Why the Many Worlds Interpretation of Quantum Mechanics Needs More Than Hilbert Space Structure

Meir Нетто and Orly Shenker

1 Introduction

In their contribution to this volume, McQueen and Vaidman argue that common sense requires that explanations in physics be not only causal¹ but also local (they give some necessary conditions for what counts as ‘local’; see end of Section 3 in their paper). Their main claim is that the MWI of quantum mechanics (originally due to Everett)^[1] provides local explanations of the outcomes of experiments that in other interpretations of quantum mechanics seem to require (some sort of) nonlocality. In this sense, they argue, the MWI restores common sense to quantum mechanics.

world there is also a trace in the inner box, which cannot be explained locally by the motion of our particle through the inner box because no trace in our world leads to our particle passing through the inner box (see McQueen and Vaidman, Chapter 3, Figure 3.5). Here the intuition is that a local explanation of the trace in the inner box (which appears as strong as the trace along path C) based on physical matters of fact in our world requires that there are traces of the particle in our world that lead to the inner box. But there are no such traces in our world. According to McQueen and Vaidman the physical facts that explain this inner trace occur in another world. In that other world, there is a continuous trace of the motion of a copy of our particle from the origin through the inner box up to detector D_s (see Figure 3.10c, McQueen and Vaidman, Chapter 3). And their point about local causation is that this motion of the copy particle in the other world leaves a trace in the inner box also in our world.

So the situation is this: (i) a copy of our particle is detected by (a copy of detector) D _s in some other world, but not in ours, and therefore this copy particle might be said to exist in that world; (ii) this copy particle leaves a continuous trace in the other world from the origin of the experiment via the (copy) inner box up to D_s (we shall omit from now on the term ‘copy’ unless it is needed); (iii) our particle does not leave a trace that leads to the inner box, so if its motion were the cause of the trace we see in the inner box, this trace would be created either by some sort of a nonlocal influence in our world (from path C to the inner box) or else by some sort of a nonlocal motion in which our particle travels along both path C and the inner box (without leaving a trace that leads to or from the inner box; (iv) to avoid this sort of nonlocality, the claim (on this proposal) is that it is the copy particle in the other world that causes (or creates or what have you) the trace in the inner box also in our world.

We grant that in the MWI, one might take the trace in the inner box in our world to be some sort of a ‘photograph’ of the other ‘parallel’ world, where in the ‘photograph’ we have a ‘record’ of a segment of the trace left locally by the copy particle in the (copy) inner box of the other world. But we don’t see that this local behavior in the other world leads to some sort of a local picture of how things play out causally in our world, even on this way of looking at the trace (i.e., as a ‘photograph’ in our world of segments of the motion of the copy particle in the other world) because also on this way of thinking we have a cross-world causation by which the ‘record’ of the trace is formed. The same points (mutatis mutandis) arise with respect to the other experiments described by McQueen and Vaidman.

The upshot is that particles in the MWI leave traces both in the world where they exist but also in other worlds in which they don’t exist. So the explanation is said to be local because spacetime splits together with the particles so that there is no influence at space-like separation within

a world, but the causal influence literally travels across ‘parallel’ worlds. This seems to us to stretch the concept of locality beyond common sense, if not beyond breaking point, even if one understands causation weakly in terms of counterfactual dependence rather than by straightforward physical interaction.

This leads to the more general question of how precisely one should understand the concept of ‘worlds’ in the MWI, which is indispensable but quite tricky. In this context it seems to us that an old problem that has been mounted against the MWI³ (called the ‘preferred basis problem’), which is believed by many to have been solved by decoherence,⁴ is still open for reasons that have escaped the literature up to now.

