The physics and metaphysics of Tychistic Bohmian Mechanics

The paper takes up Bell's (1987) “Everett (?) theory” and develops it further. The resulting theory is about the system of all particles in the universe, each located in ordinary, 3-dimensional space. This many-particle system as a whole performs random jumps through 3N-dimensional configuration space – hence “Tychistic Bohmian Mechanics” (TBM). The distribution of its spontaneous localisations in configuration space is given by the Born Rule probability measure for the universal wavefunction. Contra Bell, the theory is argued to satisfy the minimal desiderata for a Bohmian theory within the Primitive Ontology framework (for which we offer a metaphysically more perspicuous formulation than is customary). TBM's formalism is that of ordinary Bohmian Mechanics (BM), without the postulate of continuous particle trajectories and their deterministic dynamics. This “rump formalism” receives, however, a different interpretation. We defend TBM as an empirically adequate and coherent quantum theory. Objections voiced by Bell and Maudlin are rebutted. The “for all practical purposes”-classical, Everettian worlds (i.e. quasi-classical histories) exist sequentially in TBM (rather than simultaneously, as in the Everett interpretation). In a temporally coarse-grained sense, they quasi-persist. By contrast, the individual particles themselves cease to persist.


Introduction
The present paper advances a "minimally" Bohmian theory -Tychistic 1 Bohmian Mechanics (TBM)as both empirically and metaphysically adequate. It is minimally Bohmian in two senses. First, it satisfies a plausible minimum of desiderata for a theory to qualify as Bohmian; secondly, it uses only a minimum of assumptions on which the predictive success of ordinary Bohmian Mechanics (BM) rests -BM's "working posits". 2 Metaphysically, its key novelty consists in a distinctive combination of fundamental stochasticity, its many-worlds ontology, and Bohmicity (i.e. it belongs to the class of Bohmian quantum theories). 3 In a nutshell, TBM retains BM's overarching ontological framework. Its referents are particles, located in ordinary 3-dimensional space. Their positions are always determinate. In contrast to standard BM, however, TBM drops the supposition that those particles follow continuous trajectories: according to TBM, the universeunderstood as an N-particle system as a wholeperforms fundamentally stochastic jumps through configuration space. Rather than co-existing simultaneously as in the Everett interpretation, different worlds pop into existence sequentially: by hopping through configuration space, the universe instantiates (actualises) those structures in the wavefunction of the universe which correspond to Everettian worlds (i.e. quasi-classical histories, warranted by decoherence), see Fig. 1. The probability for those spontaneous materialisations is given by the Born Rule for the wavefunction of the universe.
An inchoate articulation of the theory harkens back to Bell's (1987) interpretation of Everett's many-worlds interpretation. 4 Demurring that it leads to a temporal form of solipsism, he dismissed it. The subsequent literature has largely concurred with the arguments Bell gestures at. In consequence, the theory received only marginal attention. We are unhappy with several of the received arguments in that discussion. Accordingly, our present paper pursues four goals: (1) To provide a more perspicuous formulation of the theory -TBM.
(2) To re-appraise the extant criticism of its central idea, in particular by Bell and Maudlin. (3) To argue that TBM is a close cognate of BMwhat is left of BM, once one shears the latter of all dispensable elements (and readjusts the interpretation of one of its remaining postulates)rather than a reading of the Everett interpretation. (4) To outline TBM's salient metaphysical features.
Our present aim isn't to advocate TBM as necessarily superior to BM (nor to the Everett interpretation). That would require a circumspect evaluation of all of their respective shortcomings and advantages. In particular, we don't argue that TBM scores necessarily better in terms of parsimony. Such claims call for a separate analysisone that must pay attention to different forms of parsimony/simplicity exemplified in the theories in question (and how they are supposed to trade off against each otherif one wants to regard them as truth-indicative criteria for theory selection at all). Here, we rest content with demonstrating that TBM is an empirically adequate and coherent theory. As such, it deserves a place at the table of the current discussions. Like standard BM, it falls within the framework of so-called Primitive Ontology (see x3.4 for details). Thus, TBM circumvents criticism of those who insist on compliance with that framework as a metaphysical sine qua non for any theory. Insofar as in what follows we compare TBM and other theories, it solely serves the purposes of clarification, in particular by demarcating TBM from those other theories.
The paper proceeds as follows. In §2 we review BM, recapitulating its motivation, formulation and standard interpretation. In §3 we introduce and develop TBM. §3.1. argues that an investigation of TBM is rewarding, even if one rejects it in favour of BM. §3.2 elucidates TBM's principles and interpretation. We demonstrate its empirical adequacy.
§3.3. sketches the role of probabilities in TBM, and their possible interpretations. In §3.4., we argue that TBM counts as a Bohmian theory, distinct from the Everett interpretation. We next turn to TBM's salient metaphysical features. §3.5. critically re-evaluates Bell and Maudlin's reasons for rejecting TBM. In elaborating further its many-worlds character, we show TBM to be coherent. We summarise our findings in §4.

Review of Bohmian Mechanics
This section outlines the motivation, basic postulates and received interpretation of standard BM. It's a theory about particles with determinate, deterministic trajectories, whose dynamics is constructed from the wavefunction.
Vis-a-vis the strife over the foundations of QM, especially the measurement problem, the interpretation of the Heisenberg relations and their joint culmination in the EPR "paradox", the question arises whether QM in its orthodox form (e.g. Von Neumann, 1932) is incomplete: might there exist an element of physical reality that has no counterpart in the Space is densely filled with many-worlds sequentially popping into existence. The corpuscles form a quasi-plenum. 3. On a temporally coarse-grained level, the FAPPworlds quasi-persist. description of the QM formalism (cf. Einstein et al. (1935), cited in Redhead (1987), p.71, who also discusses the EPR argument in detail)? Einstein (1949, p. 666), for instance, was " […] firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this [theory] operates with an incomplete description of physical systems." BM is an attempt to complete QM (if only in the sense of supplying the latter with a clear ontology). It offers a deterministic account of a corpuscular sub-quantum reality. From it, QM emerges in some sense, in a manner "approximately analogous […] to the statistical mechanics within the framework of classical mechanics" (ibid.).
From a non-historical perspective, BM is primarily motivated by the desire for a thoroughly ontic interpretation of QMa "quantum theory without observer" (see e.g. Bell, 1990, p. 215;Allori et al., 2008, sect. 4): it's about objective matters of fact, rather than subjective or epistemic states of experimenters. Such a form of realism had come under attack with influential anti-realist presentations of QM (for historical accounts, see e.g. Howard, 2004;Scheibe, 2007, Ch. VIII;. BM is a non-relativistic theory about N (massive, charged, etc.) particles and their continuous spatiotemporal evolution in 3-dimensional space. Their dynamics is such that the empirical content of QM, as enshrined in the Born Rule, remains unaltered.
Within a realist setting, BM thereby achieves its goal to provide a solution to the measurement problem (cf. Dürr &Teufel, 2009, p. 177;Lazarovici, 2019;Maudlin, 1995a). The latter consists in the incompatibility of the following three propositions (see e.g. Maudlin, 1995b): (A) The wavefunction of a quantum-mechanical system is complete: It specifies (directly or indirectly) all of the physical properties of a system. (B) The wavefunction evolves unitarily (in accord with e.g. the Schr€ odinger Equation). (C) Measurements, such as of an electron's spin, always have determinate outcomes (represented by the corresponding eigenstates): After a measurement, the measurement device is in a(n eigen)state either indicating spin-up or spin down. Superpositions aren't recorded.
While any two of these propositions are consistent with each other, their conjunction isn't. Schr€ odinger's famous cat paradox illustrates this. Assume that a cat's state is completely described by the wavefunction. Then, the QM formalism implies that at some point it's no longer in a determinate state of either dead or alive: the cat will be smeared out in a superposition of life and death. This seems to flout experience. BM eschews the dilemma by contesting (A): for a complete description of a system's state, its wavefunction must be supplemented by the positions the system's constituent particles occupy. (To distinguish the Bohmian particles from classical ones, we'll hereafter refer to them as "corpuscles". 5 ).
