Feynman's Path Integrals

⚡ Key Takeaways

Feynman's Path Integral reformulates quantum mechanics by assigning a complex amplitude $e^{iS[x]/\hbar}$ to every conceivable path a system can take between two configurations, then summing over all of them. The probability amplitude to go from $A$ to $B$ equals a weighted "sum over histories." It applies to non-relativistic QM, quantum field theory, statistical mechanics, and beyond. Introduced by Richard Feynman (1948), building on a 1933 remark by Paul Dirac.

💡 The Core Insight

Quantum amplitudes are obtained by superposing contributions from every path, each contributing a unit-modulus phase determined by its classical action measured in units of $\hbar$ . Where the action varies rapidly between neighboring paths, phases cancel; where it is stationary (the classical trajectory), phases reinforce. Classical mechanics is thus the constructive interference skeleton of quantum mechanics.

DefinitionThe Propagator as a Path Integral

The quantum-mechanical propagator (kernel) for a single particle of mass $m$ in potential $V(x)$ is

1

\quad K(x_b,t_b;x_a,t_a)=\int_{x(t_a)=x_a}^{x(t_b)=x_b}\mathcal{D}x(t)\;\exp\!\left(\frac{i}{\hbar}S[x(t)]\right)

Derivation hint: Insert complete position-basis resolutions of identity at infinitely many intermediate times into $\langle x_b|e^{-iH(t_b-t_a)/\hbar}|x_a\rangle$ and take the continuum limit. where the classical action is

2

\quad S[x(t)]=\int_{t_a}^{t_b} L\big(x,\dot{x},t\big)\,dt=\int_{t_a}^{t_b}\!\left(\tfrac{1}{2}m\dot{x}^2-V(x)\right)dt

Derivation hint: $L=T-V$ is the Lagrangian; the action is its time integral along a trajectory. Symbols: $x_a,x_b$ are initial/final positions; $t_a,t_b$ the corresponding times; $\hbar$ is the reduced Planck constant; $L$ the Lagrangian; $\dot{x}=dx/dt$ ; $\mathcal{D}x(t)$ denotes the (formal) functional measure over all paths with the fixed endpoints. The propagator gives the wavefunction's evolution via

3

\quad \psi(x_b,t_b)=\int_{-\infty}^{\infty}K(x_b,t_b;x_a,t_a)\,\psi(x_a,t_a)\,dx_a

Derivation hint: $K$ is the matrix element of the time-evolution operator $U(t_b,t_a)$ ; act on the initial state and insert $\int|x_a\rangle\langle x_a|dx_a=\mathbb{1}$ .

DefinitionThe Time-Sliced (Lattice) Definition of the Measure

The functional integral is defined as the $N\to\infty$ limit of a finite-dimensional integral. Partition $[t_a,t_b]$ into $N$ slices of width $\epsilon=(t_b-t_a)/N$ , with intermediate points $x_0=x_a,\,x_1,\dots,x_{N}=x_b$ :

4

\quad K=\lim_{N\to\infty}\left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2}\!\int\!\prod_{k=1}^{N-1}dx_k\;\exp\!\left[\frac{i}{\hbar}\sum_{j=0}^{N-1}\epsilon\!\left(\frac{m(x_{j+1}-x_j)^2}{2\epsilon^2}-V(x_j)\right)\right]

Derivation hint: Each short-time kernel $\langle x_{j+1}|e^{-iH\epsilon/\hbar}|x_j\rangle$ is evaluated to $O(\epsilon)$ using the Trotter splitting $e^{-iH\epsilon/\hbar}\approx e^{-iT\epsilon/\hbar}e^{-iV\epsilon/\hbar}$ and a Gaussian momentum integral; the prefactor is the normalization of each Gaussian. The prefactor $(m/2\pi i\hbar\epsilon)^{N/2}$ is the precise content of the otherwise symbolic $\mathcal{D}x$ .

