The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)

Chapter 4. (Im)probabilities

There are more things in heaven and Earth, Horatio,
Than are dreamt of in your philosophy.

—William Shakespeare, Hamlet

The Schrödinger equation gave quantum mechanics a solid mathematical foundation. But what did it mean to use a wave to describe a “solid” object like an electron? Physicists were used to classical physics, in which there is always a smooth transition from one state of motion to another. What were they to make of the bizarre quantum jumps from one energy level to the next? What was the relationship between the wave nature and the particle nature of objects? How were physicists to make sense of this absurd theory?

Half a Molecule Plus Half a Molecule Equals No Molecules

Successful explanation of the atomic line spectra provided good evidence for both the wave nature of electrons and the correctness of the Schrödinger equation. Could the wave nature of electrons be demonstrated more directly? An American experimenter, Clinton Davisson, decided to find out. Davisson realized that it would be impossible to perform a two-slit experiment for electrons like Young’s two-slit experiment that demonstrated the wave nature of light. The wavelength of electrons, according to de Broglie’s formula, was too small: You would need atom-sized slits to see electron interference. Davisson had discovered how to make crystals of metallic nickel, and he realized that an electron beam fired at such a crystal should behave in a similar manner as light behaves in the two-slit experiment. The nickel atoms in the crystal form an orderly array. When the electron beam strikes the surface of the crystal, each nickel atom reflects part of the beam. These reflected beams should combine like waves, according to de Broglie. That is, the reflected beam should display interference: At some locations the reflected waves should add (constructive interference), whereas at other locations they should cancel (destructive interference), just as in the two-slit experiment. Davisson successfully measured the interference pattern in 1927, providing the first direct experimental demonstration that particles have a wave nature.

Wave behavior of electrons may not seem so surprising. After all, no one has ever seen an individual electron; the electron is almost as much an “airy nothing” as the photon, the particle of light. Indeed, an electron is the smallest-mass particle of those that make up ordinary matter, and so is perhaps as close to nothing as a particle can get. However, similar experiments have since been performed with beams of neutrons, of atoms, even of whole molecules. Molecules are as solid as you can get: Everything we call solid is made of molecules, including your body and mine. If we can demonstrate wave behavior for molecules, the unavoidable conclusion is that everything in the universe must have a wave nature.

The largest molecules for which wave behavior has been demonstrated to date are called buckminsterfullerenes, or buckyballs, because of their shape, which resembles the geodesic domes built by architect Buckminster Fuller. Each molecule consists of 60 carbon atoms arranged in a soccer ball shape. Instead of a barrier with two slits, the experiment uses a grating having multiple slits 1/10 of a micron apart. These slits corral the buckyballs into a narrow beam. If the molecules behaved like peas shot from a peashooter, we would expect the detector to record molecules only in the direct beam path. Instead, the experimental results clearly showed the wave behavior of the molecules: spreading (diffraction) of the beam, and interference arising from the multiple slits of the grating.

We might be tempted to explain buckyball interference by assuming the molecules interact with each other in some way. Perhaps they bump into each other on the way to the detector, and that is what causes the peaks and valleys of the interference pattern. But this explanation doesn’t work: the pattern is the same even if we send the buckyballs through one at a time. Each buckyball interferes with itself, as if it somehow passes through all the slits at once. The buckyball, though, is about 100 times smaller than the slit width, so saying that it “passes through all the slits at once” is like saying that a soccer ball goes through both goals simultaneously, at opposite ends of the field.

The interference experiment reveals the fundamental conceptual difficulty in quantum mechanics, the thing that induced physicists to use words like “crazy” and “absurd.” Is an electron (or an atom, or a molecule) a particle or a wave? A single molecule can be isolated and weighed, it can collide with another molecule and bounce off, it can even be imaged using a scanning tunneling microscope. These properties make sense only if a molecule is a particle, something like a tiny billiard ball. On the other hand, the interference experiment only makes sense if the molecule can spread out like a wave in order to go through all the slits at once. Neither the wave model nor the particle model by itself can explain all of the experiments. We need a new conceptual model, something that is neither particle nor wave, something for which our everyday experience provides no analogy. That “something” is what I have been calling the quantum field. It is a field that obeys a wave equation (the Schrödinger equation) that gives us the properties of interference and so on, but it always comes in chunks—in any interaction, only a whole electron (or atom or molecule) is emitted or absorbed or detected.

