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I want to understand the following

"The Dirac equation for a charged massive fermion predicts, correctly, the existence of an antiparticle of the same mass and spin, but opposite charge, and opposite magnetic moment relative to the direction of the spin. The Dirac equation for a neutrino ν allows the existence of an antineutrino ¯ν." .

can somebody write the mathematics of it?

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The Dirac equation has negative energy solutions which are interpreted as the positively charged antiparticles of the electron. The limit of the Dirac equation for zero mass spin 1/2 particles is the (also relativistic) Weyl equation which was thought to describe the neutrinos before it was discovered that they have small mass. The mathematics of this can be found in any textbook of relativistic quantum theory. See, e.g., Landau & Lifshits' eponymous volume.

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  • $\begingroup$ I think this answer is sustaining an early attempt that Dirac invoked to interpret his eqn but it is not the one generally judged to be correct now. There are no negative energy solutions, just positive energy particle solutions and positive energy antiparticle solutions. $\endgroup$
    – Andrew Steane
    Commented Jul 11, 2023 at 18:48

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