TheoremEquivalence to the Schrödinger Equation

The kernel $K$ defined by [4] satisfies, as a function of $(x_b,t_b)$ , the time-dependent Schrödinger equation

5

\quad i\hbar\,\frac{\partial K}{\partial t_b}=-\frac{\hbar^2}{2m}\frac{\partial^2 K}{\partial x_b^2}+V(x_b)\,K,\qquad \lim_{t_b\to t_a}K=\delta(x_b-x_a)

Derivation hint: Expand the single-slice convolution [3] to first order in $\epsilon$ and second order in $\eta=x_b-x'$ ; the Gaussian moments $\langle\eta\rangle=0$ , $\langle\eta^2\rangle=i\hbar\epsilon/m$ generate the kinetic term, while the $V$ term comes directly.

ProofFrom the Operator Formalism to the Sum over Paths

Step 1 — Split the evolution operator. The amplitude is $K=\langle x_b|e^{-i\hat H(t_b-t_a)/\hbar}|x_a\rangle$ with $\hat H=\hat p^2/2m+V(\hat x)$ . Write $e^{-i\hat H T/\hbar}=\big(e^{-i\hat H\epsilon/\hbar}\big)^N$ with $\epsilon=T/N$ .

Step 2 — Trotter factorization. Since $[\hat T,\hat V]\neq 0$ , use

e^{-i\hat H\epsilon/\hbar}=e^{-i\hat T\epsilon/\hbar}e^{-i\hat V\epsilon/\hbar}+O(\epsilon^2),

whose product over $N$ slices converges (Trotter–Kato theorem) as $N\to\infty$ .

Step 3 — Insert position resolutions. Place $\int dx_k\,|x_k\rangle\langle x_k|=\mathbb{1}$ between each factor, giving $N-1$ integrals over $x_1,\dots,x_{N-1}$ .

Step 4 — Evaluate one slice. Insert a momentum resolution to handle $\hat T$ :

\langle x_{j+1}|e^{-i\hat T\epsilon/\hbar}|x_j\rangle=\int\frac{dp}{2\pi\hbar}\,e^{ip(x_{j+1}-x_j)/\hbar}e^{-ip^2\epsilon/2m\hbar}=\sqrt{\tfrac{m}{2\pi i\hbar\epsilon}}\,\exp\!\left[\tfrac{im(x_{j+1}-x_j)^2}{2\hbar\epsilon}\right].

This is a Gaussian (Fresnel) integral. The potential factor contributes $e^{-iV(x_j)\epsilon/\hbar}$ .

Step 5 — Assemble. Multiplying the $N$ slices reproduces exactly [4]. The exponent is a Riemann sum for $\int(\tfrac12 m\dot x^2-V)\,dt=S[x]$ , so in the continuum limit we obtain [1].

Step 6 — Classical limit. As $\hbar\to0$ , the integrand $e^{iS/\hbar}$ oscillates violently; by stationary phase the dominant contribution comes from paths with $\delta S=0$ , i.e. solutions of the Euler–Lagrange equation. $\blacksquare$

∎

RP

Richard P. Feynman

1918–1988

Feynman developed the path-integral formulation in his 1942 Princeton PhD thesis under John Wheeler and published it fully in 1948 ("Space-Time Approach to Non-Relativistic Quantum Mechanics," Rev. Mod. Phys.). He was inspired by a 1933 paper of Dirac suggesting that $\langle x_{t+dt}|x_t\rangle$ is analogous to $\exp(iL\,dt/\hbar)$ ; Feynman discovered the analogy was in fact a proportionality. The formulation underlies his diagrammatic approach to QED, for which he shared the 1965 Nobel Prize with Schwinger and Tomonaga. His textbook with Hibbs, Quantum Mechanics and Path Integrals (1965), remains the canonical pedagogical source.

🧠 Intuition

Imagine sending a quantum particle from $A$ to $B$ . In the double-slit experiment we already accept that the particle "explores" both slits; the path integral merely takes this to its logical extreme — the particle explores every curve in spacetime, including wild zig-zags that backtrack in space. Each path is a tiny clock hand (a phasor) of unit length pointing at angle $S[x]/\hbar$ . To find the total amplitude, lay all the clock hands tip-to-tail. For most paths a neighbor differs slightly in $S$ , so the hands point in scattered directions and cancel. But near the classical path, $S$ is stationary: a whole bundle of paths shares nearly the same phase, their clock hands align, and they add up coherently. Classical mechanics is the bright fringe of a cosmic interference pattern.