If molecules show wave behavior, and everyday objects are made of molecules, why don’t everyday objects exhibit that same behavior? Why can’t we demonstrate interference effects using a peashooter and a board with two holes drilled in it? The answer lies in the size of Planck’s constant, the new constant of nature in the Schrödinger equation. Together with the mass and the velocity of the particle, this constant determines the wavelength of the particle. Peas shot from a peashooter, for example, would have a wavelength of about 10-30 meter, much less than the width of a nucleus. To detect these interference fringes you would need to measure the position of the pea to this precision—an impossible task. In special relativity, it was the large speed of light that made special relativistic effects difficult to detect, and hence unfamiliar in our everyday experience. For quantum mechanics, it is the smallness of Planck’s constant that removes wave phenomena so far from our everyday experience. Only at the atomic level do quantum effects become large enough to detect.

Gefiting Chancy

In the 1920s, physicists struggled to understand the quantum field and what it meant. Max Born, a German physicist, realized that everything made sense if the field was related to the probability of finding the particle at a given point in space.

Suppose we modify the two-slit experiment to trap the particle after it passes the barrier by placing a box behind each slit. Fire a single particle, an electron, say, toward the slits. The quantum field, we know, passes through bothslits, and so part of the field ends up in one box and part ends up in the other box.

Born realized that the quantum field can’t be the electron itself: An electron never splits up. You never find half an electron in each box. What you find instead is that, if you repeat the experiment many times, half of the time the electron ends up in one box and half of the time in the other box.

Probabilities are always positive numbers. If something is certain to happen (for instance, death or taxes), it has probability one, that is, 100 percent chance of happening. If something (say, your teenager being home by 10 P.M. on Friday night) is certain never to happen, it has probability zero. When there is less certainty, the probability is a number between zero and one: a fair coin has probability one-half of coming up heads. You combine probabilities by adding them. For instance, the probability that the coin will come up either heads or tails is equal to: (probability of heads) + (probability of tails) = 1/2 + 1/2 = 1. The coin must come up either heads or tails, so the total probability must be one.

Now, the probability can’t be equal to the value of the quantum field, because probabilities are always positive numbers, whereas the quantum field can be positive or negative. Born discovered that the probability is equal to the square of the quantum field:

Probability = (Quantum Field)2

This relationship makes interference phenomena possible. Suppose we tried to make a theory that used only the probability, instead of the quantum field value. Think again about the two-slit experiment. When only one slit is open, there is some probability that your detector will detect an electron. Now, open the other slit. According to the rules of probability, the probability of detecting an electron is now equal to the sum: (probability that electron comes from slit 1) + (probability that electron comes from slit 2). Since probabilities are always positive numbers, this sum is always bigger than the original probability. Using only probability, we can’t explain what we found in the interference experiment: that opening the second slit can make the probability decrease.

Quantum mechanics gets around this by working with the quantum field instead of with the probability itself. We have to take the quantum field for an electron coming from slit 1 and add the quantum field for an electron coming from slit 2. Because the field values can be positive or negative, some places on the viewing screen will have a positive contribution from slit 1 and a negative contribution from slit 2, so that the sum is zero: These are the nodes (the valleys) of the interference pattern. The probability of detecting an electron at one of these nodal points is the total field value squared. Zero squared is equal to zero: We never see an electron at a node.

Adding quantum fields rather than probabilities is the fundamental feature of quantum mechanics that makes its predictions so counterintuitive. This principle is important enough to have a name: the superposition principle. The superposition principle says the way to find the probability of any outcome is to add all the quantum fields for all the possible routes to that outcome and then square the result. The superposition principle allows us to combine any two (or more) quantum states to obtain a new quantum state. For example, we can combine two energy levels of the harmonic oscillator to get a new state. The average energy of this new state lies between the energies of the original states. Measure the energy of the particle in the superposition state, though, and you will always observe one of the original energies, never the new, intermediate energy. The superposition state represents some probability of finding the particle in energy level 1 and some probability of finding the particle in energy level 2. We can adjust the probabilities at will, creating different superposition states with larger or smaller probability of either energy level, as long as the two probabilities sum to 1 (the particle must be in some energy level).

