Thank you all for so much help as I work through Zee's QFT book. Here I finally have a question about physics instead of math.
Zee makes several comments that the main thing left to do in QFT is find a way to solve
$$ Z(J)=\int\!D\varphi\,\exp\!\left[ i\int\!d^4x\,\frac{1}{2}\big[ (\partial\varphi)^2-m^2\varphi^2 \big] -\frac{\lambda}{4!}\varphi^4+J\varphi \right] .$$
Since we cannot solve it directly, even by the magical process for "discretizing and undiscretizing" infinite dimensional path integrals, Zee makes a study of two simplifed cases. He presents a "baby problem"
$$ Z_B(J)=\int_{-\infty}^\infty\!dq\,\exp\!\left[ -\frac{1}{2}m^2q^2-\frac{\lambda}{4!}q^4+Jq \right], $$
and a "child problem"
\begin{align*} Z_C(J)&=\int_{-\infty}^\infty\int_{-\infty}^\infty\dots\int_{-\infty}^\infty\!dq_1\,dq_2\,\dots dq_N\,\exp\!\left[ -\frac{1}{2}\,q\cdot A\cdot q-\frac{\lambda}{4!}q^4+J\cdot q \right]\\ &=\iint\dots\int\!dq_1\,dq_2\,\dots dq_N\,\exp\!\left[ -\frac{1}{2}\sum_{m,n=1}^N q_m A_{mn} q_n-\frac{\lambda}{4!}\sum_{k=1}^N q_k^4+\sum_{l=1}^N J_l q_l \right]. \end{align*}
Analytically, I see the difference is that $J$ and $\varphi$ are functions of a continuous variable in $Z$, a discrete variable in $Z_C$, and they are not functions of anything in $Z_B$.
Q1: What would be the physical significances of $Z_B$ and $Z_C$ respectively?
Q2: What does it mean for $q$ not to be a function of anything?
For $q$ to be a function of a discrete variable, I see that $q$ describes, essentially, the vertical displacement position of a set of oscillators located at some number of discrete $\vec x_k$. If it's not a function of anything, does that mean it is a single oscillator? How could it not be a function of time at least? Is the location of the single oscillator coded into $q$ itself so $q$ means "the oscillator at some $\vec x_0$?" If you can go into detail about the physics we are looking at here, that would be very helpful.
