by Lee Smolin
The third step is to postulate a law to describe how the system changes in time.
Before quantum physics, physicists had a simple but powerful ambition for science. At the second step we would be able to describe a system in terms that were complete, in two senses. Complete means, first of all, that a more detailed description is neither needed nor possible. Any other property the system might have would be a consequence of those already included. Additionally, the list of properties should be exactly what is needed to give precise predictions of the future. This is done using the laws. The future can be determined precisely, given our complete knowledge of the present. This is the second meaning of the description being complete.
Between Newton, in the late seventeenth century, and the invention of quantum mechanics in the 1920s, it was believed that the properties making up that complete description were the positions of all the particles and their momenta.
It might, of course, happen that we don’t know the precise positions and momenta of all the particles making up a system. The air in this room consists of around 1028 atoms and molecules, so a complete listing of their positions is impossible. We have to use a very approximate description in terms of density, pressure, and temperature. These refer to averages of the atoms’ positions and motions. Our bulk description will have to employ probabilities, and the predictions it makes will then be to some degree uncertain.
But the use of probabilities is just for our convenience, and the resulting uncertainties just express our ignorance. Behind our bulk description of a gas in terms of density and temperature, we continue to believe there is a precise description, which includes listing where every last atom is and how it is moving. We share a faith that if we had access to that description we could use the laws to predict the future precisely. That faith is based on the belief in realism—that there is an objective reality, which it is possible for us to know.
Quantum mechanics blocks this complacent ambition, because its first principle asserts we can know, at most, only half the information that would be needed to realize it.
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THE COMPLETE INFORMATION NEEDED to precisely predict the future is called a classical state. “Classical” is how we refer to physics as it was between Newton and the discovery of the quantum. It is then natural to call a specification of half of that information a quantum state. The half is arbitrary; it can be chosen to be only the momentum, or only the position, or some mixture of these, as long as half the information needed to precisely predict the future is present, and half is missing.
The quantum state is a central notion in quantum theory. A realist will want to ask: Is it real? Does a particle’s quantum state correspond precisely to the physical reality of that particle? Or is it just a convenient tool to make predictions? Perhaps the quantum state is a description, not of the particle, but only of the information we have about the particle?
We are not going to resolve these questions here. Experts disagree about them. We will soon enough have the chance to focus on these and other questions about the meaning and correctness of quantum mechanics. For now we take a pragmatic viewpoint and regard the quantum state as a tool for making predictions about the future.
A quantum state is a useful tool because it can do just that. This is our next principle:
Given the quantum state of an isolated system at one time, there is a law that will predict the precise quantum state of that system at any other time.
This law is called Rule 1. It is also sometimes called the Schrödinger equation. The principle that there is such a law is called unitarity.
Thus, while the relation between the quantum state and the behavior of an individual particle can be statistical, the theory is deterministic when it comes to how the quantum state changes in time.
As we said, quantum states with definite values of energy and momentum are represented by pure waves with exact frequency and wavelength. But these quantum states are very special. What about other quantum states, whose momenta are uncertain, so that they do not vibrate at a single frequency and with a single wavelength? More general quantum states are represented by waves with arbitrary profiles. These are sharp in neither position nor momentum, so if either of these quantities is measured, there will be uncertainties.*
There are also states of definite position and completely indefinite momentum; if we graph them, they look like spikes, which are zero everywhere except the single point where the particle is. Other states are peaked over a region of space and correspond to particles which are localized imprecisely, so we know only approximately where they are.
One way to make a general quantum state is by adding together pure waves, each with a different frequency and wavelength.
FIGURE 2. Three wave functions are illustrated showing how different kinds of states are represented. (A) shows a pure wave of a single wavelength, which corresponds to a definite momentum. The position is completely uncertain, as is required by the uncertainty principle. The spike in (B) shows a state with a definite position, but the wavelength is completely indefinite and uncertain. The intermediate case (C) is built by combining several wavelengths, so the momentum and position are both somewhat uncertain.
If we measure the energy of such a combination, we get a range of values corresponding to the different frequencies that make up the wave.
If this were music, the waves would be sound waves. A pure wave with a single frequency sounds a single note. Playing several notes simultaneously produces a chord. There is no limit to how many notes you can play at once, nor to how many quantum states can be added together.
Combining two states by adding the waves that represent them is called superposing the states. It corresponds to combining two ways the particle may have traveled to arrive at the detector. Earlier, when we divided cat and dog people into the living room and the kitchen, each room represented a quantum state, defined by a definite pet preference. When we brought everyone together in the dining room we superposed those two states.
