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Couldn't we just assume that waves have mass and momentum and can become localized? Dirac Deltas can be given a rigorous mathematical foundation but physicist do not use the Gelfand triple. Why not just assume that electrons, photons, etc., have the same dimension as space?

I know string theory does this but are there reformulations of QFT that use this idea that together with the insight that renormalization has a connection with scale invariance gets rid of all the problems that plague QFT? It just seems to me that the concept of a particle is a very unphysical idea.

Quillo
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Lina Jane
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    What name would you give to clicks in a photodector? – Andrew Sep 26 '23 at 02:26
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    I don't mean semantically, I mean mathematically why are Dirac Delta's of infinite mass density a thing? Those do not seem like physical things. Clicks in a photodetector are not actually infinitely thin ponts – Lina Jane Sep 26 '23 at 02:28
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    A little less flippantly, my point is that quantum particles exhibit discrete behavior that is not characteristic of classical waves. Namely, you can count the number of quanta that arrive at a sensitive detector. If measuring states with frequency $\omega$, you never that those states deposit less than $\hbar\omega$ worth of energy in the detector, whereas classical waves with any frequency could have any energy. – Andrew Sep 26 '23 at 02:41
  • In QFT the particles are only defined in the non-interacting infinitely separated long time limits, and do not have a presence inside the rest of the theory. As such, any problems in QFT cannot possibly come from what you are thinking. Also, it turns out that the Gelfand triple is the wrong way to solve the problems with quantum theory. Instead, the successful methods are refinements in the conception of Hilbert spaces as used in quantum theory. It is true, however, that naïve handling of Dirac delta distributions cause UV divergences, and that part has been tamed. – naturallyInconsistent Sep 26 '23 at 03:06
  • @Andrew the click of a photodetector does not need the notion of a particle for its explanation. It is the fundamental interactions (such as what takes place in the detector) that are quantized. – flippiefanus Sep 26 '23 at 03:06
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    What Lina Jane and flippiefanus said. If the detector has to absorb a whole quantum of energy over some period of time before it can detect it, why assume that the energy arrives in one glob at one instant? – J Thomas Sep 26 '23 at 03:13
  • Some worth checking questions on similar lines: https://physics.stackexchange.com/q/420486/226902, https://physics.stackexchange.com/q/713942/226902, https://physics.stackexchange.com/q/737183/226902, https://physics.stackexchange.com/q/118927/226902. About Gelfand triple (just for reference): https://physics.stackexchange.com/q/441503/226902. – Quillo Sep 26 '23 at 04:18
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    particles happen to be observable . see the track of an electron in this answer https://physics.stackexchange.com/q/237350/ – anna v Sep 26 '23 at 04:48
  • @naturallyInconsistent What refinements? – Lina Jane Sep 26 '23 at 05:07
  • @J Thomas No one competent says that the quantum particle is the infinitely localized glob of energy. The quantum particle is the quantum state of something that we describe with QFT, that may be very much unlocalized. Its "particleness" is in the discreteness of structure of spectrum (i.e. you may distinguish one-particle states, two-particle states etc) in the weakly-coupled regime and in the locality of interactions (which may be understood in terms of the fields without any talk about pointlikeness) – OON Sep 26 '23 at 06:21
  • Ok. Very useful. Thank you. But I know that the discrete spectrum comes from a bounded interval that forces the waves to have certain frequencies and wavelenghts, and by de Broigle discrete energy spectrum, but is is true that before interactions are introduced as perturbations the free field solutions consider a discrete spectrum because the solutions with a continuos energy spectrum are unphysical? Why? – Lina Jane Sep 26 '23 at 17:17
  • The discreteness of particles also comes from the range bounded by a potential. But this boundedness is not in the ordinary space but in the space of the field configurations. I.e. if you consider, a plane wave, its amplitude behaves just like a particle in the harmonic potential, which leads to a discrete spectrum of stationary states. In short, a particle is a travelling wave in the ordinary space but a staying wave in the field configuration space. – OON Sep 28 '23 at 05:50
  • You can of course have other states, like a coherent state which is not stationary and approximates a classical wave.. That would be a superposition of particle states – OON Sep 28 '23 at 05:51

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The QFT is not a theory of classical particles. However it is also not a theory of classical fields. And it has both as liming cases. Because of that it is often useful to use the language of both while understanding that you talk about something that is neither one of them.

You may approach it in the field-centric fashion. Construct free QFT as a "quantization" of the classical free field theory, then construct interacting QFT as a perturbation by a local interaction.

Or you can approach it in the particle-centric fashion. Construct Fock space as a formal state space of relativistic identical particles assuming that they have momentum and are discrete entities. Then construct their scattering amplitudes as a formal series assuming that they interact in the local way (i.e. assuming that they are in a cerain sense pointlike). Because most of the problems in QFT are scattering problems you borrow from this way of thinking even when you start in the field-centric fashion.

I can see you screaming "But what about non-perturbative QFT???" But the thing is that most of the time we don't have anything you may call non-perturbative QFT!!! Instead you have recipes based on some ideological considerations that should complete you perturbative construction. Because we have some axiomatic field-correlator constructions (e.g. CFT) or semi-classical considerations (that assume the specific classical limit to work in this parameter range) that don't rely on the perturbative expansions, we try use them. But let's be honest, we must be ready that these recipes may fail to give the appropriate non-perturbative completion.

The fun thing is that contrary to what you write about string theory, it is built in the particle-centric fashion! The things like string dualities and AdS/CFT that we use to understand string theory non-perturbatively do not rely on attempts to generalize the field-notion like string field theory.

This goes with your comment about physicists and Gelfand triples. The mathematicians go with the purified abstract constructs that are defined in a closed and self-consistent fashion. The physicsts use the constructs to describe reality. So physicst tend to the way "Let's regularize and define it as a limit, hoping that it does not depend on the regularization" because:

  1. They are only interested in obtaining results
  2. They usually are lazy to decipher mathematitician's language that differs sometimes from the their language considerably. And unlike physicist mathemations more often than not do not give "the complete idiot's explanation of my idea" in their papers.
  3. There is often an actual physical reason to consider model as a limit of regularized stuff
  4. They often work with bad models (like non-renormalizable effective QFTs) that actually have regularization-dependence and other issues. And while mathematicians with give you their abstract constructs we return to the reasons 1-3 for the physicsit to skip them.

In fact most physicists don't care even about all the complicated stuff about QM: that there is all these stuff about definitions on dense subspaces, about self-adjoint extensions of symmetric operators, about rigged Hilbert spaces etc. When they find the issues with naive ways they usually regularize the problem and take the limit and reinvent e.g. symmetric operator with multiple self-adjoint extensions in the fashion they need for their physics problem. Or find that actually this limiting definition pinpoints the specific self-adjoint extensions whereas the abstract approach do not give you the answer. I personally had such experience.

So if you are a physicst do not cling to any nice model and actually absorb all viewpoints because you don't know which one will actually be more useful in the future.

OON
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