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I About Noether Theorem

  1. Dec 13, 2017 #1
    I've been looking at the original work of Noether and I'm confused about this point. The transformation of fields and coordinates are supossed to form a group, then how the inverse of
    $$B^{\mu}=B^{\mu}(A^{\mu},\partial A^{\mu}/\partial x^{\nu},x^{\mu},\epsilon) $$
    $$y^{\mu}=y^{\mu}(A^{\mu},\partial A^{\mu}/\partial x^{\nu},x^{\mu},\epsilon) $$
    is supposed to be obtained?
    For the sake of simplicity we suppose that ##\epsilon## is a single parameter and only first derivatives of the field appear.
  2. jcsd
  3. Dec 16, 2017 #2


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    Could you write a little bit of the context? I don't know what [itex]B^\mu[/itex] is, or what kind of transformation you are talking about.

    For a simple scalar field [itex]\phi[/itex], we assume a transformation of the form: [itex]\phi \rightarrow \phi + \epsilon \psi[/itex]. This change will leave the action unchanged if its effect on the lagrangian density is a divergence:

    [itex]\mathcal{L} \rightarrow \mathcal{L} + \epsilon \partial_\mu \Lambda^\mu[/itex]

    for some vector field [itex]\Lambda^\mu[/itex]. In that case, there is a conserved current:

    [itex]J^\mu = \Lambda^\mu - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \psi[/itex]

    For this simple transformation, the inverse is pretty simple:

    • The forward transformation: [itex]T(\phi) = \phi + \epsilon \psi[/itex]
    • The inverse transformation: [itex]T^{-1}(\phi) = \phi - \epsilon \psi[/itex]
  4. Dec 16, 2017 #3
    ##B^\mu## are the transformed components of the field and ##y^\mu## are new coordinates, ##\epsilon## are parameters.
    The problem are the derivatives of the field components. If they were not present we could have inverted the original equations obtaining:
    $$ A^\mu=A^\mu(B^\mu,x^\mu,\epsilon)$$
    $$ y^\mu=A^\mu(B^\mu,x^\mu,\epsilon)$$
    However the appearance of the filed derivatives seem to create a problem for the inversion process.
  5. Dec 17, 2017 #4


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    Is there a link to a scanned copy of her paper you are addressing?
  6. Dec 17, 2017 #5
    I was talking about the original paper in the book "The Noether Theorems : Invariance and Conservation Laws in the Twenty Century" but there is also the paper
    by Barbashov and Nesterenko "Continous Symmetries in Field Theory" Fortschr. Phys. 31 (1983) 10, 535-567
  7. Dec 17, 2017 #6


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    I might give a look at it, can't guarantee it though.
  8. Dec 17, 2017 #7
    (For F. Klein, on the occasion of the fiftieth anniversary of his doctorate)
    by Emmy Noether in Gottingen
    Presented by F. Klein at the session of 26 July 1918∗"
    in page four she only mentions that that the deriatives occur in the the transformations.
    In Babashov and Nesterenko paper it is written explicitly

    Attached Files:

  9. Dec 17, 2017 #8


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