## Introduction to Quantum Theory

### by Lucien Hardy

Quantum theory was developed during the first part of the last century by studying the behaviour of small things like atoms and photons (particles of light). The word "quantum" refers to the fact that light is apparently released and absorbed in lumps or quanta. But this is not the main point of quantum theory. There are other features of the theory that mark it out as representing a radical point of departure from classical theories (theories predating quantum theory are often collectively referred to as "classical") and force us to reconsider our picture of reality. In classical theories the mathematical symbols in the theory relate in a simple way to a picture of the world that is not far removed from our everyday experience. For example, a ball flying through the air might be described in Newtonian mechanics by mathematical symbols representing its speed, its position, the direction in which it is moving, and the speed and direction in which it is spinning. These quantities relate directly to the picture (which we can quite easily visualize in our minds) of a spinning ball moving through the air. Before quantum theory came along, it was taken for granted that the quantities we use in physics relate in a simple way to some such picture of the world and it came as something of a shock that quantum theory is not obviously like that. What quantum theory provides us with is a neat and simple mathematical formalism for calculating probabilities relating to measurements we might make on quantum systems.

In fact, once stripped of all its inessential structure, quantum theory can be regarded as a generalization of classical probability theory - that theory we use to calculate the odds when rolling dice. The quantum formalism consists of various mathematical symbols. The great thing about quantum theory is that it works! So far no one has seen a deviation from the rules of quantum theory. However, unlike in the classical case (for example with the spinning ball), the symbols in the theory do not relate in any obvious way to any simple intuitive picture of reality. Knowing how to use quantum theory is a little like knowing how to perform magic but without understanding why the magic works. This leads us to ask three types of question:

• Exactly what is it that is odd about quantum theory - where does it contradict our usual intuitions about reality?
• How should we interpret quantum theory - what kind of picture of reality, if any, does it give us?
• Why is nature described by quantum theory (rather than some other possibly more sensible theory)?

Attempts to answer these questions represent the main research efforts in the foundations of quantum theory.

What Is Odd About Quantum Theory?

Let us start by considering a ball (a baseball for example) which can be in one of two boxes - box A or box B. Imagine we do not know which box it is in. In this case we are still inclined to believe it is actually in one of the two boxes while nothing is in the other box. The fact that we do not know which one is interpreted as ignorance on our part, having nothing to do with the real world. However, now imagine that rather than a ball, we have a quantum object like an atom. In this case it would be wrong, in general, to suppose that the atom was actually in one box and not the other. In quantum theory the atom can behave like it is, in some sense, in both boxes at once. Of course, if we look into the boxes to find out where the atom is (we could do this by shining bright laser light into the boxes and look for the light scattered by the atom) then we will only find it in one of the two boxes, not both, since there is only one atom. Why, then, say it can behave like it is in both boxes at once? Well, physics is ultimately a discipline which relates to experiments, so to answer this question let us consider an experiment which, as we will see, leads us to the strange conclusion that something can be in two places at the same time.

Consider a quantum particle (it could be an electron, an atom, a photon, or any other type of quantum particle). Let the particle impinge on a beam splitter. This is a device which may either transmit the particle or reflect it.

If we now place one detector in the transmitted path and another in the reflected path then only one will fire. There is only one particle and it can only be detected at one place. However, instead of placing detectors in these two paths, we can bring the two paths together at a second beam splitter, placing detectors in the rightward and upward paths of this second beam splitter. It is a consequence of quantum theory, and demonstrated many times in laboratories, that:

1. If we ensure that the two paths between the two beam splitters are equal in length, then the particle will always be detected in the rightward path of the second beam splitter.
2. If one path is longer than the other by a certain amount (equal to half the wavelength associated with the particle) then the particle will always be detected in the upward path of the second beam splitter.

