Peskin and Schroeder eqn 9.14 [closed]

Question

I am not familiar with functional integral, and in the text like $$\int D\phi D\pi \exp [i\int^T_od^4x(\pi\dot{\phi}-\frac{1}{2}\pi^2-\frac{1}{2}(\nabla \phi)^2-V(\phi))].\tag{9.14}$$ I try to compile it but get: $$\int D\phi\left[ \prod_{x,t}\int d\pi\ \exp [i\epsilon^4(\pi\dot{\phi}-\frac{1}{2}\pi^2)] \right] \exp([i\int^T_od^4x(-\frac{1}{2}(\nabla \phi)^2-V(\phi))]$$ $$=\int D\phi \left[\prod_{x,t}\sqrt{\frac{2\pi}{i\epsilon^4}} \exp [i\epsilon^4\frac{\dot{\phi}^2}{2}] \right] \exp([i\int^T_od^4x(-\frac{1}{2}(\nabla \phi)^2-V(\phi))]$$ $$=\int D\phi \left[\prod_{x,t}\sqrt{\frac{2\pi}{i\epsilon^4}}\right] \exp[i\int^T_od^4x(\frac{1}{2}\dot{\phi}^2-\frac{1}{2}(\nabla \phi)^2-V(\phi))].$$ I know things were terrible wrong, but what is it?

Answer 1

OP's infinitely many Gaussian integrations of the momentum field are essentially the correct method. The infinite product in OP's last line is usually tucked away by normalizing the path integral appropriately.