Scalar density and switching

Displayed formulas in this passage

• VII-B@1#eq:scalar-density · line 1523

  $${\mathcal{L}}_{\mathrm{sc}}(\mathbf{g},\varphi )=\frac{1}{2}{A}^{\mu \nu }(\kappa ){\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -\frac{1}{2}{m}^{2}d(\kappa ){\varphi }^2.$$

• VII-B@1#eq:scalar-switch · line 1534

  $$\int g\left({\mathcal{L}}_{\mathrm{sc}}(\eta +\kappa h,\varphi )-{\mathcal{L}}_{\mathrm{sc}}(\eta ,\varphi )\right){\mathrm{d}}^{4}x.$$

• VII-B@1#eq:full-interaction · line 1547

  $$\begin{array}{rl}V(\kappa )=& \int g\left({\mathcal{L}}_{\mathrm{sc}}(\eta +\kappa h,\varphi )-{\mathcal{L}}_{\mathrm{sc}}(\eta ,\varphi )\right){\mathrm{d}}^{4}x\\ & +\int g_{\mathrm{grav}}\left({\mathcal{L}}_{\mathrm{grav}}(\kappa )-{\mathcal{L}}_{\mathrm{grav},0}\right){\mathrm{d}}^{4}x\\ & -\kappa \int g_{\mathrm{gh}}{\bar{c}}_{\mu }{C}^{\mu }[{\mathcal{L}}_{c}h]{\mathrm{d}}^{4}x.\end{array}$$

• VII-B@1#eq:V1sc · line 1570

  $$\begin{array}{rlrl}{V_1}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_1}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_1{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_2}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_2}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_2{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_1}^{\mathrm{gh}}& =-\int g_{\mathrm{gh}}{\bar{c}}_{\mu }{C}^{\mu }[{\mathcal{L}}_{c}h]{\mathrm{d}}^{4}x,& {V_2}^{\mathrm{gh}}& =0,\\ {V_1}^{\mathrm{aux}}& =0,& {V_2}^{\mathrm{aux}}& =0.\end{array}$$

• VII-B@1#eq:V2sc · line 1570

  $$\begin{array}{rlrl}{V_1}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_1}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_1{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_2}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_2}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_2{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_1}^{\mathrm{gh}}& =-\int g_{\mathrm{gh}}{\bar{c}}_{\mu }{C}^{\mu }[{\mathcal{L}}_{c}h]{\mathrm{d}}^{4}x,& {V_2}^{\mathrm{gh}}& =0,\\ {V_1}^{\mathrm{aux}}& =0,& {V_2}^{\mathrm{aux}}& =0.\end{array}$$

• VII-B@1#eq:Vghost · line 1570

  $$\begin{array}{rlrl}{V_1}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_1}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_1{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_2}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_2}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_2{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_1}^{\mathrm{gh}}& =-\int g_{\mathrm{gh}}{\bar{c}}_{\mu }{C}^{\mu }[{\mathcal{L}}_{c}h]{\mathrm{d}}^{4}x,& {V_2}^{\mathrm{gh}}& =0,\\ {V_1}^{\mathrm{aux}}& =0,& {V_2}^{\mathrm{aux}}& =0.\end{array}$$

• VII-B@1#eq:Vaux · line 1570

  $$\begin{array}{rlrl}{V_1}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_1}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_1{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_2}^{\mathrm{sc}}& =\frac{1}{2}\int g\left({A_2}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{d}_2{\varphi }^2\right){\mathrm{d}}^{4}x,& & \\ {V_1}^{\mathrm{gh}}& =-\int g_{\mathrm{gh}}{\bar{c}}_{\mu }{C}^{\mu }[{\mathcal{L}}_{c}h]{\mathrm{d}}^{4}x,& {V_2}^{\mathrm{gh}}& =0,\\ {V_1}^{\mathrm{aux}}& =0,& {V_2}^{\mathrm{aux}}& =0.\end{array}$$