QFT: Path Integrals & Methods

Path Integrals & Methods

Quantum Field Theory » Path Integrals & Methods

Knowing what a quantum field is tells you almost nothing about how to compute with one. This page is the workshop: the path integral that packages all of QFT into a single weighted sum over field configurations, the generating functionals that turn that sum into a machine for spitting out correlation functions, the perturbative expansion of that machine into Feynman diagrams, and the effective-field-theory mindset that lets you calculate without knowing the full theory. These are the tools that produce the numbers — the electron g−2 to twelve digits, cross sections at the LHC, the running of the strong coupling.

• Sum over histories. Every field configuration contributes an amplitude $e^{iS/\hbar}$; the classical path is just where the phase is stationary.
• Diagrams are the expansion. Each Feynman diagram is one term in a power series in the coupling — propagators for lines, factors for vertices.
• One functional, all correlators. Differentiate the generating functional $Z[J]$ and every Green’s function falls out automatically.
• Calculate at the right scale. Effective field theory lets you ignore physics you cannot reach and organize the rest by powers of energy.

What You’ll Find on This Page

How the Pieces Fit Together

graph TD
    PI["Path integral: sum over field configs"] --> Z["Generating functional Z[J]"]
    Z --> W["Connected functional W[J] = -i ln Z"]
    W --> G["Effective action Γ[φc] (Legendre transform)"]
    Z --> PT["Perturbative expansion"]
    PT --> WICK["Wick's theorem"]
    WICK --> FD["Feynman diagrams"]
    FD --> AMP["Scattering amplitudes / cross sections"]
    EFT["Effective field theory"] --> PT
    style PI fill:#11998e,color:#fff
    style AMP fill:#38ef7d,color:#222
    style EFT fill:#ccf,color:#222

The Path-Integral Formulation

The path integral, due to Feynman (building on a remark of Dirac), provides an alternative to canonical quantization that is fully equivalent but often far more powerful. Instead of promoting fields to operators and imposing commutation relations, it assigns a complex amplitude to every possible field configuration and sums them all up.

Sum Over Histories

In ordinary quantum mechanics, the amplitude for a particle to travel from $x_i$ at time $t_i$ to $x_f$ at time $t_f$ is obtained by summing $e^{iS/\hbar}$ over all paths connecting the endpoints — not just the classical one:

The classical trajectory is the one where the action is stationary ($\delta S = 0$), so nearby paths interfere constructively; far from it, the rapidly oscillating phase causes cancellation. This is how the classical limit ($\hbar \to 0$) emerges: only the stationary-action path survives.

The Field-Theory Path Integral

Promoting the single coordinate $x(t)$ to a field $\phi(x)$ defined at every spacetime point, the transition amplitude between field configurations becomes a functional integral:

where the action is the spacetime integral of the Lagrangian density:

The symbol $\mathcal{D}\phi$ denotes integration over every possible field configuration — a continuous infinity of ordinary integrals, one for the value of $\phi$ at each spacetime point. This object is the master quantity from which everything else is derived.

Euclidean Formulation

The oscillatory factor $e^{iS}$ makes the Minkowski path integral only conditionally convergent. The standard cure is Wick rotation: analytically continue to imaginary time, $t \to -i\tau$. The action picks up factors of $i$ that convert it into the Euclidean action $S_E$, and the integrand becomes a real, exponentially damped weight:

This has two enormous payoffs:

1. Convergence. The integrand is a genuine probability-like weight; field configurations with large action are exponentially suppressed, making the integral well-defined and amenable to numerical evaluation (this is the basis of lattice QFT).
2. Statistical-mechanics connection. $Z_E$ is mathematically identical to the partition function of a classical statistical system, with $S_E/\hbar$ playing the role of $\beta H$. Correlation lengths map to inverse masses, phase transitions to critical phenomena, and the renormalization group is shared between the two fields. (See Statistical Mechanics.)

Generating Functionals

The path integral becomes a calculational tool the moment we couple the field to an external source $J(x)$. Differentiating with respect to that source pulls down factors of the field, so a single functional encodes every correlation function at once.

The Generating Functional Z[J]

Add a linear source term $J(x)\phi(x)$ to the action:

Functional derivatives with respect to $J$ bring down fields, and setting $J=0$ leaves the time-ordered vacuum correlation functions (Green’s functions):

This is the central identity of the functional approach: all the physical content of the theory is packed into $Z[J]$, and any correlator is extracted by mechanical differentiation.

Connected Functional W[J]

The full Green’s functions contain redundant, disconnected pieces (processes happening independently in different regions). Taking the logarithm strips these away, leaving only connected correlators:

Physically, $W[J]$ generates exactly the diagrams that cannot be split into independent pieces — the ones that carry real interaction information.

