 Solution

Write ${\int }_{\mathbf{p}}=\int d^{3}p/(2\pi {)}^3$ and $E_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}$. Insert the mode expansion of a free field for each quantity into the commutator. Only the mixed creation operator and annihilation operator terms survive. The Dirac delta function in their canonical commutation relation sets the two momenta equal, so the square-root factors multiply to $1/2$:In the second equality, each Fourier transform is the same Dirac delta function, since the integration domain is invariant under $\mathbf{p}\mapsto -\mathbf{p}$. This is the equal-time canonical commutation relation. The analogous calculation gives $[\varphi (\mathbf{x}),\varphi (\mathbf{y})]=[\pi (\mathbf{x}),\pi (\mathbf{y})]=0$, because their two mixed terms have opposite signs.