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In reading books about quantisation, there is (sometimes hidden) the claim, that quantisation is done using a pertubative approach. You look at the free field, find that it is essentially a sum of harmonic oszillators (lets here ignore problems in QED with the longitudinal degrees of freedom), you know how to build a Fock space from that, and then you treat interactions as pertubations.

What Scharf did in his book 'finite quantum electrodynamics' is to show, that the indvidual terms of the pertubative expansion is well defined if you do the right casual splitting.

In these treatments it is somehow alluded that quantisation of the non-free theory is either 'not possible' or too difficult. I am exactly wondering about that. Why can you not do canonical quantisation on the full interacting theory? Is it just, that the physical interpretation of the states in unclear? What is the problem one would need to solve?

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    $\begingroup$ You might want to read up on Haag's theorem. $\endgroup$ – Michael Seifert May 4 '18 at 17:11
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    $\begingroup$ Good question. Possible duplicate of What is the issue with interactions in QFT? and links therein. $\endgroup$ – AccidentalFourierTransform May 4 '18 at 19:57
  • $\begingroup$ @MichaelSeifert I read up (a while ago) on Haags theorem. Personally I think it even gives more foundation to my question: if the interaction free definition of Hilbert space is not suitable, shouldnt one start with the interacting theory instead with the free one? $\endgroup$ – lalala May 5 '18 at 16:10
  • $\begingroup$ I'd say it's just an illustration of the streetlight effect, but that may be a bit uncharitable. $\endgroup$ – Michael Seifert May 5 '18 at 19:05
  • $\begingroup$ @MichaelSeifert is this supposed to.be related to the question? If so could you elaborate. $\endgroup$ – lalala May 5 '18 at 19:51

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