Quantum Mechanics

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In sum, the wave structure of the electron-detector-observer system consists of two distinct branches, the A-outcome branch and the B-outcome branch. Since these two branches are relatively causally isolated from each other, we can describe them as two distinct worlds , in one of which the electron hits the detector at A and the observer sees the A-detector fire, and in the other of which the electron hits the detector at B and the observer sees the B-detector fire. This talk of worlds needs to be treated carefully, though; there is just one physical world, described by the quantum state, but because observers along with all other physical objects exhibit this branching structure, it is as if the world is constantly splitting into multiple copies.

It is not clear whether Everett himself endorsed this talk of worlds, but this is the understanding of his work that has become canonical; call it the many-worlds interpretation. According to the many-worlds interpretation, then, every physically possible outcome of a measurement actually occurs in some branch of the quantum state, but as an inhabitant of a particular branch of the state, a particular observer only sees one outcome.

This explains why, in the electron interference experiment, the outcome looks like a discrete particle even though the object that passes through the interference device is a wave; each point in the wave generates its own branch of reality when it hits the detectors, so from within each of the resulting branches it looks like the incoming object was a particle.

The main advantage of the many-worlds interpretation is that it is a realist interpretation that takes the physics of standard quantum mechanics literally. It is often met with incredulity, since it entails that people along with other objects are constantly branching into innumerable copies, but this by itself is no argument against it.

Still, the branching of people leads to philosophical difficulties concerning identity and probability, and these particularly the latter constitute genuine difficulties facing the approach. Various solutions have been developed in the literature. One might follow Derek Parfit and bite the bullet here: what fission cases like this show is that strict identity is not a useful concept for describing the relationship between people and their successors.

Or one might follow David Lewis and rescue strict identity by stipulating that a person is a four-dimensional history rather than a three dimensional object. According to this picture, there are two people two complete histories present both before and after the fission event; they initially overlap but later diverge. Identity over time is preserved, since each of the pre-split people is identical with exactly one of the post-split people. Both of these positions have been proposed as potential solutions to the problem of personal identity in a many-worlds universe.

A third solution that is sometimes mentioned is to stipulate that a person is the whole of the branching entity, so that the pre-split person is identical to both her successors, and despite our initial intuition otherwise the successors are identical to each other.


So the problem of identity admits of a number of possible solutions, and the only question is how one should try to decide between them. Indeed, one might argue that there is no need to decide between them, since the choice is a pragmatic one about the most useful language to use to describe branching persons. The problem of probability, though, is potentially more serious. As noted above, quantum mechanics makes its predictions in the form of probabilities: the square of the wavefunction amplitude in a region tells us the probability of the particle being located there.

The striking agreement of the observed distribution of outcomes with these probabilities is what underwrites our confidence in quantum mechanics. But according to the many-worlds interpretation, every outcome of a measurement actually occurs in some branch of reality, and the well-informed observer knows this. It is hard to see how to square this with the concept of probability; at first glance, it looks like every outcome has probability 1, both objectively and epistemically.

In particular, if a measurement results in two branches, one with a large squared amplitude and one with a small squared amplitude, it is hard to see why we should regard the former as more probable than the latter. But unless we can do so, the empirical success of quantum mechanics evaporates. It is worth noting, however, that the foundations of probability are poorly understood. When we roll two dice, the chance of rolling 7 is higher than the chance of rolling But there is no consensus concerning the meaning of chance claims, or concerning why the higher chance of 7 should constrain our expectations or behavior.

So perhaps a quantum branching world is in no worse shape than a classical linear world when it comes to understanding probability. We may not understand how squared wavefunction amplitude could function as chance in guiding our expectations, but perhaps that is no barrier to postulating that it does so function. A more positive approach has been developed by David Deutsch and David Wallace, arguing that given some plausible constraints on rational behavior, rational individuals should behave as if squared wavefunction amplitudes are chances.

If one combines this with a functionalist attitude towards chance—that whatever functions as chance in guiding behavior is chance—then this program promises to underwrite the contention that squared wave amplitudes are chances. However, the assumptions on which the Deutsch-Wallace argument is based can be challenged. In particular, they assume that it is irrational to care about branching per se : having two successors experiencing a given outcome is neither better nor worse than having one successor experiencing that outcome.

