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The terms like 'particle', 'quantum of light' or 'unit of energy exchange' lead to believe that photons are something discrete and sudden. Second quantization supplements this idea. For example, in second quantization, the Hamiltonian of a single state (say a particular standing wave in a cavity) can be written as

This is also the Hamiltonian for harmonic oscillator. Consequently, we can easily then write the 'wave function' of this state as $\Psi(q)$ and Hamiltonian with classical kinetic energy like $p^2$ and potential energy like $q^2$ terms. We can write this wave function as linear combination

Instead, all that is, is the field and it is subject to typical quantum wave evolution. This field couples to matter.

Let's discuss the two transverse modes of a propagating photon (there are actually 2 more, longitudinal and energy-like but that is out of scope). It is often said that photon has angular momentum of $\pm \hbar$, which corresponds to circularly polarized light particles. This leads to a spinor like representation for photon.

Say a dye molecule get's excited in our eye-receptor and it subsequently changes it's form and a nerve impulse is transmitted. Such process resembles a quantum measurement since it involves so many uncontrolled degrees of freedom in high temperatures, and a phenomenom called decoherence happends. Thus, if the photon wave function was previously $(\frac{1}{\sqrt{2}}|1> + e^{i\theta} \frac{1}{\sqrt{2}}|0>$, the effective wave function (integrating out the macroscopic degrees of freedoms) is in a discrete state with probabilities given by their amplitudes. That is why photons can be seen and hear as clicks. With a grain of salt, it is the collapse of the wave function which makes the sound :)

4. Far field and near field photons are different

It is often said that photon has definite energy and momentum which must be conserved (i.e. it follows the dispersion relation $E=\hbar k$ and one photon which hits the detector has always this (E,k). But for example, there are photonic crystals, where photon energies have band gaps and photons appear to have masses (non-linear dispersion relations). Again, one can quantize the Maxwell equations in a photonic crystal by some choice of states, and assign particles to these states. One can speak of photons here as well, and even say that they have mass since their equations of motion behave as they have mass.

Now picture more modes than just one before. The wave function is now $\Psi(q_1,q_2 \ldots q_N)$. Now imagine creating a linear combination of these modes $q'_i = \sum_j A_{ij} q_j$ to localize them as much as possible. In fact, let's localize to the extent that one mode $q'$ will correspond to a particular location in space. Now, what you have is a 'wave function' of a photon, which gives a probability amplitude of photon field at different positions of space

By limiting ourselves to N coordinates which describe a photon roughly around positions ($r_1 \ldots r_N$), we have effectively imposed an energy cutoff to our equations and everything is fine.

Now imagine extending this process to a continuum limit (far from trivial) and switching on the light-matter interaction and we have encountered the problem of renormalization and all really hard and hard core stuff.

Given all that, one want's for practial reasons and for physical intuitions sake to go back to the second quantization and talk about 1 photon in mode 15. In other words, second quantization and the talk about particles as an excitations of harmonic oscillators is all just instruments created by and for the physical intuition. But if one want's to understand what photon is, one needs to go under the hood.

The terms like 'particle', 'quantum of light' or 'unit of energy exchange' lead to believe that photons are something discrete and sudden. Second quantization supplements this idea. For example, in second quantization, the Hamiltonian of a single state (say a particular standing wave in a cavity) can be written as

This is also the Hamiltonian for a harmonic oscillator. Consequently, we can easily then write the 'wave function' of this state as $\Psi(q)$ and Hamiltonian with classical kinetic energy like $p^2$ and potential energy like $q^2$ terms. We can write this wave function as a linear combination,

Instead, all that is, is the field, and it is subject to typical quantum wave evolution. This field couples to matter.

Let's discuss the two transverse modes of a propagating photon (there are actually two more, longitudinal and energy-like, but that is out of scope). It is often said that a photon has angular momentum of $\pm \hbar$, which corresponds to circularly polarized light particles. This leads to a spinor-like representation for a photon.

