Well, here it is, the last part in what you probably thought was an interminably long series on the interpretations of Quantum Mechanics. In this part, I want to do three things – first, mention a few other interpretations that I didn’t discuss in depth, second, talk briefly about some other features of QM and how they relate to the different interpretations, and third, take a larger view of the interpretations and give you my personal opinion on the whole thing.

Here are the previous parts:

Part 2 (Collapse Interpretations)

Part 3 (De Broglie – Bohm Interpretation)

Part 4 (Everett/Many-Worlds Interpretations)

Just as a bit of review, and so that everyone has it fresh in their minds, I want to reiterate the trilemma that is at the heart of the measurement problem, and the corresponding interpretations. The problem is how to deal with puzzling states that, according to the linear dynamics of the Schrödinger Equation, should arise whenever a measurement is performed, where a person gets into a superposition of states in which she sees different results for the measurement. There are three ways of trying to deal with this:

1. Add something to the dynamics1 of Quantum Mechanics, namely a collapse postulate, that violates the Schrödinger Equation and results in one or another possible outcome being selected. This leads to the various collapse interpretations, including the Copenhagen Interpretation, consciousness-induced collapse, and GRW.

2. Add something to the ontology2 of Quantum Mechanics, over and above the wave functions of particles, that allows one or another outcome to be “picked out” as the one that occurs, even though the Schrödinger Equation remains true at all times. This leads to hidden variable interpretations, like Bohmian mechanics.

3. Don’t add anything to Quantum Mechanics, and instead accept that those strange states exist and try to make sense of what they mean. This leads to the various flavors of Everett interpretations, including the Bare Theory, Many Worlds, Many Minds, and Single Mind.

That covers the major interpretations, but there are some other approaches that I want to at least mention before moving on.

**Modal Interpretations**

In part 3, when I talked about hidden variables, I mentioned that there’s also a similar, more recent approach called “modal interpretations”. This approach was first suggested by Bas van Fraasen in the 1970s. Like hidden variables, these interpretations opt for the second prong of the trilemma, adding something to the ontology of the theory that “picks out” the result of a measurement even though the wave function never collapses.

The gist of modal interpretations is to draw a distinction between the “dynamical state” of a system and its “value state”, the notion being that the “dynamical state” represents the *possible* properties of the system and the “value state” represents its *actual* properties. It thus has something in common with “modal logic”, an extension of classical logic that includes notions of necessity and possibility. The dynamical state is supposed to evolve deterministically, according to the Schrödinger Equation, at all times, while the value state evolves probabilistically in accordance with the usual probability rules.

The trick for a modal interpretation is to then specify *which* properties of a system have “actual” values at a given time. They can’t all have actual values at all times, as that would violate the uncertainty relations. There are various ways to choose which properties have definite values, which give rise to different modal interpretations.

In the simple case where *position* is chosen as the property that has an actual value at all times, this modal interpretation becomes, essentially, identical to Bohm’s interpretation. In other words, the De Broglie – Bohm picture of particles that always have definite positions, plus wave functions that push those particles around in space, can be thought of as just one particular example of a more general class of interpretations.

I’ll be the first to admit that I don’t know all that much about modal interpretations (other than Bohm’s), but as far as I’m aware, they don’t particularly improve on Bohmian mechanics – that is to say, pretty much any issue that one might have with Bohmian mechanics could pretty easily be applied to any of the modal interpretations as well.

**The Transactional Interpretation**

John G. Cramer has proposed what is called the Transactional Interpretation. The TI has been the subject of much debate, and I think it would not be unfair to say that even its proponents have significant disagreements about what exactly it entails. The fundamental idea, though, is that all physical interactions consist of an “offer wave”, the ordinary wave function of the particle, moving forward in time, as well as a “confirmation wave”, a corresponding wave function that travels *backward* in time from another particle, with which the first one is going to interact. All particles are pictured as continously sending both offer waves into the future and confirmation waves into the past, with the amplitudes of those waves both following the Schrödinger Equation. When two particles exchange an offer and confirmation wave, the “transaction” is completed, and the particle collapses into a definite trajectory between its starting point and future interaction point. *Which* potential transaction a particle “accepts” is chosen probabilistically, following the normal square amplitude rule for the exchanged waves.

