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Number of 4-fermion, 2-derivative Lorentz invariants

  1. Sep 6, 2010 #1
    The notation below, is consistent with Wess and Bagger's https://www.amazon.com/Supersymmetry-Supergravity-Julius-Wess/dp/0691025304".

    Given a Majorana spinor field in 4D, written in 2-component notation as
    [tex] \Psi(x) = \begin{pmatrix} \psi(x) \\\\ \bar\psi(x) \end{pmatrix} ,
    \quad (\psi_\alpha)^* = \bar\psi_{\dot\alpha} \ ,
    how many linearly independent Lorentz invariants can be formed using 2n spinors and n space-time derivatives, tied together with the sigma/Pauli matrices and various metrics?

    And, more importantly for my application, how many are there modulo total derivatives?

    Is there a general (eg representation theory) approach to this type of problem?


    1. I am mainly (at the moment) concerned with the n=2 case.
    2. Due to anticommutativity, there are no such terms with n>4.
    3. This can obviously be rewritten using 4-component Majorana spinors and Dirac matrices.


    For example with n=1 the only invariant is the standard kinetic term
    [tex] \psi \sigma^a \partial_a \bar\psi
    = \psi^\alpha \sigma^a_{\alpha\dot\alpha} \partial_a \bar\psi^{\dot\alpha}
    = - (\partial_a\psi) \sigma^a \bar\psi + \text{total derivative} \ .

    For [tex] n=2 [/tex], I believe (and want to prove) that there are only 6 invariants up to total derivatives.
    Defining the matrix
    [tex] v_a{}^b = i \psi\sigma^b\partial_a\bar\psi [/tex]
    and its complex conjugate
    [tex] \bar v_a{}^b = -i (\partial_a\psi)\sigma^b\bar\psi [/tex],
    I chose the basis(?)
    [tex] (\partial^a\psi^2)(\partial_a\bar\psi^2) \ ,\; tr(v)tr(\bar v) \ , \;
    tr(v)^2\ ,\; tr(v^2)\ ,\; tr(\bar v)^2 \,\; tr(\bar v^2)\ .
    Other terms being related by (for example)
    [tex] tr(v\bar v) = tr(v)tr(\bar v)
    +\tfrac12\Big(tr(v^2)-tr(v)^2+tr(\bar v^2)-tr(\bar v)^2\Big)
    + \text{total derivative}
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 7, 2010 #2

    Does anyone have any ideas how to systematically approach such a problem?
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