Figure

Sum over paths: the classical trajectory is where neighboring path phases align

ExampleFree Particle Propagator

Given: A free particle, $V(x)=0$ , mass $m$ , from $(x_a,t_a)$ to $(x_b,t_b)$ with $T=t_b-t_a$ .

Step 1 — Classical action. The classical path is uniform motion $x_{cl}(t)=x_a+\frac{x_b-x_a}{T}(t-t_a)$ , giving

S_{cl}=\frac{m(x_b-x_a)^2}{2T}.

Step 2 — Fluctuation factor. Writing $x=x_{cl}+y$ with $y(t_a)=y(t_b)=0$ , the action separates: $S=S_{cl}+\frac{m}{2}\int\dot y^2\,dt$ . The fluctuation integral $\int\mathcal{D}y\,e^{i\frac{m}{2\hbar}\int\dot y^2}$ depends only on $T$ and yields $(m/2\pi i\hbar T)^{1/2}$ (evaluate the time-sliced Gaussians, telescoping product).

Step 3 — Combine.

K_{\text{free}}=\boxed{\left(\frac{m}{2\pi i\hbar T}\right)^{1/2}\exp\!\left[\frac{i m (x_b-x_a)^2}{2\hbar T}\right]}

Units check: $[m/\hbar T]^{1/2}$ has units $(\text{kg}\cdot\text{s}^{-1}\cdot\text{J}^{-1}\text{s}^{-1})^{1/2}=\text{m}^{-1}$ , correct for a 1-D propagator (so that $\int K\,dx$ is dimensionless). The exponent is dimensionless. ✓

ExampleHarmonic Oscillator Propagator

Given: $V(x)=\tfrac12 m\omega^2 x^2$ , time $T=t_b-t_a$ , with $\omega T\neq n\pi$ .

Step 1 — Classical action. Solving the EL equation with boundary conditions:

S_{cl}=\frac{m\omega}{2\sin\omega T}\Big[(x_a^2+x_b^2)\cos\omega T-2x_a x_b\Big].

Step 2 — Fluctuation determinant. For quadratic actions the path integral is exact (Gaussian): the fluctuation prefactor is $\left(\frac{m\omega}{2\pi i\hbar\sin\omega T}\right)^{1/2}$ , obtained from the Van Vleck–Pauli–Morette determinant $\det(-\partial^2 S_{cl}/\partial x_a\partial x_b)$ .

Step 3 — Result (Mehler kernel).

K_{\text{HO}}=\boxed{\left(\frac{m\omega}{2\pi i\hbar\sin\omega T}\right)^{1/2}\exp\!\left\{\frac{im\omega}{2\hbar\sin\omega T}\big[(x_a^2+x_b^2)\cos\omega T-2x_ax_b\big]\right\}}

Numerical sanity check: As $\omega\to0$ , $\sin\omega T\to\omega T$ and $\cos\omega T\to1$ , so the bracket $\to (x_a^2+x_b^2-2x_ax_b)=(x_b-x_a)^2$ and the prefactor $\to(m/2\pi i\hbar T)^{1/2}$ — exactly the free particle [Example 1]. ✓ The ground-state energy $E_0=\tfrac12\hbar\omega$ can be read off the small- $T$ trace, $\operatorname{Tr}K\sim e^{-i\omega T/2}\cdots$ .

⚠️ Common mistake

The "measure" $\mathcal{D}x$ is not a genuine measure. There is no countably additive, translation-invariant measure on the infinite-dimensional space of paths with $e^{iS/\hbar}$ weight (the oscillatory integral has no Lebesgue-measure realization; the relevant object is at best an oscillatory/Cameron–Martin construction). The path integral in [1] is defined by the limit in [4], and quantities like "the average velocity of a quantum path" are ill-defined — typical contributing paths are continuous but nowhere differentiable, with $\langle(\Delta x)^2\rangle\sim\hbar\epsilon/m$ , so $\Delta x/\epsilon\sim\epsilon^{-1/2}$ diverges.

