Probabilities	in	deBroglie-Bohm	Theory: Towards	a	Stochastic	Alternative (Version	0.1	beta) **	UNDER	REVIEW	–	PLEASE	DON'T	CITE	** Patrick	M.	Duerr,	Oriel	College,	University	of	Oxford,	UK,	patrick.duerr@oriel.ox.ac.uk Alexander	Ehmann,	Lingnan	University,	HK,	alexanderehmann@alexanderehmann.com 20.	January	2017 "Once	we	have	discarded	our	rooted	predilection	for	absolute	Causality,	we	shall	succeed	in overcoming	the	difficulties." E.	Schrödinger:	What	is	a	Law	of	Nature?	Science	Theory	and	Man Abstract: We	critically	examine	the	role	and	status	probabilities,	as	they	enter	via	the	Quantum	Equilibrium	Hypothesis, play in the standard, deterministic interpretation of deBroglie's and Bohm's Pilot Wave Theory (dBBT), by considering interpretations of probabilities in terms of ignorance, typicality and Humean Best Systems, respectively.	We	argue	that	there	is	an	inherent	conflict	between	dBBT	and	probabilities,	thus	construed. The	conflict	originates	in	dBBT's	deterministic	nature,	rooted	in	the	Guidance	Equation.	Inquiring	into	the	latter's role within dBBT, we find it explanatorily redundant (in particular for dBBT's solution of the	Measurement Problem, which only requires that the corpuscles possess definite positions), and subject to a number of difficulties.	Following	a	suggestion	from	Bell,	we	propose	to	abandon	the	Guidance	Equation,	whilst	retaining dBBT's	point	particle-based	Primitive	Ontology,	with	positions	as	local	beables.	The	resultant	theory,	which	we identify	as	a	stochastic,	minimally	deBroglie-Bohmian	theory,	describes	a	random	walk	through	configuration space.	Its	probabilities,	we	propose,	are	best	understood	as	dispositions	of	possible	corpuscle	configurations	to manifest	themselves.	We	subsequently	evaluate	the	merits	of	sdBBT	vis-à-vis	dBBT,	such	as	the	justification	of the	Symmetrisation	Postulate	and	the	violation	of	the	Action-Reaction	Principle. Not	only is sdBBT	an attractive	Bohmian theory that,	whilst retaining	dBBT's virtues, overcomes	many	of its shortcomings;	it	also	sparks	off	a	number	of	exciting	follow-up	questions,	such	as	a	comparison	between	sdBBT 2 and other stochastic hidden-variable theories, e.g. Nelson Stochastics, or between sdBBT and the Everett interpretation. Keywords: Bohmian Mechanics; Probabilities; Typicality; Humeanism; Bell's Formulation of the Everett Interpretation;	Primitive	Ontology Content I. Introduction	...................................................................................................................	3 II.	dBBT	and	its	discontents	................................................................................................	4 II.1.	Standard	dBBT	......................................................................................................................	4 II.2	Critical	analysis	of	dBBT's	subjectivist	probabilities	...............................................................	5 II.3.	Two	culs-de-sac	to	objective	probabilities	for	dBBT	..............................................................	9 II.3.1.	Typicality	...............................................................................................................................	9 II.3.2.	Humean	Best	Systems	........................................................................................................	14 II.3.3.	Digression:	Objectivism	and	Heisenberg	relations	.............................................................	20 III.	Stochastic	deBroglie-Bohm	Theory	(sdBBT)	.................................................................	22 III.1.	Role	of	the	Guidance	Equation	..........................................................................................	22 III.	2.	Probabilities	in	sdBBT	.......................................................................................................	28 IV:	Critical	Analysis	of	sdBBT	.............................................................................................	40 IV.1:	sdBBT	as	a	minimally	deBroglie-Bohmian	theory	...............................................................	41 IV.2:	sdBBT	and	realism	.............................................................................................................	43 IV.3:	"Temporal	Solipsism"	(Bell)	...............................................................................................	45 IV.3.1	sdBBT's	many	worlds	..........................................................................................................	45 IV.3.2	Bohm	Brains	vs.	Boltzmann	Brains	.....................................................................................	51 IV.4:	sdBBT	as	a	phenomenalist	theory?	....................................................................................	54 IV.5:	sdBBT	vs.	dBBT	..................................................................................................................	55 IV.5.1:	Metaphysical	quarrels	with	wavefunctions	in	dBBT	.........................................................	55 IV.5.2:	Justification	of	the	Symmetrisation	Postulate	...................................................................	57 IV.5.3:	Quantum	tunnelling	..........................................................................................................	59 IV.5.4:	sdBBT,	dBBT	and	relativistic	QM	.....................................................................................	61 V.	Summary	and	outlook	..................................................................................................	69 Bibliography	.....................................................................................................................	71 3 I. Introduction Vis-à-vis	the	hassle	in	the	foundations	of	quantum	mechanics	(QM),	esp.	the	measurement problem,	the	interpretation	of	the	Heisenberg	relations	and	their	joint	culmination	in	the	EPR "paradox"	(1935),	the	question	arises	whether	QM	in	its	current	form	is	incomplete:	Might there exist an element of physical reality that has no counterpart in QM?1 Einstein, 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."2 deBroglie-Bohm	Theory (dBBT) is an	attempt to complete	QM: It proffers an	account	of a deterministic	dynamics	that	describes	a	sub-quantum	particle	world,	from	which	QM	emerges –	as	Einstein	had	hoped for	– in a	manner "approximately analogous [...] to the statistical mechanics	within	the	framework	of	classical	mechanics."3 This	reference	to	statistical	mechanics	(SM)	prompts	three	questions. 1. What	does	the	asserted	analogy	between	dBBT	and	SM	consist	in? 2. To	what	extent	is	it	justified? 3. What precisely does the "statistical character" of QM consist in from the dBBT perspective?	What	is	the	role	and	status	of	probabilities	within	dBBT? Our	subsequent	pursuit	of	these	questions	will	put	its	finger	to	what	we'll	argue	to	be	dBBT's biggest shortcoming, namely the joint incompatibility of its deterministic dynamics, its probabilistic	Quantum Equilibrium	Hypothesis and its aspiration to a thoroughly objective "quantum theory	without	observer."	As a	natural and conservative resolution, following a suggestion by Bell, we propose to simply drop the deterministic dynamics, yielding a fundamentally	stochastic	deBroglie-Bohmian	"rump	theory".	The	following	paper	will	attempt to	take	up	the	cudgels	for	this	so-far	largely	neglected	theory	as	superior	to	dBBT. We'll	proceed	as	follows:	In	section	II,	we	briefly	review	(II.1)	and	critically	examine	(II.2)	the standard interpretation of dBBT and the status probability has in it, considering interpretations	in	terms	of	ignorance	as	well	as	typicality	and	Humean	Best	Systems	(II.3).	The 1	Einstein	et	al.	(1935),	cited	in	Redhead	(1987),	p.71,	who	also	discusses	the	EPR	argument	in	detail. 2	Einstein	(1949),	p.	666 3	Ibid. 4 subsequent	analysis	of	the	role	the	Guidance	Equation	plays	within	dBBT	(III.1)	suggests	an alternative stochastic reading of dBBT that dispenses with it: stochastic deBroglie-Bohm Theory	(sdBBT).	The	resulting	theory	is	irreducibly	stochastic	with	the	probabilities	of	the	Born Rule	representing	a	disposition	for	a	random	walk	through	configuration	space	(III.2).	In	IV,	we critically	examine sdBBT, its status	as	a	minimally	deBroglie-Bohmian theory (IV.1), and its status as a fundamental (rather than a merely phenomenalist) theory (IV.2). In IV.3, we address	an	objection	Bell	has	articulated	against	sdBBT,	namely	its	"temporal	solipsism",	i.e. the	temporal	discontinuity	of	the	world	according	to	sdBBT.	We	conclude	section	IV	with	the completion	of	our	comparison	of	dBBT's	and	sdBBT's	virtues,	respectively,	along	the	lines	of metaphysical	questions	regarding	the	status	of	the	wavefunction	in	both,	the	justification	of the	Symmetrisation	Postulate	and	claimed	conceptual	advantages	for	calculations	regarding quantum	tunnelling.	IV.5.4	discusses	the	issue	of	relativity	in	sdBBT	and	dBBT,	respectively.	In the last section (V),	we summarise	our	main findings and sketch	promising lines	of future enquiry. II.	dBBT	and	its	discontents II.1.	Standard	dBBT For	a	universe	with	N	corpuscles	of	mass	m" each4,	dBBT	consists	of	three	axioms: (1) The standard, non-relativistic	N-particle Schrödinger Equation (SE):	iħ % %& ψ& q, t = Hψ&(q, t) with the wavefunction	ψ&:	R23×R → C and the N-particle Hamiltonian H = − ħ 8 9:; 3 "<= ∇? 9 + V(q, t)	with	∇? = B Bqi ,	i = 1,... ,N. (2) The	Guidance	Equation	(GE),	governing	the	i-th	corpuscle's	trajectories	(i = 1,... ,N): m" HQ; H& = ħJm ∇;L L :	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	(more	on	this	in	III.2). 4	Ascribing	the	masses	to	the	corpuscles	turns	out	to	be	subtle	–	an	issue	to	which	we'll	return	in	III.2.	For	the time	being	suffice	it	here	to	cite	the	arguably	consensus	view	Esfeld	articulates	"[...][The	corpuscles]	do	not	have any	hysical	properties	over	and	above	their	being	localised	in	physical	space.	Hence,	[...]	these	objects	cannot	be considered	as	having	an	intrinsic	mass	or	an	intrinsic	charge	(or	an	intrinsic	spin).	They	do	not	have	any	intrinsic properties",	Esfeld	(2016),	p.4. 5 (3) The	Quantum	Equilibrium	Hypothesis	(QEH):	For	an	ensemble	of	identical	systems	with the	same	wavefunction	ψ,	the	corpuscles'	configurations	are	distributed	according	to the	Born	Rule	(BR),	ρ = ψ 9. The	continuity	equation	∂&ρL + divjψ = 0	obtained	from	the	SE,	with	the	probability density	ρL = ψ 9 and the probability flux	jψ = jU L U<=,...,3 = ħ 9":V Jm ψ∗∇kψ implies:	If	at	any	instant	particles	are ψ 9-distributed,	so	are	they	at	any	other	time:	If the	corpuscles	are	initially	distributed	according	to	the	BR,	they	satisfy	it	later,	too	(and vice	versa). For	the	time	being,	we'll	take	the	wavefunction	to	represent	a	holistic,	real	property	of	the system	of	N	corpuscles,	constituting	the	basic	stuff	of	reality,	located	in	physical	3-dimensional space.	More	on this in III.2.	Regarding the	wavefunction, let's	preliminarily	accept it as	an entity	that	in	some	way	"pilots"	the	corpuscles.	At	this	stage,	a	more	detailed	discussion	of	its ontological status – E.g.:	A	physical field like the	electromagnetic one?	A law-like abstract entity?	A	disposition?	–	needn't	delay	us	yet. II.2	Critical	analysis	of	dBBT's	subjectivist	probabilities Let's	first	turn	to	the	nature	of	dBBT's	probabilities.	Such	an	inquiry	can	take	two	directions: 1. Can	the	BR	somehow	be	derived/deduced,	rather	than	being	just	stipulated	by	fiat? E.g.	Valentini	et	al.,	taking	up	Bohm's	idea	from	1953,	indeed	attempt	to	explain	how a system initially in quantum non-equilibrium relaxes into quantum equilibrium – analogously to Boltzmann's H-Theorem. The results hinge on strong assumptions, however,	only	little	better	than	just	postulating	the	BR.5 2. Following	mainstream	presentations	of	dBBT,	we	therefore	include	the	BR	(or	QEH)	as an independent axiom. The question then remains: What's the status of the probabilities	the	QEH	introduces? According	to	Bell,	"[...]	the	only	use	of	probability	here	is,	as	in	classical	[SM],	to	take	account of	uncertainty	in	initial	conditions."6	In	the	following	we	assume	the	orthodox	interpretation of	dBBT's	probabilities	to	be	an	ignorance	interpretation.7 5	Cf.	Callender/Weingard	(1997),	Callender	(2006) 6	Bell	(1980),	p.	156 7	E.g.	also	Esfeld	(2016),	p.	5 6 But how convincing is the first part of Bell's answer, viz. the asserted analogy with thermodynamic	equilibrium	(i.e.	the	probability	distribution	of	the	canonical	ensemble,	ρ = e[ \ V]/Z,	with	Boltzmann's	constant	k,	the	classical	Hamiltonian/energy-function,	the	partition function	Z	and	temperature	T	of	the	equilibrated	system)?8	At	best,	we	submit,	it's	heuristic: • The system's	dynamics imposes constraints	on the	probability	distributions in	both cases.	(More	on	this	in	II.3). • "[I]n	both cases it seems	natural to try to justify these	equilibrium	distributions	by means	of	mixing-type,	convergence-to-equilibrium	arguments	[...].	[It's]	been	argued, however,	that	in	both	cases	the	ultimate	justification	for	these	probability	distributions must	be	in	terms	of	statistical	patterns	that	ensembles	of	actual	subsystems	within	a typical	individual	universe	exhibit."9	The	success	of	such	attempts	is	controversial.10 Here	the	similarities	end: • An	immediate	crucial	difference	in	terms	of	physical	significance	is	that,	whereas	our current	universe	is	far	from	thermodynamic	equilibrium/heat-death	(with	equilibrium states,	of	course,	observable	only	locally,	e.g.	in	one's	morning	café	au	lait)	the	dBBT universe	has	always	been	in	global	Quantum	Equilibrium,	which	thus	isn't	a	"quantum heat-death",	devoid	of	all	structures.11 • More importantly, whereas in the thermodynamic case, the classical-mechanical Hamiltonian (operating, with its double role as energy of the macrosystem and generator	of	the	dynamics	of its	micro-constituents,	on	both	levels) links	the	microdynamics with the macro-system's properties, dBBT's BR-probability distribution contains	only	the	wavefunction:	Although	it,	too,	occurs	on	the	"macro-",	i.e.	quantum level,	it	doesn't	genuinely	link	two	levels,	as	dBBT	includes	the	SE,	as	QM's	essence,	as an	axiom	and	hence	as	part	of	the	"micro-",	i.e.	subquantum	level	description.	In	other words,	since	from	the	dBBT	perspective	the	SE,	arguably	constituting	the	essence	of the	macro-level	QM,	is	an	axiom	of	the	micro-level	dBBT,	both	levels	are	not	clearly 8	Cf.	Goldstein	(2013),	Sect.	9 9	Ibid. 10	For	the	SM	case,	cf.	Sklar	(2015),	sect.	3. 11	Cf.	Dürr	et	al.	(2003),	Ch.	12,	13. The	global	nature	of	quantum	equilibrium	plays	a	crucial	role	for	the	dynamical	systems	analysis	within	dBBT,	cf. loc.	cit. 7 separated; QEH thus cannot properly bridge them. Rather than as a subquantum theory,	which	"completes"	QM,	this	suggests	to	regard	dBBT	as	an	alternative	theory in	its	own	right,	empirically	equivalent	to	QM.	Then,	however,	the	analogy,	which	turns on	the	idea	of	one	theory	emerging	from	the	other,	breaks	down	altogether. Let's	move	on	to	Bell's	second	assertion,	viz.	that	SM	probabilities	are	epistemic.12	This	is	a contentious point 13 – and hence ill-suited to illuminate the interpretation of dBBT's probabilities: • Popper,	for	instance,	argues	that	explanations	of	SM	in	which	epistemic	probabilities play	a	role	assert	"that	irreversibility	[as	expressed	in	the	2nd	Law	of	thermodynamics] is	a	result	of	our	ignorance	of	the	details	of	the	state	of	the	gas."	This,	he	continues, "[...]	leads	to	the	absurd	result	that	the	molecules	escape	from	our	bottle	[air-filled and then uncorked in vacuum], because we do not know all about them [...]"14 . Popper, in	short,	claims	that	epistemic	probabilities	amount	to	the	absurd	belief in telekinesis. • He also observes the incompatibility between Boltzmannian SM and epistemic probabilities: Firstly, "nescience always increases, provided we do not start with complete knowledge. But disorder, or entropy, decreases at times; according to Boltzmann,	it	fluctuates.	Secondly	nescience	does	not	increase	if	we	have	complete	or perfect	knowledge	to	start	with	[...].	Again	this	is	incompatible	with	Boltzmann's	view; for	if	ever	a	system	should	attain	perfect	order	by	a	highly	improbable	fluctuation,	it would, in all probability, immediately become disordered again, according to Boltzmann."15 • Furthermore,	Popper	maintains, the	probability-subjectivist cannot	explain the fact and the irreversibility of diffusion of the gas molecules in the example: "[The 12	According	to	Uffink	(2011),	one	should	distinguish	between	two	forms	of	non-objective	probabilities,	where an	objective	quantity	or	quality	of	an	object	corresponds	to	an	inherent	property	of	the	physical	object	itself, independent	of	any	subject's	knowledge	of	it:	Subjective	probabilities	reflect	the	strength	of	an	individual's	belief, i.e. the	degree	of subjective	certainty;	by	contrast,	epistemic	probability	assignments,	capture	an individual's certainty relative to the information available to them. In other words, epistemic probabilities express an objective	or at least inter-subjective	evaluation	of their knowledge. In the following,	we	will treat ignorance interpretations	of	probability	as	epistemic	interpretations. 13	Cf.,	e.g.	Lavis	(2011)	for	recent	defences	of	objective	probabilities	in	SM. Contrariwise,	e.g.	Frigg	(2007)	or	Uffink	(2011)	make	the	case	for	epistemic	probabilities	in	SM. 14	Popper	(1982),	pp.	109	(our	emphasis) 15	Op.	cit.,	p.	115	(Popper's	emphases) 8 subjectivist]	cannot	even	say	that	the	gas	has	in	fact	expanded.	All	he	can	say	is	that his	state	of	ignorance	has	increased	[...]."16 As Frigg points out, however, a	misconception underlies these objections:17	Espousing an epistemic	view	on	probability	doesn't imply that	our	beliefs	or lack	of	knowledge	cause	or bring about the physical facts. Epistemic probabilities only explain "why or when it is reasonable	to	expect	[gases	to	disperse,	ice	cubes	to	melt,	or	coffee	to	mix]."18 One may counter that our scientific explanations should go beyond "reasonable expectations"19;	rather,	adhering	to	a	vision	of	physics	close	to	those	of	Einstein	or	Planck20, we	should	strive	for	explanations	and	interpretations	in	purely	physical	terms	only,	with	no reference	to	subjects	and	their	epistemic	states. And indeed, dBBT expressly aspires after a realist, objectivist "quantum theory without observer"	(Popper).21	In	particular,	dBBT	intends	to	be	able	to	describe	fundamental	reality even	in	the	absence	of	any	epistemic	subjects	who	could	have	any	"reasonable	expectations", such as in the early phases of the universe. (Note that such deviations from Quantum Equilibrium	elicit	physical	effects,	which	leave	objective,	in	principle	detectable	traces	in	the cosmic microwave background. 22 ) Thus, the question still looms: How do epistemic probabilities fit into its	otherwise realist,	objectivist/subject-free framework?	We'll further pursue	that	line	of	thought	shortly. One	might	bypass	the	problem	if	dBBT's	probabilities	turned	out	not	really	to	be	probabilities directly expressing chance-related quantities, but something like constraints on all conceivable statistical initial distributions of Bohmian corpuscles. Let's ponder: How to understand the continuous BR-probability distribution ρ&b(Q) ≔ ψ(Q) 9 vis-à-vis the corpuscles' discrete distributions of the form ρd:e(Q):= = 3 δ(Q − qi)3"<= , with	qi the corpuscles' actual positions? Goldstein declares that the former (which he calls the 16	Op.cit.	p.	116	(Popper's	emphases) 17	Cf.	Frigg	(2010),	p.	30 18	Uffink	(2011),	p.	45 19	E.g.	Bunge	(1974)	for	a	thoroughgoingly	objectivist	philosophical	semantics 20	Cf.	Scheibe	(2006),	Ch.	II,	III	and	IX 21	Allori	et	al.	(2007),	sect.	8,	expressly	classify	dBBT,	alongside	GRW,	as	a	quantum	theory	without	observer. Dürr/Teufel	(2009),	pp.	177	illustrates	the	vehemence	with	which	dBBT-adherents	dismiss	subjectivism	in	the context	of	the	Measurement	Problem. 22	Cf.	Valentini	(2010) 9 "theoretical distribution")	must approximate the latter (the "empirical distribution"): "The theoretical distribution is an idealisation providing a good approximation to the empirical distribution,	ρd:e ≈ ρ&b,	in	the	limit	of	large	ensembles	of	subsystems."23 But	what's	supposed	to	be	the	physical	and	ontological	status	of	such	an	idealising	constraint? What	does	the	discrepancy	between	the	"theoretical	distribution"	and	the	actual	"empirical distribution",	which	ineluctably	arises	for	any	Bohmian	universe	of	finite	corpuscle	number, signify?	Two	oddities	obstruct	an	interpretation	of	the	QEH	as	a	contingent	boundary/initial condition of our universe: Firstly, the constraint itself evolves dynamically – against the intuition	that	boundary/initial	conditions	should	be	fixed.	Secondly,	the	Schrödinger	Equation, which governs this dynamics, is a partial differential equation. Hence, it requires the specification	of	contingent	initial	data	for	the	wavefunction.	