A superposition state is a very strange beast, for which there is no analogy in everyday experience. Suppose you wanted to paint your house red, but your spouse preferred blue. “No problem,” says the painter, “I’ll just mix the red with the blue.” You agree reluctantly, fully expecting to come home to a purple house. Instead, you find that each time you return home the house is either red or blue. It’s never purple. This isn’t the way things work in our experience. We all know that if you mix red and blue paint you get purple paint, not paint that is sometimes red and sometimes blue. But in the microworld, where quantum mechanics rules, this is how things work. Using more red paint and less blue paint doesn’t make the house redder, it’s just red on more days. On blue days, the house is just as blue as if you hadn’t used any red paint at all.

The superposition principle applies to any aspect of the particle, to any of its measurable properties. For instance, a particle’s position can be in a superposition state. We already encountered one example: the modified two-slit experiment, in which the electron could end up in either the upper box or the lower box. When we look inside the boxes, we find the electron in one box or the other—not in some location in between. The velocity, the spin, even the charge of a particle can be in a superposition state. Superpositions of different charge states will be crucial for understanding quark physics.

The quantum field embodies the principles of quantum mechanics stated in the previous chapter:

a. The motion of any particle is described by a wave, which we call the quantum field.

b. The probability for the particle to be detected at a given point is equal to the square of the quantum field at that point.

c. The quantum field changes according to the Schrödinger equation.

Now, principles (A) and (C) are essentially the same as in classical physics. The Schrödinger equation, which tells how the quantum field changes in time, is quite similar to Maxwell’s equations, which tell how the (classical) electric and magnetic fields change in time. There is no more randomness in this time evolution of the quantum field than there is in classical physics. The great departure from classical physics comes in the interpretation of the quantum field, given in (B). In classical physics, the fields were interpreted as real, physically existing entities spread throughout space. The quantum field, in contrast, is only an information wave. It doesn’t tell us where the electron is. Rather, it summarizes everything we know about the electron. Quantum mechanics doesn’t model the physical world, it reflects what we can say about the world.

The universe, it seems, does not admit of a complete description. The quantum mechanical description is the best we can do, and it gives only probabilities, not certainties.

I’m Not Sure If I’m Heisenberg

Quantum mechanics deals in probabilities. In some situations, though, the probability predicted by quantum mechanics is equal to one: There is no uncertainty about the outcome. For instance, an electron in the ground state of the hydrogen atom will be found to have the same energy each time the energy is measured. Could we avoid probabilities entirely by working only with states like this?

The Heisenberg uncertainty principle gives the answer: No! Physical quantities in quantum mechanics always come in complementary pairs: the more certainty we have about one quantity, the less certain we will be about some other quantity. The uncertainty principle is not an addition to quantum mechanics; it follows from the wave nature of particles. Suppose we try to trap a particle in a square well. Like the harmonic oscillator or the hydrogen atom, the square well has a set of discrete energy levels (calculable from the Schrödinger equation). In the figure that follows, the particle occupies a definite energy level, but its position is uncertain: it might be found anywhere in the well, or even a short distance outside the well. The particle’s velocity is also uncertain: the possible values range roughly from zero to the value determined by the wavelength in this state:


Let’s reduce the uncertainty in position by decreasing the size of the box. Squeezing the box makes the wavelength decrease. Thanks to de Broglie, we know that a smaller wavelength corresponds to a larger velocity. Decreasing the uncertainty in position thus increases the range of possible velocities of the particle. We conclude that position and velocity are complementary physical quantities in quantum mechanics: decreasing the uncertainty of one quantity increases the uncertainty of the other.

The uncertainty relation just derived is the position-momentum uncertainty principle. Momentum being simply mass times velocity, a greater range of velocity implies a greater range of momentum. Writing △x for the range of possible position measurements (that is, the width of the box) and Δp for the range of momentum measurements, the position-momentum uncertainty principle becomes △x Δp >020, where021is Planck’s constant: There can be more uncertainty in the possible measured values, but never less. Closely related is the energy-time uncertainty principle: ΔE △t >022. The more narrowly the energy is specified (△E), the more time it takes to do the measurement (△t). This uncertainty relation relates the lifetime of a decaying particle to the energy of the process causing the decay; it will come in handy in later chapters.

