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Global U(1) invariant of Dirac Lagrangian

  1. Jan 27, 2012 #1
    Does anybody know what interpretation the invariant corresponding to the global U(1) invariance of the Dirac Lagrangian is? I have always had it in my head that it's charge, but then I realized that uncharged free particles such as neutrinos satisfy this equation too! Any thoughts much appreciated.
  2. jcsd
  3. Jan 27, 2012 #2
    Depends on the specifics of your lagrangian. Think lepton number, baryon number, B-L, etc.
  4. Jan 27, 2012 #3
    Requiring the Hamiltonian to be hermitian can break this symmetry. Uncharged particles usually have a Lagrangian that looks something like [itex]\mathcal{L} = \frac{1}{2}\partial_u\phi\partial^u\phi - \frac{1}{2}\mu^2\phi^2[/itex]. In that case, the hermiticity of [itex]H[/itex] forces [itex]\phi[/itex] to be real, so there is no U(1) symmetry anymore. Was that what you meant, or did you have a specific Lagrangian in mind?
    Last edited: Jan 27, 2012
  5. Jan 27, 2012 #4
    Well, I just had in mind the Dirac lagrangian. It seems that neutrinos have mass, and if so should presumably obey this equation. But if so, what does the U(1) symmetry correspond to, given that these guys aren't charged? Thanks for your replies!
  6. Jan 27, 2012 #5
    If you just have neutrinos, it would be neutrino number. If you have something like a standard model lagrangian, it would be lepton number and baryon number (two separate U(1)s)
  7. Jan 27, 2012 #6


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    Massive neutrinos with a "Majorana mass" are not described by the Dirac lagrangian. See e.g. Srednicki's text for details.
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