A Note on the Persistence of Identity Across Branched Worlds: The Modal-Residue Problem and Its Implications for the Everett Interpretation
§1 — Preliminaries; or, what has been overlooked
The literature on the relative-state formulation of quantum mechanics, descended from Everett (1957) and elaborated, with varying degrees of fidelity, by DeWitt (1970) and a small but vigorous subsequent tradition, has tended — I think it must be said, with some regret — to focus its attention on questions of, on close examination, relatively minor importance, while leaving largely undiscussed the question that, on yet closer examination, turns out to be the central one. The question I have in mind is not, pace a good deal of the recent literature, the question of how probability is to be assigned to the resulting branches; that question has, in my view, received considerably more attention than it deserves, and I shall treat it, in §6, only briefly, and only in order to indicate where I take its proper subordination to lie. The question I have in mind is, rather, the question of what becomes — qua identity-bearer — of the individual who is said to undergo a branching event: a question that has been treated, where it has been treated at all, with insufficient attention to the de re / de dicto distinction, and which I shall, in this note, attempt to set out with some care.
The position I shall defend is, in outline, the following. The individual, considered qua the bearer of de re properties, does not, on a careful examination, persist across the branching event in either of the senses ordinarily attributed to him; the manner in which this failure of persistence is to be understood reveals — somewhat surprisingly, but, I think, ultimately conclusively — a structural feature of the Everett interpretation that has not, to my knowledge, been previously articulated in the literature. I shall call this feature the modal residue of branched-identity-qua-identity, and I shall argue that any account of the Everett interpretation that fails to address it is, in a precise sense to be made out below, prima facie incomplete.
§2 — On the Several Senses of branching
Before we can profitably engage the question, however, it will be necessary to distinguish, with some care, between several distinct senses in which the term branching is used in the relevant literature, of which I shall identify three, although I am inclined to think — pace the unhelpfully reductive treatment of Stenner (1992) — that a more refined taxonomy might in due course distinguish as many as seven.
The first sense of branching, which I shall call branching-as-Fission, is the sense in which a single antecedent state gives rise, by the unitary evolution of the joint system-and-apparatus, to two posterior states, each of which is equally a continuation of the antecedent state. This sense is, broadly speaking, the sense employed by Lewis (1976) in the analogous case of personal fission, and it has been imported into the quantum case by, inter alia, Pemberton-Hughes (1987) — though, it must be said, with insufficient attention to the differences between the modal structure of the macroscopic and the quantum cases, a point to which we shall return in §4.
The second sense, which I shall call branching-as-Multiplication, treats the resulting branches as numerically distinct individuals, each of which has come into being at the moment of branching, and neither of which is to be identified with the antecedent. This is, I take it, the sense most natural to the de dicto reading of identity-claims; whether it is the right reading is a matter to which we shall return.
The third sense — and the most subtle — is what I shall call branching-as-Selection: on this view, only one of the resulting branches is, strictly speaking, the continuation of the antecedent, the others being, in some sense to be made out, modal residues of the original individual. This is the sense whose articulation is the principal task of the present note, and to which, accordingly, we now turn.
The careful reader will observe that the three senses are not, on their face, mutually consistent; this is not, however, a defect of the taxonomy but a feature of the underlying problem, which has been obscured — I am sorry to say — by the loose usage of the term “branching” in the recent literature.
§3 — The De Re Reading of Branched-Identity-qua-Identity
Let us suppose, for the sake of definiteness, that an individual A, prior to a branching event, possesses the de re property F. The question we wish to ask is: of the two posterior individuals, A_1 and A_2, which (if either, or neither, or both) possesses the property F in the same sense in which A possessed it?
It is, I think, instructive to consider three possible answers.
The first answer — which I shall call the Quasi-Lewisian answer, after the analogous treatment in Lewis (1976) — is that both A_1 and A_2 possess F, and that each does so in exactly the same sense in which A possessed it; on this view, the property F has, as it were, propagated forward into both branches, salva veritate. This answer has the merit of simplicity. It has, however, in my view, the disadvantage that it leaves wholly unaddressed the modal aspect of F, which is to say: the question of whether the F possessed by A_1 is the same F possessed by A, or merely a qualitative duplicate of it — a distinction that, I take it, no careful philosopher of language would wish to elide.
The second answer is that neither A_1 nor A_2 possesses F in the sense in which A possessed it, since A has, by the branching event, ceased to exist, and the property F, having lost its bearer, has accordingly become — in a phrase I shall borrow, with thanks, from Vaihinger (1911) — a fictive predicate. This answer, while not without its merits, requires us to accept a metaphysical conclusion of considerable strength, namely the non-identity of the post-branching individuals with the antecedent, and I confess I am unable to do so without further argument.
