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Supersymmetric Lagrangian Transformation (Grassmann Numbers)

  1. Mar 30, 2013 #1
    I've been tasked with showing that a Lagrangian under a set of transformations changes by a time derivative. All has gone well, except I'm left with two remaining terms, that I am completely confident, aren't there by mistake (as the 16 terms that should be expected have all popped out with the correct signs etc).

    The two terms, I believe, probably cancel but I don't just want to speculate. Anyway, the two terms that remain are,

    [tex]W'' W' (ψ^*ε^* + εψ)[/tex]

    where W is the Superpotential (primes representing derivatives, w.r.t x) as a function of the spatial co-ordinate x. ψ represents the Superspace co-ordinate and epsilon represents the small change Grassmann parameter relevant to our transformation. Stars represent complex conjugates.

    Now, I will also add that ψ is a Grassmann number so we can rewrite the terms such that,

    [tex]W'' W' (ψ^*ε^* - ψε)[/tex]

    So, briefly put, do these two terms cancel one another out? If so, why?

    (also, please do not delete my post for not following the template, I didn't find it neccessary given that this is quite a small scale question)

    Thanks for your time, Physics forum!
     
    Last edited: Mar 30, 2013
  2. jcsd
  3. Mar 31, 2013 #2
    Ok... after some thought I don't think they can cancel in this way (or the rest of what I have done wouldn't make sense).

    Anyway, instead of writing everything out in full, I'm basically working the Lagrangian on page 5 of this source http://www.phys.columbia.edu/~kabat/susy/susyQM.pdf and using the transforms also on the bottom of page 6.

    The terms above come from the final term of the Lagrangian, i.e.

    [itex]ψ^*ψW''[/itex]

    The the 3rd and 7th terms when multiplying out the brackets. As I mentioned before, everything else cancelled out in the calculation or was used to show that the change in the Lagrangian is a total time derivative, I was simply left with the aforementioned two terms.
     
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