Comments on Weingarten’s preferred branch

A senior colleague asked me for thoughts on this paper describing a single-preferred-branch flavor of quantum mechanics, and I thought I’d copy them here. Tl;dr: I did not find an important new idea in it, but this paper nicely illustrates the appeal of Finkelstein’s partial-trace decoherence and the ambiguity inherent in connecting a many-worlds wavefunction to our direct observations.


We propose a method for finding an initial state vector which by ordinary Hamiltonian time evolution follows a single branch of many-worlds quantum mechanics. The resulting deterministic system appears to exhibit random behavior as a result of the successive emergence over time of information present in the initial state but not previously observed.

We start by assuming that a precise wavefunction branch structure has been specified. The idea, basically, is to randomly draw a branch at late times according to the Born probability, then to evolve it backwards in time to the beginning of the universe and take that as your initial condition. The main motivating observation is that, if we assume that all branch splittings are defined by a projective decomposition of some subsystem (‘the system’) which is recorded faithfully elsewhere (‘the environment’), then the lone preferred branch — time-evolving by itself — is an eigenstate of each of the projectors defining the splits. In a sense, Weingarten lays claim to ordered consistency [arxiv:gr-qc/9607073] by assuming partial-trace decoherenceNote on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e., a mathematical criterion for sets of consistent histories) that captures some, but not all, of the properties of the physical and dynamical process of decoherence. That’s why I’ve called it “partial-trace consistency” here and here.a   [arXiv:gr-qc/9301004]. In this way, the macrostate states stay the same as normal quantum mechanics but the microstates secretly conspire to confine the universe to a single branch.

I put proposals like this in the same category as Bohmian mechanics. They take as assumptions the initial state and unitary evolution of the universe, along with the conventional decoherence/amplification story that argues for (but never fully specifies from first principles) a fuzzy, time-dependent decomposition of the wavefunction into branches. To this they add something that “picks out” one of the branches as preferred. By Bell’s theorem, the added thing has to have at least one very unattractive quality (e.g., non-locality, superdeterminism, etc.), and then the game is to try to convince oneself that something else about it makes it attractive enough to choose on aesthetic grounds over normal quantum mechanics.In addition to Bohmian mechanics, see important examples like Kent’s late-time photodetection [arXiv:1608.04805] and the “Many Interacting Worlds” [PRX 4, 041013 (2014)].b  

Weingarten is refreshingly clear about this and correctly characterizes his proposal as a hidden variable theory. I’d say its virtue is that, on its face, it doesn’t introduce new mathematical objects like the Bohm particle. However, if we try and quantify theory elegance with something like algorithmic complexity, then the bit string b used to specify the preferred branch (which is necessary to write down the complete theory, unlike normal quantum mechanics) is an equivalently inelegant structure.

Weingarten argues (in the paragraph beginning “Each |Ψ(hj​​,t)> may be viewed…”) that this proposal solves most of the fuzziness problems associated with the decoherence story, but I’d say it just repackages them. You need to help yourself to a precise choice of splitting events (when exactly they happen, etc.) to even define the ensemble of branches |Ψ(hj​​,t)⟩, but if you assume you already have that precision, then what’s the problem? Why not just declare that the set of branches is nothing but an ensemble of potential outcomes, exactly one of which is chosen at random (according to the Born probability), thereby reducing quantum mechanics to a classical non-local stochastic theory?

Perhaps Weingarten’s issue is that Many-Worlders like Davide Wallace often embrace a fuzzy/emergent nature of branches, likening them to the fuzzy/emergent nature of a tiger, and refuse to specify an arbitrary precise definition. But then it seems Weingarten would be happy with a consistent histories interpretation, where the branches are specified precisely with projectors…whose precision, insofar as it exceeds the fuzziness inherent to the decoherence story, is just picked arbitrarily.

Indeed, despite the marked inelegance of Bohmian mechanics, it has at least one advantage over Weingarten’s proposal: the Bohm particle path is automatically precise and this requires only an initial random sample from a well-defined probability distribution (to be compared with an arbitrary and still unspecified choice of branch structure). This means that, if we accept the Bohm story, branches can be fuzzy for the same reason that we’re OK with tigers being fuzzy in a universe where we understand atomic physics precisely.

Finally, note that, in the far future, this single-branch theory shares a problem with all theories that take branches as fundamental: eventually, the universe will thermalize and branch structure must break down.