The Effective Action

One Legendre transform further isolates the one-particle-irreducible (1PI) content — diagrams that stay connected when any single internal line is cut. Define the classical field $\phi_c = \delta W/\delta J$ and form:

$\Gamma[\phi_c]$ is the effective action. Its great virtues:

• Its derivatives are the 1PI vertex functions — the irreducible building blocks from which all diagrams are assembled.
• It includes all quantum corrections to the classical action, so extremizing $\Gamma$ (rather than $S$) gives the quantum equations of motion.
• Its minimum locates the true vacuum, making it the natural tool for analyzing spontaneous symmetry breaking: the effective potential $V_{\text{eff}}(\phi_c)$ is the part of $\Gamma$ with no derivatives, and one-loop corrections to it (the Coleman–Weinberg potential) can shift the vacuum away from the classical minimum.

Gaussian Integration: the Free Theory

For a free field the action is quadratic, and the path integral reduces to an infinite-dimensional Gaussian — which we can do exactly. Writing the quadratic form as $\phi K \phi$:

The two-point function is the inverse of the kinetic operator — which is precisely the Feynman propagator:

For the Klein–Gordon field, $K = -(\Box + m^2)$ and its inverse in momentum space is the familiar

This is the cornerstone of perturbation theory: the free theory is solved in closed form, and interactions are added as a controlled expansion around it.

Perturbation Theory

Realistic theories are interacting, and almost none are exactly solvable. The strategy is to split the Lagrangian into a free (quadratic) part we can integrate exactly and an interaction part we expand in powers of the small coupling.

Expanding Around the Free Theory

Write $\mathcal{L} = \mathcal{L}0 + \mathcal{L}{\text{int}}$. The interacting generating functional can be written by pulling the interaction outside the integral as a differential operator acting on the free functional:

Expanding the exponential generates an infinite series, each term containing some number of interaction vertices acting on the free propagators inside $Z_0[J]$. This series is the perturbative expansion, and each term corresponds to a Feynman diagram.

Wick’s Theorem

To evaluate a given term we must reduce a time-ordered product of many fields into propagators. Wick’s theorem does exactly this: it rewrites a time-ordered product as a sum over all possible pairwise contractions, where each contraction is a Feynman propagator:

Each full contraction of $n$ fields (with $n$ even) into $n/2$ propagators contributes a product like $D_F(x_1-x_2)\,D_F(x_3-x_4)\cdots$. The normal-ordered terms vanish between vacuum states, so only the fully contracted pieces survive in vacuum correlators. Wick’s theorem is the bookkeeping device that turns the abstract functional expansion into a concrete, finite set of propagator products at each order — precisely the lines of a Feynman diagram.

Organizing the Expansion

Two complementary expansions are in play:

• Coupling expansion. Powers of the coupling $g$ (or $\lambda$, $e$) count vertices; weak coupling means few vertices dominate.
• Loop expansion. Powers of $\hbar$ count independent loop momenta. The number of loops is $L = I - V + 1$ for $I$ internal lines and $V$ vertices, so tree diagrams ($L=0$) are the classical/leading result and loops are successive quantum corrections.

For QED the relevant small parameter is the fine-structure constant $\alpha = e^2/4\pi \approx 1/137$, which is why the perturbative series converges so well numerically — each additional loop costs roughly a factor of $\alpha$.

Feynman Diagrams

Feynman diagrams are not mere illustrations; they are a precise, one-to-one shorthand for the terms of the perturbation series. Each diagram translates, via the Feynman rules, into a specific mathematical expression contributing to an amplitude.

The Dictionary

graph LR
    A["External line"] -->|"incoming/outgoing particle"| B["wavefunction factor"]
    C["Internal line"] -->|"virtual particle"| D["propagator i/(p²−m²+iε)"]
    E["Vertex"] -->|"interaction"| F["coupling factor, e.g. −ieγ^μ"]
    G["Closed loop"] -->|"unfixed momentum"| H["integrate ∫d⁴k/(2π)⁴"]
    style A fill:#11998e,color:#fff
    style E fill:#38ef7d,color:#222

Reading a diagram:

• External lines represent the incoming and outgoing real particles; each carries a wavefunction factor (a spinor $u(p)$, $\bar u(p)$, $v(p)$, $\bar v(p)$ for fermions, or a polarization $\varepsilon^\mu$ for photons).
• Internal lines are virtual particles — off-shell intermediate states that need not satisfy $p^2 = m^2$. Each contributes a propagator.
• Vertices encode the interaction and carry the coupling. Momentum is conserved at every vertex.
• Loops carry an undetermined internal momentum that must be integrated over; these are the source of both quantum corrections and ultraviolet divergences.

