What are the possible candidates for entities which are more fundamental than spacetime?

Question

Many physicists now believe that there are more fundamental entities than spacetime. I would like to know if this is at this point an educated guess or do we know possible candidates which replace spacetime or from which spacetime appears emergent?
Please include links to relevant papers/literature if possible.

Answer 1

Naturally, you can understand some of those, if we revise what spacetime (as a mathematical model) is. We construct it through the following steps:

Sets --> Topological spaces --> Topological Manifolds --> Differentiable Manifolds

As an intuitive picture: If I consider events to be elements(sets) of a "map", then I can tear the map apart into tiny pieces, and then give it into your hand. No information is lost.

However, you can intuitively feel, that the map is more then just the set of its elements: It is only a map, if every element is at the right place, that is, if the "neighbourhood-relations" of different elements are correct. Such "neighbourhood-relations" and the sets together define a topological space.

Topological spaces may be discrete, continous, and impossibly many other categories. If you believe that spacetime is discrete, but at least "orderly" in a certain sense, then the "more fundamental spacetime" is going to be a topological space.

Luckily, we usually assume that the topological space locally corresponds to an Euclidean space, which simplifies the description enormously. This structure is called topological manifold. It allows continuity, and continous trajectories to be defined.

If we expect the dynamics to be also smoothly defined, then we only consider special topological spaces where differentiation is possible, we call those differentiable manifolds.

Depending on what properties you do not wish to assume, you can easily go back to that step.

However, I would argue that spacetime needs to be differentiable, which -- on a semi-theological politically-incorrect thought experiment -- can lead us to conclude that spacetime has to be four-dimensional. The thought experiment is as follows: Let us assume that we are God, and we wish to create the Universe. If we are only interested in creating nice, smooth universe with differentiable-manifolds as spacetime, then we will point randomly to any possibly differentiable-manifold.

As it is known, (up to diffeormorphism) for a given dimensional topological manifold, there are a given number of distinct differentiable manifold.

For 1,2 and 3 dimensions, this number is 1. For 5 dimensions, there is finitely many. For 4 dimensions, there is infinitely many ways.

This means (putting aside some mathematical nontriviliaties, remember this is a politically-incorrect thought-ecperiment) that if we choose a random differentiable manifold, the probability that we choose a 4-dimensional manifold, is one.

We can further show, that given some local properties (e.g. linearity), the metric is locally either Galileian, Lorentz, or Euclidean. Euclidean spacetime is both unstable, and have some bad properties, so we need to only experimentally decide between locally Galilei- and locally Lorentz-manifolds.