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Discussion:
The Redundancy Argument Against Bohm's Theory
Craig Callender
Department of Philosophy, 0119, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0119, USA
1. Introduction
Advocates of the Everett interpretation of quantum mechanics have long claimed that other interpretations needlessly invoke "new physics" to solve the measurement problem. Call the argument fashioned that gives voice to this claim the Redundancy Argument, or 'Redundancy' for short. Originating right in Everett's doctoral thesis, Redundancy has recently enjoyed much attention, having been advanced and developed by a number of commentators, as well as criticized by a few others. Although versions of this argument can target collapse theories of quantum mechanics, it is usually conceived with no-collapse "hidden variable" interpretations in mind, e.g., modal and Bohmian interpretations. In particular, the argument is an attack against theories committed to both realism about the quantum state and realism about entities what Bell 1987 calls "beables" that supplement this state. Particles, fields, value states, and more have been suggested as possible ontology to supplement the quantum state. Redundancy is the argument that this supplementation is methodologically otiose, the superfluous pomp that Newton scorned.
In this brief note I argue that Redundancy is a nonstarter. Although advanced for decades by many eminent physicists and philosophers, the argument is really quite audacious. A lot of the discussion (pro and con) has focused on whether targets of the Redundancy must reject a kind of functionalism. I think this is a mistake. Functionalism may or may not be true, but the target of Redundancy isn't driven to reject it. If I'm right the real issue is much simpler: the functionalist premise of Redundancy is inextricably intertwined with whether Everettians successfully solve the measurement problem. Granting that they do, the argument isn't all that bad; indeed, it has the interesting consequence that Everettians and their targets can't really agree that they each solve the measurement problem. Yet as the storm produced by the measurement problem shows, we are very far from granting this assumption. As matters currently stand, one could argue that functionalism favors some of the targets of Redundancy. Only after the dust settles will Redundancy gain any force. Unfortunately at that point Redundancy will itself be almost redundant.
In this paper I'll sketch Redundancy, describe a number of ways around it, show how functionalism and the measurement problem are connected, and conclude with a brief discussion of the interesting question of whether wavefunction realism ought to be embraced. To keep the discussion economical, let's focus on the principal object of this argument, the deBroglie-Bohm interpretation of quantum mechanics. Most of what I say about this case will apply, mutatis mutandis, to other targets.
2. The Redundancy Argument
Redundancy comes in different forms, but the best version goes more or less as follows. Stick to ordinary nonrelativistic quantum mechanics. Consider a microscopic system (say, an electron) in a superposition of spin-1/2 states. Then as is familiar from the measurement problem, the linearity of the Schrdinger evolution guarantees that as systems interact with this system, they become entangled with this microscopic system and become superposed in this basis too. Assuming no collapse occurs, a quantum state can in principle develop that corresponds to a superposition of distinct macroscopic states. If the macroscopic system is a cat ingeniously devised to die if the spin is up and live if the spin is down, then we obtain the notorious Schrdinger cat state. The cat, in neither an eigenstate of the operators corresponding to being alive or dead, seems suspended between life and death. Or is it?
Famously, the Everettian asserts that we can nevertheless discriminate cat-patterns in the wavefunction. That is, if we carefully inspect the web of relations in the structure of the wavefunction, they believe that we will find structures that are very catlike. In the case at hand, the Everettian discerns two cats, one dead and one alive. In more complicated states with human observers, buildings, planets, and so on, they discern internally consistent (guaranteed by linearity) quasi-classical "branches" consisting of scientists seeing dead cats, scientists seeing alive cats, and so on. (The basis or set of bases for this branching is, the Everettian hopes, selected by decoherence.)
Assume what Wallace calls "Dennett's criterion", but which is perhaps better thought of as a kind of functionalism:
a macro-object is a pattern, and the existence of a pattern as a real thing depends on the usefulness of theories which admit that pattern in their ontology (2003, 93).
Granting this, what it is to be a cat is to be a cat pattern. To be a cat is just to instantiate the cat pattern in the fundamental stuff, in this case, wavefunction stuff.
By contrast, consider interpretations that similarly treat the wavefunction as part of the ontological furniture of the world and yet add some new physics, e.g., Bohmian particles. Briefly put, the Bohmian solves the nonrelativistic measurement problem by adding to the fundamental ontology particles that enjoy always-determinate positions and velocities. In the case at hand, a catlike distribution of particles develops, and this is what the Bohmian considers a cat. Despite the wavefunction not being an eigenstate of the cat-alive operator, the dynamics and initial distribution guarantee that the particle-cat is either dead or alive. Due to a decoherence-like mechanism, the uncollapsed quantum state of the system undergoes an apparent but effective collapse when the different components of the wavefunction have mostly disjoint supportwhich will happen when the different components are macroscopically distinct (more on this later). The velocities of the particles in the cat will then be governed effectively by one or other of the superposed wavepackets. In this way the Bohmian explains the occurrence of collapse.
