Path-Integral Formulation

Equivalent forms

\langle x_b|e^{-i\hat H T/\hbar}|x_a\rangle = \int \mathcal{D}x\, e^{iS/\hbar}

S[x] = \int_{t_a}^{t_b} L(x,\dot x)\,dt

\psi(x_b,t_b) = \int K(b,a)\,\psi(x_a,t_a)\,dx_a

Classical mechanics isn't overturned by quantum theory — it's the leftover when ℏ → 0 and only the least-action path survives the interference. One integral contains both worlds.

Symbol	Name	SI unit	Dimension	Range
$K(b,a)$	Propagator (kernel)output Amplitude to travel from event a to event b	m^{-1}	L^-1	—
$S$	Classical action Time integral of the Lagrangian along a path	J*s	ML^2T^-1	0 – 5e-33
$L$	Lagrangian Kinetic minus potential energy, T − V	J	ML^2T^-2	—
$\hbar$	Reduced Planck constant	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34

Where it holds

Exact in principle for non-relativistic QM and the foundation of QFT. The measure D[x] is mathematically subtle (rigorous via Wiener/Euclidean continuation to

e^{-S_E/\hbar })

. Semiclassical (stationary-phase) approximation is excellent when S >>

\hbar

; breaks down when only a few paths dominate.

Discovery

Richard Feynman (after Dirac) · 1948

As a Princeton graduate student, Feynman heard at a beer party about an obscure 1933 Dirac paper hinting that the quantum amplitude is 'analogous to' e^{iS/ℏ}. Feynman asked what 'analogous to' meant, worked through the night, and found it was *proportional to* — and that summing the exponential over all paths reproduced the Schrödinger equation exactly. He developed it fully in his 1942 thesis and 1948 paper. The same machinery, continued to imaginary time, became the backbone of quantum field theory, statistical mechanics and lattice QCD.

Real-world applications

Quantum field theory and Feynman-diagram perturbation expansions
Lattice QCD simulations of the strong force
Path-integral Monte Carlo in chemistry and condensed matter
Instanton calculations of quantum tunneling rates

Common misconceptions

The particle does not 'really' travel infinitely many paths in a literal sense — the paths are a calculational basis whose amplitudes interfere
All paths contribute with equal magnitude; only their phases differ — it is interference, not weighting, that selects classical behavior
Least action is not assumed — it emerges as the stationary-phase point where neighboring paths stop cancelling

Experimental verification

Electron double-slit and which-path interference experiments directly visualize multiple-path superposition. The Aharonov–Bohm phase, neutron interferometry, and the precise anomalous magnetic moment of the electron (computed via Feynman-diagram path integrals to 12 digits) all confirm the formulation.

Derivation

Slice the time interval T into N steps

\varepsilon = T/N

and insert complete position bases between successive short-time propagators: ⟨

x_b|e^{-i

Ĥ

T/\hbar }|x

_a⟩

= \int \prod dx_j \prod

⟨

x_{j+1}|e^{-i

Ĥ

\varepsilon /\hbar }|x_j

⟩.

For small

\varepsilon

each factor

\approx (m/2\pi i\hbar \varepsilon )^{1/2} \exp [(i/\hbar )(\tfrac{1}{2}m((x_{j+1}-x_j)/\varepsilon )^{2} - V)\varepsilon ]

.

Summing the exponents reconstructs the action

S = \int L dt

, and taking

N \to \infty

defines the functional integral

K = \int D[x] e^{

iS/

\hbar }

.

Conversely, expanding for infinitesimal time and demanding consistency reproduces

i\hbar \partial \psi /\partial t

= Ĥ

\psi

— proving the path integral and Schrödinger equation are equivalent.

What if…

What

if \hbar \to 0

?

The phase e^{iS/ $\hbar }$ oscillates infinitely fast, so only the stationary-action path survives — exactly the classical principle of least action.

What if you rotate time to imaginary values

(t \to -i\tau )

?

e^{iS/ $\hbar }$ becomes $e^{-S_E/\hbar }$ , a real positive weight — turning quantum amplitudes into a statistical-mechanics partition function. This 'Wick rotation' is what makes lattice simulations possible.

What if you block all but two paths (a double slit)?

You keep only two terms in the sum; their phase difference produces the familiar two-slit interference fringes.

1

Free-particle propagator

Given ·

V = 0

, travel

x_a \to x_b

in time T

Find · K(b, a)

Steps

Only the straight-line path is classical; the Gaussian fluctuation integral gives the prefactor
$S_cl = \tfrac{1}{2} m (x_b - x$ _a $)^{2}/T$
$K = (m/2\pi i\hbar T)^{1/2} e^{$ iS_ $cl/\hbar }$

Result ·

K = (m/2\pi i\hbar T)^{1/2} \exp [ i m (x_b - x

_a

)^{2} / (2\hbar T) ]

— a Gaussian spreading packet.

2

When does the classical limit kick in?

Given · Action S compared

to \hbar \approx 1.05e-34 J\cdot s

Find · criterion for sharp classical paths

Steps

Phase change between neighboring paths $\sim \delta S/\hbar$
$If \delta S$ >> $\hbar$ ⟹ rapid cancellation ⟹ classical path dominates
$If \delta S \sim \hbar$ ⟹ paths add coherently ⟹ quantum behavior

Result · When S >>

\hbar the

phase e^{iS/

\hbar }

spins so fast off the stationary path that all but the least-action trajectory cancel — recovering Newton. For

S \sim \hbar

(electrons, atoms) many paths survive and quantum interference is visible.

Time-Dependent Schrödinger Equation→The Schrödinger Equation→Canonical Commutation Relation→Born Rule→