Quantum mechanics is not complete without an interpretation, which is required to connect experimental observations with theory. There are at least three general categories of interpretation:
a) Quantum theory is not correct as it stands. It must either be modified to describe the process of measurement, or it must be supplemented to include the phenomenon of wavefunction collapse which we shall describe later. The "orthodox" interpretation belongs to the latter category.
b) Quantum theory is correct as it stands, but the wavefunction is not a complete description of the system. It must be supplemented by the addition of "hidden variables", i.e., the positions and velocities of all of the particles at all times. In this interpretation, the particles are always present. The wavefunction is no longer interpreted as a probability, but is the source of a quantum force which acts on the particles in addition to all of the classical forces, like the electromagnetic and gravitational forces.
c) Quantum theory is as correct and as complete as possible. This leads to the "many worlds" interpretation.
In this interpretation, there are no particles before a measurement, and the wavefunction is a complete description of the system before measurement, i.e., no other information about the system is possible. At the time of measurement, particles are observed , so the wavefunction must change from a probability wave which includes all of the possibilities that existed before the measurement to one which describes only the possibilities which are allowed by the measurement. This is called reduction, or collapse, and it is not explained by the theory. In this interpretation, the wavefunction is the only objective reality prior to a measurement.
We will first show that any system which is completely described by quantum theory cannot exhibit wavefunction reduction. In order to do this in the most efficient manner, we will use a symbolic notation which makes the description concise and precise. Do not let this frighten you--it is simply a notation, not higher mathematics. The notation will refer to a particular type of experiment with particles that have spin. The spin of a particle is related to its rotation. A macroscopic analog is a spinning top. We can say that if the top is spinning normally on a flat, smooth surface, the spin (like the top) is pointing down. If for some reason, the top flips so that it spins upside down (there are tops which do this), we can say the spin is pointing up. Particles with spin (like the electron) can have their spins pointing either up or down.
We start with an experiment in which an incoming electron is in a superposition of spin-up (+) states and spin-down (-) states. By superposition, we mean that the wavefunction is a sum of two terms, one describing the + state, and one describing the - state. This is called a "pure" state. The notation we now introduce is called the Dirac "ket" notation. Instead of writing the wavefunction simply as y as we did before, we enclose it in ket brackets and obtain ç y > . We do the same with the notation for the + and the - states, and obtain
ç y > = a ç + > + b ç - >
All this equation says is that the electron is a wave consisting of a superposition of a spin up state and a spin down state. Here, ç a ç 2 is the probability that a measurement would result in a spin-up particle, and ç b ç 2 is the probability that it would result in a spin-down particle. (These are written with absolute value signs because a and b are in general complex quantities. However, this detail need not concern us here.)
We now send this beam into a "Stern-Gerlach" apparatus. This contains a nonuniform magnetic field which causes the ç + > component of the electron to go upward and the ç - > component to go downward. Therefore, after the electron passes through the apparatus, the Schrödinger equation tells us that it is described by the pure state wavefunction
ç y > = a ç +,up > + b ç -,down>
where it is obvious that ç + > goes up and ç - > goes down. This wavefunction is not arbitrary--given the electron and the characteristics of the Stern-Gerlach apparatus, the Schrödinger equation dictates this form. We now send the two separate components of the wavefunction into a detector, which records "on" if a ç + > particle is detected and "off" if a ç - > particle is detected. (The labels "on" and "off" are purely arbitrary. They could also be called, e.g., "1" and "2".) To make this clear, a diagram is shown below.
We assume that the detector, like the rest of the system, is described by the Schrödinger equation. We must then include the states of the detector in the wavefunction, and the pure state becomes
ç y > = a ç +,up,on > + b ç -,down,off>
This leads to a very important conclusion. Any object in the system that can be described by the Schrödinger equation must be included in the superposition of terms describing the system. The Schrödinger equation always converts a pure state into a pure state. A pure state wavefunction will always be a superposition which means that there is a probability of finding the system in either state.
Reduction, or collapse, of the wavefunction requires going from a pure state consisting of a superposition to a final state consisting of only one term because the reduced wavefunction must describe the detector being in either one state or the other, but not both. Therefore, no object that can be described by the Schrödinger equation can reduce the wavefunction, i.e., make a measurement.
Now suppose that I look at the detector and that I also can be described by the Schrödinger equation. Two states are needed to describe me, which we will call me+ and me- , with the obvious connotations. The final wavefunction will be the pure state
ç y > = a ç +,up,on,me+ > + b ç -,down,off,me- >
However, if I am aware of the final state of the detector, this wavefunction cannot describe the combined system since I know that the detector is either in the "on" state or the "off" state. Something that cannot be described by quantum mechanics has reduced the wavefunction. If we assume that any physical system can be described by quantum mechanics, then reduction must have been caused by something nonphysical. The obvious nonphysical attribute that I possess is awareness.