The next step in the argument simply appeals to theoretical economy. Focus now on the component of the wavefunction to which the particles are responsive. Assuming functionalism, Everettians say, there already is a cat in that branch, a wavefunction-cat! Adding a particle-cat to a branch already containing a wavefunction-cat solves the measurement problem twice, once more than is needed. Furthermore, since the "empty wave", the wavepacket not directly guiding the Bohmian particles, is still real, the Bohmian posits just as much ontology as the Everettian before adding the particles. Bohmians, as Deutsch puts it, are Everettians in "self-denial". Redundancy concludes that one should take Occam's razor to this bloated ontology and get rid of the superfluous particles.
2. Caveats
Before going further, we should immediately note various ways in which this argument might not get off the ground long enough to make the later points relevant.
Firstly, and most obviously, the Bohmian who does not believe the wavefunction is a real field on a par with the added beables (in Bell's terminology) is not affected by this argument. There aren't wavefunction cats in the world on this view because wavefunctions aren't part of the furniture of the world. Particle-cats are therefore not redundant. This view is attractive in some ways because the wavefunction evolves in 3N-dimensional space and not 3-dimensional space. I'll return to this point in Section 5.
Secondly, as Valentini 2009 points out in detail, some versions of Bohmian mechanics modify the theory so that one gets non-Born probabilities at various times in the universe. Without particle trajectories the Everettian cannot follow suit, so the argument cannot be rerun in this case. See Valentini 2009 for the details.
Thirdly, the Bohmian may insist that he or she recovers more than standard quantum mechanics and its algebra of observables. The guidance equation, the equation for velocity that is fundamental to Bohmian mechanics, arguably generates many other "observables" to which no operator in Hilbert space corresponds. One can, for instance, invert the Bohm velocity to obtain quantities like the dwell time (i.e., the average amount of time spent by Bohm particles going between two spatial points), transmission times, escape times, and more. See Leavens 1990. If part of being a cat is having a certain dwell time, the catlike structures that Everettians find in the wavefunction aren't cats.
Fourthly, and in the same spirit as the third, the argument presupposes that the only thing that matters in theory choice in the given context is recovering quasi-classical histories. This is a somewhat subtle point that I believe Valentini 2009 is making when he talks of interpreting each theory "on its own terms". Consider the two-slit experiment. As Valentini remarks, here we do not have a multiplicity of quasi-classical histories, but we do have a single always determinate Bohm particle traveling through one or the other slit. The Bohmian world has a lot more structure to it than the Everettian one. It doesn't just get cats, scientists, planets, and so on, but it also gets particles traveling through one slit or the other. To the Everettian this structure is surplus structure. But what determines whether structure is surplus? Here we enter an epistemological thicket. If we're concerned only about the quantum observables, then this additional structure is superfluous. Yet we might have hoped for more than this from a physical theory. Philip Pearle speaks of the 'Reality Problem' rather than 'Measurement Problem'. The Measurement Problem emphasizes the problem with superpositions when they become macroscopic. But one might have been worried about superpositions or properties earlier, even at the microscopic level. If theoretical virtues are found in having a determinate microphysical world, in saying a particle really went through one of the slits, then it's prejudicial for the Everettian to assume recovering anything other than quasi-classical histories is unnecessary surplus.
Each of the above points has undeniable merits for some Bohmians. Yet the first two ways out require somewhat non-standard understandings or even modifications of Bohm's theory. The second two ways instead send us irretrievably into epistemology. Going non-standard or into epistemology are not evidence at all against these moves. They may well be right, but I do think there is a simpler response available. Unless otherwise noted, let's go along with the Everettian and choose not to exercise any of these caveats.
3. The Redundancy Argument and Functionalism
Lewis 2007a argues that the Bohmian seeking escape from Redundancy should deny functionalism. He accepts the reasoning behind the Redundancy Argument but wants to deny the functionalist premise. He notes that all interpretations of quantum mechanics have counter-intuitive consequences, and if the rejection of functionalism is counter-intuitive, this is simply one more such consequence for the Bohmian to bear. Brown and Wallace 2005 and Wallace 2008 agree that rejecting functionalism is one way to go. Since it's widely conceived as the way out, let's spell out the denial of functionalism in a little more detail.
The Everettian making the Redundancy Argument must be committed to variants of both metaphysical and theoretical functionalism. Theoretical functionalism is the claim that we define a term via the regularities of the complete science of the said entity. For mental state functionalism, the meaning of 'pain' is given by 'the state such that, if one stubs a toe, the state causes them to yell, ' and so on as prescribed by the complete science of pain. One defines a theoretical term in terms of its characteristic functional role, then one Ramsifies the term. There are many logically inequivalent ways of doing this (see Weir 2001), but we needn't get into such fine detail here. Theoretical functionalism has many fans, and not merely for mental states. Many eminent philosophers insist that every term that isn't logical or referring to a basic component of our ontology must be cashed out this way. Note that 'cat' is not a basic feature of the Everettian ontology, so there is no circularity in the Everettian strategy. We currently lack a complete theory of cats, of course, but we can spot the Everettian this oneeven if cats are especially baffling creatures.
