On page 282 of Peskin and Schroeder discussing functional quantization of scalar fields, the authors use expression 9.12, the path integral in ordinary quantum mechanics
$$U(q_a,q_b;T)= $$
$$\bigg(\prod_i\int\mathcal{D}q(t)\mathcal{D}p(t)\bigg)\exp\bigg[i\int_0^T\,dt\big(\sum_ip^i\dot{q}^i-H(q,p)\big)\bigg]\tag{9.12}$$
then since $$H=\int\,d^3x[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+V(\phi)],$$
we have
$$\langle\phi_b(\boldsymbol{x})|e^{-iHT}|\phi_a(\boldsymbol{x})\rangle=$$
$$\int\,\mathcal{D}\phi\mathcal{D}\pi\exp\bigg[i\int_0^T\,d^4x\Big(\pi\dot{\phi}-\frac{1}{2}\pi^2-\frac{1}{2}(\nabla\phi)^2-V(\phi)\Big)\bigg].\tag{9.14}$$
My question is: what does 9.14 even mean? Previously we derived 9.12 from ordinary quantum mechanics, and it has a well defined meaning, in the sense that I can in theory write down a precise equation for symbols like $\mathcal{D}p(t)$; in fact Peskin and Schroeder does it in 9.11. But how can one write down the precise meaning of $\mathcal{D}\pi$ and $\mathcal{D}\phi$?
Best Answer
Well, the path integral is a heuristic construction. Formally P&S identify the index $$ i ~=~{\bf x} $$ with a point in 3-space, so that $$ q^i(t)~=~\phi({\bf x},t) \qquad\text{and}\qquad p^i(t)~=~\pi({\bf x},t). $$ To complete the transition from point mechanics (9.12) to field theory (9.14), one often imagines that 3-space and time are discretized. Then spacetime derivatives are replaced by appropriate finite differences, and the path integral measures become $${\cal D}q ~=~\prod_{i,t} \mathrm{d}q^i(t) \qquad\text{and}\qquad {\cal D}\phi ~=~\prod_{{\bf x},t} \mathrm{d}\phi({\bf x},t).$$