Hamilton's principle/Explainers/Sum over histories

Technical version: The Feynman path integral

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

This page explains why the whole idea works, according to quantum mechanics. It uses no mathematics. The technical page it goes with is named above.

The question nobody could answer

Phases adding near the stationary path and cancelling elsewhere.

The volume has spent a long time on a rule: nature picks the path where the action is stationary.

The rule works. It reproduces Newton's laws, it explains conservation, it builds electromagnetism and gravity.

But it is a strange rule to believe. How would a ball know? To pick the path where the action is stationary, something has to compare all the paths. A ball leaving your hand does not appear to survey the alternatives.

For three centuries the honest answer was that nature does behave this way, and nobody knew why. Some people found the rule so odd that they suspected purpose behind it. Maupertuis, who proposed an early version in 1744, thought it showed the economy of a wise creator.

The actual answer arrived in 1948, and it is stranger than the puzzle.

Every path happens

The quantum answer is that the ball does not pick a path.

It takes all of them.

Every path from start to finish contributes. The sensible ones and the ridiculous ones. The straight one, the one that loops via the ceiling, the one that goes to the Moon and comes back in time for tea. All of them.

And they all contribute equally strongly. The classical path is not weighted more heavily. It is not marked out. Nothing in the rule mentions it.

What differs between paths is not strength but timing.

Timing and cancellation

Picture each path as carrying a tiny clock hand. As the path is traversed the hand rotates. How far it rotates depends on the action of that path — the same quantity the volume has been discussing all along.

At the end, add up all the hands, arrow to arrow. The size of the total tells you how likely the journey is.

Now think about what happens when you add up an enormous number of arrows pointing in random directions. They cancel. Point them every which way and the total is close to nothing.

For anything bigger than an atom, the action is huge compared with the natural quantum unit. So a tiny change of path spins the clock hand round many times over. Neighbouring paths point in wildly different directions.

They cancel. Almost every path contributes nothing, not because it is forbidden, but because a nearby path is pointing the opposite way.

Except in one place

There is one exception, and it is the whole point.

Near a path where the action is stationary, nudging the path barely changes the action. That was the definition, from Comparing paths.

Barely changing the action means barely rotating the clock hand. So all those paths point in nearly the same direction.

Arrows pointing the same way do not cancel. They add.

So the total is dominated by paths near the stationary one. Everything else destroys itself. What survives is what we call the classical path.

What this changes

The rule has not been replaced. It has been explained, and the explanation inverts it.

Before: nature picks the path where the action is stationary.

After: nature does not pick anything. It does everything at once, and stationary action is simply the place where an enormous cancellation fails to happen.

Hamilton's principle is not a law about nature's preferences. It is a fact about adding up a very large number of things that mostly cancel.

That is why the ball does not need to know. There is no survey and no choice. What looks like selection is the residue left over after interference.

It settles the word "least"

This volume has insisted throughout that the right word is stationary, not least. The quantum picture confirms it, and from an unexpected angle.

For the arrows to add up rather than cancel, all that is needed is that the action barely changes near the path. It makes no difference whether the action is smallest there, largest there, or neither.

A maximum reinforces exactly as well as a minimum does.

So the light path that takes the longest time, mentioned earlier in the volume, is a genuine path for exactly the same reason as any other. Its arrows add up. Nothing about the physics prefers small numbers.

Anyone who says "least action" is describing a special case and calling it the rule.

Where the classical picture fails

The cancellation argument depended on the action being large compared with the quantum unit. For everyday objects it is — by a factor with something like twenty-seven zeros.

For an electron it is not. The cancellation is then incomplete, other paths survive, and the object stops having a single trajectory in any useful sense.

That is not a failure of measurement or a gap in our knowledge. The electron really does go several ways, and the ways interfere. Fire electrons at a screen with two slits and you get a pattern that only makes sense if each electron went through both.

So classical mechanics is not wrong. It is what quantum mechanics looks like when the numbers are large enough for almost everything to cancel.

Where to go next

This is the end of the argument. To see how the rule was found in the first place, start at Comparing paths. To see why it fails to mean "least", go to Where the minimum fails.