by Lee Smolin
If we knew both A and B at a given time, we could precisely predict the future of the system.
We can choose to measure A or we can choose to measure B; and in each case we will succeed. But we can’t do better. We cannot choose to simultaneously measure both A and B.
As I have stated it, this is a prohibition of what we can measure; but, if we prefer, we can express it as a prohibition of what we can know about the system.
But wait, why can’t you measure A and then, at a later time, measure B? You can. But your measurement of B will render irrelevant (for the purpose of predicting the future) your past knowledge of A. One way this can happen is that after the measurement of B, the value of A is randomized. We cannot measure B without disrupting the value of A, and vice versa. Thus, if we measure A, then B, then A again, the value of A we get the second time will be random, and hence unrelated to the value we got the first time we measured A.
1. and 2. together are called the principle of non-commutativity. Two actions are said to commute if it doesn’t matter in what order we do them. If the action that is done first matters, we say they are non-commutative. It doesn’t matter (except to a few fanatics) in what order you put milk and sugar into coffee; they commute. Getting dressed involves non-commutative operations; the order in which you put on your underwear and pants matters. But it doesn’t matter which sock you put on first, or whether you put your socks on first, partway through the process, or last. So putting on socks commutes with everything except putting on shoes. (The mathematically minded will understand this as an application of algebra to topology.)
What if we allow there to be some specified amount of uncertainty in the measurement of A? Then we can measure B, but only up to some accuracy. These uncertainties are reciprocal—the better we know A, the worse we can know B, and vice versa.
For example, let’s suppose that A is the position of a particle. Then B is its momentum. Suppose we do a measurement that tells its location to within a meter. Then we can measure the momentum to a corresponding uncertainty. If we increase the uncertainty in A, then we can make the measurement of B more precise and vice versa. This gives us a principle called, not surprisingly, the uncertainty principle.*
(Uncertainty in A) × (Uncertainty in B) > a constant
Applied to position and momentum, it reads
(Uncertainty in position) × (Uncertainty in momentum) > a constant
Physics is like a college campus where every building is named after someone. The constant is named after Max Planck, and the uncertainty principle is named after Werner Heisenberg.
The uncertainty principle is quite powerful, as is shown by this important consequence. Let’s go back to the scenario in which you measure A, then you measure B, then you measure A again. As I said above, once you know the result of measuring B, the second measurement of A is randomized; it is no longer equal to the original value of A. But suppose that, just before you remeasure A, you do something to forget what the value of B was. Then the system remembers—yes, that word is the one we use to describe this situation—the original value of A.
This is called interference. It is allowed by the uncertainty principle because once you forget the measurement of B, B’s uncertainty is very large, so A’s uncertainty can be small.
But how can we undo a measurement? Let me give a fanciful example. There are many simple cases in which A and B each have two possible values. Let the systems we study be people, and let A be political identity, which we will simplify to be a binary choice: either left-wing or right. I will let B be pet preference, cat lovers versus dog lovers. We now play a game in which a person can’t have both a definite pet preference and a political identity. We go to a party where everyone has left-wing views and ask each whether they are a cat person or a dog person. We put the cat lovers in the living room and the dog lovers in the kitchen. If we go into either room and inquire about their political views, then half will now be right-wingers. That is what must happen if political identity and pet preference don’t commute.
But let’s afterward call everyone together into the dining room. We let them mingle for a while, then we go in and pick a random person. They could have come from either the living room or the kitchen, we don’t know which, so we’ve lost track of their pet preference. Then, when we ask them about politics, we find they are all left-wingers again.
These principles are entirely general. A and B are often the answers to yes/no questions. But in the original case, A was the position of an elementary particle, say an electron, and B was the momentum of the particle.
Momentum is one of those words that functions as a barrier to comprehension, so let’s take a moment to define it.
In physics we often have to refer to the speed and the direction of motion of a particle. We combine these into one quantity which we call the velocity. You can think of a particle’s velocity as an arrow that points in the direction of its motion. The faster the speed, the longer the arrow.
To survive a collision you want to experience as little force as possible. The force a truck will impart on a car is proportional to the truck’s change of speed. But it’s also proportional to the mass of the truck. You’d rather collide with a Ping-Pong ball than a truck, even if they are traveling toward you at the same speed. To express this, physicists define momentum as the product of the mass times the velocity. This is also an arrow pointing in the direction of motion, only now the length is proportional to both the speed and the mass.
Momentum is a central concept in physics because it is conserved. That means that in any processes at all, we can add up the momenta of the various particles involved at the beginning, and, no matter what happens, the resulting total momentum won’t change in time. Before, during, and after a collision, the total momentum will be the same. What happens in a collision is that momentum is exchanged from one body to another. This change of momentum is experienced as a force.
