Many systems in physics are studied by looking at a "spectral-abstraction" (i.e. use of a Fourier transform and/or eigenvectors) of some underlying first principles.

A good example is a guitar string's vibrations. The "first principles" is the force balance acting on each segment of string, form which we get the wave equation. The spectral-abstraction is describing the harmonics of the string. This makes the behavior easier to model but a computer could still solve the first-principles force-balance equation.

Non-relativistic quantum mechanics is similar. The "first principles" is that we have particles (wave-functions) which evolve via the time-dependent Schrodinger equation, and (the statistics of) everything we can observe can be deduced from the initial state and the evolution of the wave-function. The spectral abstraction is describing i.e orbitals an electron can be in. Similar to the guitar string, orbitals are eigenvectors that correspond to standing-wave solutions of the first-principles equation (we calculate eigenstates using the time-independent equation but that is just a math shortcut).

Understanding the first principles is important because they are much easier to generalize. For our guitar example, if the string was a rubber-band with a J-shaped stress-strain curve we would no-longer have harmonics. For the QM example, suppose we have many particles in a well that exert a force on the "walls" of the well and/or eachother, and we found a way to simplify the system as a single wave-function that couples to the energy landscape in a non-linear manor. In both cases we need to dig down to and use the first-principles (with modifications) to model them. Of course the first-principles are still approximations. But as long as stay in the domain where they are good approximations we can use them to make our predictions.

Unfortunately, quantum field theory uses a spectral-abstraction as it's first principles. In QM, the number of particles is fixed. But in QFT particles can be created or destroyed. For example, a creation operator would move us form a state with 10 particles to one with 11 particles much as we would describe the transition from the S1 to the S2 orbital in hydrogen. Describing a QFT system as "there are 10 particles" is like describing a QM system as "the electron is in the xyz orbital". In a fast interaction with rapid creations and annihilations and energy uncertainties on the order of the mass of the particles, statements like "at time t there are 10 particles" aren't that meaningful. But any discussion of QFT seems to use particles as it's ground truth, rather than an abstraction of something else.

Is there any "non-spectral" description of quantum field theory?

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    $\begingroup$ Comment (v1). QFT is not the theory of annihilation and creation operators. $\endgroup$ – yuggib Jan 19 '17 at 8:54
  • $\begingroup$ @yuggib: I guess my creation/annihilation example is related but not core to QFT. What I am trying to say is that particles is a spectral abstraction in QFT like states are in QM. $\endgroup$ – Kevin Kostlan Jan 19 '17 at 9:35
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    $\begingroup$ Your terminology is very unclear to me, sorry. Particles are essentially always an approximation of something/ ideal objects in physical theories. Quantum theories, both QM and QFT, share the same "core" mathematical description as theories of noncommutative probability. $\endgroup$ – yuggib Jan 19 '17 at 10:09
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    $\begingroup$ After reading your post a couple of times, I still don't understand what "spectral-abstraction" is supposed to mean, sorry. $\endgroup$ – AccidentalFourierTransform Jan 19 '17 at 12:01
  • $\begingroup$ @AccidentalFourierTransform, @ yuggib: I edited the question to explain it better. $\endgroup$ – Kevin Kostlan Jan 19 '17 at 22:23

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