Path-integral formulation in QFT

The path-integral formulation expresses quantum amplitudes as a weighted sum over all possible field configurations (or particle trajectories), with weight $e^{i S /ℏ}$ where $S$ is the classical action. In QFT it generates correlation functions, the S-matrix, and the partition function from a single functional integral $Z = \int D ϕ e^{i S [ϕ]}$ . It applies to virtually all of perturbative and non-perturbative quantum field theory, gauge theory, statistical mechanics, and string theory. Conceived by Dirac (1933), developed by Feynman (1948), and made rigorous in Euclidean form by Wiener, Kac, Symanzik, and Glimm–Jaffe.

The path-integral formulation replaces the operator-and-Hilbert-space machinery of canonical quantization with a sum over histories: every transition amplitude is an integral over all configurations connecting initial and final data, each weighted by the phase $e^{i S /ℏ}$ . In quantum field theory the "histories" are entire spacetime field configurations $ϕ (x)$ , and the central object is the generating functional $Z [J]$ , from which all Green's functions follow by functional differentiation. The formalism makes symmetries, gauge invariance, and the classical limit manifest, and it is the natural language for non-perturbative phenomena (instantons, lattice gauge theory) and for deriving Feynman rules systematically. A surprising fact: the integral is mathematically ill-defined as a Lebesgue measure (there is no translation-invariant measure on infinite-dimensional space), yet after Wick rotation to imaginary time it becomes a genuine, rigorously controlled probability measure — the bridge between QFT and statistical mechanics.

The Core Insight

A quantum amplitude is not a single trajectory but a coherent superposition over all trajectories. The classical path is merely the stationary point of the phase; quantum corrections are the spreading of the phase around it. Encode the entire theory in one functional $Z [J]$ , and quantization becomes an integration problem rather than an operator-ordering problem.

Formal Statement

Quantum-mechanical path integral

For a single particle with Lagrangian $L (q, \overset{q}{˙})$ , the propagator (transition amplitude) between position eigenstates is

⟨ q_{f}, t_{f} ∣ q_{i}, t_{i} ⟩ = \int_{q (t_{i}) = q_{i}}^{q (t_{f}) = q_{f}} D q (t) exp (\frac{i}{ℏ} \int_{t_{i}}^{t_{f}} L (q (t), \overset{q}{˙} (t)) d t) .

⟳Insert

N - 1

complete sets of position eigenstates between

N

infinitesimal time evolutions

e^{- i H Δ t /ℏ}

, evaluate each short-time kernel with the Trotter/Lie product formula, and take

N \to \infty

with

D q = lim_{N} \prod_{k} (m /2 π i ℏΔ t)^{1/2} d q_{k}

Here $D q (t)$ denotes the formal measure over all continuous paths with fixed endpoints, and $S = \int L d t$ is the classical action.

Field-theory generating functional

For a scalar field $ϕ (x)$ with action $S [ϕ] = \int d^{4} x L (ϕ, \partial_{μ} ϕ)$ coupled to an external source $J (x)$ , the generating functional of correlation functions is

Z [J] = \int D ϕ exp (\frac{i}{ℏ} (S [ϕ] + \int d^{4} x J (x) ϕ (x))) .

⟳Promote the QM time-slicing to a spacetime lattice; the configuration variable at each lattice site

ϕ (x_{n})

is integrated over

R

, and the product

\prod_{n} d ϕ (x_{n})

defines

D ϕ

in the continuum limit.

Every symbol: $ϕ$ is the integration field, $J$ the classical source, $S$ the action, $L$ the Lagrangian density, $d^{4} x = d t d^{3} x$ the spacetime measure.

:::definition[Correlation functions from $Z [J]$ ] The time-ordered $n$ -point functions are obtained by functional differentiation:

⟨ 0 ∣ T ϕ (x_{1}) \dots ϕ (x_{n}) ∣ 0 ⟩ = \frac{1}{Z [ 0 ]} \frac{( - i ℏ ) ^{n} δ ^{n} Z [ J ]}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0} .