This is an example of a general principle called the superposition principle.
Any two quantum states may be superposed together to define a third quantum state. This is done by adding together the waves that correspond to the two states. This corresponds to a physical process that forgets the property that distinguished the two.
Logically, a superposition of two states C and D communicates C or D. The person could be a dog lover or she could be a cat lover. The connector “or” means something has been forgotten. Someone might have been a C or they might have been a D, but when we’ve forgotten which, we can only say they are a C or a D.
As I have emphasized, quantum states are important because they evolve in time according to a definite rule. The relation between the quantum state and an observation is probabilistic, but the relationship between that state now and the quantum state at a different time is definite. But there is an important caveat, which is that the definite evolution rule applies only to systems that are isolated from the rest of the universe. Only in cases where the system is free from disturbances or influences from outside sources is the evolution rule deterministic.
When we make a measurement on a system, we disturb it, typically by forcing it to interact with a measuring instrument. So Rule 1 does not apply to measurements. This is true not only of measurements, but of any interaction between the system and outside forces. So is there anything special about measurements?
Measurements are special because they are where probabilities enter quantum theory.
Quantum mechanics asserts that the relationship between the quantum state and the outcome of a measurement is probabilistic. Generally, there is a range of possible outcomes of a given measurement. These will each occur with some probability, and these probabilities depend on the quantum state. In the case where we measure the position of a particle, this dependence is particularly sim
ple:
The probability of finding the particle at a particular location in space is proportional to the square of the height of the corresponding wave at that point.
This is called the Born rule, after Max Born, who proposed it.
Why the square? Probability is always positive, but waves generally oscillate between positive and negative values. But the square of a number is always positive, and it is the square that is related to probability. The important thing to remember is that the larger the magnitude, or height, of a wave, the more likely that you will find the corresponding particle there.
These last few points are key to how quantum mechanics works, so let me summarize them: The wave represents the quantum state. When we leave the system alone, it changes in time deterministically, according to Rule 1. But the quantum state is only indirectly related to what we observe when we make a measurement, and that relation is not deterministic. The relation between the quantum state and what we observe is probabilistic. Randomness enters in a fundamental way.
But, even if the quantum state gives us only probabilities for what we observe, once we get a result, there is something that is definite, because afterward you know exactly what the state is. It is the state corresponding to the result obtained by the measurement. Suppose we measure an electron’s momentum, and get the result that the electron is moving north with momentum 17 (in some units). Then, just after the measurement we know that the quantum state is NORTHWARD, MOMENTUM = 17.
This is enshrined in a second rule,* which we call Rule 2:
The outcome of a measurement can only be predicted probabilistically. But afterward, the measurement changes the quantum state of the system being measured, by putting it in the state corresponding to the result of the measurement. This is called collapse of the wave function.
For example, in our story about political and pet preferences, as soon as a person answers a question about either one, they go into the quantum state defined by having that definite preference.
Since the outcome of the measurement is probabilistic, so is the change in the quantum state dictated by Rule 2.
Once the measurement is over, the system can be considered to be isolated again and Rule 1 takes over, until the next measurement.
Rule 2 raises a whole bunch of questions.
Does the wave function collapse abruptly or does it take some time?
Does the collapse take place as soon as the system interacts with the detector? Or only later, when a record is made? Or perhaps later still, when it is perceived by a conscious mind?
Is the collapse a physical change, which means that the quantum state is real? Or is it just a change in our knowledge of the system, which means the quantum state is only a representation of that knowledge?
How does a system know a particular interaction has taken place with a detector, so that it should then, and only then, obey Rule 2?
What happens if we combine the original system and the detector into a larger system? Does Rule 1 then apply to the whole system?
These questions are all different aspects of the measurement problem.
Diverse answers have been given, which have been a source of controversy for nearly a century. We will have a lot to say about all this, once we have the full picture.
FOUR
How Quanta Share
Useful as it is under everyday circumstances to say that the world exists “out there” independent of us, that view can no longer be upheld.
—JOHN ARCHIBALD WHEELER
The superposition of quantum systems poses a grave challenge to realism. But an even more insidious set of obstacles to realism comes from how quantum mechanics describes systems which are built by combining simpler systems.
Superposition is about combining different possible states of a single system. As I said, it corresponds to “or.” Quantum mechanics also has interesting things to say about combining two different systems to make a composite system. Suppose we have an electron and a proton. Each has to begin with its own quantum state. We can combine them to make a hydrogen atom. The whole atom has its own quantum state, which is made by combining the states of its constituents. This corresponds to “and.” Each quantum state represents half the possible information needed for a complete description of its components. The joint quantum state also represents half the possible information about the atom. This leads to very interesting new phenomena.
