# Number of 4-fermion, 2-derivative Lorentz invariants

1. Sep 6, 2010

### Simon_Tyler

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
$$\Psi(x) = \begin{pmatrix} \psi(x) \\\\ \bar\psi(x) \end{pmatrix} , \quad (\psi_\alpha)^* = \bar\psi_{\dot\alpha} \ ,$$
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?

Notes:

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.

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

For $$n=2$$, I believe (and want to prove) that there are only 6 invariants up to total derivatives.
Defining the matrix
$$v_a{}^b = i \psi\sigma^b\partial_a\bar\psi$$
and its complex conjugate
$$\bar v_a{}^b = -i (\partial_a\psi)\sigma^b\bar\psi$$,
I chose the basis(?)
$$(\partial^a\psi^2)(\partial_a\bar\psi^2) \ ,\; tr(v)tr(\bar v) \ , \; tr(v)^2\ ,\; tr(v^2)\ ,\; tr(\bar v)^2 \,\; tr(\bar v^2)\ .$$
Other terms being related by (for example)
$$tr(v\bar v) = tr(v)tr(\bar v) +\tfrac12\Big(tr(v^2)-tr(v)^2+tr(\bar v^2)-tr(\bar v)^2\Big) + \text{total derivative}$$

Last edited by a moderator: May 4, 2017
2. Sep 7, 2010

### Simon_Tyler

(bump)

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