Let's follow Peskin and Schroeder section 9.2, page 282.
The Hamiltonian of a free real scalar field is
so the expression for the functional integral is
then Peskin and Schroeder say that in order to integrate over the $\pi$ you just have to complete the square which gives us
now, my question. How do you get rid of the first exponent? My teacher said something about Gaussian integrals but this doesn't convince me. This is not a regular integral, this is a functional integral, so we shouldn't use directly the formula for Gaussians here. How do you perform this integral without resorting to hand wavy arguments?