The first conundrum is what picture of QM to choose, in order to describe such a scattering. Unlike in non-relativistic QM, in RQFT the three all-known pictures are not at all equivalent. The Schrodinger picture is more or less meaningless since the Schrodinger equation is not relativistic invariant (since it contains only a time derivative, whereas a general Lorentz transformation mixes time and space coordinates). Moreover, the very notion of a state vector defined at a finite time, $|\psi(t)\rangle$, is very problematic in RQFT for many reasons. Dirac has a very interesting paper ("Quantum Electrodynamics without dead wood" published in Phys. Rev., http://journals.aps.org/pr/abstract/10.1103/PhysRev.139.B684) in which he shows that. The interaction picture does not exist in RQFT due to Haag's theorem. One is left with only the Heisenberg picture. I haven't encountered a serious treatment of scattering in the Heisenberg picture anywhere, not even in non-relativistic QM, let alone in RQFT. I would be very much interested in such a setting, namely in how to describe the state vectors at the beginning and at the end of scattering, the operators, the time evolution, and above all, how to make everything relativistically invariant. If somebody could write a book or lecture notes on such an important topic it would be a match made in heaven. Usually, the books on RQFT, and I think I consulted most of them (at least all that are in my university's library), treat the scattering problem very non-rigorously (even the treatise by Weinberg), giving a lot of hand-waving arguments in which they use at least a forbidden step, the treatment is not fully relativistic from beginning to end, etc., just to arrive at the Feynman diagrams. I've recently heard about a way to circumvent the interaction picture by using the Haag-Ruelle scattering theory. I don't know much about it since it is very technical and mathematically demanding, but my question is this: is this treatment fully and manifestly relativistic from the beginning to the end?

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    $\begingroup$ There is a book "Lectures on Quantum Field Theory" of lectures given by Dirac at the Belfer Graduate School of Science, Yeshiva University, New York in 1963-64 which says similar things to the paper cited in the question. On page 148, Dirac says, "I just do not know how to define an S-matrix working with the Heisenberg picture. ... the usual way of introducing it in field theory involves so much departure from logic that I do not see how one could take it over into a logical theory." $\endgroup$ Sep 25 '15 at 16:34
  • $\begingroup$ @StephenBlake, that's an interesting comment. Do you know if the book is online somewhere? $\endgroup$ Sep 25 '15 at 19:24
  • $\begingroup$ @Jan Lalinsky : I cannot find Dirac's lectures online. It's possible to get a copy at Abebooks abebooks.co.uk/servlet/… $\endgroup$ Sep 26 '15 at 6:10
  • $\begingroup$ @StephenBlake Dirac was wrong! I've discovered that H. Ekstein did just that, presenting the most lucid treatment of scattering theory in the Heisenberg picture in Scattering in field theory. It's a superb paper! $\endgroup$ Aug 17 '16 at 5:45
  • $\begingroup$ @AndreaBecker I've just begun to study H. Ekstein's paper and have an initial worry. Suppose the system is in state $|\psi_{H}\rangle$ in the Heisenberg picture. We are interested in measuring the observable $\hat{A}_{S}$ $\endgroup$ Aug 19 '16 at 21:44

Scattering in relativistic QFT is rigorously treated in Haag-Ruelle theory, which is based on the Heisenberg picture. See

K. Hepp, On the connection between the LSZ and Wightman quantum field theory, Comm. Math. Phys. 1 (1965), 95-111. http://projecteuclid.org/euclid.cmp/1103758732


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