In	other	words:	QEH,	construed as a contingent constraint, in turn, is subject to another contingent	meta-constraint. This seems redundant. (We'll revert to these two	peculiarities in IV.5.1)	Goldstein's suggestion yields	no	satisfactory	resolution. Let's	therefore	bite	the	bullet:	In	accordance	with	dBBT's	realist	framework,	what	we're	after is	a	way	to	accommodate	for	its	probabilities	in	an	objective	way.	What	options	are	on	the table? II.3.	Two	culs-de-sac	to	objective	probabilities	for	dBBT In	his	review	of	objective	interpretations	of	probability24,	Maudlin	discusses	two	that	appear viable	for	our	deterministic	dBBT,	viz.	typicality	and	Humean	Best	Systems.	How	do	dBBT's probabilities	fare	on	these? II.3.1.	Typicality The analogy with SM, fickle as it may be, raises the question: Might Boltzmann's own understanding of probabilities in his approach to SM bail us out?25	Here, the probability measure	figures	as	a	modal	measure	of	how	typical/common	certain	sets	of	phase-space	are. A statement involving an equilibrium macrostate with typicality measure ("t-measure", henceforth) close to unity holds typically, i.e. for the overwhelming majority of micro- 23	Goldstein	(2011),	p.	9 24	Cf.	Maudlin	(2011a) 25	Cf.	Goldstein	(2001);	Lazarovici/Reichert	(2015) 10 configurations – or equivalently, since the dynamics preserves the	measure, initial	microconfigurations: Most microscopic initial conditions evolve into indistinguishable coarsegrained	macrostates.	Out	of	all	nomologically	possible	systems,	most	behave	typically. Neither	randomness	nor	ignorance	becloud	the	sky	of	such	an	interpretation:	T-probabilities in	SM	are	compatible	with	the	determinism	inherent	in	its	underlying	micro-dynamics. Advocated	as	an	apposite	framework	for	probabilities	in	SM26,	it's	tempting	to	ponder:	Can we	transfer	the	typicality	account	to	dBBT's	probabilities?	Indeed,	standard	presentations	of dBBT27	couch	the	QEH	in	terms	of	typicality:	The	universal	wavefunction	(of	the	universe)	Ψ induces a t-measure	Plm ≔ Ψn 9 . A Law of Large Numbers then establishes that typical subsystems of a universe in Quantum Equilibrium, with corresponding "effective" wavefunction	ψ	are	distributed	according	to ψ 9,	i.e.	the	BR	as	probed	in	laboratory	contexts. More	precisely,	relaxing	the	assumption	of	the	wavefunction	being	factorisable,	a	subsystem is	said	to	have	an	effective	wavefunction	ψ,	if	the	universal	wavefunction	Ψ:X×Y → C,	with X	and	Y	denoting	the	configuration	space	of	the	subsystem	and	its	environment,	respectively, can	be	decomposed	as ∀ x, y ∈ X×Y:Ψ x, y = ψ x φ y + Ψv x, y , where	φ	and	Ψv	have	macroscopically	disjoint	y-support	and	Y ⊆ supp φ .	Subsystems	with an	effective	wavefunction	and	negligible interaction	with its	environment	can	be	shown	to satisfy	the	SE	for	ψ. For subsystems with the same wavefunction ψ , the Pl -measure, conditional on all environmental configurations Y that yield to the same effective wavefunction ψ , is determined	(independent	of	Y)	as: Pl Q = X, Y : X ∈ d~x Ψ . , Y = ψ: Y = ψ 9d~x. 26	Cf. Volchan (2006);	Maudlin (2011a); Hemmo/Shenker (2015); Lazarovici/Reichert (2015); Oldofredi et al. (2016) 27	E.g.	Dürr/Teufel	(2009),	Ch.	8.3.	We	follow	the	presentation	in	Oldofredi	et	al.	(2016),	Sect.	3 11 From	this,	a	Law	of	Large	Numbers	follows:	For	any	measurable	set	A ⊆ R2~	and	an	ensemble of	N identically prepared subsystems with the effective wavefunction	ψ and the position random	variables	X",	it	holds	that ∀ε > 0:Plm Q ∈ R2~ : 1 N χ;∈A Q − 3 "<= d2~Q A ψ(Q) 9 < ε 3→ 0. In	other	words:	The	distribution	of	corpuscles	in	sufficiently	large	ensembles	of	subsystems, each	prepared	with	the	same	effective	wavefunction	ψ,	typically	approximate	the	statistics	of the	BR,	i.e. ψ 9,	where	the	measure	of	typicality	is	given	by	the	QEH,	i.e.	Pl = Ψ 9,	with	the universal	wavefunction	Ψ. In	light	of	these	results,	Oldofredi	et	al.	announce:	"Born's	rule	is	thus	predicted	and	explained by	[dBBT]	as	a	statistical	regularity	of	typical	Bohmian	universes."28	If the	QEH	is	accepted, subsystems	necessarily	obey	the	usual	quantum	mechanical	probabilities. But	why	buy	into	the	QEH,	why	accept	Pl	as	its	t-measure?	In	SM,	stationarity	figures	in	the usual	motivation/justification	of the Lebesgue-measure (a justification, however, Frigg has argued to be fundamentally misguided29). In dBBT, the notion of equivariance30 suitably generalises stationarity: Imposing it uniquely determines Ψn 9 as the t-measure 31 that depends only locally on	Ψn and its derivatives – a far more satisfactory picture! Their importance as mathematical theorems notwithstanding, these mathematical uniqueness results	don't	answer	the	question	why	we	should	assume	QEH.	Dickson	succinctly	writes:	"It's not	at	all	obvious	why	equivariance is	a	preferred	property	of	measures	over the	possible initial	distributions.	Equivariance	is	a	dynamical	property	of	a	measure,	whereas	the	question 'Which initial distribution is the correct one?' involves no dynamics, nor is it clear why 28	Oldofredi	et	al.	(2016),	pp.15 29	Cf. Frigg (2011). One	main argument is that although the Lebesgue	measure is indeed the only	measure invariant	under	all	Hamiltonian	flows,	it's	not	clear	that	this	property	is	at	all	relevant	for	justifying	the	choice	of the	Lebesgue	measure,	since	each	system	is	governed	by	exactly	one	Hamiltonian.	For	any	specific	Hamiltonian, there	could	also	be	invariant	measures	other	than	the	Lebesgue	measure. 30	Let	Ψ	denote	the	wave	function	of	the	universe.	It	generates	the	dBBT	dynamics	in	the	form	of	the	flow	Φnl. A	Ψ dependent measure	Pl is called equivariant, if for a measurable set A the following equality holds: P&l A ≔ Pl ∘ Φnl [= A = Plm(A),	cf.	Dürr/Teufel	(2008),	Ch.	11. 31	Cf.	Goldstein/Struyve	(2007) 12 dynamical	properties	of	a	measure	are	relevant".32	Moreover,	we	submit,	typicality	accounts are	inept	to	settle	the	interpretation	of	dBBT's	probabilities: • Albeit a	measure	over	possible –	hence,	by standard	accounts33,	non-real –	worldconfigurations, the t-measure, Plm = Ψn 9 , satisfies a dynamical law – usually characteristic of real properties. (We'll revisit this argument in a different context more	in	detail	in	III.2.) The	dynamical	nature	of	the	wavefunction	has	an	important	implication:	Insofar	that typicality quantifies how common a trait is amidst the space of nomological possibilities, a typicality statement shouldn't be contingent, that is, vary across different possible worlds. But since the wavefunction, qua SE, requires the specification	of	initial	conditions	that	characterise	a	particular	world,	Plm measure	is contingent,	as	well.	Therefore	it	cannot	be	a	measure	of	typicality. • More	oddities are in the offing: Even if, as	we reported, dBBT's t-measure,	Plm = Ψn 9,	is	unique	and	to	some	extent	appears	plausible,	one	must	be	wary	of	concluding that	sets	of	measure	zero	w.r.t.	Plm (e.g.	the	set	of	initial	conditions	for	which	the	GE is	well-defined,	see	III.1)	are	small34:	Sets	of	measure	zero	in	general	needn't	be	small at all in any mundane sense: Think, for example, of the infinitely many rational numbers in the interval between 0 and 1, whose Lebesgue measure λ is zero: λ Q ∩ 0; 1 = 0. Moreover,	Frigg	stresses	that	"[...]	as	Sklar	[...]	has	pointed	out,	sets	of	measure	zero needn't	even	be	negligible,	if	sets	are	compared	with	respect	to	properties	other	than their	measures.	For	instance,	we	can	judge	the	'size'	of	a	set	by	its	cardinality	or	Baire category	rather	than	by	its	measure	which	may	lead	to	different	conclusions	about	a set's	'size'.	[...]	So	we	face	the	question	of	what	conveys	upon	measures	a	privileged status	when	it	comes	to	judging	typicality."35 • Quite	concretely,	one	may	ponder:	What	ensures	that	typical	solutions	are	actually observed?	Suppose,	we	rearrange	the	first	M ≫ 10=92	(the	latter	being	an	estimate of	the	number	of	bits	of	our	universe)	digits	of	the	(infinite)	series	of	digits	of	a	so- 32	Dickson	(1998),	p.	123	(Dickson's	emphasis) 33	In	non-standard	views,	such	as	modal	realism,	cf.	Lewis (1986)	and	dispositional	metaphysics,	cf.	Mumford (2003),	the	situation	is	different.	We'll	explore	the	prospect	of	the	latter	in	III.2. 34	Cf.	Frigg	(2011),	p.90 35	Loc.cit. 13 called	normal	number	z = d=, d9, ... , i.e.	one in	which	each	digit i occurs	equally often	(with	equal	t-measure	μ( i ) = = = ),	such	that	of	the	first	M	digits	every	other number	is	seven:	z∗ ≔ 7, d=∗, 7, d9∗ ... ,	where	the	d"∗'s	are	obtained	from	deleting	all 7's from the first	M	d" 's. Of course, the rearrangement z* doesn't change the frequency with which 7 occurs; its t-measure is preserved under finitely many permutations,	∀i: μ∗ i = μ( i ).	Yet,	by	looking	at	z∗,	it	appears	that	every	other number	is	7,	i.e.	that	μ∗ 7 = = 9 . The	immediate	lesson	is:	Without	randomisation	that	suitably	mixes	the	results,	a	high t-measure over the ensemble of possible dBBT systems doesn't explain why our observed	statistical	patterns	match	typical,	i.e.	BR-obeying	outcomes.	The	facts	of	the world	(assumed	to	be,	at	most,	countably	inifinite)	could	perfectly	well	have	the	QEH's measure	of	typicality,	whilst	still	all	empirical	evidence	could	violate	the	BR	–	if,	e.g.	a malicious	demon	decided	to	suitably	re-arrange	the	facts	of	the	world	in	a	way	that	all those	(finitely	many)	facts	available	to	us	deceive	us.	Why	preclude	such	a	demon? How	to	ensure	that	typicality	is	linked	to	empirical	results? If	the	choice	of	the	t-measure	can't	explain	the	empirical	adequacy	of	the	BR,	we	could equally	well	postulate	any	other	t-measure!	Hence,	without	a	randomisation,	the	BR loses its empirical content. But then,	what's the	point of an allegedly fundamental theory	without empirical content? So,	we	wind up	where	we started:	Whence the randomness	in	the	otherwise	deterministic	dBBT	universe? • One	way	out	of	the	preceding	problem	is	to	simply	assume	that	our	dBBT	universe	or any other system we're interested in is typical, i.e. started from typical initial conditions. But as Frigg poignantly remarks: "Whether the system	has a particular initial	condition	is	a	factual	question,	and	as	such	it	has	to	be	settled	by	an	appeal	to matters	of	fact	and	not	measures	of	sets;	[...]	we	need	an	argument	for	the	conclusion that	the	system	indeed	started	out	in	a	typical	initial	condition,	but	that	these	are	of measure	(close	to)	one	does	not	give	us	such	an	argument.	[...]	Whether	or	not	this initial	condition	is	also	typical	is	simply	irrelevant."36 One reply	would	be to argue that in the absence	of further information	about the details	of	a	system,	it's	rational	to	expect	the	system	to	be	typical. 36	Op.cit.,	p.91 14 We already encountered such an epistemic manoeuvre, and dismissed it as endangering	the	Bohmian	objectivist	agenda.	M Perhaps more to the point, even any connection between typicality and "rational expectability" is	questionable.	Conceptually, a t-measure refers to the collection	of possible worlds/systems. In essence, such a collection of systems is a canonical/Gibbsian	ensemble,	as	known	from	Gibbsian	SM.37	Thus,	its	use	is	subject	to the	same	criticism	as	in	the	Gibbsian	case.38 In particular, dBBT claims universal applicability, encompassing the universe as a whole.	The	latter	doesn't	have	any	copies,	though.	And	given	that,	for	all	we	know, there's only one universe, why should it be "rational to expect" it to be a typical member of all conceivable universes? It's true that multiverse speculations have become fashionable in contemporary cosmology – however, not without being severely	scathed39;	furthermore,	demanding	that	a	consistent	interpretation	of	dBBT's probabilities	be	perforce	committed	to	multiverse	scenarios	seems	an	unnecessarily compromising	deed	of	desperation. We conclude that it's not even clear how typicality should even matter to a substantial interpretation	of	dBBT	–	let	alone	how	it	would	settle	the	status	of	its	probabilities. II.3.2.	Humean	Best	Systems After	hitting	a	blind	alley	with typicality	accounts,	what	about the	other road to	objective chances	Maudlin	considers,	Humean	Best-System	theories	(HBSs)?40 What	underlies	them	is	Lewis'	conception	of	laws	as	the	systematisation	of	statements	about the	"Humean	Mosaic"	(i.e.	the	actual/categorical	local,	only	spatiotemporally	connected	facts making	up	the	world's	history)	that	strikes	the	best	balance	between	simplicity,	strength	(How many phenomena does it cover?) and fit (how exact are the predictions of the systematisation?). These statements can be either deterministic or probabilistic. So, HBSs 37	Cf.	Dürr/Teufel	(2008),	p.	64 38	Cf.,	e.g.	Frigg	(2008),	Ch.	3.3 39	E.g.	Ellis	(2014) 40	Maudlin	(2011a),	sect.	3;	Cf.	also	Hájek/Hoefer	(2006);	Frigg/Hoefer	(2006) 15 framework	allows	for	chances	even	in	a	fundamentally	deterministic	world41	– a	conceptual advantage	inviting	an	application	to	a	deterministic	dBBT	with	its	probabilistic	GE. What	motivates	HBSs	is	to	avoid	postulating	necessary	connections	and	irreducible	modalities (about	both	of	which	Humeans	share	their	eponym's	famous	scepticism),	as	well	as	a	close link	between	chances	and	action-guiding	credence42.	Especially	attractive	with	regard	to	dBBT is	that	HBSs	can	accommodate	for	the	QEH	as	a	statement	about	an	initial	state	of	the	universe (thanks	to	the	continuity	equation	for	the	probability	density/flux	from	the	SE),	similar	to	the Past	Hypothesis:	"[...]	[A]	stochastic	dynamics	cannot,	by	itself,	have	any	implications	about what	the	initial	state	of	the	universe	is,	since	that	state	is	not	produced	by	a	transition	from anything else. But, for the Humean, probabilistic claims are just a means of conveying information	about	empirical	distributions."43 Yet,	upon	closer	inspection	the	optimism	that	HBS	might	provide	a	satisfactory	framework	for dBBT	and	its	probabilities	starts	to	subside: • Besides	a	general	worry	that	HBSs	are	unable	to	even	provide	a	metaphysics	proper44, the	characterisation	of	laws	as	a	"best	match"	between	simplicity,	fitness	and	strength is	vague:	How	to	flesh	out	these	notions	–	and	their	relevant	"combination"	–	in	an objective,	exact	way?	One	may	feel	uneasy	about	this	flavour	of	subjectivism,	sticking to	this	quadruple	vagueness	–	and	it's	a	purely	objective,	subject-free	interpretation of	dBBT	we'd	set	out	for. • A more severe problem springs from the non-separable nature of dBBT's wavefunction. To formulate this popular argument, let's inspect Maudlin's more precise contemporary characterisation of Humeanism as the conjunction of three doctrines: 41	Cf.	Hoefer	(2011),	Frigg	(2014) 42	Cf.	Brown	(2011) 43	Maudlin	(2011a),	p.302 44	Hájek	(2011),	sect.	3,	for	instance,	voices	this	suspicion	that	HBSs	"[...]	mistake	an	idealised	epistemology	of chance	for	its	metaphysics".	And	someone	desiring	a	metaphysical	framework	for	dBBT	may	now	interject,	it's an	objective	metaphysics,	an	attempt	to	understand	and	explain	its	concepts	and	how	a	world	would	look	like	if it	were	true,	that	a	fundamental	theory	such	as	dBBT	calls	for:	"Humean	laws	[...]	can	summarise,	but	not	[...] explain"	Maudlin	(2011a),	p.	303. 16 Separability:	"The	complete	physical	state	of	the	world	is	determined	by	(supervenes on)	the	intrinsic	physical	state	of	each	spacetime	point	(or	each	point-like	object)	and the	spatio-temporal	relations	between	points."45 Physical	Statism:	"All	facts	about	a	world,	including	modal	and	nomological	facts,	are determined	by	its	total	physical	state."46 A third condition specifies the type of facts admissible in Physical Statism: "The intrinsic	physical	state	of	the	world	can	be	specified	without	mentioning	the	laws	(or chances,	or	possibilities)	that	obtain	in	the	world."47	The	Humean	thus	abstains	from invoking	irreducible	nomic,	modal,	dispositional	or	causal	facts	in	specifying	the	state of	the	world. Entangled states, e.g. the singulet state ψ"~ = = 9 ↑↓ − ↓↑ , inherent in generic quantum	theories,	violate	separability48;	in	particular,	dBBT	does,	with	(according	to	Maudlin) the	state	of	the	universe	being	specified	by	the	pair Q,Ψ .	In	short,	insofar	as	Humeanism	is committed	to	Separability	and	an	interpretation	of	dBBT	regards	the	wavefunction	as	a	real, fundamental	entity,	Maudlin	contends	that	Humeanism	and	dBBT	are	incompatible. Retaining	Maudlin's	definition	of	Humeanism,	his	argument	against	a	Humean	interpretation of	entanglement-involving	quantum	theories	can	be	circumvented	by	demoting	the	status	of the	wavefunction	to	a	non-real	(more	precisely:	non-fundamenta)	entity49.	On	Humeanism, only	the	Mosaic	and	its	elements	fundamentally	exist,	i.e.	actual,	local	matters	of	particular fact	in	spacetime;	all	other	entities	don't:	Not	having	any	correlates	in	fundamental	reality, they're	useful	fictions	in	our	theories	to	economically	summarise	or	systematise	the	patterns in	the	Mosaic,	to	algorithmically	compress	the	data	of	the	Mosaic.	The	status	of	those	nonfundamental	entities	as	useful	fictions	is	called	"Humean	supervenience". 45	Maudlin	(2007),	p.	51 46	Loc.cit. 47	Op.cit.,	p.	52 48	Loc.cit.,	Ch.	2.1 49	Cf.	Miller	(2014);	Esfeld	(2014),	Callender	(2015) 17 By	deciding	to	categorise	a	certain	term	as	only	supervenient,	the	Humean	needn't	ban	any terms	from	physical	theories	as	metaphysically	illegitimate:	Physicists	are	allowed	to	continue their	business	as	usual	–	as	long	as	they	don't	forget	these	terms	are	merely	fictional. Consequently, by declaring the wavefunction supervenient, its non-separability (more precisely,	its	non-factorisability)	becomes	innocuous	–	merely	a	mathematical	peculiarity	of	a useful fiction, not a feature of reality; the Humean tenet of Separability thus remains unharmed. In	terms	of	ontological	categories,	the	demotion	of	the	wavefunction	to	a	supervenient	fiction amounts	treating	it	as	law-like	or	"nomological",	comparable	to	the	status	of	the	Lagrangian in classical field theories. Hence, this ontological re-classification of the wavefunction (in principle,	independent	from	any	commitment	to	Humeanism)	is	known	as	the	"Nomological View". Is the	Humean	Bohmian	now	off the	hook	with resorting to the	Nomological	View?	Three objections,	we	submit,	mar	her	hopes: • Whilst	Humeanism	delimits	which	entities	count	as	fundamental	and	correspond	to something real (viz. the elements of the Mosaic), and which don't (viz. laws, irreducible	dispositions,	etc.),	it	procures	no	criteria	for	ascertaining	whether	a	certain theoretical term should be classified as representing an element of the	Mosaic or merely "nomological". In particular, declaring the wavefunction as "nomological" mandates further arguments; just decreeing it in order to preserve	Humeanism is question	begging. • Should	a	term,	by	fiat	declared	"law-like",	not	jibe	with	characteristics	all	other	bona fide laws	share,	we	are	all the	more	entitled	to	demand	good	reasons	for	that	fiat. This, we shall see in III.2, is the case for the wavefunction. Downplaying the importance	or	accuracy	of	intuitions	about	laws,	as	for	instance	Goldstein	and	Zanghì parry50,	is	a	red	herring:	The	issue	is	not	how	or	whether	to	revise	our	intuitions	about 50	Goldstein/Zanghì	(2013) 18 laws,	but that the	proponent	of the	Nomological	View	must	proffer	arguments for declaring	the	wavefunction	nomological.51 • But	let's	for	the	time	being	grant	that	the	wavefunction's	status	can	convincingly	be established	to	be	law-like.	Within	dBBT,	this	would	spawn	an	odd	hierarchy	of	law-like entities: If the wavefunction is law-like, the SE, as the law that determines the wavefunction,	would consequently be "meta-law"-like; in turn, the	GE, as the law governing	via	the	wavefunction	the	dynamics	of	the	corpuscle	positions,	would	ought to	be	seen	as	"hypo-law"-like.	Both	meta-laws	and	hypo-laws	come	in	two	flavours: dynamical (viz. the SE and the GE, resp.) and "static" (viz. the boundary/initial conditions	the	solution	of	the	SE	requires,	and	the	QEH,	resp.).	Such	an	explosion	of legislature	seems	unnecessary. One	may	try	to	fend	off	the	preceding	objections	by	construing	"nomological"	less	narrowly: In	this	vein,	Bhogal	and	Perry52	suggest	that	in	using	the	term	"law-likeness"	one	shouldn't	be deceived	by	any	received	connotations	of	laws;	rather,	nomologicality	refers	to	any,	merely supervenient	theoretical	term,	regardless	of	its	ontological	category	as	a	law,	property,	etc.	In other words: For such a liberalised Humeanism, law-likeness and non-fundamentality are synonymous.	(Consequently,	even	mass	or	charge	in	e.g.	Esfeld's	similar	proposal	for	a	liberal "Quantum	Humeanism"53,	on	which the	Mosaic consists	only	of	points in spacetime	being either	occupied	or	empty,	would	count	as	"law-like".) However, whether this liberalisation ameliorates the Humean Bohmian's predicament is questionable:	The	Humean	still	owes	us	good	arguments that the	wavefunction should	be regarded	as	a	supervenient	entity.	