More precisely, for a universe with N corpuscles of mass m i each, 6 standard BM comprises three postulates (cf. Bohm & Hiley, 1993;Holland, 1993;Dürr & Teufel, 2009;Passon, 2010;Tumulka, 2017;Goldstein, 2017 for detailed reviews): (1) The standard, non-relativistic N-particle Schr€ odinger Equation with the universal wavefunction (i.e. the wavefunction of the universe) of the N particles, and the N-particle N acting on the i-th particle. (For convenience, we'll subsequently suppress the wavefunction's time-index.) (2) The Guidance Equation (GE): It supplies the dynamics for each (i ¼ 1; …; N) corpuscle. Note that the expression on the r.h.s. depends on the configuration of all particles. (This renders BM's non-locality manifest.) Given initial positions of the corpuscles, the GE determines their positions at any other time. Existence and uniqueness of the trajectories are guaranteed under prima facie reasonable assumptions. Note that the corpuscles' velocity fields generated by the GE depend on the wavefunction: (3) The Quantum Equilibrium Hypothesis (QEH): Let Ψ denote the wavefunction of the universe. Via its associated (Born) measure jΨj 2 ; it (uniquely 7 ) induces a measure of typicality: It quantifies which (measurable) sets of corpuscle configurations Q⊆R 3N count as large ("typical"), i.e. R Q d 3N QjΨðQÞj 2 ¼ 1 À ε, for some small ε. 8 (This definition of typicality is time-independent in a suitably generalised sense ("equivariant"), see Dürr et al. (1992), sect. 7.) Given this typicality measure, one can then show that for ensembles of identically prepared subsystems of the universe, each with the wavefunction ψ, the corpuscles' configurations are typically distributed according to the Born Rule, ρ ¼ jψj 2 (ibid.; Goldstein, 2011, p. 4.4). That is: for typical, large ensembles, the Born Rule approximates the corpuscles' empirical distribution,  Brown, 1995or Brown, Elby, & Weingard, 1996. Thereby, we hope to alert toalbeit perhaps not necessarily compellingsubtleties for identifying the Bohmian "primitive stuff" with classical particles. Two salient features of the latter have been discerned (cf. Mittelstaedt, 1995;Falkenburg, 2007, Ch. 6.1): (INDEP): (a) They may be in non-interacting, uncoupled states. (b) Their initial conditions are statistically uncorrelated.(COMP): A "law of thorough-going determination" (Mittelstaedt) holds: For every property, we can predicate either it or its negation of the particle. Bohmian particles flat-out flout (INDEP). Qua BM's non-locality, they don't seem to conform to (a). Nor do they respect (b): the quantum equilibrium hypothesis (viewed as a kind of law-like statement as part of the orthodox Bohmian-package deal) imposes a statistical constraint on initial conditions; they must be distributed approximately in accordance with the (QEH). Due to the Bohmian particles' contextuality, (COMP) is also plausibly violated: according to DGZ's interpretation of the contextual variables, these don't represent properties of the system/the corpuscles; in particular, one can't meaningfully ascribe, say, angular momentum to particlesin contradiction to what "thorough-going determination" would demand.Sincere thanks to an anonymous referee for pressing us on this! 6 Ascribing the masses to the corpuscles alone isn't uncontroversial, cf. Dewdney & Brown (1995); Brown (1996a). Esfeld (2018, p. 170) denies that corpuscles possess any intrinsic mass or charge (or any intrinsic properties). However, the interference phenomena Esfeld cites don't unequivocally support that (cf. Brown et al., 1996, sect. 4). 7 More precisely: jΨj 2 is the unique, natural measure that is equivariant under the dynamics defined by the GE, see Goldstein & Struyve (2007). 8 For more on the typicality interpretation of the QEH, see Maudlin (2011); Goldstein (2011);Lazarovici & Reichert (2015); cf. Frigg (2009Frigg ( , 2011 or Valentini (2020) for a critical voice.
formalism and its empirical content. Consider the large, but (arguably) finite number of position measurements, performed in the universe's life-time on identically prepared systems with wavefunction ψ. The specific positions of those systems' corpuscles vary. But at a statistical level, their distribution is well approximated by the density jψ t j 2 : in almost all measurements, the corpuscles are roughly distributed by jψ t j 2 . In the absence of any further, i.e. more fine-grained information (information that given contemporary QM isn't available to us in principle), we mayaccording to advocates of the typicality interpretationtreat our observational-empirical data as typical, see e.g. Dürr & Struyve, 2019.) For extensions of BM to incorporate spin or external electromagnetic fields, we refer the reader to the literature (e.g. Holland, 1993, Ch. 9, 10;Norsen, 2014).
A comment on BM's ideology (in the sense of Quine) is in order: what, according to BM, are the corpuscles' properties and relations (besides their mass, charge and magnetic momentum)? All dynamical variablesmomentum, energy, spin, etc.other than position, are contextual: their values depend on which other variables are assigned definite values. Position is the only non-contextual variable: only the corpuscles' positions (and their time-derivatives, i.e. velocities) always possess a sharp value. (Thus, BM evades the Kochen-Specker no-go theorem for hidden variable theories, see e.g. Redhead, 1989, Ch. 5, 6;Held, 2018 for details.) Ontologically, therefore momentum, energy, spin, etc. are subsidiary, non-fundamental quantities: in BM, they are non-classical degrees of freedom 9 that merely codify (supervene on) the corpuscles' motion (for details, see Daumer et al., 1997;Esfeld et al., 2014;Lazarovici et al.. 2018, sect.5).
For the present purposes, we set aside the thorny issue of the status of the wavefunction in BM (cf., for instance, Esfeld et al., 2014). Suffice it to state the dilemma one faces. On the one hand, the quantum statethe putative entity to which the wavefunction refers (Maudlin, 2013)appears real at least in two regards. First, the wavefunction enters the QEH and in this (at least, purely mathematical) sense constrains the corpuscles' distribution; secondly, it also enters the GEand in this (at least, purely mathematical) sense determines their dynamics. Vis-a-vis these observations, Bell (1987, p. 128) judged: "Nobody can understand this theory, until he is willing to think of ψ as a real, objective fieldrather than just a probability amplitude." On the other hand, as Bell likewise stresses, the wavefunction is defined on 3N-dimensional configuration space. At first blush, it's unclear how to understand such an entity inhabiting this space. It's even more mysterious how it relates to and is supposed to affect the particles, inhabiting our familiar 3-dimensional space (for a survey of possible responses, see the contributions in Ney & Albert, 2013). We'll return to this dilemma in x3.2.
In summary: BM provides an objective account of the world, made up of point-like corpuscles. Their only dynamical variables are positions. At all times, their positions are determinate. Via the wavefunction, a dynamics is defined that guides the corpuscles' deterministic spatiotemporal evolution. A universal constraint on the corpuscles' initial distribution secures empirical equivalence with QM.
This provides the background against which we'll now elaborate a cousin of BM.

Tychistic Bohmian Mechanics
This section introduces and unpacks TBMthe theory that naturally emerges upon removing the GE from BM (with suitable interpretational re-adjustments). TBM can be considered the "rump Bohmian theory" of randomly materializing corpuscles which one obtains after jettisoning BM's GE (and the concomitant determinism)together with a reinterpretation of the QEH as a stochastic guidance law (rather than a typicality statement, as in BM). The corpuscles are no longer assigned continuous trajectories. But their positions remain determinate at all times. With the probability equal to the Born Rule, the N-corpuscle system as a whole localizes itself in a fundamentally stochastic process: it performs random jumps through configuration space. Can TBM deliver a coherent picture of the world? We'll answer this in the affirmative.
We'll first (3.1.) say why it's worthwhile inspecting this theory. In 3.2, we'll clarify TBM's conceptual basis and defend TBM's empirical adequacy. 3.3. discusses possible interpretations of its probabilities. In 3.4., we categorise TBM as a minimally Bohmian theory. Subsequently, we elaborate on TBM's metaphysical and epistemological adequacy: 3.5. reevaluates Bell's criticism of TBM as "solipsistic" by illuminating its manyworlds character.

Motivation
Let's first spell out the motivation of (and the intention behind) our discussion of TBM: why consider removing the GEeven if one considers BM perfectly acceptable?
In light of BM's three postulates, two questions arise: are they logically independent? Are all of them strictly necessary? 10 Here, we wish to remain conservative with respect to the established physics. That is, we want to retain as few of BM's postulates as possible, whilst keeping its spirit intact. We'll probe a different question: can the GE be excised from BM, whilst forfeiting neither empirical and metaphysical viability nor, to a reasonable extent, BM's spirit?
Such an inquiry will not only deepen our understanding of BM along two lines. It's also one prima facie plausible reaction to the empirical underdetermination of BM's dynamics.
First, imagine a reader who endorses BM in its current form. She should welcome the envisaged study. To fully appreciate BM's merits, one needs to understand the import of each of its postulatesin particular that of the GE. A crucial question then is: what (metaphysical) consequences ensue if one abandons it? Our understanding of scientific theories is considerably deepened by systematically exploring such modifications (including omissions of some) of their axioms/postulates (cf. Lehmkuhl, 2017).
But the project is also of interest to those disconcerted by one of BM's features: the GE is vastly underdetermined by any possible observational data. An infinitude of equally viable, empirically equivalent alternative dynamicsdifferent forms of the GEexist. Each generates distinct trajectories (Deotto & Ghirardi, 2002;Holland & Philippidis, 2003). In principleand by constructionit's impossible to experimentally discriminate between those options.
Such empirical under-determination obstructs a naïve realism about BM (Kukla, 1994, p. 157;cf.;Stanford, 2017): why assume one particular particle dynamicssay, the standard GErather than another, equally suitable alternative? Advocates of BM have responded that the standard GE is distinguished as the simplest choice for a dynamics (Dürr, Goldstein, & Zanghì, 1992, p. 852) that respect certain desiderata. 11 9 "Degree of freedom" here shouldn't be understood traditionally, as representing a system's properties. Mindful of Dürr et al.'s warnings of "naïve realism about operators", we use the term in a purely formal-mathematical sense, denoting the parameters that need to be specified for a full description of the system and its behaviour. Thanks to an anonymous referee for pressing us on this.
10 One particularly intriguing issue is: Is it possible to derive the Born Rule in a manner analogous to Boltzmann's H-Theorem? That is: Do the configurations of most subsystems relaxvia the corpuscle dynamicsinto a "quantum equilibrium", i.e. the distributionjψ t j 2 ? We'll not pursue further such questions (see, however, Valentini & Westman, 2005;cf. Callender & Weingard, 1997;Callender, 2007). 11 Dürr et al. demand that the guidance law to be constructed be a first order differential equation, homogeneous (of degree zero) as a function of the wavefunction, Galilei-invariant and invariant under time-reversal.
One may well question the force of this response: all supposedly natural desiderata turn out to be tenuous (Belot, 2010;Fankhauser & Dürr, 2021 x2.2). Moreover, suppose, for the sake of the argument, that they were compelling. Yet, the argument still isn't entirely convincing: it pivotally turns on mathematical simplicitythat is, simplicity of the mathematical form of the dynamics. The infinitely many variants of BM with alternative guidance equations that differ from standard BM's only by a divergence-free term (cf. Fankhauser & Dürr, 2021) don't differ in their ontology. Hence, mathematical simplicity here doesn't even imply differences with respect to ontological parsimony. But why deem mathematical simplicity 12rather than a pragmatic criterion or even a subjective, rather elusive aesthetic preferencea reliable guide to truth? To do so is controversial (see e.g. van Fraassen, 1980, esp. Ch. 4.4;Ivanova, 2014)already on inductive grounds (cf. Hossenfelder, 2018;Norton, 2018). Dürr et al.'s reliance on simplicity thus considerably detracts from their argument's force. More generally, vis-a-vis the coexistence of empirically equivalent theories one may, of course, always invoke super-empirical criteria for theory selection. The challenge for an advocate of such a strategy then is: how to justify this choice of super-empirical criteriaand, in particular, why believe that they track truth?