⟳Each functional derivative

δ / δ J (x)

brings down a factor

\frac{i}{ℏ} ϕ (x)

from the exponent in [2]; dividing by

Z [0]

cancels disconnected vacuum bubbles.

Equivalence to canonical quantization

For a Hamiltonian $H = \frac{p ^{2}}{2 m} + V (q)$ (and more generally for Lagrangians quadratic in velocities with metric-independent measure), the path-integral propagator [1] coincides with the canonical kernel $⟨ q_{f} ∣ e^{- i H (t_{f} - t_{i}) /ℏ} ∣ q_{i} ⟩$ , and the Green's functions [3] equal the Wightman/time-ordered functions of the canonically quantized theory.

Time-slicing derivation of the path integral

Step 1 — Subdivide time. Split

[t_{i}, t_{f}]

into

N

intervals of width

ϵ = (t_{f} - t_{i}) / N

and write

e^{- i H (t_{f} - t_{i}) /ℏ} = (e^{- i H ϵ /ℏ})^{N} .

⟳Exact operator identity from the semigroup property of time evolution.

Step 2 — Insert resolutions of identity. Between each factor insert

1 = \int d q_{k} ∣ q_{k} ⟩ ⟨ q_{k} ∣

and

1 = \int \frac{d p _{k}}{2 π ℏ} ∣ p_{k} ⟩ ⟨ p_{k} ∣

, giving a product of matrix elements

⟨ q_{k + 1} ∣ e^{- i H ϵ /ℏ} ∣ q_{k} ⟩

. Step 3 — Evaluate the short-time kernel. To

O (ϵ)

the Baker–Campbell–Hausdorff corrections are higher order, so

⟨ q_{k + 1} ∣ e^{- i H ϵ /ℏ} ∣ q_{k} ⟩ = \int \frac{d p _{k}}{2 π ℏ} exp [\frac{i}{ℏ} (p_{k} (q_{k + 1} - q_{k}) - ϵH (p_{k}, \overset{q}{ˉ}_{k}))] .

⟳Use

⟨ q ∣ p ⟩ = e^{i pq /ℏ}

and expand

e^{- i H ϵ /ℏ} \approx 1 - i H ϵ /ℏ

; the Weyl-ordered midpoint

\overset{q}{ˉ}_{k}

ensures correctness to

O (ϵ)

Step 4 — Gaussian $p$ -integration. For

H = p^{2} /2 m + V

, the

p_{k}

integral is Gaussian:

\int \frac{d p _{k}}{2 π ℏ} e^{\frac{i}{ℏ} (p_{k} Δ q_{k} - ϵ p_{k}^{2} /2 m)} = (\frac{m}{2 π i ℏ ϵ})^{1/2} exp [\frac{i}{ℏ} \frac{m}{2} (\frac{Δ q _{k}}{ϵ})^{2} ϵ] .

⟳Complete the square in

p_{k}

; the Fresnel integral

\int d p e^{ia p^{2}} = π / (- ia)

supplies the normalization factor.

Step 5 — Assemble and take the limit. The product of exponents builds the discretized action

\sum_{k} [\frac{m}{2} (Δ q_{k} / ϵ)^{2} - V (\overset{q}{ˉ}_{k})] ϵ \to \int L d t

, and the product of measures defines

D q

. Taking

N \to \infty

recovers [1].

■

Richard P. Feynman | 1918–1988

Feynman, building on a cryptic 1933 remark by Dirac that $⟨ q_{f} ∣ q_{i} ⟩$ "corresponds to" $e^{i S /ℏ}$ , developed the full sum-over-histories formulation in his 1942 Princeton thesis and 1948 Reviews of Modern Physics paper. He recognized that the classical limit emerges by stationary phase, that interference among paths reproduces quantum mechanics, and that the formalism generalizes seamlessly to fields, yielding the diagrammatic rules now bearing his name. His geometric, intuition-driven approach complemented the operator methods of Schwinger and the integral equations of Tomonaga, for which the three shared the 1965 Nobel Prize.