The	critic	of	a	Humean	approach	to	dBBT	may	be	happy	to concede	that	the	wavefunction	could	be	nomological,	and	that	consequently	such	a	Humean approach	to	dBBT	is	at	least	conceptually	possible	–	but	ultimately	a	critic	would	demand	more than	a	proof	of	principle.	Neither	does	a	liberal	Humeanism	prima	facie	curtail	the	unnecessary proliferation	of	nomological	entities	of	suband	superordinate	rank. 51	Callender	(2015) illustrates	this	need	for	an	explicit justification	very	clearly	by	pointing	to	Hamilton-Jacobi Theory	to	demonstrate	that	the	default	argument	for	the	Nomological	View,	viz.	the	fact	that	the	wavefunction is	defined	on	configuration	space,	isn't	cogent:	Being	defined	on	configuration	space,	isn't	a	sufficient	condition for	"nomologicality". 52	Cf.	Bhogal/Perry	(2015) 53	Cf.	Esfeld	(2014);	Esfeld	et	al.	(2015) 19 But	let's	charitably	grant	that	the	liberal	Humean	averts	the	mentioned	problems.	Still,	she faces	yet	another	counter-argument,	which	turns	on	the	Bohmian	Humean's	core	claim	that dBBT	is	indeed	the	best	systematisation	of	the	Mosaic.	Dewar	has	recently	criticised	this	claim in	detail	and	its	potential	defences54,	arguing	that	the	Humean	Bohmian	shirks	their	burden of	proof	of	his	core	claim,	with	dBBT	prevailing	also	over	other	approaches	to	QM,	including ordinary	QM itself.	Here, let's elaborate two lines	of thought aiming	at very	dBBT-specific issues. • The	first	one	targets	the	notions	of	simplicity	appealed	to	when	singling	out	the	GE	as the	allegedly	simplest	one	amongst	all	empirically	equivalent	alternative	GEs.	(We'll return	to	this	underdetermination	of	the	GE	in	detail	in	III.1).55	Ceteris	paribus,	should the	Humean	Bohmian	fail	to	demonstrate	that	the	GE	presented	in	section	II,	is	the simplest	such	choice,	his	own	Humean	standards	would	compel	him	to	abandon	dBBT in	its	orthodox	form.	Humeanism	must	show	the	simplicity	of	the	GE,	lest	it	result	in	a reductio	ad	absurdum.	Assaults	on	dBBT's	simplicity	can	come	from	three	different directions: Dürr	et	al.'s	attempts	to	justify	the	GE	by	deriving	it	from	Galilei	invariance	presuppose that	the	latter	is	a	necessary	a	priori	constraint	to	impose	on	any	viable	dynamics.56 Whence,	though,	this	distinguished	status,	especially	given	that	BM's	inertial	structure is	"Aristotelian"	(Valentini),	rather	than	Galilean,	such	as	in	Newtonian	mechanics?	An adherent of Brown's Dynamical Approach to symmetry 57 , for instance, would furthermore flat-out dismiss that justification as putting the cart before the horse, insisting	that	the	spacetime	symmetries	be	derived	from	the	dynamics/the	GE,	not	the other	way around.	Quite generally again, one	may simply doubt the plausibility of requiring	the	spacetime	symmetries	of	a	less	fundamental	theory,	namely	CM,	to	hold also	for	the	hopefully	more	fundamental	dBBT. 54	Dewar	(2016) 55	We	gloss	over	here	the	general	vagueness	in	the	notion	of	simplicity,	as	is	illustrated	by	Goldstein's	"identitybased	Bohmian	Mechanics",	discussed	by	Esfeld	et	al.	(2015),	sect.	3:	The	Guidance	Equation	of	this	identitybased	Bohmian	Mechanics	is	considerably	more	complicated	than	the	one	in	standard	dBBT,	"but	doesn't necessarily	amount	to	more	complicated	physics",	op.cit.,	p.	17.	Esfeld	et	al.	thus	seem	to	insinuate	that	in order	to	assess	how	simple	a	theory	is,	one	needs	to	differentiate	between	mathematical	and	some	"physical" simplicity,	where	the	latter	seems	to	override	the	former. 56	Dürr	et	al.	(2002),	Sect.	3 57	Brown	(2007) 20 A	third	strategy	to	undermine	the	simplicity	of	the	standard	GE	originates	in	the	nonrelativistic	limit	of	the	GE,	obtained	from	the	Dirac-Bohm	Theory	(see	III.1	and	IV.5).	It differs from the ordinary GE, despite empirical equivalence. Since the Dirac-Bohm Theory has a considerably larger strength, it seems plausible to consider its nonrelativistic	limit	as	"better"	in	the	sense	of	HBSs.	Consequently,	dBBT	with	the	standard GE	doesn't	count	as	the	Best	System;	hence,	the	Humean	should	reject	standard	BM. • The	underdermination	of	the	GE	menaces	the	Humean	Bohmian	even	more	severely: As	we'll argue in the subsequent section, the	GE is empirically inaccessible, vastly underdetermined and serves no explanatory function – a strong incentive for the Humean to simply abandon the GE, and thereby dBBT altogether, in favour of a statistical theory of random corpuscle jumps – an option	we'll explore in detail in section	III. Dewar	poignantly	adumbrates	the	same	idea:	"The	problem	is	that	our	evidence	for quantum	mechanics is (famously)	statistical in	nature. It is	not that	we	have	direct access to some small number of the Bohmian trajectories, and have successfully stitched	those	together	by	overlaying	a	wavefunction	governed	by	quantum	dynamics. What we have instead are individual but imprecise measurements of positions at particular	times.	[...]	So	what	we	have	really	woven	together	into	a	quantum	tapestry are those	probability	densities, rather than the trajectories themselves;	and	on the Bohmian's own account, those probability densities represent all that can ever be known	for	sure	about	the	trajectories."58 In	short,	a	Humean,	upon	re-examining	the	GE,	will	have	good	reasons	not	to	consider dBBT the simplest systematisation; a Humean couching of dBBT thus seems selfdefeating. After this brief tours d'horizon into	Humeanism,	we find the latter, too, a dead end for a satisfactory	framework	for	dBBT	in	general,	and	its	probabilities	in	particular. II.3.3.	Digression:	Objectivism	and	Heisenberg	relations Germane	to	dBBT's	problems	with	probabilities	and	the	conflict	with	its	objectivist	framework cropping	up,	is	the	standard	account	of	the	Heisenberg	indeterminacy	relations	in	dBBT:	For 58	Dewar	(2016),	p.13	(Dewar's	emphases) 21 roughly	known	corpuscle	positions	their	wavefunctions	are	sharply	peaked.	"[...]	The initial randomness of the particle position translates into the randomness of the particle's asymptotic	velocity, [...]	given	by	the	modulus	squared	of the	Fourier transform	of the [...] wave	packet.	That	distribution	is	all	the	more	spread	out	as	the	initial	wave	packet	is	sharply localised.	This	is	Heisenberg's	uncertainty	relation.	[...]	Quantum	Equilibrium	entails	absolute uncertainty	about	the	Bohmian	positions."59 Two	observations	regarding	this	passage	are	in	order: • An	epistemic	category,	uncertainty,	invades	here	the	otherwise	objective/subject-free account.	Other	standard	reference	on	Quantum	Equilibrium	are	even	more	explicit	in this	regard,	identifying	the	latter	as	the	"origin	of	absolute	uncertainty",	conveying	the "most detailed knowledge possible concerning the present configuration of a subsystem	(of	which	the	'observer'	or	'knower'	is	not	part	[...]."60 This starkly contrasts with the objectivism officially professed by many of dBBT's proponents. E.g. Bell red-flags "[...] some words which, however legitimate and necessary	in	application,	have	no	place	in	a	formulation	[of	QM]	with	any	pretension to	physical	precision:	system,	apparatus,	environment,	microscopic,	macroscopic,	[...] observable,	information,	measurement."61 • Secondly, note in Dürr/Teufel's cited account their remark that corpuscles' initial positions	are	randomly	distributed. Is it	actually	meaningful to	apportion	randomness	to initial	conditions?	Prima	facie, only	stochastic	processes,	e.g.	Poisson	processes	such	as	radioactive	decay,	can	display randomness;	by	contrast,	initial	conditions	are	brute	fact	data.	Moreover,	even	if	one grants the meaningfulness of randomness in initial conditions, we're back in our previous pickle: How can there be randomness in dBBT's	deterministic world? The 59	Loc.cit.,	p.	223 60	Dürr	et	al.	(2003)	,	p.	57	(the	authors'	emphases);	cf.	also	Dürr	et	al	(1995),	p.	27. Dürr et al. (2003), Ch. 11, 12, try to somewhat mitigate the charge of smuggling in epistemic/subjectivist elements:	"Whatever	we	may	reasonably	mean	by	knowledge, information,	or	certainty	–	and	what	precisely these	do	mean	is	not	at	all	an	easy	question	–	it	simply	must	be	the	case	that	experimenters,	their	measuring devices,	[...]	must	be	a	part	of	or	grounded	in	the	environment	of	these	systems.	The	possession	by	experimenters of such information must thus be reflected in correlations between the system properties to which this information	refers	and	the	features	of	the	environment	which	express	or	represent	this	information.	We	have shown,	however,	that	given	its	wave	function	there	can	be	no	correlation	between	(the	configuration	of)	a	system and	(that	of)	its	environment,	even	if	the	full	microscopic	environment	[...]	is	taken	into	account"	(loc.cit.,	p.	57). 61	Bell	(1990),	p.	34 22 same	authors	paradoxically	negate that: "It looks as if objective chance is at	work, while	in	truth	it	is	not.	There	is	no	chance."62	How	to	resolve	this	contradiction? Pledging	allegiance	to	its	objectivist	outlook,	might	perhaps	we	re-cast	dBBT	–	or	at	least,	its essential ingredients – as a stochastic/indeterministic theory and thereby resolve the conundrums? III.	Stochastic	deBroglie-Bohm	Theory	(sdBBT) III.1.	Role	of	the	Guidance	Equation Evidently,	the	GE	curtails	a	stochastic	re-conceptualisation	of	dBBT.	The	elephant	in	the	room now	is:	Might	it	be	dispensable?	We'll	argue	in	this	and	the	following	two	sections,	it's	indeed not	only	explanatorily	redundant,	but	also	spawns	a	few	sources	of	discomfort. Let's	scrutinise	first	some	of	dBBT's	often	perceived	blemishes	that	involve	the	GE: • An	immediate	inquiry	concerns	the	latter's	formal	definability. In	general,	a	wavefunction	possesses	zeros.	Consequently,	for	some	initial	conditions, the GE (Q" ∝ Jm ∇;L L ) will steer corpuscles into that set of zeros, where the GE becomes	singular.	Thus,	it	isn't	well-defined	for	all	of	configuration	space. Although	the	set	of	initial	conditions	that	lead	to	singularities	is	of	measure	zero63,	this poses no satisfactory resolution of the problem – a rehearsal the measure-zero problem	in	SM64. In	particular,	as	we	saw	in II.3.1,	sets	of	measure	zero	needn't	be negligible	or	even	small. In	consequence,	the	GE	pares	down	dBBT's	domain	of	applicability. • Another formal discontent is levelled at GE's non-uniqueness: Are different GEs possible?	Indeed,	the	orthodox	GE	given	above,	Q = j  with	the	quantum	probability flux j = ħ 9":V Jm ψ∗∇ψ U and the probability density	ρ = ψ , is but one of an infinitude of viable alternative dynamics 65 : Any other choice j′ = j + j , with a 62Dürr/Teufel	(2009)	p.64. 63	Cf.	Berndl	et	al.	(1995) 64	Cf.	Frigg	(2008),	Ch.	3.4,	esp.	pp.	125 65	Cf.	also	Deotto/Ghirardi	(2002) 23 divergence-free	vector	field	j,	div	j = 0, for	the	quantum	probability is	equally	fine and	compatible	with	the	QM	prediction.	Yet, the	resulting	trajectories	will	differ. In other	words,	the	choice	of	no	less	suitable	GEs/dynamics	for	deterministic	trajectories is	vastly	underdetermined.	Which	is	the	right	one? The	orthodox	GE	has been argued to be the simplest such choice compatible	with Galilean	and	time-reversal	invariance.66 But this is problematic: Firstly, because appeals to simplicity are glaringly handwaiving:	Why	should	nature	care	about	our	preconceptions	of	simplicity?	Secondly,	as we	saw	in	II.3.2,	the	motivation	for	demanding	Galilei	invariance	is	doubtful:	It's	true, though,	that	the	Schrödinger	Equation	turns	out	to	be	Galilei	invariant	(provided	the wavefunction	transforms	in	a	particular,	non-scalar	way),	but	why	expect	that	from	the GE,	as	well?	This	seems	particularly	in	need	of	further	arguments,	since	the	motivation of	Galilei	invariance	lies	in	the	Galilean	spacetime	structure	of	CM	and	dBBT	seeks	to supersede CM: So, why should we assume dBBT's GE to inherit CM's spacetime structure? Thirdly, relativistic generalisations of dBBT in fact speak against this putatively	simplest	choice	of	the	GE:	Plugging	the	quantum	probability	4-current	of	the Dirac	equation	(see	section	IV.5.4),	i.e.	j¡ = cψ£γ¡ψ	(with	the	Dirac	matrices	γ¡,	and the	Dirac-spinor	ψ), into	Q" = cj i ¦§ (i=1,2,3),	one	obtains	a	relativistic	GE	for	fermions. Its	non-relativistic	limit67,	however,	is	not	our	orthodox	GE	(using	spinors	which	satisfy the	Pauli	Equation68),	but	contains	a	spin-dependent	additional	divergence-free	term. What	lesson	to	draw	from	this	underdetermination	of	the	GE?	It	seems	to	betray	that the concrete trajectories per se play no role in the theory, as long as they are compatible	with	the	Schrödinger	dynamics.	dBBT's	determinism	produces	no	tangible insights	into	a	more	fundamental	subquantum	world	–	a	dead	end	Einstein	seems	to have	intuited,	when	dismissing	dBBT	in	a	letter	to	Born	form	1952	as	"too	cheap",	likely for the reason that	"we	would	not	have	discovered	statistical	mechanics	by	adding 66	Cf.	Dürr/Teufel	(2009),	Ch.	8.1 67Cf.	Holland/Philippidis	(2003).	The	difference	vis-à-vis	the	original,	non-relativistic	GE	amounts	to	exactly	such an	added	divergence-free	vector-field. 68	Cf.	Dürr/Teufel	(2009),	Ch.	8.4;	Holland	(1993),	Ch.	9 24 small	corrections	to	thermodynamics,	or	by	adding	hidden	variables	that	were	in	some way	'guided'	by	the	free	energy,	or	some	other	thermodynamic	quantity."69 We	can	substantiate	the	suspicion	that	the	GE	is	explanatorily	idle	even	further: • As mentioned earlier, dBBT is empirically indistinguishable from QM. How is this achieved? The equivalence in no way depends on the GE. 70 Let's recall the two ingredients	for	ensuring	empirical	equivalence	with	ordinary	QM: o The	BR	delivers	the	right	probability	for	finding	a	dBBT-corpuscle	at	a	certain position. For position	measurements, this coincides with the predictions of standard	QM. o dBBT's	ontology	procures	the	rest:	In	dBBT,	there	are	no	dynamical	properties other than position; what we usually interpret as such properties, e.g. momentum	or	spin,	are	only	manifestations	of	the	wavefunction	and	how	it guides the	particles	positions.	When it	comes	to	observable	effects, though, the	statistics	obeys	the	BR. In consequence, the	empirical content	of dBBT is independent	of the	deterministic trajectories	of	the	dBBT	corpuscles;	the	GE	eschews	empirical	accessibility. Remark: It's instructive to note how dBBT circumvents the two principal no-go theorems	for	hidden	variables	theories71:	Since	all	the	theoretical	work	is	done	by	the wavefunction,	which	enters	both	the	GE	and	the	BR,	and	the	latter is	"non-local" in that it involves the	non-separable/non-factorisable	wavefunction, Bell's Theorem is automatically satisfied. As for contextuality, as enforced by the Kochen-SpeckerTheorem,	it's	explicitly	implemented	in	the	Bohmian	ontology,	which	construes	only position	as	the	only	"non-contextual"	variables. • Let's	look	then	beyond	empirical	equivalence	with	QM	–	at	the	Measurement	Problem and how to interpret the macroscopic superpositions the minimally interpreted 69	Squires	(1996),	quoted	in	Passon	(2005),	fn.	11 70	The	"familiar	statistical	description	of	sub-systems	in	terms	of	effective	wave-functions	[...]	is	really	all	that matters	for	most	practical	purposes",	as	Esfeld	et	al.	(2015)	remark. 71	Wallace	(2008),	sect.	2.6.2,	cashes	out	four	empirical	constraints	on	any	hidden	variables	theories.	The	GE	only features	in	the	fourth	constraint,	which	requires	that	hidden	variables	via	their	dynamical	equation	be	affected only	by	"their"	branch	of	the	state	vector.	Crossing	out	the	GE	altogether,	this	constraint	is	trivially	satisfied. 25 quantum	formalism	seems	to	predict.	Motivating	to	a	considerable	extent	the	hidden variables	agenda, the	Measurement	Problem	more	precisely consists in the	mutual inconsistency of the three assumptions that the wavefunction is complete, that it evolves in accord with the Schrödinger dynamics and that measurements yield determinate	outcomes,	respectively.72 dBBT has been argued to its most convincing solution – by denying that the wavefunction alone completely specifies the physical state: 73 Only the pair wavefunction	and	corpuscles'	position,	(Ψ,Q),	(rather	than	the	wavefunction	alone, as e.g. in the	Many	Worlds Interpretation74) represents the complete state of the system:	Since	the	positions	of	the	corpuscles	are	definite	at	any	instance,	so	is	the	state of	each	system	–	including	the	various	pointer	positions	of	measurement	devices.	In other	words:	Although	the	wavefunction	of	a	measurement	device,	too,	in	general	is in	a	superposition,	the	value-definite/-determinate	configuration	of	the	corpuscles	of which	the	device	is	composed,	picks	out	a	definite	measurement	outcome. At	no	point	does	determinism,	i.e.	the	GE,	enter	the	stage:	All	the	work	is	done	by	the presupposed ontology with its position-definiteness and wavefunction-mediated contextuality	w.r.t.	all	other	dynamical	variables.	The	GE	thus	is	irrelevant	to	dBBT's solution	of	the	Measurement	Problem. This	is	not	to	downplay	a	related,	but	distinct	question	Maudlin	calls	the	"Problem	of Effect":	"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."75 Indeed,	the	GE	accounts	for	the	Problem	of	Effect76,	allowing	for	information	of	the measurement	to	propagate	into	the	future.	As	we	shall	argue	in	detail	in	section	IV.3, abandoning	the	GE,	and	stomaching	the	absence	of	a	solution	to	the	Problem	of	Effect is	of	no	empirical	or	pragmatic	consequence:	We	are licenced	to	dispense	with the requirement that "(t)he result of the first measurement (be) not codified in the subsequent	wavefunction	[...]".77 72Cf.,	for	instance,	Maudlin	(1995) 73Cf.	Dürr/Teufel	(2009),	Ch.	9	or	Esfeld	(2014) 74	Cf.,	for	instance,	Wallace	(2008),	Ch.	2.4 75	Maudlin	(1995),	p.	13 76	Cf.	Albert	(1992),	pp.	147 77	Maudlin	(1995),	p.	14 26 • It	has	been	argued	that the	GE in	some	sense	explains	dBBT's	peculiar	ontology, in which corpuscles have positions as their only non-contextual dynamical variables: "Bohmian	Mechanics should be regarded as a first-order theory, in	which it is the velocity	[...]	that	is	fundamental	in	that	it	is	this	quantity	that	is	specified	by	the	theory, directly	and	simply	[...].	[...]	This	is	not	to	say	that	these	second-order	concepts	[viz. acceleration, force,	work	and	energy]	play	no role in	Bohmian	mechanics; they	are emergent	notions."78 However,	an	ontology	cannot	be	simply	extracted	from	the	formalism.	It	follows	from axioms	that	must	be	postulated	separately79;	a	bare	formalism	never	provides	its	own interpretation or ontology. Consequently, the formalism cannot explain (in any deductive	sense)	the	ontology:	Rather, the	two in	some	cases	match	better	than in others,	with	some	ontological	stipulations	appearing	more	plausible	than	others. How	well	motivated,	one	may	immediately	wonder,	is	dBBT's	ontology	with	its	beable status	of	positions	and	contextuality	of	all	other	variables?	We	concur	here	with	Bell, who	praises its	naturalness	and	cogency80	without	any reference to the	GE: "[...] in physics	the	only	observations	we	must	consider	are	position	observations,	if	only	the positions	of	instrument	pointers.	It	is	a	great	merit	of	the	deBroglie-Bohm	picture	to force us to consider this fact. If you make axioms, rather than definitions and theorems,	about	the	'measurement'	of	anything	else,	then	you	commit	redundancy and	risk	inconsistency."81 • Does	the	GE	at	least	help	us	explain	and	grasp	quantum	phenomena?	One	may	well challenge that, arguing that the GE merely garnishes the QEH with a "fictitious determinism"82	which	doesn't	enhance	our	understanding: 78	Dürr	et	al.	(1995),	pp.	7	(Dürr	et	al.'s	emphasis). Originally, the	argument is intended	as	a	criticism	of	the	2nd-order	"quantum	potential" formulation	of	dBBT, espoused	e.g.	by	Holland	(1993). 79	Cf.	Bunge	(1967);	Esfeld,	passim 80	Daumer	et	al.	(1997)	elaborate	Bell's	idea	that	the	non-contextuality	of	variables	other	than	position	is	rooted in	a	naïve	realism	about	operators. 81	Bell	(1982),	p.	166 82	Englert	(2001)	in	his	review	of	the	German	version	of	Dürr/Teufel	(2009)	(D.	Dürr:	"Bohmsche	Mechanik	als Grundlage	der	Quantenmechanik",	Springer,	2001) 27 o The idiosyncrasies of dBBT's ontological framework, with its dualist wavefunction-point particle ontology, contextuality, etc., drastically depart from	any	classical	picture,	to	begin	with.83 o Also from a less philosophical angle, the "surrealistic" trajectories are misleading:	"The	Bohm	trajectory	is	[...]	macroscopically	at	variance	with	the actual, that is: observed track"84 ; semi-classical dBBT trajectories in semiclassical	situations	differ	strongly	from	classical	trajectories.85 Einstein's	objection	to	dBBT	from	1953	takes	up	this	point	(already	articulated	by	Pauli much earlier in regards to deBroglie's original pilot-wave theory from 1927) and couples	it	to	a	methodological	principle.86	He	considers	a	one-dimensional	particle	in a perfectly elastic box of length L, centred around zero. Inside the box, the corresponding	energy	eigenfunctions	are	superpositions	of	plane	waves	travelling	to the	right	and	left,	respectively: φ~ x, t = 2 ∤ n: 1 2L cos nπx L e ["ħ22 28μ¶2& 2|n: 1 2L sin nπx L e ["ħ22 28μ¶2& . The	GE	then	yields	a	vanishing	velocity	at	all	times	–	a	result	that	"contradicts	the	wellfounded requirement that in the case of a	macro-system the	motion should agree approximately	with	the	motion	following	from	classical	mechanics."87 Similarly,	in	the	case	of	the	ground	state	of	the	hydrogen	atom,	due	to	its	real-valued wavefunction	the	shell	electron,	according	to	dBBT,	is	at	rest88	–	again,	at	variance	with the	demand that in the	macro-limit	one should	approximately recover the	classical motion. For	the	moment,	we	postpone	the	discussion	of	two	further	types	of	blemishes	to	the	next subsection.	