebraic and geometric structures.  It took the early 20th century to introduce the quantum behavior of matter.

With quantum matter in hand, we can revisit Einstein's view of spacetime.  Indeed, Einstein rejected the probabilistic nature of quantum theory and this prejudice has remained for more than 100 years, for many physicists and mathematicians.  Yet with over 100 years of experimental verification, it is clear quantum theory is fundamental to our understanding of microscopic reality.

This begs the question: how does one build geometry, in the sense of Einstein's matter-driven spacetime, while also incorporating the quantum nature of matter?

We must approach this question with 20th century mathematics.  Many physicists are not aware that a revolution took place in mathematics, initiated by luminaries such as Emmy Noether and John von Neumann.  What Emmy suggested in the 1920's was to clarify the notion of generic point in an algebraic variety by using the following recipe:

1)  Start with the coordinate ring of an algebraic variety (the ring of polynomial functions defined over the variety)

2)  The maximal ideals of this ring will correspond to ordinary points of the variety (given suitable conditions are met)

3) The non-maximal prime ideals will correspond to the various generic points, for each subvariety and by taking all prime ideals, one recovers the whole collection of ordinary and generic points.

Emmy did not pursue this idea, but mathematicians such as Krull, Weil, Zariski, Serre, Chevalley, Nagata, Martineau and Grothendieck did follow the prescription and generalized to very general ring spectra.  In light of this, it is thus clear how Connes' noncommutative geometry is the more mature manifestation of Emmy Noether's vision.  The keyword here is generalized schemes.  What is a scheme? 

Definition:  A scheme is a topological space together with commutative rings for all its open sets, arising by gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets.

To generalize this to noncommutative rings, we declare:

Definition:  A noncommutative scheme is a topological space together with noncommutative rings for all its open sets, arising by gluing together spectra (spaces of prime ideals) of noncommutative rings along their open subsets.

To recover noncommutative geometry, and its spectral triples, we merely take the noncommutative ring to be a noncommutative C*-algebra.  The relevant topology is usually taken to be the Zariski topology.

Now, what does this mathematical machinery mean for physics and the study of spacetime?  The key is to view quantum theory in its abstract form.  We recall that matter is described by states in a Hilbert space and an algebra of observables acts on these states.  This algebra of observables is a noncommutative C*-algebra!

So let's consider X as our topological space of possible states of some physical system and the elements of C(X) (our noncommutative ring) are the observables for this system.  The value of an observable at a point in X is the result of our observation (which we perceive in the form of its spectrum, or eigenvalues).  The Zariski topology captures all the semidecidable properties one can decide using observations in C(X).  For example, an element in C(X) could give position as a semidecidable property, which is decided by computing the position to finite precision.

Since quantum theory has been shown to be built on noncommutative C*-algebras, it is natural to take a noncommutative scheme as our building block for spacetime.  The matter of the standard model comes in representations of the standard model gauge group SU(3) x SU(2) x U(1).  Connes has formulated a noncommutative model that incorporates the standard model symmetry by using a carefully selected C*-algebra for this spectral triple.  In light of grand unified theory, which must ultimately be re-cast in the noncommutative framework, one can consider much more general C*-algebras than that of Connes.  Also, one must keep in mind Minkowski space must be emergent in this choice of noncommutative scheme and its accompanied C*-algebra.  One might also suspect more general algebraic structures might become relevant at the big bang.  The search goes on.