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Noether's Theorem

  1. Dec 22, 2009 #1
    Hi

    I was wondering if someone would be kind enough to help me understand an example in my class notes:

    If we have a Lagrangian:

    [tex]L=m(\dot{z}\dot{z^{*}})-V(\dot{z}\dot{z^{*}})[/tex]

    where z=x+iy.

    Why does it follow that

    [tex]Q=X^{i}\frac{{\partial}L}{{\partial}\dot{q}^{i}}[/tex]

    is equal to:

    [tex]X\frac{{\partial}L}{{\partial}\dot{z}}+X^{*}\frac{{\partial}L}{{\partial}\dot{z^{*}}}[/tex]?

    I mean, mathematically that seems wrong, why are we adding the second term (the one with the complex conjugate of [tex]\dot{z}[/tex])

    Also, can I check if my understanding of the superscript 'i' is correct - does it correspond to the axes, for example i=1 corresponds to the x axis, i=2 corresponds to the y axis, etc. If z = x + iy, we are no longer talking about a 3 dimensional real space, so how are the superscripts relavent?

    And is there a reason why the superscript 'i' has gone in the second line?

    thanks
     
  2. jcsd
  3. Dec 23, 2009 #2

    Dale

    Staff: Mentor

    The superscripts are confusing to me. e^(0 i pi/2) is 1 which is the x axis in the complex plane, e^(1 i pi/2) is i which is the y axis in the complex plane, e^(2 i pi/2) is -1 which is the -x axis in the complex plane, and e^(3 i pi/2) is -i which is the -y axis in the complex plane. Are the superscripts related to that somehow?
     
  4. Dec 23, 2009 #3
    I am not sure about this, as the x,y, and z axes should be orthogonal to each other, in other words they are linearly independent - ie. you can't define z in the way that it has been defined (z=x+iy), as a linear combination of x and y...
     
  5. Dec 23, 2009 #4
    Call me crazy, but don't you need to first have a transformation of the Lagrangian before you can define a Noether current and a conserved charge?

    BTW, in my QFT class I never had to deal with a complex-valued coordinate. But I'm guessing that x and y in this context are both four vectors (and therefore z as well). I could be wrong...
     
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