Now imagine trying to explain this behaviour as if the particle went along only one of the two paths, though without us knowing which one. Imagine that, in actual fact, the particle went along the lower path while nothing went along the upper path. When the particle arrives at the second beam splitter, it has to choose whether to go up or to the right. This particle has only transversed the lower path whilst nothing has transversed the upper path and hence, the particle has no information as to whether the two paths are equal in which case it should go right, or differ by half a wavelength in which case it should go up, and hence there is no way the particle can behave in the way we actually observe in experiments. The same dilemma would apply if the particle took the upper path between the beam splitters. This means that we simply cannot think in this way. We cannot think in terms of one path being empty. There is some sense in which the particle is in two places at once.

This device, called a quantum interferometer, illustrates the central property of quantum theory that distinguishes it from classical theories. We have two possibilities - "the particle is in path A" and "the particle is in path B". In a classical theory only one of these two statements could be true at any given time, while in quantum theory, both statements can be partially true at once. This is called the superposition principle. The state of reality is given by superposing two states which are, from our usual classical way of thinking, mutually exclusive. Mathematically, we represent this by taking the two mutually exclusive statements and adding them together:

 "the atom is in box A" + "the atom is in box B"

(Actually, the formalism is a little more complicated than this since we multiply each statement by a number which represents "how much" the atom is in one or the other box.) The superposition principle can be applied in many situations. For example, we can have an atom in two boxes at once, a photon over here and, at the same time, over there, a particle having clockwise spin, and at the same time having anticlockwise spin. We are forced to think in this way because all quantum objects are subject to interference experiments like the one we just described.

Another feature of quantum theory is that it is inherently probabilistic. Before we look to see whether the atom is in one box or the other, there is no way of knowing which box it will be found in. This fact is problematic for people who believe that every effect must be fully determined by some cause. In fact, one can attempt to remove this inherent indeterminism by supposing that quantum theory is a statistical theory derivable from a deeper deterministic theory in which the true state of affairs is given by a more complete description of the world than that provided by the quantum state. Such theories are often called "hidden variable theories". These hidden variables determine, in the above example, exactly which box the atom will actually be found in. The variables are called "hidden" since, before making a measurement, we do not know what values they take.

A big problem in quantum theory arises because we do not know when to stop using the superposition principle. Imagine that rather than an atom in box A or box B we have a molecule. This is a bit bigger than an atom but still very small and so definitely quantum. What about a slightly bigger object like a single cell? What about a really big object like a baseball or a cat or even a whole galaxy? In principle, there is nothing that tells us when the superposition principle does not apply. Thus, in principle we really could imagine a cat, for example, being in a superposition of being alive and dead at the same time. In practise it would be very difficult to actually see quantum interference with very big objects like baseballs or cats (though interference has been seen with large molecules) and hence we can, for all practical purposes, ignore the possibility that there is a superposition. However, if we are interested in the "in principle" question, we cannot ignore this possibility. This is a problem because we have to explain why we do not actually see superpositions of large objects in two places at once - our everyday experience is of a macroscopic world with well localized objects. There are various approaches to solving this problem. One is to say that, for sufficiently big systems, the state "collapses" onto one or the other of the two possibilities. Thus, it goes from being in the superposed state "A" + "B" to being in one of the states "A" or "B". Such theories of quantum theory are called "collapse theories".

One of the strangest features of quantum theory is quantum entanglement. This arises when we have two physical systems which have interacted with each other. After such an interaction, the total system becomes describable only as a single entity. There is no way of describing it by completely describing its components. Entanglement is a fascinating subject from an interpretational point of view, from a mathematical point of view and from the point of view of applications. The most striking effect that arises from quantum entanglement is quantum nonlocality - first discovered by John S Bell in 1964. Bell's theorem is one of the most important results in physics and has played a large role in motivating much recent work in quantum theory. Imagine that two quantum systems interact and then separate to a great distance where measurements are made. These measurements will reveal certain correlations between the two systems. So far, this is what we expect since the systems have interacted in the past. Bell showed, however, that we cannot explain these correlations purely in terms of the previous interaction between the systems. It appears that, even though the two systems are at a great distance, they continue to talk with one another. However, this effect is not so strong that it can actually be used to send signals, so the principle that information cannot travel faster than the speed of light is safe for the time being.