3 Consequences of the Preferred Basis Problem in the MWI

The problem of the preferred basis stems from the mathematical fact that the quantum state is symmetric (or invariant) under the (infinitely many) choices of basis of Hilbert space in which it can be written. By this we mean that given the Hilbert space structure, a choice of basis in which the quantum state is described makes no difference with respect to the physical state and the facts obtained in the universe when it is in this state, it makes no difference with respect to the time evolution of the quantum state, and it makes no difference with respect to the predictions of future facts. Moreover, the standard description of local interactions (as well as the decoherence interaction) presupposes a factorization of the set of all degrees of freedom of the universe into subsets (which are the subsystems), for example, a measured system (say the spin + position of an electron in a Stern-Gerlach device), a photographic screen, an observer, and environment. This standard factorization is intuitive and justified by our experience. However, there are theoretically other factorizations: for example, into the electron, the left hemisphere of the observer’s brain + one cubic meter of air molecules in the laboratory, and the rest of the degrees of freedom of the universe. We call this the factorization-symmetry of Hilbert space, by which we mean the following: there are many (possibly an infinite number of) factorizations of the universal Hilbert space into sets of degrees of freedom (or subsystems), such that given the quantum state of the universe, all the factorizations are on equal footing; in other words, there are no facts determining a preferred factorization. But our experience corresponds (by and large) to the standard factorization and in addition also to certain local states of macroscopic systems given the standard factorization. In this sense the standard factorization and the local basis of states are preferred, but there is no deeper account of why they are preferred. In particular, the structure of the interactions does not explain this preference because it presupposes it. When one appeals to the structure of the interactions in the universe, say the decoherence interaction, or the fact that the interactions between macroscopic systems are local, one presupposes the factorization that features in our experience of the total set of degrees of freedom. This is acceptable, but we should note already at this stage that it does not explain our experience. In other factorizations, the structure of the interactions among the subsystems, induced by the same total Hamiltonian, is different.

Let us illustrate this idea by the following figure:

Figure 4.1 Branching and other structures.

On the left side of Figure 4.1 we depict the branching structure that matches our classical-like experience as described by the psi-basis, which is the basis of localized states, and the same structure as it is described by another basis, which we call the phi-basis. In these two bases, the interference terms (denoted by the thin, black lines on the left side of the figure) between the branches are small and the interaction Hamiltonian picks out the psi-basis as dynamically preferred in the sense that the interference terms in this basis are small and the states match our experience. Formally, in the phi-description of the branching on the left side, we do not add up the similar terms in the different branches. It is a fact that under the time evolution of the universal state, the interference between the branches in the psi-basis are very small; this fact is common to the two descriptions of the branching structure on the left side in terms of the psi-basis and the phi-basis. By contrast, on the right side of the figure, we depicted a different structure that (by the symmetry of bases in Hilbert space) equally exists in the same quantum state when it is written in the phi-basis. In this structure (on the left side of the figure), the interference terms (denoted by the thick, black lines) between the branches are large.

Vaidman³ and McQueen⁶ acknowledge the factorization-symmetry and the symmetry of bases in Hilbert space as well as the preferred basis problem that follows from these symmetries for the MWI:

[mathematically, one can decompose the wave function of the universe into a superposition of orthogonal components... in many other ways that will not provide a familiar world’s picture in every branch. So, critics might say that the proposal is circular: I define by fiat what I want to explain. First, a simple definition that is confirmed by observation sounds to me like a legitimate strategy. But there is also a more specific answer. The basis of the decomposition is indeed preferred.⁷

But the question is: which facts make the decomposition corresponding to our experience preferred? Or what makes it the case that our experience is described by components of the state in the preferred (psi-)basis (and factorization)? According to Vaidman and McQueen, the local structure of the interactions singles out the decomposition of states such as state 5 in terms of the psi-states in which macroscopic systems are in localized states (see the left side of Figure 4.1). But as we have argued, this claim already presupposes our experience; it does not follow from the structure of the universal Hilbert space alone. In measurements, for example, the interaction Hamiltonian depends on the position of a macroscopic pointer, or the position of ink marks on a piece of paper, or the position of neurons in our brains, and so on, and even if one disregards the decoherence interaction with the environment, the position basis, or more generally the expansion of the state in terms of narrowly peaked Gaussians in position, are preferred in the sense that these states match our experience. In our example of Figure 4.1, the psi-states (corresponding to the branch structure on the left side of the figure) are the localized states that match our experience, whereas the phi-states (depicted on the right side of the figure) are delocalized superpositions of the psi-states and do not match our experience. But what in the Hilbert space structure accounts for this asymmetry between the psi-states and the phi-states? Vaidman⁸ argues, in a way that might seem to undermine this point, that the localized states are preferred because they are stable over time:

[ujntil now I have not mentioned time evolution. Everything was considered at a particular moment. But we cannot experience anything at zero time. We need an order of 0.1 seconds to identify our experience. Thus, the world needs some finite time to be defined. The world has to be stable, at least on the scale of seconds. Locality of interactions in nature ensures that only the decomposition of wave functions corresponding to well-localized macroscopic objects can be stable. A quantum state describing the superposition of a macroscopic object in separate locations with a particular phase evolves almost immediately into a mixture that has a large component with a different phase. This obvious fact is analyzed in numerous papers using the buzzword ‘decoherence’.⁹