May	the	aforementioned	shortcomings	suffice	now	as	a	motivation	to	jettison	the redundant GE, leaving us with the dBBT ontology (position determinateness and 83	Cf.	Esfeld	et	al.	(2014) 84	Englert	et	al.	(1992),	quoted	in	Passon	(2005),	p.	8,	who	critically	discusses	this	argument.	Note,	though,	that this	discrepancy	between	the	"surrealistic"	dBBT	trajectories	and	the	"observed	tracks"	doesn't	amount	to	any empirical	deviation	of	dBBT	from	the	well-confirmed	quantum	mechanical	predictions,	cf.	ibid. 85	Cf.	Matzikin/Nurock	(2003) 86	Cf.	Myrvold	(2003),	esp.	sect.	3.1.	See	also	Holland	(1993),	Ch.	6.5.	Both	authors	criticize	these	objections. 87	Einstein	(1953),	quoted	in	Myrvold	(2003),	p.	10 88	Cf.	Holland	(1993),	Ch.	4.5. 28 contextuality) and the QEH/BR, whose conjunction alone warrants that all predictions of ordinary QM are reproduced. We dub the resultant theory "stochastic deBroglie-Bohm Theory"(sdBBT)	–	a	name	to	whose	deliberate	choice	we'll	turn	in	III.3.89 In	the	following,	let's	take	sdBBT	seriously	–	as	a	fundamentally	stochastic	micro-theory: • Corpuscles	no	longer	are	assigned	continuous,	deterministic	trajectories,	picked	out	by specifying	initial	conditions. • They	only	have	a	localisation	probability,	equal	to	the	BR-probability,	quantifying	–	in a	manner	to	be	made	precise	in	the	subsequent	subsection	–	the	corpuscles'	random walk through configuration space: The corpuscles spontaneously jump between possible	positions. • Their random localisations notwithstanding, at every instance they have definite positions.	All	other	dynamical	properties, like in	the	standard	dBBT	ontology,	which sdBBT	inherits	from	dBBT,	are	only	derivative/contextualised. III.	2.	Probabilities	in	sdBBT Let's	examine	in	more	detail	now	sdBBT's	probability	space R21,A, P ,	with	configuration space	R21 as the so-called sample space, the	σ-algebra90	of "events"	A	generated	by	R21 (i.e. the Borel set) and the Born-probability measure	dP = ψ 9d23Q. 91 We propose the following	(non-Popperian92)	propensity	interpretation93: • The sample space is the set	of	possible configurations an	N-corpuscle	universe can occupy. • As	the	most	"literal"	ontology	for	the	probabilistic	sdBBT	is	a	"world	of	propensities" (Popper),	we	propose	the	following:	Rather	than	being	uniquely	determined	in	their spatiotemporal	evolution	by	initial	conditions,	the	corpuscles	possess	only	a	tendency 89	Bell	(1987),	Ch.5	considers	the	same	idea	as	a	characterization	of	the	Everett	interpretation.	We	take	up	the question	of	whether	this	identification	is	justified	or	not	in	III.3. 90	Certain	mathematical	difficulties	prohibit	the	definability	of	the	probability	measure	on	all	partial	sets	of	the sample	space.	With	a	σ-algebra	of	events	one	constructs	a	smaller,	but	simultaneously	still	sufficiently	rich	set on	which	to	define	the	probability	measure. 91	For	all	formal/mathematical	issues,	cf.,	e.g.,	Georgii	(2009),	esp.	Ch.	1. 92 Propensity views can vary considerably w.r.t. their referents (propensities of what?), their content (propensities	for	what?)	and	their	ontological	status	(e.g.	dispositions?),	cf.	Torretti	(1990),	Ch.4. In	particular	in	the	first	two	respects	our	propensity	view	differs	from	Popper's,	which	takes	propensities	to	be dispositions	of	the	whole	experimental	setup	to	produce	long-term	frequencies. 93	We	draw	partially	on	Bunge	(2011),	Ch.	4. 29 (disposition or propensity) to spontaneously, randomly materialise or jump into existence	in	a	certain	configuration	at	a	certain	instance	in	time:	The	manifestation	of the disposition is the localised corpuscle configuration, which we register as frequencies.	(We	don't	assume	–	and	in	fact	expressly	reject,	see	IV.3	and	IV.5.2	–	the identity	of	the	corpuscles	between	different	localisations:	Corpuscles	don't	persist	in time,	lacking	diachronic	identity.)	Thus,	a	change	in	a	corpuscle's	position	no	longer requires the presence of a triggering mechanism – as a literal reading of sdBBT's formalism	seems	to	call	for. Note that the corpuscles' spontaneous localisation isn't a "collapse of the wavefunction": The disposition continues to evolve unitarily also after its manifestation. The probability measure quantifies the strength of such a propensity 94 , with a configuration of zero probability possessing minimal propensity to happen. That doesn't	mean,	though,	that	the	configuration	is	impossible.	(Recall	that	generally,	for every	continuous	probability	distribution	each	single,	possible	outcome	has	probability zero.)	Conversely,	a	configuration	with	probability	1	isn't	necessary/inevitable95;	it	only has the greatest tendency to become actualised. (Dispositionalism suggests a modification	of	the	notion	of	(nomological)	impossibility:	An	event	may	be	said	to	be impossible, iff it cannot be assigned a probability value. Adopting this notion of nomological	possibility	allows	us	to	pre-empt	a	metaphysical	objection	against	sdBBT's ontology:	namely	that	it	postulates	an	incessant	creatio	ex	nihilo,	with	the	corpuscles popping	into/out	of	existence	from/back	into	absolute	nothingness,	respectively.	Such spontaneous materialisations and de-materialisations apparently contradict the received	metaphysical principle of substance conservation, i.e. that substances can neither	spontaneously	emerge	nor	perish.96	Dispositionalism	avoids	the	conflict	with that	principle,	though,	since	firstly	the	N	corpuscles'	disposition	to	localise	themselves stretches throughout	all	of space:	The	disposition is	everywhere (not,	of	course, its manifestation).	Secondly,	the	number	of	corpuscles	is	also	conserved:	At	any	instant, there	are	always	N	actual	corpuscles.	Hence,	the	corpuscles	are	not	popping	into	/out 94	The	status	of	the	wavefunction	may	be	seen	as	the	gauge	field	(analogous	to	the	electromagnetic	4-potential in	electromagnetism)	with	the	corresponding	physical	field	(analogous	to	the	Faraday	tensor). 95	Cf.	Bunge	(2011),	sect.	6.3. 96 30 of	existence	out	of/into	absolute	nothingness	or	empty	space.	One	may,	of	course, criticise	the	idea	that	a	disposition	elicits	spontaneous,	random	manifestations	–	but that's	a	general	scepticism	of	a	dispositional	metaphysics,	not	specific	to	sdBBT.) Advocating	above	a	single-case	propensity	interpretation,	it	now	behoves	us	to	address	the canonical	objections	against	single-case	propensities	–	an	opportunity	also	to	elucidate	some of	sdBBT's	salient	features. • The Reference Class Problem 97 arises for propensity interpretations when, as in Popper's	propensity	theory	from	1957,	probabilities	are	assigned	to	an	experimental setup	of	physical	conditions	that	generate	the	observed	outcomes:	The	propensity	of an	event	is	thus	always	relative	to	the	generating	conditions.	These,	however,	can	be incorporated in	various,	different classes.	Consequently,	one	cannot	unambiguously assign	an	event	its	propensity. While the Reference Class Problem notoriously plagues frequentism and long-run propensities, sdBBT – like most variants of QM 98 – escapes it 99 : The system's wavefunction completely determines the dBB-propensities; no further facts of the system	as	"generating	conditions"	are	necessary.	Such	propensities	are	understood	as quantities	inherent	in	the	corpuscles. • Hájek	criticises	this	manoeuvre	to	circumvent	the	Reference	Class	Problem	as	exacting the price of a vacuous notion of propensity:100	Indeed, propensity approaches are often dismissed as pseudo-explanatory of the type a quack physician dishes up in Molière's Le Malade Imaginaire, when "explaining" the sleep-inducing powers of opium	through	a	virtus	dormitiva. This	complaint,	however,	rests	on	a	misunderstanding:	Quantification,	explanation	and interpretation	are	distinct	conceptual	operations. o Propensity	accounts	don't	pretend	to	be	explanations	(i.e.	answers	to:	"How can we derive the BR-probabilities from a certain metaphysical or physical 97	Cf.	Hájek	(2006) 98	Cf.,	e.g.	Galavotti	(2001) 99	Cf.	Frigg/Hoefer	(2010),	sect.	3	and	5.	The	argument	carries	over	verbatim. 100	Cf.	Hájek	(2006),	pp.	25 31 theoretical framework"	– in the	manner	of, say,	Robb's (failed) attempts to derive	the	metric	structure	from	causal	relations101). o Rather,	propensities	intend	to	interpret	and	ontologically	ground	probabilities: "What	could	the	concept	of	probability	refer	to	in	a	world	described	by	sdBBT?" In	sdBBT,	propensities	are	inherent,	dispositional,	real	quantities	–	figuring	as truthmakers	of	probabilistic	statements. o In turn, probabilities formalise and quantify the vaguer, pre-theoretical, qualitative	and	ontological	notion	of	a	propensity:	"How	to	render	the	concept of a propensity sufficiently mathematically precise?" To this end, Kolmogorow's axioms are imposed as formal desiderata, whose	motivation (esp.	that	of	σ-addivity)	shall	not	concern	us	here.	Once	imposed,	probabilistic statements, thus rendered quantitative, can subsequently be subjected to empirical	tests.102 • Arguably "devastating for views that take propensities to involve weakened or intermittent	causation"103	is	Humphreys'	Paradox.	It	boils	down	to	the	incompatibility between	inverse	conditional	probabilities,	understood	as	propensities,	and	the	timeasymmetry	of	causality. Consider	a	partition B": i ∈ I of	the	sample	space	of	physical	states	of	some	system, Ω = B""∈¿ , with	∀i ≠ k:	B" ∩ BU = ∅ , and let	A ∈ Ω be an event. Then, Bayes' Theorem	purports:	∀i ∈ I:	P B" A = P(Ã Ä;)P(Ä;) P(Ã ÄÅ)P(ÄÅ)Å∈AE .	What	does	this	signify	in	terms of	propensities? It	seems	natural	to	interpret	the	conditional	probability	P(A B")	as	the	propensity	of the system	to	undergo the transition	B" ↝	A.	This suggests to	understand	B" as the cause of the effect A , and hence conditional probabilities as representing a probabilistic	form	of	causation. The conditional probability for the inverse transition,	P B" A , definable via	Bayes' Theorem,	would	thus	quantify	the	tendency	for	the	effect	A	to	bring	about	the	cause B". 101	Cf.	Sklar	(1992),	pp.	83 102	Cf.	Georgii	(2011),	Ch.10	&11;	Gillies	(2000),	Ch.	6,7;	Beisbart	(2011),	esp.	4.2;	Suárez	(2014) 103	Eagle	(2004),	p.	402 32 In	consequence,	not	only	would	smoking	cause	lung	cancer,	but	also	conversely	would lung	cancer	cause	smoking.	This	seems	absurd:	Only	causes	produce	their	effects,	not vice versa; causal chains of events cannot be inversed. Probabilities therefore, the argument	concludes,	cannot	be	understood	in	terms	of	propensities. A	closer	inspection	of	the	paradox	is	apposite.	Three	groups	of	posits	enter	it: (i) We adopt standard probability calculus, with probability measures obeying Kolmogorow's	axioms. (ii) The second group of posits comprises assumptions about the nature of propensities	(denoted	by	P,	time-indexed	w.r.t.	t= < t9 < t2)	in	a	causal	process A(t=) ↝ B(t9) ↝ C t2 (i.e.	chain	of	events,	A,	B	and	C): a. The propensity of the process A(t=) ↝ B t9 is nontrivial: 1 > P&Ê B t9 A(t=) > 0 b. The	propensity	of	the	causal	chain	isn't	minimal	: P&Ê C t2 A t= ∧ B t9 > 0. c. In the absence	of	B, the process	A(t=) ↝ C t2 has	minimal propensity: P&Ê C t2 A t= ∧ ¬B t9 = 0. d. Future events are causally neutral/irrelevant for past events: P&Ê B t9 A t= ∧ C t2 = P&Ê B t9 A t= ∧ ¬C t2 = P&Ê B t9 A t= (iii) The last posit bridges (i) and (ii),	with an identity thesis, according to	which all propensities	P&	can	be	uniquely	associated	with	probabilities,	with	the	strength	σ of	the	propensity	being	identified	with	the	probability: P& → P&, ∀x: P& x ↦ σ P& x ≡ P& x Humphreys'	Paradox	now	consists	in	the	inconsistency	that	arises	from	the	conjunction	of (i), (ii) and (iii). Which premise therefore to relinquish in order to overcome the contradiction? In	contradistinction	to	Humphreys	himself,	we	wish	to	remain	as	conservative	as	possible, and	retain	the	standard	probability	calculus. 33 Accepting	the	causal	intuitions	of	propensities	captured	in	(ii),	Suárez	has	pled	for	rejecting the	identity	thesis104	–	in	particular,	its	first	part:	Not	all	probabilities	are	amenable	to	a propensity interpretation; some conditional probabilities in particular don't refer to factually	possible	transitions	in	the	world.	This	isn't	unfamiliar	from	other	theories	whose formalism	also treats	physically impossible situations, forbidden	by	extra-mathematical selection rules. E.g. in standard QM, N-particle wavefunctions transform either symmetrically or anti-symmetrically under permutations; formally, nothing prevents mixed-symmetric	transformation	behaviour.	Nature,	though,	doesn't	seem	to	realise	this option. In	their	defence	of	propensity	interpretations	in	GRW	collapse-theories,	Hoefer	and	Frigg have indeed abolished the identity thesis by asserting "that GRW propensities are all forward-looking in time" 105 – a "response that any advocate of objective quantum probabilities	will	wish	to	make."106 In	addition	to	abandoning	the identity	thesis (iii),	sdBBT	compels	us	to	abandon (iic): It captures the intuition that in a causal chain	A(t=) ↝ B(t9) ↝ C t2 the intermediary event	B	is	indispensable;	i.e.	in	B's	absence,	the	chain	has	minimal	propensity	to	occur.	In sdBBT,	however,	the	corpuscles'	configurations	spontaneously	and	randomly	jump	from one instant to another, independently of previous configurations; sequences of configurations	are	no	longer	causally	connected.	In	other	words,	there	exists	no	triplet	of events	that	satisfies	(iic);	in	sdBBT	(iic)	is	violated.	Humphreys'	Paradox	thus	lapses. Spurred by those doubts regarding the conceptual tenability of propensity interpretations simpliciter,	as	well	as	perhaps	by	discomfort	about	the	lack	of	direct,	empirical	falsifiability	of propensities107,	one	might	generally	baulk	at	our	invocation	of	a	metaphysics	of	dispositions. What	swayed	us?	We'll	consider	first	reasons	to	regard	the	wavefunction	real	("wavefunction 104	He	furthermore	makes	a	case	against	conversely	identifying	all	propensities	with	probabilities. 105	Frigg/Hoefer	(2007),	p.385 106	Loc.cit. 107 As well as the "inferential link" between frequencies as estimates for probabilities and the "decisiontheoretical	link"	concerning	the	role	probabilities	play	in	decisions	and	action-relevant	choices,	cf.	e.g.,	Brown (2011),	esp.	sect.	3. Such	issues	are	intimately	tied	up	with	the	debate	between	Humeanism	and	Governing	Law	Approaches,	which lies	outside	our	current	scope.	We	refer	to	e.g.	Maudlin	(2007),	esp.	Ch.	2	for	further	details. 34 realism"); we'll then argue that viewing the wavefunction more specifically as a real, dispositional	quantity	overcomes	the	key	problems	of	wavefunction	realism. Above,	we	declared	the	corpuscles'	propensity	a	real/physical	property,	represented	by	the wavefunction. This wavefunction realism naturally accounts for the wavefunction's contingency,	structural	complexity	and	time-evolution108: • The	wavefunction	has	dynamical	degrees	of	freedom	of	its	own,	governed	by	the	SE and	dependent	on initial/boundary conditions.	Varying	across	different	worlds that differ in those conditions, the wavefunction consequently represents a contingent quantity	–	as	opposed	to	absolute	objects,	which	don't	vary	across	possible	worlds, e.g. the Minkowski spacetime metric in Special Relativity. Contingency is seen as characteristic	of	real	entities	–	as	opposed	to	merely	conventional	ones. As	Brown	and	Wallace	suggestively	point	out,	"(h)istorically,	it	was	exactly	when	the gravitational and electric fields began to be attributed independent dynamics and degrees	of	freedom	that	they	were	reified:	the	Coulomb	or	Newtonian	'fields'	may	be convenient	mathematical	fictions,	but	the	Maxwell	field	or	the	dynamical	spacetime metric	are	almost	universally	accepted	as	part	of	the	ontology	of	modern	physics."109 • Its contingency also implies that	we can to some	extent control the	wavefunction, insofar	as	we	can	prepare	a	physical	system	via	initial	and	boundary	conditions.	This controllability speaks strongly against a nomological interpretation of the wavefunction110,	which conceive	of the latter as a law.	As	Goldstein/Zanghì admit: "And	laws	are	not	supposed	to	be	things	that	we	can	control	–	we're	not	God."111 108	Cf.	Brown/Wallace	(2004),	pp.	12 109	Loc.cit. 110	E.g.	Esfeld	et	al.	(2014) 111	Goldstein/Zanghì	(2013),	p.99.	Their	counterargument	is	"[...]	that	there	is	only	one	wave	function	we	should be	worrying	about,	the	fundamental	one,	the	wave	function	Ψ	of	the	universe"	(loc.cit.),	and	that	the	latter	is not controllable. Taking up their invocation of God, one needs to distinguish between two forms of Divine intervention:	On	the	one	hand,	God's	thaumaturgical	ability	to	suspend	the	extant	laws	of	nature,	and	on	the other	hand	his	weakly	and	strongly	demiurgical	ability	to	create	worlds:	The	former	refers	to	the	ability	to	create distinct nomologically possible worlds, differing only w.r.t. contingent elements, such as initial/boundary conditions, while the non-contingent ("nomological") elements remain the same; strong demiurgy refers to God's	ability	to	create	worlds	in	which	different	laws	of	nature	hold. The	kind	of	uncontrollability	on	which the intuitive	notion	of	a law turns is	weak	demiurgy. In consequence, assuming	that	He	shares	our	ontological	intuitions,	God,	too,	arguably	would	reject	the	nomological	view. 35 • Like	other	real	quantities,	such	as	the	metric	in	General	Relativity,	subject	to	a	set	of 16 coupled hyperbolic-elliptic nonlinear PDEs, sdBBT's wavefunction-dependent propensity	is	structurally	very	rich. In	both	sdBBT	and	dBBT,	this	structural	complexity	mirrors	its	ontological	importance in that the wavefunction is the only property (other than positions) that can be assigned	to	corpuscles. • sdBBT's	propensity	evolves	in	time	according	to	the	SE.	It	seems	unnatural	to	regard something	that	thus	changes	over	time	as	not	real;	in	fact,	mutability	has	even	been proposed	as	the	essential	difference	between	purely	conceptual/"Platonic"	abstracta and	physical	concreta.112 Despite	indeed	suggesting	that	the	wavefunction	represents	a	real	entity	(rather	than,	say,	a law-like	abstractum,	as	in	the	nomological	interpretation)	–	Brown	and	Wallace's	"reification" argument	remains	still	silent	on	the	ontological	category	under	which	to	subsume	the	"reified" entity:	Is	it	a	substance	or	property? For	an	answer	within	a	traditional	ontological	framework,	let's	analyse	the	extent	to	which	the wavefunction carries itself properties or not; the former would suggest its status as a substance. Given	the	dependence	of	the	wavefunction	–	as	a	solution	to	the	SE	–	on	parameters	such	as mass	or charge,	Holland	has proposed that the	wavefunction itself should	be regarded as massive,	charged	etc.113	Discussing	neutron	interferometric	thought	experiments,	Brown	et al. elaborate this proposal to attribute the state-independent, non-dynamical properties mass, charge, magnetic moment, etc. to both the wavefunction and the Bohmian corpuscles.114 One	of	the	core	features	of	the	traditional	notion	of	substance115	is	"ultimate	subjecthood" (i.e.	being	predicable	without	itself	being	predicated	of	something	else).	It	suggests	that	the wavefunction	should	be	regarded	as	a	substance,	yielding,	prima	facie,	two	substances,	Ψ	and the corpuscles. One may find this substance-dualism already unpalatable enough. 112	Cf.	Bunge	(1981) 113	Cf.	Holland	(1993),	pp.	78 114	Cf.	Brown	et	al.	(1995);	Brown	et	al.	(1996),	sect.	3	and	4 115	Cf.,	e.g.,	Kuhlmann	(2010),	Ch.	II.7 36 Ontologically	yet	more	disconcerting	is	that	in	dBBT	the	corpuscles,	albeit	ultimate	subjects	of position ascription, ignore another traditional category feature of substances, viz. independence:	Via	QEH	and the	GE, respectively, the corpuscles' initial configurations	and behaviour	are	determined	by	the	wavefunction.	(A	converse	dependence	of	the	wavefunction on the corpuscles	doesn't obtain	– an issue	we'll discuss in III.3.). In consequence, dBBT's ontology	is	a	substance-dualism	consisting	of	one	bona	fide	substance,	viz.	the	wavefunction, and	one	entity,	viz.	the	corpuscles,	whose	ontological	category	is	ambivalent;	the	fact	that	the wavefunction and the corpuscles arguably share the state-independent properties mass, charge,	etc.	further	exacerbates	this	ontological	uneasiness. We	submit,	sdBBT's	commitment	to	a	metaphysics	of	dispositions	("dispositionalism")	offers a	neater	ontological	picture:	While	concurring	with	the	aforesaid	proposal	to	ascribe	the	nondynamical properties also to the wavefunction, sdBBT needn't classify the latter as a substance: The	wavefunction represents corpuscles' disposition of spontaneously popping into existence, both localising themselves at certain sites with certain masses, charges, magnetic	momenta,	etc.	