Hamilton's principle/History/Feynman 1948
Feynman's Space-Time Approach to Non-Relativistic Quantum Mechanics appeared in 1948. It formulates quantum mechanics as a sum over every path between two points, each path weighted by a phase equal to its classical action.
It closes this volume's argument, because it explains why there is a stationary-action principle at all.
Provenance note
This is the volume's only source blocked by access rather than by unperformed work.
The published paper is behind a paywall and no open copy meeting this volume's standard was found. Its abstract is available and states the postulate in prose.
So nothing on this page is quoted from Feynman and no equation is attributed to him. Closing the gap needs institutional access to the paper, or the 1942 doctoral thesis, published in 2005, which contains the same postulate.
The row is UNCOMPUTED with block absence on Hamilton's principle/Sources.
What the paper proposes
Stated in this volume's words, from the abstract: the probability of an event that can happen 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 the reduced Planck constant.
Two features of that postulate are worth separating.
Every path contributes. Not just the classical one. Not just smooth ones. Every path between the endpoints, including paths that violate the equations of motion.
All contribute with equal magnitude. The classical path carries no extra weight. Nothing in the postulate distinguishes it, or mentions it.
Why the classical path survives
The classical path emerges from cancellation rather than from selection.
For anything much larger than an atom the action is enormous compared with the quantum unit, so a small change of path changes the phase by a large amount. Neighbouring paths then point in unrelated directions and cancel.
Near a path where the action is stationary, a small change barely alters the action, so neighbouring paths share a phase and reinforce.
That is the whole mechanism, and it inverts the classical statement. Nature does not select a path. It sums over all of them, and stationary action is where a sum of rapidly oscillating terms fails to cancel.
Hamilton's principle is a theorem about interference.
It settles the volume's terminology
The account also vindicates the word this volume has insisted on, from an independent direction.
For contributions to reinforce, the action need only be stationary. Whether it is a minimum, a maximum or neither makes no difference — a maximum reinforces exactly as a minimum does.
So the maximum-time reflection on Hamilton's principle/Applications/Fermat's principle is a real optical path for the same reason as any other. There is no sense in which nature prefers small numbers.
Every earlier argument in this volume against the word least was mathematical: counterexamples where a stationary path is not a minimum. This one is different in kind. It says the word was never the right kind of word, because nothing is being minimised by anything.
The same answer, three centuries apart
Fermat's principle drew an objection in 1662: it seemed to attribute knowledge and purpose to light, since taking the quickest route implies comparing the routes. That objection is recorded at Hamilton's principle/History/Least time before least action.
The wave theory of light answered it in the nineteenth century. A wavefront sweeps all the paths; the surviving ray is where nearby paths agree; nothing chose anything.
Feynman's answer for mechanics is the same answer. All paths are taken; almost all cancel; what survives looks like a choice and is not.
Three centuries, two subjects, one resolution. The apparent purposefulness of variational principles is what interference looks like from outside.
Maupertuis's metaphysics on Hamilton's principle/History/Maupertuis and the 1744 memoir does not merely turn out to be unnecessary. It turns out that there is nothing for it to explain.
What is not settled
The sum over paths is difficult to define rigorously. The integrals do not converge absolutely and the measure is not a measure in the usual sense. The standard route to rigour goes through imaginary time, where the object is well defined.
This volume notes the difficulty and does not treat it. It is named as further reading on Hamilton's principle/Sources.
Where the volume ends
The technical treatment is at Hamilton's principle/Applications/The Feynman path integral, and its plain-language counterpart at Hamilton's principle/Explainers/Sum over histories.
The volume opened by asking what is stationary and why the word is not least. The classical answer took a century and four conditions and remained local. The quantum answer is that stationarity is where interference survives, and that a maximum interferes as well as a minimum.
Both reject least. They reject it for reasons that look different and are the same reason at two scales.
Provenance: nothing on this page is quoted from Feynman. The postulate is paraphrased from the abstract of the 1948 paper, the only obtainable part. The account of the semiclassical limit and of the measure-theoretic difficulty is editorial summary. See Hamilton's principle/Sources.