Three Roads to Quantum Gravity

by Lee Smolin

  This has very profound implications for a whole host of issues. It means that to judge the rationality of our decisions, we do not have to pretend that there is some supernatural observer who knows everything: it is enough to demand that the different observers report what they see honestly. When this rule is followed we discover that when we and another person each have enough information to decide whether something is true or false, we always make the same decision.

  Thus, the philosophers who attempted to ground ethics and science in the ultimate judgements of an all-knowing being were mistaken. We can live rationally without having to believe in a being who sees everything. We need only believe in the ethical principle that observers should communicate honestly what they see. If we stick to this, then the fact that there will always be questions that we cannot answer need not prevent us from coming to an agreement about how to understand those aspects of our world which we share in common.

  So topos, or cosmological, logic is also the right logic for understanding the human world. It, and not Aristotle, must be the right basis for economics, sociology and political science. I am not aware that anyone in these areas has taken up topos theory and tried to make it the foundation of their subject, although George Soros’s approach to economics, which he calls the theory of reflexivity, is certainly a start in the right direction. But we should not be surprised if both cosmology and social theory point us in the same direction. They are the two sciences that cannot be formulated sensibly unless we build into their foundations the simple fact that all possible observers are inside the systems they study.

  CHAPTER 3

  MANY OBSERVERS, NOT MANY WORLDS

  So far I have said nothing at all about quantum theory. We have seen that even without it, doing cosmology requires a radical revision of our way of doing science - a revision that goes even to the foundations of logic. Any scientific form of cosmology requires a radical change in the logic we use, to take into account the fact that the observer is inside the universe. This requires us to build our theory so that from the beginning it takes into account a form of observer dependence. We must acknowledge that each observer can have only a limited amount of information about the world, and that different observers will have access to different information.

  With this important principle in mind, we may turn to the problem of how to bring quantum theory into cosmology. ‘Hold it!’, I can hear the reader saying. ‘Quantum theory is confusing enough. Now I’m being asked to think about how to apply it to the universe as a whole! Where do I get off?’ That’s understandable, but, as I shall explain in this chapter, thinking about how to apply quantum theory to the universe as a whole may make quantum physics easier, not harder, to understand. The principles we have looked at in the first two chapters may very well be the key to making quantum theory comprehensible.

  Quantum theory is puzzling because it challenges our standard ideas about the relationship between theory and observer. The theory is indeed so puzzling that there is no universally accepted physical interpretation of it. There are many different points of view about what quantum theory really asserts about reality and its relationship to the observer. The founders of quantum theory, such as Einstein, Bohr, Heisenberg and Schrödinger, could not agree on these questions. Nor is the present-day situation any better, for now we have extra points of view that those guys, smart as they were, were not imaginative enough to foresee. There is now no more agreement about what quantum theory means than when Einstein and Bohr first debated the question in the 1920s.

  It is true that there is only one mathematical formalism for the quantum theory. So physicists have no problem with going ahead and using the theory, even though they do not agree about what it means. This may seem strange, but it does happen. I have worked on projects in quantum gravity where everything went smoothly until the collaborators discovered one day over dinner that we had radically different understandings of the meaning of quantum theory. Everything went smoothly again after we had calmed down and realized that how we thought about the theory had no effect on the calculations we were doing.

  But this is no consolation to the layperson, who does not have the mathematics to fall back on. With only the concepts and principles to go on, it must be very disconcerting to discover that different physicists, in their different books, offer very different versions of the basics of quantum theory.

  Quantum cosmology helps rather than hinders because, as we are about to see, it limits the scope for possible interpretations of the quantum theory. If we stick to the principles introduced in the first two chapters, several of the approaches to the interpretation of quantum mechanics must be abandoned. Either that, or we must give up any idea that quantum theory can be applied to space and time. The principle that there is nothing outside the universe and the principle that in the future we shall know more do point to a new way of looking at quantum theory that is both simpler and more rational than many of the older ideas. As a result of applying quantum theory to cosmology, there has emerged over the last few years a new approach to the problem of the meaning of quantum theory. This is what I want to communicate in this chapter.

  Ordinary quantum theory is a theory of atoms and molecules. In the form developed originally by Bohr and Heisenberg, it required the world to be split into two parts. In one part was the system under study, which was described using the quantum theory, and in the other part lived the observer, together with whatever measuring instruments were needed to study the first system. This separation of the world into two parts is essential for the very structure of quantum mechanics. At the heart of this structure lies the superposition principle, which is one of the basic axioms of the quantum theory.

  The superposition principle is not easy to understand, because it is formulated in seemingly abstract terms. If one is not careful it can lead to a kind of mysticism in which its meaning is over-interpreted far past what the evidence calls for. So we shall be careful, and spend some time looking at the statement of this important principle.

  Let us first state it. The superposition principle says that if a quantum system can be found in one of two states, A and B, with different properties, it may also be found in a combination of them, aA + bB, where a and b are any numbers. Each such combination is called a superposition, and each is physically different.

  But what does this actually mean? Let us break it down. The first thing to understand is what physicists mean when they talk about ‘states’. This one word contains almost the full mystery of the quantum theory. Roughly, we say that the state of a physical system is its configuration at a particular moment. For example, if the system is the air in the room, its state might consist of the positions of all the molecules together with their speeds and the directions of their motions. If the system is a stock market, the state is the list of the prices of all the stocks at a particular moment. One way to say this is that a state consists of all the information needed to completely describe a system at an instant of time.

  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 aA + bB, 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 a2 and b2. However, what is most crucial to take on board is that the statement that the system is in the state aA + bB 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 aA + bB 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 qua
ntum 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.