In the Schrödinger cat paradox, I observe the cat in either the live state or the dead state, not both. If awareness reduces the wavefunction, it is either my awareness or the cat’s that does it. At this time it is an open question which of the two awarenesses it is. What I see when I open the box will be exactly the same in either case.
Because most physicists are materialists and believe that consciousness is at most an epiphenomenon, they do not like to admit that consciousness is needed to reduce the wavefunction. Rather, they prefer to think that it is some physical property of macroscopic devices that causes reduction. Of course, if that is the case, that property at present cannot be described by quantum theory, so to them, quantum theory is presently incorrect. (However, inconsistently, most do not believe that to be true, either.)
In this section, we shall assume the orthodox interpretation, and initially we shall also assume that the wavefunction describes only the physical systems, i.e., awareness is not included.
Now we suppose that we have a Stern-Gerlach experiment with two detectors instead of one, as shown in the figure below. One detector records the ç +,up > state and the other records the ç -,down > state. The detectors do not communicate with each other, and may be arbitrarily far apart. What prevents both detectors from recording an event simultaneously? This example shows that no local process can produce collapse of the wavefunction because such processes could not prevent coincidences between the detectors. Hence, we must conclude that wavefunction collapse cannot be produced by any known physical process (which are all local). (This result also can be inferred from the Bell-Aspect experiments.) Thus, any interpretation of quantum theory requiring wavefunction collapse is not consistent with a materialist philosophy.
Now suppose there are two observers, you and I, so that you observe the ç -,down > state while I observe the ç +,up > state. Then when I observe my detector to record "on", you must observe your detector to record "off". In order to insure that this is so, if consciousness collapses the wavefunction, this consciousness must be nonlocal consciousness.
Of course, our interpretation also requires that there are local, individual, sentient observers which observe the results.
This conclusion can be illustrated in a much simpler example than the experiment described above. Suppose two local (i.e., noncommunicating) observers make simultaneous observations on the same object whose color is unknown before the observation. In this case all possible colors must be represented in the wavefunction of the object before it is observed. Then why do both observers observe the same color rather than one observer observing, for example, a red object and the other observing a blue object? If consciousness collapses the wavefunction, the answer must be that the consciousness of both observers is the same consciousness.
Now let us consider the same example without reference to quantum theory. As before, let us assume that all objects are observer-created rather than existing in an objective sense, but now there are no wavefunctions before observation. It is easy to see that the consciousness of the observers must be universal consciousness if both observers are to see the same object. Thus, whenever we assume that objects appear only as mental images, not as independently existing objects, the consciousness of the individual observers must be universal consciousness. Of course, in this example, even the observers themselves must be mental images.
One reason we abandoned classical particles was because we showed they could not go through two slits at once and produce interference, whereas waves could. But interference is possible with classical particles if there is also a wave present. A theory which includes both is the hidden variable theory developed by David Bohm (1917 - 1992). This is the best developed and best known of the hidden variable models. This model is fully deterministic and assumes that the particles are classical and are subject to classical forces. They are also subject to a quantum force which is derived from a wavefunction. (To be more accurate, there is a quantum potential which is derived from the wavefunction, and the quantum force is derived from the quantum potential.) Since the particles are assumed to be classical, their positions and velocities are always definite, even before an observation. Contrary to the orthodox interpretation, the wavefunction in the hidden variables interpretation is not a complete description of the system because the particle positions are also required. In the initial state, the wavefunction specifies the actual distribution of particles in space, not just a probability. The time development of the wavefunction is then described by Schrödinger’s equation, as in ordinary quantum theory.
The wavefunction is identical with that in orthodox quantum theory and contains all parts of the waves, e.g., reflected and transmitted parts, or the parts going through different slits, even if none of the particles follow these paths. A peculiarity of the quantum force is that it can be very large even where the wavefunction is very small. Since the wavefunction, and therefore the quantum force, depends on all parts of the experimental apparatus, so do the particle trajectories, even though trajectories and apparatus may be quite distant from each other. Thus, all parts of the apparatus simultaneously affect all parts of all of the particle trajectories, no matter how distant. Because of this simultaneous effect which is independent of distance, hidden variables theory is nonlocal.
How can we reconcile the determinism of this model with our experimental observations that exact particle positions and velocities cannot be predicted? The answer is that, although in principle the particle trajectories are completely determined in this theory, in practice they are strangely chaotic and, within the precision that a particle can be located in its initial state, the location of the particle in the final state can be given only probabilistically. Because the exact trajectories can never be known, this is called a hidden variables theory.
