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Pair production is where an energetic photon on its interaction with strong electric field surrounding a nucleus produces an electron-positron pair. Annihilation is its converse where an electron-positron pair produces two photons on interaction with nucleus.

But how do these happen? I mean, why do they require strong electric field and nuclei?

(Is strong nuclear force necessary for this process?)

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To start with you are stating as annihilation, pair production! , so you have to edit. "annihilation is when an electron and a positron disappear into two photons", and the two photons are necessary for momentum conservation. –  anna v Aug 9 '12 at 12:09

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Annihilation can happen when all the quantum numbers of two colliding particles add up to zero. It might be electron on positron, proton on antiproton, neutron on antineutron , quark on antiquark etc.The force responsible depends on the possible interactions of the annihilating particles.

In the case of electron positron annihilation it is primarily the electromagnetic force that is involved and so one gets two photons as an output, usually, in order to conserve quantum numbers and momentum .( a single photon would not conserve spin also as the spin of the electron positron system is even). An annihilation into four photons is very much suppressed by the 1/137 coupling constant entering each photon vertex.

In the case of proton antiproton the main force is the strong force and the products are various hadrons , mesons which conserve quantum numbers, as it is the quarks and antiquarks that disappear and rearange into mesons.

Annihilation does not require the presence of other fields.

Pair production by a single photon needs an external field in order to conserve momentum as @KarsusRen states in his answer. The interaction is electromagnetic.One can think of this as photon photon scattering, where one of the photons is virtual and comes out of the field of the nucleus. Gluons are not free so one cannot observe free creation of antiproton proton pairs, but the diagrams exist.

Here is an interesting measurement of off shell gammma gamma collisions, both photons off shell, generating a proton antiproton pair, which shows how far one can go with the concept of pair production and annihilation,

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Electron and positron can annihilate in free space, but a single photon cannot be turned into electron-positron pair because conservation of energy-momentum cannot be satisfied. The nucleus in pair production absorbs some of the momentum.

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Annihilation of a particle and it's anti-particle always produces at least two photons. –  John Rennie Aug 9 '12 at 11:12

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