This is, to put it mildly, a very outlandish idea. The motivation is to remove the non-locality of QM (or, at least, make that non-locality unobjectionable) by introducing this notion of retrocausality. It’s also claimed that it eliminates any problems associated with the collapse of the wave function, because here collapse doesn’t occur at any specific moment, but rather occurs all at once along the whole trajectory of the particle. But it does this at the expense of introducing a kind of pseudo-time in which the transaction is made. If one strips this interpretation of its notions of retrocausality and pseudo-time, it seems to me that it is no different from something like Bohmian mechanics, containing both wave functions and particles with definite positions/trajectories.

You can probably tell that I don’t think too much of this interpretation, though I confess that my understanding of it is far from perfect. It seems to me that it introduces several highly questionable notions with little to show for it. In any case, it remains a fairly niche view.

**Stochastic Interpretations**

All of the interpretations I’ve talked about so far assume that Quantum Mechanics3, as a fundamental theory of the universe, is true – that is, that it is not an approximation to some other, underlying theory in the way that classical mechanics is an approximation to Quantum Mechanics. An interesting but little-known approach, though, is the Stochastic Mechanics proposed by Edward Nelson. Instead of starting with the Schrödinger Equation as an axiom, Nelson assumes that there is some more fundamental dynamics at work, with particles being pushed around on a fundamental level by a random, or at least chaotic, diffusion process. The Schrödinger Equation would then not be a fundamental law, but would be derived from the chaotic diffusion processes, in a way similar to that in which macroscopic thermodynamics is derived, stochastically, from the microscopic motions of atoms and molecules.

What Nelson ends up with is something very much like the De Broglie – Bohm pilot wave theory. But it is conceptually very different. In both cases, particles are real things with determinate positions, being pushed around. In Bohm’s case, they are being pushed around by the wave functions, which are real, physical things that are governed by the Schrödinger Equation. In Nelson’s case, they are being pushed around by the chaotic motions of some underlying medium that fills space, and because of the way those motions work, the particles end up being very likely to follow trajectories very similar to those predicted by Bohm’s theory. The wave function, in this view, is something like temperature or pressure in thermodynamics – if you zoom in far enough, it doesn’t really exist, but it’s an effective abstraction that appears to follow definite laws on a more macroscopic level. If this were the case, then in principle it ought to be possible to experimentally observe violations of Schrödinger’s Equation, but one could suppose that those violations will be so small as to be *practically* unobservable.

However, as it turns out there are difficulties in deriving the fully general Schrödinger Equation from simple stochastic assumptions; one needs to add in some *ad hoc* conditions to get things right. That’s not necessarily a problem, but the argument for a stochastic interpretation would be a lot stronger if the Schrödinger Equation could be derived from it more naturally. Moreover, it should be noted that no deviations from the exact Schrödinger Equation have ever been observed. It seems somewhat premature to seek a new fundamental theory when the one we have shows no signs of breaking down.

**Quantum Logic**

In the 1930s, Garret Birkhoff and John von Neumann pointed out a strong analogy between the mathematics of observable properties in Quantum Mechanics and non-classical systems of logic – that is, various formal systems of propositional logic that differ from standard, Boolean logic. In the 1950s and 1960s, several philosophers suggested that the lesson from the apparent paradoxes of Quantum Mechanics was that QM provided empirical evidence that classical logic itself was flawed.

The question of the status of logic is a thorny philosophical issue. It has traditionally been supposed that logic is analytic and *a priori* – in other words, that logic itself is not the kind of thing that is subject to empirical inquiry. An analogy might be made to language – you can ask whether a sentence in a given language is true, or false, or nonsensical, but you can’t ask whether a *language* is true or false. However, the fact that alternative systems of logic can be constructed invites the question of what justifies us in believing that Boolean logic is the correct system for understanding the world. W.V.O. Quine, Hans Reichenbach, and Hilary Putnam all put forward arguments to the effect that Quantum Mechanics shows that Boolean logic is *not* the correct system.