Say a dye molecule gets excited in our eye-receptor, and it subsequently changes its form, and a nerve impulse is transmitted. Such a process resembles a quantum measurement since it involves so many uncontrolled degrees of freedom in high temperatures, and a phenomenon called decoherence happens. Thus, if the photon wave function was previously $(\frac{1}{\sqrt{2}}|1> + e^{i\theta} \frac{1}{\sqrt{2}}|0>$, the effective wave function (integrating out the macroscopic degrees of freedoms) is in a discrete state with probabilities given by their amplitudes. That is why photons can be seen and heard as clicks. With a grain of salt, it is the collapse of the wave function which makes the sound :)

4. Far-field and near-field photons are different

It is often said that a photon has definite energy and momentum which must be conserved (i.e. it follows the dispersion relation $E=\hbar k$ and one photon which hits the detector has always this (E,k). But for example, there are photonic crystals, where photon energies have band gaps and photons appear to have masses (non-linear dispersion relations). Again, one can quantize the Maxwell equations in a photonic crystal by some choice of states, and assign particles to these states. One can speak of photons here as well, and even say that they have mass since their equations of motion behave as they have mass.

Now picture more modes than just one before. The wave function is now $\Psi(q_1,q_2 \ldots q_N)$. Now imagine creating a linear combination of these modes $q'_i = \sum_j A_{ij} q_j$ to localize them as much as possible. In fact, let's localize to the extent that one mode $q'$ will correspond to a particular location in space. Now you have a 'wave unction' of a photon, which gives a probability amplitude of the photon field at different positions of space.

By limiting ourselves to N coordinates which describe a photon roughly around positions ($r_1 \ldots r_N$), we have effectively imposed an energy cutoff to our equations and everything is fine.

Now imagine extending this process to a continuum limit (far from trivial) and switching on the light-matter interaction, and we have encountered the problem of renormalization and all really hard and hardcore stuff.

Given all that, one wants for practical reasons and for physical intuitions sake to go back to the second quantization and talk about one photon in mode 15. In other words, second quantization and the talk about particles as an excitations of harmonic oscillators is all just instruments created by and for the physical intuition. But if one wants to understand what a photon is, one needs to go under the hood.

I am annoyed by the definitions of photon as described in the question. Below is what I think. That is of course no new physics, and every interpretation is subjective. I will go through this by introducing few antithesis.

I am annoyed by the definitions of photon as described in the question. It is not that they are wrong, but because I was mislead by them almost as if they were preventing me to understand what a photon is. Below is what I think now. That is of course no new physics, and every interpretation is subjective. I will go through this by introducing few antithesis.

However, in some applications, it better to analyze only linearly polarized photons ($\Psi_{x,y}(q) = \frac{1}{2} (\Psi_L(q) \pm i \Psi_L(q))$). Now, it is easy to see, that just like electron spin, one has chosen just a preferred frame of reference and there is nothing extremely special about the choice of these discrete coordinates. (Of course, there is something special in choice of coordinates: the physical intuition to describe a problem well.) But in fact, I think that even the transversenss of a polarization is a choice of reference.

It is often said that photon has discrete energy and momentum (i.e. it follows the dispersion relation $E=\hbar k$ and one photon which hits the detector has always this (E,k). But for example, there are photonic crystals, where photon energies have band gaps and photons appear to have masses (non-linear dispersion relations). Again, one can quantize the Maxwell equations in a photonic crystal by some choice of states, and assign particles to these states. One can speak of photons here as well.

By limiting ourselves to N discrete positions, we have effectively imposed a cutoff to our equations and everything is fine.

However, in some applications, it better to analyze only linearly polarized photons ($\Psi_{x,y}(q) = \frac{1}{2} (\Psi_L(q) \pm i \Psi_R(q))$). Now, it is easy to see, that just like electron spin, one has chosen just a preferred frame of reference and there is nothing extremely special about the choice of these discrete coordinates. (Of course, there is something special in choice of coordinates: the physical intuition to describe a problem well.) But in fact, I think that even the transverseness of a polarization is a choice of reference.

It is often said that photon has definite energy and momentum which must be conserved (i.e. it follows the dispersion relation $E=\hbar k$ and one photon which hits the detector has always this (E,k). But for example, there are photonic crystals, where photon energies have band gaps and photons appear to have masses (non-linear dispersion relations). Again, one can quantize the Maxwell equations in a photonic crystal by some choice of states, and assign particles to these states. One can speak of photons here as well, and even say that they have mass since their equations of motion behave as they have mass.

By limiting ourselves to N coordinates which describe a photon roughly around positions ($r_1 \ldots r_N$), we have effectively imposed an energy cutoff to our equations and everything is fine.