Feynman’s theory

According to quantum physics, the outcomes of physical processes cannot be predicted with certainty because they are not determined with certainty.

Instead, nature determines the future state of a system through a process that is fundamentally uncertain.

Nature does not dictate the outcome of any process or experiment, even in the simplest of situations.

Rather, it allows a number of different eventualities, each with a certain likelihood of being realized.

It is, to paraphrase Einstein, as if God throws the dice before deciding the result of every physical process.

That idea bothered Einstein and is why he later became critical of it.

Quantum physics might seem to undermine the idea that nature is governed by laws. But that is not the case.

Instead, it creates a new form of determinism: Given the state of a system at some time, the laws of nature determine the probabilities of various futures and pasts instead of determining the future and past with certainty.

This is distasteful to some. But this agrees with experiment, not their own preconceived notions.

Science demands that a theory be testable.

If the probabilistic nature of quantum physics predictions means that it was impossible to confirm those predictions, then quantum theories would not qualify as valid theories.

But despite the probabilistic nature of their predictions, we can still test quantum theories.

For instance, we can repeat an experiment many times and confirm that the frequency of various outcomes conforms to the probabilities predicted.

The Buckyball Experiment

Quantum physics tells us that nothing is ever located at a definite point because if it were, the uncertainty in momentum would have to be infinite.

In fact, each particle has some probability of being found anywhere in the universe.

So even if the chances of finding a given electron within the double-slit apparatus are very high, there will always be some chance that it could be found instead:

• on the far side of the star Alpha Centauri, or
• in the pie at your office cafeteria.

So if you kick a quantum buckyball and let it fly, no amount of skill or knowledge will allow you to say exactly where it will land.

But if you repeat that experiment many times, the data you obtain will reflect the probability of finding the ball at various locations.

Experimenters have confirmed that the results of such experiments agree with the theory’s predictions.

Probabilities in quantum physics are not like probabilities in Newtonian physics, or in everyday life.

We can compare the patterns:

• built up by the steady stream of buckyballs fired at a screen
• built up by the pattern of holes by players aiming for the bull’s-eye on a dartboard

The chances of a dart landing near the center are greatest. They diminish as you go farther out.

As with the buckyballs, any given dart can land anywhere. Over time, a pattern of holes that reflects the underlying probabilities will emerge.

We might compare this to everyday probabilities and say that a dart has a certain probability of landing in various spots.

But the landing is based on the conditions of its launch.

We could improve our description if we knew exactly how the player released the dart, its angle, spin, velocity, and so forth.

In principle, then, we could predict where the dart will land with a precision as great as we desire.*

But probabilities in quantum theories are different. They reflect a fundamental randomness in nature.

The quantum model of nature encompasses principles that contradict:

• our everyday experience
• our intuitive concept of reality*

The Double-slit Experiment

Richard Feynman once wrote, “I think I can safely say that nobody understands quantum mechanics.”

In the 1940’s, he was intrigued how the interference pattern in the double-slit experiment arises.

When we fire molecules with both slits open, the pattern that emerges is different from the pattern with just 1 slit open. Instead, when both slits are open, we find a series of light and dark bands.

• The dark bands are regions where no particles land.

That means that particles that would have landed in the area of the dark band if, say, only slit 1 was open, do not land there when slit 2 is also open.

It seems as if, somewhere on their journey from source to screen, the particles acquire information about both slits.

That kind of behavior is different from how balls behave in everyday life.

• A ball would follow a path through one of the slits and be unaffected by the situation at the other.

According to Newtonian physics, each particle follows a single well-defined route from its source to the screen.

• It cannot detour along the way.

But according to the quantum model, the particle has no definite position when it is between the starting point and the endpoint.

Feynman realized that particles take every possible path between those points. Feynman asserted that this makes quantum physics different from Newtonian physics.

The situation at both slits matters because particles take every path simultaneously!

Feynman formulated its math as the Feynman sum over histories. This reproduces all the laws of quantum physics.

In Feynman’s theory the mathematics and physical picture are different from that of the original formulation of quantum physics.

• But the predictions are the same.

In the double-slit experiment, Feynman’s ideas mean the particles take paths that go through only one slit or only the other

paths that thread through the first slit, back out through the second slit, and then through the first again;

paths that visit the restaurant that serves that great curried shrimp, and then circle Jupiter a few times before heading home; even paths that go across the universe and back.

He thinks that this explains how the particle acquires the information on which slits are open.

If a slit is open, the particle takes paths through it.

When both slits are open, the paths in which the particle travels through one slit can interfere with the paths in which it travels through the other, causing the interference.

Feynman’s formulation has proved more useful than the original one.

Imagine a simple process wherein a particle begins at some location A and moves freely.

In the Newtonian model, that particle will follow a straight line.

After a certain precise time passes, the particle will be at some precise location B along that line.

In Feynman’s model, a quantum particle samples every path connecting A and B. It collects a number called a phase for each path.

That phase represents the position in the cycle of a wave, that is, whether the wave is at a crest or trough or some precise position in between.

When you add together the waves from all the paths, you get the “probability amplitude” that the particle, starting at A, will reach B.

The square of that probability amplitude then gives the correct probability that the particle will reach B.

The phase that each individual path contributes to the Feynman sum (and hence to the probability of going from A to B) can be visualized as an arrow that is of fixed length but can point in any direction.

To add 2 phases, you place the arrow representing one phase at the end of the arrow representing the other, to get a new arrow representing the sum.

To add more phases, you simply continue the process.

Note that when the phases line up, the arrow representing the total can be quite long.

But if they point in different directions, they tend to cancel when you add them, leaving you with not much of an arrow at all. The idea is illustrated in the figures below.

To carry out Feynman’s prescription for calculating the probability amplitude that a particle beginning at a location A will end up at a location B, you add the phases, or arrows, associated with every path connecting A and B. There are an infinite number of paths, which makes the mathematics a bit complicated, but it works. Some of the paths are pictured below.

Feynman’s theory gives an especially clear picture of how a Newtonian world picture can arise from quantum physics, which seems very different. According to Feynman’s theory, the phases associated with each path depend upon Planck’s constant.

The theory dictates that because Planck’s constant is so small, when you add the contribution from paths that are close to each other the phases normally vary wildly, and so, as in the figure above, they tend to add to zero.

But the theory also shows that there are certain paths for which the phases have a tendency to line up, and so those paths are favored; that is, they make a larger contribution to the observed behavior of the particle.

It turns out that for large objects, paths very similar to the path predicted by Newton’s will have similar phases and add up to give by far the largest contribution to the sum, and so the only destination that has a probability effectively greater than zero is the destination predicted by Newtonian theory, and that destination has a probability that is very nearly one.

Hence large objects move just as Newton’s theory predicts they will.

So far we have discussed Feynman’s ideas in the context of the double-slit experiment. In that experiment particles are fired toward a wall with slits, and we measure the location, on a screen placed beyond the wall, at which the particles end up. More generally, instead of just a single particle Feynman’s theory allows us to predict the probable outcomes of a “system,” which could be a particle, a set of particles, or even the entire universe. Between the initial state of the system and our later measurement of its properties, those properties evolve in some way, which physicists call the system’s history. In the double-slit experiment, for example, the history of the particle is simply its path.

Just as for the double-slit experiment the chance of observing the particle to land at any given point depends upon all the paths that could have gotten it there, Feynman showed that, for a general system, the probability of any observation is constructed from all the possible histories that could have led to that observation. Because of that his method is called the “sum over histories” or “alternative histories” formulation of quantum physics.