Dear Felix Lev,
Thanks for an interesting treatment of some problems with quantum field theory (QFT). Of particular interest is your discussion about the fact that "their [local fields] products at the same point are poorly defined." This, in my mind, is an example of Dirac's complaint about "physical ideas that were not correctly incorporated into the theory", resulting in "no sound mathematical foundation."
If one assumes (as I do) that particles are not points, then the infinities that arise from mathematical points should not be taken too seriously.
You clearly and concisely observe that "The interaction Lagrangians where the fields interact at the same points is the main source of difficulties and inconsistencies in QFT", followed by your question as to whether this notion is needed at all.
Whereas you treat relativistic QFT and general relativity, I work the other end of the universe in my essay, The Nature of the Wave Function, in the sense that I treat non-relativistic quantum mechanics and the weak field approximation to general relativity. At first reading I am unable to bridge the gap between these two extremes, but I found your end fascinating and hope that you obtain something of value from my essay.
Your derivation of the cosmological acceleration and its connection to quantum theory is fascinating, as is your problem of finding a symmetry algebra that reproduces Newton's gravity for two free particles. Yours is a fresh view and a joy to read.
Edwin Eugene Klingman