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A simple conceptual question today: is it true that QFT can only be approached in a non-perturbative way only through the functional methods (like 1/N), while in the Hamiltonian formalism we can only do the perturbation theory?

In QM, the functional methods are, of course, widely-known for finding the non-perturbative corrections 1. However, there also exist certain methods based on the Hamiltonian formalism. For example, studying the resolvent operator of the Schrodinger equation 2. Alternatively, one can employ the Uniform WKB quantisation condition 3.

I know that in QFT people also construct the trans-series expansions. But what methods do they use?..

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  • $\begingroup$ My attitude is that in the Hamiltonian formulation the space of states is much too gigantic to make tractable calculations. I have seen people in the mid-80's trying to apply variational methods to get at the ground state wave function of the fields. Anyway, I do find that it is certainly useful to think about it for conceptual reasons. $\endgroup$ – QuantumDot Oct 13 '16 at 18:45
  • $\begingroup$ This review paper might be relevant, in particular section 2 which discusses full, rigorous constructions of (low-dimensional) interacting QFTs in the Hamiltonian formalism. $\endgroup$ – Luzanne Oct 13 '16 at 21:04

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