# Photon Statistics

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Photon statistics
In a given mode with an average number of photons ##\bar{n}##, the photons are distributed around their average according to the formula

$$p_n = e^{-\bar{n}} \frac{\bar{n}^n}{n!}$$

The justification of this formula in quantum field theory involves considering field operators acting on a space with an indefinite metric, because it involves relations like ##[A_\mu, \dot{A}_\nu] = -ig_{\mu\nu} \delta(x - y)##. The S matrix involved in these calculations must leave the positive metric physical subspace invariant.

I wonder what is the right way to think about Fock spaces with an indefinite metric? These spaces occur in string theory because of the commutation rules like ##[\alpha_\mu, \alpha_\nu] = ig_{\mu\nu}##. The n! in the above formula is a consequence of bose statistics, and the ##\bar{n}^n## is a consequence of the fact that (a+)n ~ nn/2, but the factor ## e^{-\bar{n}}## is mysterious. The mass shell condition in string theory involves relations of the form m2 ~ N, where m2 is a kind of hamiltonian on Hilbert spaces with indefinite metric. What is the significance of Hilbert spaces with an indefinite metric in QFT and in string theory?

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Mentor
In a given mode

Mode of what? What physical situation are we talking about here? There are lots of different possible photon states that apply to different physical situations.

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Like a momentum mode. a+(k) creates a photon in the mode k. This is a general formula that applies in any situation.

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Mentor
Like a momentum mode.

Momentum alone doesn't specify a photon's state. You also need its polarization.

If you are ignoring polarization, then I don't understand the distribution formula you are using: distribution of what? A momentum mode as you are defining it is a Fock state, i.e., an eigenstate of photon number. It doesn't have a "distribution" of anything.

A reference to where you are getting all this from would help.

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This derivation is given in Itzykson and Zuber. The steps are clearly given, but I'm not entirely sure how to think about this derivation. The partition function is Z[j] = 〈0|Tei∫d4x j(x)φ(x)|0〉. This framework of QFT involves expressions of the form

e∫d4x∫d4y j(x)j(y)D(x - y)

and also in momentum space ei∫d4p j(p)D(p, -p)j(-p). In an expression like j(x)j(y)D(x - y), the current j(x) would be the amplitude to emit a particle at x or absorb an antiparticle at x, so this is just Feynman's way of thinking where you multiply the amplitudes to emit, to propagate and then to absorb. D(p, -p) is the amplitude to "propagate with momentum p." These types of expressions are closely linked to the 'equation of motion' D(p, -p)j(p) = φ(p) which you get by transforming it into momentum space, also the Lagrangian can also be written as L = j(x)φ(x) in some sense using the equation of motion to replace (p2 - m2)φ(x) with j(x) etc. The propagator is D(x - y) = 〈0|Tφ(x)φ(y)|0〉 which also involves the vev of a time ordered product. But I am not entirely sure how to interpret things like φc(x) = ∫d4y D(x - y)j(y) (which is guess is some version of "amplitude to find at x is amplitude to emit at y times amplitude to propagate from y to x, summed over all y"). But there are more expressions like

φ(x) = φin(x) + ∫d4y Dret(x - y)j(y)

φ(x) = φout(x) + ∫d4y Dadv(x - y)j(y)

And so on, which I haven't yet tried to imagine the way Feynman would imagine them. The expression for the S matrix that Itzykson and Zuber gives contains a factor exp ∫∫[φ-inj, φ+inj]. If someone can tell me about this factor in Itzykson and Zuber's expression for the S-matrix, I would appreciate it.

[I think one important thing in the above is the fact that φ(x) ~ √2E. Even when you think of φ as a spacetime field, its square must be interpreted as the energy density in space. You can write seemingly space-time symmetric equations like [φ(x), φ(y)] = D(x - y), but as I said this is slightly misleading since φ2 is the space density of particles. ]

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Mentor
Itzykson and Zuber

Can you give a more specific reference? This is not a source I'm familiar with.

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Itzykson and Zuber's "Quantum Field Theory". It is in chapter 4.

Mentor
Itzykson and Zuber's "Quantum Field Theory". It is in chapter 4.

Unfortunately I don't have this reference so my ability to help will be limited. Other experts in this forum might have it.

The "distribution" you gave in the OP looks like the formula for the expectation value of the photon number operator in a coherent state. But a coherent state is very different from a Fock state.

