How to Teach Physics to Your Dog

by CHAD ORZEL

  You could be more specific about the measurement time, by measuring for only half a second, say, and counting one tail wag, but then the frequency uncertainty becomes much larger: f = 2.0 ± 0.2 Hz. You can’t have a small uncertainty in both the frequency of an oscillation and the time when it was measured.

  The same logic applies to measuring the frequency of light, though the oscillations are much too fast to count by hand. We directly see this uncertainty principle in action in the interaction between light and atoms. We know from chapters 2 and 3 that atoms will be found only in certain allowed energy states, and that they move between these states by absorbing or emitting photons.

  When an atom moves from a high-energy state to a lower-energy state, the frequency of the emitted photon is determined by the energy difference between the two states. That energy difference has some uncertainty, though, that depends on how much time the atom spent in the higher-energy state. Two identical atoms placed in the same high-energy state can thus emit photons with very slightly different frequencies.

  The difference is tiny—for typical atoms, it’s about one hundred- millionth of the frequency of the photons. This can be measured using lasers, though, and this tiny frequency uncertainty limits our ability to make certain measurements of atomic properties.

  “So it takes you awhile to count things. Big deal. What’s that have to do with bunnies made of cheese?”

  “The frequency-counting example is just one example of a more general principle. Energy-time uncertainty holds no matter what form the energy is in.”

  “Why?”

  “Well, all forms of energy have to be equivalent, because you can convert one form of energy to another. So, if you have a photon of uncertain energy, and you use it to start an electron moving, the kinetic energy of that electron must be uncertain.”

  “I still don’t see what this has to do with bunnies.”

  “We know from Einstein’s theory of relativity that mass and energy are equivalent—”

  “E = mc2!”

  “Exactly. Since mass is just another form of energy, you can convert energy into mass and mass into energy. Mass has uncertainty just like all the other forms of energy, and that uncertainty is related to how long the mass stuck around.”

  “So a bunny made of cheese would have an uncertain mass?”

  “Right. If it was around for only a short time, the uncertainty could be very large—for something like a top quark, which sticks around for only 10-25 seconds, the quantum uncertainty due to that lifetime is close to 1% of the total mass.”

  “But if it was around for a long time, the mass uncertainty would be small? I want a bunny made of cheese with a small mass uncertainty!”

  “Good luck with that. A bunny made of cheese would have an awfully short lifetime around you.”

  “Ooh. Good point.”

  WHEN THE HUMANS ARE AWAY . . . : VIRTUAL PARTICLES

  How does this get us bunnies made of cheese? Well, let’s think about applying this uncertainty principle to empty space. If we look at some small region over a long period of time, we can be quite confident that it is empty. Over a short interval, though, we can’t say for certain that it isn’t empty. The space might contain some particles, and in quantum mechanics, that means it will.

  Uncertainty about the emptiness of space isn’t as strange as it may seem at first. If a physicist or a stage magician gives a dog a box to inspect at leisure, she can conclusively state that the box is empty. She can sniff in all the corners, check for false bottoms, and make absolutely sure that there’s nothing hiding in some little recess. If she’s allowed only a brief peek or a quick sniff inside the box, though, she can’t be as confident that the box is empty. There might be something tucked into a corner that she wasn’t able to detect in that short time.

  The amount of time needed to determine whether the box is empty also depends on the size of the thing you might expect to find. You don’t need to look for very long to determine whether the box contains Professor Schrödinger’s famous cat, but if you’re attempting to rule out the presence of a much smaller object—a crumb of a dog treat, say—a more thorough inspection is required, and that takes time.*

  The same idea applies to empty space in quantum physics, via the energy-time uncertainty relationship. When we look at an empty box over a long period of time, we can measure its energy content with a small uncertainty, and know that there is only zero-point energy—no particles are in the box. If we look over only a short interval, however, the uncertainty in the energy can be quite large. Since energy is equivalent to mass through Einstein’s famous E = mc 2, this means that we can’t be certain that the box doesn’t contain any particles. And as with Schrödinger’s cat, if we don’t know the exact state of what’s in the box, it’s in a superposition of all the allowed states. The cat is both alive and dead, and the box is both empty and full of all manner of particles, at the same time.

  To put it another way, the box can contain some particles, as long as they appear and disappear fast enough that we don’t see them directly. But how can we arrange for particles to disappear so conveniently? Ordinary particles don’t disappear when we’re not looking, unless they’re edible and there’s a dog in the area.

  Particles can disappear from the box provided they come in pairs, one matter and one antimatter. Every particle in the universe has an antimatter equivalent with the same mass and the opposite charge—the antiparticle for an electron is a positron, the antiparticle for a proton is an antiproton, and so on. When a particle of ordinary matter comes into contact with its anti-particle, the two annihilate each other, converting their mass into energy.

