Dear Felix,
You present a very interesting and useful point of view. Particularly striking is the lack of necessity for the notion of gravitational interaction between elementary particles. A few questions and remarks:
1. As you point out, the usual approach (e.g. Weinberg) to QFT is "symmetries first," where the symmetries arise from a "spacetime" background that is taken for granted. Your approach seems to be "algebras first," which eliminates the primary status of spacetime. Now, as you know, Alain Connes and his collaborators working in the field of noncommutative geometry begin with Hilbert spaces and operator algebras too. If it's possible to do so in a few sentences, could you briefly contrast your approach to theirs?
2. You mention (page 6) that "there is no reason to believe that [Galilei and Poincare symmetries] are exact symmetries of nature," and I strongly agree with this point. However, I am not sure if you mean that some different symmetries (such as dS or AdS) are "exact symmetries," or if you mean that the symmetry interpretation of covariance is ultimately only an approximation. For instance, you mention (in regard to GFQT) that "sooner or later quantum theory will be discrete and finite," and this would seem to definitely rule out Lie group symmetries in the fundamental picture. Do you expect finite symmetry groups in GFQT, or is group symmetry replaced with something else?
3. Following question 2, I will remark that I personally believe the symmetry interpretation of covariance breaks down at the fundamental level, and an alternative interpretation in terms of order theory takes over (if you are interested, you may see my essay On the Foundational Assumptions of Modern Physics for more details on this).
4. As you know, there is a version of QM that appears superficially to be diametrically opposed to your thesis, namely the interpretation "spacetime first," where I take the terminology from the title of Feynman's 1948 paper on what is now known as the path integral or sum-over-histories version. Of course, this approach is very general and does not actually require spacetime in the traditional sense; any partially ordered set will do (where the partial order supplants time in defining the action, which is used to weight the various paths.) I am wondering what your general view of this approach is. Do you think it is ultimately wrongheaded, or does it play some role?
5. Following question 4, I will reveal that my favorite approach to quantum gravity is a version of the sum-over-histories method, which I would call "relations first," or "interactions first." The partial orders used are causal orders, and Schrodinger equations, operator algebras, etc. emerge via path sums. I know this method is quite different from your approach, but I would still be grateful for any remarks you might have about it. In particular, if you know some good reason why it absolutely won't work, it would save me a lot of trouble!
Thanks for the great read! Take care,
Ben Dribus