Let us consider again people with two incompatible properties, political views and pet preference. Let’s suppose Anna and Beth share an apartment. They talk about getting a pet. Individually, Anna is a cat lover and so is Beth. The state of their couple is just the combination of these. Each has a definite pet preference, so each has indefinite political views. If asked for her political preference, each will have a 50 percent chance of answering left and a 50 percent chance of answering right. So, if asked about politics, half the time they will discover they agree and half the time they will discover they disagree. In the state in question, in which they each separately have a definite pet preference, their political views are random and uncorrelated. Anna stating her political views has no effect on Beth’s views.
Quantum physics also allows us to define states for the couple in which all their individual views are indefinite, but we can have definite knowledge of how their views relate. An important example of such a state is one in which the only thing that is certain is that, if we ask Anna and Beth the same question, they will disagree. This state is called CONTRARY. In this state you can ask them both any question, and whatever one asserts, the other will assert the opposite. Yet it is impossible to predict their individual answers.
CONTRARY is an example of a surprising phenomenon, which is that quantum states exist for two particles in which we know something about how the particles are related to each other, but nothing about each particle individually. We call such states entangled. The phenomenon of entanglement is something new, which comes into physics with the quantum and has no classical analogue.
The information that they will disagree, whatever question they are asked, adds up to exactly half the information that would be needed to predict their actual answers. The other half is about their individual responses. So in the CONTRARY state, we know nothing about their individual views, and everything about how their views correlate. Hence, when in the CONTRARY state, Anna and Beth share a property which is not just the sum of properties they have individually.
The couple spend the evening together and wake up in the CONTRARY state. They each go off to work. Over lunch Anna’s colleagues will ask her about either politics or pets. They decide only at the last minute which question to ask. Afterward they record which question was asked of Anna and what she said. Beth’s colleagues do the same. This is repeated every day for a year, after which the two sets of colleagues meet at a conference and compare notes. What do they discover?
Half the time, Anna and Beth will have been asked different questions. Let’s ignore these cases and look only at the days when they were asked the same question. In 100 percent of these cases, their answers disagreed with each other. This is in spite of the fact that, looked at individually, each of their answers appears to have been completely random.
As I’ve described this, it would not be hard to explain. All that is needed is that each morning over breakfast the couple toss a coin to decide who will give which answers, if asked. But there are analogous stories in which we study pairs of photons, rather than pairs of people. We can put pairs of photons in the CONTRARY state and measure various properties of them. Whenever we ask each the same question, they disagree. But we can show that this cannot be explained by any agreement established in advance of our asking. This was proved in an important paper, written by the Irish physicist John Bell, in 1964.
In the case of photons, we ask not about political or pet preferences, but about polarization. An electromagnetic wave consists
of oscillating electric and magnetic fields. The oscillations are perpendicular to the direction in which the wave is traveling. These oscillations define a plane, which jumps around as the fields oscillate. We say that light is polarized when the electric field oscillates steadily in a particular plane. Individual photons that pass through a polarized lens, such as are common in sunglasses, have a well-defined polarization.
FIGURE 3. This figure shows what we mean by saying that electromagnetic radiation can be polarized. Here are traces of two waves moving through the electric field, in the absence of external currents and charges. Note that the electric field points perpendicular to the direction of motion of the wave. The oscillations of the field, together with the direction of motion, define a plane in three-dimensional space. This is called the plane of polarization. We show two planes of polarization, perpendicular to one another.
We can produce pairs of photons that together have polarizations in the CONTRARY state. To show this we let them travel in opposite directions till they are far from each other, then we put in their paths polarized glass, which they either pass through or not. In the state CONTRARY, if the glasses have the same plane of polarization, one of the two photons will pass through the glass, but the other won’t. Which one passes, however, is random because in the state CONTRARY their individual properties are completely uncertain.
We can also swivel one glass, which rotates the plane of polarization to one side. The two polarizers are now at different angles. Now some of the time both photons pass. How frequently both pass depends on the angle between the polarizers. When the angle between the two polarizers is zero, we are asking the same question on each side and it never happens that both pass. Let us then rotate one polarizer a bit, so that they are asking slightly different questions. Now in a few instances photons pass on both sides. We ask about how the proportion of cases in which they both pass increases as we vary the angle between the two planes of polarization.