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{{Plebeian|Hamilton's principle/Explainers/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 ==

[[File:OPL-HP-F-15.png|thumb|right|400px|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
[[Hamilton's principle/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 ==

[[File:OPL-HP-F-50.png|thumb|right|400px|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 <math>\hbar</math>.

In symbols, the amplitude to go from <math>(q_a, t_a)</math> to
<math>(q_b, t_b)</math> is

:<math>K(b,a) = \sum_{\text{all paths}}
\exp\!\left( \frac{i}{\hbar}\,S[q] \right)
\qquad\qquad\mathrm{(1)}</math>

with <math>S[q] = \int_{t_a}^{t_b} L\,dt</math> the classical action of
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]
— Hamilton's <math>S</math>, not the abbreviated <math>W</math>.

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

'''Except where <math>\delta S = 0</math>.''' 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:

{| class="wikitable"
! !! classical reading !! path-integral reading
|-
| status of <math>\delta S = 0</math> || a law || a stationary-phase condition
|-
| other paths || do not occur || occur, and cancel
|-
| why this path || postulated || surviving interference
|-
| <math>\hbar</math> || 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
[[Hamilton's principle/Applications/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 <math>[t_a, t_b]</math> into
<math>N</math> steps of length <math>\epsilon</math>, approximate each path by
its values <math>q_1, \ldots, q_{N-1}</math> at the intermediate times, and
integrate over each:

:<math>K(b,a) = \lim_{N\to\infty} \frac{1}{A}\int\cdots\int
\exp\!\left( \frac{i}{\hbar}\sum_{j=0}^{N-1}
S(q_{j+1}, q_j) \right)
\frac{dq_1}{A}\cdots\frac{dq_{N-1}}{A}
\qquad\qquad\mathrm{(2)}</math>

with <math>A</math> a normalisation depending on <math>\epsilon</math>. For a
free particle <math>A = \sqrt{2\pi i\hbar\epsilon/m}</math>.

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 <math>\epsilon</math>. Writing
<math>\psi(q, t+\epsilon) = \int K\,\psi(q', t)\,dq'</math> and expanding the
action for a single step,

:<math>S \approx \epsilon\left[
\frac{m}{2}\left( \frac{q - q'}{\epsilon} \right)^2 - V(q) \right]
\qquad\qquad\mathrm{(3)}</math>

The Gaussian factor concentrates <math>q - q'</math> within
<math>\sqrt{\hbar\epsilon/m}</math> of zero, so expanding <math>\psi</math> to
second order in that displacement and to first order in <math>\epsilon</math>,
and doing the Gaussian integrals, gives

:<math>i\hbar\,\frac{\partial\psi}{\partial t}
= -\frac{\hbar^2}{2m}\,\frac{\partial^2\psi}{\partial q^2} + V\psi
\qquad\qquad\mathrm{(4)}</math>

The path integral and the Schrödinger equation are equivalent formulations.

Note where the second derivative comes from: the displacement scales as
<math>\sqrt{\epsilon}</math>, not <math>\epsilon</math>, 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

:<math>K \approx \sqrt{\frac{1}{2\pi i\hbar}\,
\frac{\partial^2 S_{\text{cl}}}{\partial q_a \partial q_b}}\;
\exp\!\left( \frac{i}{\hbar} S_{\text{cl}} - \frac{i\pi\nu}{2} \right)
\qquad\qquad\mathrm{(5)}</math>

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

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

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
[[Hamilton's principle/Applications/The optical-mechanical analogy]].

<math>\nu</math> is the '''Maslov index''': the number of conjugate points the
path crosses. Each crossing contributes a phase of <math>-\pi/2</math>. This is
the phase correction promised on
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Applications/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.

<blockquote>
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 [[Hamilton's principle/Sources]] is UNCOMPUTED, block: absence.
</blockquote>

== Next ==

[[Hamilton's principle/Explainers/Sum over histories]] gives this page's
plain-language counterpart.

== References ==

<references />

← Older revision		Revision as of 04:58, 16 September 2026
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	published version is behind a paywall and no open copy meeting this volume's		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		sourcing standard was located; the row is UNCOMPUTED with block '''absence''' on
	[[Hamilton's principle/Sources]]. Its abstract is available and states the		[[Hamilton's principle/Sources\|Sources]]. Its abstract is available and states the
	postulate in prose.		postulate in prose.

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	with <math>S[q] = \int_{t_a}^{t_b} L\,dt</math> the classical action of		with <math>S[q] = \int_{t_a}^{t_b} L\,dt</math> the classical action of
	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]]
	— Hamilton's <math>S</math>, not the abbreviated <math>W</math>.		— Hamilton's <math>S</math>, not the abbreviated <math>W</math>.

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	stationary. A maximum or a saddle interferes constructively exactly as a minimum		stationary. A maximum or a saddle interferes constructively exactly as a minimum
	does. The maximum-time reflection of		does. The maximum-time reflection of
	[[Hamilton's principle/Applications/Fermat's principle]] is a real optical path		[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]] is a real optical path
	for precisely this reason, and the volume's terminology is vindicated from an		for precisely this reason, and the volume's terminology is vindicated from an
	unexpected direction.		unexpected direction.
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	<math>S_{\text{cl}}</math> is the action evaluated on the actual motion — exactly		<math>S_{\text{cl}}</math> is the action evaluated on the actual motion — exactly
	Hamilton's characteristic function of		Hamilton's characteristic function of
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]], and		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]], and
	the second derivative in the prefactor is a derivative of his relation (C.),		the second derivative in the prefactor is a derivative of his relation (C.),
	<math>\partial S/\partial q = p</math>.		<math>\partial S/\partial q = p</math>.
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	refocus and the second derivative blows up. That is the same breakdown as the		refocus and the second derivative blows up. That is the same breakdown as the
	diverging amplitude in the eikonal approximation on		diverging amplitude in the eikonal approximation on
	[[Hamilton's principle/Applications/The optical-mechanical analogy]].		[[Hamilton's principle/Applications/The optical-mechanical analogy\|the optical-mechanical analogy]].

	<math>\nu</math> is the '''Maslov index''': the number of conjugate points the		<math>\nu</math> is the '''Maslov index''': the number of conjugate points the
	path crosses. Each crossing contributes a phase of <math>-\pi/2</math>. This is		path crosses. Each crossing contributes a phase of <math>-\pi/2</math>. This is
	the phase correction promised on		the phase correction promised on
	[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],		[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points\|the Jacobi condition and conjugate points]],
	and it makes the conjugate-point count a physically measurable quantity rather		and it makes the conjugate-point count a physically measurable quantity rather
	than a technical condition in a sufficiency theorem.		than a technical condition in a sufficiency theorem.
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	by the argument of that page — then (5) is a sum over them, and they interfere.		by the argument of that page — then (5) is a sum over them, and they interfere.
	Gravitational lensing on		Gravitational lensing on
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]] is the		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]] is the
	astronomical case.		astronomical case.

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	postulate is paraphrased from the abstract of the 1948 paper, the only part		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		obtainable; equations (1) to (5) are standard and are derived or stated here. The
	Feynman row on [[Hamilton's principle/Sources]] is UNCOMPUTED, block: absence.		Feynman row on [[Hamilton's principle/Sources\|Sources]] is UNCOMPUTED, block: absence.
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Explainers/Sum over histories]] gives this page's		[[Hamilton's principle/Explainers/Sum over histories\|Sum over histories]] gives this page's
	plain-language counterpart.		plain-language counterpart.