What's really important is the metaphysical version of functionalism. Pains, cats, etc., are identified and realized by their functional roles. There isn't anything more to being a pain or a cat than its functional role in the sciences of pains and cats, respectively. A pain or a cat is merely a node or property of a node in a vast structural network. It is this feature of functionalism that plays a central role in Lewis' argument.
Lewis 2007a claims that Bohm's theory must presuppose the rejection of functionalism. The reason is that Bohm's theory must prefer particle-cats to wavefunction-cats, yet this preference is anathema to functionalism. Functionalists don't tie themselves to particular types of fundamental stuff. Tables are whatever entities there are that realize the table functional role. They can thus be made of maple, pine, mahogany, etc. Similarly, cats are whatever realize the cat role, and functionalists shouldn't be too picky about whether particles, fields, superstrings or wavefunctions instantiate this feline role. For this reason Lewis argues that Bohm's theory presupposes the falsity of functionalism, and it is therefore not fair, he thinks, for the Everettian to assume it in an argument against the Bohmian.
Let's refine Lewis' claim. The issue is not really functionalism, but rather chauvinism. Ever since its inception in philosophy of mind philosophers realized that functionalism faces a "chauvinism/liberalism" problem (see Block 1980). If no constraints on the type of realizing substance are allowed, then there are famous examples of the people of China in their entirety, swarms of bees, the economy of Bolivia, and many other strange entities, instantiating the same functional role my neurons do when I'm in love. For these fleeting moments do we want to say that the Bolivian economy (etc.) is in love? Some functionalists are willing to bite the bullet here, but others prefer a more chauvinistic version. Putnam, for instance, once ruled out such entities as having mental states by the criterion that the entities fulfilling the function cannot themselves satisfy the role for having beliefs; so since Chinese people, Bolivian economic units, and so on, can themselves be in love, these higher-order networks didn't count. How one picks the inputs and outputs stipulates a point along the liberalism/chauvinism spectrum. A function is a set of pairs (inputs, outputs) that associates with every possible input a unique output. What are the inputs? For mental state functionalism, are they a system's sensory and motor mechanisms, or rather the external objects themselves? If we think we can have the same intentional states in different environments, then the functionalist may be tempted to tie the inputs to our sensory and motor mechanisms. But doing so nudges us toward chauvinism, for understanding the inputs and outputs this way may rule out all sorts of systems. Every functionalist must make a decision about what not to be functionalist about.
The question, therefore, is not exactly whether the Bohmian must abandon functionalism, but merely how chauvinist he or she wants to be. The issue then turns to whether there is room for a principled chauvinistic reply. There may well be. As mentioned earlier, suppose the choice of inputs and outputs described a functional role not discernible in the wavefunction. To be specific, suppose the functional role made reference to the amount of time a microscopic entity spent in a certain region, the dwell time. Such a functional role would more or less pick out Bohmian particles. Admittedly, this choice would be doubly controversial. Questions would arise about whether aspects of the wavefunction already satisfied these roles and about the motivation for these descriptions. Yet at least this is an example of a possibly well-motivated chauvinism that would favor Bohmian mechanics.
Second, and more important, suppose that there is no well-motivated Bohmian reason for chauvinism. Then surely the Bohmian cannot rest satisfied with Lewis' reply on their behalf. The proponents of the Redundancy Argument have a real point. If you believe in wavefunctions, and you think Everettians solve the measurement problem, then Bohmians have an embarrassing plethora of cats. True, one can chauvinistically claim one doesn't allow wavefunction-cats. Yet this begs the question: why on earth not? Without a motivation for chauvinism, no answer is forthcoming. Lewis' Bohmian can protest that her right not to incriminate herself has been revoked when the Everettian invokes liberal functionalism. However, such protests are unmoving when one has no reason for the right in the first place. Prima facie, there is nothing in the nature of a cat that picks out one type of fundamental stuff rather than another.
4. Redundancy Rejected
Although I have sympathy with Lewis' and Valentini's reactions, I have always thought the Redundancy Argument merited a far simpler and more direct response. The Redundancy Argument only has force if the Everett interpretation is a genuine solution to the measurement problem. Lewis grants this. He wants to tease apart the various challenges the theory faces from whether it solves the measurement problem, believing that focusing on these worries for Everett is not fair game. However, the Bohmian wanting to rebut the Redundancy Argument without abandoning functionalism has to go on the offensive. Lewis' conciliatory approach allows that cat patterns in the wavefunction constitute cats. For the Bohmian to escape the problem, he need question the assumption that cat patterns in the wavefunction are really cats.
Obviously, whether Everettians have solved the measurement problem is a huge topic, not something to be decided in a small note. Here I wish only to show that solving a major problem for Everettians, the so-called probability problem, is highly relevant to the question of functionalism and the Redundancy Argument. One can read Valentini 2009 as attempting a similar move by challenging the Everettians hope that decoherence picks out well-defined quasi-classical histories. Either problem could serve my purpose. I pick the probability issue because it's simpler and more widely acknowledged, even by Everettians, as a major difficulty.