In Quantum Mechanics, we encounter what seem to be puzzling situations – for instance, on a naïve reading of the two-slit experiment, one might say that all of the following logical possibilities are ruled out:

– The photon cannot have gone through just the right slit (because then it would not have created an interference pattern)

– The photon cannot have gone through just the left slit (because then it would also not have created an interference pattern)

– The photon cannot have gone through both slits (because when it shows up on the screen, it is a point-like particle with a single, definite position)

– The photon cannot have gone through neither slit (because we saw it show up on the screen)

All the interpretations we’ve talked about so far resolve this problem by saying that there are some false assumptions in its set-up – depending on the interpretation, they’d say that the photon did go through both slits as a wave but then collapsed to a single position when it hit the screen, or that the photon itself went through only one slit but its associated wave function went through both slits, etc. But the Quantum Logic approach instead seeks to resolve the problem by saying that classical logic is wrong in insisting that those are four mutually exclusive possibilities. The hope is that with an approach like this, the measurement problem never even comes up, because the measurement problem assumes that classical logic applies.

The philosophical issue at the heart of this – i.e. what justification classical logic, or any system of logic, could have – is a really interesting one, I think. But as an interpretation of QM, I think that most people today would agree that Quantum Logic is not very compelling. As we’ve seen, there are several perfectly good ways to resolve the measurement problem without deviating from classical logic at all. To reject our whole system of logic over it strikes me as a rather extreme overreaction.

**Decoherence**

There’s something of a pernicious myth to the effect that determinate outcomes can be generated without ever violating the Schrödinger Equation or adding anything to the theory, through a process called decoherence. The claim is, in other words, that even without positing a collapse process as a fundamental part of the theory, the linear dynamics of QM will, all by themselves, lead to an “effective collapse” when a measurement is made, that will leave the system in, or nearly in, a definite state – and that, therefore, the measurement problem doesn’t really exist at all. And a surprisingly large number of smart people (people who are way, way smarter than me) have repeated this claim.

Now, decoherence is a real thing, and it is an important concept in quantum physics. Very loosely speaking, decoherence happens when one part of a particle’s wave function becomes entangled with its environment in a different way from another part, and because of this difference, the two terms in the wave function can no longer interfere with one another. Typically, to observe Quantum Mechanical phenomena, we need to work with an isolated system; as the system interacts with its environment and its wave function acquires complicated entanglements with that environment, quantum behavior tends to disappear.

Decoherence plays several big roles. For one thing, it is a big piece of the explanation for how apparently classical behavior arises in macroscopic collections of particles that are, on a fundamental level, governed by QM. It’s also what makes it so difficult to empirically test whether wave function collapse has occurred. I mentioned in part 2 that one could test whether collapse has occurred after an electron has interacted with a measuring device by measuring certain properties of the combined electron/device system. However, in practice, the macroscopic measuring device will be strongly coupled to its environment. Just to take one example, if the pointer moves in one direction, it will become entangled with the air molecules around it in one way, if it moves in the other direction, it will become entangled with them in a different way. And to test for whether collapse has occurred, we need to measure a property of that whole entangled system, including any air molecules that interact with the pointer, any photons that bounce off of it, and so on.

Decoherence is also the reason that in interpretations without collapse, like Bohm’s theory or the Everett interpretations, the different branches of the wave function typically don’t interact with each other once they differ on a macroscopic level. Consider the two-slit experiment again. One part of the photon’s wave function goes through one slit and one part goes through the other. If they have no other interactions, then these two parts can interfere with each other on the other side and produce the famous interference pattern. However, let’s say you place a measuring device on each of the slits so that you can tell which one the photon passed through. If you do that, you’ll find that half the time, it goes through one slit and half the time the other – but, plot twist, the interference pattern will *not* appear. Instead, you’ll just get two bright spots, one from each slit. Collapse interpretations will tell you that this is because your measurement of which slit the photon went through collapsed the wave function. It had to “choose” one slit to go through, and because it only went through one slit, there was nothing for it to interfere with on the other side. But in Bohm’s theory or any of the Everettian interpretations, collapse never happens, and both branches of the wave function continue to exist. So why don’t they interact with each other and produce the interference pattern? It’s because one branch of the photon’s wave function is now entangled with one measurement device and one branch with the other, so when they get to the other side of the slits, they can no longer just add up in the simple, coherent way required to produce the pattern.

