# Field quantization and photon number operator

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## Main Question or Discussion Point

[Moderator's note: This thread is spun off from a previous thread since it was getting into material too technical for the original thread. The quote at the top of this post is from the previous thread.]

He starts with the Planck spectrum of black-body radiation which rightfully needs field quantization, i.e., the photon picture. Then he uses the naive billiard-ball photon picture all the time although in a very beautiful section he writes all the arguments against it, including the point that both Compton and photoelectric effects are explainable through the semiclassical approximation and explicitly (and rightly!) stating that both effects do not necessarily prove the necessity of field quantization. So, why the heck is he using the wrong intuitions although obviously knowing much better?
Field quantization doesn't require a photon picture. A measurement device that creates a record that corresponds to the Planck spectrum just has to be modeled by an operator that has (in the given state) that probability distribution associated with its continuous spectrum (or else by a POVM). That doesn't require a photon picture.
That some operators have a discrete (or continuous) spectrum does not mean that the world is discrete (or continuous). Sometimes measurement results are discrete, which have to be modeled as discrete, and sometimes measurement results are continuous, which have to be modeled as continuous, and there are tools available for both (of course, because otherwise we would use a different mathematics). Where there is always continuity is in the evolution of probability densities over time for an operator that has a given spectrum, or, for one example, as we move a screen behind a double slit backward and forward so that an interference pattern changes, and I take it to be the dominance of that continuity that leads us to model using a (quantum) field theory.

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vanhees71
Gold Member
2019 Award
Field quantization doesn't require a photon picture. A measurement device that creates a record that corresponds to the Planck spectrum just has to be modeled by an operator that has (in the given state) that probability distribution associated with its continuous spectrum (or else by a POVM). That doesn't require a photon picture.
That some operators have a discrete (or continuous) spectrum does not mean that the world is discrete (or continuous). Sometimes measurement results are discrete, which have to be modeled as discrete, and sometimes measurement results are continuous, which have to be modeled as continuous, and there are tools available for both (of course, because otherwise we would use a different mathematics). Where there is always continuity is in the evolution of probability densities over time for an operator that has a given spectrum, or, for one example, as we move a screen behind a double slit backward and forward so that an interference pattern changes, and I take it to be the dominance of that continuity that leads us to model using a (quantum) field theory.
Maybe I've fallen in my own trap here. Perhaps, I should have said that the derivation of the Planck distribution requires field quantization. Photons are only adequately defined by QED, and a photon is defined via the single-photon Fock states of the quantized electromagnetic field.

The Planck spectrum is simply described by the canonical equilibrium operator
$$\hat{\rho}=\frac{1}{Z} \exp(-\beta \hat{H}),$$
with the partion sum $Z$ and $\beta=1/(k_{\text{B}} T)$ the inverse temperature.

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Maybe I've fallen in my own trap here. Perhaps, I should have said that the derivation of the Planck distribution requires field quantization. Photons are only adequately defined by QED, and a photon is defined via the single-photon Fock states of the quantized electromagnetic field.
A first difficulty is that the number operator $\hat N$ is a global operator, so we can never say from using local measurements what number of photons there are in a given state.
To indicate a second difficulty, I have to first develop a particular mathematical perspective on QFT (apologies if this is well-known to you). Quantized electromagnetism "done right" uses "test functions" (though from a signal analysis perspective we could call what are called test functions in the mathematical physics literature either "modulation functions" or "window functions", depending on what they are used for), so that we modulate the globally defined vacuum |0〉 using a smeared field operator, giving us $F$f|0〉, not using something like $F$μν(x)|0〉, parameterized by position, or the fourier transform $\tilde F$μν(k)|0〉, parameterized by wave-number, both of which have infinite norm so neither is in the Hilbert space (instead of talking about a "modulation" f, we can also talk of QFT being about sending a "message" f, using the QFT vacuum as a carrier for the message, which more intuitively has to be finite in duration; the point of QFT is that these modulations are modulations of probability densities, so they're more complicated than the modulations of a simple carrier frequency that we're used to when we talk about FM or AM modulated radio waves, and mathematically of a different kind).
The second difficulty, then, is that $F$f|0〉 is a single-photon state for any choice of f whatsoever. If one takes a messaging or modulation perspective, there's no obviously most natural choice of message or modulation, so there's no single candidate for a "photon".
Although I led off from a statement of yours, obviously I've developed it in a very specific way. This isn't what one much finds in textbooks (although worries about the nature of particles/photons in QFT have been widespread in the philosophy of QFT literature for about the last 20 years, and one finds such concerns in algebraic QFT), and it's unlikely you'll find this helpful for the teaching teachers class you're about to do. This more for the future, I guess, as an evolving rehearsal for an interpretation of QM/QFT as a signal analysis formalism.