The probability problem is twofold. First, given that all possible outcomes actually happen during an Everettian measurement, is it even coherent to say that the chance of obtaining (say) spin-up in a particular direction is 75%? Second, assuming it is coherent, why should one assume the chances are in accord with Born's rule? Both questions are, it is fair to say, very open. See Barrett 200x for an introduction to the issue and some Everettian responses and Greaves 2006 for recent work on the problem.
Lewis 2007a wants to think that Everett solves the measurement problem, i.e., gets cats, but still may have a problem with understanding Born's rule, i.e., getting these cats statistically distributed in the right way. My claim is simply that unless the cats are statistically distributed the right way, they aren't cats. That is, the Born statistical distribution is part of the cat-constituting functional role. Failing to obtain it means failing to get what the functionalist judges a cat.
One cannot "bracket" this problem. Lewis 2007a wants to say that the Everettian gets cats because of functionalism but simply may not get the statistical distribution of cats right. This position doesn't make sense. The cat patterns in the wavefunction only get to be cat patterns in the wavefunction if they are statistically distributed the right way. The 'getting the cat' problem, i.e., the measurement problem, and the 'probability problem' are inextricably linked.
Imagine the following argument between behaviorists and psychofunctionalists about mental states. Behaviorists can be understood as functionalists who restrict the inputs and outputs to observable sensory states. Psychofunctionalists are functionalists who restrict the inputs and outputs to sensory states plus other mental states singled out by our best theories in cognitive science. Suppose psychofunctionalism is the correct theory of mind. Well, still there exist those patterns among behavioral inputs and outputs that the behaviorist was talking about. They didn't go away. A psychofunctionalist mind still may realize behavioral functional roles corresponding to what the behaviorist considered mental states. I may satisfy the psychofunctional definition of 'believing 2+2=4' and the behavioral definition of 'believing 2+2=4'. Suppose the behaviorist then mounted a redundancy style argument against psychofunctionalism. The right response is not chauvinism, but to show that the behaviorist certainly has some nerve in thinking their definition is really of the belief that 2+2=4. We claim the behaviorist-mental states don't count as mental states by showing that they aren't capable of as much explanation and prediction as the ones defined by the psychofunctionalist. They aren't up to the job.
We should conceive of the current debate the same way. Consider three different sketches of Ramsey sentences for cats. Assume similar Ramsey sentences exist for "tails", "legs", and so on, so they all obtain their interpretation "at once." Here are the sentences:
'x is a cat' iff (x(x has four legs and x has a tail)
'x is a cat' iff (x(x has four legs and x has a tail and x purrs when stroked behind the ears and x bites when tail is pulled)
'x is a cat' iff ((x has four legs and x [same as b] and x is probabilistically distributed in accordance with Born's rule)
Is there any question which Ramsey sentence wins, i.e., is best for explanation and prediction? No, Ramsey sentence c is clearly the best. a doesn't tell us much about what cats do. b tells us more. But both a and b are hopeless in making the predictions we expect from quantum mechanics. Neither type of cat need obey quantum mechanics.
Everettians have jumped the gun. If the Everettian can answer the probability objection, then they get to espouse Redundancy. Until then, the Bohmian can respond to the Everettian as the psychofunctionalist above responds to the behaviorist. From the Bohmian perspective, it's not the assumption of functionalism that is the problem, it's the assumption that Everettians get cats statistically distributed according to Born's rule that is the problem. Some Bohmians are Bohmians just because of the probability problem (and its cognates) for Everett. From this perspective one wants to say that of course any theory that solves the measurement problem looks like it offers superfluous machinery when compared to any non-solution masquerading as a solution. And in the event that Everettians discharge all their obligations, there are still all those caveats to deal with from Section 2. Since some of those involve coming up with essentially an epistemology of theory choice, when the conditions are finally met for the Everettian to mount Redundancy there will be little point for they will have won.
5. Is the Wavefunction an Object?
As mentioned, there is a way of understanding Bohm's theory that makes it immune to Redundancy even if Everettians can satisfactorily answer the various problems with their theory. The way is for the Bohmian to reject the interpretation of the wavefunction as being part of the ontological furniture of the world. If not part of the theory's ontology, there is of course no redundancy. The only cats are particle-cats, not particle-cats and wavefunction-cats. Despite the disapproval that meets this position by supporters and detractors of Bohm (Brown and Wallace 2005, Lewis 2007a,2007b, and Valentini 2009 all consider and reject this option), I've always found it the natural interpretation of Bohm's theory. From this perspective, Redundancy simply has no purchase, and the Bohmian, if she pleases, can turn theoretical economy in her favor by questioning the need for a large multiplicity of quasi-classicla objects.
