Lecture 24

4/28/94

The Measurement Problem, Part II

Last time, we saw that Bohr proposed an interpretation of the quantum mechanical formalism, the Copenhagen interpretation, that has become the received view among physicists. Today we will look more closely at the picture that view gives us of physical reality and the measurement process in particular. Then we will examine several alternative interpretations of measurement, proceeding from the least plausible to the most plausible alternatives.

The Copenhagen Understanding Of Measurement

According to Bohr, it only makes sense to attribute values to observable quantities of a physical system when system is being measured in a particular way. Descriptions of physical systems therefore only make sense relative to particular contexts of measurement. (This is Bohr's solution to the puzzling wave-particle duality exhibited by entities such as photons and electrons: the "wave" and "particle" aspects of these entities are "complementary," in the sense that it is physically impossible to construct a measuring device that will measure both aspects simultaneously. Bohr concluded that from a physical standpoint it only makes sense to speak about the "wave" or "particle" aspects of quantum entities as existing relative to particular measurement procedures.) One consequence of Bohr's view is that one cannot even ask what a physical system is like between measurements, since any answer to this question would necessarily have to describe what the physical system is like, independent of any particular context of measurement.

On Bohr's view (which is generally accepted as the "orthodox" interpretation of quantum mechanics even though it is often misunderstood), the world is divided into two realms of existence, that of quantum systems, which behave according to the formalism of quantum mechanics and do not have definite observable values outside the context of measurement, and of "classical" measuring devices, which always have definite values but are not described within quantum mechanics itself. The line between the two realms is arbitrary: at any given time, one can consider a part of the world that serves as a "measuring instrument" (such as the Stern-Gerlach device) either as a quantum system that interacts with other quantum systems according to the deterministic laws governing the state vector (Schrödinger's equation) or as a measuring device that behaves "classically" (i.e., always has definite observable properties) though indeterministically.

There are several difficulties with this view, which together constitute the "measurement problem." To begin with, the orthodox interpretation gives no principled reason why physics should not be able to give a complete description of the measurement process. After all, a measuring device (such as a Stern-Gerlach magnet) is a physical system, and in performing a measurement it simply interacts with another physical system such as a photon or an electron. However, (a) there seems to be no principled reason why one particular kind of physical interaction is either indescribable within quantum physics itself (as Bohr suggests), or is not subject to the same laws (e.g., Schrödinger's equation) that governs all other physical interactions, and (b) the orthodox view offers no precise characterization that would mark off those physical interactions that are measurements from those physical interactions that are not. Indeed, the orthodox interpretation claims that whether a certain physical interaction is a "measurement" is arbitrary, i.e., a matter of choice on the part of the theorist modeling the interaction. However, this hardly seems satisfactory from a physical standpoint. It does not seem to be a matter of mere convenience where we are to draw the line between the "classical" realm of measuring devices and the realm of those physical systems that obey the deterministic laws of that formalism (in particular, Schrödinger's equation). For example, Schrödinger pointed out that the orthodox interpretation allows for inconsistent descriptions of the state of macroscopic systems, depending on whether we consider them measuring devices. For example, suppose that you placed a cat in an enclosed box along with a device that will release poisonous gas if (and only if) a Geiger counter measures that a certain radium atom has decayed. According to the quantum mechanical formalism, the radium atom is in a superposition of decaying and not decaying, and since there is a correlation between the state of the radium atom and the Geiger counter, and between the state of the Geigen counter and the state of the cat, the cat should also be in a superposition, specifically, a superposition of being alive and dead, if we do not consider the cat to be a measuring device. If the orthodox interpretation is correct, however, this would mean that there is no fact of the matter about whether the cat is alive or dead, if we consider the cat not to be a measuring device. On the other hand, if we consider the cat to be a measuring device, then according to the orthodox interpretation, the cat will either be definitely alive or definitely dead. However, it certainly does not seem to be a matter of "arbitrary" choice whether (a) there is no fact of the matter about whether the cat is alive or dead before we look into the box, or (b) the cat is definitely alive or dead before we look into the box.