Thus,	sdBBT's	corpuscles	are	the	only	substances.	They	are	ascribed properties, which fall into two inter-related types – the disposition of spontaneous materialisation, represented	by the	wavefunction, and the corresponding	manifestation	of that	disposition	of	the	corpuscles	to	randomly	localise	themselves	at	certain	positions,	with certain state-independent properties. That the wavefunction is defined on configuration space, and that the corpuscles and the wavefunction can both be attributed the stateindependent properties, now simply reflects the ontological dependence between dispositions	and	their	manifestations. To some, our subscription to such a "revisionary	metaphysics" (Strawson) that posits also dispositions,	rather	than	exclusively	categorical	properties,	may	seem	appalling,	e.g.	on	the grounds	of	dispositions	defying	direct	empirical	access	–	an	objection,	however,	that	applies generically	to	theoretical	terms	and	runs	into	the	pitfalls	already	the	Logical	Positivists	had	to face	with	their	observational/theoretical	dichotomy.116 Dispositionalism has merits both specifically in the germane context of GRW theories117 (arguments	that	carry	over	verbatim	to	sdBBT),	as	well	as	generally	in	its	own	right.	Amongst 116	Cf.,	for	instance,	Suppe	(1974),	esp.	pp.	85 117	Cf.	Dorato/Esfeld	(2010);	for	a	slightly	different	view,	cf.	Frigg/Hoefer	(2006),	esp.	sect.	5 37 them,	the	literature,	to	which	we	refer	for	details118,	extolls	the	following:	"[...]	a	clear	sense in	which quantum systems in entangled states possess properties even in the absence of definite values; [...] a clear transition from quantum to classical properties; [...] a clear transition from quantum to classical structures; and [...] [the grounding of] the arrow of time."119 The crucial advantage of dispositionalism in our context at hand, we submit, is that the dispositional character of the wavefunction can straightforwardly account for the 3Ndimensionality	of	the	configuration	space,	on	which	the	wavefunction	is	defined.120 By	means	of	contrast,	consider	two	alternate	proposals: • "Configuration Space Realism" regards the wavefunction as a real object in configuration	space.	It	faces	"the	Problem	of	Perception":	If	the	fundamental	quantum world lives in	3Ndimensional	configuration	space,	how is this	compatible	with	the world	ostensibly	unfolding	itself	in	3D	physical	space?	How	to	adjudicate	between	the rivalling	claims	of	reality	being	3N-dimensional	vs.	3-dimensional? • "Multifield Realism" tries to avoid the Problem of Perception by regarding both configuration space and 3-space as equally real; the wavefunction, acting like an invisible hand on the particles and fields by guiding their dynamics, encodes a multitude of fields in ordinary 3D-space.	Multifield Realism faces "the Problem of Communication", though: How does the Multifield Wavefunction impart on the objects	in	3-space	the	relevant	physical	information?	What's	the	mechanism	by	which it	affects	them? How	does	sdBBT	ward	off	these	problems	and	position	itself? The	above	description	of	the	Problem	of	Perception	implicitly	presumed	that	wavefunction realism	entails	that	the	wavefunction	is	"fundamental"	in	the	sense	of	a	substantial,	physical field,	i.e.	a	spatially	extended	body,	which	hence	indeed	one	would	expect	to	be	defined	on	3space. Recall, however, our previous remark that wavefunction realism only asserts the reality, not necessarily the ontological fundamentality/substantiality of the wavefunction. 118	Cf.,	for	instance,	Choi/Fara	(2012)	or	Esfeld	(2008) 119	Dorato/Esfeld	(2010),	p.	41 120	Cf.	Suárez	(2015). 38 SdBBT thus remains wavefunction realist, whilst conceiving of the wavefunction as a (dispositional) property. Consequently, the intuition that the wavefunction, no longer understood as a substantial/physical field, must be defined on physical space loses its plausibility; the	wavefunction	represents	a	property	of the	N-corpuscle	universe.	Since the ontologically subordinate wavefunction and the ontologically primary corpuscles don't compete	for	fundamentality,	the	spaces	they	each	inhabit	–	configuration	space	vs.	3-space, respectively	–	can	peacefully	coexist. Put	differently,	our	actual	perceptions	are	based	on	the	actual	positions	the	corpuscles	occupy in	3-space;	these	positions	in	turn	are	the	manifestations	of	the	disposition,	represented	by the	wavefunction.	Consequently,	the	compartmentalisation	of	reality	into	dispositions	(where the	wavefunction	belongs)	and	their	manifestations	(where	all	actual	configurations	belong, including	those	we	detect	and	observe)	resolves	the	Problem	of	Perception. The	Problem	of Communication is overcome in tandem: The	wavefunction and the actual configurations	fall	into	different	ontological	compartments.	Consequently,	the	wavefunction can't	act	on	the	configurations	in	any	causal	form. Dispositionalism	avoids	the	conflict	with	that	principle,	though,	since	firstly	the	N	corpuscles' disposition to localise themselves stretches throughout all of space: The disposition is everywhere (not, of course, its	manifestation). Secondly, the	number	of corpuscles is also conserved:	At	any	instants,	there	are	always	N	actual	corpuscles.	Hence,	the	corpuscles	are not	popping	into	/out	of	existence	out	of/into	absolute	nothingness	or	empty	space. Let's	close	this	subsection	with	a	remark	on	the	global	nature	of	the	disposition	at	hand.	The wavefunction	Ψ& , representing the propensity of the N-corpuscle universe to pop into existence	with	a	certain	configuration	Q:= Q=,... ,Q3 ∈ R23, is	a	holistic	(non-separable, but	not	non-local)	property	of	the	whole	system.	It	doesn't	supervene	on	the	properties	of each individual corpuscles.121	Rather, the propensity	PlÏ Q¦ ∈ Q of the i-th corpuscle to localise itself in the region	Q ⊂ R2 in physical 3D space, derives from the global, usually entangled	wavefunction	via: 121	Cf.	also	Esfeld	et	al.	(2014) 39 PlÏ Q" ∈ Q = d2Q¦ Ψ& Q 9 3 ¦<= ¦Ò" Q . Deferring	for	the	moment	a	detailed	discussion	of	the	non-separability	of	the	wavefunction	to section	IV.5.4,	let's	see	how	our	proposed	interpretation	of	sdBBT's	wavefunction	as	a	holistic disposition	overcomes	the	alleged	inability	to	account	for	interactions	that	Esfeld	and	Gisin have recently diagnosed for primitive ontologies, with discrete beables whose temporal sequences	are	no	longer	continuous122	("flash	ontologies"): According to	Maudlin,	one form	of the	problem,	as it arises in	GRW-flash, is that "(f)or to accept	this	theory	is	to	accept	that	microscopic	reality	is	nothing	at	all	like	what	we	took	it	to be	–	not	even	the	parts	that	we	naively	accepted	as	'revealed'	to	us	by	microscopes.	The	balland-stick	models	of	DNA	–	which	do	an	admirable	job	of	accounting	for	cellular	behaviour	– would be wildly	misleading: a strand of DNA	would show up in space-time as a sparsely scattered	set	of	flashes,	which	would	hardly	suggest,	over	reasonable	time	periods,	a	double helix.	Accepting	the	flash	ontology	entails	rejecting	the	space-time	picture	of	cellular	structure that	has	guided	the	great	advances	in	medicine	and	biochemistry.	The	theory	does,	of	course, account	for	the	practical	success	of	that	picture."123	In	other	words,	Maudlin	appeals	to	the realist principle that the accounts the various disciplines envision of their scrutinised fragments	of	reality	be	mutually	consistent,	rather	than	contradict	each	other;	the	GRW-flash theory,	Maudlin's	objection	asserts, flouts this	principle:	According to	GRW-flash theory,	a DNA	strand	materialises	approximately	only	once	per	day,	whereas	the	DNA	model	biologists successfully	utilise	in	their	research	assumes	that	the	DNA	strand	persists. Despite its likewise "flashy" ontology, sdBBT obeys	Maudlin's principle: As a result of the entire	N-corpuscle universe performing a random jump through configuration space, the macroscopic objects of our stochastic Bohmian world are indeed composed of actual corpuscles. To	be	sure,	sdBBT	radically	revises	our	picture	of	the	world,	namely	implying	that we	most	probably	only	exist	for	one	instant	in	the	whole	history	of	the	universe	(more	on	this "temporal solipsism" in section IV.3); surprisingly, though, as we shall see, this doesn't 122	Esfeld/Gisin	(2014),	sect.	5,	whom	we	follow	here. 123	Maudlin	(2011b),	p.	257 40 contradict	the	picture	of	the	world	we	get	from	other	sciences,	as	long	as	this	is	understood as	the	best	guess	of	how	the	world	looks,	given	our	current	evidence. Arguably a more serious version of the problem concerns measurements: What is a measurement device supposed to interact with, if the measured quantum object, say a trapped	electron,	only	materialises	once	in	a	blue	moon?	Esfeld	and	Gisin	write:	"Let	us	take for	granted	that	the	flash	ontology	can	account	for	macroscopic	objects	such	as	measuring apparatuses in terms of 'galaxies of flashes'. But on the flash ontology, there is nothing between	the	source	of	the	experiment	and	the	measurement	apparatus.	In	other	words,	there is	nothing	with	which	the	apparatus	could	interact:	there	is	no	particle	that	enters	it,	no	mass density	and	no	field	that	gets	in	touch	with	it,	either."124	They	conclude	that	the	flash	ontology is	too	sparse	to account	for	interactions.	How	to	get	out	of	this	pickle?	The	previous	argument is	of	no	avail in this regard:	Although	at	each instant	of time	our	ambient	macrocosmos is made	up	of	actual	corpuscles,	positing	an	interaction	between	these	corpuscles	would	amount to	an	instantaneous	action-at-a-distance,	which	not	only	falls	short	of	a	plausible	mechanism, but	also	rubs	against	the	relativity	of	simultaneity	at	the	heart	of	Special	Relativity.125 Our	dispositionalist	interpretation	of	sdBBT	resolves	the	problem:	The	dispositional	nature	of the wavefunction warrants that both the	measurement system and the quantum system indeed	always	exist,	albeit	not	necessarily	actualiter. To	establish the	connection	between measurement	system	and	observed	system,	now	the	wavefunction's	holistic	nature	comes into	play:	Both systems	are fundamentally inseparable,	with	only the	wavefunction	of the compound	system	being	the	complete	quantum	state,	representing	the	holistic	disposition	of the	pair	(configuration	of	the	measurement	apparatus,	configuration	of	the	measured	system) popping	into	existence,	with	its	statistical	correlations	as	brute	facts	of	the	manifestation	of the	holistic	disposition. IV:	Critical	Analysis	of	sdBBT After	expounding	in	the	previous	section	some	conceptual	and	metaphysical	idiosyncrasies	of sdBBT	and	addressing	some	enquiries	regarding	its	consistency,	let's	now	turn	to	two	sorts	of further	qualms: 124	Esfeld/Gisin	(2014),	p.	12 125	E.g.	Maudlin	(2011) 41 • A	first	one	concerns	the	taxonomisation	of	sdBBT	as	a	deBroglie-Bohmian	theory:	Does the family resemblance between dBBT and sdBBT sanctify or do the differences contest	it? • Such	a	comparison	evokes	a	second	question: Is sdBBT	merely	a	phenomenological theory,	and	hence	not	on	a	par	with	theories,	such	as	dBBT,	that lay	claim	to	being fundamental?	And	furthermore,	if	we	grant	that	sdBBT	is	a	fundamental	theory,	what are	its	decisive	advantages	over	dBBT	that	might	justify	taking	sdBBT	indeed	seriously? IV.1:	sdBBT	as	a	minimally	deBroglie-Bohmian	theory A	reader	who	has	charitably	followed	our	discussion	of	sdBBT	so	far	might	yet	hesitate:	To what	extent	may	sdBBT	still	be	classified	as	a	deBroglie-Bohmian	theory?	Is	sdBBT	sailing	under false	colours? Note	first	that	sdBBT	differs	from	ordinary	QM	in	its	ontological	premises,	all	of	which	sdBBT shares	with	dBBT	(except	for	determinism):	In	QM,	no	pair	of	conjugated	dynamical	variables is	ontologically	(or	otherwise)	preferred	over	another;	in	particular,	position	and	momentum are	on	a	par126.	(In	the	case	of	Everett's	Many	World	Interpretation	of	QM,	this	symmetry,	i.e. the	absence	of	any	intrinsically	distinguished	dynamical	variables,	has	occasionally	led	to	the charge of the so-called Problem	of Preferred Basis.) By contrast, sdBBT, albeit empirically equivalent to standard QM, breaks this symmetry of all pairs of conjugated variables by distinguishing	position	as	the	sole	"beable"	(Bell), i.e.	the	dynamical	variable	that is	always value-definite/-determinate;	furthermore,	position	is	the	only	non-contextual	variable,	i.e.	in both dBBT and sdBBT corpuscles have no dynamical properties other than position. In consequence, Bell's identification 127 of sdBBT as a version of Everett's Many Worlds Interpretation	of	QM	is	mistaken. This	distinction	of	position	as	a	beable	can	be	seen	to	be	motivated	physically	by	the	fact	that decoherence	– the	entanglement	of systems	with their ambient	environment	–	acts	as	an effective superselection rule,	with	position	as	a	preferred	variable, relatively stable/robust 126 Pauli and Heisenberg reiterated this difference in their criticism of dBBT as introducing an "artificial asymmetry",	cf.	Myrvold	(2003),	sect.	3,	who	also	critically	evaluates	it. 127	Contra	Bell	(1987),	Ch.	5,	for	whom	"keeping	the	instantaneous	configurations,	but	discarding	the	trajectory is	the	essential	[...]	of	the	theory	of	Everett",	loc.cit.,	p.	133. By	contrast,	Daumer	et	al.,	cf.	the	chart	on	p.	393	implicitly	distinguish	between	the	Everett	interpretation	and sdBBT	(despite	labeling	the	latter	"Bell's	reformulation	of	Everett's	theory",	cf.	op.	cit.,	fn	13). 42 under interference, and in	particular, entanglement	effects, and commuting	with	all other observables. Conversely, such	a	distinction	by stipulation	has explanatory surplus value: It supplies a clear primitive ontology	whose	merits, such as an advantage in the debate on scientific	realism,	have	been	extolled	elsewhere.128 In sdBBT,	we just	dismantled from the	orthodox	dBBT the	"superfluous 'ideological superstructure'"129	that	undermined	the	consistency	of	an	objectivist	interpretation	of	all	its	three axioms	and	was	the	culprit,	subject	of	a	number	of	further	complaints.	Sharing	the	ontological framework	of	dBBT,	which	we	deem	the	hallmark	of	the	deBroglie-Bohmian	theory	paradigm, sdBBT is thus fully-fledged "deBroglie-Bohmian", of which it may be seen as a minimal version.130	(N.B.:	Other	theories	might	as	well	count	as	minimally	deBroglie-Bohmian	theories, Nelson	Stochastics	being	a	potential	candidate,	see	sect.	VI.)	More	formally,	we	stipulate	that a theory	be called	deBroglie-Bohmian, iff it's a so-called "primitive-ontology theory"131	(as opposed	to	a	so-called	"wavefunction	ontology"	theory,	such	as	the	Everett	interpretation) with	the	following	features132: (1) Its	objects	are	particles	or	corpuscles	–	as	opposed	to	e.g.	primitive	ontology	theories with strings, flashes or matter fields as fundamental stuff. (N.B.: Wavefunction ontology	theories	postulate	the	wavefunction	as	the	only	fundamental	constituent	of reality,	with	no	need	for	stuff	over	and	above	it.133) 128	Cf.	Allori	et	al.	(2008);	Allori	(2013,	2016) 129	Heisenberg	(1953),	quoted	in	Myrvold	(2003),	p.	12 130	Fine	(1996)	comes	to	the	same	conclusion	and	encourages	a	realist	understanding	of	sdBBT	(avant	la	letter): "At	the	heart	of	Bohmian	mechanics	ist	the	wavefunction	and	determinate	particle	positions,	and	perhaps	we need	be	realists	about	nothing	else",	p.	249. 131	Cf.	Allori	et	al.	(2008);	Allori	(2015) 132	Cf.	also	Dürr	et	al.'s	criteria	for	a	theory	to	count	as	deBroglie-Bohmian:	"A	Bohmian	theory	should	be	based upon a clear ontology, the primitive ontology, corresponding roughly to Bell's local beables. [...] For the nonrelativistic	theory	[...]	the	primitive	ontology	is	given	by	particles	described	by	their	positions,	but	we	see	no compelling	reason	to	insist	upon	this	ontology	for	a	relativistic	extension	of	Bohmian	Mechanics.	[...]	There	should be	a	quantum	state,	a	wave	function,	that	evolves	according	to	the	unitary	quantum	evolution	and	whose	role	is to	somehow	generate the	motion for the	variables	describing the	primitive	ontology.	The	predictions	should agree	(at	least	approximately)	with	those	of	orthodox	quantum	theory	–	at	least	to	the	extent	that	the	latter	are unambiguous."	Dürr	et	al.	(1995),	sect.	5 133	It	deserves	to	be	pointed	out	that	despite	some	important	similarities,	viz.	the	assumptions	that	the	world	is made	up	of	discrete	events	in	spacetime	and	that	these	are	stripped	of	their	spatiotemporal	continuity,	the ontology	of	the	GRW	flash	theory	differs	(contra	e.g.	Esfeld	et	al.	(2015),	p.5)	from	sdBBT's	in	that	the	latter	is	a particle-based	primitive	ontology,	whereas	the	former	falls	under	wavefunction	ontology	approaches;	by contradistinction,	with	its	interpretation	of	the	wavefunction	as	a	mass	density	field	of	matter	in	physical space,	the	GRW	matter	theory	is	a	field-based	primitive	ontology,	cf.,	for	instance,	Esfeld	et	al.	(2015).	In	short, sdBBT's	stuff	consists	of	particles,	GRWf's	in	events	of	the	wavefunction. 43 (2) The	positions of these corpuscles have	beable status – as opposed to e.g. fermion number	in	Bell's	merely	Bohm-like	lattice	Quantum	Field	Theory.134 (3) To secure compatibility with the Kochen-Specker-Theorem, all dynamical variables other	than	position	are	contextual. (4) It's	fully	empirically	equivalent	with	QM	–	as	opposed	to	objective	collapse	theories, which	tamper	with	the	Schrödinger	dynamics. Within this family of theories, sdBBT seems to be the ontologically most parsimonious member. IV.2:	sdBBT	and	realism One might object: Wasn't one of the key points of the hidden-variables agenda to reinaugurate	realism	by	dispensing	with	QM's	probabilities?	E.g.,	according	to	von	Neumann, the	refusal	to	accept	indeterminism	inherent	in	QM	underlies	the	search	for	hidden	variable theories.135	So,	doesn't	sdBBT,	as	an	irreducibly	stochastic	theory,	defeat	its	purpose? The	question	rests	on	a	misunderstanding	that	conflates	determinism	and	realism.	Dürr	and Teufel	rectify	this	view:	"It	is	often	said	that	the	aim	of	(dBBT)	is	to	restore	determinism	in	the quantum world. That is false. [...] What is 'out there' could just as well be governed by stochastic	laws	[...].	A	realistic	quantum	theory	is	a	quantum	theory	which	spells	out	what	it	is about.	(dBBT)	is	a	realistic	quantum	theory.	It	happens	to	be	deterministic	[...].	The	merit	of (dBBT)	is	not	determinism,	but	the	refutation	of	all	claims	that	quantum	mechanics	cannot	be reconciled	with	a realistic	description	of reality."136	In short: In giving	up	dBBT's "fictitious determinism",	we	needn't	succumb	to	anti-realism. Note, however, as Fine points out, that dBBT compels us to disentangle the two claims underlying	classical realism137: firstly, the	metaphysical	one, that there	exists	an	observerindependent world; and secondly, the epistemological one, that our measurements and 134	Cf.	Dürr	et	al.	(2004) 135	Von	Neumann	(1932),	Ch.	III.2 136	Dürr/Teufel	(2009),	pp.	10	(our	emphasis). In	the	same	spirit,	Goldstein	et	al.	declare	that	the	goal	of	dBBT	"[...]	is	to	replace	the	measurement	postulate	of standard	quantum	mechanics	with	postulates that refer to	electrons	and	nuclei instead	of	observers,	axioms from	which	the	measurement	rules	can	be	derived",	Goldstein	et	al.	(2009),	p.	11. 137	Cf.	Fine	(1996),	Sect.	6 44 observations	disclose	features	of	the	world	as	they	had	existed	before	and	independent	of	the measurement	act. While	dBBT	and	sdBBT	doubtlessly	both	embrace the	metaphysical component	of classical realism ("objectivism"), a measurement act in dBBT – itself a physical process – can occasionally	disturb	the	state	of	the	measured	system,	so	that	its	pre-measurement	state	and the	state	revealed	through	the	measurement	differ:	Measurements	can	be	invasive,	and	not merely passive records of the state of affairs. Already Bohm, in his treatment of the 1dimensional particle in a box (see Sect. III.3) acknowledged this turning away from the epistemological	component	of	classical	realism:	"According	to	the	Bohmian	prescription	for velocity, the unmeasured particle is actually standing still! Measurement disturbs the situation,	freeing	the	wave	function,	which	guides	the	particle	into	motion."138 A	generic	feature	of	theories	in	which	the	wavefunction	plays	a	prominent	role	for	the	state of	a	system,	the	interaction	with	a	measurement	device	also	in	sdBBT	can	considerably	affect the	wavefunction of the	measured system:	Measurements thus can no longer be seen as passively	reading	out	pre-existing	values.	