To begin, I think it very odd to judge the wavefunction to be part of the basic ontology of the Bohmian world. The wavefunction lives in 3n-dimensional space, where n is the number of particles. (Nonseparable wavefunctions for even two particles cannot be understood as evolving in physical space.). Those thinking it real conceive of it as a field over a 3n-dimensional configuration space, one specifying amplitude and phase values at each point of this space. Being very high-dimensional, n-dependent, and more, the space and the field on it seem much more like the abstract state spaces of classical mechanics and classical field theory than anything typically posited as physical space and physical fieldsespecially when we think about the gauge transformations allowable when transforming phase values. This field has all the earmarks of being part of the mathematical machinery of the theory as opposed to the ontological furniture of it. So why do so many think it must correspond to an object?
J.S. Bell, perhaps the greatest supporter of Bohm's theory, wrote in favor of wavefunction realism:
No one can understand this theory until he is willing to think of as a real objective eld rather than just a probability amplitude. Even though it propagates not in 3-space but in 3N -space. (Bell 1981, 128)
Bell doesn't elaborate on his reasons for this claim. To me it's not clear that he thought of the wavefunction as an object in the world. Surely he is right that to confuse the wavefunction with a probability amplitude is to think it's essentially tied to probability when its relation to probability is better off derived. And if we conceive of probabilities as epistemic, then he is again right to draw a contrast between the field being "objective" and it being epistemic as it is often conceived in some Copenhagen-inspired theories. One can even agree that that it aids the understanding of Bohm's theory to conceive of the wavefunction as real. Yet the argument for it as an heuristic aid doesn't constitute a recommendation that the wavefunction be part of the furniture of the world.
Perhaps Bell and others have in mind the explanatory benefits that come with the reality of the wavefunction. Why did that particle in 3-space move? Because the wavefunction in 3n-space so governed it. Why are the outcomes of measurement on one wing of a Bell experiment correlated with those on the other? Because the wavefunction so arranged matters. However, these explanatory features don't require the wavefunction to be part of the furniture of the world. As DGZ 1997 emphasize, the wavefunction can simply be thought of as part of the laws of nature. The laws govern the particles' motion, just as they do in classical mechanics. All the explanatory benefits hinted at above accrue to this position as well. And if one doesn't like the somewhat mysterious talk of governing, one can replace it (without loss, I think) by a more empiricist-friendly theory of laws. On this view, what there is in the nonrelativistic Bohmian world is a 3-dimensional space populated with Bohmian particles. The best way to compactly and yet powerfully summarize their motions is via the guidance equation for their velocities, and that equation that takes the wavefunction as input. The ontology of the Bohmian world differs from that of the classical world in certain subtle ways, and the laws turn out to be more complicated, but that's the best that can be done with particles that behave so strangely. Hence, we should treat the wavefunction just as we do the Hamiltonian. Both are functions in Schrdinger's equation, and neither are defined over 3-dimensional space. Just as no one asks where the Hamiltonian lives and insists that we reify it, so the question of where the wavefunction lives is a category error on this view.
I believe that there are good reasons for the Bohmian to conceive the theory this way. If we are not to have an "interaction problem" akin to the one for Cartesian mind-body dualism, we must demote either the particles' ontological status (and believe in only a Bohm "world particle" in 3n space) or demote the wavefunction's ontological status (and believe only in the particles in 3-dimensional space). So the question is which view is superior.
Consider the "3n" interpretation. The Bohmian world consists of a 3n-dimensional space populated with a single 3n-dimensional Bohmian 'world particle', the particle that encodes all the information about the apparent n particles. On this view, 3-dimensional particles, people and planets are all emergent features of the evolution of the world particle. While elegant in certain respects, this is a deeply strange view. It's committed, for instance, to the counterfactual that if there were one more electron in the universe then the world would have more spatial dimensions. And it ends up denying the fundamental reality of all the entities that gave rise to the theory in the first place.
Apart from being odd, such a theory also seems explanatorily inferior to its rival in some ways. Think of all the evidence of the 3-dimensionality of the world. Forget about "seeing" the world as 3-dimensional I don't think we do that in any straightforward sense. I mean instead to consider all the places in our physics wherein 3-dimensonality is deeply immersed; for instance, symmetry and conservation principles. Very basic rotation, translation and reflection symmetries are connected with the 3-dimensionality of space, which are in turn (minus the discrete symmetries) related to the many conservation principles we have. Think also of all the arguments from Kant to present times picking out 3-dimensions as physically special, e.g., that stable orbits are possible only in 3-dimensions. Even though I believe such arguments fail to establish their conclusions (see Callender 2005), it seems to me undeniable that the 3-dimensionality of space is deeply entrenched in our entire systematization of physics. In all of this, the 3-dimensionality of space plays a crucial role in explaining physical features. The 3n-dimensional view can (arguably) recover all of this, but only by placing special constraints on the Hamiltonian (Albert 1996). As I calculate the explanatory benefits and costs, the 3-dimensional view comes out on top.
