Lesson 7.5 · 7. Quantum Field Theory

Functional Methods in QFT

The canonical formulation describes fields as operators evolving in time. The functional integral takes another view: it sums over every possible field configuration. This language makes symmetries, Feynman diagrams, and effective field theory especially transparent.

An integral over functions

An ordinary integral sums over numbers. A functional integral sums over entire functions $\phi(x)$. The symbol $\mathcal{D}\phi$ formally denotes a measure on this configuration space. In practice, the problem is first defined on a lattice or with another regulator, and then one studies the continuum limit.

Amplitude and Action

For a scalar field, the transition amplitude takes the form:

Z=\int\mathcal{D}\phi\,e^{iS[\phi]/\hbar}

Every field history contributes a phase fixed by its action. When $S\gg\hbar$, neighboring configurations cancel except near stationary points with $\delta S=0$. Classical field equations emerge by stationary phase.

The Generating Functional

Introduce an external source $J(x)$:

Z[J]=\int\mathcal{D}\phi\,\exp\left[\frac{i}{\hbar}\left(S[\phi]+\int d^4x\,J(x)\phi(x)\right)\right]

Functional derivatives with respect to $J$ insert fields into the integral:

\langle0|T\phi(x_1)\cdots\phi(x_n)|0\rangle= \left.\frac{1}{Z[0]}\left(\frac{\hbar}{i}\right)^n \frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}

These correlation functions are the objects from which the LSZ formula extracts scattering amplitudes.

Free Theory and the Propagator

For a quadratic action, the integral is Gaussian and can be computed exactly. The Feynman propagator $\Delta_F$ appears in the result:

Z_0[J]=Z_0[0]\exp\left[-\frac{i}{2\hbar}\int d^4x\,d^4y\,J(x)\Delta_F(x-y)J(y)\right]

The propagator is the inverse of the quadratic differential operator with the appropriate causal prescription. It is not the path of a hidden particle; it is a correlation function.

Interactions and Diagrams

For an interaction $\lambda\phi^4/4!$, expand the exponential in powers of $\lambda$. Wick's theorem organizes all contractions. Each contraction gives a propagator, each interaction term gives a vertex, and the combinatorics supplies symmetry factors. Feynman diagrams are therefore a graphical ledger of perturbation theory.

Connected, irreducible, and effective

$W[J]=-i\hbar\ln Z[J]$ generates connected diagrams. Its Legendre transform, the effective action $\Gamma[\phi_c]$, generates one-particle-irreducible vertices. The equation $\delta\Gamma/\delta\phi_c=0$ contains the equations of motion corrected by quantum effects.

Wick Rotation

The oscillating real-time integral is often difficult to define. Setting $t=-i\tau$ formally gives Euclidean time:

Z_E=\int\mathcal{D}\phi\,e^{-S_E[\phi]/\hbar}

The weight then resembles a Boltzmann distribution. This connection enables lattice gauge-theory simulations. Returning to real time requires precise analytic conditions, and sign problems can make some theories numerically very difficult.

Ward Identities

A change of variables in the functional integral leaves $Z$ invariant if the measure and action respect the symmetry. This produces identities among correlation functions. For gauge symmetry, Ward-Takahashi identities enforce current conservation and constrain renormalization. If the measure is not invariant, a quantum anomaly can appear.

Exercises

Explain why the logarithm of $Z[J]$ selects connected diagrams.
What physical distinction separates a propagator from a classical trajectory?
Why is a Euclidean weight better suited to Monte Carlo methods?

Key Takeaways

The functional integral sums over field configurations weighted by $e^{iS/\hbar}$.
The generating functional produces correlation functions through functional differentiation.
Feynman diagrams organize a perturbative expansion; they are not literal histories.
The effective action collects quantum corrections to classical equations.
Wick rotation connects quantum field theory and statistical physics under controlled analytic assumptions.

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