The justification of the expectation value of photon number for a coherent state has nothing to do with any "indefinite metric" in any textbook treatment of the subject that I have seen. It is a simple application of the standard QM formula for the expectation value of an operator given a state.

Perhaps there is more context in that specific textbook that I am unaware of.

In a given mode with an average number of photons ##\bar{n}##, the photons are distributed around their average according to the formula

$$p_n = e^{-\bar{n}} \frac{\bar{n}^n}{n!}$$

I highly doubt that the book says or intends to say that. I do not own that book (and the google preview does not allow me to read chapter 4), but the table of content explicitly states that chapter 4 is about a "quantized electromagnetic field interacting with a classical source". This is a very relevant constraint and excludes a lot of nonclassical light fields (and in fact also some classical ones).

The n! in the above formula is a consequence of bose statistics, and the ##\bar{n}^n## is a consequence of the fact that (a+)n ~ nn/2, but the factor ## e^{-\bar{n}}## is mysterious.

Indeed the equation given above is not general, but valid for coherent states only. As coherent states are the most classical states there are, this matches the name of the chapter quite nicely. I do not know how the booko proceeds exactly, but maybe it helps to have a look at the simpler non-QFT vanilla quantum optics way of deriving the Poissonian distribution first, check where the exponential term comes from and then to construct the QFT analogue.

In standard quantum optics, one wants to be able to describe rather classical fields as well. In classical physics a measurement is fully noninvasive, which is not possible in a quantum treatment as the detection of a photon will destroy it. However, the closest thing is a state ##|\alpha\rangle## which does not change upon annihilation of a photon. One may describe ##|\alpha\rangle## in the Fock state basis, so that ##|\alpha\rangle=\sum_{n=0}^\infty c_n |n\rangle##
So one requires:
$$\hat{a}|\alpha\rangle=\alpha |\alpha\rangle$$
or equivalently
$$\sum_{N=1}^\infty \sqrt{n} c_n|n-1\rangle=\alpha \sum_{n=0}^\infty c_n |n\rangle.$$

As the Fock states form an orthonormal basis, the equation must be fulfilled for each term of the sum, so you get e.g.:
$$\sqrt{n} c_n=\alpha c_{n-1}.$$
This gives you a recursion formula for ##c_n##:
$$c_n=\frac{\alpha^n}{\sqrt{n!}} c_0.$$
One may then get ##c_0## by considering thatone should have ##\langle \alpha|\alpha\rangle=1:##
$$\langle \alpha|\alpha\rangle=|c_0|^2 \sum_{n=0}^\infty \frac{|\alpha|^{2 n}}{n!}=|c_0|^2 e^{|\alpha|^2},$$
so you get:
$$c_0=e^{-\frac{|\alpha|^2}{2}}.$$
This is where the exponential term enters. ##\frac{\alpha}{\sqrt{2}}## is the amplitude of a classical light field, so ##\frac{|\alpha|^2}{2}## corresponds to ##\bar{n}##.

The reasoning in the QFT approach should not be too different for classical sources. The math will of course differ a bit.

PeterDonis and vanhees71
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Summary:: Photon statistics

In a given mode with an average number of photons ##\bar{n}##, the photons are distributed around their average according to the formula

$$p_n = e^{-\bar{n}} \frac{\bar{n}^n}{n!}$$

The justification of this formula in quantum field theory involves considering field operators acting on a space with an indefinite metric, because it involves relations like ##[A_\mu, \dot{A}_\nu] = -ig_{\mu\nu} \delta(x - y)##. The S matrix involved in these calculations must leave the positive metric physical subspace invariant.

I wonder what is the right way to think about Fock spaces with an indefinite metric? These spaces occur in string theory because of the commutation rules like ##[\alpha_\mu, \alpha_\nu] = ig_{\mu\nu}##. The n! in the above formula is a consequence of bose statistics, and the ##\bar{n}^n## is a consequence of the fact that (a+)n ~ nn/2, but the factor ## e^{-\bar{n}}## is mysterious. The mass shell condition in string theory involves relations of the form m2 ~ N, where m2 is a kind of hamiltonian on Hilbert spaces with indefinite metric. What is the significance of Hilbert spaces with an indefinite metric in QFT and in string theory?
You have, of course, to specify the state of the electromagnetic field. From your Poisson distrbution I deduce that you talk about a single-mode coherent state of the photon field. For details, see

