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The Dirac equation is given by:

$$ (i\partial \!\!\!/ - m) \psi(x) = 0$$

It famously predicted positrons by suggesting negative energy solutions. First it was solved with the 'Dirac sea' and positrons as holes in the sea.

Now I'm wondering since muons can be considered as more massive electrons (and tau-particles even more massive versions) does the dirac equation suggest the existence of muons as a kind of solution? Is there a different equation that makes it natural that there exists 3 different versions of electrons with unequal mass?

Qmechanic
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bananenheld
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2 Answers2

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It does not predict muons or tauons. They are still solutions but that doesn't mean it predicts them. What is special about the Dirac equation is that it predicts 2 pairs of solutions: anti-particles and particles for each solution. So there are electrons and antielectrons, muons and anti-muons, tauons and anti-tauons.

Saying the Dirac equation predicts muons and tauons is like saying Newton's second law:

$$F=ma$$

predicts there is a particle with mass 2 kg.

Habouz
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At the time when Dirac first published his famous equation, only electrons were known to physicists. That's why it was thought to describe electrons.

The mass term in the Dirac equation is just the parameter, it could have any value (not only the measured values of electron, muon or tau masses, but just any value from $0$ to infinity).

In the Standard model all 3 leptons are initially massless and indistinguishable. For some (yet unknown) reason only Higgs boson can distinguish, say, electron and muon. It couples stronger to muon, thus giving it more mass.

There is an empirical formula that establishes relationship for the masses of leptons. But it has nothing to do with Dirac equation.

Murod Abdukhakimov
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