In classical physics, knowing the state of the universe means knowing the exact position and velocity of every particle. According to the uncertainty principle(s), such knowledge is impossible. The more precisely we constrain the one, the more uncertain the other becomes. Complete knowledge of the classical state of even a single particle can never be obtained. The best we can do is to determine its quantum state, which only gives the probabilities of any of the measurable quantities.

The Dice-Playing God

For physicists from the time of Newton until the early twentieth century, the goal of physics had been the same—to predict the future. Consider your body: It is made of particles (atoms) interacting via fields (electromagnetic and gravitational). In principle, according to classical physics, a physicist could discover the complete state of your body (the position and velocity of every atom together with the field information), then use the equations of physics to calculate your every move—what time you will go to bed, when you will wake up, what your next words will be. This seems to make the universe a rather depressing place for a human—you are under the compulsion of the laws of physics. You have no choice about your next words, your next meal. Even if we cannot, in practice, calculate what you will do next, either because we don’t yet know all the laws of physics, because we can’t determine the exact location of every atom in your body, or because our computers can’t handle the calculation, the very idea that such a calculation is possible in principle seems to leave the universe a very soulless place. Where is there any room for individual choice, for free will, in this picture? A human, it seems, is a mere robot following a complicated computer program.

Quantum mechanics completely undermined this mechanistic view of the universe, by removing not one but two of its foundations. First, according to the Heisenberg uncertainty principle, it is impossible, even in principle, to determine the exact position and velocity of each particle in your body. The best that can be done, even for a single particle, is to determine the quantum state of the particle, which necessarily leaves some uncertainty about its position or velocity. Second, the laws of physics are not deterministic but probabilistic: given the (quantum) state of your body, only the probabilities of different behaviors could be predicted. Quantum mechanics, it seems, releases us from robothood.

This radical change of perspective took a long time to be absorbed by the scientific community. The rules of the game had changed: The goal was no longer to predict the future completely, but to learn as much as can be known about the future, namely, the probabilities of some range of possible outcomes. Albert Einstein famously rejected the new view: “God does not play dice,” he said. He claimed that quantum mechanics was incomplete; some deeper theory would be able to predict outcomes with certainty.

Think about rolling dice. If we knew the precise angle of throw, the exact effect of air resistance, and the friction with the table on which the die lands, then, in principle, we could predict with certainty which number will come up. In practice, the calculation is much too difficult, so the best we can do is a statement of the probability of each outcome: one in six (1/6) for each of the numbers on the die. Rolling dice is not inherently random; the outcome only seems random because of our ignorance of the little details, the hidden variables (like launch angle and friction) that determine the outcome of the roll.

Some laws of physics are probabilistic for the same reason as for the dice. Take, for example, the atomic model of gases. By averaging the motions of a huge number of particles, one can derive the ideal gas law and many other properties of the gas. Because the theory deals with the average properties, though, many questions we might ask about the individual molecules can only be answered in terms of probabilities. The atomic model can give us the probability that a particular gas molecule has a certain velocity, but we can’t determine its actual velocity without knowing its entire history: its original velocity when it first entered the container, the angle and speed of every collision with the walls or the other molecules. If we knew those hidden variables, though, we could determine the exact velocity of the molecule. Einstein, and other physicists since him, wondered if quantum mechanics could be like rolling dice. Perhaps it only gives probabilities because we are ignorant of some of the hidden variables. Perhaps it is only an average theory, like the atomic model. Perhaps a deeper theory, one that included those variables, would give definite predictions.