But it is not clear that this is a matter of rationality any more than the question of whether having several happy children is better than having one happy child. A further worry about the many-words theory that has been largely put to rest concerns the ontological status of the worlds. It has been argued that the postulation of many worlds is ontologically profligate. However, the current consensus is that worlds are emergent entities just like tables and chairs, and talk of worlds is just a convenient way of talking about the features of the quantum state.

On this view, the many-worlds interpretation involves no entities over and above those represented by the quantum state, and as such is ontologically parsimonious. There remains the residual worry that the number of branches depends sensitively on mathematical choices about how to represent the quantum state.

Wallace, however, embraces this indeterminacy, arguing that even though the many-worlds universe is a branching one, there is no well-defined number of branches that it has. If tenable, this goes some way towards resolving the above concern about the rationality of caring about branching per se : if there is no number of branches, then it is irrational to care about it.

The many-worlds interpretation would have us believe that we are mistaken when we think that a quantum measurement results in a unique outcome; in fact such a measurement results in multiple outcomes occurring on multiple branches of reality. But perhaps that is too much to swallow, or perhaps the problems concerning identity and probability mentioned above are insuperable.

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In that case, one is led to the conclusion that quantum mechanics is incomplete, since there is nothing in the quantum state that picks out one of the many possible measurement results as the single actual measurement result. If this view is correct, then quantum mechanics stands in need of completion via the addition of extra variables describing the actual state of the world. These additional variables are commonly known as hidden variables. However, a theorem proved by John Bell in shows that, subject to certain plausible assumptions, no such hidden-variable completion of quantum mechanics is possible.

One version of the proof concerns the properties of a pair of particles. Each particle has a property called spin: when the spin of the particle is measured in some direction, one either gets the result up or down. According to the hidden variable approach, the particles have determinate spin values for each of the three measurement directions prior to measurement. The question is how to ascribe spin values to particles to reproduce the predictions of quantum mechanics.

And what Bell proved is that there is no way to do this; the task is impossible. Bell concluded instead that one of the assumptions he relied on in his proof must be false. First, Bell assumed locality —that the result of a measurement performed on one particle cannot influence the properties of the other particle.

This seems secure because the measurements on the two particles can be widely separated, so that a signal carrying such an influence would have to travel faster than light. Second, Bell assumed independence —that the properties of the particles are independent of which measurements will be performed on them. This assumption too seems secure, because the choice of measurement can be made using a randomizing device or the free will of the experimenter. Despite the apparent security of his assumptions, Bell knew when he proved his theorem that a hidden-variable completion of quantum mechanics had been explicitly constructed by David Bohm in Bohm assumed that in addition to the wave described by the quantum state, there is also a set of particles whose positions are given by the hidden variables.

The wave pushes the particles around according to a new dynamical law formulated by Bohm, and the law is such that if the particle positions are initially statistically distributed according to the squared amplitude of the wave, then they are always distributed in this way. In an electron interference experiment, then, the existence of the wave explains the interference effect, the existence of the particles explains why each electron is observed at a precise location, and the new Bohmian law explains why the probability of observing an electron at a given location is given by the squared amplitude of the wave.

Nevertheless, the name has stuck. The new law introduced by Bohm is explicitly non-local: the motion of each particle is determined in part by the positions of all the other particles at that instant. Bell recognized this possibility. Bohm chooses positions as the properties described by the hidden variables of his theory.

His reason for this is that it is plausible that it is the positions of things that we directly observe, and hence completing quantum mechanics via positions suffices to ensure that measurements have unique outcomes. But it is possible to construct measurements in which the outcome is recorded in some property other than position.

As a response to this possibility, one might suggest adding hidden variables describing every property of the particles simultaneously, rather than just their positions. However, a theorem proved by Kochen and Specker in shows that no such theory can reproduce the predictions of quantum mechanics.

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A final way to accommodate such measurements within a hidden variable theory is to make it a contingent matter which properties of a system are ascribed determinate values at a particular time. That is, rather than supplementing the wavefunction with variables describing a fixed property the positions of things , one can let the wavefunction state itself determine which properties of the system are described by the hidden variables at that time. The idea is that the algorithm for ascribing hidden variables to a system is such that whenever a measurement is performed, the algorithm ascribes a determinate value to the property recording the outcome of the measurement.

Such theories are known as modal theories. In the modal case, the rule for deciding which properties of the system are made determinate depends on the complete wavefunction state at a particular instant, and this allows a measurement on one particle to affect the properties ascribed to another particle, however distant. As mentioned above, one can solve this problem by supplementing special relativity with a preferred standard of simultaneity.

