# Born rule for photons: it works, but it shouldn't?

We can observe double-slit diffraction with photons, with light of such low intensity that only one photon is ever in flight at one time. On a sensitive CCD, each photon is observed at exactly one pixel. This all seems like standard quantum mechanics. There is a probability of detecting the photon at any given pixel, and this probability is proportional to the square of the field that you would calculate classically. This smells exactly like the Born rule (probability proportional to the square of the wavefunction), and the psychological experience of doing such an experiment is well described by the Copenhagen interpretation and its wavefunction collapse. As usual in quantum mechanics, we get quantum-mechanical correlations: if pixel A gets hit, pixel B is guaranteed not to be hit.

It's highly successful, but Peierls 1979 offers an argument that it's wrong. "...[T]he analogy between light and matter has very severe limitations... [T]here can be no classical field theory for electrons, and no classical particle dynamics for photons." If there were to be a classical particle theory for photons, there would have to be a probability of finding a photon within a given volume element. "Such an expression would have to behave like a density, i.e., it should be the time component of a four-vector." This density would have to come from squaring the fields. But squaring a tensor always gives a tensor of even rank, which can't be a four-vector.

At this point I feel like the bumblebee who is told that learned professors of aerodynamics have done the math, and it's impossible for him to fly. If there is such a fundamental objection to applying the Born rule to photons, then why does it work so well when I apply it to examples like double-slit diffraction? By doing so, am I making some approximation that would sometimes be invalid? It's hard to see how it could not give the right answer in such an example, since by the correspondence principle we have to recover a smooth diffraction pattern in the limit of large particle numbers.

I might be willing to believe that there is "no classical particle dynamics for photons." After all, I can roll up a bunch of fermions into a ball and play tennis with it, whereas I can't do that with photons. But Peierls seems to be making a much stronger claim that I can't apply the Born rule in order to make the connection with the classical wave theory.

[EDIT] I spent some more time tracking down references on this topic. There is a complete and freely accessible review paper on the photon wavefunction, Birula 2005. This is a longer and more polished presentation than Birula 1994, and it does a better job of explaining the physics and laying out the history, which goes back to 1907 (see WP, Riemann-Silberstein vector , and Newton 1949). Basically the way one evades Peierls' no-go theorem is by tinkering with some of the assumptions of quantum mechanics. You give up on having a position operator, accept that localization is frame-dependent, redefine the inner product, and define the position-space probability density in terms of a double integral.

Related:

What equation describes the wavefunction of a single photon?

Amplitude of an electromagnetic wave containing a single photon

Iwo Bialynicki-Birula, "On the wave function of the photon," 1994 -- available by googling

Iwo Bialynicki-Birula, "Photon wave function," 2005, http://arxiv.org/abs/quant-ph/0508202

Newton T D and Wigner E P 1949 Localized states for elementary systems Rev. Mod. Phys. 21 400 -- available for free at http://rmp.aps.org/abstract/RMP/v21/i3/p400_1

Peierls, Surprises in theoretical physics, 1979, p. 10 -- peep-show possibly available at http://www.amazon.com/Surprises-Theoretical-Physics-Rudolf-Peierls/dp/0691082421/ref=sr_1_1?ie=UTF8&qid=1370287972

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I guess you've read Marcella's paper? – twistor59 Jun 3 '13 at 19:50
"I feel like the bumblebee who is told that learned professors of aerodynamics have done the math, and it's impossible for him to fly" Nice. – dmckee Jun 3 '13 at 19:52
@twistor59: No, I hadn't, +1. But the paper doesn't even explicitly talk about what type of particle is being considered. He mentions Young, which would imply photons, but he also talks about the Schrodinger equation...? – Ben Crowell Jun 3 '13 at 19:56
Oh I just found an anti-Marcella paper. I'll have to read that one now... – twistor59 Jun 3 '13 at 19:56
@BenCrowell Yeah I think it's just a generic Schroedinger equation describing a single quantum particle and making use of a position operator/wavefunction (so a bit cheating for photons!) – twistor59 Jun 3 '13 at 19:59

You need quantum field theory to describe light as it is clearly a relativistic system. Photons are excitations of the electromagnetic (EM) field which is a spin 1 boson field. In the limit of large numbers of photons the usual classical limit is obtained, i.e. one can describe the field evolution classically to a good approximation. So it is not correct to think of the EM field as simply a quantum mechanical wave function of a photon which one could then apply Born's rule to.

However, in the classical limit the energy density of the EM field is proportional to the square of the magnitude of the EM field. So if we have a monochromatic EM field, the energy of each photon will be a constant proportional to the frequency of the EM wave and so the photon number density will be proportional to the square of the magnitude of the EM field.

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The classical limit for light is a wave theory. The (quantum) amplitude of the wavefunction wave becomes the (classical) amplitude of the wave (e.g. the magnitude of the electric field), and the (quantum) expected number of photons in a volume becomes the (classical) intensity of the wave (e.g. the squared electric field).

Of course it is correct to map the intensity of a classical electromagnetic wave to the probability of measuring a photon in a particular spot.

It seems to me like Peierls is saying what I said above: The classical limit of photons is a wave theory, not a particle theory. By "particle theory" I mean that there would be a large number of classical particles called photons that are traveling along individual trajectories. There are many arguments that a particle theory cannot work for classical light -- the one that Peierls is using is related to the Lorentz transformation properties:

If there were a classical particle theory of photons, then you would have an observable quantity called "number of photons per volume", and you would expect it to Lorentz-transform in a certain way. But the intensity of a classical light wave does not Lorentz-transform that way. QED. (By contrast, an electron wavefunction DOES transform that way.)

Peierls is not saying anything about the quantum theory of light, or how the observables are defined. I really don't think Peierls is denying that the expected value of the number of photons is proportional to the square of the amplitude of the light-wave. At least that's how I read the excerpts.

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Unfortunately this doesn't really clarify the issue for me. It seems like what we need is a concrete example of applying the Born rule in two frames of reference and coming up with a contradiction. – Ben Crowell Jun 3 '13 at 21:18
It turns out that Peierls does give an explicit example of what goes wrong when you change frames of reference. In a standing wave, unlike a plane wave, we don't have $|E|=|B|$. Therefore if you have a detector moving relative to the boundaries, the fields transform, and in the detector's frame the energy density has interference fringes whose positions are shifted. (He refers to these as "Lippman fringes," which is obscure but can be googled.) – Ben Crowell Jun 12 '13 at 4:12
Lubos has an interesting article in his blog of how photons build up the classical field motls.blogspot.com/2011/11/… – anna v Jun 15 '13 at 5:22