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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.


    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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