Since classical particles exist in this hidden variables interpretation, there is no wavefunction collapse, and therefore it is not necessary to introduce consciousness into the interpretation. Hence, hidden variables is consistent with a materialist philosophy.
There are problems with this theory. It is very difficult to make calculations with it and it is not known whether a relativistic theory can be made from it. The quantum force is independent of the positions and velocities of the particles and is unaffected by their motion, whereas the particles are directly affected by the quantum force. This kind of asymmetry is not easily accepted by physicists. The fact that the quantum force does not fall off with distance also disturbs many physicists.
Regardless of the problems with the theory, there are important philosophical implications that can be drawn from it. In those cases where calculations are possible, the results from it agree in every detail with those from orthodox quantum theory. This is not surprising because the theory was constructed to do so. Now we must ask the question, if two radically different theories both give results which agree with experiment, which is the correct theory? In this particular case, is reality probabilistic or deterministic? Even though experiment cannot distinguish between the two theories, there are profound metaphysical implications to choosing between them because the orthodox interpretation requires consciousness, and is therefore consistent with a dualist or idealist philosophy, while the hidden variables interpretation does not, and is thus consistent with a materialist philosophy.
The physics community has effectively made a choice by almost completely ignoring the Bohm theory. The reasons are that orthodox quantum theory can be made relativistic (resulting in quantum field theory) with results that are as accurate as experiment can determine. The orthodox theory is much simpler and lends itself to a wide variety of calculations. Its probabilistic interpretation no longer bothers physicists and it does not have the problems mentioned that the Bohm theory has. By and large, most physicists use the theory as a mathematical description of reality while ignoring the problems in describing and understanding wavefunction collapse.
As mentioned above, consciousness was not a part of Bohm’s original hidden variables theory. However, he later extended it to his quantum theory of fields, and from this generalized it to include speculations about the nature of mind, matter, and consciousness. He called this a theory of the implicate order, and we shall encounter it in Chapter 8.
This interpretation was invented by Everett in 1957 so that cosmologists could apply quantum theory to the entire universe at the time of its origin. According to accepted cosmology, the universe exploded from a point at the time of the big bang, approximately 12 billion years ago. Early on, the universe was so tiny and its density was so high that its gravitational forces were enormously high. In such conditions, gravity cannot be treated classically; so it must be described quantum mechanically. Even though as yet we have no such quantum theory of gravity, we do know that the initial universe must be described by a wavefunction. The universe by definition includes everything, so there can be no outside observers. Without any observers, there can be no wavefunction collapse, so quantum theory is assumed to be correct without any corrections or additions.
Let us now look at the Stern-Gerlach experiment in the light of the many-worlds interpretation. We return to the wavefunction which describes my observation of the detector:
ç y > = a ç +,up,on,me+ > + b ç -,down,off,me- >
There can be no reduction of the wavefunction now. Both terms must describe reality. The many-worlds interpretation says that at the moment of an observation, the world splits, or branches, and that both branches continue after the observation. There is a me in both branches. The many-worlds interpretation maintains that in each branch, the me in that branch is aware of only the observation that it made. Since in my world, I am aware of only one result, I exist only in that branch. In the other branch, the other me is aware of the other result. The two branches do not communicate with each other, so the two me’s are unaware of each other.
[Technical note: Assuming all of this to be true, what then is the interpretation of a and b ? The probabilistic interpretation of quantum theory says that ç a ç 2 and ç b ç 2 are the statistical probabilities of each outcome. These probabilities can be measured only by making many measurements on identical systems. What can they mean here when we have only one universe? De Witt in 1970 proposed the following interpretation. In the first trial of such an experiment, both branches resulted from the observation. If I now make many measurements with my apparatus in my branch, I will measure probabilities that agree with ç a ç 2 and ç b ç 2. At each measurement, there will be another branching which will result in this me being in one branch, and another me being in another branch. If each of these other me’s continues the measurements, he will also measure probabilities which agree with ç a ç 2 and ç b ç 2.]
It is easy to see how the number of branches rapidly proliferates as the observations continue. In addition, most observations on most types of systems will result in not only two branches, but many more, as many as are allowed by Schrödinger’s equation. In fact, the number of branches is usually infinite. It is clear that, while the many-worlds interpretation is very economical in terms of the number of concepts required in the theory, it is grossly extravagant in terms of the complexity of the world it describes. It is this feature which many physicists find hard to accept.
In the many-worlds interpretation, after a branching, I am in only this branch, and I observe only this branch. As far as I am concerned, the other branches are not materialized. The advantage of many-worlds is that the unobserved branches can still be described by wavefunctions even though they are unobserved. Thus, quantum theory does not require any mysterious reduction mechanism to get rid of the unobserved wavefunctions, even though some mysterious mechanism is required to materialize my branch. Cosmologists think this mysterious mechanism could be epiphenomenal consciousness which arose after the wavefunction evolved into enough complexity. If we stipulate that the unobserved branches remain unmaterialized, the many-worlds and orthodox interpretations are very similar, and for our purposes can be considered to be equivalent.