Most physicists ignored Einstein’s objections and accepted the arguments of Niels Bohr and others that quantum mechanics was complete. In the meantime, quantum mechanics racked up one successful prediction after another. Then, in 1967, John S. Bell published a paper that resolved the issue. Bell proved that any theory that involved hidden variables as envisioned by Einstein, and that excluded faster-than-light effects, must make predictions that conflict with the predictions of quantum mechanics. Quantum mechanics cannot simply be incomplete in the way that the atomic model is incomplete. If the hidden variable idea is right, then either quantum mechanics or special relativity is wrong. Bell’s paper received little attention at first, but in 1969 it was realized that it was possible to directly test the conclusion experimentally. Since then, many further tests have been done, and quantum mechanics has won resoundingly. Bell’s paper is now recognized as a fundamental advance in our understanding of quantum mechanics.

Let’s delve a little more deeply into the meaning of Bell’s discovery. Here’s one of the basic equations of classical physics for the motion of an object with constant velocity: x = vt. In this equation, x represents the position of the object (measured from some reference point that we label x = 0), t represents the elapsed time, and v stands for the velocity of the object. For instance, a car traveling at 60 miles per hour will cover, in two hours, a distance x = vt = (60 miles/hour) (2 hours) = 120 miles. This may seem very straightforward and simple, but we need to recognize that something very subtle has taken place. We have taken an equation (x = vt), an abstract mathematical expression, and related it to a physical phenomenon (motion of an object), by way of an interpretational scheme (x ↔ position of object, t ↔ elapsed time, v ↔ velocity of object). This was the great triumph of physics from Galileo in the sixteenth century through Maxwell in the nineteenth: that physical phenomena could be transformed, via some interpretational scheme, into mathematical relationships.

In cases like the simple equation x = vt, the interpretation is straightforward. We have an intuitive notion of the concepts position of the object and time elapsed, and we can make those ideas more precise if necessary. For example, position of the car could be specified as the distance from the tip of mile marker 1 to the license plate on the rear bumper, measured in miles. But even in classical physics, the interpretation scheme can be quite complicated and removed from everyday experience. The electric field of Maxwell’s equations, for example, is not something that can be experienced directly. We need an interpretational scheme that goes something like this: The electric field at a location represents the force that would be felt by an object carrying one unit of charge if placed at that location, and if the introduction of that charge didn’t affect the distribution of the other charges that created the field. The interpretation here involves a hypothetical situation, the introduction of an additional electric charge, which may be difficult if not impossible to accomplish in practice. If, for example, we are concerned with the electric field due to a charged conducting sphere, we know that introducing an additional charge in the vicinity of the sphere will cause the charges on the sphere to move around. There is no practical way to “nail down” the charges as required by the interpretational scheme. Still, this interpretation of the electric field is extremely useful; in fact, it works so well that physicists have no hesitation in thinking of the electric field as a real, physical entity, onto-logically on a par with atoms, molecules, dust specks, and cars. The electric field (more precisely, a change in the electric field) moves with a measurable speed: the speed of light. The field also carries energy; for instance, the electromagnetic (light) energy reflected from this page carries energy to your eyes, which convert the light energy into nerve impulses that your brain can decode. Suppose we wanted to avoid the electric field concept entirely, and use equations that refer only to positions and velocities of “real” objects: stars, planets, rocks, and particles. Then we would find that energy disappears from the light bulb illuminating this book, reappears briefly on the page, then disappears again before reappearing in the rods and cones of your eyeball. Now, there is nothing logically wrong with an interpretation that avoids the electric field concept entirely, in which energy leaps into and out of existence like this. However, the electric field concept is so useful, and the conservation of energy so compelling theoretically and so well established experimentally that it seems eminently reasonable to consider both energy and electric fields as real in the same sense that atoms (and cars) are real.

What about quantum mechanics? Is the quantum field real? True, it describes the motion of matter (the electron, say) and therefore energy (the electron’s kinetic energy, for example) from one place to another and so, like the electric field, would seem to be real enough. But the probability interpretation makes this view difficult to maintain.

Consider again the modified two-slit experiment discussed earlier. The incoming quantum field splits into two parts; there is an equal probability for the electron to end up in either of the two boxes that we placed behind the screen. Now suppose we insert a detector behind one of the slits. As the quantum field passes the detector, the detector either clicks, registering that a particle was in that box, or it doesn’t click, in which case the particle must have been in the other box. There is no longer a 50-50 probability. The probability is now 100 percent for one of the two boxes and 0 percent for the other box. This is sometimes called the collapse of the wave function: Part of the quantum field disappears and the quantum field elsewhere instantaneously changes. Now imagine extending the two boxes in different directions. You can make them as long as you want, so the two arms of the apparatus could be separated by miles, or light-years, when the collapse takes place. If the quantum field is a real physical object, we seem to have a violation of special relativity—a physical effect that travels faster than the speed of light.

