The quantum object

Builder/output: quantum_object.py, P14_quantum_object.json.

Action. The action is computed directly in transformed variables:

with $\Psi =-i\int {\bar{c}}_{\mu }(2D^{\mu }[h]-b^{\mu })$. By order, $X$ contains $K^{\prime }c,\bar{c}c,b^2,bh,b{\bar{c}}^{\ast \prime }$ at ${\kappa }^0$; $K^{\prime }ch,\bar{c}ch,c^{\ast \prime }cc,{\varphi }^{\ast \prime }c\varphi$ at ${\kappa }^1$; and nothing at ${\kappa }^2$. $S_{\varphi }$ contributes $\varphi \varphi$, $h\varphi \varphi$, $hh\varphi \varphi$. There are no switch or insertion fields: the baseline is $S_{\mathrm{inv}}$, not $S_D$.

Fields and sources. The integrated fields are $h_{(mn)}$ (10 independent components), $c^{\mu }$, ${\bar{c}}_{\mu }$, $\varphi$ and $b^{\mu }$. The external antifields are ${K^{\prime }}_{(mn)}$ (${\sigma }_h=+1$), ${c^{\ast \prime }}_{\mu }$, ${\bar{c}}^{\ast \prime \mu }$ and ${\varphi }^{\ast \prime }$; $b^{\ast \prime }$ is absent because $sb=0$.

{$b$ is retained and integrated.} Four reasons:

• $b$ occurs only in ${\kappa }^0$ bilinears, so it has no interaction vertex;

• $M_{bb}$ is constant and invertible ($\det =-16$);

• the full $(h,b)$ inverse is verified, with Schur complement $q^{2}P$;

• eliminating $b$ (computed) would make $S_{\Psi }$ quadratic in ${\bar{c}}^{\ast \prime }$, whereas retaining it keeps $S_{\Psi }$ linear in every external antifield.

${\bar{c}}^{\ast \prime }$ couples only through ${\bar{c}}^{\ast \prime }b$, which has one quantum leg and is tree-level and one-particle reducible. So the 1PI ${\Gamma }_1$ is independent of ${\bar{c}}^{\ast \prime }$ and $b^{\ast \prime }$, while $K^{\prime }$, $c^{\ast \prime }$ and ${\varphi }^{\ast \prime }$ enter through $K^{\prime }ch$, $c^{\ast \prime }cc$ and ${\varphi }^{\ast \prime }c\varphi$.

1PI convention. The weight is $e^{i{S}_{\Psi }/\hslash }$ and ${\Gamma }_{\Psi }=S_{\Psi }+\hslash {\Gamma }_1+O({\hslash }^2)$, with the Legendre transform taken in the fields only (antifields are spectators). ${\Gamma }_1=\frac{i}{2}\mathrm{STr}\log {S^{(2)}}_{\Psi }=-i\log Z{|}_{1PI}$, so that $i{\Gamma }_1$ is the sum of 1PI one-loop graphs (Gate 0; the Berezin/Wick shadow below).

Vacuum and one-point. The vacuum is normalised by ${\Gamma }_1[0]=0$. The one-point function (D14a–c) is retained: no tadpole cancellation and no background shift are imposed, and no value is assigned.

Fourier, propagators and i0. We use $\partial \to ik$ with incoming momenta. The propagators are $⟨hh⟩=i(P^{-1}{)}_{ab}/(q^2+i0)$ (de Donder; independent components), $⟨c^{\mu }{\bar{c}}_{\rho }⟩=i({M_0}^{-1}{)}_{\mu \rho }=-{\delta }_{\mu \rho }/(q^2+i0)$ and $⟨\varphi \varphi ⟩=i/(q^2+i0)$, all of them inverses of the verified Hessians; the $i0$ is Feynman throughout.

Bookkeeping. Each $n$-leg vertex is $O({\kappa }^{n-2})$. At one loop, ${\Gamma }_1$ at ${\kappa }^k$ has exactly $k$ external legs (${\sum }_v(n_v-2)=E$), and this is checked against the prefactor of every exported integrand.

Pullback, not theorem. ${\Gamma }_{\Psi }(\Phi ,{\Phi }^{\ast \prime })=\Gamma (\Phi ,{\Phi }^{\ast \prime }-\partial \Psi /\partial \Phi )$ is used only as the classical pullback (verified at tree level). No quantum gauge-change theorem is asserted.