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How can a massless particle such a photon be the result of electron-positron annihilation? What about the law of conservation of energy? Is a valid explanation that the pair's energy transforms itself into the kinetic energy of the photon, not its mass?

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Just as a general note: types of energy are not conserved, only total energy. The first place students usually encounter this is in in-elastic collisions: the kinetic energy of the products is not equal to the kinetic energy of the initial bits. There is no apriori reason to think that the mass of the reactants needs to be expressed as mass in the products. Now $\sum m_i = \sum m_f$ does hold in day to day life at the human scale, but you're asking about things at a very different size and energy density. –  dmckee May 14 '12 at 18:00
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Photons are massless but they do have momentum. Check the relativistic energy-momentum equation: $E^2 = p^2 c^2 + m^2 c^4$. Even when $m=0$ particle can have energy.

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The annihilation process is $e^-+e^+\to 2\gamma$. In the rest frame of the system, the spatial momenta of the particles before and after the event are opposite to each other, but their kinetic energies (the 0-component of the relativistic 4-momentum) add up and all four of them are equal.

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