How Should We Interpret Quantum Theory?

There are many interpretations of quantum theory. At the risk of committing a gross over-simplification, we can say that they fall into two basic categories. There are realist/ontological interpretations and positivist/instrumentalist interpretations. Realism is the idea that the real world exists independently of us. To provide an ontology is to provide a picture of the world. So, in realist/ontological interpretations, an attempt is made to come up with a picture of an independently existing world which goes beyond what we actually see (that is to say it goes beyond the detector clicks and meter readings we record in our laboratory notebooks). Positivism is the doctrine that there is nothing beyond what we directly observe. An instrumentalist emphasizes the role of instruments (in our case, the laboratory equipment) in the description of any phenomena. Positivist/instrumentalist interpretations are not so concerned with developing a picture of an independently existing world. Rather, an attempt is made to develop a consistent way of thinking about the various detector clicks, meter readings, and other effects.

The realist/ontological interpretations can best be summarized by where they fall on the following table:

 hidden variables no hidden variables collapse (no major interpretations here) collapse models (GRW, ....) no collapse the pilot-wave model (de Broglie, Bohm) the many worlds interpretation

In collapse models the evolution of the state is chosen such that the state will collapse whenever a superposition of a sufficiently large object occurs. This means that the macroscopic world does consist of objects with definite properties and explains why, when we look, we do not have the experience of seeing a baseball in two places at once. The quantum state can be regarded as a real existing object which, in the limit of the macroscopic world, accords with our usual experience. There have been various attempts to build explicit collapse models with some degree of success, though without guidance from some overriding principle, such models are ad hoc. It has been suggested by Penrose and others that gravitational mechanisms should induce collapse, but this idea has yet to lead to a fully worked out model. If collapse really happens then this should be detectable in experiments and there is currently some effort being made to do this. Unfortunately, it may be a long time before experimental techniques are sufficiently refined to detect collapse - if it happens.

In the pilot wave model, the usual quantum description of the state (as a state vector or wave function) is supplemented by an additional variable actually specifying the positions of the particles. In the interferometer we can say that there is a wave that goes along both paths whilst the particle only goes along one path. The wave collects information about both paths and the particle uses this information to decide which way to go at the second interferometer (hence the "pilot-wave"). This interpretation of quantum theory has various advantages. For example, it explains why the particle is only detected in one place at once and it provides an almost classical picture of what is happening at the microscopic level. However, to do this it is necessary to introduce variables which go beyond quantum theory and detract from much of its simplicity and elegance.

The third option on the table is when we have no collapses and no hidden variables. In this case, we really do get superpositions of macroscopic objects. We need a way to explain why we do not actually see this. The solution proposed by what is variously called the many worlds interpretation, the Everett interpretation (after Hugh Everett who invented it), and the relative state interpretation (this was Everett's name for it and, despite being less popular is a more accurate description of Everett's idea) is that we can regard the world as splitting when a macroscopic superposition occurs. In each branch one of the macroscopic possibilities happens. In an individual world, macroscopic objects have well defined locations but when all the worlds are taken together, it is still the case that there are superpositions of macroscopic objects. One way of thinking about the splitting is that it is merely an artefact introduced to help us understand the whole picture. Consider the analogy of drawing a grid on a map so we can describe features on the map relative to the grid. The grid does not actually exist. Likewise we can argue that there is, in Everett's interpretation, only one universe but by introducing a splitting we can explain the appearance of the world relative to a particular observer. There are many problems with this interpretation. The most obvious is that it represents an extremely radical new view of the universe that many of us feel uncomfortable with. But, even if we can swallow this, there remain technical problems. One is that there are many ways to introduce the splitting into many worlds and yet we need to pick out some preferred way of doing this. Another more serious problem is probability. Each world in the splitting has a weighting attached to it which is supposed to represent the probability of that world. A probability is usually regarded as a probability of some particular possible outcome actually happening. However, in the Everett picture all outcomes are realized and so it is not clear what this probability is a probability of.

