The Functional Integral and Generating Functional

The canonical route quantizes fields by imposing commutators. The path-integral route is the alternative promised in Postulate 9: define the theory by summing $e^{iS}$ over all field configurations. It generalizes the single-particle path integral of QM to fields, packages all correlation functions into a single generating functional $Z[J]$, and re-derives the Feynman rules with symmetry factors automatic. This is the entry point to the effective action, Ward identities, and the modern treatment of gauge theories.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

From the QM path integral to fields

The QM path integral sums over trajectories $x(t)$ weighted by $e^{iS[x]}$. A field theory has a dynamical variable $\varphi (\mathbf{x})$ at every spatial point, so the sum is over field configurations $\varphi (x)$ on all of spacetime:

$$\int \mathcal{D}x(t)e^{iS[x]}⟶\int \mathcal{D}\varphi (x)e^{iS[\varphi ]},S[\varphi ]=\int d^{4}x\mathcal{L}.$$

The measure $\mathcal{D}\varphi ={\prod }_{x}d\varphi (x)$ is a formal product over all spacetime points; it is defined rigorously only after regularization (e.g. on a lattice), but perturbation theory needs only Gaussian integrals, which are unambiguous.

The generating functional (Definition)

Couple the field to an external source $J(x)$ and integrate:

$$\boxed{Z[J]=\int \mathcal{D}\varphi \exp \left(iS[\varphi ]+i\int d^{4}xJ(x)\varphi (x)\right).}$$

$Z[J]$ is the generating functional for correlation functions: differentiating with respect to the source brings down factors of $\varphi$, and setting $J=0$ gives the vacuum time-ordered correlators of Postulate 9:

$$\boxed{⟨0|T\varphi (x_1)\cdots \varphi (x_n)|0⟩=\frac{1}{Z[0]}\left(\frac{1}{i}\frac{\delta }{\delta J(x_1)}\right)\cdots \left(\frac{1}{i}\frac{\delta }{\delta J(x_n)}\right)Z[J]{|}_{J=0}.}$$

The functional derivative $\frac{\delta }{\delta J(x)}$ is the continuum tool introduced with functional derivatives, obeying $\frac{\delta J(y)}{\delta J(x)}={\delta }^4(x-y)$. The division by $Z[0]$ removes disconnected vacuum bubbles automatically — the diagrammatic statement that only connected diagrams contribute.

The free-field generating functional

For the free scalar $S_0=\int d^{4}x[\frac{1}{2}(\partial \varphi {)}^2-\frac{1}{2}{m}^2{\varphi }^2]$, the integral is Gaussian. Integrating by parts and completing the square in $\varphi$ gives, up to the field-independent constant $Z_0[0]$,

$$\boxed{Z_0[J]=Z_0[0]\exp \left(-\frac{1}{2}\int d^{4}x{d}^{4}yJ(x)D_F(x-y)J(y)\right),}$$

where $D_F$ is exactly the Feynman propagator — here it appears as the inverse of the kinetic operator $(\square +m^2)$, with the $iϵ$ from the requirement that the Gaussian integral converge. Two functional derivatives recover the two-point function,

$$⟨0|T\varphi (x)\varphi (y)|0⟩=D_F(x-y),$$

and $2n$ derivatives reproduce Wick's theorem — the sum over all pairings of the $2n$ points into propagators — with the pairing combinatorics generated automatically by the product rule for functional differentiation. This is the path-integral proof of Wick's theorem.

Perturbation theory and Feynman rules, functionally

Split the action $S=S_0+S_{\text{int}}$ and pull the interaction outside the Gaussian integral by trading each $\varphi$ for a source derivative:

$$Z[J]=\exp \left(i{S}_{\text{int}}\left[\frac{1}{i}\frac{\delta }{\delta J}\right]\right)Z_0[J].$$

Expanding both exponentials in powers of the coupling generates precisely the Feynman diagrams: each factor of $S_{\text{int}}$ is a vertex, each contraction from $Z_0[J]$ a propagator. The decisive advantage over the canonical Dyson-series derivation is that the symmetry factors — the error-prone $1/S$ divisions — emerge automatically from the combinatorics of functional differentiation, rather than having to be inserted by hand.

Euclidean rotation and the statistical-mechanics analogy

The oscillatory weight $e^{iS}$ makes the Lorentzian integral only conditionally convergent. Wick rotation to imaginary time $t\to -i\tau$ turns it into a real, exponentially damped weight:

$$iS\stackrel{t=-i\tau }{\to }-S_E,Z_E[J]=\int \mathcal{D}\varphi e^{-S_E[\varphi ]+\int J\varphi },$$

with the Euclidean action $S_E\ge 0$. This is the same $iϵ$/Wick-rotation seed noted for the propagator and discussed in the QM path integral. The Euclidean functional integral is identical in form to the partition function of classical statistical mechanics in four dimensions:

$$Z_E⟷\sum _{\text{configs}}{e}^{-\beta H}=Z_{\text{stat}},$$

with $S_E$ playing the role of $\beta H$. This analogy is not a curiosity — it is the foundation of lattice field theory (Monte-Carlo evaluation of $Z_E$), the Wilsonian renormalization group (borrowed wholesale from critical phenomena), and finite-temperature QFT (compactified Euclidean time).

Why the path integral is worth it

The two formulations are equivalent (they compute the same correlators), but the path integral is the more powerful organizing principle for everything downstream.

Summary

• $Z[J]=\int \mathcal{D}\varphi e^{iS+i\int J\varphi }$ generates all correlators by source differentiation.
• Free case: $Z_0[J]=Z_0[0]e^{-\frac{1}{2}\int J{D}_{F}J}$ — Wick's theorem, automatically.
• Interactions: $\exp (i{S}_{\text{int}}[\delta /i\delta J])$ generates Feynman diagrams with symmetry factors built in.
• Wick rotation $\to$ Euclidean $Z_E$, identical in form to a statistical partition function.

Where this leads

• Fermions need anticommuting integration variables: Grassmann integration.
• The 1PI generating functional and effective potential: the effective action.
• Symmetries of the measure become Ward–Takahashi identities; their failure is an anomaly.
• Gauge fixing in the functional integral: Faddeev–Popov.
• The analytic-continuation meaning of Wick rotation ($t\to -i\tau$ as a contour rotation): math/complex-analysis: analytic continuation.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 9.1–9.2.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 9.
• Zee, Quantum Field Theory in a Nutshell, Ch. I.
• Srednicki, Quantum Field Theory, Ch. 6–9.