But this is widely regarded as an ad hoc and unwarranted addition to an otherwise elegant and well-confirmed physical theory. Indeed, the same charge is often levelled at the hidden variables themselves; they are an ad hoc and unwarranted addition to quantum mechanics. If hidden variable theories turn out to be the only viable interpretations of quantum mechanics, though, the force of this charge is reduced considerably.

Nevertheless, it may be possible to construct a hidden variable theory that does not violate locality. Since one can choose the measurements however one likes, it is initially hard to see how this assumption could be violated. But there are a couple of ways it might be done. First, one could simply accept that there are brute, uncaused correlations in the world.

There is no causal link in either direction between my choice of which measurement to perform on a currently distant particle and its properties, but nevertheless there is a correlation between them. This approach requires giving up on the common cause principle—the principle that a correlation between two events indicates either that one causes the other or that they share a cause. However, there is little consensus concerning this principle anyway. A second approach is to postulate a common cause for the correlation—a past event that causally influences both the choice of measurement and the properties of the particle.

But absent some massive unseen conspiracy on the part of the universe, one can frequently ensure that there is no common cause in the past by isolating the measuring device from external influences. However, the measuring device and the particle to be measured will certainly interact in the future , namely when the measurement occurs. It has been proposed that this future event can constitute the causal link explaining the correlation between the particle properties and the measurements to be performed on them.

This requires that later events can cause earlier events—that causation can operate backwards in time as well as forwards in time. For this reason, the approach is known as the retrocausal approach. The retrocausal approach allows correlations between distant events to be explained without instantaneous action at a distance, since a combination of ordinary causal links and retrocausal links can amount to a causal chain that carries an influence between simultaneous distant events.

No absolute standard of simultaneity is required by such explanations, and hence retrocausal hidden variable theories are more easily reconciled with special relativity than non-local hidden variable theories.

Where Quantum Probability Comes From

Retrocausal theories vary in their ontological presuppositions. But it may be possible to make do with the particles alone, with the wavefunction representing our knowledge of the particle positions rather than the state of a real object. The idea is that the interaction between the causal influences on the particles from the past and from the future can explain all the quantum phenomena we observe, including interference. However, at present this is just a promising research program; no explicit dynamical laws for such a theory have been formulated.

Hidden variable theories attempt to complete quantum mechanics by positing extra ontology in addition to or perhaps instead of the wavefunction. For example, if a measuring device fed a spin-up particle leads to a spin-up reading, and a measuring device fed a spin-down particle leads to a spin-down reading, then a measuring device fed a particle whose state is a sum of spin-up and spin-down states will end up in a state which is a sum of reading spin-up and reading spin-down. This is the multiplicity of measurement outcomes embraced by the many-worlds interpretation.

To avoid sums of distinct measurement outcomes, one needs to modify the basic dynamical equation of the quantum mechanics equation so that it is non-linear. In particular, for each particle in a system there is a small chance per unit time of the wavefunction undergoing a process in which it is instantly and discontinuously localized in the coordinates of that particle.

The localization process multiplies the wave state by a narrow Gaussian bell curve , so that if the wave was initially spread out in the coordinates of the particle in question, it ends up concentrated around a particular point. The point on which this collapse process is centered is random, with a probability distribution given by the square of the pre-collapse wave amplitude averaged over the Gaussian collapse curve.

The way this works is as follows. The collapse rate for a single particle is very low—about one collapse per hundred million years. But macroscopic objects contain on the order of a trillion trillion particles, so we should expect about ten million collapses per second for such an object. Furthermore, in solid objects the positions of those particles are strongly correlated with each other, so a collapse in the coordinates of any particle in the object has the effect of localizing the wavefunction in the coordinates of every particle in the object.

This means that if the wavefunction of a macroscopic object is spread over a number of distinct locations, it very quickly collapses to a state in which its wavefunction is highly localized around one location. In the case of electron interference, then, each electron passes through the apparatus in the form of a spread-out wave.

The collapse process is vanishingly unlikely to affect this wave, which is important, as its spread-out nature is essential to the explanation of interference: wave components traveling distinct paths must be able to come together and either reinforce each other or cancel each other out. But when the electron is detected, its position is indicated by something we can directly observe, for example, by the location of a macroscopic pointer.

