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I am trying to give concise motivation (to math students) for why we have QFT as the fundamental theory of matter and forces, I may start like this:

  1. Schrodinger's equation (in the sense of wavefunction, not Hilbert space) with fine-structure corrections to Hamiltonian, gave a pretty good theory for the observed Hydrogen spectrum. Yet it involved ad-hoc theories to describe spin, and likewise could not be fundamental as it is not coming from a relativistic classical mechanics.

  2. Dirac and others sought out a relativistic wave-equation with probabilistic interpretation. Mathematical spinors were discovered and a fundamental description for a single electron could be said to have been found already. The previous work on Hydrogen is now re-producable in a covariant way. Historically, the negative energy solution lead to the hole theory, which eventually gave way to the concept of positrons.

At this point, what are some key faults/computations/philosophies that lead physics to turn towards quantum fields rather than wavefunctions?

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    $\begingroup$ For a start, even relativistic QM doesn’t allow for the creation and annihilation of particles, which pretty much rules out a truly coherent picture of spectroscopy! $\endgroup$
    – Matt Hanson
    Commented Apr 24 at 20:38
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    $\begingroup$ The introduction of David Tong's notes does a good job of explaining some of the reasons: non-conservation of particle-number and behavior of indistinguishable quantum particles. $\endgroup$
    – march
    Commented Apr 24 at 22:17
  • $\begingroup$ Thanks for these nice perspectives: so it seems that the originators of quantum mechanics must have known at the outset that the Schrodinger equation wasn’t even close to the end of the story, as one would like to explicitly describe the photons which show up to us as spectral lines? (Is this what you mean by coherent?) Or was there somehow a school of thought that knowing the energy and other quantum numbers of a system was all there was to know? $\endgroup$
    – Integral fan
    Commented Apr 25 at 2:02
  • $\begingroup$ They knew there was a problem because there was no way to account for photons appearing and disappearing “as needed” in calculations, which kicked off the search for a quantum field theory called quantum electrodynamics that allows particles to be described as localized excitations in a quantum field. Essentially, the quantum field has a harmonic oscillator attached to every point due to the evenly spaced energy levels and ladder operator methods lending themselves nicely to a particle number interpretation. $\endgroup$
    – Matt Hanson
    Commented Apr 25 at 3:12
  • $\begingroup$ Okay that makes sense: sounds like a QED-type worldview was already I suppose present from the outset: for example, Einstein calculated his coefficients without even a theory of quantum mechanics using the "as needed" photons, Dirac in 1927 gave the first fundamental description of these photons as the spring box of harmonic oscillators we now well know as a quantized free field. I think at least from the perspective of radiation and understanding that electrons must be spinors in the mathematical sense, the jump to QED is then quite natural. Thank you both for your help! $\endgroup$
    – Integral fan
    Commented Apr 25 at 7:41

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