Footnotes

(↵ returns to text)

  1. Note on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e., a mathematical criterion for sets of consistent histories) that captures some, but not all, of the properties of the physical and dynamical process of decoherence. That’s why I’ve called it “partial-trace consistency” here and here.
  2. In addition to Bohmian mechanics, see important examples like Kent’s late-time photodetection [arXiv:1608.04805] and the “Many Interacting Worlds” [PRX 4, 041013 (2014)].
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4 Comments

  1. Pingback: Various Topics in Interpretation of Quantum Mechanics | Not Even Wrong

  2. Hi Jess,

    1. I agree that the fuzziness of many-worlds + decoherence is simply repackaged. But that repacking is actually the point of the whole thing. The paper starts with the hypothesis that the intrinsically approximate character of the many-worlds + decoherence characterization of branching makes its branches, by themselves, implausible candidates for the substance of reality. The idea is to find something else with precise evolution rules that can be written in underneath to which the macro-reality of many-worlds + decoherence can then be viewed as an approximation.

    2. The environment bits are not additional degrees of freedom in the same class as Bohm trajectories. They were meant, in a particular model, to be re labelings of degrees of freedom already present in the system. In any case, an updated version of the paper was posted a while back without the bit vectors.

    3. Two problems with consistent histories from my point of view. First, for many (most?) systems, if you require consistent histories criteria to be fulfilled exactly, you will have no takers at all, so no macro-reality. Second, there are sets of 4 propositions such that every pair is consistent, but a particular subset of 3 of them can not be consistent.

    4. Two responses also to your comment that eventual thermalization of the universe is a problem for the proposal. Whether records of the branch structure of the universe persist into thermalization depends of how the records are encoded. Also, the idea is that the observable branch structure of the universe is not primary. It is supposed to be an approximate macro feature of an underlying ensemble of initial states. So if it becomes harder to notice at some late time in history unclear to me why that’s a problem.

    Best,
    Don

    • Hi Don,

      Thanks very much for your comments, and sorry for the big delay in getting back to you. Also, sorry if my response runs long; I think these issues are both important and subtle, so I tend to belabor things.

      > 1. …the intrinsically approximate character of the many-worlds + decoherence characterization of branching makes its branches, by themselves, implausible candidates for the substance of reality. The idea is to find something else with precise evolution rules that can be written in underneath to which the macro-reality of many-worlds + decoherence can then be viewed as an approximation.

      But if we assume that branches are understood approximately [1] and we just want to pick something that is precise, why not just arbitrarily choose a precise branch structure? I interpret your proposal to be defined by a two step process: First, pick a precise branch structure arbitrarily from the set of all branch structures that are compatible with the range of ambiguity inherent in the smooth process of decoherence in the the wavefunction of conventional quantum mechanics (henceforth “the traditional wavefunction”). Second, pick one of those branches and evolve it backward in time to t=0, then declare that to be the real world. So why not just stop after the first step?

      You might retort that under your ontological hypothesis (that the only fundamental object is the preferred branch) the traditional wavefunction with non-realized branches is just a human constructions. And indeed, we can’t rule this out. But my response is that, until we can write down a preferred-branch theory without reference to the traditional wavefunction, then we also ought to also consider the alternative ontology of precise branches. These do have precise evolution rules, which personally I prefer because the inelegance is transparent.

      It’s true that the evolution of a single preferred branch is smooth (in time) compared to the weird discrete-time nature of branching, which you emphasize in the good new paragraph from the updated version of your paper (“One piece of the formulation of quantum mechanics we now propose remains approximate, but another has become exact….”) but this is achieved by massive microscopic conspiracy. I would characterize this as merely obscuring the inelegance using an implicit definition that draws on an assumed branch structure with discrete times. Note that this criticism would not apply if you had an alternate way of defining the smoothly-evolving preferred branch.

      > 2….The environment bits are not additional degrees of freedom in the same class as Bohm trajectories.

      When I said “…the bit string b used to specify the preferred branch … is an equivalently inelegant structure”, I definitely didn’t mean to suggest that the environmental bits were additional dynamical degrees of freedom like the Bohm particle. My point is just that, from an information-theoretic point of view, fully specifying your theory requires a writing down big chunk of entropy (i.e., some way of identifying the preferred branch from all others) which is not present in normal quantum mechanics.

      Since I understand your proposal to be about repackaging for elegance rather than increasing the observational explanatory power of the theory, this is not a big deal.

      3….First, for many (most?)  systems, if you require consistent histories criteria to be fulfilled exactly, you will have no takers at all, so no macro-reality.

      It’s true that a set of histories specified using mathematically-simple-to-define projectors is unlikely to be exactly consistent, but there will exist an exactly consistent set of histories that is close enough to be observationally indistinguishable. (See J. N. McElwaine, PRA 53, 2021 (1996), especially the first three paragraphs in Sec. II and references therein.) Relying on this sort of existence argument without actually specifying the consistent set is definitely a flaw, but it applies equally well to a preferred-branch theory.

      >…Second, there are sets of 4 propositions such that every pair is consistent, but a particular subset of 3 of them can not be consistent.   


