When it comes to scattering in QED it seems only scattering cross sections and decay rates are calculated. Why is that does anyone calculate the actual evolution of the fields themselves like how the fields evolve throughout a scattering process and the associated total electron density to visualize the dynamics via wavepackets. For example when 2 electrons scatter off each other is it possible to compute the evolution of the fields in terms of Gaussian Wavepackets which scatter off each other of course using perturbation theory and visualizing it in terms of the total electron density or the charge density. Also can the scattering amplitudes be used to construct the evolution of the fields themselves if so why isnt it ever mentioned is it because constructing the evolution is a trivial exercise maybe. This bugs me because for a theory that is so praised and can supposedly describe Electromagnetic Fields at the fundamental level it seems awfully limited in what it can do. For the record I dont know any QFT I simply looked at it a bit out of curiosity of whats the big deal about QFT and found myself kinda puzzled so please forgive me if I made any mistakes. I simply want to know if its possible to describe the actual dynamics of scattering processes in QED with Feynman Diagrams via Wavepacket scattering or only cross sections can be computed.
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$\begingroup$ Particle number is not conserved in quantum field theory. The mental image that you have in mind is a low energy approximation that is incorrect even for a single electron with zero momentum: the theory predicts an anomalous magnetic moment even in that case. For a bound system of an electron and a positron we get a lifetime prediction and the emission of photons. For two scattering electrons we also get the emission of photons (in a different context we call this "Bremsstrahlung"). In other words, one can't reduce the theory to a simple one or two body problem. $\endgroup$– FlatterMannCommented Jun 20, 2023 at 6:13
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$\begingroup$ I was saying the evolution of the fields themselves of whatever particle number and the total particle density of ALL the particles even if the particle number changes isnt there a way to visualize the time dependent charge distribution of the particles. $\endgroup$– user366596Commented Jun 20, 2023 at 6:15
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1$\begingroup$ Thinking that quantum fields “have an evolution” is like thinking that quantum particles “have a trajectory”. Quantum particles don’t have trajectories because they don’t have a well-defined position and momentum at each instant. Quantum fields also have uncertainty relations. The more certainty you have about what the field is at some point, the less certainty you have of how fast it is changing there, and vice versa. $\endgroup$– GhosterCommented Jun 20, 2023 at 7:17
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1$\begingroup$ What you can think about is the probability amplitude of a particular evolution from an initial configuration to a final configuration. This is the basis of the Feynman path integral approach to quantum field theory. The amplitude is simply proportional to $e^{iS/\hbar}$ where $S$ is the action functional. The difficulty is that there are an infinite number of evolutions to take into account, since the path integral approach says that they all happen. $\endgroup$– GhosterCommented Jun 20, 2023 at 7:19
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$\begingroup$ I was saying similar to how in regular QM you have the total electron density which is a function of 3 variables despite the system having 3N coordinates. A way to describe the evolution of that quantity and by evolution of the quantum fields i meant of the states sorry if i was imprecise. But i meant the evolution of the states AND the total electron density $\endgroup$– user366596Commented Jun 20, 2023 at 7:28
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What is happening is that the actual real-world evolution is so horrendously difficult, that we have been reduced to only be able to talk about the infinite time results. If the mathematics was not so difficult, we would have loved to have visualisable simulations of the time evolution. Instead, every perturbation term that we naïvely wish to compute, is infinite, and there is just no way to even begin generating an acceptable approximate picture of how the fields evolve in time. We are only able to follow strict rules and extract some predictions here and there.
In a comment, you talk about the charge distribution. This thing is such a mess, that even basic quantum mechanics, with wavefunctions, we cannot define the charge distribution properly either. This is why we talk about Born effective charges, etc. The problem is just so completely hopeless that we are left with broken hacks here and there all over the place.
One learns humility in the face of great difficulties.
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$\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$– Buzz ♦Commented Jun 21, 2023 at 13:06