So decoherence is an important phenomenon. But, as I said, the further claim is sometimes made that it solves the measurement problem. The argument goes something like this: when, say, an electron becomes entangled with a measurement device, all the various, complicated entanglements of that measurement device with the environment cause the system to undergo decoherence, suppressing any specifically Quantum Mechanical phenomena (like superpositions) and rendering the system effectively classical; it undergoes an “effective collapse”, in which one of the branches of the wave is highly suppressed. Thus, we never get into those troubling states in which measurements don’t have outcomes.

It’s relatively easy to show that this argument doesn’t hold up. It was, after all, a fairly straightforward line of reasoning that led us to conclude that, if we don’t add anything to the theory, we *must* end up in those troubling states. Let’s rehearse that line of reasoning again:

– We know that we can build a measuring device such that whenever an electron in the state |x-up> enters it, the measuring device shows “up”, and whenever an electron in the state |x-down> enters it, the measuring device shows “down”.

– We know that we can prepare an electron in the state 1/√ 2 |x-up> + 1/√ 2 |x-down>

– From the linearity of the dynamics, we can conclude that when an electron in that superposed state enters the measuring device, the measuring device will end up in a superposition of states where it shows “up” and where it shows “down”.

And that’s that. Nothing about decoherence can change the fact that reliable measuring devices exist and that the linearity of the Schrödinger Equation guarantees that such devices will end up in macroscopic superpositions. If decoherence *did* prevent those macroscopic superpositions, then it could be argued that the measurement problem would be entirely obviated. But it doesn’t.

**Locality and Relativity**

One of the major surprises that Quantum Mechanics had in store for physicists in the early 20th century was that it appeared to be *non-local*. Locality is, roughly speaking, the principle that things in one location cannot physically affect things in a different location without that effect somehow being transmitted through the space between them. In particular, the Special Theory of Relativity imposes a strong locality condition; since the speed limit of the universe is the speed of light, there can be no causal relationship between events unless a signal travelling at the speed of light had time to pass from one to the other. Since Quantum Mechanics predicts things like entanglement, where the result of a measurement at one location has an instantaneous effect on the state of a particle at another location, it seems to violate the principle of locality pretty blatantly. This is the “spooky action at a distance” that was one of the things Einstein most strongly objected to about QM.

Let’s note two things, though. First, the principle of locality is not some god-given rule. There’s no reason the universe *couldn’t* be fundamentally non-local. Sure, all previous theories of physics had obeyed a principle of locality, but since we’re saying that those previous theories were, when you come down to it, *wrong*, there’s no reason to suppose that locality must be a fundamental truth of the universe.

Second, it’s worth noting that there are some pretty stringent limitations on these non-local effects, across all interpretations of QM. In particular, it can be shown that despite this spooky action at a distance showing up in the formalism of the theory, it’s actually impossible to use this effect to transmit information. The fact that a non-local effect has occurred can only be deduced if the results of measurements at separate locations are later brought together and compared.

It’s worth taking a moment to notice that locality, or the lack thereof, plays a rather different role in different interpretations of Quantum Mechanics.

In your standard collapse interpretations, non-locality comes in because the collapse of the wave function occurs instantaneously. It doesn’t matter how “spread out” in space a particle’s wave function is; when whatever condition occurs that triggers collapse4, the *whole* wave function collapses all at once. And if multiple particles are entangled, a collapse for any one particle causes the whole multi-particle wave function to collapse, regardless of how far away those other particles are.

In Bohmian Mechanics, one could argue that the non-locality is more egregious, even though collapse never occurs in this interpretation. Here, the particles are thought of as being pushed around by their wave functions, according to the Guiding Equation. In particular, under the Guiding Equation, the velocity of a particle depends on the value of the universal wave function *at the actual locations of all the other particles*. In collapse interpretations, non-local effects occur at discrete moments when wave function collapse happens. In Bohmian Mechanics, non-local effects are *continuously* at play, since the velocity of each particle is directly determined by the positions (and wave functions) of *all* the particles with which it is entangled.