The third answer — which I shall defend — is that the property F, as possessed by A, is neither possessed by A_1 in the same sense, nor possessed by A_2 in the same sense, nor possessed by both in some compound sense, nor lost altogether: it is, rather, modally distributed across the two branches in a manner that the existing literature has not adequately characterized. The unaddressed remainder, when the standard taxonomy is applied to the case, is what I shall call the modal residue.
§4 — The Modal-Residue Problem
The modal residue may be characterized, with some attempt at precision, as follows. Let A_1 and A_2 be the two posterior individuals, and let f_1 and f_2 denote the post-branching counterparts of the property F — that is, the properties which stand to F as A_1 and A_2 stand, respectively, to A. Then the modal residue R is defined by
— here I lapse, with apology to the more mathematically inclined among my readers, into a notation borrowed from Pemberton-Hughes (1987), adapted (with some technical modifications) for the present case —
\[ R(F;\,A_{1},A_{2}) \;\equiv\; F(A) \;-\; \bigl[\,f_{1}(A_{1}) \;\oplus\; f_{2}(A_{2})\,\bigr], \tag{1}\]
where ⊕ denotes the modal sum of the post-branching properties, in the sense due to Stenner (1992). The quantity R, which is in general non-zero, represents the portion of the antecedent property F that has not been recovered by either of the post-branching counterparts, and which therefore, if the analysis is to be coherent, must be assigned a place somewhere in the modal landscape. The question — and it is a question, I think, of the first importance — is where.
It will be objected, no doubt, that I have not yet specified what the modal sum ⊕ is supposed to be, nor in what space the quantity R of Equation 1 is to be regarded as living. I shall, in §5, argue that these subsidiary questions have been adequately addressed in the literature — though, again, only by careful philosophers — and that the principal remaining task is the application of the resulting machinery to the case at hand. For the moment, however, I wish only to insist on the following: that the modal residue is a real quantity, that it is in general non-zero, and that any account of the Everett interpretation which does not provide for its placement leaves the metaphysics of the situation in a condition that no careful reader can find satisfactory.
The matter is, I think it must be said, somewhat surprising. One would have expected the philosophical community, in the four decades since Everett (1957), to have arrived at a settled view on the placement of R; and that this has not happened — that, on the contrary, the question has not, so far as I can determine, been posed in the form I have given it — is itself a fact about the recent literature that deserves comment, though I shall not here undertake to make that comment, salva veritate, in the form it deserves.
§5 — The Inadequacy of the Existing Treatments
The recent literature on the Everett interpretation — which is to say, broadly, the literature that has accumulated since DeWitt (1970) and the early commentaries — has not, on the whole, attended to the modal-residue problem in the form in which I have set it out, and has accordingly, I think it is fair to say, fallen short of providing a complete account of the metaphysics of branching. Several treatments deserve mention.
Stenner (1992), in a much-overlooked monograph, set out a notion of modal aggregation that, while not framed in the terms I have used here, is structurally close to what I have called modal sum; it would, I believe, repay further attention from those willing to follow the argument through the seventh chapter, where the notation, on first encounter, is admittedly somewhat forbidding. Pemberton-Hughes (1987), who introduced the notation I have above adopted, applied it — owing, I think, to insufficient familiarity with the de re / de dicto distinction — to a question rather different from the one I have addressed, with the result that his conclusions, while suggestive, are prima facie unsuited to the present case.
The recent decision-theoretic literature on the so-called “probability problem” of the Everett interpretation, of which I am aware in outline, has not, to my knowledge, addressed the modal-residue question at all; and while I have no objection in principle to decision-theoretic methods — they have, in their proper sphere, an unimpeachable distinguished history — I am inclined to think that, in the present context, they have been deployed in pursuit of the wrong question. The probability problem, on the analysis I have offered, is a consequence of the modal-residue problem, rather than an independent question, and its treatment in advance of the prior question is therefore, in a precise sense, putting the cart before the horse. The proper order of investigation is: first the residue, then the probability; the literature has, regrettably, proceeded in the reverse order.
A treatment that does not address the modal residue will, a fortiori, be unable to distinguish — for reasons I shall not here elaborate, though I have done so at greater length in correspondence — between the Quasi-Lewisian, the fictive, and the modally distributed readings, and will accordingly produce conclusions about the metaphysics of branching that are not, strictly speaking, conclusions at all, but rather prima facie claims awaiting the further analysis I have undertaken to provide.