Empirically equivalent, genuinely distinct theories are in fact rare (Norton, 2008). This raises a double worry: are they merely notational variants of the same theory, or does (at least) one of them posit superfluous structure? The former case is exemplified by the duality between Heisenberg's matrix mechanics and Schr€ odinger's wave mechanics (for caveats, see Muller, 1997a;1997b). Germane for us is the latter case: could it be that all variants of BM share a common coreand that we should only be realists about this common core (cf. Le Bihan & Read, 2018)? Different variants of BM 13 differ primarily over the (in principle undetectable) corpuscle trajectories. It's therefore natural to contemplate whether one can dispense with them altogether. This would in some sense attenuate the challenge of underdetermination: it would efface the postulated key differences between different versions of BM as illusory.
Our study of TBM explores the viability of such a "'common core'strategy" (Le Bihan & Read) in the case of BM: it would require, of course, that the "common core"-theory be both empirically adequate, and that it admit of a coherent interpretation. This we affirm. Thereby, TBM is shown to be a prima facie serious rival to BM vis-a-vis the latter's empirical underdetermination. 14 In fact, selective realisma cautious form of realism that has emerged from detailed analyses of historical challenges (see e.g. Kitcher, 1995;Psillos, 1999, esp. Ch. 5&6;Vickers, 2017Vickers, , 2018Vickers, , 2019)suggests that our realist commitment is only warranted towards the "working posits" of successful theories, i.e. those parts indispensable for their predictive and explanatory success. The remaining "idle posits" don't merit realist commitment. Ifas we maintain -TBM is empirically adequate and metaphysically coherent, it's conceivable that the GE counts as an "idle posit". In that case, following the selective realist's suggestion, realism about the GEand by implication, BM in its entiretywouldn't be justified.
We refrain, however, from any verdict as to whether TBM is indeed superior to BM. Only a detailed comparison of all of its explanatory successes (and, arguably, super-empirical virtues) can decide this. Here, we merely provide a proof of principle: to discard TBM in favour of BM calls for non-trivial arguments.
In sum: Irrespective of parsimony considerations that might commend the resulting theory, exploring whether one can still make sense of the formalism of BM without the GE promises to shed light on the latter's function, both physical and metaphysical.
With this prospective pay-off in mind, let's now plunge into the theory. To maintain TBM's adequacy, we'll rectify two misapprehensions about it that have impeded a wider consideration of Bell's proposal: one concerns Bell's metaphysically opaque presentation; the other concerns his worry that TBM exhibits a temporal form of solipsism, which he and others deem problematic. We'll address both points by firstly delineating a more perspicuous formulation, and secondly by demonstrating that TBM doesn't entail temporal solipsism. Bell's concern turns out to be unfounded. To these issues we turn now.

Basics and empirical adequacy
In this section, we'll unravel some of TBM's central conceptual structure by clarifying the status of its corpuscles' persistence. Notwithstanding their lack of persistence in TBM, we'll argue for TBM's empirical adequacy.
First, let's briefly dwell on persistence of corpuscles within BM. Here, the GE fulfils a metaphysical function: it ensures the corpuscles' persistence. With no GE, TBM's corpuscles are no longer guaranteed to persist. Should this faze us? The worry splits into two components.
(1) The first revolves around empirical coherence (or "epistemic stability"): without the corpuscles' persistence, does TBM undermine the reliability of its own empirical evidence (cf. Barrett, 1996)? To address this worry, we need some conceptual preparations.
(2) The lack of persistence prompts a second worry: is the GE necessary for empirical adequacy within BM? We contend that it isn't.
Persistence is closely tied up with the measurement problem. The latter encompasses in fact two distinct problems. The first concerns how to account for the determinacy (value-definiteness) of measurement outcomes. BM achieves this solely in virtue of determinate corpuscle positions (determined, of course, by the wavefunction). Persistence per se is irrelevant for this measurement problem. But it plays a role in a related, other measurement problemthe "Problem of Effect" (Maudlin): "The result of a measurement […] has predictive power for the future: after the first measurement is completed, we are in a position to know more about the outcome of the second than we could before the first measurement was made" (Maudlin, 1995a,b, p. 13).
In BM, the GE accounts for the Problem of Effect: it allows information of the measurement to propagate into the future. As measurements effectively (albeit not actually) induce a collapse of the wavefunction (e.g. Dürr & Teufel, 2009, p. 175;Lazarovici, 2019), repeated (sufficiently non-invasive) measurements yield the same outcomes. In our interactions with reality, the stability and temporal continuity appear to be robust empirical phenomena.
Removing the GE from standard BM disconnects the past from the present. This threatens TBM's empirical adequacy: TBM appears to flatout contradict the aforesaid stability and temporal continuity. If thus the past and the present are no longer connected, why are measurements recorded at different times and places mutually consistent? Shouldn't we rather expect, say, the datings of organisms in our phylogenetic past via 12 It deserves to be pointed out that there are other forms of simplicity, not necessarily compatible with mathematical simplicity (Bunge, 1963). Even if one believes in simplicity as a guide to truth, it's far from clear that mathematical simplicityhowever that term may be made precise or objective (in particular: objective in the sense of formalism/representation-independent!)is the relevant form of simplicity (cf. also Barrett 1999, p. 156 on different types of simplicity). 13 We'll not be concerned with Bohm's original quantum potential theory (championed also by Holland, 1993). Due to the latter's ontological differences, we deem it a theory distinct from the first-order theories under consideration. To these we'll refer as "variants of BM". 14 This isn't to deny, of course, the existence of other serious rival theories: they likewise constitute prima facie plausible reactions to BM's underdetermination. Goldstein's Identity-Based Bohmian Mechanics is a particularly interesting such alternative, due to its qualitative parsimony in the sense of Lewis (1973, pp. 87). In the present paper, we refrain from any further evaluation oflet alone arbitration betweenthose empirically equivalent theories along their superempirical virtues. fossils and molecular clocks, respectively, to diverge?
For TBM, the question is thrown into sharp relief. The actualisation of configurations is stochastically independent. That is, for measurable regions Q; Q 0 ⊆R 3N , one stipulates that the joint probability factorises: .That is: the actualisation of one configuration at some instant in time doesn't affect the probability for another configuration at some other instant in time. Accordingly, any link between generic actual configurations is cut. All memory of an antecedent configuration is erased in a jump. It seems absurdly improbable that both you and your spouse hold consistent memories of, say, your childrens' names. Prima facie, TBM appears to predict that our memories should tell of disparate pasts, more bizarre than surrealists' paintings or Borges' City of the Immortals. Doesn't this seal TBM's fate as hopelessly inadequate? To glean how that danger is warded off, we'll avail ourselves of three ingredients: first, to represent (for convenience) the N-corpuscle universe's total configuration in 3N-dimensional configuration space 15 ; secondly, to hone our understanding of empirical adequacy, borrowing Barbour's notion of 'time-capsules'; and thirdly, to hone our understanding of an empirically adequate theory's acceptability in terms of empirical coherence.
In order to defend TBM's acceptability, it will be advantageous to adopt the perspective of the N-particle system's total configuration. For readability, we'll henceforth refer to the N-particle system as a whole, comprised of all corpuscles in the universe as the totality of corpuscles (TOC). On TBM's interpretation of the QEH, we stipulate, it's the TOC that performs random jumps through configuration space. At any instant, it always occupies a definite configuration. By implication, the corpuscles' positions, too, are always definite.
Talk of such simultaneous jumps may sound a little mysterious: how do the corpuscles "coordinate" or "synchronise" their behaviour? Two responses are possibledepending on one's penchants for a metaphysically thin or thick interpretation of the wavefunction. (We'll revert to this topic in greater detail in x3.4.) Consider first a metaphysically thin reading, for instance, a Humean (e.g. Bhogal & Perry, 2017) or nomological stance towards the wavefunction, according to which a system's statistical behaviour is thoughteconomically encoded or summarised in the wavefunction. On a metaphysically thin interpretation of the wavefunction, that's all there is to say about it; all further metaphysical commitments are refrained from. Hence, the simultaneous jumps of the corpuscles aren't coordinated or synchronised in that they aren't explained or caused by anything deeper; the universe's N corpuscles positions at any time just are what they areand we can effectively describe them as a random walk tracing out the amplitude square of the N corpuscle universe's wavefunction (see Fig. 1).
A metaphysically thick perspective on the wavefunction yields a different response. From this perspective, a presupposition of the question becomes important: that the corpuscles are ontologically independent, and that therefore the behaviour of the system must ultimately be explained on the level of their individual properties, interactions and spatial distribution. Due to its peculiar nature as an essentially nonclassical system, the TOC and its constituent corpuscles exhibit an unfamiliar mutual dependence. On the one hand, the corpuscles are mereological parts of the former. This compositional dependence is familiar. Yet, there also exists a distinctively quantum, converse dependence: the TOC can't be ontologically reduced to the corpuscles. Rather, it's holistic in the following sense (cf. Esfeld, 1998Esfeld, , 2003: the corpuscles stand in a salient relationentanglement of the quantum statesthat doesn't supervene on their individual properties and spatial arrangements. It characterises the system as a quantum (rather than classical) system. Empirically, entanglement manifests itself in the correlations in virtue of which the Bell Inequalities are violated (see Bell, 2004, esp. Ch. 2,4, and 8;Howard, 1989Howard, , 1992Brown, 2005, Appendix B1-B3 for details).