  In practical terms, this means that a particle and its anti-particle can pop into existence from nothing inside the box. This temporarily increases the energy inside the box by a small amount—an electron-positron pair increases the energy by two electron masses multiplied by the speed of light squared—but as long as they mutually annihilate in a short enough time, there’s no problem, because the uncertainty in the energy is large enough to cover two extra particles.*

  How long do we need to look to rule out any possibility of the box containing a particular type of particle? Well, we need the uncertainty in the energy to be smaller than the energy that the particle has from Einstein’s E = mc2, so the time will depend on the mass. To make the energy uncertainty small enough to rule out the possibility of the box containing one electron and one positron, we need only to look at the box for 10-21 seconds. That’s a billionth of a trillionth of a second—slightly less than the time required for light to travel from one side of an atom to the other. A proton and antiproton have a greater mass, and would last even less time—10-24 seconds or less. A bunny with a mass of a kilogram (whether it’s made of cheese or something else), would last for 10-52 seconds,* or 0.00000000000000000 00000000000000000000000000000000001 s.

  “I think you’re making this much more complicated than it needs to be. How hard is it to measure zero particles?”

  “This, from a dog who has to sniff the entire kitchen floor every night, on the off chance that we dropped a molecule of food while making dinner?”

  “But you do drop food sometimes, and I like your food better than mine.”

  “The point is, it’s really difficult to measure zero. You can never really say that something has a definite value of zero, only that the value is no bigger than the uncertainty associated with your measurement. Some people spend their whole scientific careers measuring things to be zero, with better and better precision.”

  “That sounds pretty depressing.”

  “It does require a certain personality type. Anyway, that’s kind of a side issue, because we know that virtual particles exist. We can detect their effects.”

  “Wait, if these things only stick around for such a short time, how do you detect them?”

  “We can’t detect them, not directly. We know they exist because we can see the effects they have on other particl
es. Virtual particles popping in and out of existence change the way that real particles interact with one another.”

  “How does that work?”

  “Well . . .”

  EVERY PICTURE TELLS A STORY: FEYNMAN DIAGRAMS AND QED

  Virtual particles, particles that appear and disappear too quickly to be seen directly, may seem too fanciful for a serious scientific theory. In fact, they are absolutely critical to the theory known as quantum electrodynamics or QED, which describes the interaction between light and matter at the most fundamental level. QED describes all interactions between electrons, or between electrons and electric or magnetic fields, in terms of electrons absorbing and emitting photons.

  The best-known form of QED relies on pictures called Feynman diagrams, after the noted physicist and colorful character Richard Feynman, who invented them as a sort of calculational shortcut. These diagrams represent complex calculations in the form of pictures that tell a story about what happens as particles interact.* The simplest Feynman diagram for an electron interacting with an electric or magnetic field looks like this:

  The straight lines represent an electron moving through some region of space, and the squiggly line represents a photon from the electromagnetic field. In this diagram, time flows from the bottom of the picture to the top of the picture, while the horizontal direction indicates motion through space, so the diagram itself is a story in pictures: “Once upon a time, there was an electron, which interacted with a photon, and changed its direction of motion.”

  You might think that this little story, boring as it is, is the complete picture of what happens when an electron interacts with light, but it turns out there are myriad other possibilities, thanks to virtual particles. For example, we can have diagrams like these:

  These tell much more eventful stories: on the left, our electron is moving along through space, and just before it interacts with the photon from the electromagnetic field, it emits a virtual photon* and changes direction. Then it absorbs the real photon from the field, changes direction again, then reabsorbs the virtual photon it emitted earlier. The right diagram is even stranger: the virtual photon spontaneously turns into an electron-positron pair,† which then mutually annihilate, turning back into a photon, which gets reabsorbed.

  These diagrams show the odd relationship between virtual particles and the normal laws of physics. The virtual particles seem to disobey some basic rules of physics—a real electron could never move fast enough to get back in front of a photon to reabsorb it—but virtual particles can break the rules as long as they’re not around long enough to violate energy-time uncertainty. It’s similar to the relationship between a dog and furniture: when humans are looking, being on the couch is strictly forbidden, but as long as there are no humans around, and she gets off before they see her, it’s a great place to nap.

  Each of these Feynman diagrams represents a tiny slice of the story of what might happen to an electron interacting with a photon from an electric or magnetic field. Each of these diagrams also stands for a calculation in QED giving the final energy of the electron, and the likelihood of the events depicted. As you increase the number of virtual particles, the diagrams become much less likely, but they remain possible, as long as they happen very quickly.

  The diagrams on the previous page only involve electrons, positrons, and photons, but any type of particle can turn up as a virtual particle. The likelihood of a particle appearing decreases as its mass increases, so a virtual proton-antiproton pair is much less likely than a virtual electron-positron pair (a proton has almost 2,000 times the mass of an electron), but in principle, there is no limit. If you wait long enough, you should expect any and every type of object showing up as a virtual particle—even bunnies made of cheese.

  “Okay, so virtual particles affect the way electrons interact with photons. Big deal. Why should I care?”

