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${}^\dagger$ For more details on this, I refer you to this excellent answer by ACuriousMind to another question of yours.

${}^\dagger$ For more details on this, I refer you to this excellent answer by ACuriousMind to another question of yours.

So yes, in a sense, you are right that we don't know what a photon is. But this [formal] obstacle doesn't prevent us from making predictions, like the case of pair production that worries you. The key point is that we don't know what happens during the interaction, we cannot know it and we don't need to know it. We only need to compare the asymptotic states before and after the interaction. In order to do that, we need to perform some approximation, usually a perturbative expansion (that results in Feynman diagrams, the wrongly-called "virtual particles" and all that). The comparation between in and out states, encoded in the $S$ matrix, is enough to predict decay rates, cross sections and branching ratios for any process you can imagine. And those observables are the only ones that we can measure. In conclusion, the things that you can't precisely define are the things that you can't experimentally verify.

So yes, in a sense, you are right that we don't know what a photon is. But this [formal] obstacle doesn't prevent us from making predictions, like the case of pair production that worries you. The key point is that we don't know what happens during the interaction, we cannot know it and we don't need to know it. We only need to compare the asymptotic states before and after the interaction. In order to do that, we need to perform some approximation, usually a perturbative expansion (that results in Feynman diagrams, the wrongly-called "virtual particles" and all that). The comparation between in and out states, encoded in the $S$ matrix${}^\dagger$, is enough to predict decay rates, cross sections and branching ratios for any process you can imagine. And those observables are the only ones that we can measure. In conclusion, the things that you can't precisely define are the things that you can't experimentally verify.

${}^\dagger$ For more details on this, I refer you to this excellent answer by ACuriousMind to another question of yours.

But this type of Lagrangians presents a huge drawback: we don't know how to quantize them in an exact way. Things get messy with interactions. We only can talk, with some rigor of asymptotic states: states long before or long after any interactions resemble those of the free fields. Therefore, the real photon is the excitation of thequantum electromagnetic potential that in the limit $t\to \pm\infty$ tends to the free photon as defined above.

This answer is only a sketch, a complete answer would require me to write a book on the topic. If you want to know more, I encourage you to read any book on QFT, like Peskin & Schroeder, Weinberg, Srednicki, etc.

But this type of Lagrangians presents a huge drawback: we don't know how to quantize them in an exact way.^* Things get messy with interactions. We only can talk, with some rigor of asymptotic states: states long before or long after any interactions resemble those of the free fields. Therefore, the real photon is the excitation of the quantum electromagnetic potential that in the limit $t\to \pm\infty$ tends to the free photon as defined above.

This answer is only a sketch, a complete answer would require me to write a book on the topic. If you want to know more, I encourage you to read any book on QFT, like Peskin & Schroeder, Weinberg, Srednicki, etc.

^* In an interacting theory, the classical equations of motion are non-linear, and can't be solved using a Fourier expansion that produces creation and annihilation operators. In the path integral formulation, we only know how to solve Gaussian integrals (i.e. free fields). To solve the path integrals for interacting fields we still need approximate methods like perturbative expansions or lattice QFT. According to Peskin & Schroeder:

No exactly solvable interacting field theories are known in more than two spacetime dimensions, and even there the solvable models involve special symmetries and considerable technical complication.