      Yes, without further conditions on the propositions (represented mathematically by projectors) that go into histories, it’s impossible to uniquely identify any preferred set of histories or, indeed, any single true proposition. Another pathology (related to the one you mention) is contrary inferences: given an initial state and some observed final data (e.g., the outcome of an experiment), there will exist two incompatible sets of consistent histories such that P is true with certainty in one set, Q is true with certainty in the other, where P and Q are commuting contrary propositions: PQ = 0 = QP.

      For exactly these reason, I agree with folks like Kent, Dowker, Bassi, Ghirardi, Okon, and Sudarsky that there is a set-selection problem, i.e., consistent histories needs to be augmented with a criterion for picking out (at least approximately) a preferred set. (This is equivalent to a full precise specification of branch structure, and is basically the maximal generalization of the decoherence program.) I would characterize consistent histories as a language for making classical logic statements about wavefunction branches rather than a complete theory.

      > 4…Whether records of the branch structure of the universe persist into thermalization depends of how the records are encoded.

      I predict that if you try to track records through the period of thermalization, you will either find they dissolve or you will be forced to distort your definition of records to become meaningless In particular, records will become completely delocalized, so that measuring the “environment” (or the “system”) would require a joint measurement of the entire universe. I am extremely interested in how to usefully and mathematically generalize the concept of records to one that makes sense at late times, so please let me know if you disagree.

      > … the idea is that the observable branch structure of the universe is not primary. It is supposed to be an approximate macro feature of an underlying ensemble of initial states. So if it becomes harder to notice at some late time in history unclear to me why that’s a problem.

      It’s worrying because you’re privileging (without explanation) some indeterminate intermediate time period that lies between now and heat death. Here’s what I mean:

      If we were to look around at the observationally accessible macro features at noon today, the simplest (or otherwise most likely) possible quantum state of the universe consistent with those features would be the traditional wavefunction which is known to branch, i.e., to develop superpositions of distinct macro features later in time.

      You are suggesting that instead we should consider a very different state which is consistent with current macro features but which also evolves through a sequence of states that are each consistent with individual macro configurations (i.e., no macro superposition) at their respective times. The way you implicitly construct this preferred state is by assuming we can distinguish the orthonormal set of macro-feature eigenstates at some final time — i.e., that the branch structure is at least approximately understood and defined for the traditional wavefunction — and then just choosing to privilege one branch.

      The problem is that there is no final time just before thermalization. And if you pick a time long before thermalization, then you’ll get macroscopic superpositions following that time in your preferred branch.

      I claim branches in the traditional wavefunction (which are inferred through decoherence theory) will start to smoothly dissolve into each other as heat death approaches, so that each is a joint eigenstates of fewer and fewer macro observables. Similarly, I conjecture that if you tried to go beyond decoherence theory and define a more fine-grained branch structure for the traditional wavefunction, you’d find either that (A) it was unstable from one time step to the next or (B) you had to simply fix macroscopically interpretable branches arbitrarily at some single preferred time prior to heat death and then evolve the branches forward in time without caring about the fact that they didn’t retain any recognizable records or other macro interpretation.

      Best,
      Jess

      [1] PS.: Since it is my personal passion project, let me emphasize that no one yet has a general method for obtaining, given a candidate wavefunction of the universes (or even of just a large many-body system), the branch decomposition in Eq. (5), |\Psi(t)\rangle = \sum_h |\Psi(h,t)\rangle, or even an approximation thereto. Rather, all we have are a collection of toy models where the decomposition is obvious/intuitive, and we extrapolate that it’s possible to find Eq. (5) for the wavefunction of the universe up to an error that is not detectable “for all practical purposes” (FAPP). (This is in contrast to the Bohmian approach, where a simple principle is declared that exactly specifies the ensemble of possibilities, i.e. the probability distribution for the Bohm particle position.) This is mostly a separate issue from my main critique of your paper, so I am assuming for the sake of discussion that a well-defined procedure for finding Eq. (5) exists up to a small error.

  3. Also, as a separate issue, note that the “branch decomposition ambiguity” you describe in Eq. (2) is not quite right. The branches you identify are essentially determined by the Schmidt decomposition (equivalently, the singular-value decomposition), and this decomposition is only ambiguous when the coefficients 1/\sqrt{2} in the state are exactly equal, which is a set of (Haar-)measure zero in Hilbert space. Otherwise, the decomposition into c_1|s_1\rangle|m_1\rangle+c_2|s_2\rangle|m_2\rangle is unique. The real issue with this story, which is partially solved by decoherence and redundant records (aka “quantum Darwinism”), is that the basis determined by the Schmidt decomposition is highly unstable and does not generically correspond to macroscopic outcomes. See D. Page, arXiv:1108.2709 and my related comments.

    (I know you’re just trying to recall a standard story for the reader here rather than offer an authoritative treatment, but I think this is sufficiently off the mark that you risk misleading people.)

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