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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 Γ(A μ,ψ,ψ˜,b,c,χ)\Gamma(A_\mu,\psi,\tilde\psi,b,c,\chi). They are simply derived by demanding that the BRST variations

(1)δ BRSTb =−1ξ(∂⋅A−ξ 1/2χ) δ BRSTA μ =∂ μc δ BRSTχ =ξ −1/2∂ μ∂ μc δ BRSTψ =iecψ δ BRSTψ˜ =−iecψ˜ δ BRSTc =0\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 Γ\Gamma: δ BRSTΓ=0 \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 ℒ=−Z A4F μνF μν+Z ψ(iψ †σ¯⋅(∂−ieA)ψ+iψ˜ †σ¯⋅(∂+ieA)ψ˜−Z mm(ψψ˜+ψ †ψ˜ †))+ℒ GF+ℒ gh \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 ℒ GF+ℒ gh =δ BRST12(b(∂⋅A+ξ 1/2χ)) =−12ξ(∂⋅A) 2+12χ 2−b∂ μ∂ μc \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).)

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