I am now in Oxford for the Quantum Theory and Nature of Reality conference. In their white paper, Andreas Doring and Chris Isham write about Quantum Field Theory (QFT):
Notwithstanding the success of standard quantum theory in atomic, molecular and solid-state physics, there are good reasons for wanting to see beyond it. For example, relativistic QFT is still plagued with ultra-violet divergences, and even free fields encounter the self-energy problem. No matter how sophisticated the renormalisation procedure that is adopted, the fact remains that relativistic QFT is fundamentally flawed.
This ultra-violet problem is clearly linked to the continuum model for space and time...
I may be missing their point but I would like to offer a possible alternative perspective: the divergence problems with renormalisation are not a problem with QFT but rather merely reflect the stratified nature of reality (discussed in my white paper) i.e. there is a hierarchy of energy scales and with each there emerge new interactions and phenomena which are described by some effective theory). In quantum field theory one is always working with some effective Hamiltonian/Lagrangian which describes interactions at some energy scale. To prevent divergences associated with renormalisation one introduces some high energy (or short wavelength) cutoff. This just reflects that there is other physics which becomes relevant at higher energies and is not explicitly contained with the starting Lagrangian or in its renormalised version which emerges from high energy processes.
I would argue that these problems are not unique to quantum field theory. They are also present for the renormalisation theory of classical phase transitions. In that case they also just reflect the