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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?

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    $\begingroup$ Standard Gaussian integral - complete the square. $\endgroup$ Commented Sep 21, 2022 at 14:32
  • $\begingroup$ @SeanE.Lake I am familiar with the Standard Gaussian integral where the limit of $x$ goes from $-\infty$ to $\infty$. What I am puzzled is the $\mathcal{D}\pi$, it’s limit and other things as I point in the last paragraph of my post. Thanks! $\endgroup$
    – Daren
    Commented Sep 21, 2022 at 15:00
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    $\begingroup$ $\mathcal{D}\pi = \prod_x \mathrm{d}\pi(x)$ and the limits are all negative infinity to infinity. In other words, you're integrating over the space of all functions with support on the specified part of space-time (specified in the integral inside the exponential function). $\endgroup$ Commented Sep 21, 2022 at 15:04
  • $\begingroup$ I just want to clarify the integration should have $\sqrt{2\pi}$, is it dropped due to it is just a phase change? $\endgroup$
    – Li Chiyan
    Commented Dec 21, 2022 at 4:24
  • $\begingroup$ @LiChiyan Which integration do you mean? $\endgroup$
    – Daren
    Commented Dec 27, 2022 at 11:13

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Briefly, there are 2 issues:

  1. 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.)

  2. 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 ${\cal N}^{-1}$.

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