An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 3.6

The Dirac Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · fierz-transformations

Problem

Fierz transformations. Let uiu_i, i=1,…,4i = 1, \ldots, 4, be four 4-component Dirac spinors. In the text, we proved the Fierz rearrangement formulae (3.78) and (3.79). The first of these formulae can be written in 4-component notation as uˉ1γμ(1+γ52)u2uˉ3γμ(1+γ52)u4=−uˉ1γμ(1+γ52)u4uˉ3γμ(1+γ52)u2.\bar{u}_1 \gamma^\mu \left(\frac{1+\gamma^5}{2}\right) u_2 \bar{u}_3 \gamma_\mu \left(\frac{1+\gamma^5}{2}\right) u_4 = -\bar{u}_1 \gamma^\mu \left(\frac{1+\gamma^5}{2}\right) u_4 \bar{u}_3 \gamma_\mu \left(\frac{1+\gamma^5}{2}\right) u_2. In fact, there are similar rearrangement formulae for any product (uˉ1ΓAu2)(uˉ3ΓBu4),(\bar{u}_1 \Gamma^A u_2)(\bar{u}_3 \Gamma^B u_4), where ΓA,ΓB\Gamma^A, \Gamma^B are any of the 16 combinations of Dirac matrices listed in Section 3.4. (a) To begin, normalize the 16 matrices ΓA\Gamma^A to the convention tr[ΓAΓB]=4δAB.\mathrm{tr}[\Gamma^A \Gamma^B] = 4\delta^{AB}. This gives ΓA={1,γ0,iγj,…}\Gamma^A = \{1, \gamma^0, i\gamma^j, \ldots\}; write all 16 elements of this set. (b) Write the general Fierz identity as an equation (uˉ1ΓAu2)(uˉ3ΓBu4)=∑C,DCABCD(uˉ1ΓCu4)(uˉ3ΓDu2),(\bar{u}_1 \Gamma^A u_2)(\bar{u}_3 \Gamma^B u_4) = \sum_{C,D} C^{AB}{}_{CD} (\bar{u}_1 \Gamma^C u_4)(\bar{u}_3 \Gamma^D u_2), with unknown coefficients CABCDC^{AB}{}_{CD}. Using the completeness of the 16 ΓA\Gamma^A matrices, show that CABCD=116tr[ΓCΓAΓDΓB].C^{AB}{}_{CD} = \frac{1}{16} \mathrm{tr}[\Gamma^C \Gamma^A \Gamma^D \Gamma^B]. (c) Work out explicitly the Fierz transformation laws for the products (uˉ1u2)(uˉ3u4)(\bar{u}_1 u_2)(\bar{u}_3 u_4) and (uˉ1γμu2)(uˉ3γμu4)(\bar{u}_1 \gamma^\mu u_2)(\bar{u}_3 \gamma_\mu u_4).

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