Thursday, February 16, 2023

Is space-time really doomed?

The Twistor Grasmann approach has led to the idea that space-time  is doomed. This idea is strongly  advocated by Nima Arkani Hamed. Should one really throw out this notion or should one sit down for a moment  and think carefully before doing anything so dramatic?

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).  

  1. 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.  
  2. 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.
  3. 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).
  But what about Minkowski space? Should one try to get rid of it too?  Poincare and conformal  symmetries are fundamental in gauge theories and twistors code for these symmetries.  There are no twistors without 4-D Minkowski space.  

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?

  1. 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?  
  2. 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.  
  3. 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 the development of the ideas about twistor lift of TGD, see for instance this, this,  this,   and this.

For a summary of earlier postings see this">Latest progress in TGD.

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