Fear of Physics (14 page)

The important point about all this is that I can consider the “probabilities” associated with two different paths, each of which, when taken separately, might yield a nonzero final probability but, when added together, cancel out to give a zero result. This is exactly what can happen for the electrons traveling to the phosphorescent screen through the two slits. If I consider a certain point on the screen, and I cover up one slit, I find that there is a nonzero probability for the electron to take a trajectory from A to B through this slit and get to the screen:

Similarly, there is a nonzero probability for it to go from A to B if I cover up the other slit:

However, if I allow both possible paths, the final probability to go from A to B, based on the sum of the quantum-mechanical “probabilities” for each path, can be zero:

The physical manifestation of this is simple. If I cover up either slit, I see a bright spot develop on the screen at B as I send electrons through one by one. However, if I leave both open, the screen remains dark at B.
Even though only one electron at a time travels to the screen, the probability of arriving at B depends upon both paths being available, as if the electron somehow “goes through”
both slits!
When we check to see if this is the case, by putting a detector at either slit, we find that each electron goes through one or the other, but now the spot at B is bright! Detecting the electron, forcing it to betray its presence, to make a choice if you will, has the same effect as closing one of the slits: It changes the rules!

I have taken the trouble to describe this in some detail not just to introduce you to a phenomenon that is fundamental to the way the world is at atomic scales. Rather, I want to describe how, hanging on tenaciously to this wild, but proven result, and to the implications of special relativity, we are forced into consequences that even those who first predicted them found difficult to accept. But physics progresses by pushing proven theories to their extremes, not by abandoning them just because the going gets tough.

If we are to believe that electrons, as they travel about, “explore” all trajectories available to them, impervious to our ability to check up on them, we must accept the possibility that some of these trajectories might be “impossible” if only they were measurable. In particular, since even repeated careful measurements of a particle’s position at successive times cannot establish unambiguously its velocity between these times, by the uncertainty principle, it may be that for very short times a particle might travel faster than the speed of light. Now it is one of the fundamental consequences of the special relationship between space and time that Einstein proposed—in order, I remind you, to reconcile the required constancy of the speed of light for all observers—that nothing can be measured to travel faster than the speed of light.

We are now led to the famous question: If a tree falls in the forest and no one is there to hear it, does it make a sound? Or, perhaps more pertinent to this discussion: If an elementary particle whose average measured velocity is less than the speed of light momentarily travels faster than the speed of light during an interval
so small that I cannot directly measure it, can this have any observable consequences? The answer is yes in both cases.

Special relativity so closely connects space and time that it constrains velocities, which associate a distance traveled with a certain time. It is an inevitable consequence of the new connections between space and time imposed by relativity that, were an object to be measured to travel faster than the speed of light by one set of observers, it could be measured by other observers to be going
backward
in time! This is one of the reasons such travel is forbidden (otherwise causality could be violated, and for example, as all science fiction writers are aware, such unacceptable possibilities could occur as shooting my grandmother before I was born!). Now quantum mechanics seems to imply that particles can, in principle, travel faster than light for intervals so small I cannot measure their velocity. As long as I cannot measure such velocities directly, there is no operational violation of special relativity. However, if the quantum theory is to remain consistent with special relativity, then during these intervals, even if I cannot measure them, such particles must be able to
behave
as though they were traveling backward in time.

What does this mean, practically? Well, let’s draw the trajectory of the electron as seen by a hypothetical observer who might watch this momentary jump in time:

If such an observer were recording her observations at the three times labeled 1, 2, and 3, she would measure one particle at time 1,
three
particles at 2, and one particle again at 3. In other words, the number of particles present at any one time would not be fixed! Sometimes there may be one electron moving along its merry way, and at other times this electron may be accompanied by two others, albeit one of which apparently is moving backward in time.

But what does it mean to say an electron is moving backward in time? Well, I know a particle is an electron by measuring its mass and its electric charge, the latter of which is negative. The electron traveling from position B at one time to position A at an earlier time represents the flow of negative charge from left to right as I go backward in time. For an observer who is herself moving forward in time, as observers tend to do, this will be recorded as the flow of
positive
charge from right to left. Thus, our observer will indeed measure three particles present at times between 1 and 3, all of which will appear to move forward in time, but one of these particles, which will have the same mass as the electron, will have a positive charge. In this case, the series of events in the previous diagram would be pictured instead as:

From this viewpoint, the picture is just a little less strange. At time 1, the observer sees an electron moving from left to right. At position A, at a time between 1 and 2, the observer suddenly sees an additional pair of particles appear from nothing. One of the particles, with positive charge, moves off to the left, and the other, with negative charge, moves off to the right. Some time later, at position B, the positive-charged particle and the original electron collide and disappear, leaving only the “new” electron continuing to move on its merry way from left to right.

Now, as I said, no observer can actually measure the velocity of the original electron for the time interval between 1 and 3, and thus no physical observer could directly measure this apparent spontaneous “creation” of particles from nothing, just as no observer will ever measure the original particle to be traveling greater than the speed of light. But whether or not we can measure it directly, if the laws of quantum mechanics allow for this possibility, they must, if they are to be made consistent with special relativity, allow for the spontaneous creation and annihilation of pairs of particles on time scales so short that we cannot measure their presence directly. We call such particles
virtual particles.
And as I described in chapter 1, processes such as those pictured above may not be directly observable, but they do leave an indirect signature on processes that are directly observable, as Bethe and his colleagues were able to calculate.

 

 

The equation that amalgamated the laws of quantum mechanics as they apply to electrons with special relativity was first written down in 1928 by the laconic British physicist Paul Adrian Maurice Dirac, one of the group that helped discover the laws of quantum mechanics several years earlier, and the eventual Lucasian Professor of Mathematics, a position now held by Hawking
and earlier occupied by Newton. This combined theory, called quantum electrodynamics, which formed the subject of the famous Shelter Island meeting, was only fully understood some twenty years later, due to the work of Feynman and others.

No two physicists could have been more different than Dirac and Feynman. As much as Feynman was an extrovert, so much was Dirac an introvert. The middle child of a Swiss teacher of French in Bristol, England, young Paul was made to follow his father’s rule to address him only in French, in order that the boy learn that language. Since Paul could not express himself well in French, he chose to remain silent, an inclination that would remain with him for the rest of his life. It is said (and may be true) that Niels Bohr, the most famous physicist of his day, and director of the institute in Copenhagen where Dirac went to work after receiving his Ph.D. at Cambridge, went to visit Lord Rutherford, the British physicist, some time after Dirac’s arrival. He complained about this new young researcher, who had not said anything since his arrival. Rutherford countered by telling Bohr a story along the following lines: A man walks into a store wanting to buy a parrot. The clerk shows him three birds. The first is a splendid yellow and white, and has a vocabulary of 300 words. When asked the price, the clerk replies, $5,000. The second bird was even more colorful than the first, and spoke well in four languages! Again the man asked the price, and was told that this bird could be purchased for $25,000. The man then spied the third bird, which was somewhat ragged, sitting in his cage. He asked the clerk how many foreign languages this bird could speak, and was told, “none.” Feeling budget-conscious, the man expectantly asked how much this bird was. “$100,000” was the response. Incredulous, the man said, “What? This bird is nowhere near as colorful as the first, and nowhere near as conversant as the second.
How on earth can you see fit to charge so much?” Whereupon the clerk smiled politely and replied, “This bird thinks!”