In consequence, from the vantage point of a metaphysically thick stance towards the wavefunction, one shouldn't say that the corpuscles (instantaneously) influence each other, when jumping simultaneously. Such a causal account requires that the causal relata be distinct. That is, they mustn't stand in any ontological dependence relation, such as supervenience, grounding, etc. This isn't the case here: the TOC possesses a property that doesn't supervene on the properties of its constituents 16the quantum state, represented by their N-particle wavefunction; in virtue of it its constituent corpuscles lose one form of ontological independence (but not all, see e.g. Tahko & Lowe, 2015). Therefore, the corpuscles' simultaneous jumping is better understood as a fact of the holistic system they form (rather than a mysteriously synchronised action of independent corpuscles). 17 Irrespective of whether one adopts a metaphysically thin or thick interpretation of the wavefunction: the fundamentally random localisation process of TBM's TOC replaces the deterministic trajectories of BM's corpuscles. The TOC's jumps are stochastically independent. The QEHinterpreted now as a stochastic lawfurnishes the probability measure for this localisation process: the probability density for the universei.e. the TOC formed by all corpusclesto localise itself spontaneously at Q :¼ ðQ 1 ; …; Q N Þ 2 R 3N is P Ψt ðQÞ ¼ jΨ t ðQÞj 2 , with the universal wavefunction Ψ t . In other words: while we keep BM's QEH as a formal postulate, we no longer interpret it in terms of typicality. Instead, in TBM, it takes over the role of an irreducibly stochastic guidance law for the corpuscles.
(QEH) TBM The N-corpuscle universe (TOC) locates itself spontaneously in a measurable set of configurations Q⊆R 3N , with a probability given by R Q d 3N QjΨðQÞj 2 : 15 Albeit perhaps reminiscent of the move in Albert (1996) (anticipated by Bell, 1987, sect. D) we'd like to stress that we don't subscribe to any of Albert's further interpretative/metaphysical commitments, especially concerning his "marvellous point" or "super-particle". Our use of 3N-dimensional configuration space solely serves as a formal representation of the N-particle universe's total state. 16 A few advocates of BM may baulk at this, due to their allegiance to the socalled nomological interpretation of the wavefunction (see e.g. Dürr et al., 2013, Ch. 11-13). It takes an ontologically deflationist stance towards the status of the wavefunction: rather than representing a physical entity (what, following Maudlin, we called the "quantum state"), it merely codifies the corpuscles' dynamics. The wavefunction's status resembles more that of a Hamiltonian or Lagrangian in classical mechanics than that of, say, an electron. In that vein, the wavefunction should be primarily understood through its dynamical role for the corpuscles' motion. (An extreme form of this nomological interpretation is Esfeld's quantum Humeanism, to which we'll come further below.) If one thus construes the wavefunction nomologically, one will be inclined to reject the ontological holism and (partial) dependence of the corpuscles: our above reasoning for them crucially involved taking the corpuscles' quantum state ontologically serious. That consequence worries us little. First, we regard the nomological interpretation as (at best) an option for BM (or any other quantum theory)not a universally compelling stance towards the wavefunction (not even for BM). Furthermore, the nomological interpretation has incurred scathing criticism from various authors (e.g. Wallace & Brown, 2004;Ney, & Philipps, 2013;Sol e & Hoefer, 2019). We largely concur with their assessments. Accordingly, we remain skeptical of the nomological interpretation. But, of course, readers are free to adopt it: they can safely skip our elaborations of our proposed ontologically meaty interpretation of the wavefunction as a holistic disposition. 17 That is, with this holistic construal, the TOC view eschews the interaction (or "communication") problem of the "two-space reading" of BM (Ney, 2012, p. 535)the difficulty to conceptualise how the wavefunction (as an inhabitant of 3N-dimensional configuration space) can influence the corpuscles (as inhabitants of 3-dimensional "physical" space). Note that any "two space" dualism is straightforwardly avoided: there are no two distinct substances whose interaction needs to be accounted for: the TOC and the corpuscles aren't ontologically independent entities.
To complete our defence of TBM's empirical adequacy, we need a second ingredienta more precise notion of what we are after: what does empirical adequacy amount to? According to van Fraassen's standard definition, a theory is empirically adequate, if the observable phenomena can be embedded into it: "(T)he structures which can be described in experimental and measurement reports we can call appearances: The theory is empirically adequate, if it has some model such that all appearances are isomorphic to empirical substructures of that model" (van Fraassen, 1980, p. 64).
How does this translate in the case of TBM and its empirical adequacy? For the moment, we postpone the question of how van Fraassen's definition applies to TBM's probabilistic nature (and whether the given definition of empirical adequacy suffices to make TBM acceptable). We'll return to that particular problem shortly.
To gauge TBM's empirical adequacy (or rather: its non-probabilistic component), we must heed what according to Bell was the (in his eyes largely overlooked) "really novel element in the Everett theory" -"a repudiation of the concept of the 'past' which could be considered in the same liberating tradition as Einstein's repudiation of absolute simultaneity" (Bell, 1987, p. 118): "We have no access to the past. We have only our 'memories' and 'records'. But these memories and records are in fact present phenomena" (op.cit., p. 136). Elsewhere, Bell states even more clearly: "For we have no access to the past, but only to memories, and these memories are just part of the instantaneous configuration of the world" (Bell, 1976, p. 16, our emphasis). In other words: to save the appearancesi.e. to achieve empirical adequacyit suffices to be able to explain why corpuscle configurations are actualised that contain structures corresponding to the records (in particular of measurements, but more mundane "traces of the past", such as a diary, a scar or a glacier) and brain states involved in memory, abundant in our world.
Characteristic of such records and (non-pathological) memories is that they allow for consistent causal stories. In TBM, a causal link between two arbitrary configurations of the universe evidently can't be had. (In this regard, i.e. the absence of a causal link between configurations, it resembles the GRW flash theory; we'll revert to the differences between the two theories with respect to their ontology, what kind of primitive stuff they posit, in x3.4.). With the aforementioned "novel element in the Everett theory"the "repudiation of the concept of the 'past'"this needn't make us despond over TBM's empirical adequacy: it's preserved, if configurations of the universe permit of consistent quasi 18 -causal stories. For TBM to be empirically adequate, it must give rise to worlds like ours, i.e. with the following two features (cf. Barrett, 1996): (DEF) Measurement outcomesand properties of macro-objectsmust possess sharp (definite) values.
(REC) Records, including memories, must be mutually consistent.
(DEF) is equivalent to a solution to the first measurement problem mentioned in x3.2. With positions as its local beables, we already saw how TBM accomplishes this. What about (REC)? Is it satisfied? Barbour (1999, p. 31, Ch. 2, 3, 17-19;) evocatively dubbed configurations achieving thisi.e. configurations "that (appear) to contain mutually consistent records of processes that took place in a past in accordance with certain laws" -'time-capsules'. An example are configurations that instantiate our planet. Its multifarious strata, fossil records and rock formations, tell a mutually compatible, robustly regular story with uniform structural layers: "The story of antiquity of the Earth and of its creation from supernova debristhe stardust from which we believe we ourselves are madeis a story of patient inference built upon patient inference based upon marks and structures of rocks. On this rock the Earth in all its glorythe geologists have built the history of the world, the universe even" (op.cit., p. 33).
In short: the standard definition of empirical adequacy requires that in TBM amongst the actualised configurations there be time-capsules. But the cited standard definition of empirical adequacy can't be the whole story. It brackets TBM's probabilistic (statistical) nature. For probabilistic theories, we must supplement van Fraassen's definition (as van Fraassen (1980, Ch. 4) does himself): for a probabilistic theory to be acceptable, it must be also empirically coherent (or: "epistemically stable", cf. Myrvold, 2016;Carroll, 2017). That is, it must guarantee that we may trust our epistemically accessible empirical data. Else, it would undermine its own evidential basisthe empirical data on which belief in the theory's truth rests.
TBM is acceptable (i.e. empirically adequate and coherent), if highprobability time-capsules are TBM's empirical substructures into which the observable phenomena can be embedded. Is this requirement satisfied? Prima facie, prospects appear forlorn: time capsules form only a miniscule fraction of all configurations which the TOC could visit during its erratic wanderings. (Recall: with every random jump, the TOC always lands on exactly one, unique point in configuration space.) This answer, however, overlooks that one mustn't naively count configurations: they aren't equidistributed. The distribution afforded by the QEHconstrued in TBM as a probability measureinvolves the structurally rich wavefunction of the universe. In particular, it contains a mechanism for the universe to ferret out time-capsules (op.cit., Ch. 20)decoherence (see e.g. Bacciagaluppi, 2012;Schlosshauer, 2014;Wallace, 2011).
Entrenched in the wavefunction is the decoherence-induced branching structure. The details are familiar from the literature on the Everett interpretation; we needn't rehearse them here (e.g. Zeh, 2003Zeh, , 2000Wallace, 2002Wallace, , 2003Wallace, , 2010Wallace, , 2011Wallace, , 2012aWallace, , 2012b. Note, however, that nothing extraneous is borrowed: Everettian branching is no prerogative of the Everett interpretation. A direct consequence of decoherence, it's a generic feature of any theory respecting the standard quantum formalism as Everettians are keen to stress. Wavefunctions of subsystems of the universe, large enough to accommodate observers, have the functional structure of classical worlds -"for all practical purposes (FAPP)" (Bell). Corpuscles that occupy them realise FAPP-classical worlds. In particular, macro-scale interference effects are suppressed in them.
With its jumps, the TOC traces out the universal wavefunction. It thereby traces out the Everettian branching structure, too: with overwhelming probability the TOC lands on a point in configuration space, perched on a FAPP-classical world-branch. The latter is thereby actualised: a FAPP-world materialises. Each such world is FAPP-uniformly structured, admitting of FAPP-causally consistent histories. In other words: the TOC has a preponderant probability to actualise timecapsules.
This completes the vindication of TBM's empirical adequacy and coherence (contra Barrett, 1996, sect. 4): due to decoherence, the TOC has an overwhelmingly high probability to actualise FAPP-classical worlds. They allow for consistent histories governed by FAPP-causal laws. Historical evidence, including memories, contained in them is mutually FAPP-consistent.
