## July 30, 2024

### The Zinn-Justin Equation

A note from my QFT class. Finally, I understand what Batalin-Vilkovisky anti-fields are for.

The Ward-Takahashi Identities are central to understanding the renormalization of QED. They are an (infinite tower of) constraints satisfied by the vertex functions in the 1PI generating functional $\Gamma(A_\mu,\psi,\tilde\psi,b,c,\chi)$. They are simply derived by demanding that the BRST variations

(1)$\begin{split} \delta_{\text{BRST}} b&= -\frac{1}{\xi}(\partial\cdot A-\xi^{1/2}\chi)\\ \delta_{\text{BRST}} A_\mu&= \partial_\mu c\\ \delta_{\text{BRST}} \chi &= \xi^{-1/2} \partial^\mu\partial_\mu c\\ \delta_{\text{BRST}} \psi &= i e c\psi\\ \delta_{\text{BRST}} \tilde{\psi} &= -i e c\tilde{\psi}\\ \delta_{\text{BRST}} c &= 0 \end{split}$

annihilate $\Gamma$: $\delta_{\text{BRST}}\Gamma=0$ (Here, by a slight abuse of notation, I’m using the same symbol to denote the sources in the 1PI generating functional and the corresponding renormalized fields in the renormalized action $\mathcal{L}= -\frac{Z_A}{4}F_{\mu\nu}F^{\mu\nu} + Z_\psi \left(i\psi^\dagger \overline{\sigma}\cdot(\partial-i e A)\psi+ i\tilde{\psi}^\dagger \overline{\sigma}\cdot(\partial+i e A)\tilde{\psi} -Z_m m(\psi\tilde{\psi}+\psi^\dagger\tilde{\psi}^\dagger) \right) +\mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}$ where $\begin{split} \mathcal{L}_{\text{GF}}+\mathcal{L}_{\text{gh}}&= \delta_{\text{BRST}}\frac{1}{2}\left(b(\partial\cdot A+\xi^{1/2}\chi)\right)\\ &=-\frac{1}{2\xi} (\partial\cdot A)^2+ \frac{1}{2}\chi^2 - b\partial^\mu\partial_\mu c \end{split}$ They both transform under BRST by (1).)

Posted by distler at 1:56 PM | Permalink | Followups (1)