Feynman Rules for QED

The interaction in quantum electrodynamics is the single term $\mathcal{L}{\text{int}} = -e\,\bar\psi\gamma^\mu\psi\,A\mu$, giving a remarkably compact set of rules.

Vertex factor (one photon, one incoming and one outgoing electron line):

Electron (fermion) propagator:

Photon propagator (Feynman gauge):

Closed fermion loop: include a factor of $(-1)$ and a trace over the Dirac indices around the loop.

Loop integration: integrate each unfixed loop momentum with $\displaystyle\int \frac{d^4k}{(2\pi)^4}$.

Assembling these factors according to the topology of the diagram, then summing over all diagrams at the desired order, yields the invariant amplitude $i\mathcal{M}$.

From Amplitude to Observable

The amplitude $\mathcal{M}$ feeds directly into measurable quantities. For a $2 \to 2$ scattering in the centre-of-mass frame, the differential cross section is

A Worked Amplitude

To make the machinery concrete, consider electron–muon scattering, $e^- \mu^- \to e^- \mu^-$, at lowest (tree) order in QED. Because the electron and muon are distinct particles, there is a single tree diagram: the two fermion lines exchange one virtual photon.

Step 1 — Identify the diagram. Incoming electron $p_1$ and muon $p_2$; outgoing electron $p_3$ and muon $p_4$. A virtual photon of momentum $q = p_1 - p_3$ connects the two vertices. Momentum conservation: $q = p_1 - p_3 = p_4 - p_2$.

Step 2 — Apply the rules. Write down the factors:

• Electron line: $\bar u(p_3)\,(-ie\gamma^\mu)\,u(p_1)$
• Muon line: $\bar u(p_4)\,(-ie\gamma^\nu)\,u(p_2)$
• Photon propagator: $\dfrac{-ig_{\mu\nu}}{q^2}$

Step 3 — Assemble the amplitude.

which simplifies to

Step 4 — Square and average over spins. Using the spin sums $\sum_s u(p)\bar u(p) = \not{p} + m$, the spin-averaged squared amplitude becomes a product of Dirac traces:

Evaluating the traces (using $\text{Tr}[\gamma^\mu\gamma^\nu] = 4g^{\mu\nu}$ and the four-gamma trace identity) and expressing the result in Mandelstam variables $s,t,u$ gives, in the high-energy limit $m_e, m_\mu \to 0$:

with $t = q^2$. Step 5 — Insert into the cross-section formula above to obtain $d\sigma/d\Omega$. This is the entire pipeline — diagram to rule to trace to observable — that underlies every QED prediction; adding the muon’s internal structure or moving to identical particles (Bhabha, Møller scattering) only changes which diagrams appear.

Loops & Functional Methods

Beyond tree level, diagrams contain closed loops with unconstrained internal momenta. These loops carry the genuine quantum corrections — and the divergences that motivate renormalization.

Loop Integrals

A one-loop correction requires integrating over the loop momentum. The QED electron self-energy, for instance, is

and the vertex correction is

These integrals typically diverge in the ultraviolet. Handling them — via dimensional regularization, counterterms, and the renormalization group — is the subject of the renormalization page. The vertex correction $\Lambda^\mu$ is also exactly what produces the famous anomalous magnetic moment, the electron $g-2$, agreeing with experiment to twelve digits.

Practical Techniques for Loops

A handful of standard tools turn loop integrals into tractable expressions:

• Feynman parameters. Combine denominators into a single quadratic via $\frac{1}{AB} = \int_0^1 dx\,[xA + (1-x)B]^{-2}$, then shift the loop momentum to complete the square.
• Wick rotation. Rotate $k^0 \to ik^0_E$ to turn the Minkowski integral into a convergent Euclidean one.
• Dimensional regularization. Compute in $d = 4 - \varepsilon$ dimensions, isolating divergences as $1/\varepsilon$ poles while preserving gauge invariance.
• Integration by parts (IBP). Use $\int d^d k\,\partial_\mu[k^\mu f(k)] = 0$ to reduce a large family of integrals to a small basis of master integrals — the workhorse of modern multi-loop calculations.

Functional Identities: Ward & Schwinger–Dyson

The functional formalism does more than reproduce diagrams; it yields exact, all-orders relations.

