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I recently read that the photon "decay" (if you can call it that) in an external field occurs only in an electric field but not in a magnetic field. The reason being that the Euler-Heisenberg Lagrangian is real-valued for the magnetic field whereas it has a non-vanishing imaginary part in the case of an electric field (the calculation is quite lengthy, so I won't repeat it here). That explanation is great, but I was wondering if there is also an intuitive physical reason why photons only produce $e^- e^+$-pairs in an electric field?

I find this effect quite fascinating as electric fields and magnetic fields usually behave so similarly. After all, you can change from one to the other by a simple Lorentz boost (at least to some extent).

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For some easy intuition, think about what happens in classical electrodynamics if you have a pair of equal-mass, opposite-charge particles initially moving parallel to each other in a background electric or magnetic field.

A background electric field tends to make the particles accelerate away from each other. If the background electric field is strong enough, it will overcome the particles' mutual attraction and pull them away from each other permanently, with a distance that increases indefinitely.

A background magnetic field tends to make the particles move in circles, one clockwise and the other counterclockwise. Thus the magnetic field only pulls the particles away from each other temporarily, and then pushes them back toward each other, repeating indefinitely. Making the magnetic field stronger only makes those circles even tighter.

That's only a classical analogy, of course. Pair-production doesn't occur in classical electrodynamics. But the background electric/magnetic fields have similar tendencies in quantum electrodynamics, so the intuition is still valid: pair production occurs only in a field that tends to pull the positive/negative charges far enough away from each other.

The question requested intuition, but for completeness I'll mention the mathematical review

Section 1.3.1, cases (1) and (2) on pages 10-11 highlights the same mathematical observation that was described in the question.

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