Transforming [product] with Fierz Transformations

[Lester]

If I have a product like

<br /> \bar\ psi\gamma^\mu\psi\bar\psi\gamma_\mu\psi<br /> &lt;br /&gt; &lt;br /&gt; how can i rearrange with Fierz transformations?

 

[samalkhaiat]

I assume, you meant to write

<br /> V(4,2;3,1) = \bar{\psi}(4) \gamma_{a} \psi(2) \bar{\psi}(3) \gamma^{a} \psi(1)<br />

The Fietz transformation is

V(4,2;3,1) = \left( - S + \frac{1}{2} V - \frac{1}{2} A + P \right) (4,1;3,2)

where

S(4,1;3,2) = \bar{\psi}(4) \psi(1) \bar{\psi}(3) \psi(2)

<br /> A(4,1;3,2) = \bar{\psi}(4) \gamma_{5} \gamma_{a} \psi(1) \bar{\psi}(3) \gamma^{a}\gamma_{5} \psi(2)<br />

<br /> P(4,1;3,2) = \bar{\psi}(4) \gamma_{5} \psi(1) \bar{\psi}(3) \gamma_{5} \psi(2)<br />

and similar one for V(4,1;3,2). See page 160-162 in Itzykson & Zuber; QFT.

regards

sam

 

[Lester]

Thanks a lot, Sam. This is what I was looking for.

Jon