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In quantum electrodynamics (QED), for example, we can calculate any scattering amplitude using the Feynman diagram perturbative expansion. That is, we can calculate the matrix element $\langle i | f \rangle$, where $|i\rangle$ represents some collection of particles coming in from $t = -\infty$ with specified momenta, and $|f\rangle$ represents some collection of particles going out to $t = +\infty$ with specified momenta.

In contrast, the derivation of the Lamb shift appears somewhat ad hoc: it does not seem to be an instance of a general technique, nor does it suggest any general approach to computing properties of bound states in QED. The derivation of the Casimir effect also seems ad hoc.

In non-relativistic quantum mechanics, one writes down the Schrödinger equation and then solves it, and the wave function tells you everything you need to know about the system. Does there exist a general technique in quantum field theory (QFT) for calculating anything other than scattering amplitudes?

innisfree
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Brian Bi
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    An analogous thing to 'just write down the Schrodinger equation' in QM might be 'just write down the Schwinger-Dyson equations' or 'just evaluate the path integral'. Those would be really clumsy ways to derive the Casimir effect or Lamb shift, but so is using the Schrodinger equation for QM. An analogously large question would be "what is the general technique to make a fast algorithm for any problem?" – knzhou Mar 09 '17 at 01:39
  • http://physics.stackexchange.com/q/13997/ – Count Iblis Mar 09 '17 at 01:39
  • @knzhou However with the Schrödinger equation we can use perturbation theory to get an approximation to any desired degree of accuracy, and often only a few terms are necessary. If the Schwinger-Dyson equations can be used in a similar way, and you're interested in elaborating, I think you can write an answer. – Brian Bi Mar 09 '17 at 04:15
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    That highly depends on what exactly it is that you want to compute "other than scattering amplitudes". If you want expectation values of observables in thermal states you're going to do something different than if you want to do real-time non-equilibrium time evolution, which again is different from doing lattice theory, etc... I think there is no single answer to this question simply because both the objects you're interested in and the techniques that're appropriate for your problem are so varied. "QFT" is a very broad field, it's like asking for a technique to "solve mechanics problems". – ACuriousMind Mar 09 '17 at 13:10
  • In principle, the Schrödinger equation describes the time evolution of probability amplitudes in any quantum mechanical theory (with a Hamiltonian). So the general method for QFT is... one writes down the Schrödinger equation and solves it. But good luck with that :) The entire machinery of QFT (path integrals, perturbation theory etc.) exists precisely because the Schrödinger equation is completely intractable for anything but a handful of toy problems. – Mark Mitchison Mar 10 '17 at 00:50
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    @MarkMitchison my understanding is that that is not doable in QFT because nobody actually knows how to characterize the Hilbert space in an interacting theory. – Brian Bi Mar 10 '17 at 00:51
  • It's also not doable in non-relativistic QM. The Schrödinger equation is just the formal starting point, you can rarely just sit down and solve it. And there are different approximations and techniques appropriate for different problems, as in every field of physics. – Mark Mitchison Mar 10 '17 at 01:10
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    @MarkMitchison At least in principle the Schrödinger equation allows you to reduce the problem to a mathematical problem, which can be solved using whatever mathematical techniques exist for solving partial differential equations. Implicitly, this requires us to be able to write down the Hamiltonian explicitly with respect to a particular basis. In QFT it seems like we can't write down the Hamiltonian explicitly. So what can we do? – Brian Bi Mar 10 '17 at 01:38
  • @Brian Bi, I was unaware of this issue regarding the Hilbert space in interacting QFTs. I recommend you reword your question along those lines. Also, besides scattering amplitudes that particle physics is concerned with, many body physics methods within condensed matter focuses on understanding more than scattering processes. – KF Gauss Mar 10 '17 at 04:33

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