Where are quantised states in QFT?

Question

Perhaps the most striking fact of quantum mechanics, and where the name comes from, is the fact that the energy of a quantum system is generally quantized (for bound states), i.e. it can only take discrete values.

Now, being QFT the generalisation of classical FT to be consistent with QM, I would expect it to show some kind of "quantumness" (energies or momenta that are quantized)$^1$. But, as far as I know, there are no such things. We always consider a continuum of energies. So my question is the following: Why is this the case? And, maybe most interestingly, how does the quantization of energy (or of the states) arise from QFT?

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$^1$ - I know that FT is quantised by imposing the canonical commutation relations on the fields. My question is not about that. I only care about bound states.

Answer 1

Particles are quanta, or discrete packets of energy, in a quantum field. The fact that we see photons (for example), with a minimum energy $E=hf$ for a given frequency $f$, and not a continuum of field energy for a given frequency, is one of the most striking quantum features of quantum field theory, and responsible for may of the early experimental puzzles, like the blackbody radiation spectrum and photoelectric effect.

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Below I'm including some comments that seemed to be helpful to the OP, since comments sometimes get deleted.

You don't need quantum field theory to compute the energy levels for an electron in a hydrogen atom (although you can). The energy levels for an electron bound to hydrogen are quantized because the electron is in a bound state -- meaning it does not have enough energy to escape to infinity. This is a general phenomenon, where bound states lead to quantized energy levels. On the other hand, scattering states, which involve particles coming in from infinity and interacting, can have a continuum of energies (in both quantum mechanics and qft). One of the main (...) applications of relativistic QFT is to compute scattering amplitudes in particle collider experiments, which is maybe why you might naively associate QFT with a continuum of energies. However, bound states also exist in QFT, with discrete energies. For example, the eightfold way was a way of classifying particle states in strong-interaction physics, assuming the various particles were excitations of some underlying field. (We have a different fundamental picture now but this still works as an approximation).

Answer 2

Perhaps the most striking fact of quantum mechanics, and where the name comes from, is the fact that the energy of a quantum system is generally quantized (for bound states), i.e. it can only take discrete values.

It somewhat misleading to say that the energy is "generally quantized." The energies of certain bound states (such as the familiar states of the electron in a Coulomb potential) are quantized. But, even for the simple case of an electron in a Coulomb potential, there are energy eigenstates that do not have quantized energies--these are the positive energy states, and they form a continuum.

Yes, I know, we do not usually talk about these positive-energy states in our introductory quantum mechanics courses, but nevertheless, they exist and do not have quantized energies...

Now, being QFT the generalisation of classical FT to be consistent with QM, I would expect it to show some kind of "quantumness" (energies or momenta that are quantized). But, as far as I know, there are no such things.

One reason QFT is useful is because it allows us to deal with systems that have a changing particle number. If you want to work with a single electron in a Coulomb potential, you can happily restrict yourself to the $N=1$ part of the Fock space and recover all your favorite results from "ordinary" quantum mechanics.

We always consider a continuum of energies.

We do not always do this.

So my question is the following: Why is this the case?

It is not the case.

Answer 3

Perhaps it is important to add one clarification to the already good answers here. Although the focus, as presented by the OP, is the difference between QM and QFT, it is important to point out that the discreteness of bound state is not a defining property of quantum physics.

Even an optical fibre which is a purely classical device has discrete modes. It is simply a property of certain potentials that they admit discrete solutions.

There exists a perfectly similar classical manifestation of the "quantum" harmonic oscillator in the form of the gradient index lens. Its equation of motion is formally the same as that of the quantum harmonic oscillator and therefore has the same discrete solutions. We can thus conclude that it is not the discreteness of these solutions that make a system a quantum system.