Wigner's Idealism: Consciousness As The Cause Of Collapse

Physicist E. Wigner argued against Bohr's view that the distinction between measurement and "mere" physical interaction could be made arbitrarily by trying to emphasize its counterintuitive consequences. Suppose that you put one of Wigner's friends in the box with the cat. The "measurement" you make at a given time is to ask Wigner's friend if the cat is dead or alive. If we consider your friend as part of the experimental setup, quantum mechanics predicts that before you ask Wigner's friend whether the cat is dead or alive, he is in a superposition of definitely believing the cat is dead and definitely believing that the cat is alive. Wigner argued that this was an absurd consequence of Bohr's view. People simply do not exist in superposed belief-states. Wigner's solution was that, contrary to what Bohr claimed, there is a natural division between what constitutes a measurement and what does not--the presence of a conscious observer. Wigner's friend is conscious; thus, he can by an act of observation cause a collapse of the wave function. (Alternately, if we consider the cat to be a conscious being, the cat would be definitely alive or definitely dead even before Wigner's friend looked to see how the cat was doing.)

Few physicists have accepted Wigner's explanation of what constitutes a measurement, though his view has "trickled down" to some of the (mainly poor quality) popular literature on quantum mechanics (especially the type of literature that sees a direct connection between quantum mechanics and Eastern religious mysticism). The basic source of resistance to Wigner's idealism is that it requires that physicists solve many philosophical problems they'd like to avoid, such as whether cats are conscious beings. More seriously, Wigner's view requires a division of the world into two realms, one occupied by conscious beings who are not subject to the laws of physics but who can somehow miraculously disrupt the ordinary deterministic evolution of the physical systems, and the other by the physical systems themselves, which evolve deterministically until a conscious being takes a look at what's going on. This is hardly the type of conceptual foundation needed for a rigorous discipline such as physics.

The Many-Worlds Interpretation

Another view, first developed by H. Everett (a graduate student at Princeton) in his Ph.D. thesis, is called the many-worlds interpretation. It was later developed further by another physicist, B. de Witt. This view states that there is no collapse when a measurement occurs; instead, at each such point where a measurement occurs the universe "branches" into separate, complete worlds, a separate world for every possible outcome that measurement could have. In each branch, it looks like there the measuring devices indeterministically take on a definite value, and the empirical frequencies that occur upon repeated measurement in almost every branch converge on the probabilities predicted by the Projection Postulate. The deterministically evolving wave function correctly describes the quantum mechanical state of the universe as a whole, including all of its branches.

Everett's view has several advantages, despite its conceptual peculiarity. First, it can be developed with mathematical precision. Second, it postulates that the universe as a whole evolves deterministically according to Schrödinger's equation, so that you only need only type of evolution, not Schrödinger's equation plus collapse during a measurement. Third, the notion of a measurement (which is needed to specify where the branching occurs) can be spelled out precisely and non-arbitrarily as a certain type of physical interaction. Fourth, it makes sense to speak of the quantum state of the universe as a whole on Everett's view, which is essential if you want to use quantum mechanical laws to develop cosmological theories. By contrast, on the orthodox, Copenhagen view, quantum mechanics can never describe the universe as a whole since quantum mechanical description is always relativized to an arbitrarily specified measuring device, which is not itself given a quantum mechanical description.

On the other hand, the many worlds interpretation has several serious shortcomings. First, it's simply weird conceptually. Second, it does not account for the fact that we never experience anything like a branching of the world. How is it that our experience is unified (i.e., how is it that my experience follows one particular path in the branching universe and not others)? Third, as noted above, the many worlds interpretation predicts that there will be worlds where the observed empirical frequencies of repeated measurements will not fit the predictions of the quantum mechanical formalism. In other words, if the theory is true, there will be worlds at which it looks as if the theory is false! Finally, the theory does not seem to give a clear sense to locutions such as "the probability of this electron going up after passing through this Stern-Gerlach device is 1/2." What exactly does that number 1/2 mean, if the universe simply branches into two worlds, in one of which the electron goes up and in the other of which it goes down? The same branching would occur, after all, if the electron were in a state where the probability of its going up were 3/4 and the probability of its going down were 1/4.

Because of problems like this, the many worlds interpretation has gained much notoriety among physicists, but almost no one accepts it as a plausible interpretation.

