Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I U(1) Invarience Lagrangians

  1. Oct 31, 2016 #1

    bhobba

    User Avatar
    Science Advisor
    Gold Member

    Hi

    On page 176 of Physics from Symmetry it says (note 9)

    If we assume Ψ describes our particle directly in some way what would U(1) the transformed solution Ψ' = e^iθ Ψ be which is equally allowed describe.

    He is speaking of allowed solutions of Lagrangian's. Its true for all Lagrangian's I know but why is it true generally? Or is he only speaking of the usual Lagrangian's?

    Thanks
    Bill
     
  2. jcsd
  3. Nov 1, 2016 #2

    Vanadium 50

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor

    Because the only thing that's observable is the probabiliy density, Ψ†Ψ. Ψ'† = e^-iθ Ψ†, i.e. puts a minus sign in front of the phase. The two phases multiply to 1.
     
  4. Nov 1, 2016 #3

    bhobba

    User Avatar
    Science Advisor
    Gold Member

    That's true - but he hadn't reached that point (ie interpreting Ψ) so maybe he was thinking of something else - but what :confused::confused::confused::confused::confused::confused::confused::confused:

    Thanks
    Bill
     
  5. Nov 1, 2016 #4
    Would it correspond to a change of the basis?
     
  6. Nov 1, 2016 #5

    bhobba

    User Avatar
    Science Advisor
    Gold Member

    I am now thinking ts simply because its a generator.

    Thanks
    Bill
     
  7. Nov 2, 2016 #6

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    For the non-relativistic case (Schrödinger, Pauli equations) it's the existence of a cintinuity equation, ensuring the conservation of the norm of the wave function in the time evolution. This enables the probability interpretation via Born's rule, because ##|\psi|^2## is the density in the continuity equation.
     
  8. Nov 4, 2016 #7

    strangerep

    User Avatar
    Science Advisor

    For the benefit of other readers who don't have the book, the context is:

    I think the way he argues this is little better than hand-waving gobbledegook. The question has no chance of being answered unless one first converts the words "in some way" to a precise meaning.

    He does not mention "interferometer", nor "Aharonov-Bohm", nor even "diffraction" or "double slit" anywhere in the book, hence does not have to confront the misleading superficiality of his exposition.

    Imho, he's just referring to the trivial result that a symmetry of the Lagrangian will necessarily be mirrored somehow (possibly trivially) in the space of solutions of the equations of motion (since the eom are unchanged by those symmetry transformations).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: U(1) Invarience Lagrangians
Loading...