This	is	vividly	demonstrated	by	the	Quantum	Zeno Effect,	where the	physical interaction	with	a	measurement	device inhibits the	decay	of	an unstable particle. But already thoroughly classical systems, once they reach a certain complexity, can flout the epistemological claim of classical realism, as the butterfly effect popularly	illustrates:	Any	tiny	perturbation	of	any	observation	qua	physical	interaction	with the	observational instruments	may	be	amplified, so that	what the	presence	measurement reveals	about	the	system's	state	can	considerably	differ	from	the	state	in	which	the	system would	have	been	in	isolation,	i.e.	without	the	measurement.	(It's	in	fact	such	an	–	in	essence classically optical – perturbation of the system through the	measurement apparatus that Heisenberg	invoked	in	his	microscope	thought	experiment	as	an	argument	or	explanation	for his	uncertainty	relations.) In	light	of	the	fact,	however,	that	not	even	classical	physics	always	epistemologically	conforms to	classical	realism,	it	seems	that	epistemologically	the	latter	demands	too	much.	It	therefore seems	well	motivated	to	rest	content	with	the	merely	metaphysical	realism,	which	both	sdBBT and	dBBT	exhibit. 138	Loc.cit.,	pp.	245 45 IV.3:	"Temporal	Solipsism"	(Bell) IV.3.1	sdBBT's	many	worlds More	disquieting	is	sdBBT's	departure	from	another	traditional	metaphysical	intuition,	namely the temporal continuity of configurations: Since the corpuscles perform random jumps through	configuration	space, the	connection	between	their	past	and future	states through continuous	trajectories	is	severed.	By	contradistinction,	in	dBBT,	micro-	(and,	a	fortiori,	macro- )	configurations	are	continuously	linked	in	time. More	precisely,	Bell	apprehends	that	sdBBT	posits	many	worlds,	which	"[...]	exist,	not	at	the same time, but one after another."139	Indeed, an sdBBT universe typically keeps jumping between	even	macroscopically	distinct	configurations.	Any	minute,	thus,	a	universe	could	pop into existence in which dinosaurs aren't extinct yet. Strictly speaking, each such "world" endures only for infinitesimal instants of time – a peculiarity Bell lampoons as "temporal solipsism".140	Yet,	striking	our	common	sense	intuitions	as	absurd,	as	sdBBT	doubtlessly	does in this regard, bears little argumentative	weight – especially not if the theory is expressly conceded empirical adequacy141, i.e. being experimentally indistinguishable from QM; as Lewis	is	reported	to	have	said:	"I	do	not	know	how	to	refute	an	incredulous	stare." Indeed,	Bell	hints	at	a	more	systematic	reason142,	worrying	that	"[...] if (sdBBT)	were	taken seriously,	it	would	hardly	be	possible	to	take	anything	else	seriously."143	This	can	be	explicated in	two	ways: • One source of Bell's worry may be that sdBBT astronomically exacerbates our uncertainty	of	any	past	or	future	we're	capable	to	reconstruct	or	anticipate.	Since	the state of the	world is the outcome of an irreducibly stochastic jump, it's extremely probable	that	already	in	the	next	second	the	universe	will	visit	a	region	of	configuration space corresponding to a world in which not even the Milky Way has formed. Conversely,	it	seems	overwhelmingly	improbable	that	the	world	in	which	our	fiancées have	just	accepted	our	marriage	proposal	still	exists	tomorrow!	How	can	we	trust	that 139	Allori	(2015),	p.17 140	Cf.	Bell	(1987),	pp	135 141	E.g.	Allori (2015),	p.17;	Allori (2016),	esp. sect.	5.1,	6.1;	For	general criteria for the	empirical	adequacy	of primitive	ontological	approaches	to	QM,	cf.	Allori	et	al.,	(2007),	sect.7.3 142	Cf.	loc.cit.,	sect.	8.1. 143	Bell	(1987),	p.136 46 our	memories	and	future	anticipations	retro-/predict	the	real,	actual future?	SdBBT simply	thwarts	this	trust,	transforming	it	even	into	utter	unlikeliness. • Bell	now	links	this	maddening	uncertainty,	rooted	in	the	objective	randomness	of	the world,	to	the	reliability	of	our	current	knowledge	of	the	world:	In	particular,	how	can we	trust	that	our	memories	anticipate	the	real	future?	In	other	words,	he	assumes	a tight connection between the uncertainty about our world's existence and the relevance	of information:	How	can	our	current	knowledge	be	relevant for	decisions whose	consequences	lie	in	a	future	that	might	–	with	overwhelming	probability	–	not happen? If we assume that memories and knowledge supervene on current microphysical states, then the loss of temporal continuity entails the possibility of worlds	spontaneously	popped	into	existence	and	containing	archaeological	evidence, neuronal	configurations,	etc.	that	tell	of	a	past	which	has	never	occurred:	We	might suddenly	wind	up	in	a	world	in	which	we	actually	remember	yesterday's	unpleasant encounter	with	a	stegosaurus	in	our	bathroom!	Temporal	solipsism	thus,	Bell	suggests, evinces	a	form	of	solipsism	even	more	radical	in	that	regard	than	the	traditional	one: Not	only	is	everything	outside	our	own	minds	unreliable,	but	also	everything	outside the	present	content	of	our	minds,	i.e.	momentary	perceptions. • As a third reason to reject sdBBT, Bell accuses its temporal solipsism of being irrefutable.	We	construe	this	as	a	methodological	ramification	of	the	conjunction	of the	previously	claimed	unreliability	of	our	memories	and	the	doubts	regarding	their relevancy for future-directed actions: If contemporaneous configurations are no longer reliably linked to past configurations, with memories, archaeological and historical	records	being	"entirely	unreliable"144,	the	same	must	apply	to	measurements –	or even	perceptions: They, too, forfeit their reliability and relevancy. SdBBT thus becomes	untestable,	and	hence	arguably	unscientific. How	lethal	is	Bell's	criticism?	Our	strategy	will	be	to	block	it	by	denying	that	sdBBT's	stochastic jumping	between	worlds	in	any	way	impinges	on	the	relevancy	of	the	information	available	to us. 144	Allori	(2015),	p.	20 47 To	elaborate	that	thought,	we	first	have	to	clarify	the	Achilles'	heel	of	Bell's	above	criticism, namely	its	lax	notion	of	a	"world". A sideway glance at Everett's Many	Worlds Interpretation will prove rewarding: How do Everettians	define	worlds?	145	According to	Wallace, a "world" isn't a fundamental, but an emergent	concept,	defined	only	pragmatically	as	the	components	of	the	wave	function	of	the universe	in	some	decomposition	into	dynamically	approximately	independent	wavepackets, narrowly localised in	momentum	and	position space, and	hence	approximately stable and robust. These components ("branches") and their trajectories are picked out by decoherence146,	which	also	enables	their	re-identification	over	time.	Such	worlds	approximate well	"for	all	practical	purposes"	(FAPP)	classical,	Newtonian	trajectories.	As	Wallace	succinctly puts it:	"And	if	there	are	multiple	wave-packets,	the	system	is	dynamically isomorphic	to	a collection	of	independent	classical	systems."147 In the following, "worlds"	will always refer to corpuscle configurations in such emergent, pragmatically	defined	"branches". Neither decoherence nor the functionalism employed in the above pragmatic concept of world	are	committed	to	any	specific	interpretation	of	QM.	Thus,	with	impunity	we	may	also avail ourselves	of them to clarify sdBBT's	many	worlds character,	which	Bell had correctly diagnosed: Whereas in Everett all such worlds exist simultaneously, in sdBBT, each is spontaneously realised only one at a time, randomly "selected" by the configuration into which	the	"supercorpuscle" jumps.	The	Everettian	picture	thus	resembles	an ill-attuned	TV that	displays	several	channels	simultaneously148;	the	corresponding	sdBBT	picture	is	that	of	a TV	randomly	switching	between	different	channels. Note that on this understanding of a world, sdBBT's temporal solipsism involves "jumps between worlds", not merely jumps between (macro-)configurations simpliciter: Being associated	with	decoherence-induced	wavepackets,	which	one	can	track	through	time,	worlds display	some	form	of	diachronic	stability	(we'll	elaborate	on	this	in	a	moment);	consequently, jumps	between	worlds amount to the random	actualisation	of	parts of	multiple "parallel" 145	Cf.	Vaidman	(2014)	and	Wallace	(2001,	2011),	whom	we	follow. 146	Bacciagaluppi	(2012) 147	Wallace	(2011),	p.	11 148	We	take	this	simile	from	Allori	(2013),	sect.	4.4 48 world	histories.	The focus	on jumps	between	configurations	simpliciter	prescinds from	any trans-temporal,	historical	aspects. By	definition,	"our	memories"	refer	to	macro-configurations	that	belong	to	a	world	indexically pointed	out	by	the	speaker.	Everettians	would	paraphrase	this	as	the	indexical	selection	of	the branch	of	the	wavefunction	that	the	observer	inhabits.	In	the	following,	we'll	also	borrow	from the Everett literature the notion of personal identity, distinguishing between multiple (actualised	or	non-actualised)	copies	of	the	present	authors	in	various	worlds,	and	"us",	i.e. particular	(and	actualised)	copies,	associated	with	a	particular	past.149 Memories	(barring	psychologically	flawed	ones),	archaeological	evidence,	etc.	are	parts	of	the history of that world. Since any such world behaves FAPP-classically, and in particular is causally FAPP-closed, the histories we reconstruct from our available, present knowledge about	the	world	will	be	causally	closed:	All	the	data	within	one	world, i.e.	our	perceptions, memories and future	anticipations, are consistent and	even coherent,	with	no	a-causal or otherwise absurd irregularities. In short, any snapshot	of the	history	of a	world (including perceptions,	memories,	etc.)	will	appear	completely	normal	–	FAPP-indistinguishable	from	a suitable snapshot of a classical world. Consequently, assuming that QM is empirically adequate,	for	all	practical	purposes	no	empirical	evidence	(including	measurement	records) consistent	with	QM	can	ever	contradict	sdBBT,	which,	after	all,	simply	consists	in	positing	a stochastic occurrence of such snapshots of world histories, allowed by the universal wavefunction. In short, sdBBT is capable of accounting for all empirical phenomena QM predicts. Whether a	world is actualised at other points in time or not, is a separate question; if a nanosecond	later	a	world	is	actualised	with	completely	different	historical	records,	then	simply a different world has jumped into existence, notwithstanding our linguistic habit to then indexically	refer	to	that	world,	too,	as	"ours".	In	short,	the	reply	to	Bell's	question:	"How	can a	temporal	solipsist	take	seriously	what	he	or	she	remembers	and	perceives?"	is	simply	that	– with the usual epistemological and methodological caution – on that basis we reconstruct/anticipate	our	world. 149	Saunders/Wallace	(2008),	esp.	Section	2.1 49 What,	then,	about	the	relevance	of	our	current	memories	and	perceptions	for	future-directed actions? Doesn't sdBBT's temporal solipsism undermine any such relevancy of current information	for	future-directed	actions?	Why	contract	a	life	insurance	in	this	world,	when	the latter's	persistence	already	until	next	Saturday	is	doubtful?	Rephrasing	our	reconstruction	of Bell's	criticism,	such	an	argument	assumes	that	the	following	two	questions	are	related: 1. Does	our	world	actually	persist	"without	interruptions"?	Or	does	the	world	we	(believe to)	inhabit	fade	in/out	of	actuality	or	existence	only	for	finite	periods	or	moments,	with times of non-actuality/non-existence in-between? We'll call this the "Problem of Persistence". 2. Is	data	within	one	world	–	namely the	one	we	happen	to inhabit	at	some	arbitrary instant	–	relevant	to	actions?	Should	we	pursue	our	duties,	dreams	and	hopes	that pertain to a still-to-come future of our world? We'll call this the "Problem of Relevancy". We	deny	that	the	Problem	of	Persistence	has	any	logical	bearing	on	the	Problem	of	Relevancy: The	temporal	discontinuity	of	our	actual	world	doesn't	impinge	on	the	relevancy	of	taking	an action	(or	adopting	a	certain	belief)	now,	on	the	basis	of	current	knowledge. • Even	if	the	event	I	remember	or	as	a	result	of	which	I'm	taking	action	(or	adopt	a	belief) wasn't/isn't actualised, it still belongs to	my world. (Recall the above stance on personal identity	we imported from	the	Everett literature.) If at	any later	point	my world	again	pops	into	existence,	I	will	face	the	consequences	of	forbearing	a	necessary action	–	or	conversely,	harvest	the	fruits	of	having	done	it. In	that	respect, I	should act/believe:	If	I	exist	again,	then	I	benefit	from	doing	A.	If	I	don't	do	A,	and	I	don't	exist again,	I	won't	have	the	chance	to	regret	having	done	A,	either. • More formally, let's tentatively take an action	A relative to a certain goal	G to be relevant, represented by the holding of the relation	r(. , . ) if and only if A is an appropriate	means	to	reach	G,	i.e.	ceteris	paribus	brings	about	the	intended	goal:	G → A:	r(G, A) ⟺	G → A. This	definition	of	r	remains	silent	on	the	actuality	α(W)	of	the	world	W	to	which	G belongs,	i.e.	that	α(W)	holds,	such	that	G ∈ W. Such reference is straightforwardly implemented by stipulating that an action A relative to goal G be considered relevant, i.e. that r A, G holds, iff G isn't an 50 appropriate	means	for	A,	i.e.	¬ G → A ,	only	if	the	G-containing	world	W	isn't	actual, either	¬α(W).	That	is:	r G, A ⟺ ¬ G → A → ¬α(W).	By	contraposition,	thus: r G, A ⟺ α W → G → A . This	bi-conditional	remains	true,	even	if	the	world	W	isn't	actual,	¬α(W). In	conclusion, the	relevancy	of	an	action	relative	to	a	goal is independent from	the actuality	of	a	world	in	which	the	goal	exists. By decoupling the reliability of our	memories or future anticipations from the ontological Problem	of	Persistence,	Bell's	worry	of sdBBT	undermining the relevancy	of	our actions is blocked. Consequently,	also	his	claim	of	sdBBT's	irrefutability	becomes	moot	–	as	far	as	the	reliability of	measurements is	the issue:	They	are just	as	reliable	and	relevant	as	any	other	historical records. As far as the issue is irrefutability, it's either trivially true – or beside the point:	Given its empirical	equivalence	with	ordinary	QM,	any	violation	of	a	quantum-mechanical	prediction would refute sdBBT, too. Insofar, however, that Bell criticises that a defence of temporal solipsism	can't	resort	to	empirical	arguments,	the	objection	is	beside	the	point:	Firstly,	should empirical	indistinguishability	be	the	issue,	it	suffices	to	recall	that	under-determination	by	the empirical data is a generic scientific phenomenon150, for which one cannot reproach any specific	theory.	Secondly,	should	the	issue	be	the	need	for	trans-empirical	arguments,	then	to assess empirically indistinguishable theories, one needs trans-empirical criteria, such as internal	consistency,	relationship	to	other	theories	or	explanatory	value.151 Temporal solipsism	certainly	goes	against	our	metaphysical	grain,	prompting the	question: Why	take	sdBBT	seriously?	Why	assume	that	sdBBT	offers	an	account	closer	to	reality	than others?	E.g.	Collins	requires	that	"in	order	to	have	sufficient	reason	to	believe	that	a	theory	is (approximately) true, the theory must at least offer some truth-indicating theoretical advantage	–	that	is,	make	some	explanatory	progress	–	over	its	phenomenalist	counterpart 150	Cf.,	for	instance,	Stanford	(2013) 151	Cf.	Esfeld	(2014) 51 [i.e.	sdBBT]."152	We'll	postpone	a	critical	discussion	of	whether	Collins'	classification	of	sdBBT as	"phenomenalist"	is	justified	or	not,	to	section	IV.4. Relevant	for	this	section	is	that	his	epistemological	critique	of	dBBT	concludes	that	vis-à-vis sdBBT,	"we	do	not	have	sufficient	reason	to	believe	Bohm's	theory"153.	This	concords	with	the results	of	our	analysis	so	far,	which	this	section	reaffirmed	by	rebutting	Bell's	methodological criticism	of	sdBBT,	and	vindicating	sdBBT's	empirical	adequacy	and	conceptual	consistency: The	picture	of	the	world,	i.e.	the	idea	of	us	being	trapped	in	an	infinitesimal	"time	capsule" (Julian	Barbour),	may	be	uncanny;	but,	we	argued,	it	can	satisfactorily	account	for	all	available empirical	data	and	their	role	as	reliable	evidence. IV.3.2	Bohm	Brains	vs.	Boltzmann	Brains It's instructive to compare sdBBT's	multiverse nature, rooted in its temporal solipsism, to Boltzmann	Brains	scenarios	in	SM,	the	possibility	of	worlds	randomly	jumping	into	existence, filled	with	memories	and	records	of	a	non-existent	past. Boltzmann Brain scenarios arise in the context of Boltzmann's H-theoretical attempts to explain	the	entropy	increase,	captured	in	the	2nd	Law	of	Thermodynamics.	Not	hinging	on	a preferred direction of time, his argument also holds for time-reversal. (Loschmidt's Umkehreinwand). Consequently, low entropy states most likely are random fluctuations amidst	thermal	equilibrium,	islands	in	an	ambient	high-entropy	ocean. This	consequence	also	should	apply	to	our	brains,	which	then	are	likewise	to	be	seen	as	low entropy	fluctuations:	"We	pop	into	existence	as	thermal	fluctuations	with	our	brains	full	of memories	of	a	nonexistent	past.	[...]	In	the	Boltzmann	brain	scenario,	you	most	likely	came into	being	by	a	fluctuation	in	this	instant.	So,	your	recollection	of	reading	the	sentence	before this	one	is	just	as	fabricated	as	every	other	memory."154 How	to	respond	to	this	bizarre	speculation?155 152	Collins	(1996),	p.	266 153	Loc.cit. 154	Norton	(2015),	p.	3 155	We	shall	gloss	over	the	controversial	question	whether	such	an	interpretation	of	the	H-theorem	in	terms	of a	theory	of	random	fluctuations	is	justified	or	not,	two	important	issues	in	this	context	being	the	identification of	the	uniform	measure	with	a	probability	measure	and	the	status	of	ergodicity. Lazarovici/Reichert	(2014)	argue	for	an	interpretation	in	terms	of	typicality,	instead. 52 Norton	has	pointed	out	that	in	addition	to	Boltzmann's	argument that	a	non-equilibrium	state arose	with	some,	albeit	small	probability	from	a	fluctuation	away	from	equilibrium	requires an	extra	step,	namely	the	existence	of	a	particular	low	entropy	state,	i.e.	a	brain	(including	its memories)	just	like	ours.	He	contemplates	two	main	conceivable	arguments	for	this	extra	step: The	first	one	appeals	to	Poincaré	recurrence,	which	states	that	eventually	the	evolution	of	a thermal	system	will	return	to	an	arbitrarily	small	neighbourhood	of	its	starting	point.	However, "(t)he	recurrence	argument	cannot	give	us	isolated	brains	that	exist	momentarily,	since	we would	have	to	assume	their	possibility	in	a	[sic]	the	first	place."156	Hence,	Norton	discards	this argument as circular. Furthermore, he remarks that the Poincaré recurrence theorem presupposes antecedent conditions, such as finite phase space volume and finitely many degrees	of	freedom,	likely	not	to	be	met	for	our	universe. The second argument appeals to the time-reversal invariance of the underlying microdynamics:	Since	an isolated	brain	of	a	sentient	observer	can	degrade	thermally into	a final equilibrium	heat	death,	by	time-reversal,	a	state	of	thermal	equilibrium	can	also	evolve	–	no matter	how	improbably	–	into	an	isolated	brain	of	a	sentient	being.	Norton	also	repudiates this argument, pointing out that the time-reversal invariance of CM's	micro-dynamics no longer	needs	to	carry	through	from	a	more	fundamental	quantum	perspective,	with	the	weak interaction (mediating radioactivity and hence arguably also biologically non-negligible) violating time-reversal invariance and "[...] the standard account of quantum mechanics (involving	a	time-irreversible	collapse	of	a	quantum	wave	on	measurement.	If	that	collapse	is a	real	process,	then	the	microphysics	is	very	far	from	time	reversal	invariant."157 How	do	Norton's	objections	against	Boltzmann	Brain	scenarios	fare	in	the	sdBBT	case?	We submit,	they	don't	carry	over. Poincaré	recurrence	isn't	satisfied	for	generic	quantum	theories	(the	quantum	counterparts assume discrete energy spectra, a far too restrictive assumption for the general case158), hence, it	can't	be invoked.	Neither	does	the	time-reversibility	argument	work in	the	sdBBT case:	Although	Norton's	second	objection	to	it,	in	terms	of	a	time-irreversible	collapse	of	the 156	Norton	(2015),	p.	5 157	op.cit.,	p.	8 158	Cf.	Wallace	(2013) 53 wavefunction,	is	blocked	in	sdBBT,	as	a	non-collapse	theory,	the	time-irreversibility	of	weak interactions	persists. The	pivotal	reason,	however,	for	the	futility	of	the	argument	invoking	Poincaré	recurrence	was the	"initiation	problem"	(Norton):	"While	the	time	reversed	constitution	of	brains	is	possible, without	a	Poincaré	recurrence	theorem,	what	assurance	is	that	this	particular	fluctuation	will be	initiated,	even	if	only	in	probability?"159	In	contrast	to	the	Boltzmann	Brains	case,	sdBBT overcomes the Initiation Problem by its fundamental stochasticity: A "Bohm Brain", indiscernible from ours, complete with spurious memories, could indeed just happen to randomly	materialise. Besides	the	above	problems	related	to	the	generation	of	a	Boltzmann	Brain,	which	sdBBT	thus seems	to	be	able	to	escape,	Norton	elaborates	a	second,	arguably	more	detrimental	argument against Boltzmann Brains: "Our memories are of a relatively orderly past, full of regular occurrences	conforming	to	discoverable	laws.	