In both interpretations, consciousness is required, in many-worlds to materialize my branch, and in the orthodox interpretation to collapse the wavefunction. However, so far we have seen nothing in either interpretation that intrinsically requires the nondual consciousness of idealism rather than the dualistic mind of mind-matter dualism. The mind of dualism could collapse a wavefunction or materialize a branch just as the consciousness of idealism could. On the other hand, in mind-matter dualism, in neither interpretation can there be a material world without consciousness, and in neither interpretation would consciousness have anything to be aware of without the material world. Thus, consciousness and matter seem to be essential to each other. This leads us to speculate that they may even be the same thing.
Even before we reached this chapter, we saw in Section 4.3 that the Bell-Aspect experiments show that reality is nonlocal. This in itself does not imply nonlocal causality, which would violate Einstein’s special theory of relativity, but simply that reality can support nonlocal correlations. Causality itself is still believed to be local (Einstein locality) and this means that all physical processes are also still believed to be local.
If this is so, and if we then assume that wavefunction collapse operates in the Bell-Aspect experiments, we are forced to conclude that wavefunction collapse is a process that violates local causality. Thus, starting with the simple assumption that the physical world must obey Einstein locality, a careful theoretical and experimental examination of quantum processes leads to the conclusion that they cannot be physical! Physics cannot explain what we thought were purely physical processes! (This is reminiscent of Gödel’s theorem which we discussed in Section 5.6.) Thus, we must now begin to question our assumptions about the reality of space and time. We shall say more about this in Section 7.1.
As we have seen above and also in Section 6.4, if it is consciousness that collapses the wavefunction (or that materializes a branch), then consciousness must be nonphysical. If it is nonlocal universal consciousness as we have inferred in Section 6.5, then we are faced with some other far-reaching conclusions. As we have seen, what two individual observers see is determined by universal consciousness, not by any kind of individual consciousness that might exist. This applies to all of our sensory perceptions without exception. Since everything we perceive is determined by universal consciousness, it makes no sense to say that there is a material world independent of consciousness. Thus the dualism of mind and matter is excluded.
It is only a small step now to suppose that, if all of our sensory perceptions are determined by universal consciousness, then all of our thoughts and feelings are also, because there is no intrinsic difference between them (as we shall see in Chapters 9 and 22). If all experiences are determined by universal consciousness, then we must conclude that nothing in our lives that we consider to be "ours" as individuals is truly ours. If everything flows from universal consciousness, "our" lives are not our lives at all but are lives of universal consciousness. "My" consciousness cannot really be mine, nor can there be any free will if none of "our" thoughts is ours. Even the thought that I exist is not mine. With these astounding conclusions, we are forced to ask the questions, "Do I really exist?", and, "What am I, really?" We shall consider these questions later in the course.
In Section 4.1, we alluded to the possibility that the wavefunction is not a physical wave but is merely an algorithm for calculating the probabilities for certain prescribed events to occur. If this is so, there is no real quantum wave either before or after an observation. The wavefunction would reflect only our knowledge of a situation and nothing more.
A few physicists hold to this viewpoint because it avoids the paradox of nonlocal collapse mentioned above. These physicists do not deny the possibility of the existence of an objective reality independent of what observers perceive. However, as we discussed in Section 1.1, it is clear that the existence of such a reality can never be proved nor disproved, and can be only a metaphysical assumption. If we assume that there is such an objective reality, our understanding of its nature is limited by the kinds of experiments we do and by the type of theory that we use to interpret them. Our present picture of the microscopic world as consisting of atoms, molecules and elementary particles is determined in an essential way by these limits. Radically different kinds of experiments and theories might produce a radically different kind of picture.
Careful examination of our present picture of physical reality seems to lead to either paradoxes or hidden variables. This is readily seen whenever an interference pattern is measured, a simple example being the two-slit experiment described in Section 4.1. Interference suggests that physical waves are interfering, whether or not they are identified with the wavefunction. Even though a particular interference pattern can simply be the result of a particular experimental setup without the need for postulating physical waves, our understanding is greatly enhanced if we assume that the pattern is indeed caused by physical waves. If these are not identified with the wavefunction (in which case we would consider them to be hidden variables), they must have properties very closely related to those of the wavefunction because they produce the same kind of interference pattern that the wavefunction would produce were it a physical object. Yet, to identify them with the wavefunction leads to the paradox of nonlocal collapse. Perhaps this is Nature’s way of hinting to us that there is no such thing as objective, physical reality.