Despite the apparent naturalness of this position, many are persuaded that the 3-dimensional interpretation is a nonstarter. Perhaps the best way to see why is to point out certain disanalogies with the Hamiltonian. Earlier I said that the wavefunction is analogous to the Hamiltonian, but there are some prima facie differences that ought to be considered. In the literature, these differences tend to seize the day.
Brown and Wallace best capture the worries. They try to force the reification of the wavefunction upon the Bohmian. To this goal, they marshal various intuitions in favor of regarding the wavefunction as part of the ontology. Unlike the Newtonian gravitational potential (which also doesn't live in 3-space), they write, the wavefunction
is contingent (equivalently, it has dynamical degrees of freedom independent of the corpuscles); it evolves over time; it is structurally overwhelmingly more complex (the Newtonian potential can be written in closed form in a line; there is not the slightest possibility of writing a closed form for the wave-function of the Universe.) We dont pretend to offer a systematic theory of which mathematical entities in physical theories should be reified. But we do claim that the decision is not to be made by fiat, and that some combination of contingency, complexity and time evolution seems to be a requirement. (41)
The wavefunction is contingent, evolves with time, and complex. If we consider the Hamiltonian, it is usually conceived as part of the background structure of the theory, and so, non-contingent. Of course the Hamiltonian can evolve with time, so there need be no difference there. And the Hamiltonian, unlike the wavefunction of the universe, is allegedly not as complex as the wavefunction of the universe.
Before responding directly, let me emphasis something Brown and Wallace say. They don't pretend to offer a systematic theory of what to reify and what not to in an arbitrary physical theory. They simply think of these three features as guides or symptoms of what to reify. I want to go a little further by reminding the reader that there are no logical rules whatsoever telling one what bits of a theory one must reify. One arguably needs to reify enough to provide an ontology that is capable of best explaining what we observe. All manner of factors may play a role in determining this. Contingency, complexity and time evolution individually or collectively are neither necessary nor sufficient features of what scientists reify.
More directly, these three features, especially when the comparison is between the Hamiltonian and wavefunction, appear quite weak reasons to reify. Regarding complexity, are we really to believe that the wavefunction of the universe is necessarily more complex than the Hamiltonian of the universe? And how do we define 'complexity' here, and in what language? Complexity seems a hopeless case. Evolution in time? As mentioned, the Hamiltonian can evolve or not in time. And in general, there are plenty of variables that evolve in time that we wouldn't dream of reifying. Contingency? In one sense I might agree that contingency contributes to a compulsion to reify, but in the precise sense defined here (i.e., having dynamical degrees of freedom independent of the particles) I feel no compulsion.
I do agree with Brown and Wallace on this much: if one asked most physicists why they consider wavefunctions real, if they do, they might point to the fact that they change with time and that they prepare different systems with different wavefunctions, and are in this sense contingent. A Bohmian explanation of these intuitions may be supererogatory, but it would be nice all the same.
Interestingly, the Bohmian can explain these two intuitions. From the Bohmian perspective, strictly speaking, there is only one wavefunction, the universal wavefunction. In Bohmian eyes, the wavefunctions that physicists use in practice, i.e., to represent subsystems of the universe prepared in laboratories, are actually what Bohmians call effective wavefunctions, not the universal wavefunction. On the Bohm theory, the universal wavefunction never collapses; nevertheless, an effective collapse can occur. That is, whenroughlyenough decoherence has occurred between the system and its environment, the wavefunction's components become widely separated (have approximately disjoint macroscopic support) in configuration space. The actual configuration of particles then must "choose" which part of the wavefunction will largely guide it, much as a cork floating in a river must choose a piece of the river when it meets a fork. Like the cork, the actual configuration is then sensitive to the piece it choose, until and unless the "rivers" rejoin (which is monstrously improbably due to the huge numbers of degrees of freedom that would have to conspire for this). When this effective collapse occurs, thanks to the existence of an actual configuration of particles, the Bohmian defines the effective wavefunction. The effective wavefunction is completely determined by the actual configuration and the wavefunction of the universe. If we want to explain intuitions arising from practice, it is to the effective wavefunction that we must look. When we do, we find precisely what we need.
Suppose the wavefunction of the universe is not contingent, part of the laws of nature. Still, thanks to the initial distribution of particles (which is contingent), different effective wavefunctions for different subsystems will be definable in this world. The effective wavefunction for a system prepared in a laboratory depends on the actual distribution of particles in that laband these could have been differently distributed. Bohmian mechanics therefore predicts the kind of contingency that we find, even if the universal wavefunction is not contingent.
A similar kind of explanation can hold for dynamics. Many Bohmians have pointed out that in the context of canonical quantum gravity one can have a wavefunction of the universe that does not evolve but which nonetheless generates non-trivial evolution (see Goldstein and Teufel 2001 and references therein). In particular, consider the program in quantum gravity known as canonical quantum gravity. Its defining equation is the Wheeler-DeWitt equation. Suppose ((g) is a solution to the static Wheeler-DeWitt equation, where g is a spatial three-metric. It is easily shown that despite this wavefunction being static, g can still change with time in accord with the Bohm guidance equation so long as ((g) is complex. But one can also show that in the semi-classical case effective-like wavefunctions for subsystems of the universe will approximately satisfy the time-dependent Schrdinger equation (Callender and Weingard 1996; Goldstein and Teufel 2001). These effective wavefunctions are the wavefunctions physicists "see" evolving in laboratory situations. So even if the wavefunction is static, still one will see effective wavefunctions evolving.
