Path integrals and generating functions

Use $\hbar=c=1$ and Lorentzian metric $(+---)$. Break a time interval into small steps and insert complete position bases between short-time evolution operators. For ordinary quantum mechanics this motivates integrals over intermediate positions weighted by complex phases. Discretizing space as well gives finitely many field coordinates at a finite cutoff. The continuum symbol $\mathcal D\phi$ summarizes a limit of such constructions; it is not an ordinary uniform measure on every imaginable field.

The Lorentzian factor $e^{iS}$ has no positive-probability interpretation. Contributions can cancel by interference, and the bare oscillatory integral may need an $i\epsilon$ prescription or a specified integration contour. A vacuum generating functional also needs its state and boundary prescription. Dividing by the zero-source functional removes a normalization factor; it does not by itself prove convergence or define every interacting continuum field theory.

Differentiating with respect to $J(x)$ brings down a field insertion and a factor of $i$. Repeating this produces time-ordered vacuum correlations in the appropriate vacuum path integral. The source is a device for organizing questions; setting it to zero after differentiation returns the undriven theory. For a free action, completing the Gaussian square determines all correlations from the two-point function, reproducing the pairing structure used in Wick’s theorem.

Expanding an interaction inside the integral generates the same vertex insertions and contractions as operator perturbation theory. A useful consistency check is to reproduce a simple low-order amplitude in both languages under matching conventions. The path integral is therefore more than a new metaphor: it changes the organization of a calculation. Yet it does not make arbitrary strong-coupling integrals easy, and a formal rearrangement does not establish convergence of the rearranged series.

Under suitable analyticity and boundary assumptions, rotating to imaginary time turns a stable scalar theory into a Euclidean weight $e^{-S_E}$. A positive real weight can be sampled numerically in many bosonic lattice models. This is not an automatic operation for every contour, state, or theory. Fermion determinants and finite-density systems can introduce sign or complex-weight problems, and recovering real-time quantities from Euclidean data is often ill-conditioned.

A finite lattice provides a regulator and a concrete finite-dimensional integral. Removing its spacing while holding physical scales fixed requires tuning and a justified continuum limit. The existence of the finite integral does not guarantee that desired limit exists or is interacting. Gauge theories also need control of gauge redundancy; gauge fixing and its Jacobian enter continuum treatments. These issues connect the elegant generating-function notation to the renormalization and nonperturbative questions of the next module.