Part 1l

Feynman’s theory

According to quantum physics, no matter how much information we obtain or how powerful our computing abilities, the outcomes of physical processes cannot be predicted with certainty because they are not determined with certainty.

Instead, given the initial state of a system, nature determines its future state through a process that is fundamentally uncertain. In other words, 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 so even though he was one of the fathers of quantum physics, 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 leads us to accept 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 rather than determining the future and past with certainty. Though that is distasteful to some, scientists must accept theories that agree with experiment, not their own preconceived notions.

What science does demand of a theory is that it be testable. If the probabilistic nature of the predictions of quantum physics meant 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.

Consider 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, according to quantum physics, 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 shepherd’s pie at your office cafeteria. As a result, if you kick a quantum buckyball and let it fly, no amount of skill or knowledge will allow you to say in advance 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, and experimenters have confirmed that the results of such experiments agree with the theory’s predictions.

It is important to realize that probabilities in quantum physics are not like probabilities in Newtonian physics, or in everyday life. We can understand this by comparing the patterns built up by the steady stream of buckyballs fired at a screen to the pattern of holes built up by players aiming for the bull’s-eye on a dartboard. Unless the players have consumed too much beer, the chances of a dart landing near the center are greatest, and diminish as you go farther out. As with the buckyballs, any given dart can land anywhere, and over time a pattern of holes that reflects the underlying probabilities will emerge.

In everyday life we might reflect that situation by saying that a dart has a certain probability of landing in various spots; but if we say that, unlike the case of the buckyballs, it is only because our knowledge of the conditions of its launch is incomplete.

We could improve our description if we knew exactly the manner in which 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. Our use of probabilistic terms to describe the outcome of events in everyday life is therefore a reflection not of the intrinsic nature of the process but only of our ignorance of certain aspects of it.

Probabilities in quantum theories are different. They reflect a fundamental randomness in nature. The quantum model of nature encompasses principles that contradict not only our everyday

experience but our intuitive concept of reality. Those who find those principles weird or difficult to believe are in good company, the company of great physicists such as Einstein and even Feynman, whose description of quantum theory we will soon present. In fact, Feynman once wrote, “I think I can safely say that nobody understands quantum mechanics.” But quantum physics agrees with observation. It has never failed a test, and it has been tested more than any other theory in science.

In the 1940s Richard Feynman had a startling insight regarding the difference between the quantum and Newtonian worlds. Feynman was intrigued by the question of how the interference pattern in the double-slit experiment arises. Recall that the pattern we find when we fire molecules with both slits open is not the sum of the patterns we find when we run the experiment twice, once with just one slit open, and once with only the other open. Instead, when both slits are open we find a series of light and dark bands, the latter being regions in which no particles land. That means that particles that would have landed in the area of the dark band if, say, only slit one was open, do not land there when slit two 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 drastically different from the way things seem to behave in everyday life, in which a ball would follow a path through one of the slits and be unaffected by the situation at the other.

According to Newtonian physics—and to the way the experiment would work if we did it with soccer balls instead of molecules—each particle follows a single well-defined route from its source to the screen. There is no room in this picture for a detour in which the particle visits the neighborhood of each slit along the way.

According to the quantum model, however, the particle is said to have no definite position during the time it is between the starting point and the endpoint.

Feynman realized one does not have to interpret that to mean that particles take no path as they travel between source and screen. It could mean instead that particles take every possible path connecting those points. This, Feynman asserted, is what makes quantum physics different from Newtonian physics.

The situation at both slits matters because, rather than following a single definite path, particles take every path, and they take them all simultaneously! That sounds like science fiction, but it isn’t. Feynman formulated a mathematical expression—the Feynman sum over histories—that reflects this idea and 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.

This, in Feynman’s view, explains how the particle acquires the information about 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.

It might sound nutty, but for the purposes of most fundamental physics done today—and for the purposes of this book—Feynman’s formulation has proved more useful than the original one.

Feynman’s view of quantum reality is crucial in understanding the theories we will soon present, so it is worth taking some time to get a feeling for how it works. Imagine a simple process in which 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, we will find the particle at some precise location B along that line.

In Feynman’s model a quantum particle samples every path connecting A and B, collecting 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. Feynman’s mathematical prescription for calculating that phase showed that 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 two 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.


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