Brown and Wallace complain that quantum gravity is speculative, that the Wheeler-DeWitt equation has technical problems, and so on. They're right. It's all very speculative. Yet the challenge demanded such speculation. The fact remains that even if we agree to play their game and insist on finding particular symptoms of what needs reifyingwhich isn't ultimately defensible anywayand assume the worst-case scenario, that the wavefunction of the universe is static and the Hamiltonian timeless, and furthermore, confine ourselves to real world proposals in quantum gravity, even then we can rigorously show that effective wavefunctions have the desired properties. To find fault at this point and object that maybe Bohmians can't produce a similar result with different unnamed speculative physics is a criticism Bohmians can live with.
In sum, I've argued that even with all the major presuppositions of the argument granted, Redundancy doesn't yet pose a problem for Bohmians. Moreover, what is arguably the most natural way of understanding Bohm's theory makes Redundancy simply inapplicable. Either way, Redundancy shouldn't worry the Bohmian.
References
Albert, D. 1996. "Elementary Quantum Metaphysics". In: Bohmian Mechanics and Quantum Theory: An Appraisal, (eds.) Cushing, J.T., Fine, A., and Goldstein, S. Springer, 277-284.
Barrett, J. A. 1999. The Quantum Mechanics of Minds and Worlds, Oxford University Press: Oxford.
Bell, J.S. 1987. Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press.
Block, N. 1980. "Troubles with Functionalism", in Readings in the Philosophy of Psychology, Volumes 1 and 2. Cambridge, MA: Harvard University Press.
Brown, H. R. and Wallace, D. 2005. Solving the Measurement Problem: de Broglie-Bohm loses out to Everett, Foundations of Physics 35: 517540.
Callender, C. and Weingard, R. 1996. "Time, Bohm's Theory and Quantum Cosmology Philosophy of Science 63, 470-74.
Callender, C. 2005. An Answer in Search of a Question: Proofs of the Tri-Dimensionality of Space Studies in History & Philosophy of Modern Physics 36, 113-136.
Cohen, J. and Callender, C. 2009. "The Better Best System Theory of Lawhood", Philosophical Studies, forthcoming.
Drr, D., Goldstein, S., and Zanghi, N. 1992, "Quantum Equilibrium and the
Origin of Absolute Uncertainty", Journal of Statistical Physics 67(5/6), 843907.
Drr, D., Goldstein, S., and Zanghi, N. 1997. Bohmian Mechanics and the
Meaning of the Wave Function, in R. S. Cohen, M. Horne, and J. Stachel (eds.),
Experimental Metaphysics: Quantum Mechanical Studies for Abner Shimony, vol. 1, Boston Studies in the Philosophy of Science 193. Dordrecht: Kluwer, 2538.
Goldstein, S. and Teufel, S. 2001. "Quantum Spacetime without Observers: Ontological Clarity and the Conceptual Foundations of Quantum Gravity", in Physics meets Philosophy at the Planck Scale, edited by C. Callender and N. Huggett, Cambridge University Press, 275-289.
Deutsch, D. 1996. Comment on Lockwood, British Journal for the Philosophy of Science 47: 222228.
DeWitt, B.S. and Graham, R.N. (eds.) 1973. The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press.
Greaves, H. 2006. "Probability in the Everett Interpretation". Philosophical Compass.
Leavens, C.R. 1990. "Transmission, Reection and Dwell Times within Bohm s Causal Interpretation of Quantum Mechanics". Solid State Communications 74, 923.
Lewis, P. 2007a. Empty Waves in Bohmian Quantum Mechanics , British Journal for the Philosophy of Science 58(4), 787-803.
Lewis, P. 2007b. "How Bohm's Theory Solves the Measurement Problem", Philosophy of Science 74, 749760.
Rey, G. 1997. Contemporary Philosophy of Mind. Oxford: Blackwell Publishers.
Valentini, A. 2009. "De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial?" To appear in: Everett and his Critics, eds. S. W. Saunders et al., Oxford University Press.
Wallace, D. 2003, Everett and Structure, Studies in the History and Philosophy of Modern Physics 34: 87105.
Wallace, D. 2008. "Philosophy of Quantum Mechanics" In: The Ashgate Companion to Contemporary Philosophy of Physics, ed. D. Rickles, Ashgate Press.
Weir, A. 2001. "More Trouble for Functionalism". Proceedings of the Aristotelian Society 101 (3):267-293.
Zeh, H. D. 1999. Why Bohms Quantum Theory?, Foundations of Physics Letters 12: 197200.