Nothing	forces	a	Boltzmann	brain	to	have	such regular	memories.	[...]	(I)t	is	vastly	improbable	that	a	Boltzmann	brain	might	have	memories of	a	regular	past	just	like	ours."160	In	other	words,	to	the	mind	of	a	Boltzmann	Brain,	the	world, complete with memories and archeological evidence, ought to resemble rather Borges' Ciudad	de	los	Inmortales	than,	say,	the	rock	city	of	Petra.	Consequently,	the	uniformity	of	the world	we	experience	–	i.e.	a	world	that	seems	to	admit	of	perfectly	coherent	consistent	stories –	empirically	contradicts	what	Boltzmann	Brain	scenarios	would	have	us	expect,	namely	"batty brains" (Norton), typically hallucinating chaotic worlds. In short, Boltzmann Brains are empirically	inadequate:	The	theory	would	predict	batty	memories	and	perceptions. Of course, one could just stipulate that a "mind-bogglingly rare" (Myrvold) fluctuation generated	our	brains,	such	that	our	world	simply	happens	to	appear	regular	to	us.	But	on	what grounds	should	we	believe	in	such	a	miraculous	coincidence	–	other	than	in	order	to	restore empirical	adequacy	of	our	initial	hypothesis	in	an	ad-hoc	way?	It's	exactly	this	(and	only	this) move	that	would	render	the	whole	Boltzmann	Brain	scenario	self-refuting,	since	it	relies	on 159	Norton	(2015),	p.7 160	op.	cit.,	pp.	8 54 science,	whilst	simultaneously	reducing	all	empirical	evidence,	on	which	our	trust	in	science rests,	to	illusions,	thereby	undermining	the	very	reliability	of	science.161 sdBBT's	Bohmian	Brains	avoid	this	self-refuting	dodge,	as	we	saw:	The	worlds	that	randomly pop into existence are by construction regular, with for all practical purposes uniform histories;	unlike	in	the	case	of	Boltzmannian	Brains,	sdBBT	predicts	that	batty	brains	are	very unlikely	–	in	line	with	sdBBT's	empirical	adequacy. IV.4:	sdBBT	as	a	phenomenalist	theory? Let's	briefly,	however,	pause	to	gainsay	Collins'	filing	of	sdBBT	as	a	phenomenological	theory, i.e.	"an	account	that	merely	takes	the	phenomenological	laws	of	quantum	mechanics	as	its unexplained	given."162	This	characterisation	is	objectionable: • Firstly,	as	we reiterated	above, sdBBT	differs in important	ways from	ordinary	QM, conceptually	going	far	beyond	any	account	of	QM	simpliciter. • Secondly:	What	is	actually	meant	by	"phenomenological	laws",	such	as	Hubble's	Law, describing	the	expansion	of	the	universe	with	a	(highly idealised!) linear	correlation between the distance and observed recession velocities of far-away galaxies? On common notions, phenomenological models link observable properties, without providing	any	mechanism	in	terms	of	explanations	from	fundamental	first	principles.163 (In the case of Hubble's Law, of course, after Hubble's formulation of his phenomenological law,	Lemaître	succeeded in	deriving it	as	an	approximation from General Relativity, promoting thereby its status from phenomenological law to (approximate)	theorem.) SdBBT	sits	squarely	with	such	a	notion	of	"phenomenological":	SdBBT	doesn't	restrict itself to more or less directly observable properties: This is evident for the wavefunction,	but	also	applies	to	the	corpuscle	positions	–	and	the	contextual	nature of	all	other	dynamical	variables. 161	E.g.	Myrvold	(2014)	for	such	a	criticism	of	Boltzmann	Brains	(echoing	Bell's	that	we	saw	in	the	preceding subsection). 162	Op.cit.,	p.	265 163	E.g.	Bunge	(1997) 55 • Moreover,	with	its	clearly	delineated,	objective	ontology	of	the	basic	constituents	of reality,	sdBBT	qualifies	as	a	candidate	for	a	fundamental	account	of	reality	no	less	apt than	dBBT	or	other	primitive	ontology	approaches.164 In	other	words:	sdBBT,	as	a	primitive	ontology	in	general,	and	as	a	theory	with	copious advantages	over	dBBT,	offers	much	to	justify	the	claim	that	it	actually	provides	first principles,	and	thereby	skirts	also	the	second	part	of	the	above	characterisation	of	a phenomenological	theory. IV.5:	sdBBT	vs.	dBBT The	case	for	sdBBT's	advantages	over	dBBT	can indeed	be	further	strengthened in	at least three	regards: 1. Firstly,	with	now	all	the	conceptual	paraphernalia	in	our	hands,	we	can	see	how	sdBBT overcomes	three	often	perceived	ontological	shortcomings	of	dBBT,	one	related	to	the status	of	empty	wavefunctions,	and	the	other	two	related	to	the	odd	double	role	of dBBT's	wavefunction. 2. Secondly,	we	need to address an intriguing	objection that dBBT's	GE after all	does possess	explanatory	surplus	value	–	viz.	by	allowing	a justification	of the	Symmetry Postulate for composite N-corpuscle systems. An objection pulling in the same direction is that dBBT	offers some conceptual advantages for the treatment of i.a. tunnelling	times. sdBBT,	we'll	argue,	will	prove	no	less	successful	w.r.t.	these	features. 3. Thirdly,	we'll	argue	that	sdBBT's	prospects	of	a	satisfactory,	relativistic	treatment	look promising,	with	dBBT's	essential	impediments	being	removed	in	sdBBT. IV.5.1:	Metaphysical	quarrels	with	wavefunctions	in	dBBT Having	deepened	our	understanding	of	sdBBT's	conceptual	and	metaphysical	structure,	we're now	in	a	position	to	resume	our	previous	discussion	of	what	many	have	perceived	as	dBBT's ontological	shortcomings.	We'll	see	how	sdBBT	overcomes	them. • In dBBT, by specifying initial corpuscle positions one selects one of the possible trajectories as the	actual	one. The	positions thus	distinguish those	branches	of the wavefunction	that	encode	the	actual	state	of	the	system.	Yet,	all	other	branches	of	the 164	Cf.	Allori	(2013ab,	2015ab,	2016) 56 wavefunction, into which no trajectories lead, are equally real: Myriads of empty wavefunction	branches,	which	in	general	no	longer	affect	the	system,	thus	populate space – an unsavoury feature.165	It's unclear what their status is supposed to be; furthermore,	if	they	are	irrelevant	for	the	dynamics	of	the	corpuscles,	then	one	would rather	dispense	with	them. SdBBT	removes	this	blemish:	Since	the	modulo	square	of	the	wavefunction	represents the	corpuscles'	localisation	probability,	conceived	of	as	a	real	property,	the	notion	of an	"empty	wavefunction"	loses	its	meaning	–	or	at least its	sting:	Corpuscles	simply have	a	propensity	to	localise	themselves	anywhere.	Whether	this	propensity	actually manifests	itself	or	not,	has	no	bearing	on	its	reality. Of course, the price for this solution is the postulate of a disposition,	which by its metaphysical nature is empirically rather elusive. However, we argued in III.2, a number	of	advantages	commend	dispositionalism,	compensating	for	this	elusiveness. • Another	ontological	oddity	has	struck	many	of	dBBT's	detractors:	While	via	the	GE	the wavefunction guides the trajectories of the corpuscles, the latter don't conversely influence	the	wavefunction.	The	Action-Reaction-Principle	thus	seems	violated.166	This is	seen	as	undesirable:	Rather,	a	substance	should	only	be	a	potential	agens,	i.e.	act,	if it	can	equally	be	a	potential	patiens,	i.e.	be	in	turn	acted	upon.	Einstein,	for	instance, valued this Action-Reaction-Principle as a key (and, vis-à-vis Special Relativity (SR), novel)	virtue	of	General	Relativity167,	in	which,	as	Wheeler	famously	put	it,	spacetime tells	matter	how	to	move,	whereas	matter	tells	spacetime	how	to	curve. With	the	abolition	of	deterministic	trajectories	in	sdBBT,	the	wavefunction	no	longer acts	on	the	corpuscles.	Of	course,	they	still	have	the	dispositional	property	to	localise themselves,	but the relation	between	dispositions	and	manifestations is	not	one	of action, usually understood as changes in substances interdepend: Objects simply possess	a	property;	the	meaning	of	a	property	acting	on	the	substance	that	bears	it	to 165	Brown/Wallace	(2005)	and	Brown	(2009)	ratch	up	this	unease	with	empty	wavefunctions	to	the	charge	that dBBT	is	"Everett	in	denial":	In	essence	they	argue	that	dBBT	merely	posits	corpuscles	on	top	of	the	branches	of the	multiverse,	as	"mere	epiphenomenal	'pointers',	being	relegated	to	picking	out	one	of	the	many	branches, defined	by	decoherence,	while	the	real	story	–	dynamically	and	ontologically	– is	being	told	by	the	unfolding evolution	of	those	branches",	op.cit.,	pp.8.	This	indexical	role	is	then	criticized	as	an	ad-hoc	ingredient.	Callender (n.d.)	critically	examines	this	"redundancy	argument". 166	Cf.	Brown/Anand	(1995) 167	Cf.	Brown/Lehmkuhl	(2013) 57 us	appears	obscure.	By	dint	of	the	dispositional	nature	of	the	wavefunction,	the	validity of	the	Action	Reaction	Principle	remains	unscathed. • At	the	heart	of	dBBT's	trouble	with	the	Action-Reaction	Principle	lies	the	double	role of	the	wavefunction	in	both	guiding	the	corpuscles'	trajectories	and	in	representing probabilities	for	their	localisation	–	two	roles	that	are	logically	distinct.	This	double	role makes	dBBT	look	contrived.168 Eliminating	the	GE	trivially	disposes	of	the	double	nature,	tout	court. IV.5.2:	Justification	of	the	Symmetrisation	Postulate Linking	such	metaphysical	aspects	and	what	might	be	seen	as	explanatory,	even	predictive power,	Brown	et	al.	have	claimed	that,	whereas	standard	QM	must	posit	it	as	a	contingent axiom 169 , via the topological approach, dBBT can naturally justify the Symmetrisation Postulate for identical particles. 170 It states that wavefunctions,	Ψ(Q,... ,QN), of bosonic (fermionic) N-particle systems behave (anti-)symmetrically under permutations π"¦ of particles, labelled i and j:	π"¦Ψ Q1,... ,QN = ±Ψ Q1,... ,QN . Such an astonishing claim would	contradict the redundancy	of the	GE, from	which	sdBBT	draws its	appeal.	With	our grasp	of	sdBBT's	conceptual	structure	sharpened,	we're	now	in	a	position	to	address	the	claim. We	submit,	sdBBT	is	equally	able	to	motivate	the	Symmetrisation	Postulate. The technical details of the topological approach,	which turns on the non-trivial topology (multiple	connectedness)	of	the	reduced	configuration	space	(more	on	that	below)	shall	not detain	us	here;	instead,	we'll	focus	on	the	crucial	step,	viz.	the	removal	of	the	"coincidence points" Δ ≔ Q:= Q1,... ,QN ∈ R21: ∃i ≠ j: Qi = Qj from configuration space R21 . Brown et al. claim dBBT naturally justifies this removal. Let's recapitulate their chain of reasoning: (1) The	corpuscles	being identical (more	precisely: indistinguishable), their index labels possess	no	intrinsic	meaning.	Hence,	since	the	configuration	Q	of	corpuscle	positions completely specifies the actual state of a system, for a given wavefunction	Ψ , a 168	E.g.	Timpson	(2011),	pp.	14 169	In relativistic QFT, however, the Symmetrisation Postulate, i.e. the connection between spin and (anti-) symmetry	can	be	proven	as	a	theorem,	cf.	Streater/Wightman	(1964),	Ch.	4.4 170	Cf.	Brown	et	al.	(1998). 58 permutation	π ∈ Σ1 of indices shouldn't alter the state	of the system: πQ,Ψ and Q,Ψ are	physically	equivalent.	The	redundancy	of	configuration	space	can	thus	be purged by transition to the reduced configuration space	D ≔ R23/Σ1 , where configurations that differ from one another only by a permutation are identified, preserving	the	physical	information. (2) The GE being 1st-order, corpuscles coincide either always or never:	D\Δ	and	Δ	are invariant	under	the	action	of	the	dBBT	dynamics. a. This implies that, consistently	with the	dBBT	dynamics,	we	may remove the coincidence	points	Δ	from	reduced	configuration	space	D. b. This	removal	"[...]	seems	physically	well	motivated,	since	they	correspond	to motions for which [the involved corpuscles] coincide for all times – which would	appear	as	the	motion	of	one	particle	of	M-fold	mass	and	charge."171 (3) The removal doesn't affect dBBT's observable/statistical predictions, for the set of coincidence	points	is	of	measure	zero, d21Qã Ψ 9 = 0. Let's	analyse	these	arguments.	What's	the	role	of	the	GE	in	them? • (1)	only	requires	the	position	definiteness	and	dBBT	ontology;	hence,	it	carries	over	to sdBBT. • (3) is a	mathematical fact, independent	of the	GE: In sdBBT, too, the removal	of	Δ doesn't	affect	the	observations. • (2),	of	course,	lapses	in	sdBBT. Does	that	ruin	the	motivation	for	removing	Δ? • The	first	part	of	(2)	is	only	needed	to	ensure	consistency	with	the	GE.	With	no	GE	in sdBBT,	we	trivially	needn't	worry	about	consistency	with	it. • Regarding	the	second	part,	note	first	that	(2b)	doesn't	hinge	on	the	GE	specifically,	only on	it	being	1st	order. More	to	the	point:	Why	is	it	actually	implausible	to	regard	two	particles	with	the	same spatiotemporal	trajectories	as	two?	The	tacit	metaphysical	premise	in	the	background is a spatiotemporal principium individuationis (PI) 172 , according to which 171	Loc.cit.,	p.7 172	Schopenhauer	uses	this	term	in	connection	with	Kant,	e.g.	Kant	(1781),	A263. 59 spatiotemporal distinctness constitutes individuality/identity and thus grounds numerical	distinctness.	In	other	words,	(2b)	assumes	that	spatiotemporal	distinctness is a necessary condition for individuating two	otherwise indistinguishable	particles. And	since	the	number	of	particles	is	fixed	in	a	deBroglie-Bohmian	theory,	the	removal of	Δ	follows. However, in the stochastic world of sdBBT, corpuscles no longer have continuous spatiotemporal	paths:	They	localise/delocalise	spontaneously	at	random	points.	One thus	faces	two	options:	Either	one	continues	to	adhere	to	the	spatiotemporal	PI	and allows for multiply occupied spacetime points, thereby indeed forgoing the justification	for	removing	Δ.	Equally	well,	however,	given	that	nowhere	do	Brown	et al. explicate the above (PI), let alone argue for it, one could adapt an alternate principium individuationis (PI*), arguably	more natural to a discontinuous	world of corpuscle jumps: Two corpuscles are identical, if they coincide at one point in spacetime. PI* then directly motivates the removal of coincidence points, thereby replacing	(2b). In	short,	what	does	the	essential	work	in	the	topological	motivation	for	the	Symmetrisation Postulate	for	dBBT	are	a	dBBT-independent,	measure-theoretical	statement,	and	a	Principle of	Individuality.	Modifying	the	latter	in	a	plausible	manner,	we	conclude	that	sdBBT	is	no	less apt	to	motivate	it. IV.5.3:	Quantum	tunnelling Our	claim	of	the	GE's	redundancy	is	likewise	disputed	by	asserted	conceptual	advantages	in experimental contexts whose treatment doesn't fit comfortably within standard QM, e.g. regarding	dwell	and	tunnelling	times,	escape	times	and	escape	positions,	scattering	theory and	quantum	chaos.173 Each such application deserves an investigation in its own right; here, we'll exemplarily examine	the	arguably	clearest	case,	viz.	quantum	tunnelling174	–	a	phenomenon	important, e.g., for	describing	α-decay:	Quantum	particles	can	penetrate	a	potential	barrier	classically insurmountable,	with	the	particles'	energy	being	below	the	potential	barrier.	The	question then	naturally	arises:	How	much time	does it take	a	particle to	cross the	barrier in such	a 173	Cf.	Goldstein	(2013),	Sect.	15,	also	for	further	literature	on	these	issues. 174	Cf.	Holland	(1993),	Ch.	11	and	Passon	(2010),	Ch.	7.5,	whom	we	closely	follow. 60 scenario?	Ordinary	QM	boycotts	the	very	question:	Firstly,	time	is	not	an	observable,	but	a parameter; secondly, standard QM doesn't assign particles definite trajectories, depriving statements	about	how	long	it	takes	a	particle	to	traverse	a	certain	distance	of	any	immediate meaning.	In	order	to	define	such	time	scales,	one	must	therefore	employ	surrogate	methods that	only	use	well-defined	operators175,	by	tracking	wave	packets	via	the	evolution	of	their maxima.	The	results	obtained	from	different	methods,	however,	turn	out	to	be	not	always mutually	compatible. In dBBT, it is claimed, a straightforward, unambiguous picture emerges: The GE pilots corpuscles	that	start	from	some	initial	positions	deeply	inside	the	barrier	and	beyond;	other corpuscles	with	different	initial	positions	get	reflected.	At	each	moment,	the	corpuscles	have both,	a	definite	position	Q	and	velocity	Q,	for	simplicity	assumed	to	be	time-independent.	Let each	α-th	particle	with initial position	Q (ä) reach	a given	point	Q at time	t(ä). (In general, solutions	of	the	GE,	Q = Q(t; Q ä ),	won't	be	invertible	in	closed	form.)	For	the	corpuscle	with initial position	Q (ä) it	will take the time	Δt(QÃ, QÄ; Q ä )	to travel from sites	QÃ to	QÄ , on opposite	ends	of	the	potentials	barrier	each.	Since,	according	to	QEH,	the	initial	positions	are unknown,	only	their	statistical	distribution, ψ 9,	the	observable,	average	"dwell-time"	for	a tunnelling	process, ΔtÃÄ ,	can	be	readily	determined	as: ΔtÃÄ = dQ ä ψ(Q ä ) 9 Δt(QÃ, QÄ; Q ä ) å . The	deterministic	trajectories	play	a	crucial	heuristic	role	here,	suggesting	an	intuitive	way	to define measurable quantities. But might this advantage be merely illusory? Could the (empirically	elusive)	trajectories	make	certain	choices	of	how	to	define,	say,	dwell-time,	only appear	more	natural	than	others?	The	alternative	constructions	of	dwell-time	variables,	e.g. via	the	evolution	of	wave	packets,	are	still	equally	legitimate	and	adequate. But	let's	grant	for	the	sake	of	the	argument	that	the	conceptual	advantage	is	real.	Then,	sdBBT is	no	worse	off	than	dBBT,	suggesting	an	equally	appealing	instruction	of	how	to	define	dwelltime,	viz.	as	the	average	time	one	needs	to	wait	until	a	corpuscle	localises	itself	first	at	QÃ	and then	at	QÄ. 175	Cf.	Landauer/Martin	(1994)	and	Chiao	(1998) 61 In	conclusion, for tunnelling times, the	conceptual	advantages	accredited to	dBBT	are	of	a heuristic	nature;	should	one	indeed	deem	such	heuristic	guidance	an	advantage,	sdBBT's	more thrifty	primitive	ontology	is	no	less	apt	to	provide	it. IV.5.4:	sdBBT,	dBBT	and	relativistic	QM Pride	of	place	in	our	comparative	analysis	of	sdBBT	and	dBBT	shall	be	a	glance	at	the	most frequent criticism of dBBT – namely the (still largely unresolved) issue of its relativistic generalisation.	The locus	of those	problems	with	a relativistic	dBBT,	we'll presently	argue, again	lies	in	the	GE.	Dispensing	with	the	latter	thus	makes	sdBBT	an	attractive	alternative	to dBBT:	We submit that sdBBT, indeed, is free from non-locality, i.e. a spooky action-at-adistance,	but	exhibits	non-separability. Let's briefly recall the situation in ordinary QM with the collapse postulate and its relativisation.	There,	the	following	quandary	looms176:	Either	the	collapse	is	instantaneous	(as standard	formulations	seem	to	suggest)	–	highly	problematic	for	SR,	as	superluminal	signal transfer arguably gives rise to paradoxes, involving causal loops. Furthermore, the instantaneous	propagation	of	an	effect	privileges	a reference frame	– in conflict	with	SR's Principle of Relativity, which postulates the equivalence of all reference frames 177 ; alternatively,	contemporary	QM	needs	to	be	modified	ad-hoc	so	as	to	suitable	account	for	a collapse	mechanism.	Either	choice	seems	hard	to	swallow. By denying the existence of the collapse, non-collapse theories, such as dBBT, avoid this dilemma. The price dBBT has to pay, though, is its manifest action-at-a-distance: Each corpuscle's	velocity	field	depends	on	the	configuration	of	all	other,	even	space-like	separated corpuscles,	which	in	light	of	SR	one	wouldn't	expect	to	exert	any	influence	upon	each	other. Not	only	should	the intimated	conflict	between	an	action-at-a-distance	be	fleshed	out,	but also,	one	may	object	that	dBBT,	as	introduced	so	far,	is	a	non-relativistic	theory	–	and	hence unsurprisingly conflicting with SR. Therefore, let's examine some expressly relativistic proposals. 176	Cf.	Maudlin	(2011b) 177	E.g.	for	Popper,	an	action-at-a-distance,	as	a	possible	interpretation	of	an	instantaneous	collapse	"[...]	would mean	that	we	have	to	give	up	Einstein's	interpretation	of	special	relativity	and	return	to	Lorentz's	interpretation and	with	it	to	Newton's	absolute	space	and	time",	Popper	(1982),	p.	29. 62 Associated	with the	Klein-Gordon	Equation for the	complex scalar function	φ,	describing	a spin	0-particle	of	mass	m	and	charge	q	in	the	external	electromagnetic	potential	A¡, m9 + ∂¡ + iqA¡ ∂¡ + iqA¡ φ Q, t = 0, is	the	conserved	4-current	j¡ = = 9: φ∗∂¡ − 2qA¡ φ,	satisfying	∂¡j¡ = 0. Based	on	j¡,	one	now	may	be	tempted	to	stipulate	a	GE	for	the	Klein-Gordon	scalar,	analogous to	the	non-relativistic	case,	via	Hè ; H& = ¦ ; ¦§ .	