Hamilton's principle/History/Feynman 1948 - Revision history

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	The row is UNCOMPUTED with block '''absence''' on		The row is UNCOMPUTED with block '''absence''' on
	[[Hamilton's principle/Sources]].		[[Hamilton's principle/Sources\|Sources]].

	== What the paper proposes ==		== What the paper proposes ==
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	So the maximum-time reflection on		So the maximum-time reflection on
	[[Hamilton's principle/Applications/Fermat's principle]] is a real optical path for		[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]] is a real optical path for
	the same reason as any other. There is no sense in which nature prefers small		the same reason as any other. There is no sense in which nature prefers small
	numbers.		numbers.
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	purpose to light, since taking the quickest route implies comparing the routes.		purpose to light, since taking the quickest route implies comparing the routes.
	That objection is recorded at		That objection is recorded at
	[[Hamilton's principle/History/Least time before least action]].		[[Hamilton's principle/History/Least time before least action\|Least time before least action]].

	The wave theory of light answered it in the nineteenth century. A wavefront sweeps		The wave theory of light answered it in the nineteenth century. A wavefront sweeps
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	Maupertuis's metaphysics on		Maupertuis's metaphysics on
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]] does not merely		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]] does not merely
	turn out to be unnecessary. It turns out that there is nothing for it to explain.		turn out to be unnecessary. It turns out that there is nothing for it to explain.

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	This volume notes the difficulty and does not treat it. It is named as further		This volume notes the difficulty and does not treat it. It is named as further
	reading on [[Hamilton's principle/Sources]].		reading on [[Hamilton's principle/Sources\|Sources]].

	== Where the volume ends ==		== Where the volume ends ==

	The technical treatment is at		The technical treatment is at
	[[Hamilton's principle/Applications/The Feynman path integral]], and its		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]], and its
	plain-language counterpart at		plain-language counterpart at
	[[Hamilton's principle/Explainers/Sum over histories]].		[[Hamilton's principle/Explainers/Sum over histories\|Sum over histories]].

	The volume opened by asking what is stationary and why the word is not ''least''.		The volume opened by asking what is stationary and why the word is not ''least''.
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	paraphrased from the abstract of the 1948 paper, the only obtainable part. The		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		account of the semiclassical limit and of the measure-theoretic difficulty is
	editorial summary. See [[Hamilton's principle/Sources]].		editorial summary. See [[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

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			{{DISPLAYTITLE:Feynman 1948}}
	Feynman's ''Space-Time Approach to Non-Relativistic Quantum Mechanics'' appeared in		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,		1948. It formulates quantum mechanics as a sum over every path between two points,
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	editorial summary. See [[Hamilton's principle/Sources]].		editorial summary. See [[Hamilton's principle/Sources]].
	</blockquote>		</blockquote>

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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 ==

[[File:OPL-HP-F-58.png|thumb|right|400px|The two long gaps in the subject's history.]]

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.

<blockquote>
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]].
</blockquote>