Probabilities feature pivotally in this defence of TBM's acceptability. Its empirical coherenceour trust in the reliability of empirical datawas vouchsafed probabilistically: according to TBM, empirically adequate configurations with mutually consistent records are actualised with overwhelming probability. But how to understand this reference to probability? What is the status of probabilities in TBM? How to interpret them? We'll sketch this in the next section.

Probabilities in TBM
In x2, we reported that in BM identical copies of a system reproduce the same statisticsthose of the Born Rule. To better understand this result, we'll now state more carefully the QEH and its relation to the 18 "Pseudo-causal" would be equally apt, since the appearance of such causal stories isn't warranted at a fundamental level (see x3.5). Thanks to an anonymous referee for stressing this. statistics of subsystems (following Dürr, Goldstein, & Zanghì, 1992.) We'll then sketch possible interpretations of TBM's probabilities. Let's first refine our terminology. A subsystem of the universe (i.e. a proper subset of the corpuscles constituting the TOC) is said to have an "effective" wavefunction ψ, if the universal wavefunction Ψ : X Â Y → C, with X and Y denoting the configuration space of the subsystem and its environment, respectively, can be decomposed as 8ðx; yÞ 2 X Â Y : Ψðx; yÞ ¼ ψðxÞφðyÞ þ Ψ ? ðx; yÞ: Here φ and Ψ ? have macroscopically disjoint y-support and Y⊆ suppðφÞ. Subsystems with an effective wavefunction and negligible interaction with their environment can be shown to satisfy the SE for ψ.
Recall that the QEH supplies a probability measure P Ψ ðd 3N QÞ ¼ jΨðQÞj 2 d 3N Q on the configuration space R 3N . In contrast to BM, this isn't interpreted as a typicality measure in TBM, as we'll see presently.
First, though, let's see how this probability measure for the universe induces a probability measure for subsystems. Consider subsystems with the same wavefunction ψ. Let their number of corpuscles be M. The P Ψ -measure, conditional on all environmental configurations Y that yield the same effective wavefunction ψ, is then determined (independently of Y) as: From this, a Law of Large Numbers follows: For any measurable set A⊆R 3M and an ensemble of n identically prepared subsystems with the effective wavefunction ψ and the position (at time t) random variables X i ðtÞ, it holds that This brings us back to the notion of probabilities in TBM. 19 They constitute the core of TBM's counterpart of standard BM's QEH: in essence, it frees the latter from its interpretation in terms of typicality. But we still owe the reader a positive interpretation of those probabilities. With TBM aspiring to an objective account of reality, a fundamentally objective interpretation seems most natural. 20 Both 21 standard objective interpretations of probability are applicable, we submitpropensity and Humean Best System approaches. 22 On a propensity interpretation (e.g. Bunge, 2011, Ch. 4;Su arez 2007Su arez , 2009Su arez , 2013Su arez , 2014Su arez , 2016, probabilities in TBM quantify an inherent, physical tendency (propensity/disposition) of the universe to randomly materialise a certain configuration: The universe is irreducibly chancy. Such a propensity interpretation lends itself to TBM. As a fundamental theory of the whole universe, its probabilities don't depend on other factors. Absent extraneous triggering conditions, the TOC spontaneously jumps through configuration space. The universe's propensity manifests itself in the series of actual configurations. On TBM's propensity interpretation, the universal wavefunction represents this propensity.
On the other hand, a Humean Best System interpretation takes probabilities to be theoretical terms that meet the following criteria (e.g. Hoefer, 2011): (1) They satisfy the axioms of the probability calculus.
(2) They are suitably related to credences (i.e. rational agents' degrees of belief) such that the Principal Principle holds.
(3) They are invoked in the best systematisation of the empirical categorical facts of the universe.
The "best systematisation" here is understood as the one that strikes the best balance between simplicity, "fit" (empirical accuracy) and "strength" (empirical scope).
TBM's amplitude square of the universal wavefunction satisfies those criteria (1)-(3): the first one by construction; the second one due to its empirical adequacy (via its explanation of the Born Rule, as discussed above); and the third one due to its extraordinary simplicity. Hence, a Humean Best Systems approach to TBM's probabilities is both natural and attractive.
In conclusion: befitting a "quantum theory without observers", objective probabilities are possible in TBM. We'll not arbitrate amongst the options (including non-objective ones). Now the conceptual tolls are in place to launch into a distinctive metaphysical feature of TBMits many-worldliness. Before discussing it, it will prove useful to present TBM as a minimally Bohmian theory in the following section.

TBM as a minimally Bohmian theory
We christened the theory introduced so far "Tychistic Bohmian Mechanics". What justifies its classification as Bohmian? We'll now legitimise this claim. To that end, we'll offer a metaphysically precise formulation of the Primitive Ontology (PO) paradigm.
Our concern here isn't merely terminological. The section serves four main purposes. First, it provides a metaphysically perspicuous characterisation of the theory's ontology and ideologyin particular with respect to notions of fundamentality and ontological dependence. Secondly, our results poignantly pinpoint the regards in which TBM differs from the Everett interpretation and the GRW flash theory. Thirdly, by showing that TBM can be classified as a PO theory, we seek to by-pass a 19 At first glance, the status of probabilities resembles that of the Everett interpretation (e.g. Greaves, 2006;Saunders, 2021;Wallace, 2012b, Part II): according to both TBM and the Everett interpretation, it seems, all physically possible branches are realised. In the latter this indeed is strictly true: all branches are actualised simultaneously in a perfectly deterministic mannergiving rise to the so-called "Incoherence Problem" of Everettian probabilities. In TBM, the situation is slightly different in three regardsa difference that (depending on how deleterious one regards the Incoherence Problem) one may deem an advantage over the Everett interpretation. First, TBM is a fundamentally stochastic/indeterministic theory (to be elaborated in the main text). Secondly, fundamentally, in TBM only one branch is actualised at any given instance. (TBM's many-worldliness emerges at a temporally coarse-grained, non-fundamental level. We'll expand on this issue in greater detail in x3.5.) Thirdly, only at an effective levelfor all intents and purposesare all branches actualised in TBM within any finite time interval (TBM's "indefinite world-rate", as we'll call this phenomenon in x3.5): this doesn't occur with nomological necessityas it does (ex hypothesi) in the Everett interpretation; the actualisation of all branches within any finite time interval in TBM is merely overwhelmingly probable.We thank an anonymous referee for pressing us on this. 20 There exists a weaker notion of objective probabilities À "epistemic" probabilities (e.g. Uffink, 2011). Arguably, they are compatible with the agenda of a quantum theory without observer. For our purposes, it suffices to show that TBM allows of objective probabilities sensu stricto. 21 We pass over frequentism for two reasons. First, its defects are legion, rendering it not a particularly auspicious interpretation of probability to begin with (see e.g. H ajek, 1997, 2009 for extensive surveys). Secondly, due to TBM's many-worldliness, we expect a frequentist approach to TBM's probabilities to face similar challenges as in the case of the Everett interpretation (cf. Wallace, 2012, Ch. 4.5-4.7 divide in the foundations of quantum communitythat between wavefunction realists and PO (more below), with the latter insisting on a theory's Primitive Ontology status as a metaphysical conditio sine qua non. Fourthly, as a spin-off, the prerequisite terminological clarifications will in turn, we hope, help sharpen the core doctrine of that frameworkand thereby advance the debate. Let's ponder: is TBM sailing under false colours? Is it perhaps not a Bohmian theory? Prima facie, a glaring difference might suggest so. First, the spatiotemporal evolution of BM's corpuscles is deterministic. TBM's corpuscles, by contrast, make random jumps. Closely related is a second issue, related to persistence: whereas BM's corpuscles exist continuously (cf. Esfeld, 2017), TBM's corpuscles spontaneously (dis-)locate themselves.
Both differences we deem benign. Echoing Bohm, Dürr and Teufel declare determinism inessential for Bohmian theories (cf. also Oldofredi, 2020 for a recent, explicit case for the compatibility between stochasticity and "Bohmicity"): "It is often said that the aim of Bohmian Mechanics is to restore determinism in the quantum world. That is false. […] What is 'out there' could just as well be governed by stochastic laws […].
Following Dürr et al. (1995), sect. 5; see also Allori et al., 2006, sect. 6), a theory qualifies as Bohmian iff it's a theory about entities in 3-dimensional space that satisfies the following criteria: (1) The theory's objects are corpuscles (particles)rather than, say, flashes, strings or matter fields. To them, all ordinary mattercats and moleculesis reducible.
(2) At all times, the corpuscles have definite positionrather than, say, a definite fermion number density or bosonic field configurations. 23 Both intellectual fathers of BM, deBroglie and Bohm, expressly intended it as an interpretation of standard QM, not an alternative theory (cf., for instance, Bohm & Hiley, 1993, Ch. 1). This physical conservatism is reflected in Dürr et al.'s third requirement for a Bohmian theory: (3) It's fully empirically equivalent with QM. By contrast, objective collapse theories modify the Schr€ odinger dynamics. In principle, this yields empirical deviations from QM.
Only with respect to (1) might one have non-trivial queries. Compliance with (1) is ordinarily phrased in terms of the Primitive Ontology (PO) framework (see e.g. Allori et al., 2008;Allori, 2013bAllori, , 2015. Does TBM fall under it? Besides being a purely objectively interpreted "quantum theory without observer", this would amount to requiring that TBM be a theory fundamentally about entities located in 3-dimensional space that form the constituents of all macroscopic objects of our everyday experience. (PO theories more generally aren't limited to particle-theories: whatever fundamental entities -flashes, matter fields, etc.they posit, the only requirement is that they be located in 3-dimensional space.) It's a little obscure what exactly this (standard) formulation of a PO theory actually demands. The reference to fundamentality admits of multiple readings (cf. Oldofredi, 2021, for an illuminating perspective). Fundamentality is an ambiguous notion that may denote a number of distinct concepts (see Tahko, 2018). Most germane here are twofundamentality as a complete minimal basis, and as ontological independence, respectively.
