This is likely a naive question, following up on something @vanhees71 posted some time ago in another thread:

My question is the following - if we take an electron that has, for example, absorbed a photon, is the portion of the wavefunction representing the electron in a lower energy state ever exactly zero (and hence the probability of finding it in the lower energy state also zero)?

If so, this to me would imply that any change of the wavefunction from exactly zero to non-zero would mean that, were we able to detect it, one could measure the electron in the higher energy state and then measure the electron in the lower energy state (and with a photon produced of course) essentially instantaneously. That is, if the wavefunction changes from zero to non-zero, would this not imply that the measured position of the electron, say, could change in a superluminal manner?

Clearly the above is incorrect, since QFT respects special relativity - I'm just wondering how the change in electron energies (and the exact place/time at which a photon is emitted) in QFT squares with special relativity. @vanhees71 has provided part of the answer above, I was just looking for further clarification (since I think I'm stuck in the 'particle' picture).

The above is probably phrased very poorly: how about this: if there is a region where the probability of finding an electron is exactly zero (and nonzero elsewhere), and then the probability in said region becomes nonzero at some point, does this change not imply that the electron has in some sense 'moved' (i.e., there has been a change in mass distribution) ... in the QFT scenario, how can this always be consistent with special relativity?

Sorry for being the only one to post in the thread. One must at least be able to say in QFT that the wavefunction always evolves in a manner consistent with special relativity, correct?

Apologies for the confusion all - any help appreciated.

In relativistic QT "wave functions" have only a very limited meaning. The naive wave-function picture a la Schrödinger wave functions (position representation of non-relativistic QT) is applicable only close to the non-relativistic limit and particularly as long as the particle numbers don't change. That's the very reason why we use relativistic QFT to formulate relativistic QT: It provides the most elegant description of particle-number changing processes, which are very likely to happen when you collide particles at relativistic energies.

However, the theory of interacting fields is very complicated. It already starts with the definition of states allowing the interpretation of particles. It's simple in the non-interacting theory, because there you can define creation and annihilation operators to expand the fields in terms of energy-momentum eigenmodes, and this admits the construction of the free-particle Fock space in terms of occupation numbers.

Now what's observed in typical scattering experiments are processes where you shoot two "particles" created far away from each other and prepared in a way to have pretty well-defined momenta (and sometimes polarization, but that's not done that often; an example are the experiments with spin-polarized protons at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Lab (BNL)), so that they can be considered as non-interacting free particles. Then they meet at some predefined experimental site (a detector) and interact. In this state of the collision, it is in general not possible to interpret the corresponding "transient states" as particles. At later times, however, particles (usually many newly produced ones in the collision) get far away from each other again, and you can again analyze these in the sense of the "free-particle states" defined via the non-interacting theory.

In this sense you can interpret the physics content of QFTs as descriptions of "particles" only in the sense of "asymptotic free particles", but it's a less trivial matter than it seems. Look, e.g., in Peskin and Schroeder or other textbooks like Weinberg, where they discuss the S-matrix and cross sections, the LSZ-reduction formula. It's also well worth to check the old classic by Bjorken and Drell (forget volume 1, which delivers the handwaving attempt to formulate relativistic QT as relativistic QM, which is very confusing knowing the modern way to formulate it in terms of QFT, but volume 2 is particularly nice concerning this issue of defining S-matrix elements out of the formalism of QFT).

In a relativistic classical field theory nothing propates faster than light that is not allowed to do so by construction. That was already a question raised as early as 1907 by Willy Wien concerning the propagation of electromagnetic waves in matter, where you can have, in frequency regions, where socalled anomalous dispersion occurs, phase and group velocities greater than the speed of light. The answer was provided by Arnold Sommerfeld also in 1907 in a very short note, using elegantly Fourier integrals in the complex plane. The upshot is that in this region the physical sense of the socalled saddle-point approximation breaks down and thus the formal group velocity looses it's physical meaning too. Within the simple classical models the front velocity of the em. wave is always ##c## (speed of light in vacuum), i.e., there is no violation of Einstein causality in classical electrodynamics. Sommerfeld and Brillouin later worked out the details of wave propagation in dielectric media with very interesting results about the details of the wave form of the "transient states", of coursing none of them violating the laws of relativistic physics.

Also in the usual relativistic local QFTs (among them the Standard Model of particle physics), there is nothing that can violate the laws of relativistic physics by construction. The assumption of microcausality (particularly that the Hamilton density commutes with all local observables for space-like arguments, i.e., ##[\mathcal{H}(x),\mathcal{A}(y)]=0## for ##(x-y)\cdot (x-y)<0## (in the usual west-coast convention of the Minkowski product with ##(\eta_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)##) ensures the validity of the linked-cluster theorem for S-matrix elements, i.e., space-like separated experiments lead to uncorrelated results. This thus not exclude the well-observed far-distance correlations described by entangled observables (e.g., the detection of the polarization state of two polarization-entangled photons at far distant places), but they cannot be revealed by space-like separated experiments only, but to see the correlations, as the validity linked-cluster principle shows, one nessecarily has to compare measurement protocols afterwards. So there cannot be "faster-than-light communication" via entangled quantum channels either.

Thanks @vanhees71! One last item: in looking at a text about this (Applications of Electrodynamics in Theoretical Physics and Astrophysics, page 200) - there is a section on group propagation (classical sense) that refers to the "precursor" region before the leading front (as also propagating at c). I've never seen the term before (which appears to be related to the higher frequency components of the Fourier spectrum of the signal) ... is this "precursor" definition standard? (as something in before the leading front of the signal?) I thought the "front" was the start of the signal, so to speak.

This refers to the above cited papers by Sommerfeld and Brillouin. There's both a Sommerfeld and a Brillouin precursor. This describes with some precision the transient state, i.e., what happens when a finite wave train reaches the dielectric medium and how the time evolution to the stationary state, where you have a plane wave in the medium, happens. It's a masterpiece in the application of complex function theory.

Unfortunately the original papers are in German, but you find this discussed also in Sommerfeld's textbook on optics (A. Sommerfeld, Lectures on Theoretical Physics, vol. 4 (Optics)) and, of course, in J. D. Jackson, Classical Electrodynamics.