  “The photons in those Feynman diagrams could represent any sort of electric or magnetic interaction. Those diagrams could be describing an electron interacting with another electron, or with a proton, and that happens all the time.”

  “Okay, I have no idea what you’re talking about.”

  “Well, when you’re dealing with QED, you talk about interactions in terms of the exchange of particles. Two electrons that repel each other do so by passing a photon from one to the other—one emits a photon, which travels over to the other, and gets absorbed. The absorption and emission change the momentum, and we see that as a force pushing the two particles apart.”

  “That sounds complicated. Why would you think about it that way?”

  “It turns out to be more convenient, mathematically. You can also look at it as a natural way to incorporate the fact that nothing can travel faster than light. In the classical picture of electric forces developed before Einstein, when you change the position of one electron, the force on the other should change instantaneously, no matter how far away it is. That goes against relativity, which says that there’s no way to transmit anything faster than light.”

  “Oh. That’s a problem.”

  “That’s right. If we think of the forces as arising from photons which are passed from one particle to another, though, that takes care of the problem. The photons travel at the speed of light, and the force doesn’t change until the photons get there.”

  “So, the photons in those diagrams . . .”

  “They could be real photons, due to the electron interacting with a magnet or an electric field, or they could be photons emitted by another electron. Either way, the effect is the same—the presence of virtual particles changes the interaction, and we can detect that.”

  “You still haven’t explained how to detect it, though.”

  “I’m getting there. If you’d stop interrupting . . .”

  THE MOST PRECISELY TESTED THEORY IN HISTORY: EXPERIMENTAL VERIFICATION OF QED

  Virtual particles come and go in an extremely short time, faster than we can directly observe. We know they exist, though, because interactions with virtual particles change the way electrons interact. The effect is tiny, but we can measure it, and it agrees with experimental observations to fourteen decimal places. Not only do these experiments confirm that QED is correct, they may provide a way to detect the existence of new subatomic particles without using billion-dollar particle accelerators.

  How does this work? Well, because the virtual particles are around for such a short time, we don’t know which of the many possibilities took place for any given electron. Quantum mechanics tells us that if we don’t know the exact state of a particle, it exists in a combination of all the possible states—a superposition state like those we’ve discussed in previous chapters. So, when physicists calculate the effect of an electron interacting with an electric or magnetic field, they need to include all the possible Feynman diagrams describing the process.*

  In a certain sense, as the electron passes from point A to point B, it follows all the possible paths from A to B at the same time. It absorbs a single real photon, but also emits and reab-sorbs a virtual photon, and that virtual photon does and doesn’t turn into an electron-positron pair, and so on. All of these processes are possible, so they all contribute to the superposition of Feynman diagrams.

  Another way of thinking about this is to note that we never see a real electron interacting with a single photon; rather, what we see is the cumulative effect of a great many repeated interactions. If we could watch each of these in detail, we would see that most of the time, the electron simply absorbs the photon, with no virtual funny business. Maybe one time in ten thousand, it will emit a virtual photon. One virtual photon in ten thousand will create a virtual electron-positron pair, and so on. Each time you get one of these different processes, it changes the total energy by a small amount, and that needs to be taken into account.

  In this view, the electron is like a dog walking down the street. The chances of the dog stopping to sniff any given plant are very small,* but there are dozens of p
lants along the side of any suburban street, and one of them is bound to smell fascinating, and demand further investigation. The human holding the leash will need to take that delay into account when planning the walk.

  We don’t have the ability to watch each of the interactions of a real electron in sufficient detail to keep track of just how many times virtual particles were involved in the interaction, any more than we can predict exactly which plants the dog will stop to sniff, but we can model the effect that this has on the interaction. Adding up Feynman diagrams can be thought of as calculating the average change in the energy of the electron from absorbing one real photon. We then describe the cumulative effect of many photon absorptions, which may or may not involve virtual particles, as a series of absorptions with the same average energy, in the same way that we can describe the progress of a dog by saying that he will stop three times a block on average. It may be five times on some blocks, and only one time on others, but over the entire walk, the effect is the same as if he stopped three times on every block.

  Whether you think of it as a superposition of all possible paths at the same time, or an average interaction being used to cover the details of many individual interactions, the effect of virtual particles shows up when we look at an electron interacting with a magnetic field. Electrons (and all other material particles) have a property called “spin” that makes them act like tiny little magnets, with north and south poles.* The energy of an electron whose poles are aligned with the magnetic field is very slightly different than the energy of an electron whose poles point in the opposite direction.

  The energy difference between these states in a magnetic field depends on a number called the gyromagnetic ratio, or g-factor, of the electron, which basically tells you how big a magnet you get for a given amount of “spin.” The simplest quantum theory of the electron says that the value of this ratio should be exactly 2, which is what you would get if there were no virtual particles. Thanks to the contributions of virtual particles, the actual value is very slightly higher.