A complete minimal basis captures the idea of a foundation of our physical reality: what the demiurge had to create in order to determine everything else that exists. That is, a complete minimal basis is the smallest possible set of entitiesnot necessarily restricted to objectsthat determines the rest of reality. An entity is then fundamental in this sense ("fundamental CMB "), iff it belongs to a complete minimal basis.
This terminology allows us to formulate the criteria for PO theories more precisely: (PO1) The theory's stuff and its properties (e.g. mass, position, etc., represented by the so-called "primitive variables") are fundamental CMB for a complete minimal basis for material reality. In particular, they constitute all macro-objects, such as ouds and axolotls.
(PO2) The various elements comprising the theory's stuff are also fundamental ODEP .
A comment on (PO1) is in order. It doesn't assert that what is represented by the primitive variablesstuff and its propertiesexhausts necessarily all of reality. Other entities might exist, such as numbers, universals or laws; but they'd be immaterial (see e.g. Allori, 2018, sect. 4;cf. also Allori, 2013acf. also Allori, , 2013bcf. also Allori, , 2015. This doesn't mean that only material entities are important. A complete theoretical description of the world must also account for the behaviour of immaterial entities. This is implemented via the so-called non-primitive variables, such as momenta in Classical Mechanics. Their role is to determine how matter interacts and moves. Advocates of the PO paradigm standardly take an ontologically deflationary stance towards such variables (cf. Chen, 2019, sect. 3, and esp. 4). (A routine option is a so-called "nomological interpretation".) They needn't deny their reality (only their material reality). But the role of non-primitive variables, those authors suggest, is primarily to be understood as incorporating the dynamics of primitive variables. 24 For a theory to qualify as a PO theory, no "thicker" realism towards them is mandatory (cf. Allori et al., 2008, sect. 4: "In [BM or the GRW theories], the only reason the wave function is of any interest at all is that it is relevant to the behaviour of the [Primitive Ontology]. Roughly speaking, the wave function [as a paradigmatic example of a non-primitive variable, our addition] tells the matter how to move".). We'll therefore take a theory to be a PO theory, iff the theory, when equipped with an ontologically thin (e.g. nomological) attitude towards the non-primitive 23 Other choices for (1) and (2) lead to merely Bohm-like generalisations/extensions of BM, whichnotwithstanding non-trivial differencescontinue to exhibit close similarities, such as family resemblance, certain senses of reduction in certain limits, etc. (cf. Passon, 2006, sect. 4). It lies outside of the present paper's scope to tackle the fascinating (and largely open) question regarding inter-theory relations that underlies such a distinction: what are criteria for classifying a theory that is sufficiently different from another (think of quantum field theories and (non-)relativistic quantum mechanics) as the latter's generalisation? BM and its Bohm-like generalisations would provide an ideal case study.
variables, satisfies the two above criteria (or their modifications, see below). 25 (PO2) also deserves elucidation. As we mentioned above, fundamentality ODEP makes reference to ontological dependence. Which form is relevant here? At first blush, one might be inclined to forbid all forms of dependence. But such absolute/unrestricted fundamentality ODEP strikes us as too austere for two reasons. First, it includes mereological independence (mereological simplicity). A PO theory's stuff then wouldn't be allowed to have any parts. This choice would throw out the baby with the bathwater: it rules out the GRWm theory with its "gunky" (i.e. infinitely divisible into smaller parts, cf. Arntzenius & Hawthorne, 2005; matter density. In the literature, however, the GRWm theory is unanimously classified as a PO theory. To relax the demanded sense of fundamentality ODEPby restricting it to only some forms of ontological dependenceseems apposite already on purely metaphysical grounds (Tahko, 2018, sect. 1.1).
The ontological independence we are after should be especially tight: a PO theory's stuff is supposed not to be "reducible to more elementary notions" (Oldofredi & Esfeld, 2018, p. 11). Elsewhere, we argue that the pertinent sense of reducibility is best understood in terms of identity (in the sense in which the physicalist envisages mental states to be identical to physical/brain states), grounding or functional role (in the sense in which, according to functionalism, to be in a mental state is conceptualised as possessing a brain state that plays the functional role of that mental state). We hence reformulate (PO2) as follows: (PO2*) The theory's stuff mustn't be ontologically reducible to entities that are immaterial or not located in 3-dimensional space, in the sense of functional reduction, grounding or (type or token) identity.
Two advantages commend this refined characterisationi.e. the conjunction of (PO1) and (PO2*)of the Primitive Ontologist's core tenet. First, it recovers the classification of the paradigm examples of Primitive Ontological theories in the literature (viz. classical mechanics, classical electromagnetism, standard BM, GRWf, and GRWm), as well as of the paradigm counterexamples (in particular, the Everett interpretation, as we'll see in detail presently). Secondly, it captures what we take to be the intuition underlying the pertinent authors' insistence on irreducibility: in theories not satisfying (PO2*), what appears to be spatiotemporally located stuff, really is something else, e.g. patterns in the wavefunction. This mimics the physicalist's hunch that mental states really are brain states. 26 How now does TBM fare vis-a-vis those two criteria for PO theories? That TBM satisfies (PO1) is straightforward. On TBM, the 3-dimensional corpuscles clearly belong to the complete minimal basis. If now we adopt an ontologically thinsay, either a Humeanistic (recall x3.3) or nomologicalstance towards the wavefunction (in complete analogy to BM), the complete minimal basis for material reality solely consists of the corpuscles and their respective positions. 27 They are fundamental CMB . Compliance with (PO2*), too, is immediate to see: just as in standard BM, TBM's corpuscles aren't really anything else, not residing in spacetime. TBM thus satisfies both criteria for a PO theory.
As a particle-based PO theory, fully equivalent with standard QM, TBM consequently is a card-carrying member of the family of Bohmian theories (cf. Barrett, 1996;1999, Ch. 5.1 for a similar classification of "Bell's Everett (?) theory"). Fine (1996, p. 249) encourages a bolder conclusion: "At the heart of Bohmian mechanics is the wave function and determinate particle positions, and perhaps we need be realists about nothing else." Indeed, TBM only needs two postulatesthe Schr€ odinger Equation (SE) and the (re-interpreted) Quantum Equilibrium Hypothesis (QEH TBM )whereas standard BM needs three. 28 (This claim is, of course, predicated on the premise that Valentini's agenda to dispense with the QEH as an independent assumption in standard BM isn't successful, see fn. 10.) We therefore don't quite agree with Goldstein (2017, sect. 4) who touts standard BM as "[…] the minimal completion of Schr€ odinger's equation, for a nonrelativistic system of particles, to a theory describing a genuine motion of particles".
Our classification of TBM as a (minimally) Bohmian theory contradicts Bell's own classification. 29 Originally, he had proposed something like TBM as a (the most?) plausible version of Everett's Many Worlds Interpretation of QM (see Bell 1987Bell , 1976. 30 Before turning to the differences between TBM and the contemporary Everett interpretation, let's dwell on the differences between Bell's theory and TBM. Indeed, at first blush the two look indiscernibly similar. A crucial detail is easy to overlook: the way worlds are defined in each. We simply imported the notion from the Everettian literature. Worlds or quasi-classical branches, then, are coarse-grained notions. Bell explicitly rejects such notions (see Bell 1987, (B) and (D)), since they are "meaningful […] only on some ill-defined macroscopic level"a level of description recourse to which, Bell prescribes, should be banned, when defining a (fundamental) theory's fundamental concepts. Therefore, he diagnoses temporal solipsism: having discarded the Everettian branching-structure, Bell is left with isolated, individual configurations, actualised at one given point in timewith no link to prior configurations. (Such a link would come about by either an extremely improbable random coincidence or by the Everettian branching structure. But since Bell rejects the latter in the context of interpreting the theory's fundamental postulates, effectively all ties to prior configurations are cut.) In consequence, he doesn't distinguish between worlds and configurations. (Although he isn't explicit about this, it becomes clear in statements such as "Thus instantaneous classical configurations x are supposed to exist, and to be distributed in the comparison class of possible worlds with probability |ψ| 2 " (p. 133).
While it's true that at the level of the theories' fundamental postulates, TBM and Bell's Everett (?) theory are the same, they differ in their stance towards worlds: we (following the Everettians) embrace them as real (albeit emergent/non-fundamental) structures; Bell jettisons them wholesale. Doesn't this sameness with respect to 25 With this slightly cumbersome qualification we wish to make our characterisation of PO theories interpretatively flexible: it shouldn't be wedded ab initio to one particular interpretation of, say, the wavefunction (as a paradigmatic candidate for a non-primitive variable). Otherin particular nonnomological or even ontologically thickinterpretations of the wavefunction (such as a multi-field view, see Hubert & Romano, 2018) should be compatible with a theory's status as a PO theory. 26 It's grist to our mills that Esfeld (2019b) explicitly seems to have this sense of fundamentality in mind (albeit in a slightly different context), cf. Lam & Wüthrich, 2018. 27 To preempt misunderstandings, we hasten to add that nothing compels us to adopt such an interpretation; it's merely an option. That's all we need to establish TBM's status as a PO theory. As we argued in x3.2 and x3.3, an ontologically thick interpretation of the wavefunction in terms of a disposition is equally viable (and attractive on independent groundsviz. as a solution to the twospace reading, cf. fn. 17.). 28 Nelson Stochastics might come to mind as another candidate for a minimally Bohmian theory (see e.g. Bacciagaluppi, 2005). It dispenses with the SE by assuming that the particles' position is described by a (stochastic) diffusion process. (A special choice for the diffusion coefficient yields BM.) However, Nelson Stochastics doesn't fully recover the equivalence with QM. To restore that, additional constraints need to be imposed, casting into doubt the claim of its being minimally Bohmian.Two other candidate minimally Bohmian theories are Sebens' "Newtonian QM" (2015) and Goldstein et al.'s (2005 a,b) "Identity-Based Bohmian Mechanics". A comparison with TBM would be particularly interesting especially with respect to explicating theprima facie differentsenses in which they can lay claim to being simple or parsimonious. Unfortunately, this lies outside of the present paper's ambit. 29 To be fair, though, Bell (1987, p. 133) introduces his theory as "simply […] the pilot wave theory [i.e. BM] without trajectories". 30 By contrast, Daumer et al. (1996), p. 393; fn 13 distinguish between the Everett interpretation and Bell's Everett (?) Theory; ditto Allori et al. (2008), sect.6. P.M. Duerr, A. Ehmann Studies in History and Philosophy of Science 90 (2021) 168-183 fundamental postulates imply that TBM and the "Everett (?) theory" are identical? Not necessarily: it's not clear that theory individuation/ inequivalencewhether we should regard two theories as distinct, rather than merely notational variants of each othershould be based solely upon the theory's fundamental postulates. Regrettably, there is no consensus in the literature on sufficient criteria for theory individuation. Observational/empirical equivalence is a little controversial necessary criterion for two theories to count as equivalent. Butwhat counts as the empirical substructure of a theory is a delicate business. It's here that non-fundamental/coarse-grained concepts might become important: they, after all, delimit the observationally distinguishable from the observationally indistinguishable. (This is particularly clear in van Fraassen's account of empirical adequacy, and his overarching semantic view of theories.) To us, caution seems prudent; we therefore opt for the (tentative) distinctness of TBM and Everett's theory.
