Global U(1) invariant of Dirac Lagrangian

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Discussion Overview

The discussion revolves around the interpretation of the invariant associated with the global U(1) invariance of the Dirac Lagrangian. Participants explore the implications of this invariance, particularly in relation to charged and uncharged particles, including neutrinos.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that the invariant corresponds to charge but questions this in light of uncharged particles like neutrinos.
  • Another participant notes that the interpretation depends on the specifics of the Lagrangian, mentioning quantities like lepton number and baryon number.
  • A participant points out that requiring the Hamiltonian to be hermitian can break the U(1) symmetry, particularly for uncharged particles, and asks for clarification on the specific Lagrangian in question.
  • One participant asserts that if only neutrinos are considered, the invariant would correspond to neutrino number, while in a standard model context, it would relate to lepton and baryon numbers.
  • A later reply mentions that massive neutrinos with a "Majorana mass" are not described by the Dirac Lagrangian, referencing a specific text for further details.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the U(1) invariant, with no consensus reached on its meaning in the context of neutrinos and other particles.

Contextual Notes

The discussion highlights the dependence on the specific form of the Lagrangian and the implications of hermiticity on symmetry, but does not resolve these complexities.

metroplex021
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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.
 
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Depends on the specifics of your lagrangian. Think lepton number, baryon number, B-L, etc.
 
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?
 
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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!
 
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)
 
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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