On Peskin and Schroeder's QFT book page 282, the book considered functional quantization of scalar field.
First, begin with
$$\left\langle\phi_b(\mathbf{x})\left|e^{-i H T}\right| \phi_a(\mathbf{x})\right\rangle=\int \mathcal{D} \phi \mathcal{D} \pi \exp \left[i \int_0^T d^4 x\left(\pi \dot{\phi}-\frac{1}{2} \pi^2-\frac{1}{2}(\nabla \phi)^2-V(\phi)\right)\right] $$
Since the exponent is quadratic in $\pi$, the book evaluates the $\mathcal{D}\pi$ integral and obtains
$$\left\langle\phi_b(\mathbf{x})\left|e^{-i H T}\right| \phi_a(\mathbf{x})\right\rangle=\int \mathcal{D} \phi \exp \left[i \int_0^T d^4 x \mathcal{L}\right]. \tag{9.14} $$
There needs to be some complicated coefficients, but the book omit here.
I am puzzled how this integral finished? ie.
$$\int \mathcal{D} \pi \exp \left[i \int _ { 0 } ^ { T } d ^ { 4 } x \left(\pi \dot{\phi}-\frac{1}{2} \pi^2\right)\right] $$
Since now the integral argument is $\pi$, which is a function, how to understand it's upper and lower limit? Also, their have a term $i \int_0^T d^4 x$ inside the exponent. So how to understand this integral?
Best Answer
Briefly, there are 2 issues:
It is safest to Wick rotate $t_E=it_M$ to make the Gaussian integrals exponentially damped rather than oscillatory. (NB: Don't also Wick rotate the momentum field $\pi_M=i\pi_E$, cf. my related Phys.SE answer here.)
Truncate spacetime to a finite large box and discretize it. The result of the Gaussians integrations will be a number ${\cal N}$ that doesn't depend on any physically important parameters, but diverge in the continuum limit. Define the path integral measure ${\cal D}\pi$ to contain the reciprocal constant ${\cal N}^{-1}$.