This goes along the tradition of Everett’s original argument from 1957. But why, for example, does the world or our experience have to be stable, as Vaidman and McQueen require? Of course, as a matter of empirical fact, the world as we experience it is stable (or at least has been stable up to now). But the MWI should derive this fact from its fundamental postulates and laws, not assume it. Here, as we mentioned earlier, evolutionary arguments to the effect that stability of the preferred states is essential for survival come in.¹⁰ The idea is that because our experience is associated with components of the universal state, the components need be stable over time for biological systems to evolve and survive along the branch structure defined by these components. It is true that in biology the standard description of evolutionary survival is in terms of adaptive systems that are immersed in some environment that survive stably over time. But how does this condition become a constraint on fundamental physics, which is compatible also with universes in which there are no biological systems at all? After all, whether or not there is experience of our kind that is stable over time or, for that matter, whether or not certain biological kinds survive need not be a factor that determines whether or not something is real. If it is true that a stable basis is a condition for survival, this fact should be accounted for by fundamental physics, or else it should be added to the Hilbert space structure.

Let us suppose (for the sake of the argument) that stability of components of the universal state in some basis is indeed a condition for survival. But given the MWI the universal state now, in the present moment, is some superposition in the psi-basis: why do we not experience now the unstable phi-basis in each component of which we are in superpositions of localized states? In this case, if we grant the evolutionary argument, we would presumably cease to evolve as experiencing agents, and we would not be around to ask questions about our experience. But this is just bad luck for us. Why should the laws of physics care about our luck or our evolution in the first place? Again, perhaps it is true that a stable basis is a condition for survival, but if so, some additional structure backing this up and breaking the basis-symmetry of Hilbert space is needed. Note that the situation here is essentially the same as in the standard view about classical statistical mechanics, according to which, the past hypothesis of low entropy is added in order to break the time symmetry of the equations of motion and account for the increase of entropy only toward the future.¹¹

Our conclusion applies also to the versions of the MWI that rely on decoherence to define the preferred basis.¹² One might say that the most obvious justification of choosing the decoherence basis (or the localized states basis) as preferred (as well as the corresponding factorization) to define the worlds or the branching is that in this basis the on-diagonal elements of the reduced states of the macroscopic systems that appear in our experience can be directly interpreted as the relative frequencies of the states of these systems in our experience. Of course, this can be done (and this is what is usually done)! But our point is that this ‘obvious’ addition requires adding structure to Hilbert space beyond the structure given by the quantum state. That is, one must add structure to Hilbert space that will underlie the facts that make it the case that we have this experience. In particular, Wallace¹³ introduces a high-level law (he calls Dennett’s criterion) for the emergence of the worlds (or our experience), the role of which is to make some patterns in the quantum state real. This high-level criterion is based on functionalist ideas in the foundations of the special sciences. However, in his influential paper on the functional- state hypothesis, Putnam has already noted that “the functional-state hypothesis is not incompatible with dualism!”¹⁴ Moreover, it is provable (regardless of quantum mechanics) that if a functionally defined property is not identical to a micro-physical property (i.e., in quantum mechanics, of the quantum state), then the functional-state hypothesis implies that any token of the quantum state from which the functional property emerges must itself have some nonphysical property.¹⁵ So functionalism is not only compatible with dualism; it entails additional «owphysical structure.

Vaidman and McQueen are aware of the crucial difference with respect to the preferred basis problem and the account of our experience that holds between, on the one hand, the MWI and, on the other hand, collapse and hidden variable theories (like, respectively, the GRW theory and Bohm’s theory¹⁶) in which the additional laws that are added to the Hilbert space structure (respectively, GRW collapses or flashes, Bohm- ian trajectories) account for our familiar macroscopic experience. In the GRW theory the states that we experience are singled out by the flashes or the collapses of the wave function and in Bohm’s theory by the trajectories in 3-D space (or perhaps in 3N space; this is debated in the literature).¹⁷ But in the MWI if one only presupposes the Hilbert space structure, there is no account for why in the first place we experience the components of the universal state in the preferred basis rather than in some other (stable or not) basis or why we do not experience the entire superposition despite the fact that our brain states are superposed in the way depicted in Figure 4.1. One has to accept that our experience corresponds to the localized psi-states familiar from classical mechanics as a brute fact. It seems to us that Vaidman acknowledges this point when he says, “In quantum mechanics without collapse we must add a postulate to connect to our experience, because mathematics does not provide a (unique) picture corresponding to what we see around us.”¹⁸