https://en.wikipedia.org/wiki/Coherent_state#Coherent_states_in_quantum_optics

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Summary:: Photon statistics

In a given mode with an average number of photons ##\bar{n}##, the photons are distributed around their average according to the formula

$$p_n = e^{-\bar{n}} \frac{\bar{n}^n}{n!}$$

The justification of this formula in quantum field theory involves considering field operators acting on a space with an indefinite metric, because it involves relations like ##[A_\mu, \dot{A}_\nu] = -ig_{\mu\nu} \delta(x - y)##. The S matrix involved in these calculations must leave the positive metric physical subspace invariant.

I wonder what is the right way to think about Fock spaces with an indefinite metric? These spaces occur in string theory because of the commutation rules like ##[\alpha_\mu, \alpha_\nu] = ig_{\mu\nu}##. The n! in the above formula is a consequence of bose statistics, and the ##\bar{n}^n## is a consequence of the fact that (a+)n ~ nn/2, but the factor ## e^{-\bar{n}}## is mysterious. The mass shell condition in string theory involves relations of the form m2 ~ N, where m2 is a kind of hamiltonian on Hilbert spaces with indefinite metric. What is the significance of Hilbert spaces with an indefinite metric in QFT and in string theory?
Indefinite metric has not much to do with directly measurable quantities such as the formula for ##p_n## above. Such measurable quantities can also be obtained without the indefinite metric. In quantum theories with gauge symmetries, the indefinite metric is just a mathematical trick to make equations formally look Lorentz covariant. A more natural approach is to fix gauge completely so that the indefinite metric never appears, in which case the measurable quantities are easier to compute, but then Lorentz covariance is not obvious.

vanhees71
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There are 1/2E on-shell states in the interval dp. I think this is related to regularizing path integrals.

The Poisson distribution is an inherent property of the idea of a photon. At least this seems to be Bohr's view in his complementarity arguments. I don't think it is something that applies only in special situations. The coherent state stuff may be correct, but it might give one the wrong impression. The QFT view in Itzykson and Zuber suggests that it is a far more general thing.

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HomogenousCow
There’s a chapter in Peskin & Schroeder in which the same result is derived by summing diagrams.

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There are 1/2E on-shell states in the interval dp. I think this is related to regularizing path integrals.

The Poisson distribution is an inherent property of the idea of a photon. At least this seems to be Bohr's view in his complementarity arguments. I don't think it is something that applies only in special situations. The coherent state stuff may be correct, but it might give one the wrong impression. The QFT view in Itzykson and Zuber suggests that it is a far more general thing.
The Poisson distribution has nothing to do with photons (Fock states of the electromagnetic field) but everything with coherent states. That's why you have a fluctuation in photon number to begin with. Photon Fock states are by definition states of determined photon number.

In HEP QFT textbooks the issue occurs when it comes to the various soft-photon problems, i.e., to get rid of infrared and collinear divergences. The most simple problem, where this comes up is bremsstrahlung, where you get an IR divergent result. The reason is that the usually assumed plane waves are not the real asymptotic free states, because photons are massless and thus the electromagnetic interaction long-ranged. That's because the Coulomb potential goes only with ##1/r##, such that the usual arguments for the asymptotic behavior of the wave functions of scattering theory don't hold. This can be already seen in Coulomb scattering of non-relativistic quantum theory, where you can solve the two-body elastic-scattering problem of two charged particles exactly. It turns out that the plane waves are modified by unusual phase factors so that the Rutherford scattering formula turns out to be exact also in the quantum-mechanical treatment but also that the usual asymptotics for short-range interactions is modified. The total Coulomb cross section is also divergent in the forward-scattering region.

The true asymptotic states of charged particles are thus no plain waves but states physically meaning that a true asymptotic particle state is equivalent to a naive "bare" particles surrounded by a coherent em. field, a socalled infraparticle. A very intuitive and didactical treatment with the path-integral formalism is

P. Kulish and L. Faddeev, Asymptotic conditions and infrared
divergences in quantum electrodynamics, Theor. Math. Phys.
4, 745 (1970), https://doi.org/10.1007/BF01066485.

The result is equivalent to the standard soft-photon resummation techniques as discussed very nicely in

S. Weinberg, The Quantum Theory of Fields, Vol. 1