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The explanations of stimulated emission which I have found all describe the phenomenon in terms of non-relativistic quantum mechanics. How might you describe it in a field theory such as QED? In particular, could you say that stimulated emission occurs on tree level as

                                          enter image description here

or is this a different phenomenon altogether? If the above diagram properly describes stimulated emission, then would the absolute-value squared of this diagram be equal or proportional to Einstein's coefficient $B_{21}$?

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    $\begingroup$ Just a comment on terminology: 'Old quantum theory' usually refers to the expressions used prior to Heisenberg and Shroedinger's work, including Bohr, Einstein, and many others. 25 years of wandering in the wilderness, with a few good hits, and lots of misses. $\endgroup$ – Peter Diehr Apr 13 '16 at 21:09
  • $\begingroup$ @Peter Diehr - Thanks for the clarification. I edited my question accordingly. $\endgroup$ – David C. Apr 14 '16 at 2:37
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    $\begingroup$ Not sure I understand the kind of answer you want. I would have said it's same both in quantum mechanics and QFT. Consider any quantum process you like that produces a photon. It could be any Feynman diagram with a photon as an outgoing external line. Stimulated emission is an enhancement of the amplitude that happens when the process does not happen in vacuum but instead in a state with some number, say $n$, of photons with the same quantum numbers as the produced photon. $\endgroup$ – user2309840 Apr 16 '16 at 3:01
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    $\begingroup$ @DavidC. Have you seen the book Advanced Quantum Mechanics by J.J. Sakurai and Quantum Field theory by Itzykson and Zuber? I'm not sure but I think, they discuss both spontaneous and stimulated emission in QED. Have a look at them. $\endgroup$ – SRS Dec 26 '16 at 11:48
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    $\begingroup$ The Feynman diagram that you drew (electron emits soft photons while propogating) seems like a soft Bremsstrahlung process. $\endgroup$ – Exhaustive Mar 8 '17 at 9:02
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Section 4.1.3 of the 1980 QFT book by Itzykson & Zuber contains a discussion of stimulated emission for a quantized electromagnetic field interacting with a classical source. However, this is not a full QED treatment since matter is treated classically.

Stimulated emission in full QED can be studied only in the context of bound states or long-living resonance. Bound states and resonances are usually not well-described in textbooks. Chapter 10 of the 1980 QFT book by Itzykson & Zuber contains a discussion of the Bethe-Salpeter equation for bound states. Weinberg's 1995 QFT treatise treats bound states in Chapter 14 of Volume 1, deriving the Dirac equation (from which one proceeds as you described). On p.560 he remarks ''It must be said that the theory of relativistic effects and radiative corrections in bound states is not yet in entirely satisfactory shape.'' This is an euphemism for ''... is still a poorly understood mess''. See also https://physics.stackexchange.com/a/27623/7924

Today, 23 years later, it is not much better. Probably the main reason is that ambitious people prefer to work on grand projects possibly meriting a future Nobel prize rather than redoing old stuff in a more satisfying way.

More background: Effective field theories for QED bound states are discussed by Labelle 1998 (preprint at https://arxiv.org/abs/hep-ph/9608491). For QCD, a Dyson-Schwinger approach (to QCD) is studied by Roberts and Williams in a 1994 paper (preprint at https://arxiv.org/abs/hep-ph/9403224), and a semiclassical approach by Hoyer is discussed in a 2011 paper at https://arxiv.org/abs/1106.1420 and (on the tree level) in a 2016 paper at https://arxiv.org/abs/1605.01532.

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    $\begingroup$ Do you know any good and recent review on this problem of bound-state in QFT ? I find it fascinating that it might be so poorly understood. $\endgroup$ – Adam May 24 '18 at 14:30
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    $\begingroup$ That is fascinating indeed. I guess my tree level diagram is essentially wrong or at least incomplete, then. Thanks for this information; I'm inspired and you answered half my question. If anyone feels inclined to remedy this poorly understood mess and figures out a field theory description of stimulated emission—I may not be able to give you a Nobel prize but I will at least say that you answered the rest of my question and did field theory a great and needed service :) $\endgroup$ – David C. May 25 '18 at 15:12
  • $\begingroup$ @Adam: I added references $\endgroup$ – Arnold Neumaier May 25 '18 at 16:17
  • $\begingroup$ Great answer and nice references! Just one question: You say "Stimulated emission in full QED can be studied only in the context of bound states." -> why? Can you not have a long lived resonance state instead of a true bound state? $\endgroup$ – Wolpertinger May 28 '18 at 14:35
  • $\begingroup$ @Wolpertinger: Resonances pose the same problems. I updated the answer accordingly. $\endgroup$ – Arnold Neumaier May 28 '18 at 15:28

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