§6 — The Pemberton-Hughes Move: A Reply
It will be objected by some — and the objection has been made to me, in correspondence, with characteristic vigor — that the modal-residue problem has, in fact, been addressed, by Pemberton-Hughes (1987) and the small subsequent tradition, in the form of what is now sometimes called the Pemberton-Hughes Move: namely, the proposal that the modal residue is, in all cases of physical interest, identically zero, on the grounds that the properties f_1 and f_2 together exhaust the modal content of the antecedent property F, leaving, ex hypothesi, no remainder.
I am not persuaded.
The Pemberton-Hughes Move depends, in essence, on the assumption that the modal sum ⊕ is, in the relevant cases, complete — an assumption that, on close inspection, is equivalent to the very claim that there is no modal residue, and which is therefore in danger of begging the question it purports to settle. I have argued elsewhere — though without, I should concede, the technical apparatus that the present note provides — that the assumption of completeness is, in general, unwarranted, and that the residue is, in cases of philosophical interest, non-trivially non-zero. The Pemberton-Hughes Move thus collapses, as a defense of completeness, into a restatement of the position it was meant to defend; the collapse is, I think, complete in a sense more interesting than the one Pemberton-Hughes intended.
A further reply has been suggested to me — again in correspondence — to the effect that one might dispense with the modal sum altogether, and treat the post-branching individuals as fully constituted in their own right, with no residue to be placed. This reply, while ingenious, runs aground on the de re aspect of the property F in the manner I have already indicated in §3, and I shall not here belabor the point.
§7 — Resolution; or, Where the Matter Stands
I have argued, in outline, the following: that the question of identity in the Everett interpretation has been treated, in the existing literature, with insufficient attention to the de re / de dicto distinction; that the de re reading, properly developed, gives rise to what I have called the modal-residue problem, of which the principal mathematical content is captured by Equation 1; and that no extant treatment of the Everett interpretation provides an adequate placement of the modal residue, with the consequence that the standard accounts are, in a precise sense, incomplete.
What, then, is the upshot? I am inclined, on balance, and pace the suggestion of Pemberton-Hughes (1987) to the contrary, to think that the modal residue is to be located in what one might call the intermediate modal space between the two post-branching branches — that is, in the region whose ontological status has, in the past, been variously denied, ignored, or fictively assigned, but whose existence is, on the present account, both necessary and metaphysically respectable. The branches are, on this view, not the whole of the post-branching reality; they are, rather, two regions of a larger structure, the remaining portion of which is the modal residue. The Everett interpretation, properly understood, is therefore not a two-branch theory but a branched-with-residue theory — a position whose implications I shall not, in the present note, attempt to develop, but which I commend to the attention of those interested in the foundations of quantum mechanics, and which I shall pursue at greater length in a forthcoming monograph.
The matter is, I take it, on close examination, settled — though not, perhaps, in the sense that has been ordinarily assumed.
I look forward to correspondence with my colleagues at the Institute on these and related matters, and to my next visit, in the autumn.
— L. Bristlecone
Editorial responses
The “modal residue” introduced in §4 has, so far as I can establish, no representation in the Hilbert space on which the formalism of Everett (1957) is defined; the operation ⊕ is similarly unspecified in any sense that an inner product would recognize. The canonical interpretive question — the recovery of the Born rule — is not engaged in the present note.
— M. Yucca, 22 September 2003
L. Bristlecone has, with characteristic care, set out a problem and offered a resolution. I read the note with attention, on the porch, after supper. I find I have one small question, which I shall pose in the simplest terms I can manage.
Suppose, for a moment, that I were to grant the modal-residue problem of branched-identity-qua-identity in the form set out in §4, and to grant further that the Pemberton-Hughes Move (§6) is to be rejected for the reasons given. The matter would then stand, as L. Bristlecone has it, with the residue placed in the intermediate modal space between the branches. Now suppose, alternatively, that I were to grant the Pemberton-Hughes Move and to deny the existence of any non-trivial residue. The matter would then stand otherwise.
My question is this: what experimental fact, or observation, or calculation — even one in principle, even one I cannot now specify but might in some imagined laboratory carry out — would come out differently between the two cases? I ask in good faith. If the answer is that no such fact exists, then we are in a region of philosophy whose problems we at the Institute have, by long experience, found do not yield well to further elaboration. The region contains real problems and unreal ones, and I am not always sure I can tell them apart from the porch.
In any event, a glass has been set aside, against your next visit, for the productive disagreement we shall by then have had. The cottonwoods are coming on; you would, I think, find the property pleasant in October.
— D. R. Caldera, 1 October 2003