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In spontaneous parametric down-conversion, a single photon interacts with a nonlinear medium to produce a pair of lower energy photons. It is observed that energy, momentum, and orbital-angular-momentum are conserved in the process. However, I have seen arguments invoking Furry's theorem (see https://www.maths.tcd.ie/~fionn/qft/furry.pdf) stating that it is not possible for a photon to get destroyed to produce two photons. What is the key insight that I am missing here?

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The theorem you've linked to is true - in vacuum.

SPDC happens in a non-linear medium which breaks the symmetries that underlie the hypotheses of that theorem. As such, there is no contradiction between the two.

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  • $\begingroup$ I suspect there is something more because the process involves a single free photon at initial time, a finite interaction for a finite time, and then two free photons. As the process is parametric, there is no net transfer of energy, momentum, or angular momentum to the nonlinear medium. So if I look at the system only at the initial and final times as a unitary scattering process, I am not sure how the nonlinearity alone can rescue the process from the verdict of Furry's theorem. $\endgroup$ – Girish Jan 24 '19 at 9:24
  • $\begingroup$ Moreover, the nonlinear medium is also electromagnetic in nature. So although the medium does not possess spatial inversion symmetry, it does possess charge conjugation symmetry. Please correct me if I am wrong. $\endgroup$ – Girish Jan 24 '19 at 9:28
  • $\begingroup$ Unless the medium is a positronium crystal, or something similar, then it certainly does not possess charge-conjugation symmetry. (Hint: if it has electrons but it doesn't have positrons, then it's $C$-asymmetric.) But that's not really necessary - the lack of inversion symmetry is sufficient. $\endgroup$ – Emilio Pisanty Jan 24 '19 at 11:09

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