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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