One must ask what one means with space-time as a fundamental entity. Does one mean space-time as a non-dynamical entity as Minkowsky space of special relativity or space-time of general relativity? These are very different things.
There are good motivations for trying to get rid of space-time of general relativity (GRT).
- Poincare symmetries are fundamental for quantum field theories and are lost in general relativity: this is an easily identifiable reason for the failure of quantization of GRT.
- Both twistor structure and spin structure exist only under strong additional conditions for general space-time. Both are fundamental and should exist in some sense.
- One of the latest problems is that exotic differentiable structures typically exists and there are a lot of them - this happens just in 4-D case! This implies time-like loops and problems with causality in the case of general spacetime (see this).
Should we keep Minkowski space but replace the space-time of general relativity with something for which an analog of twistor space and twistor structure exists?
- Remarkably, the twistor space as S2 bundle with Kaehler structure exists only for M4 (and CP2) (see this). Could one use string theory as a guideline and identify space-times as 4-D surfaces in 8-D H=M4×CP2 having twistor space T(H), which is a 12-D product of twistor spaces of M4 and CP2 having a Kahler structure?
- Space-time as a 4-surface in H would have twistor and spin structures and metric, which are induced from those of H. This would give the exact Poincare invariance lost in GRT. The 6-D Twistor space of space-time surface would be a 6-D surface in T(H) and preferred extremal of the 6-D Kaehler action: dimensional reduction would give S2 bundle over space-time surface and action would decompose to 4-D Kahler action and volume term to which cosmological constant can be assigned.
- 4-D general coordinate invariance forces holography so that 3-D surface fixes space-time surface almost uniquely. One gets rid of path integral. Quantum states would be superpositions of holographic space-time surfaces/their 6-D twistor spaces with S2 fiber.
For a summary of earlier postings see this">Latest progress in TGD.