The inherited completion in differentiable form

Displayed formulas in this passage

• XIII-A@1#eq:completion-display · line 770

  $$\begin{array}{rl}{X_1}^{\mu }[r]& =-{E_0}^{\mathrm{R}}{\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \mu }[r]),\\ p_1[r;\varphi ]& =-{E_m}^{\mathrm{R}}\left({\partial }_{\mu }(g{a_1}^{\mu \nu }[r]{\partial }_{\nu }\varphi )+m^{2}g{b}_1[r]\varphi \right),\\ {X_2}^{\mu }& =-{E_0}^{\mathrm{R}}\left({\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \beta }[\gamma ]{\partial }_{\beta }{X}^{\mu })+{\partial }_{\alpha }({\sigma }_{\epsilon }(a_1[k]+a_2[\gamma ]{)}^{\alpha \mu })\right),\\ p_2& =-{E_m}^{\mathrm{R}}\left(V_{1,g}[\gamma ]p+{\partial }_{\mu }(g(a_1[k]+a_2[\gamma ]{)}^{\mu \nu }{\partial }_{\nu }\varphi )+m^{2}g(b_1[k]+b_2[\gamma ])\varphi \right).\end{array}$$

• XIII-A@1#eq:O-expanded · line 793

  $$\mathcal{O}=\int f\left(p_2-X^{\mu }{\partial }_{\mu }p+(X^{\nu }{\partial }_{\nu }{X}^{\mu }-{X_2}^{\mu }){\partial }_{\mu }\varphi +\frac{1}{2}{X}^{\mu }{X}^{\nu }{\partial }_{\mu }{\partial }_{\nu }\varphi \right){\mathrm{d}}^{4}x.$$

• XIII-A@1#eq:quadratic-derivatives · line 813

  $$\begin{array}{rl}{\dot{b}}_2& =\frac{1}{4}({\mathrm{tr}}_{\eta }\gamma )({\mathrm{tr}}_{\eta }z)-\frac{1}{2}{\gamma }_{\alpha \beta }{z}^{\alpha \beta },\\ {\dot{a_2}}^{\mu \nu }& =z^{\mu }{}_{\alpha }{\gamma }^{\alpha \nu }+{\gamma }^{\mu }{}_{\alpha }{z}^{\alpha \nu }-\frac{1}{2}\left(({\mathrm{tr}}_{\eta }z){\gamma }^{\mu \nu }+({\mathrm{tr}}_{\eta }\gamma )z^{\mu \nu }\right)+{\dot{b}}_2{\eta }^{\mu \nu },\\ {\hat{a}}^{\mu \nu }& ={a_1}^{\mu \nu }[\dot{k}]+{\dot{a_2}}^{\mu \nu },\hat{b}=b_1[\dot{k}]+{\dot{b}}_2.\end{array}$$

• unlabeled · line 750

  $${a_1}^{\mu \nu }[r]=\frac{1}{2}{\eta }^{\mu \nu }{\mathrm{tr}}_{\eta }r-r^{\mu \nu },b_1[r]=\frac{1}{2}{\mathrm{tr}}_{\eta }r,$$

• unlabeled · line 754

  $$b_2[\gamma ]=\frac{1}{8}({\mathrm{tr}}_{\eta }\gamma {)}^2-\frac{1}{4}{\gamma }_{\alpha \beta }{\gamma }^{\alpha \beta },$$

• unlabeled · line 758

  $${a_2}^{\mu \nu }[\gamma ]={\gamma }^{\mu }{}_{\alpha }{\gamma }^{\alpha \nu }-\frac{1}{2}({\mathrm{tr}}_{\eta }\gamma ){\gamma }^{\mu \nu }+b_2[\gamma ]{\eta }^{\mu \nu }.$$

• unlabeled · line 765

  $$X=X_1[\gamma ],p=p_1[\gamma ;\varphi ],k=r_{\mathrm{D}}[\gamma ]+r_g[\varphi ].$$

• unlabeled · line 804

  $$z=\mathcal{R}c,Z=X_1[z],p_z=p_1[z;\varphi ],\dot{k}={r_{\mathrm{D}}^{\prime }}_{\gamma }[z]=2r_{\mathrm{D}}(\gamma ,z).$$