To measure the location of the electron, then, the position of the pointer must become correlated with the position of the electron. Since the wave representing the electron is spread out, the wave representing the pointer will initially be spread out too. But within a fraction of a second, the spontaneous collapse process will localize the pointer and the electron to a well-defined position, producing the unique measurement outcome we observe. The spontaneous collapse approach is related to earlier proposals for example, by John von Neumann that the measurement process itself causes the collapse that reduces the multitude of pre-measurement wave branches to the single observed outcome.

This mechanism is crucial; without it, as we have seen, there is no way for the measurement process to generate a unique outcome. In the electron interference case, particle behavior emerges during measurement; the measured system exhibits only wave-like behavior prior to measurement. Strictly speaking, to say that a system contains n particles is just to say that its wave representation has 3 n dimensions, and to single out one of those particles is really just to focus attention on the form of the wave in three of those dimensions.

An immediate difficulty that faces the GRW theory is that the localization of the wave induced by collapse is not perfect. The collapse process multiplies the wave by a Gaussian, a function which is strongly peaked around its center but which is non-zero everywhere. No part of the pre-collapse wavefunction is driven to zero by this process; if the wavefunction represents a set of possible measurement results, the wave component corresponding to one result becomes large and the wave component corresponding to the others become small, but they do not disappear.

Since one motivation for adopting a spontaneous collapse theory is the perceived failure of the many-worlds interpretation to recover probability claims, it cannot be argued that the small terms are intrinsically improbable. Instead, it looks like the GRW spontaneous collapse process fails to ensure that measurements have unique outcomes. David Albert has argued that this makes the three-dimensional world of experience illusory.

A third difficulty with the GRW theory is that the collapse process acts instantaneously on spatially separated parts of the system; it instantly multiplies the wavefunction everywhere by a Gaussian. One way of responding to these difficulties, advocated by Ghirardi, is to postulate a three-dimensional mass distribution in addition to and determined by the wavefunction, such that our experience is determined directly by the mass distribution rather than the wavefunction.

This responds to the second difficulty, since the mass distribution that we directly experience is three-dimensional, and hence our experience of a three-dimensional world is veridical. It may also go some way towards resolving the first difficulty, since the mass density corresponding to non-actual measurement outcomes is likely to be negligible relative to the background mass density surrounding the actual measurement outcome the mass density of air, for example. On this proposal, the small wave terms corresponding to non-actual measurement outcomes can be understood in a straightforwardly probabilistic way: there is only a small chance that a flash will be associated with such a term, and so only a small chance that the non-actual measurement outcome will be realized.

The flashes are located in three-dimensional space, so there is no worry that three-dimensionality is an illusion. And since the flashes, unlike the wavefunction, are located at space-time points , it is easier to envision a reconciliation between the flashy theory and special relativity. Here are some prominent ones. Like spontaneous collapse theories, the consistent histories approach incorporates successive localizations of the wavefunction.

But unlike spontaneous collapse theories, these localizations are not regarded as physical events, but just as a means of picking out a particular history of the system in question as actual, much as hidden variables pick out a particular history as actual. If the localizations all constrain the position of a particle, then the history picked out resembles a Bohmian trajectory.

But the consistent histories approach also allows localizations to constrain properties other than position, resulting in a more general class of possible histories. However, not all such sets of histories can be ascribed consistent probabilities: notably, interference effects often prevent the assignment of probabilities obeying the standard axioms to histories.

However, for systems that interact strongly with their environment, interference effects are rapidly suppressed; this phenomenon is called decoherence. Decoherent histories can be ascribed consistent probabilities—hence the two alternative names of this approach. It is assumed that only consistent sets of histories can describe the world, but other than this consistency requirement, there is no restriction on the kinds of histories that are allowed. Indeed, Griffiths maintains that there is no unique set of possible histories: there are many ways of constructing sets of possible histories, where one among each set is actual, even if the alternative actualities so produced describe the world in mutually incompatible ways.

Absent a many worlds ontology, however, some have worried about how such a plurality of true descriptions of the world could be coherent. Gell-Mann and Hartle respond to such concerns by arguing that organisms evolve to exploit the relative predictability of one among the competing sets of histories.

The transactional interpretation, initially developed by John Cramer, also incorporates elements of both collapse and hidden variable approaches. It starts from the observation that some versions of the dynamical equation of quantum mechanics admit wave-like solutions traveling backward in time as well as forward in time. Typically the former solutions are ignored, but the transactional interpretation retains them. The formation of a transaction is somewhat reminiscent of the spontaneous collapse of the wavefunction, but due to the retrocausal nature of the theory, one might conclude that the wavefunction never exists in a pre-collapse form, since the completed transaction exists as a timeless element in the history of the universe.

