Path Integrals in Field Theory

Path integrals offer a second, independent route to quantum field theory. Instead of canonical quantization, you sum over all field configurations weighted by e^{iS[φ]}. This approach is manifestly Lorentz covariant, extends naturally to fermions via Grassmann integrals, and makes the connection to statistical mechanics transparent. The generating functional Z[J] encodes all correlation functions of the theory.

Key Concepts

Z = ∫Dφ e^{iS[φ]} — sum over all field configurations, each weighted by e^{iS}
Generating functional Z[J] = ∫Dφ exp(i∫(ℒ+Jφ)d⁴x) with external source J(x)
W[J] = −i ln Z[J] generates only connected Green's functions
Effective action Γ[φ_cl] = W[J] − ∫Jφ_cl generates 1PI (one-particle-irreducible) diagrams
Fermionic path integrals use anticommuting Grassmann numbers: ∫Dψ̄Dψ e^{−ψ̄Aψ} = det A
Wick rotation t → −iτ maps QFT to Euclidean field theory, connecting to statistical mechanics

Key Equations

Path integral

Z=\int\!\mathcal{D}\phi\;e^{\,iS[\phi]}

Generating functional

Z[J]=\int\!\mathcal{D}\phi\;\exp\!\Bigl[i\!\int\!d^4x\bigl(\mathcal{L}+J\phi\bigr)\Bigr]

Connected Green's functions

W[J]=-i\ln Z[J],\quad G^{(n)}_c=\frac{\delta^n W}{i^n\,\delta J(x_1)\cdots\delta J(x_n)}\bigg|_{J=0}

Effective action

\Gamma[\phi_{\rm cl}]=W[J]-\int\!d^4x\,J(x)\phi_{\rm cl}(x)

Worked Example

Example Problem

Problem

The free-field path integral gives Z₀[J] ∝ exp(½∫d⁴x d⁴y J(x)ΔF(x−y)J(y)). Take two functional derivatives δ/iδJ to recover the Feynman propagator.

Solution

δZ₀/iδJ(x₁) = Z₀[J]·∫d⁴y ΔF(x₁−y)J(y). Differentiating again with 1/iδJ(x₂) picks out y=x₂ via δ(x₂−y). Setting J=0 gives δ²Z₀/(i²δJδJ)|_{J=0} = ΔF(x₁−x₂) = ⟨0|T{φ(x₁)φ(x₂)}|0⟩.

Key Takeaways

Z = ∫Dφ e^{iS} is an equivalent formulation of QFT that is manifestly Lorentz covariant and naturally handles gauge redundancy
W[J] = −i ln Z[J] generates connected diagrams; Γ[φ_cl] generates 1PI diagrams (the "quantum action")
Euclidean path integrals (Wick rotation) reveal the deep parallel between QFT and statistical mechanics — the partition function is the same object
Grassmann path integrals for fermions produce determinants: ∫DψDψ̄ e^{−ψ̄Aψ} = det A (vs. (det A)^{−½} for bosons)