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This post claims that there is no real photon (particle) with a plane wave solution well-defined momentum state).

It makes sense somehow to me. I can think of several arguments:

  1. The plane wave solution doesn't have well-defined probability as it exists in infinite space with equal probability. We will never be able to detect those particles.

  2. Our probability interpretation breaks down with the plane wave solution. It's not square-integrable.

Even though it seems non-sense, we apply quantization rule to the free particle which makes quantization as the imaginary process which doesn't exist in reality.

Besides 'it works' argument, do we have any explanation why this quantization process makes sense?

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  • $\begingroup$ Quantization doesn't require plane waves. Why would it? Is there a specific source or a specific line of reasoning that make you think it does? (Knowing this might help people shape their answers.) $\endgroup$ Jul 24, 2020 at 13:04
  • $\begingroup$ @ChiralAnomaly What do you mean it's not required? We build the S matrix with the free particle states. And Feynman diagram is drawn with the free particles for entry and exit. If they are not quantized, how do we consider any quantities? $\endgroup$ Jul 26, 2020 at 0:46
  • $\begingroup$ I meant that quantization doesn't require using plane waves. Free particles (or non-free particles) don't require using plane waves, either. The question seems to assume that problems with plane waves imply problems with quantization or with particles, but that's not the case. Plane waves are problematic, but quantization and particles don't rely on plane waves. We can describe both quantization and particles and the S-matrix without ever referring to plane waves at all. $\endgroup$ Jul 26, 2020 at 2:53

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All models are wrong, but some are useful

Much of physics is inductive theories, which build description of complex phenomena from small elements. Neither photon, nor a point-like particle in free space (the basis of the Newtonian mechanics), nor a harmonic oscillator, nor plane waves exist. However, they are simple and convenient building blocks for developing theories that successfully model natural phenomena. Note that none of these theories tries to take account of everything - it would be meaningless - the question is that of keeping what is important, and omitting whatever is inessential.

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Besides 'it works' argument, do we have any explanation why this quantization process makes sense?

Quantum field theory (QFT) is used for interacting elementary particles, and also as a tool/model for other quantum mechanical states (condensed matter for example).

Your premise:

we apply quantization rule to the free particle

Is wrong:

QFT was not developed to study free particles, because as you say the plane wave solutions cannot describe the probability of seeing a particle at ( x,y,x,t). To describe with fields a single particle one has to use the wave packet solution,.

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Besides 'it works' argument, do we have any explanation why this quantization process makes sense?

The "it works" is reserved for the Feynman diagram use of the QFT tool/model, .

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