Path Integral Formulation — Mathematics Knowledge System

Overview

The path integral formulation, developed by Richard Feynman in the 1940s, provides an alternative to the operator-based (Heisenberg/Schrödinger) approaches to quantum mechanics. Instead of evolving a state vector via the Schrödinger equation, it computes quantum amplitudes by summing over all possible classical paths connecting two spacetime points, each weighted by a phase factor exp(iS/ℏ) where S is the classical action. This 'sum over histories' perspective yields powerful computational tools, unifies quantum mechanics with statistical mechanics via Wick rotation, and forms the cornerstone of modern quantum field theory.

Intuition

Imagine a particle traveling from point A to point B. In classical mechanics it follows the single path of least action. In quantum mechanics, the particle 'explores' every possible path simultaneously. The probability amplitude is obtained by adding up contributions from all paths, but each path contributes not with a real weight but with a complex phase e^(iS/ℏ). Paths near the classical trajectory have slowly varying phases that add constructively; wildly non-classical paths have rapidly oscillating phases that cancel. In the ℏ → 0 limit, only the classical path survives — recovering classical mechanics.

Formal Definition

Definition

For a particle with Lagrangian L(q, q̇, t), the propagator (transition amplitude) from position qᵢ at time tᵢ to qf at time tf is defined as the functional integral over all paths q(t) satisfying the boundary conditions.

K(q_f, t_f; q_i, t_i) = \int_{q(t_i)=q_i}^{q(t_f)=q_f} \mathcal{D}[q(t)]\, e^{\frac{i}{\hbar} S[q]}

Feynman propagator

S[q] = \int_{t_i}^{t_f} L(q(t), \dot{q}(t), t)\, dt

Classical action functional

\mathcal{D}[q] = \lim_{N\to\infty} \prod_{k=1}^{N-1} \frac{dq_k}{C}

Discretised path-integral measure (heuristic)

K(q_f, t_f; q_i, t_i) = \langle q_f | e^{-iH(t_f - t_i)/\hbar} | q_i \rangle

Equivalence to operator formalism

Notation

Notation	Meaning
$S[q]$	Action functional of a path q(t)
$\mathcal{D}[q]$	Formal path-integral measure
$K(q_f,t_f;q_i,t_i)$	Quantum propagator / kernel
$Z = \int \mathcal{D}[q]\,e^{-S_E[q]/\hbar}$	Euclidean partition function (after Wick rotation)

Properties

Composition (Markov) property

K(q_f, t_f; q_i, t_i) = \int K(q_f, t_f; q_m, t_m)\, K(q_m, t_m; q_i, t_i)\, dq_m

Semiclassical approximation

K \approx A\, e^{iS_{\mathrm{cl}}/\hbar} \quad (\hbar \to 0)

Wick rotation to statistical mechanics

Z_{\mathrm{stat}} = \mathrm{Tr}\,e^{-\beta H} = \int \mathcal{D}[q]\,e^{-S_E[q]/\hbar}\big|_{\beta = (k_B T)^{-1}}

Worked Examples

1
For a free particle L = mq̇²/2, so S[q] = (m/2)∫q̇² dt. Discretise the time interval into N steps of size ε = (tf-tᵢ)/N.
$S_N = \frac{m}{2\varepsilon}\sum_{k=0}^{N-1}(q_{k+1}-q_k)^2$
2
The resulting Gaussian integrals over intermediate positions q₁, …, q_{N-1} can be evaluated successively using the standard result ∫e^{-ax²} dx = √(π/a).
$\int_{-\infty}^{\infty} e^{-a(q-q')^2} e^{-a(q'-q'')^2} dq' = \sqrt{\frac{\pi}{2a}}\, e^{-a(q-q'')^2/2}$
3
After N-1 Gaussian integrations the result is the exact free-particle propagator.
$K_0(q_f, t_f; q_i, t_i) = \sqrt{\frac{m}{2\pi i \hbar (t_f - t_i)}} \exp\!\left(\frac{im(q_f - q_i)^2}{2\hbar(t_f - t_i)}\right)$

✓ Answer

The free-particle propagator is the Gaussian kernel above, confirming the path-integral reproduces the Schrödinger-equation result.

Practice Problems

Hardfree response

Show that the harmonic oscillator propagator obtained from the path integral matches the known result K(qf, tf; qᵢ, tᵢ) = √(mω / 2πiℏ sin ωT) exp(imω[(qf² + qᵢ²)cos ωT - 2qf qᵢ] / 2ℏ sin ωT) where T = tf - tᵢ.

Hardfree response

Describe how the Euclidean path integral is related to the canonical partition function of statistical mechanics.

Common Mistakes

Common Mistake

Believing the path integral measure 𝒟[q] is a rigorous mathematical measure on all paths

In Minkowski signature the path integral measure is not rigorously defined as a measure in the functional-analysis sense. The Euclidean version (Wiener measure) is rigorously defined. The Minkowski integral is treated as a formal object justified by its correct results.

Common Mistake

Thinking only smooth paths contribute

The dominant paths in the Wiener/Euclidean measure are nowhere-differentiable (Brownian) paths. The classical smooth path contributes at the level of the saddle-point approximation but is a set of measure zero.

Quiz

In the path integral, the classical path corresponds to:

The path with the largest actionThe path where the action is stationary (δS = 0)The path with the shortest lengthThe path with the fastest travel time

Wick rotation transforms the path integral into:

A Fourier transformA Laplace transformA Euclidean path integral related to the statistical partition functionA Legendre transform

Historical Background

Dirac observed in 1933 that the quantum mechanical transition amplitude is related to exp(iS/ℏ) for the classical action S. Feynman, inspired by this remark during his PhD work under Wheeler, developed the full formalism in his 1948 paper. The rigorous mathematical underpinning — Wiener measure for the Euclidean (imaginary-time) version — had been established earlier by Norbert Wiener in the theory of Brownian motion. The Minkowski-signature path integral remains formally defined rather than rigorously constructed in full generality.

1933
Dirac notes the role of exp(iS/ℏ) in quantum amplitudes
Paul Dirac
1948
Feynman publishes the path-integral formulation of quantum mechanics
Richard Feynman
1965
Feynman and Hibbs publish the textbook treatment including applications
Richard Feynman, Albert Hibbs
1974
Path integrals become central to lattice gauge theory (Wilson)
Kenneth Wilson

Summary

The path integral represents the quantum propagator as a sum over all paths weighted by exp(iS[q]/ℏ).
In the semiclassical limit ℏ → 0 the stationary-phase approximation recovers the classical trajectory and WKB formula.
Wick rotation t → -iτ maps the Minkowski path integral to a Euclidean one, connecting it to the statistical-mechanics partition function.
The formalism is the foundation of quantum field theory; Feynman diagrams arise from perturbative expansion of the path integral.

References

BookFeynman, R.P. & Hibbs, A.R. — Quantum Mechanics and Path Integrals (1965), McGraw-Hill
BookZinn-Justin, J. — Quantum Field Theory and Critical Phenomena, 4th ed. (2002), Oxford University Press
WebsiteWikipedia — Path integral formulation
WebsitenLab — path integral