Classical free action variations and normalization

For the scalar action, differentiating its density at $\varphi +\epsilon h$ gives $\delta S_{\varphi }=-{\int }_{{\mathbb{R}}^4}({\eta }^{ab}{\mathrm{\partial }}_a\varphi {\mathrm{\partial }}_{b}h+{\mu }^2\varphi h)d^{4}x\mathrm{.}$ For each coordinate, the product ${\mathrm{\partial }}_a\varphi h$ has compact support; one-dimensional integration by parts on that coordinate has zero boundary term, and Fubini [F4] permits integrating the remaining coordinates. Hence $\delta S_{\varphi }={\int }_{{\mathbb{R}}^4}h({\eta }^{ab}{\mathrm{\partial }}_a{\mathrm{\partial }}_b\varphi -{\mu }^2\varphi )d^{4}x={\int }_{{\mathbb{R}}^4}h(\mathrm{\square }-{\mu }^2)\varphi d^{4}x\mathrm{.}$

Put $R_D=(i{\gamma }^a{\mathrm{\partial }}_a-\mu )\psi$. From [F1], ${\beta }^{†}=\beta$ and ${\beta }^2=I_4$. Each ${\sigma }_j$ is Hermitian, so ${\alpha }_j^{†}={\alpha }_j$; the off-diagonal blocks give ${\alpha }_j\beta =-\beta {\alpha }_j$. Hence ${\gamma }^0=\beta$ is Hermitian and $({\gamma }^j{)}^{†}=(\beta {\alpha }_j{)}^{†}={\alpha }_j\beta =-\beta {\alpha }_j=-{\gamma }^j$, so $({\gamma }^a{)}^{†}\beta =\beta {\gamma }^a$ for every $a$. Varying $\psi$ and $\bar{\psi }={\psi }^{†}\beta$ together along a real parameter gives $\delta S_D={\int }_{{\mathbb{R}}^4}\bar{\chi }{R}_D+\bar{\psi }(i{\gamma }^a{\mathrm{\partial }}_a-\mu )\chi d^{4}x\mathrm{.}$ Integrating the second term by parts coordinatewise, with zero boundary terms since $\chi$ is compactly supported, yields $\delta S_D={\int }_{{\mathbb{R}}^4}\bar{\chi }{R}_D+(-i{\mathrm{\partial }}_a\bar{\psi }{\gamma }^a-\mu \bar{\psi })\chi d^{4}x=2\mathrm{Re}{\int }_{{\mathbb{R}}^4}{\chi }^{†}\beta R_D{d}^{4}x,$ because $R_D^{†}\beta =-i{\mathrm{\partial }}_a\bar{\psi }{\gamma }^a-\mu \bar{\psi }$.

If $(\mathrm{\square }-{\mu }^2)\varphi =0$, step 1.1 makes every variation vanish. Conversely, if the smooth residual $R_{\varphi }=(\mathrm{\square }-{\mu }^2)\varphi$ is nonzero at a point, continuity gives a ball on which it has one strict sign; choosing the nonnegative bump from [F3] inside that ball gives $\int h{R}_{\varphi }\mathrm{\ne }0$. Thus scalar stationarity for all compactly supported real $h$ is equivalent to the Klein–Gordon equation.

If $R_D=0$, step 1.2 gives stationarity. Conversely, if $R_D(x_0)\mathrm{\ne }0$, continuity gives a ball on which $\mathrm{\mid }{R}_D\mathrm{\mid }>0$. Choose a nonnegative bump $\rho$ supported there and positive on a smaller ball [F3], and take the admissible variation $\chi =\rho \beta R_D$. Then $\delta S_D=2{\int }_{{\mathbb{R}}^4}\rho \mathrm{\mid }{R}_D{\mathrm{\mid }}^2{d}^{4}x>0,$ contradicting stationarity. Thus $R_D=0$ pointwise. Its adjoint is $-i{\mathrm{\partial }}_a\bar{\psi }{\gamma }^a-\mu \bar{\psi }=0$, and conversely that adjoint equation implies $R_D=0$ since $\beta$ is invertible.