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Mathematics of Quantization and Quantum Fields by Christian Gérard, Jan Derezinski

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22

Interacting bosonic fields

The usual formalism of interacting relativistic quantum field theory is purely perturbative and leads to formal, typically divergent expansions. It is natural to ask whether behind these expansions there exists a non-perturbative theory acting on a Hilbert space and satisfying some natural axioms (such as the Wightman or Haag–Kastler axioms). It is not difficult to give a whole list of models of increasing difficulty, well defined perturbatively, whose non-perturbative construction seems conceivable. There were times when it was hoped that by constructing them one by one we would eventually reach models in dimension 4 relevant for particle physics. The branch of mathematical physics devoted to constructing these ...

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