Return now to what, following Bell, may look like a natural identification -TBM and the (contemporary) Everett interpretation. This identification can be opposed for resting on a spurious identification of what constitutes the supposed essence of the Everett interpretation. As we saw in x3.2 (and will examine more closely in x3.5), TBM's ontology is many-worldly: it contains many worlds. It's therefore tempting to consider TBM a variant of the Everett interpretation. Butalbeit absent in prime specimens of Primitive Ontology theories, such as BM or the GRW flash theoryalso some Primitive Ontology theories display manyworldliness, e.g. the GRW matter theory (e.g. Allori et al., 2008) or Schr€ odinger's many-world theory (see e.g. Allori et al., 2011). 31 The absence of many-worldliness thus can only be a contingent feature of a PO theory. (NB: One should strictly distinguish between many-worldliness and Everettianity, i.e. the classification of a theory as Everettian. Many-worldliness denotes an ontological feature of a given, interpreted theory: the presence/absence of multiple, synchronous existing worlds amongst the theory's (not necessarily fundamental) ontology. Everettianity, by contradistinction, denotes a particular interpretative scheme for quantum theories, based on the interpretative principles paradigmatically invoked in the Everett interpretation of QM, as canonised by Wallace (2012b;. 3233 ).
For Bell (1987, p. 133), "keeping the instantaneous configurations, but discarding the trajectory, is the essential […] of the theory of Everett". Most contemporary Everettians, however, will gainsay this as a faithful reconstruction of their views (cf., for instance, Wallace, 2012b;Vaidman, 2014).
For one, they refuse to postulate any corpuscles (or any other stuff) over and above the wavefunction. 34 In this spirit, Wallace (2012b), p. 38, for instance, writes with respect to the standard quantum formalism: "The 'Everett interpretation of quantum mechanics' is just quantum mechanics itself, 'interpreted' the same way we have always interpreted scientific theories in the past: as modelling the world." In other words, the Everett interpretation violates (PO1): only the wavefunction is fundamental CMB for our material reality; 3-dimensional particles aren't. Corpuscles À or three-dimensional objects more generallyonly exist as structural patterns in the wavefunction. In this sense, the Everett interpretation also violates (PO2*): all three-dimensional objects are identical with structural patterns in the wavefunction. That is, they merely have counterparts, playing roughly the same functional role at some non-fundamental, coarse-grained, effective description (see Wallace's explicit invocation of structuralism and Dennettian functionalism in Wallace, 2012b, Ch. 2).
Contemporary Everettians will disown what Bell takes to be the essence of the Everett interpretationthe commitment to instantaneous configurations: Everettians explicitly forgo a preferred decomposition into orthogonal states. Its detractors perceive this as a flawthe so-called "problem of a preferred basis". Conversely, Pauli and Heisenberg, for instance, rebuke BM for foisting an "artificial asymmetry" on position and momentum (see e.g. Myrvold, 2003, sect. 3). In QM simpliciterwith the Everett interpretation as a conceivable interpretationboth are on a par. In other words: even critics of the Everett interpretation controvert Bell's identification! Let's also push back against another misidentification of TBM's ontology: according to Esfeld (private communication), it's identical to the GRW flash ontology. The latter's "flashes" (or "hits") are elementary events, the centres of the spontaneous collapse of the wavefunction. Thus definedabsent a collapse of the wavefunction in TBMthe "flashes" or "hits" have no direct counterpart in TBM. Nonetheless, , p. 173) descries a salient ontological analogy between GRWf and TBM in the following: "The GRWf ontology of single, discrete events can be considered as a particle ontology without the trajectories so that what remains of the particles are isolated events in space-time." Underlying Esfeld's verdict is his "Quantum Humeanism" or "Humeanism without intrinsic properties" (see e.g. Esfeld, 2014a;Esfeld et al., 2017). According to this proposal, fundamentally only primitive stuff exists without any further specified qualitiesa kind of otherwise featureless prima materia/ὑλη πρωτη, the different chunks of which are individuated only via spatial (or spatiotemporal) relations. These chunks or matter points lack any intrinsic properties. Mass, charge, etc. are merely formal parameters introduced in order to account for the occupants' spatiotemporal evolution. On this view, then, non-permanent corpuscles and flashes coincide. But Esfeld's Quantum Humeanism will strike many as inordinately radical (for a critique, see Wilson, 2018). 35 To say the least, it's not the only option for an interpretative framework.
For those who don't subscribe to Esfeld's Quantum Humeanism, discriminating between TBM's and the GRW flash ontology is straightforward. Events aren't things: our weddings and our wives fall (for better or worse) into different metaphysical categories or kinds (cf. Casati & Varzi, 2014, esp. sect.1.1). Furthermore, we don't take it to be essential for Bohmian corpuscles to possess continuous trajectories: it's possible for 31 In fact, TBM is the Bohmianthat is, corpuscle-basedcousin of Schr€ odinger's many-world theory: the latter's referent is a primitive matter density field. 32 Cf. also Wallace, 2020, p. 85: "what should we expect from an 'interpretation' of quantum mechanics? Here is one natural answer: we should expect an interpretative recipe, a set of instructions which tells us, for any given quantum theory, how to understand that theory." 33 Although much work remains to be done in order to flesh out those principles (cf. Conroy, 2016), salient features (exemplified at least to some extent) arguably include: (1) the absence of additional postulates that go beyond the theory's working posits (e.g. a preferred basis or the introduction of an ad-hoc collapse mechanism); (2) a "literal" realist interpretation of the formalism; (3) quantum probabilities as primitive branch weights, rather than dispositions/propensities; (4) a "wavefunction pattern ontology": the wavefunction's fundamentality CMB and, concomitantly, a structuralist/functionalist approach to higher-level/emergent entities. 34 One of the essential points of contention between "wavefunction ontologists"such as Everettiansand Primitive Ontologists lies in their respective attitudes towards functionalism (cf. Wallace, 2008, x2.6.7).The former rest content with the existence of an entityviz. the quantum statethat possesses, at some effective, higher level of description, the approximate dynamical structure (functional description) of semi-classical objects. (cf. Brown, 2009;Brown & Wallace, 2004). By contrast, Primitive Ontologists demand that one must posit independent realisers of this dynamical structurestuff, over and above the wavefunction: This stuff instantiates the functional roles of FAPP-classical objects. Furthermore, Primitive Ontologists insist that those realisers live in ordinary 3-dimensional space (cf. Maudlin, 2010Maudlin, , 2012Maudlin, , 2013.Underlying these conflicting ontological doctrines seems to be a semantic thesis concerning how abstract, formal terms are imbued with a physical interpretation (see Dewar, 2017, p. 7). 35 One may even go a step farther and reprimand the GRW flashesand Esfeld's proposal with themas metaphysical monstrosities (cf. Myrvold, 2017, esp. 6.3): denuded of any intrinsic properties, don't they suspiciously resemble Lockean bare substrata? Bohmian corpuscles to randomly relocate. To our minds, this merely reflects the corpuscles' non-classical nature (cf. Falkenburg, 2007, Ch. 6). 36 To sum up: we argued that TBM is a bona fide minimally Bohmian theory. That is, it's a position-based corpuscle theory with a Primitive Ontology (in the most plausible construal of that framework) and empirically equivalent to standard QM. We rejected both Bell's classification of (his variant of) TBM as Everettian, as well as Esfeld's identification of TBM's ontology with the GRW flash ontology.
Bell had already contemplated something like TBM. Why did he reject it? For an answer, we must scrutinise TBM's many-worlds character.
3.5. Temporal solipsism? -Quasi-persistent many-worlds! This section will rebut Bell's criticism that TBM is temporally solipsistic: according to Bell, it's overwhelmingly probable that our world only exists for one single infinitesimal moment. 37 This diagnosis is specious, however: in any interval of time, our world pops into existence infinitely many timesas does every other quasi-classical world.
