Dear Cristi,
I find your answers enlightening. Thanks for the explanations. You do have an excellent grasp of the issues. Interested readers should follow the link provided to your answers.
Dear Ray,
Part of a response I gave to Cristi Stoica relates to your statement that, "satisfying the Coleman-Mandula theorem is the crux of that balance."
Cristi made the point that, "in Quantum Theory the time evolution is unitary, hence the information is preserved." I agree but think the following relevant:
Martinus Veltman notes that Feynman rules are derived using the U-matrix, even though formal proofs exist that the U-matrix does not exist. (Diagrammatica, p.183). The U-matrix is unitary by construction, and implies conservation of probability, probability being "the link between the formalism and observed data." In my mind, this leaves some room for 'free will' in the universe, (with consequences for information) but I have not pursued the U-matrix much farther than that. Veltman claims the U-matrix and the equations of motion are to be replaced with the S-matrix, in which the interaction Hamiltonian determines the vertex structure.
The Coleman-Mandula theorem, (according to Wikipedia) states that "the only conserved quantities in a "realistic" theory with a mass gap, apart from the generators of the Poincare group, must be Lorentz scalars." But this seems to constrain only symmetries of the S-matrix itself, not "spontaneously broken symmetries which don't show up directly on the S-matrix limit."
As the 'scattering' matrix is used to make sense of particle collisions, this seems reasonable, but 'scattering' of particles is a very artificial (if necessary) way of studying particles, that may attach undue importance to symmetry and, as I've noted in my essay, leads to a Lagrangian that is based on inventing fields, whether or not those fields actually exist in nature. If they can be solved for then they are considered in some way 'real', and this leads, IMHO, to much of the confusion today.
Veltman states that "unitarity, Lorentz invariance, locality, etc, are in some sense interchangeable." This seems problematical in light of today's push to banish locality from QM.
I don't claim to understand the solution to these problems, just to note that there seems to be some circular logic going on, and I'm not sure that logic is preserved around a complete loop of the circle.
This is part of the reason I start with the logic of one field, and work from there, ignoring, for the most part, the established formalism's of QM and GR if they don't map 100 percent into my model in a way that will satisfy experts in either field.
Edwin Eugene Klingman