Haag's Theorem & Wightman Axioms: Solving Problems in QFT?

[Denis]

but in an approximation much finer than what can be calculated numerically. Thus a theory is needed how to match these widely differing scales. If you call this mess conceptually clean we are light years away in the use of such terms.

No. We do not need a theory for this, all we need is computations, approximate computations. There is no need for different theories, one well-defined theory is sufficient, and for this well-defined theory a lattice theory with periodic boundary conditions is a good candidate.

You need a theory to match a theory with some lattice approximation of that theory?

What I call conceptually clean is lattice theory with periodic boundary conditions. This is a well-defined theory, everything finite. This characterization of the lattice theory as conceptually clean does not depend on the lattice spacing and the size. Because "conceptually clean" is about concepts, not about our ability to compute something.

Approximate computations may be messy, they are in fact always messy.

What makes the difference is if there is a well-defined theory which one attempts to approximate - which is the case if that theory is a lattice theory - or if one attempts to approximate something which is not even well-defined - which is the case in continuous Lorentz-covariant field theory, even in the nice renormalizable case.

 

[A. Neumaier]

What I call conceptually clean is lattice theory with periodic boundary conditions. This is a well-defined theory, everything finite.

But in this conceptually clean theory everything of interest has been cleaned away.

There is neither a notion of bound states nor a notion of scattering matrix, nor a notion of resonances, nor a notion of particles, nor a notion of angular momentum, nor a notion of canonical commutation relations.

Nothing is left of all the stuff needed to make conceptual sense of experimental data.