Here is Everett: "Our main criticism of this view [Bohm's theory] is on the grounds of simplicity - if one desires to hold the view that is a real field then the associated particle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory" (DeWitt and Graham 1973, section VI). Commentators developing and endorsing Everett's argument include Brown and Wallace 2005, Deutsch 1996, Wallace 2008, Zeh 1999. Those criticizing the argument include Lewis 2007a, 2007b, and Valentini 2009.
Dennett is usually associated with instrumentalism in the philosophy of mind, i.e., the idea that taking others to have mental states (even if they don't really have them) is often useful. However, I think that most Everettians would prefer to say that a cat really is a cat-pattern in a branch of the wavefunction.
Strictly speaking, Bohmian cats and Everettian cats can realize very different functions. If so, it remains open to the Bohmian to insist that his function best fits cats. A function is defined as a single-valued total relation between elements of an input set X and elements of an output set Y. Suppose the input and output sets consist in the fundamental ontology of each theory; then, because the relations among the elements in each set differ mathematically, they can realize different functions. An intuitive way to notice the difference is simply to note that Bohm particles go through one or the other slit in the two-slit experiment whereas wavefunctions do not. The particles thus have relations such as 'having been through the right slit' that wavefunctions do not. So if part of being Fuzzy's cat whisker is having a particle in it that much earlier went through the right slit, then wavefunctions cannot realize being Fuzzy's whisker. The advocate of Redundancy will rightly complain that we shouldn't use the fundamental ontology in our input and output sets, for the science of cats will not deliver such fine-grained definitions of cats or whiskers. So this way out is open only to Bohmians with an inordinate faith that feline science will produce specifically sub-quantum particulate definitions of cats.
Lewis 2007b amends his advice: the Bohmian can respond by denying functionalism or by denying the cat pattern is instantiated in the wavefunction. I'm sympathetic with the latter choice, but I'm not quite sure that I follow Lewis' reasoning. He asks: what pattern is present when we superimpose horizontal stripes upon vertical ones? Lewis argues that we could say that it is a pattern of squares or a pattern of horizontal stripes and a pattern of vertical stripes. Neither seems forced upon us, he says. Similarly, are we forced to pick out cat patterns in the massively complicated superposed Schrdinger cat state? No, says Lewis; we have a choice to make but nothing in Everett compels us to choose the Everettian way. That seems right, but I'm not sure I see the problem for Everettians. There is no choice to make. Given a functional specification of a cat, cats are either present in the wavefunction or not. Same goes for dogs, people, and even functionally-specified gruesome entities. Stripes and squares are both present in plaid shirts. We don't choose one or the other. With regard to the wavefunction, so long as cats are among the patterns present, I don't see a problem for the Everettian. Perhaps Lewis is instead pointing out that the function realized by a particle-cats is different than the function realized by a wavefunction cat. Since they are represented by quite different mathematical entities, that is surely correct. But then the considerations of footnote 3 apply.
Of course, it could be that functionalism in any of the forms needed to drive this argument is simply false. Then there is nothing at issue. Certainly the philosophical literature has witnessed no shortage of challenges to functionalism. Amusingly, one recent criticism (Weir 2001) is that some types of functionalism entail, in the mental case, a multiplicity of qualitatively identical mental states in each individual. If correct and transferred to the current scenario, then even each Everett branch would contain redundant cats. But let's put such worries aside and assume that some form of theoretical and metaphysical functionalism is sustainable, and furthermore, that we don't want to give it up.
Superstring theory has been attacked recently by a number of opponents. But imagine a chauvinistic functionalist critique: cats can be anything but superstrings. Even the theory's more ardent detractors would disparage this objection.
Even Brown and Wallace, defenders of Redundancy, seem to agree with this. They call this the most "principled" response, and they seek to answer it by defending the Everett theory from the preferred basis objection.
Georges Rey 1997 (pp. 99-103) describes four types of phenomenon that behaviorism does not predict but that functionalism does predict and explain.
For thetheory of laws I prefer, see Cohen and Callender 2009.
Let Qt be the actual configuration of particles in the universe at a time. In a composite system Qt=(Xt,Yt), where X is the actual subsystem of interest and Y is the actual environment. Then a natural definition for a subsystems wavefunction is what DGZ 1992 call a conditional wavefunction (t(x)= (t(x,Yt), where we calculate the universal wavefunction in the actual configuration of the environment. The conditional wavefunction will not in general evolve according to the Schrdinger equation. However, when certain further conditions are met (corresponding to the universal wavefunction evolving into a wide separation of components in the configuration space of the entire system--see DGZ for details) it will evolve according to the Schrdinger equation. When this is the case it is called an effective wavefunction. Note that an effective wavefunction obtains whenever the orthodox formalism assigns a wavefunction.
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2The Redundancy Argument Against Bohmian Mechanics 1. Introduction 2. The Redundancy Argument 2. Caveats1 3. The Redundancy Argument and Functionalism 4. Redundancy Rejected& 5. Is the Wavefunction an Object? ReferencesTitle Headings
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