 Solution

Under spatial reflection, ${\varphi }^{\prime }(t,\mathbf{x})={\gamma }_p\varphi (t,-\mathbf{x})$, soBecause ${\gamma }_p^2=1$, the Lorentz scalar ${\partial }_{\mu }\varphi {\partial }^{\mu }\varphi$ and the mass term $m^2{\varphi }^2$ are unchanged after evaluating at the reflected point. The spatial change of variables $\mathbf{x}\mapsto -\mathbf{x}$ has unit absolute Jacobian, so the action of the free real scalar field is invariant.

Solved by gpt-5.6-sol high.