Schwinger–Dyson equations are the quantum equations of motion. They follow from the fact that a path integral is invariant under a shift of the integration variable, $\phi \to \phi + \delta\phi$:

They form an infinite, coupled hierarchy relating $n$-point to $(n+1)$-point functions — the non-perturbative skeleton of the theory.

Ward–Takahashi identities are the special case enforced by a symmetry of the action. In QED, gauge invariance ties the vertex function to the electron propagator:

These identities guarantee that gauge invariance survives renormalization (e.g. relating the charge and field renormalization constants), and they protect the photon from acquiring a mass. They are a powerful consistency check on any calculation.

Gauge Fixing and BRST

Naively, the photon path integral diverges because gauge-equivalent configurations are summed redundantly. The Faddeev–Popov procedure factors out this redundancy by inserting a gauge-fixing condition, at the cost of introducing ghost fields (anticommuting scalars) in non-abelian theories. The whole gauge-fixed action then possesses a residual rigid fermionic symmetry — BRST symmetry — which encodes the original gauge invariance at the quantum level and guarantees unitarity of physical amplitudes. BRST is the modern, systematic way to quantize Yang–Mills theories within the path integral.

Effective Field Theory as a Calculational Tool

Effective field theory (EFT) is less a specific model than a way of thinking that makes QFT usable: you never need the complete theory of everything to compute a low-energy process — you only need the right degrees of freedom at the scale you care about.

The Core Idea

If a process occurs at energy $E$ well below some heavy scale $\Lambda$, the heavy physics cannot be produced directly. Its effects appear only indirectly, suppressed by powers of $E/\Lambda$. EFT makes this systematic: integrate out the heavy fields and write the most general local Lagrangian for the light fields consistent with the symmetries, organized as an expansion in $1/\Lambda$:

Each operator $\mathcal{O}_i$ of mass dimension $n_i$ comes with a dimensionless Wilson coefficient $c_i$. Higher-dimension operators are suppressed by more powers of $\Lambda$, so at any desired accuracy only finitely many operators matter. This is power counting, and it is what makes a non-renormalizable theory perfectly predictive at low energies.

Why It Works

• Decoupling. Heavy particles decouple from low-energy physics, leaving only their imprint in the Wilson coefficients. You can compute without ever resolving the heavy sector.
• Predictivity with a cutoff. A theory with infinitely many couplings is still predictive: truncating the $1/\Lambda$ expansion at finite order leaves a finite, controlled error of order $(E/\Lambda)^{n}$.
• Matching and running. Wilson coefficients are fixed by matching the EFT to the full theory at the scale $\Lambda$, then run down to the experimental scale using the renormalization group — automatically resumming large logarithms $\ln(\Lambda/E)$.

EFT in Practice

The classic example is Fermi’s four-fermion theory of beta decay: long before the $W$ boson was known, the weak interaction could be described by a contact interaction with coupling $G_F \sim g^2/m_W^2$. The full electroweak theory “matches onto” this EFT in the limit $q^2 \ll m_W^2$, with the heavy $W$ propagator $\sim 1/(q^2 - m_W^2) \to -1/m_W^2$ shrinking to a point. Crucially, the entire Standard Model is itself almost certainly an EFT — the leading terms of a more complete theory whose new physics lives at a scale we have not yet reached. (See Effective Field Theory as a Calculational Tool below.)

Key Takeaways

• The path integral is the master tool. Summing $e^{iS/\hbar}$ over all field configurations reproduces all of QFT and connects directly to statistical mechanics after Wick rotation.
• Generating functionals automate correlators. $Z[J]$ gives all Green’s functions, $W[J]$ the connected ones, and the effective action $\Gamma[\phi_c]$ the 1PI vertices and the quantum vacuum.
• Diagrams are the perturbation series. Wick’s theorem turns the expansion into propagator products; each Feynman diagram is one term, with rules for lines, vertices, and loops.
• Amplitudes become observables. Squaring $\mathcal{M}$, averaging over spins, and inserting phase space yields cross sections and decay rates an experiment can test.
• Functional identities are exact. Schwinger–Dyson and Ward–Takahashi relations hold to all orders, protecting gauge invariance and constraining every calculation.
• EFT lets you calculate at the right scale. Integrate out heavy physics, organize by powers of $E/\Lambda$, and compute predictively without knowing the complete theory.

See Also

• Quantum Field Theory — fields, gauge symmetry, the Standard Model, and renormalization.
• Quantum Mechanics — the non-relativistic foundation the path integral generalizes.
• Statistical Mechanics — the Euclidean path integral is a statistical partition function.
• Relativity — special relativity makes the action Lorentz-invariant.
• String Theory — worldsheet path integrals extend these methods to extended objects.
• Physics Hub — browse all physics topics.