In	the	absence	of	an	external	field,	for	a	plane	wave of	positive	energy,	this	appears	reasonable,	with	the	4-current	future-pointing,	j > 0,	and time-like,	j¡j¡ > 0. In the general case, though, such a proposal isn't viable for two	main reasons	178:	Firstly,	j	is	of	indefinite-sign,	not	even	for	free	solutions	of	positive	energy	states, and hence defies a particle or probability density. Secondly,	j¡ generically isn't time-like. Holland concisely comments: "A theory of	material objects in which an initially time-like, future-pointing	trajectory	may	pass	through	the light	cone	to	become	space-like,	and	even move	backwards	in	time,	is	clearly	unacceptable."179 In	conclusion,	the	whole	idea	of	a	dBBT	version	of	the	Klein-Gordon	Theory	can't	get	off	the ground, for the Klein-Gordon Theory doesn't admit of a satisfactory single-particle interpretation to begin with. (Besides of some formal difficulties with suitable position operators	and localizability in relativistic	QM, in	general,	Holland reminds the reader that, unless	supplemented	by	further	ad-hoc	constraints,	Lorentz-covariant	wave	equations,	such as the Klein-Gordon Equation, are well-known to exhibit superluminal transmission and backwards-causation – both consequences highly problematic in their own right, if not downright	paradoxical.) The current absence of a consistent particle interpretation of the Klein-Gordon Equation seems	to	preclude	a	satisfactory	deBroglie-Bohmian	theory,	with	its	commitment	to	a	particlebased	primitive	ontology.180 178	Holland	(1993),	Ch.	12.1 179	Op.cit.,	p.	500 180	More	satisfactory field-based	bosonic theories	exist, cf.	e.g.	Dürr	et	al. (2004);	Nikolic (2005).	But they	go beyond	our	current	scope	–	in	two	regards:	Firstly,	in	that	we	are	focussing	on	deBroglie-Bohmian	approaches, narrowly	construed	as	primitive	ontologies	based	on	particles/corpuscles	with	position	as	beable;	secondly,	we restrict our analysis to the domain of relativistic/non-relativistic QM. An extension to quantum field theory deserves	an	investigation	in	its	own	right. 63 How does the situation look for fermions? Here, we'll argue, sdBBT will unbosom its advantages over dBBT w.r.t. a relativistic extension. Exemplarily, we'll now analyse the arguably	most	successful	proposal for	a	dBBT	of	Dirac-particles,	viz.	so-called	Hypersurface Bohm-Dirac	Models.181 Bohm	himself	proposed	to	derive	the	relativistic	counterpart	of	the	GE	from	the	N-fermion Dirac	Equation	(for	ease	of	notation,	ħ = c = 1)	for	each	particle	k ∈ Nê3: iγU∂& + iγU" ∂" + eγU" A" QU, t − eγUΦ QU, t − m 3 ¦<= "<=,9,2 ψ Q, t = 0 Here,	the	γU ¡denotes	the	μ-th	Dirac	matrix,	acting	on	the	k-th	particle	(in	the	following,	we'll use the following conventions:	γU ≔ γU" "<=,9,2 and	γ  =⊗U<=3 γU ),	m and e denote the mass	and	the	charge	of	the	Dirac	particles,	respectively,	ψ:	R23×R → Cí ⨂3	the	N-particle spinor	and	A¡ = Φ,A the	electromagnetic	potentials	of	the	external	field. The	probability	4-current	for	each	particle	k, jU ¡ = ρU, jU = ψγψ, ψ ⊗¦<=U[= γ¦⨂γU⨂⊗¦<U[=3 γ¦ψ Q,& , with	the	adjoint	spinor	ψ = ψ£γ,	is	conserved,	∂¡j¡ = 0. In	complete	analogy	to	the	nonrelativistic	GE,	we	obtain	the	relativistic	GE	(rGE)	for	the	k-th	corpuscle	from	the	temporal	and spatial	components	of	the	4-current: dQU dt = jU ρU Q,& = ψ⊗¦<=U[= γ¦⨂γU⨂⊗¦<U[=3 γ¦ψ ψγψ (Q,&) In	the	single-particle	case,	N = 1,	this	rGE	reduces	to	(μ = 0,... ,3): dQ¡ dt = ψγ¡ψ ψγψ (Q,&) 181	Cf.	Dürr	et	al.	(2013) 64 It's Lorentz invariant: Due to the physical insignificance of re-scaling the time parameterisation, one obtains, for some parametrisation s of the particle's worldline, a manifestly	covariant	and	geometric	reformulation: dQ¡ ds ∥ ψγ ¡ψ. The	complicacies	surge	in	the	many-particles	case,	N > 1.	Then,	the	rGE,	by	using	a	common time for all particles, stipulates	a	hyperplane	of simultaneity	of all	N corpuscles	on	whose positions	the	l.h.s.	of	the	rGE	depends,	thereby	defining	a	distinguished	reference	frame	K.	In particular,	the	relativistic	QEH,	with	the	probability	density,	i.e.	the	0-component,	of	the	above Dirac	4-current,	no	longer	holds	in	all	frames.	Although	K	turns	out	not	to	be	detectable,	i.e. experimentally,	all	reference	frames	remain	indistinguishable182,	the	postulate	of	an	absolute reference frame violates SR's Principle of Relativity,	which declares the equivalence of all reference	frames.	In	short:	While	the	statistical	predictions	coincide	with	those	of	standard Dirac	theory,	on	the	level	of	the	individual	particles	SR	is	violated. One might try to dodge the absolute simultaneity by moving to a multi-time wavefunction	ψ(Q=, t=; ...Q3, t3)	on R2×R ⨂3 , which assigns each particle a time of its own	and	satisfies	the	multi-time	Dirac	Equation: iγU ¡ ∂¡,U9 − ieA¡ − m ψ = 0. And	indeed,	the	rGE,	constructed	via	the	above	scheme	from	the	4-current,	turns	out	to	be Lorentz invariant for factorisable states, i.e. multi-time wavefunctions of the form ψ Q=, t=; ...Q3, t3 = φU(tU, QU)3U<= . What	about	entangled/non-factorisable	states?	The	resulting	4-current	of	the	k-th	particle, jU ¡ Q=, t=; ...Q3, t3 = ψ⊗¦<=U[= γ¦⨂γU ¡⨂⊗¦<U[=3 γ¦ψ QÊ,&Ê;...Qó,&ó then likewise is no longer	separable.	For	a	viable	GE,	we	thus	need	to	connect	the	velocity	of	the	k-th	particle with	the	N-1	tuples	(t¦, Q¦),	with	j ≠ k.	One	could	now	consider	a	hyperplane	Σô on	which t= = t9 = ⋯ = t3,	the	multi-time	Dirac	Equation	then	reduces	to	the	familiar	one.	Let	η¡	be 182	Cf.	Berndl	et	al.	(1996);	Dürr	et	al.	(1999) 65 an	observer	at	rest	in	this	reference	frame	K	thus	distinguished,	η¡ ⊥ Σô.	Then,	the	4-current becomes: jU ¡ Q=, t=; ...Q3, t3 = ψ⊗¦<=U[= γ¦øηø ⨂γU ¡⨂⊗¦<U[=3 γ¦øηø ψ QÊ,&Ê;...Qó,&ó . The	resulting	rGE	for	the	k-th	particle	position	QU	is: dQU ¡ dt = jU ¡ jUøηø . The	probability	density	in	the	denominator	is	now	obviously	a	scalar,	but	furthermore	easily verified	to	be	independent	of	the	particle	index.	Consequently,	a	re-parametrisation	yields	the manifestly	covariant	rGE: dQU ¡ ds = jU ¡ Q(ù) . But	how	to	construct	the	preferred	reference	frame?	Which	vector	field	η¡	to	take?	Lest	the thus	attained	Lorentz	covariance	be	hazarded,	η¡	must	be	built	via	a	Lorentz	invariant	law.	A popular	proposal	is	to	choose	the	rest	frame	of	the	total	energy-momentum	of	the	universe, P¡ = dσø(x) Ψ t¡ø Ψû , with the wavefunction of the universe	Ψ , the total energymomentum	tensor	t¡ø	in	the	Heisenberg	picture	and	S	an	arbitrary	space-like	hypersurface. (Due	to	the	conservation	of	energy-momentum	and	Stoke's	Theorem,	P¡	doesn't	depend	on S.)	The	preferred	vector	field	then	is	η¡ ≔ ý þ ýþ . Have	we	thus	finally	achieved	a	satisfactorily	relativistic	dBBT?	Although	Lorentz	covariance has	been	restored,	the	model	still	lacks	what	Bell	calls	"serious	Lorentz	invariance":	It	needs to postulate extra structure in spacetime – an addition SR per se doesn't warrant. Furthermore, it	deserves to	be	pointed	out that it's	not	clear	how	to	extend	the	model to incorporate	interaction	potentials. In	conclusion,	the	quest	for	a	rGE	heaves	on	us	the	burden	of	extra	structure	in	spacetime;	SR simpliciter	and	dBBT	seem	incompatible.183 183	Cf.	also	Maudlin	(1996),	Sect.	2 66 Where	now	does	sdBBT	stand?	Two	points	are	pertinent: Firstly,	as	compared	to	standard	QM,	sdBBT,	as	a	non-collapse	theory,	avoids	the	full-front collision	between	instantaneous	collapses	of	the	wavefunction	and	Relativity,	as	well	as	the awkwardness of an ad-hoc	modification of some principles of QM so as to incorporate a collapse.	But	isn't	the	spontaneous	manifestation	of	a	corpuscle	configuration	tantamount	to form	of	instantaneous	collapse?	This	is	mistaken:	The	dispositional	wavefunction	continues	to evolve unitarily according to the Schrödinger dynamic, even after a manifestation. The (absence of a) manifestation of a disposition has no bearing on the disposition and its evolution	as	such. As	compared	to	dBBT,	with	the	abolition	of	a	GE,	the	need	for	preferred	reference	frames	or extra structure lapses	–	and thereby	dBBT's	obstacle for "serious Lorentz covariance".	But what about an action-at-a-distance, stemming from QM's allegedly inherent non-locality, which,	as	Bell	argued,	originates	in	the	fact	"(t)hat	the	[wavefunction]	[...]	propagates	not	in ordinary	three-space,	but	in	a	multidimensional	configuration	space[...]"?184	Indeed,	actionsat-a-distance would threaten the compatibility with SR in a manner resembling what we witnessed in the Dirac-Bohm case. However, Bell's diagnosis of QM's innate non-locality, which	the	violation	of	his	famous	inequalities	are	supposed	to	attest	to,	is	too	hasty. Bell	famously	elucidates	locality	(or	"local	causality")	for	two	measurement	outcomes	A	and	B of	spacelike	separated	measurements,	with	measurement	settings	a	and	b	as	follows:	"[...] once	all	the	possible	common	causes	of	the	two	events	are	taken	into	account	(which,	guided by classical relativistic intuitions, we take to reside in their joint past), we expect the probability	distributions for the	measurement	outcomes to	be independent	and	no longer display any correlations."185 Formally, with	λ denoting all relevant causal factors, i.e. the common	causes	in	the	overlap	of	the	past	cones	of	the	measurement	events,	the	probability distribution pa/b , expressing the correlations between the measurement outcomes, dependent	only	locally	on	the	measurement	setups	a/b,	factorises: pa,b A ∧ B λ = pa A λ pb B λ . 184	Bell	(1987),	p.	115 185	Brown	(2005),	p.184 67 From	this	Bell	derives	his	inequalities,	which	QM	generically	violates	for	entangled	states. However,	Brown	remarks	that	"[...]	failure	of	local	causality	in	Bell's	sense	does	not	entail	the presence	of	non-local	causes.	In	arriving	at	the	requirement	of	factorisability	it	is	necessary	to assume something like Reichenbach's principle of the common cause; namely that if correlations	are	not	due	to	a	direct	causal	link	between	two	events,	then	they	must	be	due	to common	causes,	such	causes	having	been	identified	when	conditionalization	of	the	probability distribution results in statistical independence."186 The violation of quantum correlations satisfying	Bell's	local	causality	thus	could	simply	imply	that	quantum	correlations	aren't	always apt	for	causal	explanations;	violation	of	local	causality	on	its	own	needn't	imply	non-locality in	the	guise	of	an	instantaneous	action-at-a-distance. sdBBT	takes	that	route:	It	blocks	the	very	formulation	of	Bell's	local	causality	by	rejecting	the Principle of Common Cause (whose scope of validity remains contented on independent grounds,	also	in	classical	physics187).	Recall	from	our	discussion	of	Humphreys'	Paradox	that we	embraced	from	the	outset	sdBBT's	stochastic,	a-causal	nature. This	a-causality is rooted in sdBBT's temporal solipsism, i.e. the	absence	of	any	diachronic identity	of	corpuscles.	It	defies	both	sdBBT's	locality	and	non-locality.	Not	persisting	beyond an infinitesimal instant of time, corpuscles can neither themselves traverse arbitrary distances, nor can they act on each other: An influence denotes a correlation between a change	in	the	state	of	one	thing	and	a	change	in	the	state	of	another.	But,	lacking	diachronic identity,	no	pair	of	successive	positions	occupied	by	corpuscles	can	be	attributed	to	the	same corpuscle;	no	corpuscle	at	a	different,	later	position	can	be	re-identified	with	the	corpuscle	at an	earlier	stage.	Conceptually,	no	corpuscle	can	change:	The	possibility	of	mutual	influence	or action	thus	is	blocked.	sdBBT	is	neither	an	action-at-a-distance	nor	a	local	theory! How	then	to	understand	entangled	states?	We	submit	that	sdBBT	simply	postulates	brute	fact statistical	correlations	that	reflect	the	wavefunction's	non-separability. More	in	detail,	following	Einstein's	view	on	QM188,	entangled	corpuscles	can	be	understood "literally"	–	namely	as	failure	of	separability	of	their	propensities:	Separated	subsystems	of	a 186	Op.cit.	p.	185 187	Cf.	Arntzenius	(2010) 188	Cf.	Howard	(1985,	1989) 68 joint system typically no longer possess separate wavefunctions. The wavefunction thus represents	a	holistic,	dispositional	property.	When	performing	a	measurement,	we	register correlations	between	manifestations	of	these	non-separable	dispositions.	The	correlations,	by themselves	relativistically innocuous,	then	are	brute	facts	of	a	fundamental,	non-separable reality.	Only	when	trying	to	further	explain	them	in	terms	of	a	common	cause	does	the	pickle regarding	compatibility	with	SR	start	–	but	there	is	neither	need	nor	justification	(beyond	the merely heuristic fecundity) to impose the a priori demand all correlations require explanation.189	Fine	offers	a	helpful	analogy:	"[Such	a	demand]	is	like	the	ideal	that	was	passed on in the dynamical tradition from Aristotle to Newton, that motion as such requires explanation."190 It	deserves	to	be	mentioned	how	sdBBT	escapes	Einstein's	transcendental	criticism	of	nonseparability. 191 Distinguishing meticulously between non-separability and non-locality, he argued, that	were	QM	complete,	we'd	have to abandon separability, i.e. the independent existence	of	separated	systems;	and	"(i)f	this	axiom	were	to	be	completely	abolished,	the	idea of	the	existence	of	(quasi-)	enclosed	systems,	and	thereby	the	postulation	of	laws	which	can be	checked	empirically	in	the	accepted	sense,	would	become	impossible."192	sdBBT	blocks	this worry	in	a	twofold	way:	Firstly,	QM	isn't	regarded	as	complete:	As	a	minimal	extension	of	QM, sdBBT	supplements	it	by	the	primitive	ontology	in	terms	of	corpuscles	and	their	positions	as beables. Secondly, the actual configurations of one subsystem	exist indeed independently from those	of a remote subsystem: The actual, local configurations completely define the actual state of each subsystem; nonetheless, there exist correlations between them that betoken	the	non-separable	disposition	of	the	joint	system. Finally, one might worry about a metaphysical incompatibility between SR and sdBBT's stochasticity.	Arguably,	SR	is	best	understood	in	terms	of	a	Block	Universe	View193,	with	no temporal	part	of	an	object's	4-dimensional	worldline	being	ontologically	distinguished	as	"the present"	and	the	present	only	an	indexically	designated	point	on	the	worldline.	By	contrast, 189	Cf.	Fine	(1989)	for	an	elaboration	of	this	for	QM 190	Op.cit.,	p.	192 191	Cf.	Howard	(1985,1989);	also	Brown	(2005),	Appendix	B3 192	Einstein	(1948),	quoted	in	Brown	(2005),	p.	187 193	Cf.	e.g.	Petkov	(2009),	esp.	Ch.	5	and	6;	Cf.,	however,	Dickson	(1989),	Ch.	8	for	a	criticism 69 the	strong	intuition	prevails,	articulated	e.g.	by	Popper194,	that	in	an	indeterministic	world	the future	is	"open",	i.e.	un-determined,	while	the	past	is	"fixed",	unalterable;	consequently,	the present	is	the	ontologically	distinguished	moment,	where	pure	potentiality	of	the	open	future turns into the congealed facticity of the past. By introducing such an ontologically distinguished	hypersurface	that	defines	the	present,	this	("Growing	Block	Universe")	view	has been argued to collide with SR. Consequently, as a stochastic and hence indeterministic theory,	sdBBT	entails	a	Growing	Block	Universe	Theory	of	Time	–	in	conflict	with	SR. The	conflict	can	be	resolved	by	challenging	the	claim	that	stochastic	theories	are	committed to	the	kind	of	indeterminism	that	implies	an	open	future	in	the	above	sense.	Dispositionalism offers	a straightforward	alternative	conceptualisation	of indeterminism:	The	4-dimensional picture	of	actual	reality	in	such	an	indeterministic	world	isn't	that	of	continuous	trajectories of	corpuscles	in	Minkowski	spacetime;	rather,	the	corpuscles	are	distributed	in	spacetime	like dust.195	In	other	words:	Whereas	the	disposition	evolves	deterministically,	its	manifestation takes	the	form	of	dust	sprinkled	over	spacetime,	their	distribution	typically	approximating	the relativistic analogues to the Born Rule, relative to an observer. Such a view is effortlessly compatible	with	a	Block	Universe	View,	and	might	aptly	be	called	a	Dust	Universe	View.	(A perhaps	helpful	paraphrase	is	that	in	the	Dust	Universe	View,	the	4-dimensional	spacetime isn't	pervaded	by	continuous	particle trajectories; rather, the	"dust	of	events" is randomly distributed.) In	summary,	we	have	seen	that	-within	the	framework	(and	confines)	of	relativistic	QMsdBBT suggests considerable advantages over dBBT; a comparative evaluation of quantum field theoretic	extensions	of	both	therefore	seems	promising.196 V.	Summary	and	outlook We	started	our	investigation	with	examining	the	conceptual	tension	between	ambitions	for an inveterately objectivist "quantum theory without observers", the probabilistic QEH, guaranteeing dBBT's empirical adequacy, and its determinism, as introduced via the GE. 194	Cf.	Popper	(1988),	Ch.	III,	sect.	18.	Popper	adduces	this	"argument	from	the	asymmetry	of	the	past	and	the future"	to	support	his	indeterminism. 195	Cf.	also	Petkov	(2009),	Ch.	10,	who	calls	this	view	"4-atomism". 196	It	seems	plausible,	however,	that	the	deBroglie-Bohmian	framework,	narrowly	understood	as	a	particle-based primitive	ontology	with	position	as	beables,	needs	to	give	way	to,	say,	a	field-based	primitive	ontology	with	e.g. fermion	number,	as	Bell's	model	for	a	Hamiltonian	QFT	suggests,	cf.	Bell	(1984). 70 Further	analysis	identified	the	latter	as	the	source	of	even	more	problems,	related	to	dBBT's formal definability, uniqueness, the violation of the Action Reaction Principle and the contrived-looking double role of the wavefunction. Fortunately, the GE turned out to be redundant: All explanatory work w.r.t. solving the Measurement Problem, empirical equivalence	with	QM	or	the	natural	justification	of	the	Symmetrisation	Postulate,	is	done	by dBBT's particle primitive ontology with position as beables and contextuality of all other dynamical	variables.	This	suggested	to	excise	the	GE,	whilst	keeping	dBBT's	aforementioned ontological framework, yielding a realist, albeit fundamentally stochastic/indeterministic theory – sdBBT – the probabilities of which we proposed to construe as an	N-corpuscle universe's	disposition	to	spontaneously/randomly	actualise	a	certain	configuration.	We	tried to	make	the	case	that	sdBBT	counts	as	a	minimally	deBroglie-Bohmian	theory,	deserving	to	be taken	seriously	as	a	potentially	fundamental	account	of	microscopic	reality.	It	turned	out	to be	a	many	worlds	theory	–	however	with	the	arguably	slightly	eccentric	feature	of	"temporal solipsism": FAPP-classical worlds typically exist actualiter only for an instant, before absconding	again	into	potentiality.	Future	research	regarding	sdBBT,	especially	along	the	lines of	its	relativistic	field-theoretical	extension,	is	promising. Its	relation,	too,	to	other	theories	sparks	off	a	number	of	exciting	questions: • Above	we	presented	sdBBT	as	a	minimally	deBroglie-Bohmian	theory	and	argued	for its	superiority	vis-à-vis	dBBT.	Another	theory	seems	to	qualify	equally	well	as	minimally deBroglie-Bohmian – namely Nelson Stochastics, which aims to derive the wavefunction and its Schrödinger dynamics from a classical Wiener-process in configuration space, subject to certain natural constraints. 197 It would be highly interesting	to	compare	sdBBT	to	Nelson	Stochastics:	Might	sdBBT	prove	advantageous also	vis-à-vis	other	minimally	deBroglie-Bohmian	Theories? • More	generally,	how	does	sdBBT	fare	vis-à-vis	other	primitive	ontology	approaches	to QM? 198 Of special interest is a comparison with "Schrödinger's Many Worlds Interpretation" (Allori et al. 199 ), an arguably minimally "continuous-matter-field 197	Cf.	Bacciagaluppi	(2005)	for	a	review,	as	well	as	a	comparison	with	dBBT. 198	Cf.	Esfeld	(2014)	for	"guidelines	for	an	assessment	of	the	proposals".	He	argues	for	the	superiority	of	dBBT. Given	our	result	of	sdBBT's	superiority	over	dBBT,	sdBBT's	superiority	seems	to	follow	straightforwardly,	but	a direct	comparison	might	yield	instructive	details. 199	Cf.	Allori	et	al.	(2014) 71 primitive ontology" (Allori et al.). It can be viewed as an Everettian many worlds counterpart	to	sdBBT,	in	which	temporal	solipsism	is	overcome. • Perhaps	even	more	alluring	will	be	the	comparison	with	proposals	from	the	rivalling "wavefunction	ontology" (Allori) camp,	a strictly	monist approach	which	postulates nothing	(e.g.	corpuscles, fields,	etc.)	on	top	of	the	wavefunction.	Of	special interest will,	of	course,	be	the	comparison	with	Everett's	Many	Worlds	Interpretation. Besides	its	many	worlds	character	being	perceived	as	metaphysically	too	sumptuous, criticism	of	the	Many	Worlds	Interpretation	has	tended	to	focus	on	the	role	and	status of	probabilities	and	the	Problem	of	the	Preferred	Basis,	both	of	which	sdBBT	eschews ab	initio. Acknowledgement We owe thanks to Harvey Brown (Oxford, UK), Michael Esfeld (Lausanne, CH), Dustin Lazarovici (Lausanne,	CH),	Niels	Linnemann (Geneva,	CH),	Vera	Matarese (HKU,	HK),	Oliver Pooley	(Oxford,	UK),	James	Read	(Oxford,	UK),	Thomas	Møller-Nielsen	(Oxford,	UK),	Darrell Rowbottom	(Lingnan,	HK)	Simon	Saunders	(Oxford,	UK)	and	Jiji	Zhang	(Lingnan,	HK)	for	helpful discussions	and	feedback	on	earlier	versions	of	the	manuscript. Patrick	Duerr	gratefully	acknowledges	the	generous	financial	support	of	the	British	Society	for the	Philosophy	of	Science. Alexander	Ehmann	gratefully	acknowledges	the	generous	financial	support	of	the	Research Grants	Council	through	the	Hong	Kong	PhD	Fellowship	Scheme. Bibliography Albert,	D.	(1992):	"Quantum	Mechanics	and	Experience",	Harvard	University	Press Allori,	V.	et	al.	(2007):	"On	the	common	structure	of	Bohmian	Mechanics	and	the	GhirardiRimini-Theory",	https://arxiv.org/abs/quant-ph/0603027 Allori,	V.	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