The Path Integral Formulation

The core idea: add up every path

Classical mechanics answers "how does a ball get from A to B?" with a single trajectory — the one that extremizes the action. Quantum mechanics, in Feynman's telling, refuses to choose. To compute the amplitude to go from A to B, you consider every conceivable path: the straight one, the wiggly one, the one that loops back on itself, even the one that momentarily runs faster than light. Each path x(t) is assigned a complex number of unit magnitude,

amplitude of one path = e^(iS[x]/ℏ)

where S[x] is the classical action of that path. The total amplitude is the sum (an integral, really) of all these unit-length arrows in the complex plane:

K(b, a) = Σ_all paths e^(iS[x]/ℏ)  →  ∫ Dx(t) e^(iS[x]/ℏ)

The probability to arrive at B is |K(b, a)|². Every path contributes the same magnitude (1) — what differs is the phase. All of quantum interference is contained in how those phases add or cancel.

The propagator and its symbols

The central object is the propagator (or kernel) K, the amplitude to be found at (x_b, t_b) given a start at (x_a, t_a):

K(b, a) = ⟨x_b, t_b | x_a, t_a⟩ = ∫ Dx(t) exp( (i/ℏ) S[x(t)] )

Symbol	Meaning	Units (SI)
K(b, a)	Propagator / kernel — amplitude from a to b	m⁻¹ (1D), so |K|² is a probability density
S[x(t)]	Action functional of a path, S = ∫ L dt	J·s (same units as ℏ)
L	Lagrangian, L = T − V = ½mẋ² − V(x)	J
ℏ	Reduced Planck constant, h/2π	1.055 × 10⁻³⁴ J·s
∫ Dx(t)	Functional (path) integral over all paths with fixed endpoints	—
ψ(x, t)	Wavefunction; propagates via ψ(b) = ∫ K(b,a) ψ(a) dx_a	m⁻¹ᐟ² (1D)

Because S has units of J·s exactly matching ℏ, the ratio S/ℏ is dimensionless — a pure angle in radians. That is what makes e^(iS/ℏ) a legitimate phase.

Why the classical path wins: stationary phase

Here is the beautiful part. For a baseball, the action of the actual trajectory is on the order of joule-seconds — say S ~ 1 J·s. Divide by ℏ ~ 10⁻³⁴ J·s and you get S/ℏ ~ 10³⁴ radians. Nudge the path a hair and S changes by a tiny amount — but a tiny amount times 10³⁴ is still an enormous number of radians, so the phase e^(iS/ℏ) whirls around the complex circle countless times. Neighbouring paths therefore point in random directions and cancel.

The single exception is near the path where the action is stationary:

δS = 0   (first variation vanishes → Euler–Lagrange equations)

There, to first order, varying the path does not change S, so neighbouring paths share nearly the same phase and add constructively. That stationary path is exactly the classical trajectory picked out by the principle of least (stationary) action. So classical mechanics is not overthrown by quantum mechanics — it emerges as the ℏ → 0 limit, where only the constructively-interfering neighbourhood of the classical path survives. The width of that surviving band scales as √ℏ, which is why quantum "fuzz" is invisible for macroscopic S.

Worked example: the free-particle propagator

For a free particle (V = 0) in one dimension the path integral can be done in closed form because the action is quadratic and the Gaussian integrals converge. The result is one of the cleanest formulas in quantum mechanics:

K(x_b, t; x_a, 0) = √( m / (2πiℏt) ) · exp( i m (x_b − x_a)² / (2ℏt) )

Read off the physics: the phase is exactly the classical action of a free particle covering distance (x_b − x_a) in time t, namely S_cl = m(x_b − x_a)²/2t. Only the classical action appears in the exponent — for quadratic actions the fluctuations contribute only the prefactor √(m/2πiℏt). The 1/√t spreading is the free-particle wavepacket dispersing. For a harmonic oscillator the same method gives K ∝ exp(iS_cl/ℏ) with S_cl = (mω/2sin ωT)[(x_a²+x_b²)cos ωT − 2x_a x_b], again just the classical action times a fluctuation prefactor.