⚠️ Common mistake

Operator ordering ambiguities hide in the time slicing. For Lagrangians with velocity-dependent potentials or curved configuration spaces, the naive midpoint vs. endpoint discretization of $V$ and the measure produce different quantum Hamiltonians. The choice of discretization is physics, not bookkeeping; it corresponds to a choice of operator ordering ( $pq$ vs. $qp$ vs. Weyl).

Counterexample

Naively replacing the sum by the classical path alone is wrong even semiclassically. One might think $K\approx e^{iS_{cl}/\hbar}$ . But the prefactor (Van Vleck determinant) and the Maslov index — a phase $e^{-i\pi\mu/2}$ accumulated each time the classical path passes a focal/caustic point — are essential. For the harmonic oscillator at $\omega T=\pi$ the classical action diverges and the propagator collapses to $\delta(x_a+x_b)$ (a perfect refocusing); dropping the Maslov correction gives the wrong sign of the amplitude past each caustic. The classical action alone is necessary but never sufficient.

connection

The path integral is woven into the rest of theoretical physics:

<a class="wiki-link" href="/wiki/ndli-iit-kgp-ph11001-physics-schrdinger-equation">Schrödinger Equation</a> — equivalent dynamics; [5] shows the kernel satisfies it, so the two formulations are interderivable.
Wick Rotation — substituting $t\to-i\tau$ turns $e^{iS/\hbar}$ into the real, positive Boltzmann weight $e^{-S_E/\hbar}$ , mapping quantum mechanics onto Statistical Mechanics and Brownian motion (Feynman–Kac formula).
<a class="wiki-link" href="/wiki/quantum-field-theory">Quantum Field Theory</a> — promoting $x(t)$ to a field $\phi(x,t)$ gives the generating functional $Z[J]=\int\mathcal{D}\phi\,e^{iS[\phi]+i\int J\phi}$ , the cornerstone of Feynman Diagrams.
Gauge Theory — gauge invariance and the Faddeev–Popov ghost prescription are most transparent in the functional integral.
Saddle Point Approximation — the $\hbar\to0$ expansion is stationary phase / steepest descent applied to a functional.

🔬 Where this appears

⏳ History

- 1923–26 — Hamilton–Jacobi theory and the optical-mechanical analogy prefigure the action's central role; de Broglie and Schrödinger build wave mechanics.

- 1933 — Dirac's paper "The Lagrangian in Quantum Mechanics" notes that

\langle q_{t+dt}|q_t\rangle

corresponds to

\exp(iL\,dt/\hbar)

.

- 1941–42 — Feynman, prompted by Dirac's word "corresponds," shows in his Princeton thesis (advisor Wheeler) it is a genuine proportionality and builds the full sum-over-paths.

- 1948 — Feynman publishes "Space-Time Approach to Non-Relativistic Quantum Mechanics," the canonical reference for [1].

- 1949 — Feynman applies the method to QED; Dyson proves the equivalence of Feynman, Schwinger, and Tomonaga formulations.

- 1951 — Feynman's "operator calculus" and the disentangling of time-ordered exponentials; coupling to external sources.

- 1955 — Polaron solution; Kac formalizes the Euclidean version (Feynman–Kac formula, rooting it in rigorous probability).

- 1965 — Feynman & Hibbs textbook; Nobel Prize to Feynman, Schwinger, Tomonaga.

- 1967–72 — Faddeev–Popov quantization of gauge fields and 't Hooft's renormalizability proofs, all in the functional-integral language, secure its dominance in QFT.

open-problem

Rigorous foundations remain incomplete in the Lorentzian (real-time) regime. The Euclidean path integral is mathematically respectable (Wiener measure, Osterwalder–Schrader reconstruction), but the oscillatory Minkowski integral $\int\mathcal{D}x\,e^{iS/\hbar}$ has no measure-theoretic definition in general; one relies on analytic continuation, which can fail for theories with sign problems or complex actions (finite-density QCD, real-time dynamics). The fermion sign problem obstructs Monte Carlo evaluation and is NP-hard in general. The correct contour of integration for gravity — where the Euclidean Einstein–Hilbert action is unbounded below (the conformal-factor problem) — and the precise meaning of "summing over geometries and topologies" remain open. Lefschetz-thimble and complex-Langevin methods, and the constructive program (Glimm–Jaffe) for interacting fields beyond two and three dimensions, are active frontiers.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also