However, there is a problem with using this idea in quantum theory, because we are not able to measure at the
same time both the position and the motion of a particle. Heisenberg’s uncertainty principle asserts that we can only ever measure accurately either the position or the direction and speed of motion of a particle. For the moment, don’t worry about why this should be. It is part of the mystery - and to be honest, no one really knows how it comes about. But let us look at its consequences.
If we cannot determine both the position and the motion of a particle, then the above definition of ‘state’ is no use to us. There may or may not be something in reality corresponding to the exact state, which comprises both the position and the motion, but, according to the uncertainty principle, even if it exists in some ideal sense it would not be a quantity we could observe. So in quantum theory we modify the concept of a state so that it refers only to as complete a description as may be given, subject to the restriction coming from the uncertainty principle. Since we cannot measure both the position and the motion, the possible states of the system can involve either a description of its exact position, or of its exact motion, but not both.
Perhaps this seems a bit abstract. It may also be hard to think about, because the mind rebels: it is hard to work one’s way through to the logical consequences of a principle like the uncertainty principle when one’s first response is simply to disbelieve it. I myself do not really believe it, and I do not think I am the only physicist who feels this way. But I persist in using it because it is a necessary part of the only theory I know that explains the main observed facts about atoms, molecules and the elementary particles.
So, if I want to speak about atoms without contradicting the uncertainty principle, I must conceive of states as being described by only some of the information I might be seeking. This is the first hard thing about states. As a state contains only part of the information about a system, there must be some rationale for that information being selected. However, although the uncertainty principle limits how much information a state can have, it does not tell us how it is decided which information to include and which to leave out.
There can be several reasons for this choice. It can have to
do with the history of the system. It can have to do with the context the system now finds itself in, for example with how it is connected to, or correlated with, other things in the universe. Or it can have to do with a choice we, the observer, have made. If we choose to measure different quantities, or even in some circumstances to ask different questions, this can have an effect on the state. In all these cases the state of a system is not just a property of that system at a given time, but involves some element outside the present system, having to do either with its past or with its present context.
We are now ready to talk about the superposition principle. What could it possibly mean to say that if a system can be in state A or state B, it can also be in a combination of them, which we write as
a
A +
b
B, where a and b are numbers?
It is perhaps best to consider an example. Think of a mouse. From the point of view of a cat, there are two kinds of mice - tasty and yukky. The difference is a mystery to us, but you can be sure that any cat can tell them apart. The problem is that the only way to tell is to taste one. From the point of view of ordinary feline experience, any mouse is one or the other. But according to quantum theory this is a very coarse approximation to the way the world actually is. A real mouse, as opposed to the idealized version that Newtonian physics offers, will generally be in a state that is neither tasty nor yukky. It will instead have a probability that, if tasted, it will be one or the other - say, an 80 per cent chance of being tasty. This state of being suspended in between two states is not, according to quantum theory, anything to do with our influence - it really is neither one thing nor the other. The state may be anywhere along a whole continuum of possible situations, each of which is described by a quantum state. Such a quantum state is described by its having a certain propensity to be tasty and another propensity to be yukky; in other words, it is a superposition of two states - the states of purely tasty and purely yukky. This superimposed state is described mathematically by adding a certain amount of one to the other. The proportions of each are related to the probabilities that when bitten, the poor mouse will prove to be tasty or not.
This sounds crazy, and even thirty years after learning it I cannot describe this situation without a feeling of misgiving. Surely there must be a better way to understand what is going on here! Embarrassing though it is to admit it, no one has yet found a way to make sense of it that is both more comprehensible and elegant. (There are alternatives, but they are either comprehensible and inelegant, or the reverse.) However, there is a lot of experimental evidence for the superposition principle, including the double slit experiment and the Einstein-Podolsky-Rosen experiment. Interested readers can find these discussed in many popular books, some of which are included in the reading list at the end of this book.
The problem with quantum theory is that nothing in our experience behaves in the way the theory describes. All our perceptions are either of one thing or another - A or B, tasty or yukky. We never perceive combinations of them, such as a × tasty + b × yukky. Quantum theory takes this into account. It says that what we observe will be tasty a certain proportion of the time, and yukky the rest of the time. The relative probabilities of us observing these two possibilities are given by the relative magnitudes of
a
2
and
b
2
. However, what is most crucial to take on board is that the statement that the system is in the state
a
A +
b
B does not mean that it is either A or B, with some probability of being A and some other probability of being B. That is what we see if we observe it, but that is not what it is. We know this because the superposition
a
A +
b
B can have properties that neither tasty nor yukky have by themselves.
There is a paradox here. Were my cat to be described in the language of quantum theory, after tasting the mouse she would experience either tasty or yukky. But according to quantum mechanics she would not be in a definite state of happy or displeased. She would go into a superposition of two states which mirrors the possible states of the mouse. She would be suspended in a superposition of a happy state and an annoyed-for-having-bitten-into-a-yukky-mouse state.
So the cat experiences herself in a definite state, but in the light of quantum theory I must see her in a superposition.
Now, what happens if I observe my cat? I shall certainly experience a purr or a scratch. But shall I definitely be in one of these two possible states? I cannot imagine that I should not experience one or the other. I cannot imagine even what it would mean to experience anything other than one or the other. But if I am described in the language of quantum theory, I too, along with the mouse and the cat, will be in a superposition of two different states. In one of them the mouse was tasty, the cat was happy and I heard a purr. In the other the mouse is yukky, the cat is angry and I am nursing a scratch.
