This quantum world/Feynman route/Principle of least action

Principle of least action

If an object travels from $A$ to $B,$ it travels along all paths from $A$ to $B,$ in the same sense in which an electron goes through both slits. Then how is it that a big thing (such as a planet, a tennis ball, or a mosquito) appears to move along a single well-defined path?

There are at least two reasons. One of them is that the bigger an object is, the harder it is to satisfy the conditions stipulated by Rule $B.$ Another reason is that even if these conditions are satisfied, the likelihood of finding an object of mass $m$ where according to the laws of classical physics it should not be, decreases as $m$ increases.

To see this, we need to take account of the fact that it is strictly impossible to check whether an object that has travelled from $A$ to $B,$ has done so along a mathematically precise path $𝒞.$ Let us make the half realistic assumption that what we can check is whether an object has travelled from $A$ to $B$ within a a narrow bundle of paths — the paths contained in a narrow tube $𝒯.$ The probability of finding that it has, is the absolute square of the path integral $I(𝒯)=\underset{𝒯}{\int }𝒟𝒞e^{(i/\hslash )S[𝒞]},$ which sums over the paths contained in $𝒯.$

Let us assume that there is exactly one path from $A$ to $B$ for which $S[𝒞]$ is stationary: its length does not change if we vary the path ever so slightly, no matter how. In other words, we assume that there is exactly one geodesic. Let's call it $𝒢,$ and let's assume it lies in $𝒯.$

No matter how rapidly the phase $S[𝒞]/\hslash$ changes under variation of a generic path $𝒞,$ it will be stationary at $𝒢.$ This means, loosly speaking, that a large number of paths near $𝒢$ contribute to $I(𝒯)$ with almost equal phases. As a consequence, the magnitude of the sum of the corresponding phase factors $e^{(i/\hslash )S[𝒞]}$ is large.

If $S[𝒞]/\hslash$ is not stationary at $𝒞,$ all depends on how rapidly it changes under variation of $𝒞.$ If it changes sufficiently rapidly, the phases associated with paths near $𝒞$ are more or less equally distributed over the interval $[0,2\pi ],$ so that the corresponding phase factors add up to a complex number of comparatively small magnitude. In the limit $S[𝒞]/\hslash \to \mathrm{\infty },$ the only significant contributions to $I(𝒯)$ come from paths in the infinitesimal neighborhood of $𝒢.$

We have assumed that $𝒢$ lies in $𝒯.$ If it does not, and if $S[𝒞]/\hslash$ changes sufficiently rapidly, the phases associated with paths near any path in $𝒯$ are more or less equally distributed over the interval $[0,2\pi ],$ so that in the limit $S[𝒞]/\hslash \to \mathrm{\infty }$ there are no significant contributions to $I(𝒯).$

For a free particle, as you will remember, $S[𝒞]=-m{c}^{2}s[𝒞].$ From this we gather that the likelihood of finding a freely moving object where according to the laws of classical physics it should not be, decreases as its mass increases. Since for sufficiently massive objects the contributions to the action due to influences on their motion are small compared to $|-m{c}^{2}s[𝒞]|,$ this is equally true of objects that are not moving freely.

What, then, are the laws of classical physics?

They are what the laws of quantum physics degenerate into in the limit $\hslash \to 0.$ In this limit, as you will gather from the above, the probability of finding that a particle has traveled within a tube (however narrow) containing a geodesic, is 1, and the probability of finding that a particle has traveled within a tube (however wide) not containing a geodesic, is 0. Thus we may state the laws of classical physics (for a single "point mass", to begin with) by saying that it follows a geodesic of the geometry defined by $dS.$

This is readily generalized. The propagator for a system with $n$ degrees of freedom — such as an $m$-particle system with $n=3m$ degrees of freedom — is

$⟨{𝒫}_f,t_f|{𝒫}_i,t_i⟩=\int 𝒟𝒞e^{(i/\hslash )S[𝒞]},$

where ${𝒫}_i$ and ${𝒫}_f$ are the system's respective configurations at the initial time $t_i$ and the final time $t_f,$ and the integral sums over all paths in the system's $n+1$-dimensional configuration spacetime leading from $({𝒫}_i,t_i)$ to $({𝒫}_f,t_f).$ In this case, too, the corresponding classical system follows a geodesic of the geometry defined bythe action differential $dS,$ which now depends on $n$ spatial coordinates, one time coordinate, and the corresponding $n+1$ differentials.

The statement that a classical system follows a geodesic of the geometry defined by its action, is often referred to as the principle of least action. A more appropriate name is principle of stationary action.

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