The main positivist/instrumentalist interpretation is the Copenhagen interpretation which was developed by Niels Bohr and colleagues working in Copenhagen. Bohr emphasizes that all experiments must be described in classical language. Thus, we have to make statements like "the pointer is pointing at the number 5" or "the ball was found in box A". Any laboratory notebook consists of many statements just like this. Bohr then argues that quantum theory is simply a consistent way of dealing with such statements. Often one can tell a story about what happened in an experiment such as "the photon was reflected at the beam splitter and travelled along path A, finally being detected at detector A". In fact, we did not see the particle being reflected nor travelling along path A. All we know is that the detector in path A fired. In an interference experiment we would not be so keen to tell a story about which path the particle travelled along. In this case we might tell a different type of story in which a wave travelled along both paths, these two waves being brought back together at the second beam splitter. The type of story we are likely to tell depends on the experimental context. Bohr describes such descriptions as complementary and he describes his interpretation of quantum theory as complementarity. It is not clear whether these "stories" should be regarded as anything more than just stories, or whether we should think of them as corresponding to what is actually happening in the world. The advantage of the Copenhagen interpretation is that it provides a consistent way of talking about quantum experiments. However, since it requires all physical statements to be related to concrete classically described apparatuses, it is difficult to arrive at a picture of what is happening in the world independent of any measurements that may be made.

Why Quantum Theory?

We know from experimental evidence that quantum theory works very well. But we also know that it is a very strange physical theory. This motivates the obvious question: why is nature described by quantum theory? How do we begin to answer a question like this? Usually explanation in physics is by means of appealing to some set of deep, simple and intuitive principles. The principles of quantum theory, whilst mathematically simple, are rather abstract and cannot be described as intuitive. It is instructive to compare this with the situation in Einstein's theory of special relativity which follows from two principles: the laws of physics are the same in all frames of reference and the speed of light is a constant for all observers. Is it possible to find a similar set of simple and intuitively reasonable principles for quantum theory? This question has been around for a while. One approach has been to reconsider logic. It can be argued that quantum theory forces us to modify the rules of logic. In this case, we could attempt to derive quantum theory from a modified set of logic rules. There has been a tremendous amount of work on quantum logic (which is, by now, a field in its own right). However, it is fair to say that, so far, no simple approach has emerged. Another approach stems from the observation that quantum theory is, when stripped of inessential structure, simply a new type of probability theory. Hence, we can consider what a reasonable theory of probability might look like. One of our researchers, Lucien Hardy, has shown that we can obtain quantum theory from five principles in this probability context. Four of the five principles are true in classical probability theory and hence it is the remaining principle that gives rise to quantum theory. To understand this remaining principle, consider a system which can be in the state 0 or the state 1 (a bit). In classical physics the only way to go from the 0 state to the 1 state is to jump - one moment the state is 0, the next it is 1. But in physics we prefer things to change in a gradual fashion. The remaining principle (which gives rise to quantum theory) demands that there exist a way of gradually transforming the state. This implies that there exist an infinity of states between 0 and 1. These "in between" states are just the usual quantum superpositions.

The future of quantum theory?

Quantum theory remains a deeply mysterious subject. Progress in understanding it will come from theoretical, philosophical and experimental developments. On the experimental side we might, one day, expect to see some deviation from the laws of quantum theory in the laboratory. Perhaps we will observe collapse for macroscopic objects. Perhaps we will see a violation of quantum theory at a much more fundamental level. At the theoretical end we need new tools for gaining insight into the theory. By understanding more deeply the reasons why nature is described by quantum theory we might be able to go beyond the theory. At a philosophical level, we might come to a better understanding of the world and our place in it if we can decode what the equations of quantum theory are trying to tell us.

© Lucien Hardy is an associate of the Perimeter Institute for Theoretical Physics