Energy is another conserved quantity. The total energy of a system of particles never changes in time. When particles interact, one may gain energy while the rest lose energy. But the total energy remains the same; none is created or destroyed.
FIGURE 1. A truck carries much more momentum than a Ping-Pong ball going the same velocity, because its mass is so much greater, and the momentum is the product of the mass and the velocity.
Energy and momentum are related. We won’t need the exact relation, but we need to know that a particle that is moving freely, and has an exact value of momentum, also has an exact energy.
The uncertainty principle then says that we can’t know both the position and momentum of a body at the same time. This means we can’t make a precise prediction of its future, because to do so we would need to know both where something is and how fast and in what direction it is moving, with complete accuracy.
If we want to develop an intuition about how quantum particles behave, we will need to be able to visualize a particle with a definite position, but, because of the uncertainty principle, no definite momentum or velocity. This is not hard: visualize the particle being somewhere momentarily. In the next moment it will also be somewhere definite, just somewhere else. Because its momentum is indefinite, it jumps around randomly.
But how do we visualize a particle with a definite momentum, but a completely indefinite position? This seems more challenging. If you look for it, you have an equal chance of finding it anywhere. So it is completely spread out. But how do we visualize its definite momentum?
The answer is that a particle with a definite momentum, but a completely indefinite position, can be visualized as a wave. And not just any wave, but a pure wave, one which vibrates at a single frequency.
A wave can be characterized by two numbers. One is its frequency; this is the number of times per second that it oscillates. The other is the distance between the peaks, which is known as the wavelength. These are related in the
following way: if you multiply these two numbers together, you get the speed at which the wave is traveling. Thus a wave which oscillates with a single frequency will also have a definite wavelength.
Quantum mechanics asserts that the momentum of the particle and the wavelength of the wave that represents it are related in a simple way, which is that they are inversely proportional. That is,
wavelength = h/momentum
h is the same Planck’s constant that came into the uncertainty relations.
Let us assume for a moment that no force acts on our particle, perhaps because it is very far from everything else. In the absence of forces, a particle with a definite momentum also has a definite energy. That energy is in turn related to the frequency of the wave, in that they vary proportionately.
Energy = h × frequency
These relations and correspondences are universal. Everything in the quantum world can be viewed as both a wave and a particle. This is a direct consequence of the basic principle that we can measure the particle’s position or measure its momentum, but we cannot measure both at the same time.
When we wish to measure its position, we visualize it as a particle, localized, but just momentarily, at a point in space. The momentum is completely uncertain, so the next moment, if we look again, we will find it has randomly jumped somewhere else. It can’t remain in one place because, if it did, it would have a definite value of momentum, namely zero.
If, on the other hand, we choose to measure the particle’s momentum, we will discover it has some definite value. It is nowhere in particular, so we visualize it as a wave, but one with a definite wavelength and frequency, according to the relations just mentioned.
What is so crazily fabulous about this is that waves and particles are quite different. A particle always has a definite position, localized somewhere in space. Its motion traces out a path through space, what we call its trajectory. Moreover, according to Newtonian physics, at each moment a particle also has a definite velocity and, consequently, a definite momentum. A wave is almost the opposite. It is delocalized; it spreads out as it travels, occupying all the space available to it.
But now we are learning that waves and particles are different sides of a duality, that is, different ways of visualizing one reality. A single reality with a dual nature: a duality of waves and particles.
A quantum particle can have a position. We ask where it is, and we will find it somewhere. But a quantum particle never has a trajectory, because, if we know where it is, where it will be next is completely uncertain. We must get used to thinking of particles at definite positions which are not points on trajectories. Similarly, if we measure a momentum we will always find a value. But then it’s a wave, spread out everywhere. Where we will find the particle, if we next measure its position, is completely uncertain.
This scheme, it must be admitted, has an incredible elegance. But what is most compelling is its universality. It applies to light, it applies to electrons, and it applies to all the other elementary particles known. It applies to combinations of those particles, such as atoms and molecules. It has worked successfully to describe the motions of large molecules, such as buckyballs and proteins. There is no case of an experiment that was sensitive enough to reveal the quantum nature of an object, but failed to do so. At least so far, size and complexity provide no limit. We do not yet know if the wave-particle duality applies to people or cats or planets or stars, but there is no reason known why it definitely can’t.
In all these cases the effect is the same: we can only know half of what we would need to know to precisely predict the future.
THREE
How Quanta Change
In the first lecture of his course on quantum mechanics, my teacher Herbert Bernstein asserted that physics is the science of everything. Our goal in physics is to find the most general laws of nature, from which the multitude of phenomena exhibited by nature may all be explained.