Hence some have questioned the extent to which the story involving forwards and backwards waves constitutes a genuine explanation of transaction formation, raising questions about the tenability of the transactional interpretation as a description of the quantum world. Ruth Kastner responds to these challenges by developing a possibilist transactional interpretation, embedding the transactional interpretation in a dynamic picture of time in which multiple future possibilities evolve to a single present actuality.

Relational interpretations, such as those developed by David Mermin and by Carlo Rovelli, take quantum mechanics to be about the relations between systems rather than the properties of the individual systems themselves. But whereas Everettians typically say that a relation such as an observer seeing a particular measurement result holds on the basis of the properties of the observer and of the measured system within a branch, Mermin denies that there are such relata; rather, the relation itself is fundamental.

Hence this is not a many worlds interpretation, since world-relative properties provide the relata that relational interpretations deny. Without such relata, though, it is hard to understand relational quantum mechanics as a description of a single world either. However, citing analogies with spatiotemporal properties in relativistic theories, Rovelli insists that it is enough that quantum mechanics ascribe properties to a system relative to the state of a second system for example, an observer. They develop rules of quantum credence by analogy with the rules of classical information theory, expressing the difference between quantum systems and classical systems in informational terms, for example in terms of an unavoidable loss of information associated with a quantum measurement.

Most notably, measurement outcomes cannot be predicted with perfect confidence, even in principle. This feature is what led Albert Einstein to complain about God playing dice with the universe. Researchers continue to argue over the best way to think about quantum mechanics. All of them share the feature that they lean on the idea of probability in a fundamental way. Like many subtle concepts, probability starts out with a seemingly straightforward, commonsensical meaning, which becomes trickier the closer we look at it. We know how to handle the mathematics of probability, thanks to the work of the Russian mathematician Andrey Kolmogorov and others.

Probabilities are real numbers between zero and one, inclusive; the probabilities of all independent events add up to one; and so on. There are numerous approaches to defining probability, but we can distinguish between two broad classes. An example of an objective approach to probability is frequentism, which defines probability as the frequency with which things happen over many trials, as in our coin-tossing example. Bayesians imagine that rational creatures in states of incomplete information walk around with credences for every proposition you can imagine, updating them continually as new data comes in.

Interestingly, different approaches to quantum mechanics invoke different meanings of probability in central ways. Thinking about quantum mechanics helps illuminate probability, and vice versa. Each of these represents a way of solving the measurement problem of quantum mechanics.

The collapse itself is unpredictable; the wave function assigns a number to each possible outcome, and the probability of observing that outcome is equal to the value of the wave function squared. When exactly does it occur? Why are measurements seemingly different from ordinary evolution?

Dynamical-collapse theories offer perhaps the most straightforward resolution to the measurement problem. Such collapses are so rare that we would never observe one for a single particle, but in a macroscopic object made of many particles, collapses happen all the time. All the particles in a large system will be entangled with each other, so that when just one of them localizes in space, the rest are brought along for the ride.

Probability in such models is fundamental and objective. There is absolutely nothing about the present that precisely determines the future. Dynamical-collapse theories fit perfectly into an old-fashioned frequentist view of probability. What happens next is unknowable, and all we can say is what the long-term frequency of different outcomes will be. Pilot-wave theories tell a very different story. Here, nothing is truly random; the quantum state evolves deterministically, just as the classical state did for Newton.

The new element is the concept of hidden variables, such as the actual positions of particles, in addition to the traditional wave function. The particles are what we actually observe, while the wave function serves merely to guide them. We can prepare a wave function so that we know it exactly, but we only learn about the hidden variables by observing them. The best we can do is to admit our ignorance and introduce a probability distribution over their possible values. Probability in pilot-wave theories, in other words, is entirely subjective.

It characterizes our knowledge, not an objective frequency of occurrences over time. A full-powered Laplace demon that knew both the wave function and all the hidden variables could predict the future exactly, but a hobbled version that only knew the wave function would still have to make probabilistic predictions.

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Then we have many-worlds. Many-worlds quantum mechanics has the simplest formulation of all the alternatives. There are no collapses and no additional variables. The answer is that the combined system of observer and object evolves into an entangled superposition.