Is There a Missing Link Between Spacetime and Fields in Mathematics?

[jlcd]

Separate questions:

1. What is the mathematical formalism where one can transform between field and geometry or they both being emergence?

2. What is the mathematical formalism that can describe QFT but not using the concept of fields nor particles. What are they called and current attempts at this? For example. In Faraday times, Electric fields are flux lines. Then in QM, Electric Fields became potentials. In QFT, Electric Fields became exchange of virtual photons. Is there another formalism where electric fields were described as another dynamics? One closer to perhaps the language of GR? How is the programme progressing if there was one?

 

[jambaugh]

This is actually the research area I have been working on, off and on, since my grad school days.

To break out of a field theory you have to re-unify the fiber bundle structure. That is to say recognize a unified space or manifold out of which the current fibration of gauge field (fiber) over space-time (base) emerged. There are different ways of treating this unified construct, is it parametric or physical?

An example in that direction is your Kaluza-Klein type theories. Recall that Kaluza proposed extending space-time by one (compactified) dimension and one got a theory not very unlike electromagnetism in curved space-time. Since it is an extension of GR it is still technically still a field theory. You can go further (with more dimensions to account for other gauge forces) and embed the curved manifold in a fixed larger dimensional space and eliminate the fields as such. This is --I believe-- a way of understanding the direction taken by String/Brane theories. And it is --I believe-- the wrong approach. One is making the above mentioned manifold structure more physical.

I believe that the other answer is to "kinda" take the Kaluza-Klein unification in reverse. But that means we first have to rethink basic gauge field theory. You can recognize the 2nd quantization step in QFT as a form of quantification, that is of going from the single bodied system to the many. In the process you are invoking a particular choice of statistic. There's bosonic and fermionic quantification as you construct Bose-Einstein fields or Fermi-Dirac ones. One must consider a non field-theoretic quantification. The possibilities are somewhat open ended. I've been playing with alternative B-E quantification with a relativized vacuum/particle count structure. It is nicely finite dimensional and regular but I don't have space-time properly represented in the construction yet. I know somewhat how it will fit in and Unruh radiation is a natural inevitable consequence (since frame relativity must involve vacuum relativity) but it is very far yet from a physical theory.

Anyway this second direction is treating space-time and thence the unified space-time-gauge bundle structure as parametric. You don't see it as such because the work gets done, not with the manifold but within the algebraic structure of the relativity group. When Einstein "unfibrated" the bundle of space 3-fibers over the time 1-base, he indeed unified space-time but the attention was paid to the (physical interpretation of the) relativity group which went from the Euclidean group of velocity frame translations and rotations to the Lorentz group of space-time pseudo-rotations. Time had always been parametric and now spatial coordinates are as well. The conceptual objects are not particles moving in space as parameterized by time coordinate but rather space-time events. With GR, and specifically the geometric interpretation of GR space-time becomes physical again as one speaks of Einstein's equations as being the dynamics of the space-time manifold. But it is a dualistic theory as one has both space-time and the matter fields within it to attend to. As I mentioned you can go Kaluza-Klein and reunify but I think the better alternative is to back off the physicalization of space-time. Keep it parametric (as in group parameters for translation groups). But now what? If I knew exactly how to do this I'd be a bit more well known in the physics circles.

And of course there's probably the truly correct approach out there which is a "none of the above" alternative.

 

[jlcd]

Could there be extra experimental consequence of this?
In Special Relativity, anything that moves faster than light can have some frames in contradiction with temporal flow. But if you can transform fields and geometry and they becoming mere shadows of a third theory where they are just emergence.. could this third theory allow you travel faster than light?

Logically. It doesn't make sense for the universe to be too big and taking so long to travel to other galaxies. So there must be a way.. that is by transcending fields and geometry or wrapping them together and let that new something transcend them to travel FTL. Remember the Cosmic Inflation occurs faster than light.. because it is not something within spacetime that moves FTL but spacetime itself. So perhaps there is something akin to this in the new theory. What forbid this and what is the current research programme about this?

 

[friend]

Dr. Frederic P. Schuller wrote a paper on how to get the gravity action from the matter (field?) action. See here and here. He also has a video series here.

 

[jlcd]

friend and Jambaugh.. please give other references like "information" (I don't know what is the right word to use).. something akin to information being more primary and spacetime and fields just emergence.. if "information" is not the right concept.. please tell me the right word..

Well.. since spacetime is smooth geometry and fields are operators in fock space. It may not make much sense to make them interrelate alone. It's like trying to figure out why a man and woman can give birth to a child by studying the man and a dog and theorizing all sorts of ways they can combine. The missing link is the woman. Therefore in spacetime and fields.. maybe there is another missing link? Who are the scientists who believe this? please mention them. Thanks a lot.