Sum over histories

Intuition

Compose the $N$ short-time propagators and let $N$ grow without limit. The $N-1$ integrals over the intermediate positions become an integral over every path $x(t)$ from $a$ to $b$ , and the phases of the steps add up to the classical action of the path, $S[x]=\int(\tfrac12m\dot{x}^{2}-V)dt$ . The result is Feynman’s formula of 1948, which Dirac had glimpsed in 1933: the propagator is the sum, over all histories of the particle, of $e^{iS/\hbar}$ . Every path counts, however wild, and every path counts with the same weight: only its phase differs. There is no single trajectory in quantum mechanics, but there is a trajectory in the sum that matters most, the classical one, and the next lesson shows why.

A light ray from a lamp to your eye seems to take one path, the fastest; in Feynman’s account of light it tries every path, and the paths near the fastest one arrive in step while the others cancel. Particles do the same, with the action in place of the travel time.

A few of the histories from $a$ to $b$ of a free particle, time across and position up. The propagator adds $e^{iS/\hbar}$ over all of them, each with the same size; the straight classical path, solid, is where the action is stationary.

As the number of slices grows without limit, with the action $S[x]=\int_{t_{a}}^{t_{b}}\bigl(\tfrac12m\dot{x}^{2}-V(x)\bigr)\,dt$ of each path from to :

K(x_{b},t_{b};x_{a},t_{a})=\int\mathcal{D}x\;e^{iS[x]/\hbar}

Properties

$\int\mathcal{D}x$ means the limit of $\bigl(\frac{m}{2\pi i\hbar\varepsilon}\bigr)^{N/2}\int dx_{1}\cdots dx_{N-1}$ as with .

The path integral

K(b;a)=\lim_{N\to\infty}\Bigl(\frac{m}{2\pi i\hbar\varepsilon}\Bigr)^{N/2}\int\prod_{k=1}^{N-1}dx_{k}\;e^{iS_{N}/\hbar}

Write the evolution over $T$ as $N$ evolutions over $\varepsilon$ , insert the completeness of position states between them, and put in the short-time propagator for each. The phases add, and their sum is a Riemann sum for the action of the broken path through the chosen positions; the limit is the action of the path.

Proof steps

e^{-i\hat{H}T/\hbar}=\bigl(e^{-i\hat{H}\varepsilon/\hbar}\bigr)^{N},\qquad N\varepsilon=T

The evolution over $T$ as $N$ short evolutions.

K=\int dx_{1}\cdots dx_{N-1}\prod_{k=0}^{N-1}\bra{x_{k+1}}e^{-i\hat{H}\varepsilon/\hbar}\ket{x_{k}}

Insert $\int\ket{x_{k}}\bra{x_{k}}dx_{k}$ between the factors, with and .

\bra{x_{k+1}}e^{-i\hat{H}\varepsilon/\hbar}\ket{x_{k}}\approx\sqrt{\frac{m}{2\pi i\hbar\varepsilon}}\,e^{i\varepsilon L_{k}/\hbar}

The short-time propagator, with $L_{k}=\frac{m}{2}\bigl(\frac{x_{k+1}-x_{k}}{\varepsilon}\bigr)^{2}-V(x_{k})$ .

S_{N}=\sum_{k=0}^{N-1}\varepsilon\,L_{k}\ \xrightarrow{\ N\to\infty\ }\ \int_{t_{a}}^{t_{b}}L\,dt=S[x]

The phases add to a Riemann sum for the action of the broken path.

Applications

Practice

Every Path Counts

The propagator is the sum over all paths from $a$ to $b$ of $e^{iS/\hbar}$ . Every path has the same weight; only its phase, set by its action, differs.

K=\int\mathcal{D}x\;e^{iS[x]/\hbar}

Try it

In the path integral, how are the paths weighted?