The violation is only apparent, however. Many years of careful study of such situations have revealed that this kind of instantaneous rearrangement of the quantum field cannot be used to send a signal faster than the speed of light. It is only through our interpretation of the quantum field as a physical object that a faster-than-light effect seems to arise. If we want to avoid faster-than-light effects (illusory though they may be), we are forced to declare that the quantum field is not a real, physical object. But then, what is it? Recall that the field tells us the probability of finding the electron at a particular position. If the quantum field is not a physical entity, perhaps it is a mathematical device that encodes all our knowledge about the electron. It is an information wave. In this view, the particle (the electron) is the independently existing physical entity, and the quantum field summarizes everything we know about the particle—the combined effect of everything that has occurred in the particle’s past. Then, the collapse of the quantum field involves no physical effect that moves faster than the speed of light; it is merely a reshuffling of information that occurs whenever new information is obtained, as when Monty Hall opens Door Number Three to show that the grand prize is not there, and the probabilities of the prize being behind Doors One and Two immediately change.

The main question in this information interpretation of the quantum field is this: How does the electron know where it should be? If the quantum field is not a physical object, why is the electron’s probability always governed by the quantum field? “It just is” is a possible, but not very satisfactory, answer. We would rather have a theory that tells us how the real, physical objects behave than one that is a mere calculation tool for probable outcomes.

Let’s suppose, then, that there is a deeper theory of the electron. Perhaps the electron has some unknown properties that determine its motion, or perhaps it interacts with a guiding wave that tells it where to go. This is where Bell’s result comes in. We can say with complete confidence that, no matter what unknown properties or guiding waves we introduce, the new theory will either involve physical effects that move faster than the speed of light, or else it will be in conflict with the results of quantum mechanics. We see there are three possible options:

1. Quantum mechanics may be violated. Since Bell’s paper was published, many experiments have searched for such violations. Some early experiments seemed to show that, indeed, quantum mechanics was flawed, but all of the more recent and more accurate experiments have resoundingly supported quantum mechanics.

2. Special relativity may be violated, but in such a way that no signals can be sent faster than light speed. A small group of physicists is trying to develop theories along these lines. Unless quantum mechanics is proven wrong by experiments (see option 1), these theories must agree in the end with quantum mechanics, so the attitude of the majority of physicist to these attempts is “Why bother?”

3. We can avoid questions of interpretation by throwing up our hands and saying “It just is” the way quantum mechanics says it is. This is the approach taken by most working physicists today.

This is the rather sorry state of our understanding of quantum mechanics. As the theory (or rather, its extension to relativistic quantum field theory) continues to pass one experimental test after another, we still lack an agreed-upon interpretational scheme for the theory. Think about our earlier example using the equation x = vt. The interpretational scheme involved here is this:


What about quantum mechanics? The equation we use is the Schrödinger equation:


Here, the interpretational scheme is as follows:


For the first time in physics, we have an equation that allows us to describe the behavior of objects in the universe with astounding accuracy, but for which one of the mathematical objects of the theory, the quantum field ψ, apparently does not correspond to any known physical quantity.

The implications are deep and far-reaching, not just for physicists and their practice, but also for how we understand the world around us. The old view of a clockwork universe is comforting in its comprehensibility. It is easy for a physicist to imagine taking apart the universe like a clockmaker taking apart a clock, pulling electrons out of atoms like pulling a gear from among its neighbors. Each piece fits together with those around it, and by understanding the pieces and how they interact, the physicist can understand the whole. On the other hand, this is a cold, mechanical view of the world: every event, even every human action and emotion, is determined by the interactions of the constituent pieces, and those interactions are fixed by the preceding interactions, and so on, back to the beginning of the universe.

