I am studying the quantization of the electromagnetic field. My text quantizes by changing amplitudes to ladder operators, by putting in an action and by imposing bosonic commutation relations upon the ladder operators.

But why does one know that photons are not fermionic? Only later one finds out that the particles have spin 1.

Does this come from some experimental realization? Or does the fermionic path not lead anywhere?

  • $\begingroup$ Interesting question. Can you argue that photons obey Bose-Einstein statistics from Planck's solution to the blackbody radiation problem? If so your question would be among the first answered in quantum mechanics (though the historical development was more complicated than that). $\endgroup$
    – rob
    Jun 18 '15 at 13:23
  • $\begingroup$ It's essentially just experimental. But if you want two electrons to interact, you don't have a lot of choice for the intermediate's spin. $\endgroup$ Jun 18 '15 at 13:24
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/81414/2451 $\endgroup$
    – Qmechanic
    Jun 18 '15 at 14:22
  • $\begingroup$ One should look at the EM radio wave as describe by Maxwell, specifically his Quaternionic form (Vol 2, Pt IV, Ch IX, Art 618-9, pp236-8 ). Then note that Quaternions have the up-down flip that is not normally considered in the vector representations (non-commutative / differential). Now consider how it is mapped... and which came first. I'm thinking that a clean powerful radio wave isn't far off a BEC in many ways. All the photons must be in lock-step (what is the temperature of a radio wave anyway (rhet)?) $\endgroup$ Dec 5 '17 at 22:33

"Spin" is dictated by the representation of the Loretz group the field the particle is a quantum of transforms in. Half-integer spins are fermions, integer spins are bosons by the spin-statistics theorem, where representations of the Lorentz group are labeled by two numbers $(s_1,s_2)$, whose sum is what we call spin in this context. The significance of the sum is that only representations with integer $s_1+s_2$ are proper linear representations, while the half-integer ones are only projective representations.

Since the electromagnetic four-potential is a four-vector, it transforms in the four-vector representation $(\frac{1}{2},\frac{1}{2})$-representation, which has integer spin, and hence photons are bosons. It is really just the kind of field we are quantizing - scalar, vector, spinor - that dictates the spin, and it is not true that we "only find out later" that photons have spin 1 (although some texts may make it seem like that), we know that from the start.

  • $\begingroup$ I read the "only later one finds" as a historical statement; the spin-statistics theorem appeared fairly late in the story. $\endgroup$
    – rob
    Jun 18 '15 at 21:37

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