Furry's theorem

In quantum electrodynamics, fermion-loop amplitudes with an odd number of photon vertices vanish after loop orientations are summed, as required by charge-conjugation symmetry.

Connect

Charge conjugation: A transformation that replaces particles with antiparticles and reverses their charges. Reversing charge maps each fermion-loop orientation to its partner with the opposite sign.
Quantum electrodynamics: The quantum field theory describing electromagnetic interactions among charged particles. Furry's theorem is a selection rule within this theory's perturbative amplitudes.
Photon-photon scattering: A process in which photons scatter from one another through quantum interactions. A fermion loop cannot generate a three-photon amplitude, but four-photon scattering is allowed.
Charge-conjugation parity: The eigenvalue of a state or operator under charge conjugation when it is a symmetry. The odd-photon rule follows from the photon field's negative charge-conjugation parity.
Feynman diagram: A graphical representation of terms in a perturbative calculation of particle interactions. The theorem cancels selected closed-loop diagrams when their orientations are paired.
Dirac field: A relativistic spinor field describing spin-one-half particles such as electrons and positrons. Its charge-conjugation transformation underlies the fermion-loop pairing.
Light-by-light scattering: Photon-photon scattering mediated by charged particles in quantum electrodynamics. Its leading fermion-loop contribution uses four photon vertices, avoiding the odd-vertex prohibition.
Landau–Yang theorem: A theorem forbidding a massive spin-one particle from decaying into two identical on-shell photons. It also forbids amplitudes, but uses angular momentum and photon identity rather than loop-orientation cancellation.
Fermion loop: A closed sequence of fermion propagators in a Feynman diagram. The theorem constrains loops whose vertices couple to an odd number of photons.
Electromagnetic current: The conserved current associated with electric charge in an electromagnetic theory. Photon vertices couple to this current, whose charge-conjugation parity is odd.