The quantum mechanical universe is very different from the clockwork universe, although the manner in which it differs depends on which of the three previous options is chosen. I will ignore the first possibility, that quantum mechanics is simply wrong. Quantum mechanics, and its descendant, relativistic quantum field theory, which is built on the framework of quantum mechanics, are simply too well supported by experiment. Even if the specifics of the Standard Model prove incorrect, the framework of quantum mechanics will certainly stand for many years to come.

The second option, of instantaneous (or at least faster-than-light) communication between particles, leads in the direction of a mystical, New Age view of the universe that makes some physicists uneasy. The argument goes like this: Because of the strange, non-local nature of the quantum field that we saw earlier, any two electrons that interact carry a strange sort of correlation, an instantaneous connection that can be ascribed neither to a property that the electron has of itself nor to a communication (in the usual sense) between the electrons. But before these two electrons interacted, they interacted with other electrons, and before that with other electrons. It seems unavoidable that all the electrons in the universe are caught in this web of interactions, so that any given electron (or indeed any other particle) has a kind of mystical, instantaneous connection with every other object in the universe. This is the view of quantum mechanics that led to comparisons with Eastern religions and was popularized in the 1970s and 1980s in books like Fritjof Capra’s The Tao of Physics. This view, or at least the connection with religious mysticism, is usually derided by physics writers today, but it remains true that in some approaches to quantum mechanics, these long-distance connections are real and physically important.

The third option denies physical reality to the long-distance correlations, in the sense that there is no separate, physical field or other entity postulated to account for the correlations. However, by doing so, it leaves the correlations completely unexplained. Quantum mechanics can tell us what correlations to expect (and, lo and behold, the predicted correlations are found experimentally), but it leaves us utterly without a mechanism, without an answer to the question “Why?” It is not that this view denies the existence of an independent physical reality; it is just that nothing in our theoretical framework can be identified with that reality. The quantum mechanical formalism tells how to compute the probabilities for the outcome of any experiment, but it is only a calculational tool that magically produces the right answer; the mathematics is not a direct reflection of the underlying physical reality as it was for Newton and Maxwell.

Philosophically, the most astonishing thing about quantum mechanics is the extent to which it protects us from the existential despair of the clockwork universe. Nor is it necessary to speculate on the possibility of long-distance correlations providing a route for mystical spiritual interconnectedness, or other metaphysical mumbo-jumbo. In order to speak of a “quantum state” at all, we need to have a repeatable sequence of physical processes that produce the state. This is how a quantum state is defined. It is simply impossible to apply the quantum state concept to the unique configuration of electrons, protons, neutrons, and photons that is a human being. No repeatable process will produce another copy of me at the moment of writing this sentence, or of you at the moment of reading it. Furthermore, even if such a description were possible, quantum mechanics gives us only the probabilities of various outcomes. Given the quantum state of the reader at, say, noon today, we could (in principle—such a thing is beyond the capabilities of modern computers the way the Andromeda galaxy is beyond your neighbor’s house) compute a 6 percent probability that you will have spaghetti for dinner tomorrow, 3percent probability of macaroni and cheese, 5 percent for hamburger.... Such predictions are no assault on free will; similar predictions could be made simply by looking at your past eating habits. It is almost as though the rules of the universe were designed to protect our free will.

Physicists are split on which of the three views of quantum mechanics they believe. Many, perhaps most, prefer not to worry about the problem as long as they can calculate the answers they need. A few brave souls struggle with the questions of interpretation or construct competing models that act as foils against which quantum mechanics can be tested. It is clear, though, that quantum mechanics replaces the clockwork universe, in which the future can be determined to any desired precision, with a world of unpredictability. Not just complex systems like the weather or the brain, but the simplest imaginable systems, a single particle, for instance, are subject to this randomness.

Quantum mechanics was not the end of the story, however: problems both theoretical and experimental remained. On the theoretical side, quantum mechanics takes a Newtonian view of space and time—it is incompatible with special relativity. On the experimental side, the details of the atomic line spectra turned up surprises. Some lines split into two or more lines when new techniques allowed a closer inspection. Other lines were shifted slightly from their expected positions. Small though these differences were, they would spark a new revolution. The grand synthesis that finally harmonized special relativity and quantum mechanics would also solve the experimental puzzles and lay the foundation for the Standard Model. That synthesis is called relativistic quantum field theory.