As mentioned above, Bell adumbrated TBM (or something very similar, with the key difference consisting in the definition of worlds, see our preceding commentfor ease of readability, we'll here use TBM and Bell's "Everett (?) theory" interchangeably). He correctly realizes that in TBM configurations of the universe at two arbitrary instants are no longer causally connected: "Thus in our interpretation of the Everett theory [read: TBM], there is no association of the particular present with any particular past" (Bell, 1987, p. 135). Despite conceding TBM's empirical adequacy, Bell (op.cit.,p. 136) deems this consequence fatal: "Everett's [read: TBM's] replacement of the past by memories is a radical solipsism extending to the temporal dimension the replacement of everything outside my head by my impressions, of ordinary solipsism or positivism. Solipsism cannot be refuted. But if such a theory were taken seriously, it would hardly be possible to take anything else seriously." If the universe randomly hops through the unfathomable vastness of configuration space, at first blush it appears astronomically unlikely that within our lifetime our world will ever pop into existence again. One might hence believe that, according to TBM, we only endure for this one single moment. Our hopes for any moment beyond the present would then degenerate into illusions. Maudlin (2016, p. 324) poignantly writes: "The Everett (?) theory [read: TBM] is a sort of physical blueprint for a Cartesian demon, but one where the subject is deceived about the past (and even her own past)." What makes this (alleged) feature pernicious? Two reasons are hinted at: first, it's supposed to undermine the rationality of our belief in TBM; secondly and more generally, temporal solipsism is supposed to be incoherent.
Vis-a-vis the puny probability that our world existed a minute ago and will continue to do so for the next one, Bell demurs: how still to trust our memories or anticipations of the future? Bell denies that we could. Our quotidian practices, his argument goes, presuppose the reliability of memories and future anticipations. Therefore solipsism, temporal and metaphysical, despite being irrefutable in principle, is pragmatically unviable (in Kantian terminology: it violates a regulative principle): "It is always interesting to find that solipsists, when they have children, have life insurance" (Bell, 1987, p. 136).
More generally, Bell intimates that TBM's temporal solipsism, on the one hand, and metaphysical solipsism on the other share the same problems. Arguably foremost amongst them is that of incoherence (see e.g. Thornton, 2004, sect. 4). In the case of metaphysical solipsism, in so far as its articulated as a rational thesis, its advocate must resort to language and logic. But the solipsist's arguments forfeit their intersubjective force (and plausibly even their comprehensibility): after all, she disputes that anything exists outside of her own mind. This vitiates all her arguments ab initio.
The analogue of the incoherence argument in the case of temporal solipsism generalises the concern about practical reliability of memories and future anticipations: if a theory entails that all the relevant evidence in its favour is deceptive, on what grounds can we still rationally believe it? Bell (1987, p. 136) likens the situation to a Young Earth Creationists' response to contradictory empirical evidence: "The theory was that of the creation of the world in 4004 BC.
[…] The trees would be created with annular rings, although the corresponding number of years had not elapsed.
[…] The rocks would be typical rocks, some occurring in strata and bearing fossilsof creatures that had never lived." In other words: According to Bell, TBM resembles Young Earth Creationism in that it invalidates its own empirical evidence. Both are therefore incoherent.
Fortunately, TBM can be salvaged. Bell's concerns about temporal solipsism don't carry over: TBM isn't temporally solipsistic. What Bell and other commentators have overlookedarguably, due to neglecting that worlds are a coarse-grained concept, see the above difference between Bell's Everett (?) theory and TBMis a peculiarity of TBM's many-worlds character. It exhibits a "stochastically successive many-worldliness": within any finite time span, our world is actualised for (uncountably) infinitely many instants. This ensues from the following considerations.
Let ðW ; t 0 Þ denote the event that the FAPP-classical macro-world W (recall 3.2) is actualised at t ¼ t 0 .
The probability for ðW ; t 0 Þ is: Macro-worlds are only FAPP-defined. They are coarse-grained concepts. This translates into them occupying finite "volume" in configuration space. Albeit tiny, the probability for W 's actualisation is therefore always non-zero: 8t 2 R : 0 < P Ψ ½ðW ; tÞ ≪ 1: Recall that probability distributions needn't have a well-defined expectation value. (The canonical example is the Cauchy distribution: it has neither finite expectation value nor variance.) Thereby, the formulation of laws of large numbers is blocked. This is the case here, too, as follows from the following simple argument.
Consider the compact time interval I⊂R. Defined as a function on it, P Ψ ½ðW ; :Þ : I → 0; 1 is continuous. It follows that its minimum is nonzero: P min ðW ; IÞ : ¼ inf t2I fP Ψ ½ðW ; tÞg > 0: Now choose from I randomly N points t 1 ; …; t N 2 I: These are the (discrete) blinks of His eyes God is willing to devote to the universe "below".
TBM's TOC is memoryless (in the mathematical-technical sense): its positions during those moments are stochastically independent. Hence, the expectation value for W popping into existence at least one moment out of the t 1 ; …; t N is: E Ψ ½W ; t 1 ; …; t N : ¼ X i P Ψ ½ðW ; t i Þ: 36 We thank Michael Esfeld (Lausanne) for an illuminating discussion. 37 We don't take Bell's main objection to be the inaccuracy of our memories and records of the past: that they don't track the complete, real history as disclosed, as it were, from the God's eye view. (We'll address that criticism later on in the main text, too.) Our reading of Bell is, we believe, buttressed by his repeated emphasis (p.133, p. 135, p. 136) on the fact that our only access to other times is via present data, and that the inherent link to those times is cutto the effect (we take him to conclude) that we are epistemically trapped in the moment.
It's bounded from below: E Ψ ½W ; t 1 ; …; t N ! X i inf t2I fP Ψ ½ðW ; tÞg ¼ NP min ðW ; IÞ > 0: How often should God expect then to behold the world W during I? It depends on how frequently He's willing to peep. If His attention is unlimited, so is the expectation value for the actualisation of W : E Ψ ½W ; t 1 ; …; t N ! N→∞ ∞: In short: on TBM, our world ceases to exist continuously. Nonetheless, in any finite interval, it pops into existence infinitely many infinitesimal instants. This holds for all macro-worlds (Everettian branches): they are actualised infinitely many times within each secondsuccessively. In the Everett interpretation, all worlds coexist simultaneously. By contradistinction, in TBM each world is realised only one at a time, randomly "selected". The Everettian multiverse picture thus resembles an illattuned TV displaying several channels at once. TBM's manyworldliness resembles that of a TV randomly and rapidly switching between different channels. 38 That is, the temporal order in which TBM's worlds are actualised is random. (Recall from x3.2 that the TOC's jumps through configuration space are stochastically independent.) TBM's stochastically successive many-worldliness extricates the theory from temporal solipsism: our existence isn't restricted to one single moment. We continue to existalbeit not continuously. Our memories and future anticipations are reliable: they permit inferences to our macroworld's history; within each atto-second, infinitely many temporal fragments of it are realised. On TBM, historical records aren't illusory or deceptive: they correctly describe a branch of reality (quite literally: an Everettian branchthat is, a quasi-classical history). This isn't to say that the FAPP-classical reconstructions of our world's history from those records are fundamentally correct. They are merely higher-level/emergent descriptions. 39 And most importantly, of course: there isn't just one such quasi-classical world, but staggeringly many. The fundamental picture is that of a stochastically successive multi-verse.
Quantum phenomenainterference experiments in particularevince structure beyond each FAPP-classical world. Being empirically adequate, TBM is capable of accommodating those quantum phenomena: On TBM, they grant us glimpses into TBM's stochastically successive many-worlds.
In sum: we defended TBM against Bell's accusation of a form of solipsism that renders it incoherent. TBM is a coherent many-worlds theory: Decohered macro-worlds successively pop into existence for an instant; within any arbitrary time interval each world exists infinitely many times, with the exact temporal sequence being random. No longer persisting continuously, we still exist "densely" in time.
The panels in Fig. 1 visualise this result. The first shows the concentration of the probability density induced by the universal wavefunction (yellow). For simplicity, it's assumed to branch only into two FAPPworlds. The other three panels show the actualised configurations of the universe (red dots) from a God's eye view. The increasing number of configurations in each picture is supposed to convey a feeling for the limit process that leads to TBM's indefinite world rate. The frequency with which snapshots are taken (or: "God blinks") increases from left to right.

Summary
We started with a review of standard Bohmian Mechanics (BM). We didn't question its viability and merits. Still, we proposed, it's worthwhile inquiring into the consequences of removing its Guidance Equation: first, this deepens our understanding of the role of the Guidance Equation in BM, and secondly, probing its (formal) eliminability is a prima facie natural response to its empirical underdeterminationto be sure: not necessarily the only option.
We showed that, based on the Quantum Equilibrium Hypothesis and the Schr€ odinger Equation alone (with an appropriate re-interpretation of the former), one can articulate an empirically adequate and metaphysically coherent theory -Tychistic Bohmian Mechanics (TBM). Pace Bell, we classified it as a Bohmian theory: 1. Its referents are (massive, charged, etc.) corpuscles (particles) in ordinary, three-dimensional space. 2. At all times, they have determinate positions. All other dynamical variables are contextual. 3. It's fully equivalent with QM. 4. All macroobjects of everyday experience are composed of TBM's corpuscles.
While ontologically deflationary interpretations of the wavefunction (such as the nomological one) are certainly possible, we showed that in TBM an ontologically thicker interpretation is both viable and naturalone in terms of an irreducibly dispositional, holistic property of the whole N-corpuscle system.
According to TBM, the N-corpuscle system that constitutes our universethe TOCperforms irreducibly random jumps through configuration space. The TOC's (objective) localisation probability density is given by the modulo square of the wavefunction of the universe. Due to decoherence, it's concentrated over configurations in FAPP-classical Everettian branches (worlds). During any finite time interval, each world is actualised (almost certainly) uncountably infinitely many times in random order. Their ratios (almost certainly) match their probability distribution. This guarantees TBM's empirical adequacy and coherence. Although the worldsincluding macro-objects such as ourselvescease to exist continuously at a fundamental level, in a temporally coarse-grained sense they persist. At this higher level of description, all FAPP-classical worlds co-exist. At TBM's fundamental level of description, each world pops into existence only one at a time.
If TBM is a viable minimally Bohmian theoryminimally Bohmian in the sense that it satisfies the minimal desiderata for a Bohmian theoryone would like to know: how does it fare vis-a-vis BM and other Primitive Ontology theories? This question must be taken up elsewhere, e.g. along Esfeld's (2014b) guidelines for evaluating Primitive Ontological quantum theories.