Perhaps for Vaidman and McQueen adding a postulate, such as the locality of the interactions in our universe, which as we argued presupposes our experience, is nevertheless more justified than the GRW collapses or Bohm’s trajectories. However, it follows from our argument that contrary to the received wisdom, the MWI is not more parsimonious, and therefore it has no advantage over other theories that solve the measurement problem (such as Bohm’s theory; or the collapse theory by Ghirardi, Rimini, and Weber; or the many minds theory of Albert and

Loewer.¹⁹ All these theories introduce additional laws or structure and additional elements of reality over and above the Schrodinger equation for the quantum state, and as we argued in this paper, the MWI is no exception in this regard. They all solve the measurement problem by changing drastically quantum mechanics, for good or for worse. Here, obviously, different questions may come up, such as the compatibility of the extra laws with relativity theory. But Ockham’s razor does not cut in favor of the MWI.

Many often reject the MWI on the grounds that the multiplicity of the worlds is extravagant. This does not strike us as a good argument; it seems to us that none of the interpretations of quantum mechanics is common- sensical. Although the set of common sense beliefs is not uniquely and sharply delineated and is often given by examples that appear to be psychologically irresistible and intuitively true, each and every interpretation of quantum mechanics is strongly incompatible with some of the most central common sense beliefs. In this sense quantum mechanics in all its interpretations shutters our common sense, if one takes it to be true. The result is that naive realism leads to physics and physics, if true, shows that naive realism is false. Therefore, naive realism, if true, is false; therefore, it is false.²⁰ The question arises: how do the common sense beliefs come about in a quantum-mechanical world and what justifies relying on empirical evidence that we understand commonsensically as conforming quantum mechanics to begin with? This question is addressed and answered by Shenker.²¹

Acknowledgment

We thank Lev Vaidman, Kelvin McQueen, Christoph Lehner, and Guy Hetzroni for discussions of the MWI. We also thank an anonymous reviewer for helpful comments. The research on which this paper is supported by the Israel Science Foundation (ISF), grant number 1148/2018.

Notes

• 1. McQueen and Vaidman’s argument is meant to be independent of any specific view about causation; see Ben-Menahem (2018) for the linkage between causation and locality (and other related concepts, e.g., determinism).
• 2. Everett (1957).
• 3. There are many versions of the MWI: see Everett’s (1957) ‘relative-state’ formulation and later versions, for example: DeWitt (1970), Zeh (1973, 2001), Deutsch (1985), Zurek (1993), Saunders (1995), Vaidman (1998,2014), and Wallace (2012). Our argument applies to all the versions.
• 4. For decoherence, see Zurek (1993) and Joos et al. (2003).
• 5. Vaidman (2014, 2019).
• 6. Private correspondence.
• 7. Vaidman (2019,100).
• 8. Vaidman (2019).
• 9. Vaidman (2019,100).

• 10. Vaidman, McQueen, private correspondence; it seems to us that this is also Zurek’s (1993) view.
• 11. See, for example, Feynman (1965).
• 12. See Zurek (1993) and Zeh (2001).
• 13. Wallace (2012).
• 14. Putnam (1975, 436).
• 15. See Нетто and Shenker (2019, Forthcoming).
• 16. See, respectively, Ghirardi, Rimini, and Weber (1986) and Bohm (1952).
• 17. See Ney and Albert (2013).
• 18. Vaidman (2019, 98).
• 19. Bohm (1952), Ghirardi, Rimini, and Weber (1986), see Bell (1987), and Albert and Loewer (1988).
• 20. Russell (1940,15).
• 21. Shenker (2020).

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• [1] The Causal Role of the Worlds We accept here the necessary conditions assumed by McQueen and Vaidman on what counts as local, and we grant (for the sake of the argument)their position that fundamental physics describes causal processes. To seewhat is at stake here, consider, for example, what happens in the experiment of the nested MZI (Figure 3.5 in McQueen and Vaidman, Chapter 3). They admit that the locality of the explanation is restored not bylooking at what happens in one or another ‘parallel’ world but rather inall parallel worlds taken together. But what does this exactly mean? Inthe MWI it turns out that physical facts in one world depend not only onwhatever happens in that world but rather on interactions literally occurring in other parallel worlds so that causation might be spatio-temporallylocal but only in virtue of these otherworldly interactions. Let us see howthis idea plays out. Take, for example, the world in which the particle is detected by D,(in the nested MZI, see Figures 3.5 and 3.10 in McQueen and Vaidman,Chapter 3). Call this world, our world. The explanation McQueen andVaidman suggest of what causes D, to click in our world is that thereis a continuous trace of the motion of the particle in our world fromthe origin of the experiment along path C up to D2. However, in our