Finally, what about Everett-type interpretations? Well, any version of Everett’s thesis that involves one universe literally splitting into multiple universes is pretty extravagantly non-local. After all, this view claims that my measurement of an electron causes the entire universe to instantaneously split in two. However, since the Schrödinger Equation itself does not violate locality, interpretations like the Bare Theory and Many Minds *do* turn out to be local.

This has some implications for porting these interpretations over to Quantum Field Theory.

Way back at the beginning of part 1, I mentioned that Quantum Mechanics itself is a non-relativistic theory, meaning that, like Newtonian Mechanics, it gets things wrong as the velocities involved get close to the speed of light. The version of Quantum Mechanics that incorporates relativity is called Quantum Field Theory. I said there that, because QFT is significantly more complex, we would just talk about non-relativistic Quantum Mechanics, and I assured you that for the most part the issues remain the same (in particular, the measurement problem still exists) and the interpretations of QM mostly work as interpretations of QFT as well. And that’s true, as long as you pay attention to that “almost” in there.

Since Everett-type interpretations, apart from the branching version of Many Worlds, are local, they really can be taken over directly into Quantum Field Theory with no changes. Even the metaphysically extravagant Many Worlds Interpretation could be made to conform with QFT by positing that the splitting between universes propagates through space at the speed of light.

Objective collapse interpretations, like GRW, on the other hand, require some modifications to comply with special relativity. That work is still ongoing, but it seems promising. The crux of the issue is that in Special Relativity, there is no absolute notion of simultaneity between two events. Two events might appear simultaneous to one observer, but to an observer with a different relative velocity – in a different “frame”, in the parlance of relativity, they will not be simultaneous. Further, there is no objectively “correct” frame in Special Relativity. So it makes no sense to say that things happen simultaneously in different places; you can only say that they happen simultaneously in such and such a frame of reference.

Obviously, this is in tension with the notion of a wave function collapse occuring instantaneously throughout space. Some more sophisticated way of specifying the conditions for wave function collapse is required, one that applies equally well to *any* frame of reference. In physics jargon, this requirement is called “Lorentz-invariance”. As far as I know, no full relativistic version of GRW yet exists, but the skeleton of such a theory has been worked out, and there seems to be no reason to think it won’t work.

Bohmian mechanics is a little trickier. Again, the problem is that of how to define simultaneity. In Bohm’s theory, the velocity of a particle at a given time depends on the value of the wave function at remote locations at that same moment. This relies pretty critically on a notion of “the same moment”, which doesn’t exist in relativity. And this turns out to be a fairly thorny issue. It looks like there is no way to adapt Bohm’s theory to make it truly Lorentz-invariant. It fundamentally *requires* the existence of a “preferred frame” of rest – in which particle velocities depend, instantaneously, on the configuration of all the other particles and wave functions in the universe.

However, a kind of pseudo-relativistic version of Bohmian mechanics *can* be constructed in which, even though there is a preferred rest frame, it turns out to be impossible for any experiment to determine *what* frame that is. So even though the theory must be written down using the concept of a fixed, absolute system of coordinates relative to which the velocities of particles are defined, it is impossible to empirically test what the absolute velocity of a particle is. Just as in true relativistic theories, all that one can measure is the velocity of a particle relative to some other particle. There is, then, a kind of observational Lorentz-invariance in a Bohmian version of QFT. Whether that is *enough* is perhaps an open question. But since our reasons for believing in Special Relativity are, as with any physical theory, fundamentally observational, it seems to me at least that there is no reason to demand more than observational compliance with relativity.

**So where does all of this leave us?**

We’ve covered a lot of territory pretty quickly in this series, so I want to step back and talk about what this all, like, means, man. Of course, since there’s nothing like a consensus on what this all means, this will amount to me giving you my personal opinion about the whole issue of the interpretations, and no doubt rambling on a little bit.

Obviously, we’re dealing with the intersection of science and philosophy here. Philosophical ideas and issues have been pretty much inextricable from Quantum Mechanics since the 1920s. I think that part of this is historical accident; it just so happened that QM was discovered at a time when logical positivism was ascendant, and so terms like “measurement” and “observer” came to be featured so prominently in the way QM was talked about. But part of it, I think, comes from the fact that, as QM shows, the universe just works rather differently on a fundamental level than what we’re used to. Perhaps it isn’t surprising, then, that it challenges some of the notions that we once took for granted.