The Ghirardi-Rimini-Weber (GRW) Interpretation

In 1986, three physicists (G.C. Ghirardi, A. Rimini, and T. Weber) proposed a way of accounting for the fact that macroscopic objects (such as Stern-Gerlach devices, cats, and human beings) are never observed in superpositions, whereas microscopic systems (such as photons and electrons) are. (Their views were developed further in 1987 by physicist John Bell.) According to the Ghirardi-Rimini-Weber (GRW) interpretation, there is a very small probability (one in a trillion) that the wave functions for the positions of isolated, individual particles will collapse spontaneously at any given moment. When particles couple together to form an object, the tiny probabilities of spontaneous collapse quickly add up for the system as a whole, since when one particle collapses so does every particle to which that particle is coupled. Moreover, since even a microscopic bacterium is composed of trillions and trillions of subatomic particles, the probability that a macroscopic object will have a definite spatial configuration at any given moment is vanishingly close to 100 percent. Thus, GRW can explain in a mathematically precise way why we never observe macroscopic objects in superpositions, but often observe interference effects due to superposition when we're looking at isolated subatomic particles. Moreover, they too can give a precise and non-arbitrary characterization of the measurement process as a type of physical interaction.

There are two problems with the GRW interpretation. First, though subatomic particles are localized to some degree (in that the collapse turns the wave function representing the position of those particles into a narrowly focused bell curve), they never do have precise positions. The tails of the bell curve never vanish, extending to infinity. (Draw diagram on board.) Thus, GRW never give us particles with definite positions or even with a small but spatially extended positions: instead, what we get are particles having "mostly" small but spatially extended positions. Importantly, this is also true of macroscopic objects (such as Stern-Gerlach devices, cats, and human beings) that are composed of subatomic particles. In other words, according to GRW you are "mostly" in this room, but there's a vanishingly small part of you at every other point in the universe, no matter how distant! Second, GRW predicts that energy is not conserved when spontaneous collapses occur. While the total violation of conservation of energy predicted by GRW is too small to be observed, even over the lifetime of the universe, it does discard a feature of physical theory that many physicists consider to be conceptually essential.

Bohm's Interpretation

In 1952, physicist David Bohm formulated a complete alternative to standard (non-relativistic) quantum mechanics. Bohm's theory was put into a relatively simple and elegant mathematical form by John Bell in 1982. Bohm's theory makes the same predictions as does standard (non-relativistic) quantum mechanics but describes a classical, deterministic world that consists of particles with definite positions. The way that Bohm does this is by postulating that besides particles, there is a quantum force that moves the particles around. The physically real quantum force, which is represented mathematically by Schrödinger's wave equation, pushes the particles around so that they behave exactly as standard quantum mechanics predicts. Bohm's theory is deterministic: if you knew the initial configuration of every particle in the universe, applying Bohm's theory would allow you to predict with certainty every subsequent position of every particle in the universe. However, there's a catch: the universe is set up so that it is a physical (rather than a merely practical) impossibility for us to know the configuration of particles in the universe. Thus, from our point of view the world behaves just as if it's indeterministic (though the apparent indeterminism is simply a matter of our ignorance); also, though the wave function governing the particle's motion never collapses, the particle moves around so that it looks as if measurement causes it to collapse.

Despite its elegance and conceptual clarity, Bohm's theory has not been generally accepted by physicists, for two reasons. (1) Physicists can't use Bohm's theory, since it's impossible to know the configuration of all particles in the universe at any given time. Because of our ignorance of this configuration, the physical theory we have to use to generate predictions is standard quantum mechanics. Why then bother with Bohm's theory at all? (2) What is more important, in Bohm's theory all the particles in the universe are intimately connected so that every particle instantaneously affects the quantum force governing the motions of the other particles. Many physicists (Einstein included) saw this feature of Bohm's theory as a throwback to the type of "action at a distance" expunged by relativity theory. Moreover, Bohm's theory assumes that simultaneity is absolute and so is inconsistent with relativity theory in its present form. (There is no Bohmian counterpart to relativistic quantum mechanics, though people are currently working on producing one.)

Conclusion

Each of the various interpretations of quantum mechanics we have examined, from the orthodox Copenhagen interpretation to the four non-orthodox interpretations (idealistic, many-worlds, GRW, and Bohm) has some sort of conceptual shortcoming. This is what makes the philosophy of quantum mechanics so interesting: it shows us that the fact that experiments agree with the predictions of quantum mechanics, under any of these interpretations, indicates one thing with certainty--the world we live in is a very weird place!