Quantum mechanics explains the widest variety of phenomena of any theory so far. At the same time, it greatly restricts the questions that can be asked of any particular phenomenon. We have already encountered one kind of limitation: that we can know only half what we would need to know about a system to make precise predictions for its future. As a result, we must give up describing exactly what goes on in individual atoms in favor of statistical predictions, which apply only to averages taken over many cases. Hence, to believe in quantum theory we must give up the ambition to precisely predict the future.
Most physicists have given up those ambitions in the face of the success of quantum mechanics. But I believe that this is shortsighted and there is a deeper level of reality to be discovered, the mastery of which will restore our ambitions for a complete understanding of nature.
Another restriction limits the range of quantum theory. We can express this in a principle I call the subsystem principle:
Any system quantum mechanics applies to must be a subsystem of a larger system.
One reason for this is that quantum mechanics refers only to physical quantities which are measured by measuring instruments, and these must be outside the system being studied. Further, the results of these measurements are perceived and recorded by observers, who are also not part of the system being studied.
Most of us approach science with the naive expectation that it will tell us what is real. We can follow John Bell and call a real property of a system a beable: it is part of what is. Bell coined the word as a contrast to the term observables, which is what anti-realists want out of a theory.
“Observables” and “beables” are loaded terms, whose use can signify allegiance to a side of the debate between realism and anti-realism. An observable is a quantity produced by an experiment or an observation. There is no commitment to believe it corresponded to something that exists apart from the measurement or had a value before the measurement. Anti-realists use this term to emphasize that the quantities quantum physicists measure need have no existence apart from, or prior to, our observation of them. Realists use John Bell’s term “beable” to refer to the reality that they believe exists whether we measure something or not.
Most scientific explanations, whether of the flights of cannonballs or of birds and bees, speak in terms of beables.
But not quantum mechanics! As Heisenberg and Bohr insisted, quantum mechanics speaks not in terms of what is, but only of what has been observed. There is, according to them, no useful talk about beables in the atomic domain; instead, quantum mechanics deals only in observables.
To measure an atom’s observables, we impose on it a large, macroscopic instrument. By definition, that device is not part of the system whose observables we are studying. Nor is the observer.
Therefore, to be described in the language of quantum mechanics, a system must be part of a larger system that includes the observer and her measuring instruments. Hence our subsystem principle.
Most applications of quantum theory are to atoms and molecules or other tiny systems; in these cases the restriction is irrelevant. But some of us have the ambition to describe the whole universe. We feel that is the ultimate goal of science. However, the universe as a whole is not, by definition, part of a larger system. The subsystem principle frustrates our hope to have a theory of the whole universe.
There is a subtle but key difference between the idea that quantum mechanics is the theory of everything, and the hope of extending quantum theory to include the whole universe. What Professor Bernstein meant by his claim is that physics is the root of the correct description of everything—each considered as a subsystem of the whole. It is very different to imagine applying quantum theory to the entire universe, which would mean including us observers inside the system being studied, and our measuring instruments.
Over the last century several attempts were made to extend quantum mechanics to a theory of the whole universe. We will meet one of these later on; a
part of our overall argument is that these attempts fail.
For one thing, making the observer a part of the system being described raises tricky questions of self-reference. It is not even clear that an observer can give a complete self-description, because the act of observing or describing yourself changes you.
But there are deeper reasons why quantum mechanics cannot be extended to a theory of the whole universe.
In several of my books (namely The Life of the Cosmos, Time Reborn, and The Singular Universe and the Reality of Time, written with Roberto Mangabeira Unger), I investigate the question of how physics may be extended to give a theory of the whole universe. I conclude that a theory of the whole universe must differ in several crucial aspects from any of the physical theories so far developed, including quantum mechanics. All these theories only make sense when interpreted as descriptions of a portion of the universe.
Indeed, the fact that quantum mechanics only makes sense when read as a theory of a part of the universe is, by itself, a sufficient reason for regarding quantum mechanics as incomplete. One thing we may ask of a theory that completes quantum theory is that it makes sense when extended to a description of the universe as a whole.
However, this is not the only line of thought that leads to the conclusion that quantum mechanics is incomplete. Other concerns and difficulties had far more influence on how the subject has evolved historically. For the time being, I will ignore the cosmological issues and focus on more immediate challenges.
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THE PROCESS OF APPLYING general laws to a specific physical system has three steps.
First, we specify the physical system we want to study.
The second step is to describe that system at a moment of time in terms of a list of properties. If the system is made of particles, the properties will include the positions and momenta of those particles. If it is made of waves, then we give their wavelengths and frequencies. And so on. These listed properties make up the state of the system.