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For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've come to a serious roadblock when it comes to learning QFT.

Unlike QM or classical mechanics, formalizations of QFT (Wightman axioms, say) seem to be formulated without an explicit description of how to pass from the formalized mathematical system to any sort of physical description of things. Instead, it seems that books merely give very particular calculations that are a step removed from the formalism. The following questions will hopefully make it clearer what I mean by this.

Does QFT provide a generic description of how states evolve in time? Or does it only describe this for very special states (for example, $S$-matrices only tell about in and out states consisting of free, noninteracting particles to my understanding)? If QFT does provide a general description, how does this fit into the overall formalism? If not every state in this formalism corresponds to free particles, how do classical observables behave when applied to these spaces? If QFT can only account for these types of scattering processes, why bother with the formalism of fields in the first place? It seems that most of the calculations revolve around deriving a Hamiltonian (density) from the field and simply using that. Why not formulate, say, the Wightman axioms in terms of this Hamiltonian density rather than the field itself?

I'm sure the questions above present quite a grotesque understanding of QFT; but unfortunately, the texts I've read seem only to provide an unexplained axiomatization in isolation, then go over scattering processes.

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We have a model (the standard model) that is formulated as a particular QFT, that seems to describe every experiment we've ever done on Earth (excluding anything gravitational). My point being that everything is in principle "accounted for" by QFT. The problem is just that when you zoom out and study low-energy physics, like tennis racquets and tennis balls, it is very complicated to solve the relevant equations, due to the presence of many particles in the system. The reason that scattering is talked about so much in QFT textbooks is because that's the one type of physical quantity that

  1. We can actually perform experimentally (e.g. set up particle colliders)

  2. We can actually calculate theoretically (e.g. calculate cross-sections)

I should mention that formally speaking, we don't know any examples of interacting QFTs in 4-dimensions. We have pretty good guesses of examples that should exist (e.g. QCD), but nobody knows how to formalise them. That being said, the reason that the Wightman axioms are formulated the way that they are is that standard lore tells us that all the physical information of a QFT is contained in correlation functions of fields. If you specify the correlation functions, you specify the QFT.

Just to give examples of other physical quantities that are contained in the correlation functions:

  • The two-point function contains information about the masses of all bound states in the theory. If you have any resonances (unstable states), they also show up, and you can extract their half-lives from the two-point function. (I'm talking about the Spectral Density Function)
  • Three-point functions then tell you matrix elements between different states of interest. (I'm really speaking as a lattice-QCD person here, I'm imagining the operators inserted at large euclidean separations)

You could focus more on Hamiltonians if you like. When it comes to real-time simulation of QFTs, a numeric approach is to discretise space-time on a lattice, in which case you can write down an explicit Hamiltonian and perform evolution (e.g. Kogut-Susskind tells us how to do this for QCD). In practice this is difficult to actually numerically simulate due to the very large hilbert space involved.

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    $\begingroup$ The problem isn't "many particles". Even single molecules can't be rigorously modeled with QFT. $\endgroup$
    – John Doty
    Sep 11, 2023 at 19:39
  • $\begingroup$ @JohnDoty Yes sorry, by "many particles" I was thinking in the case of lattice QCD, large valence quark number (e.g. we can do three-quarks very well (proton, neutron physics), we can kind of do six-quarks (deuterium), but anything more than that and it's way beyond our current capacity. Except in special cases (arxiv.org/abs/2307.15014)). This is of course very, very, very far away from simulating a tennis ball $\endgroup$ Sep 11, 2023 at 22:12
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    $\begingroup$ Yes. So QFT doesn't actually "describe every experiment we've ever done on Earth". Yes, with enough handwaving you can pretend, but in reality few experiments are accessible to QFT. $\endgroup$
    – John Doty
    Sep 11, 2023 at 22:20
  • $\begingroup$ @JohnDoty Sure, I mean that "we have no indication that anything is inconsistent with the Standard Model, other than gravity" (where I include neutrino masses in the Standard Model) (there are of course various tensions, but I would still say nothing conclusive) $\endgroup$ Sep 11, 2023 at 23:14
  • $\begingroup$ Oh really? Dark matter? Dark energy? Or just everyday physics. How would you know if the Standard Model was consistent with the appearance of a rainbow in any rigorous sense? $\endgroup$
    – John Doty
    Sep 11, 2023 at 23:26
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Quantum field theory is "just" a specific kind of quantum mechanics. Philosophically, you can think of relativistic quantum field theory in different ways that lead you to the same place, such as:

  • A quantum theory of an indefinite number of identical particles (which we need to describe interacting relativistic particles),
  • A quantum theory where the fundamental degrees of freedom are fields living on spacetime.
  • A low energy description of local interactions with a given set of symmetries (aka, Wilsonian effective field theory).

Given that quantum field theory is just "a type of" quantum mechanics, anything you can do in quantum mechanics, you can also in quantum field theory, at least in principle.

The main reason that scattering calculations are emphasized in field theory books, is that scattering experiments are the primary source of experimental data, for physics involving relativistic particles. Also, there is a well established perturbative framework for computing scattering amplitudes, that forms the basis for more sophisticated techniques.

The main reason to use fields, as opposed to a Hamiltonian density, is that fields encode local interactions. In relativity, non-local interactions would lead to a non-causal theory, so it is important that interactions occur at well defined points in spacetime. A Lagrangian built out of fields encodes locality. There is more about this in Weinberg, Volume 1, under the guise of the "cluster decomposition principle."

However, there are other quantities and effects that can be calculated in field theory, beyond scattering, including:

  • Power spectra of inflationary perturbations, that (if inflation is true) become the initial seeds of structure in the Universe and which we observe in the cosmic microwave background
  • Properties of phase transitions (when field theory is applied to condensed matter systems, which are usually not relativistic)
  • Tunneling effects, including the probability for false vacuum decay to occur
  • Asymptotic freedom and confinement (the latter needs experimental data + lattice QCD to justify)
  • The Lamb Shift of Hydrogen

The main issue with quantum field theory, compared to quantum mechanics of a fixed and relatively small number of particles, is that it is complicated to handle many particles. (Incidentally, that's not to say that quantum mechanics of a small number of particles is easy -- quantum mechanics problems with interactions between a few particles can quickly become intractable). The analogue of a few line calculation in single particle quantum mechanics may take many pages in quantum field theory. Part of this is kind of a trivial complication of dealing with more degrees of freedom However, part of this is "physically deep" in the sense that stuff can happen in quantum field theories that don't happen in quantum theories of a fixed number of particles, such as spontaneous symmetry breaking in the ground state. Books on field theory reflect this by quickly becoming very technical as they need more sophisticated tools to make sense of calculations, which are overkill in non-relativistic quantum mechanics.

Anyway, this is all to say that:

  • Quantum field theory is a "type" of quantum mechanics. Anything you can do in a general quantum theory (such as describe the time evolution of a state), you can also do in a quantum field theory.
  • Books focus on scattering for historical and experimental reasons, not logical ones. (Meaning, there's nothing about quantum field theory that says you logically must study scattering amplitudes, not that experiments aren't important, since of course they are).
  • Fields are used because they encode locality of interactions.
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