There is, however, a tendency – particularly among scientists – to dismiss the whole issue of the interpretations of QM as a philosophical one (which, for many physicists, is tantamount to saying “a meaningless one”). And I think this is a mistake, for two reasons. First, the interpretations are not all empirically equivalent. Different theories of collapse are in principle distinguishable from each other, and all of these are distinguishable from theories without collapse. As I discussed earlier, it is in *practice* very difficult to perform the necessary measurements to make those distinctions. But very difficult does not mean impossible, and there are a number of experimental avenues being explored. In particular, in the near future it will probably be possible to perform what are essentially two-slit interference experiments using not photons or electrons but macromolecules consisting of large numbers of particles. This should allow us to start to put experimental constraints on objective collapse theories like GRW.

The second reason that I think it’s not right to dismiss interpretational issues as purely philsophical is that this conflates two separate questions. The central philosophical question of metaphysical realism vs. positivism is one that is logically prior to, and distinct from, the issue of what particular physical theory we ought to believe. It is often said that, for instance, the Copenhagen Interpretation is a positivist understanding of Quantum Mechanics, while the de Broglie – Bohm theory is a realist understanding. But I don’t think that’s true, even if certainly those philosophical schools influenced the people who developed the respective interpretations. One could be a realist about *any* theory – the question of the interpretations, then, is one of *what* one takes to be real entitites, and how those real entities behave. Similarly, one could take a positivist view of *any* theory, and look at it as an algorithm for describing and predicting empirical experience. The question then is which theory best does that. And, conversely, I would also maintain5 that nothing about the particulars of any physical theory can necessitate either philosophical position. I would be extremely skeptical of anyone who claims that, say, Quantum Mechanics proves that reality is observer-dependent, or that it proves that classical logic is wrong, or any such thing. Those philosophical questions are more basic than the particulars of any physical theory, and they aren’t the sort of question that is to be answered by physics.

If it turns out that a collapse-based interpretation is correct, then the issue can, and maybe will, be empirically settled. But what if we manage to perform those experiments and we find no evidence for the collapse of the wave function? How then are we to decide between the various non-collapse theories – Bohmian mechanics, or a modal interpretation, or Many Worlds, or Many Minds? The answer, it seems to me, is that we *can’t* decide between them. Now, if you’re a realist, that might be rather troubling. That means that the universe might actually consist of a single particle moving about in a very high-dimensional space; or, it might consist of nothing but wave functions, splitting and separating into countless equally real universes. The laws of the universe might be completely deterministic if Bohm’s theory is the right one, or there might be inherent, irreducible probabilities, if something like Many Minds is right. There might be instantaneous cause and effect between remote locations, or the universe might truly obey the principle of locality. And if you think those are meaningful questions, then, alas, they are questions that it turns out to be *impossible* to know the answer to. If you’re a metaphysical realist – if you believe that there’s an external reality out there that is what it is, and that’s independent of anyone’s experiences, then I think the hard lesson of these foundational questions is that the *truth* about that reality is unknowable.

As for me, I’m a positivist. I think that the purpose of science is to describe and predict the data, and I am skeptical that phrases like “external, observer-independent reality” are really meaningful. As I see it, then, Bohmian Mechanics, Everettian interpretations, and any other theories without collapse, are all exactly the same theory, once you get down to it. They all make exactly the same observational predictions; therefore, I’m content to say that, however different they might sound on the surface, these theories all *mean* exactly the same thing. They are the same theory written down in different languages, if you will. And so it turns out that things like locality, or determinism, are not basic truths of the universe but are more like artifacts of the particular way in which one chooses to write things down.

Well, that’s about all I’ve got to say. Talking about this stuff in such a brief format is hard, but I hope that this all made some sense and perhaps was even interesting. I certainly enjoyed writing it. I should mention, again, that I drew pretty heavily on David Albert’s book *Quantum Mechanics and Experience* in writing this. As I said before, it’s a book that requires some dedication to get through, but if you’re interested in this stuff and not afraid of a little math, it’s one that I’d definitely recommend. Anyway, if you’ve made it this far, thanks for reading!