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QFT Supersymmetry problem

  1. Oct 18, 2005 #1
    Hi,

    I'm trying to do problem 3.5 of Peskin & Schroeder and I don't know where to start.

    First of all,

    I need to get the hermitian conjugate of the following expression

    [tex]\delta \chi = \epsilon F + \sigma^\mu \partial_\mu \phi \sigma^2 \epsilon^\ast[/tex]

    where [itex]\epsilon[/itex] is a 2 component-spinor of grassmann numbers, F a complex scalar field [itex]\sigma^\mu = (I,\sigma^i)[/itex] for [itex]i=1,...,3[/itex] and the [itex]\sigma^i[/itex] are the Pauli matrices, [itex]\phi[/itex] is a complex scalar field.

    I think the hermitian conjugate would be something like

    [tex]\delta \chi^\dagger = \epsilon^\dagger F^\ast + \epsilon^T \sigma^2 \sigma^\mu \partial_\mu \phi^\ast[/tex]

    Am I right?

    Thanks

    Guillaume

    Moderator note: I took the liberty of editing in your LaTeX tags.

    -TM
     
    Last edited by a moderator: Oct 18, 2005
  2. jcsd
  3. Oct 18, 2005 #2

    vanesch

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    If you want to, I did this problem and you can find it on my homepage (but that's a spoiler of course).

    Also, you can use LaTeX here, you just need to surround your LateX with the instuctions tex (between square brackets) and /tex (also between square brackets). It will be much more readable that way!
     
  4. Oct 18, 2005 #3
    I had a look at your solution and there is something I don't understand. At the end of the first page you wrote
    [tex]
    $ i \chi^\dagger \bar{\sigma}^\mu \sigma^\nu (\partial_\nu \partial_\mu \phi ) \sigma^2 \epsilon^\ast = i\epsilon^\dagger \sigma^2 \chi^\ast (\partial^\mu \partial_\mu \phi)$
    [/tex]
    This seems to imply that [tex]$ \bar{\sigma}^\mu \sigma^\nu = g^{\mu \nu}$
    [/tex]!?
    I tought that
    [tex]
    $\{ \bar{\sigma}^\mu, \sigma^\nu \} = g^{\mu \nu}$
    [/tex]
    where the brackets denote the anticommutator. Am I wrong?

    P.S. The command \bar{\sigma} doesn't seem to work in the brackets.... sorry about that.
     
    Last edited: Oct 19, 2005
  5. Oct 18, 2005 #4

    vanesch

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    A quick reply:
    I didn't check (did this long ago !) it completely, but as [tex]\mu,\nu[/tex] are summed with the partial derivatives, that symmetrises the expression, no ?
     
  6. Oct 19, 2005 #5

    dextercioby

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    It looks okay. The Pauli matrices are hermitean (they form a basis in [itex] \mbox{su(2)} [/itex], up to a 1/2 ) and involution on the Grassmann algebra goes, under hermitean conjugation (seen as simultaneous involution and transposing), into transposing.

    I dunno how Vanesch came up with a second space-time derivative...

    Daniel.
     
  7. Feb 20, 2007 #6
    I have a very similar question:
    I want to find the hermitian conjugate of
    [tex] \epsilon \sigma^\mu \partial_\mu \psi [/tex]
    where psi and epsilon are 2 component spinors of grassmann variables.
    In that case I think the hermitian conjugate should be:
    [tex] -\partial_\mu \psi^\dagger \sigma^\mu \epsilon^\dagger [/tex]
    My main concern is whether a minus sign arises in a hermitian conjugate when commuting the epsilon past the psi.
     
  8. Feb 20, 2007 #7
    I realized that there is a mistake in the last post as the psi inthe first expression should be a (psi)^dagger and a psi in the second however my question remains.
     
  9. Feb 21, 2007 #8

    dextercioby

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    If the Grassmann parity of the the spinors is 1, then they anticommute, so the "-" sign occurs.
     
  10. Feb 21, 2007 #9
    Actually I think that even when they anticommute there should not be a - sign, as when taking the hermitian conjugate we are not commuting them but using the definition of the hermitian conjugate so I would say
    [tex] (\epsilon \psi)^\dagger = \psi^\dagger \epsilon^\dagger [/tex]
    but please correct me if I am wrong
     
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