Hamilton's principle/Applications/The Feynman path integral

Plain-language version: Sum over histories

The path integral formulation of quantum mechanics assigns an amplitude to every path between two points and adds them. It closes this volume's argument: the action, which in classical mechanics is a quantity made stationary by the one true path, becomes in quantum mechanics the phase carried by every possible path. Classical mechanics is then not a separate theory but the limit in which all phases but one cancel.

Provenance note

Phases adding near the stationary path and cancelling elsewhere.

Feynman's 1948 paper could not be obtained in full text for this volume. The published version is behind a paywall and no open copy meeting this volume's sourcing standard was located; the row is UNCOMPUTED with block absence on Sources. Its abstract is available and states the postulate in prose.

Consequently no equation on this page is attributed to Feynman and nothing is quoted from him. The postulate is stated in this volume's words, and the derivations are standard textbook ones performed here. Closing the gap requires institutional access to the paper, or the 1942 doctoral thesis, published in 2005, where the same postulate appears.

The postulate

Rays crossing a focus.

Stated in this volume's words, from the abstract of the 1948 paper: the probability of an event that can occur in several ways is the squared magnitude of a sum of complex contributions, one from each way; the contribution of a single path is an exponential whose imaginary phase is the classical action for that path, in units of $\hslash$.

In symbols, the amplitude to go from $(q_a,t_a)$ to $(q_b,t_b)$ is

$K(b,a)=\sum _{\text{all paths}}\exp (\frac{i}{\hslash }S[q])\mathrm{(}\mathrm{1}\mathrm{)}$

with $S[q]={\int }_{t_a}^{t_b}Ldt$ the classical action of Abbreviated action: Maupertuis versus Hamilton — Hamilton's $S$, not the abbreviated $W$.

Every path contributes, including paths that are nowhere differentiable and paths that violate the equations of motion. All contribute with the same magnitude and differ only in phase. That is the striking content of (1): the classical path is not weighted more heavily, it is not selected, and nothing in the postulate refers to it.

How the classical path emerges

Consider two neighbouring paths differing by $\delta q$. Their phases differ by $\delta S/\hslash$. For macroscopic systems $S$ is enormous compared with $\hslash$ — a gram-scale object over a second gives $S/\hslash \sim 10^{27}$ — so a small change of path produces an enormous change of phase, and neighbouring contributions cancel.

Except where $\delta S=0$. Near a path for which the action is stationary, neighbouring paths have the same phase to first order and add constructively rather than cancelling. The sum is dominated by a neighbourhood of the stationary path.

So Hamilton's principle is recovered, and its status changes completely:

	classical reading	path-integral reading
status of $\delta S=0$	a law	a stationary-phase condition
other paths	do not occur	occur, and cancel
why this path	postulated	surviving interference
$\hslash$	absent	sets the cancellation scale

This answers a question the classical theory cannot ask. Why should nature extremise anything? The classical answer is that it does. The path-integral answer is that nature does not extremise: it sums, and stationary phase is where a sum of rapidly oscillating terms fails to cancel. The variational principle is a theorem about interference.

Note also that the reasoning explains why stationary and not least is the right word. Constructive interference requires only that the phase be stationary. A maximum or a saddle interferes constructively exactly as a minimum does. The maximum-time reflection of Fermat's principle is a real optical path for precisely this reason, and the volume's terminology is vindicated from an unexpected direction.

Making the sum meaningful

"Sum over all paths" needs definition. Divide $[t_a,t_b]$ into $N$ steps of length $ϵ$, approximate each path by its values $q_1,\dots ,q_{N-1}$ at the intermediate times, and integrate over each:

$K(b,a)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{A}\int \cdots \int \exp (\frac{i}{\hslash }\sum _{j=0}^{N-1}S(q_{j+1},q_j))\frac{d{q}_1}{A}\cdots \frac{d{q}_{N-1}}{A}\mathrm{(}\mathrm{2}\mathrm{)}$

with $A$ a normalisation depending on $ϵ$. For a free particle $A=\sqrt{2\pi i\hslash ϵ/m}$.

The limit is delicate. The integrals do not converge absolutely, the measure is not a measure in the sense of analysis, and making (2) rigorous is a genuine mathematical problem. The Euclidean version, obtained by rotating to imaginary time, is a well-defined Wiener measure and is the standard route to rigour. This volume notes the difficulty and does not treat it.

Recovering Schrödinger's equation

Take one short step of duration $ϵ$. Writing $\psi (q,t+ϵ)=\int K\psi (q^{\prime },t)d{q}^{\prime }$ and expanding the action for a single step,

$S\approx ϵ[\frac{m}{2}{\left(\frac{q-q^{\prime }}{ϵ}\right)}^2-V(q)]\mathrm{(}\mathrm{3}\mathrm{)}$

The Gaussian factor concentrates $q-q^{\prime }$ within $\sqrt{\hslash ϵ/m}$ of zero, so expanding $\psi$ to second order in that displacement and to first order in $ϵ$, and doing the Gaussian integrals, gives

$i\hslash \frac{\partial \psi }{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial q^2}+V\psi \mathrm{(}\mathrm{4}\mathrm{)}$

The path integral and the Schrödinger equation are equivalent formulations.

Note where the second derivative comes from: the displacement scales as $\sqrt{ϵ}$, not $ϵ$, so the second-order term in the expansion survives the limit alongside the first-order time derivative. Quantum paths are of unbounded variation — a typical contributing path is continuous and nowhere differentiable — and this is that fact showing up in the arithmetic.

The semiclassical limit and conjugate points

Evaluating (1) by stationary phase around the classical path gives

$K\approx \sqrt{\frac{1}{2\pi i\hslash }\frac{{\partial }^2{S}_{\text{cl}}}{\partial q_a\partial q_b}}\exp (\frac{i}{\hslash }{S}_{\text{cl}}-\frac{i\pi \nu }{2})\mathrm{(}\mathrm{5}\mathrm{)}$

Three features of (5) close loops opened earlier in the volume.

$S_{\text{cl}}$ is the action evaluated on the actual motion — exactly Hamilton's characteristic function of Action as a function of its endpoints, and the second derivative in the prefactor is a derivative of his relation (C.), $\partial S/\partial q=p$.

The prefactor diverges at conjugate points, where neighbouring extremals refocus and the second derivative blows up. That is the same breakdown as the diverging amplitude in the eikonal approximation on the optical-mechanical analogy.

$\nu$ is the Maslov index: the number of conjugate points the path crosses. Each crossing contributes a phase of $-\pi /2$. This is the phase correction promised on the Jacobi condition and conjugate points, and it makes the conjugate-point count a physically measurable quantity rather than a technical condition in a sufficiency theorem.

If more than one classical path connects the endpoints — past a conjugate point, by the argument of that page — then (5) is a sum over them, and they interfere. Gravitational lensing on the Einstein-Hilbert action is the astronomical case.

Where the volume ends

The argument has come round. The volume opened by asking what is stationary and why the word is not least. The classical answer, developed over the mathematics tracks, is that the first variation vanishes and nothing more follows without further conditions. The quantum answer is that stationarity is where interference survives, and that a maximum interferes as well as a minimum.

Both answers reject least, and they reject it for different reasons that turn out to be the same reason seen at two scales.

Provenance: no text or equation on this page is attributed to Feynman. The postulate is paraphrased from the abstract of the 1948 paper, the only part obtainable; equations (1) to (5) are standard and are derived or stated here. The Feynman row on Sources is UNCOMPUTED, block: absence.

Next

Sum over histories gives this page's plain-language counterpart.