Building Intuition

Imagine launching a particle from $q_{i}$ at $t_{i}$ and asking for the amplitude to arrive at $q_{f}$ . Classically only one path matters — the one extremizing $S$ . Quantum mechanically the particle "tries" every path, each contributing a unit-modulus phasor $e^{i S /ℏ}$ . Where the action varies rapidly between neighboring paths, phasors point in all directions and cancel. Near the classical path, $S$ is stationary, so neighboring phasors align and add coherently. Thus the classical trajectory is the locus of constructive interference, and $ℏ$ sets the scale over which this coherence persists: as $ℏ \to 0$ , only the stationary path survives (stationary-phase approximation), recovering classical mechanics. In QFT the same picture holds with whole field histories replacing single-particle paths; the vacuum is the dominant stationary configuration, and particles are quantized fluctuations around it.

Path integral: interference of histories between fixed endpoints. The classical path (gold) is the stationary point of the action where neighboring phases reinforce.

Worked Examples

Free-particle propagator

Given: A free particle of mass

m

V = 0

, propagating from

q_{i}

q_{f}

in time

T = t_{f} - t_{i}

. Step 1 — Identify the action. The classical path is a straight line; its action is

S_{c l} = \frac{m ( q _{f} - q _{i} ) ^{2}}{2 T}

. Step 2 — Gaussian fluctuations. Write

q (t) = q_{c l} (t) + η (t)

with

η (t_{i}) = η (t_{f}) = 0

. Since

S

is quadratic, fluctuations decouple and give a factor independent of endpoints. Evaluating the discretized product of Fresnel integrals (or the Van Vleck determinant) yields the prefactor

(m /2 π i ℏ T)^{1/2}

. Step 3 — Assemble.

K (q_{f}, q_{i}; T) = (\frac{m}{2 π i ℏ T})^{1/2} exp (\frac{im ( q _{f} - q _{i} ) ^{2}}{2ℏ T}) .

⟳For purely quadratic actions the path integral equals

e^{i S_{c l} /ℏ}

times the fluctuation determinant; the determinant for the free particle is the chained Gaussian product giving the displayed normalization.

Boxed answer:

K = \frac{m}{2 π i ℏ T} e^{im (q_{f} - q_{i})^{2} /2ℏ T}

, with units of

[length]^{- 1}

in 1D (so that

\int K d q_{i} = 1

in the appropriate sense). Numerical check: For

m = 1 kg

q_{f} - q_{i} = 1 m

T = 1 s

, the phase is

S_{c l} /ℏ = 0.5/ (1.055 \times 1 0^{- 34}) \approx 4.7 \times 1 0^{33}

rad — an enormously rapid oscillation, illustrating why macroscopic objects follow the single classical path.

Euclidean partition function of the harmonic oscillator

Given: Oscillator

H = \frac{p ^{2}}{2 m} + \frac{1}{2} m ω^{2} q^{2}

. Wick-rotate

t \to - i τ

and impose periodicity

q (β ℏ) = q (0)

to compute

Z = Tr e^{- β H}

. Step 1 — Euclidean action. Under

t \to - i τ

i S \to - S_{E}

with

S_{E} = \int_{0}^{β ℏ} d τ [\frac{m}{2} \overset{q}{˙}^{2} + \frac{1}{2} m ω^{2} q^{2}]

. Step 2 — Fourier modes. Expand

q (τ) = \sum_{n} c_{n} e^{i ν_{n} τ}

with Matsubara frequencies

ν_{n} = 2 π n / (β ℏ)

. The action diagonalizes:

S_{E} = \frac{m β ℏ}{2} \sum_{n} (ν_{n}^{2} + ω^{2}) ∣ c_{n} ∣^{2}

. Step 3 — Gaussian integral over modes.

Z = n \prod \frac{1}{ν _{n}^{2} + ω ^{2}} \propto \frac{1}{2 sinh ( β ℏ ω /2 )} .