A. Neumaier
2019 Award
A first difficulty is that the number operator is a global operator, so we can never say from using local measurements what number of photons there are in a given state.
In interacting QED, the number operator does not exist at all; it cannot be renormalized!

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In interacting QED, the number operator does not exist at all; it cannot be renormalized!
Fair enough. I should have been more explicit that that comment applies only to free fields. I'll take it that not being able to count the number of photons, even globally, as well as it not being possible to count photons locally, is also good enough to establish how tenuous the idea of photons is in QFT.

vanhees71
Gold Member
2019 Award
A first difficulty is that the number operator $\hat N$ is a global operator, so we can never say from using local measurements what number of photons there are in a given state.
To indicate a second difficulty, I have to first develop a particular mathematical perspective on QFT (apologies if this is well-known to you). Quantized electromagnetism "done right" uses "test functions" (though from a signal analysis perspective we could call what are called test functions in the mathematical physics literature either "modulation functions" or "window functions", depending on what they are used for), so that we modulate the globally defined vacuum |0〉 using a smeared field operator, giving us $F$f|0〉, not using something like $F$μν(x)|0〉, parameterized by position, or the fourier transform $\tilde F$μν(k)|0〉, parameterized by wave-number, both of which have infinite norm so neither is in the Hilbert space (instead of talking about a "modulation" f, we can also talk of QFT being about sending a "message" f, using the QFT vacuum as a carrier for the message, which more intuitively has to be finite in duration; the point of QFT is that these modulations are modulations of probability densities, so they're more complicated than the modulations of a simple carrier frequency that we're used to when we talk about FM or AM modulated radio waves, and mathematically of a different kind).
The second difficulty, then, is that $F$f|0〉 is a single-photon state for any choice of f whatsoever. If one takes a messaging or modulation perspective, there's no obviously most natural choice of message or modulation, so there's no single candidate for a "photon".
Although I led off from a statement of yours, obviously I've developed it in a very specific way. This isn't what one much finds in textbooks (although worries about the nature of particles/photons in QFT have been widespread in the philosophy of QFT literature for about the last 20 years, and one finds such concerns in algebraic QFT), and it's unlikely you'll find this helpful for the teaching teachers class you're about to do. This more for the future, I guess, as an evolving rehearsal for an interpretation of QM/QFT as a signal analysis formalism.
Of course, for the teachers there's no relativistic QFT lecture. However, I think that you should not teach wrong qualitative concepts like photons as little bullets. On a qualitative level, as is used to motivate non-relativistic QM from the historical development, which indeed starts with the quantum aspects of light with Planck and Einstein, you do not need any concept of localization. What is taught in this prepedeutics is nothing more than the quantized absorption-emission processes for field energy and momentum with matter (charged particles), and this can be taught without taking recourse to localized-particle pictures for photons.

Of course you are right, as in classical electrodynamics plane waves (momentum eigenstates in the quantum laugauge) of electromagnetic fields do not describe physical entities but are used for the decomposition of physical fields into modes, i.e., they are a mathematical tool. Indeed only such single-photon states that are normalizable to 1 represent physical states (that holds throughout all QM; momentum eigenstates do not belong to Hilbert space but are distribution valued, as is made very clear by fact that they are only "normalizable to a $\delta$ distribution").

The fact that there are very many single-photon states, depending on the "test function" you "smear" the plane-wave modes is not a bug but a feature since it reflects the observed variety of photons being present in nature.

vanhees71
Gold Member
2019 Award
In interacting QED, the number operator does not exist at all; it cannot be renormalized!
Well, that's a bit too strict, although it may be right from a strict mathematical point of view.

For the Planck spectrum you do not need more than the quantization of the free em. field, starting as always in a finite box in configuration space with convenient periodic boundary conditions. Then you get momentum eigenmodes (plane waves) with discrete momentum eigenvalues and normalizable momentum eigenstates. Then renormalization of the relevant quantities like energy, momentum, and angular momentum of the e.m. field boils down to the normal-ordering description making the vacuum state the eigenstate where these quantities vanish.

Of course, what's a well-defined lorentz-covariant quantity is indeed not any kind of photon number (density) but the energy spectrum, and it can be easily derived from the generating functional
$$Z[\beta]=\mathrm{Tr} \exp[-\sum_{\vec{q},\lambda} \beta(\vec{q},\lambda) |\vec{q}| \hat{N}(\vec{p},\lambda)].$$
This leads to the Bose-Einstein distribution for the $N$ (or rather the energy). In the infinite-volume limit that's the Planck spectrum
$$E \frac{\mathrm{d} N}{\mathrm{d}^3 \vec{q}} = \frac{1}{(2 \pi)^3} \frac{g}{\exp(\beta |\vec{q}|)-1}, \quad \beta=\frac{1}{T}.$$
For the details, extended to an explicitly Lorentz covariant treatment, see (sorry, it's in German)