The two completion legs

Displayed formulas in this passage

• XVI-B@1#eq:coefficients · line 1170

  $$\begin{array}{rlrl}{a_1}^{\mu \nu }[r]& =\frac{1}{2}{\eta }^{\mu \nu }{\mathrm{tr}}_{\eta }r-r^{\mu \nu },& b_1[r]& =\frac{1}{2}{\mathrm{tr}}_{\eta }r,\\ {X_1}^{\mu }[r]& =-{E_0}^{\mathrm{R}}{\partial }_{\nu }(\sigma {a_1}^{\nu \mu }[r]),& & \\ V_{1,g}[r]\varphi & ={\partial }_{\mu }(g{a_1}^{\mu \nu }[r]{\partial }_{\nu }\varphi )+m^{2}g{b}_1[r]\varphi ,& & \\ {\varphi }_1[r;\varphi ]& =-{E_m}^{\mathrm{R}}(V_{1,g}[r]\varphi ).& & \end{array}$$

• XVI-B@1#eq:coordinate-leg · line 1170

  $$\begin{array}{rlrl}{a_1}^{\mu \nu }[r]& =\frac{1}{2}{\eta }^{\mu \nu }{\mathrm{tr}}_{\eta }r-r^{\mu \nu },& b_1[r]& =\frac{1}{2}{\mathrm{tr}}_{\eta }r,\\ {X_1}^{\mu }[r]& =-{E_0}^{\mathrm{R}}{\partial }_{\nu }(\sigma {a_1}^{\nu \mu }[r]),& & \\ V_{1,g}[r]\varphi & ={\partial }_{\mu }(g{a_1}^{\mu \nu }[r]{\partial }_{\nu }\varphi )+m^{2}g{b}_1[r]\varphi ,& & \\ {\varphi }_1[r;\varphi ]& =-{E_m}^{\mathrm{R}}(V_{1,g}[r]\varphi ).& & \end{array}$$

• XVI-B@1#eq:scalar-forcing · line 1170

  $$\begin{array}{rlrl}{a_1}^{\mu \nu }[r]& =\frac{1}{2}{\eta }^{\mu \nu }{\mathrm{tr}}_{\eta }r-r^{\mu \nu },& b_1[r]& =\frac{1}{2}{\mathrm{tr}}_{\eta }r,\\ {X_1}^{\mu }[r]& =-{E_0}^{\mathrm{R}}{\partial }_{\nu }(\sigma {a_1}^{\nu \mu }[r]),& & \\ V_{1,g}[r]\varphi & ={\partial }_{\mu }(g{a_1}^{\mu \nu }[r]{\partial }_{\nu }\varphi )+m^{2}g{b}_1[r]\varphi ,& & \\ {\varphi }_1[r;\varphi ]& =-{E_m}^{\mathrm{R}}(V_{1,g}[r]\varphi ).& & \end{array}$$

• XVI-B@1#eq:scalar-leg · line 1170

  $$\begin{array}{rlrl}{a_1}^{\mu \nu }[r]& =\frac{1}{2}{\eta }^{\mu \nu }{\mathrm{tr}}_{\eta }r-r^{\mu \nu },& b_1[r]& =\frac{1}{2}{\mathrm{tr}}_{\eta }r,\\ {X_1}^{\mu }[r]& =-{E_0}^{\mathrm{R}}{\partial }_{\nu }(\sigma {a_1}^{\nu \mu }[r]),& & \\ V_{1,g}[r]\varphi & ={\partial }_{\mu }(g{a_1}^{\mu \nu }[r]{\partial }_{\nu }\varphi )+m^{2}g{b}_1[r]\varphi ,& & \\ {\varphi }_1[r;\varphi ]& =-{E_m}^{\mathrm{R}}(V_{1,g}[r]\varphi ).& & \end{array}$$

• XVI-B@1#eq:Aone · line 1198

  $${\mathcal{A}}_1[f,g](r;\varphi )=\int f(x)({\varphi }_1[r;\varphi ](x)-{X_1}^{\mu }[r](x){\partial }_{\mu }\varphi (x)){\mathrm{d}}^{4}x.$$

• XVI-B@1#eq:CD · line 1207

  $${\mathcal{C}}_{3,\mathrm{D}}(f,g)[\gamma ,\varphi ]={\mathcal{A}}_1[f,g](u_{\mathrm{D}}[\gamma ];\varphi )$$

• unlabeled · line 1190

  $$f\in {C_c}^{\infty }({M_T}^{\circ }),g\in {C_c}^{\infty }({M_T}^{\circ };[0,1]),$$