⟳Each mode contributes a Gaussian integral

\propto (ν_{n}^{2} + ω^{2})^{- 1/2}

; use the product formula

\prod_{n \geq 1} (1 + x^{2} / n^{2}) = sinh (π x) / (π x)

to resum, fixing the

n = 0

and normalization factors by the

ω \to 0

free limit.

Boxed answer:

Z = \frac{1}{2 sinh ( β ℏ ω /2 )} = \frac{e ^{- β ℏ ω /2}}{1 - e ^{- β ℏ ω}}

. Numerical check: This reproduces

Z = \sum_{n = 0}^{\infty} e^{- β ℏ ω (n + 1/2)}

exactly, confirming the path integral recovers the canonical spectrum

E_{n} = ℏ ω (n + \frac{1}{2})

. For

β ℏ ω = 1

Z = 1/ (2 sinh 0.5) = 0.958

Common Misconceptions

The measure $D ϕ$ is not a true measure in Lorentzian (real-time) QFT. There exists no countably additive, translation-invariant measure on an infinite-dimensional space, and the oscillatory integrand

e^{i S /ℏ}

does not converge absolutely. The real-time path integral is a formal, perturbatively-defined object. Rigor is achieved only after Wick rotation to Euclidean signature, where

e^{i S} \to e^{- S_{E}}

becomes a damped, genuinely integrable weight (the Wiener/Osterwalder–Schrader construction). Treating

D ϕ e^{i S}

as a literal Lebesgue integral leads to false manipulations. Operator ordering is hidden in the measure and the midpoint prescription. The naive replacement "

L = p \overset{q}{˙} - H

" obscures that the short-time kernel [5] requires a specific discretization (Weyl/midpoint) to reproduce the correct quantum Hamiltonian. For Lagrangians with velocity-dependent or position-dependent masses (curved configuration space), additional

O (ℏ^{2})

terms appear and the measure is non-trivial — a fact silently assumed away in the flat-space examples. A path integral that "looks" Gaussian but is not the canonical answer. Consider quantizing a particle on a circle of radius

R

. Naively integrating over

q \in R

with

e^{i S /ℏ}

gives the free-line propagator [7]. But the correct amplitude must respect the topology

q \sim q + 2 π R

, so one must sum over winding sectors:

K_{circle} = w \in Z \sum (\frac{m}{2 π i ℏ T})^{1/2} exp (\frac{im ( q _{f} - q _{i} + 2 π R w ) ^{2}}{2ℏ T}) .

⟳The covering-space construction lifts the circle to

R

; each homotopy class of paths is labeled by its winding number

w

and contributes independently.

Omitting the sum gives a wrong spectrum (continuous instead of $E_{n} \propto n^{2} / R^{2}$ ). This shows that " $\int D q e^{i S}$ " is incomplete without specifying the topology of configuration space and the sum over homotopy classes — a subtlety invisible in the formal expression.

Connections

The path integral is the connective tissue of modern theoretical physics. Via Wick rotation it maps onto the partition function of Statistical Mechanics, so a $d$ -dimensional QFT equals a $d$ -dimensional classical statistical system — the foundation of Lattice Gauge Theory. Functional differentiation of $Z [J]$ generates the Feynman Diagrams of Perturbation Theory in QFT. The Legendre transform of $W = ln Z$ yields the Effective Action $Γ [ϕ]$ , organizing the loop expansion and spontaneous symmetry breaking. For gauge fields the naive measure overcounts, requiring the Faddeev–Popov Procedure and ghosts. Non-perturbative saddle points of the Euclidean action are Instantons, and the formalism underlies the Renormalization Group through Wilsonian integration of momentum shells.