What makes the theory consistent is that our different states are correlated. My being happy goes along with the happiness of the cat and the tastiness of the mouse. If an observer queries both me and the cat, our answers will be consistent, and they will even be consistent with the observer’s experience if she tastes the mouse. But none of us is in a definite state. According to quantum theory, we are all in a superposition of the two possible correlated states. The root of the apparent paradox is that my own experience is of one thing or the other, but the description of me that would be given in quantum theory by another observer has me most often in a superposition which is none of the things I actually experience.
There are a few possible resolutions of this mystery. One is that I am simply mistaken about the impossibility of superpositions of mental states. In fact, if the usual formalism of quantum mechanics is to be applied to me, as a physical system, this must be the case. But if a human being can be in a superposition of quantum states, should the same not be true of the planet Earth? The solar system? The Galaxy? In fact, why should it not be a physical possibility that the whole universe is in a superposition of quantum states? Since the 1960s there have been a series of efforts to treat the whole universe in the same way as we treat quantum states of atoms. In these descriptions of the universe in terms of quantum states, it is assumed that the universe may as easily be put into quantum superpositions as can states of photons and electrons. This subject can therefore be called ‘conventional quantum cosmology’ to distinguish it from other approaches
to combining quantum theory and cosmology that we shall come to.
In my opinion, conventional quantum cosmology has not been a success. Perhaps this is too harsh a judgement. Several of the people I most respect in the field disagree with this. My own views on the matter have been shaped by experience as much as reflection. By chance I was part of the discovery of the first actual solutions to the equations that define a quantum theory of cosmology. These are called the Wheeler-DeWitt equations or the quantum constraints equations. The solutions to these equations define quantum states that are meant to describe the whole universe.
Working first with one friend, Ted Jacobson, then with another, Carlo Rovelli, I found an infinite number of solutions to these equations in the late 1980s. This was very surprising, as very few of the equations of theoretical physics can be solved exactly. One day in February 1986, Ted and I, working in Santa Barbara, set out to find approximate solutions to the equations of quantum cosmology, which we had been able to simplify thanks to some beautiful results obtained by two friends, Amitaba Sen and Abhay Ashtekar. All of a sudden we realized that our second or third guess, which we had written on the blackboard in front of us, solved the equations exactly. We tried to compute a term that would measure how much our results were in error, but there was no error term. At first we looked for our mistake, then all of a sudden we saw that the expression we had written on the blackboard was spot on: an exact solution of the full equations of quantum gravity. I still remember vividly the blackboard, and that it was sunny and Ted was wearing a T-shirt (then again, it is always sunny in Santa Barbara and Ted always wears a T-shirt). This was the first step of a journey that took ten years, sometimes exhilarating and often aggravating years, before we understood what we had really found in those few minutes.
Among the things we had to struggle with were the implications of the fact that the observer in quantum cosmology is inside the universe. The problem is that in all the usual interpretations of quantum theory the observer is assumed to be outside the system. That cannot be so in cosmology. This is
our principle and, as I’ve emphasized before, this is the whole point. If we do not take it into account, whatever we may do is not relevant to a real theory of cosmology.
Several different proposals for making sense of the quantum theory of the whole universe had been put forward by pioneers of the subject such as Francis Everett and Charles Misner. We were certainly aware of them. For many years young theoretical physicists have amused themselves by debating the merits and absurdities of the different proposals made for quantum cosmology. At first this feels fantastic - one is wrestling with the very foundations of science. I used to look at the older people and wonder why they never seemed to spend their time this way. After a while I understood: one could only go around the five or six possible positions a few dozen times before the game got very boring. Something was missing.
So we did not exactly relish the idea of taking on this problem. Indeed, at least for me, solving equations rather than worrying about foundations was a deliberate strategy to try to do something that could lead to real progress. I had spent much of my college years staring at the corner of my room, wondering about what was real in the quantum world. That was good for then; now I wanted to do something more positive. But this was different, for in a flash we had obtained an infinite number of absolutely genuine solutions to the real equations of quantum gravity. And if a few were very simple, most were exceedingly complex - as complex as the most complicated knot one could imagine (for they indeed had something to do with tying knots, but we shall come to that later on).
No one had ever had to, or been able to, contemplate the meaning of these equations in anything other than very drastic approximations. In these approximations the complexity and wonder of the universe is cut down to one or two variables, such as how big the universe is and how fast it is expanding. It is very easy to forget one’s place and fall for the fantasy that one is outside the universe, having reduced the history of the universe to a game as simple as playing with a yo-yo. (No, actually simpler, for we never would have been
able to attack something as complicated as a real yo-yo. The equations we used to model what we optimistically called ‘quantum cosmology’ were something like a description of a really stupid yo-yo, one that can only go up and down, never forward or back or to the left or right.)
What is needed is an interpretation of the states of quantum theory that allows the observer to be part of the quantum system. One of the ideas on the table was presented by Hugh Everett in his hugely influential Ph.D. thesis of 1957. He invented a method called the relative state interpretation which allows you to do something very interesting. If you know exactly what question you want to ask, and can express it in the language of the quantum theory, then you can deduce the probabilities of different answers, even if the measuring instruments are part of the quantum system. This is a step forward, but we have still not really eliminated the special role that observations have in the theory. In particular this applies equally to an infinite set of questions that may be asked, all of which are mathematically equivalent from the point of view of the theory. There is nothing in the theory that tells us why the observations we make, in terms of big objects that appear to have definite positions and motions, are special. There is nothing to distinguish the world we experience from an infinite number of other worlds made up of complicated superpositions of things in our world.