Deriving the Schrödinger equation from the path integral

Feynman's 1948 paper shows the formulations are equivalent by propagating ψ over one infinitesimal slice of time ε. Over that slice,

ψ(x, t + ε) = ∫ K_ε(x, y) ψ(y, t) dy,   K_ε ≈ A · exp( (iε/ℏ) [ (m/2)((x−y)/ε)² − V(x) ] )

The kinetic phase (im/2ℏε)(x−y)² oscillates so fast that only y within √(ℏε/m) of x contributes. Set η = y − x, expand ψ(y, t) = ψ(x, t) + η ψ' + ½η²ψ'' , and evaluate the two Gaussian moments ∫e^(imη²/2ℏε)dη = √(2πiℏε/m) and ∫η²e^(imη²/2ℏε)dη = (iℏε/m)·√(2πiℏε/m). Fixing the normalization A = √(m/2πiℏε), matching the O(ε⁰) term, and collecting the O(ε¹) terms gives

iℏ ∂ψ/∂t = − (ℏ²/2m) ∂²ψ/∂x² + V(x) ψ

— the time-dependent Schrödinger equation. The ∂²/∂x² term arises directly from the Gaussian spread η² ~ ℏε/m of the kinetic phase. This is the sense in which the path integral is quantum mechanics, not an approximation to it.

The double slit, re-read as a sum over paths

The path integral gives the double-slit experiment its cleanest reading. The electron does not "go through" one slit — the amplitude at the screen is the sum over all paths, and the paths through slit 1 and slit 2 are simply the two dominant families. Their actions differ by the extra path length, so their phases differ; where the path-length difference is an integer number of de Broglie wavelengths λ = h/p the two amplitudes add (bright fringe) and where it is a half-integer they cancel (dark fringe). The fringe spacing on a screen a distance L behind slits separated by d is Δy = λL/d — identical to the wave-optics answer. Open a third slit and you add a third path family; the machinery is unchanged. Feynman called this "the only mystery" of quantum mechanics, and the path integral turns the mystery into a bookkeeping rule: add the amplitudes, then square.

From particle to field: why QFT is built on it

Replace the particle's coordinate x(t) by a whole field configuration φ(x) defined at every point of spacetime, and the sum over paths becomes a functional integral over all field configurations:

Z = ∫ Dφ exp( (i/ℏ) S[φ] ),   S[φ] = ∫ d⁴x  ℒ(φ, ∂φ)

This generating functional Z produces every correlation function, and expanding it in the coupling constant generates Feynman diagrams order by order. Its advantages are why it dominates modern high-energy physics: Lorentz invariance and gauge symmetry are manifest in the action; particle creation and annihilation are automatic; gauge fixing has a clean treatment (Faddeev–Popov, BRST). Rotate time to imaginary values, t → −iτ (a Wick rotation), and the oscillating weight e^(iS/ℏ) becomes a real, positive Boltzmann weight e^(−S_E/ℏ) — turning Z into the partition function of statistical mechanics. That single move enables lattice QCD, the renormalization group, and the whole bridge between quantum field theory and critical phenomena.

Three equivalent pictures of quantum mechanics

Formulation	Central object	Dynamics	Best for
Schrödinger (wave, 1926)	Wavefunction ψ(x, t)	iℏ ∂ψ/∂t = Ĥψ	Bound states, spectra, intuition
Heisenberg (matrix, 1925)	Operators Â(t)	dÂ/dt = (i/ℏ)[Ĥ, Â]	Symmetries, algebra, spin
Feynman (path integral, 1948)	Propagator K = ∫ Dx e^(iS/ℏ)	Sum over all histories	Classical limit, QFT, gauge theory, lattice

All three give identical predictions; Dirac's 1933 note that the amplitude "corresponds to" e^(iS/ℏ) planted the seed Feynman turned into a full formulation in his Princeton thesis and the 1948 Reviews of Modern Physics paper.

Common misconceptions

• "The particle actually takes all the paths." The paths are a calculational device, not observable trajectories. Measuring which path is taken destroys the interference. Only the summed amplitude — and its squared magnitude — is physical.
• "Only the classical path matters." It dominates when S ≫ ℏ, but for a quantum system all paths matter; the near-classical band has finite width ~√ℏ, and away from the classical limit the "off-shell" paths carry the interference.
• "The paths are smooth curves." The dominant contributing paths are nowhere differentiable and fractal-like, with |Δx| ~ √Δt, so their velocity diverges and their length is infinite. They are not classical trajectories with wiggles.
• "e^(iS/ℏ) can be tweaked to any weight." Every path has amplitude of magnitude exactly 1; only the phase varies. Weighting by e^(−S) instead of e^(iS/ℏ) is the Euclidean (Wick-rotated) theory, a different setup used for statistical mechanics and convergence.
• "Using Celsius, or forgetting the units of S." S/ℏ must be dimensionless. If you drop the ℏ or use action not in J·s, the phase is meaningless. In natural units ℏ = 1 and the exponent is just iS.
• "The path integral needs an interpretation of measurement." It is agnostic: it reproduces the Born rule for probabilities but says nothing extra about collapse. It is compatible with Copenhagen, many-worlds, or Bohmian readings alike.