Real-World Applications

Lattice QCD — The Euclidean path integral on a discretized spacetime, evaluated by Monte Carlo importance sampling of $e^{- S_{E}}$ , gives first-principles hadron masses, the QCD phase diagram, and matrix elements feeding particle-physics experiments (e.g., $g - 2$ of the muon).
Quantum field theory of the Standard Model — All scattering cross sections at the LHC are computed from $Z [J]$ via Feynman rules derived from the path integral, including electroweak and QCD radiative corrections.
Condensed matter / many-body theory — Coherent-state and Grassmann path integrals describe superconductivity (BCS), the fractional quantum Hall effect (Chern–Simons), and quantum phase transitions.
Quantum tunneling and decay rates — Instanton (bounce) solutions of the Euclidean path integral yield the exponential rate $Γ \sim e^{- S_{E} /ℏ}$ for false-vacuum decay, relevant to cosmological phase transitions and electroweak vacuum stability.
Stochastic processes & finance — The Wiener-measure path integral (Feynman–Kac formula) prices options (Black–Scholes as imaginary-time Schrödinger) and solves Fokker–Planck equations.
Quantum gravity & cosmology — The Hartle–Hawking "no-boundary" wavefunction and Euclidean gravity path integrals estimate the entropy of black holes via the Gibbons–Hawking saddle.
Worldline & string theory — First-quantized amplitudes are literal path integrals over particle worldlines and string worldsheets, the defining formulation of string perturbation theory.

Historical Development

1923–1933 — Wiener & Dirac. Norbert Wiener constructs the rigorous measure on Brownian paths (Wiener measure). Dirac's 1933 paper The Lagrangian in Quantum Mechanics notes the formal correspondence $⟨ q_{t + d t} ∣ q_{t} ⟩ \sim e^{i L d t /ℏ}$ .
1942 — Feynman's thesis. Princeton dissertation under Wheeler develops the sum-over-histories principle, motivated by the Wheeler–Feynman absorber theory.
1948 — Feynman's RMP paper. Space-Time Approach to Non-Relativistic Quantum Mechanics publishes the full path-integral formulation and its equivalence to the Schrödinger equation.
1949 — Feynman diagrams. Space-Time Approach to Quantum Electrodynamics derives diagrammatic rules; Dyson proves equivalence with Schwinger–Tomonaga operator methods.
1949 — Feynman–Kac formula. Mark Kac recasts the Euclidean path integral as an expectation over Wiener measure, giving the first rigorous handle.
1954 — Matthews–Salam. Functional-integral formulation of QFT generating functionals and Green's functions is systematized.
1967 — Faddeev–Popov. Gauge-fixing the path integral via ghost determinants makes non-Abelian gauge theory quantizable.
1972–1975 — Symanzik, Nelson, Osterwalder–Schrader. Euclidean constructive QFT: the OS axioms specify exactly when a Euclidean measure reconstructs a Lorentzian Wightman theory.
1974 — Wilson lattice gauge theory. Discretized Euclidean path integral enables non-perturbative QCD and confinement studies; Glimm–Jaffe build rigorous low-dimensional models.

Subtleties & Limits

A fully rigorous, non-perturbative definition of the interacting path integral in four spacetime dimensions remains open: constructive QFT has succeeded for $ϕ^{4}$ in $d \leq 3$ and for gauge theories only on the lattice, while continuum 4D Yang–Mills with a mass gap is a Clay Millennium Problem. The Lorentzian (real-time) measure has no rigorous status outside perturbation theory, and Wick rotation can fail when the Euclidean action is unbounded below — most acutely in quantum gravity, where the Einstein–Hilbert action's conformal mode makes $e^{- S_{E}}$ diverge, motivating complex contour (Picard–Lefschetz / Lorentzian) approaches and the "conformal factor problem." Even at the saddle level, the sum over topologies and over Gribov copies in gauge theory is poorly understood, and the proper measure for theories with constraints, anomalies, or chiral fermions (lattice doubling, the Nielsen–Ninomiya theorem) continues to drive active research.

Path-integral formulation in QFT

The Core Insight

Formal Statement

Quantum-mechanical path integral

Field-theory generating functional

Equivalence to canonical quantization

Time-slicing derivation of the path integral

Richard P. Feynman | 1918–1988

Building Intuition

Path integral: interference of histories between fixed endpoints. The classical path (gold) is the stationary point of the action where neighboring phases reinforce.

Worked Examples

Free-particle propagator

Euclidean partition function of the harmonic oscillator

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads