Abstract
GRW theory offers precise laws for the collapse of the wave function. These collapses are characterized by two new constants, \(\lambda \) and \(\sigma \). Recent work has put experimental upper bounds on the collapse rate, \(\lambda \). Lower bounds on \(\lambda \) have been more controversial since GRW begins to take on a many-worlds character for small values of \(\lambda \). Here I examine GRW in this odd region of parameter space where collapse events act as natural disasters that destroy branches of the wave function along with their occupants. Our continued survival provides evidence that we don’t live in a universe like that. I offer a quantitative analysis of how such evidence can be used to assess versions of GRW with small collapse rates in an effort to move towards more principled and experimentally-informed lower bounds for \(\lambda \).
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Notes
It has been suggested that different particles might collapse at different rates depending on their masses (Pearle and Squires 1994). The analysis presented here could be applied to such a formulation.
There has been some debate over whether the destruction of other branches is successful; see the literature on the problem of tails. Here I assume that the problem can be solved. If it cannot, GRW is not a viable solution to the measurement problem. In particular, I will assume that if collapse chooses one part of the state and massively shrinks the rest, it is not merely improbable to find oneself in a part of the state that was not fortunate enough to be the center of the collapse, it is impossible. There is no life in those other parts soon after collapse.
This ensures that, in general, a single collapse will be sufficient to destroy branches in which the measurement turned out differently.
This would be typical if we choose \(\lambda \) to be on the order of \(10^{-33} \text {s}^{-1}\) and assume that there are about \(10^{30}\) fundamental particles brought into an entangled superposition by the experiment [using (2.2)].
Although I expect that this straightforward account of theory confirmation applies to the cases under discussion, one might reasonably be concerned. The situations considered involve self-locating uncertainty (see Sebens and Carroll 2014; Vaidman 2014a, Sect. 4.2) and Bayesian conditionalization must be somehow modified to handle such cases (see Arntzenius 2003). Some modifications will vindicate the use of conditionalization here, others will not. To avoid controversy, I focus primarily on the probability of the evidence given the theory and not the posterior probabilities that result from updating on the evidence. I approach the problem from the familiar diachronic perspective, taking one’s previous beliefs and evidence to together determine what one’s current beliefs should be. Alternatively the problem could be approached synchronically, taking one’s evidence together with what Meacham (2010) calls an “epistemic kernel” to determine what one’s current beliefs should be (there are several competing ways of implementing this approach; see Manley 2014).
What wonderful theory succeeds in recovering the Born rule, as is demanded of the theory I’ve called “QM”? This will be a matter of disagreement. Let QM stand in for your favorite theory, whichever you think recovers the right probabilities, be it MWI, GRW\(_{\lambda = 10^{-16} \text {s}^{-1}}\), Bohmian mechanics, or something else. GRW\(_{\lambda = 10^{-16} \text {s}^{-1}}\) predicts deviations from the Born rule for certain yet-to-be-conducted experiments involving, e.g., macroscopic superpositions (which, even if perfectly isolated from the environment, would be predicted to be unstable). However, for the already-conducted experiments typically taken to provide support for quantum mechanics the predictions should (approximately) match those of the Born rule (setting aside the concerns raised in Sect. 6).
It’s fine if the predictions for certain future experiments diverge (see Footnote 7) since the data might (for all we know) support GRW over alternative formulations of quantum mechanics.
More realistically, \(N_S\) would increase as a function of time.
This is an optimistic estimate. In fact there will usually be many worlds corresponding to each outcome and thus even when a collapse is centered on the right outcome \(O_i\), one’s world might well be destroyed.
Here, to keep things simple, it is assumed that life on the branches not selected by collapse ends immediately (setting aside the possibility of delayed death mentioned in the previous section and Footnote 5).
Two clarifications: First, the proposition signified by “\( O_i \& \Delta t\)” in (4.6) should be understood as the indexical claim “I am alive \(\Delta t\) after the experiment and in my world the result of the experiment is \(O_i\).” not the weaker claim that “There exists a copy of me who is alive \(\Delta t\) after the experiment and in a world where the result of the experiment was \(O_i\).” Long after the experiment, the probability of the second claim is given by \(|\left\langle O_i | \Psi \right\rangle |^2\) since it is just the probability that the GRW collapse will select outcome \(O_i\). Thus the weaker claim might appear friendlier to GRW with small \(\lambda \). Why focus on the stronger claim? The weaker claim does not take into account one’s full (indexical) evidence and using it to update probabilities in GRW and MWI leads to unacceptable results (see Footnote 24).
Second, the probability of “\( O_i \& \Delta t\)” (stronger version) is difficult (perhaps impossible) to assess before the measurement since it is unclear whether one, all, or none of the post-branching copies are identical to the original experimenter. Fortunately, we can focus on the probability assigned to “\( O_i \& \Delta t\)” immediately after branching. Since the experimenter doesn’t yet know which branch they are on or whether they will survive, it makes sense to assign probabilities at this point. Further, these later probabilities are what matter for theory confirmation as these are the probabilities assigned to the evidence right before the evidence is acquired.
For the closest analogy, imagine that each cell of your prison is occupied by a copy of you that resulted from a 1 to 100 fission midday on New Year’s Eve.
In this case, the fission in Footnote 14 should not be supposed.
That is, looking at the probabilities derived from the collapse process this is what one should expect. As discussed earlier in this section, if one performs experiments, survives for a long time, and doesn’t look at the outcome, the probabilities that should be assigned to the different possible outcomes are not the standard Born rule probabilities. Seeing a sequence that fits these expectations should shift one’s credence towards GRW with small lambda—but, only after the theory has been significantly disconfirmed by one’s survival. Thus unlike Firing Squad and Prison Poisoning, there is in fact a way to get a piece of evidence that points towards the dangerous hypothesis. Still, one’s total course of experience will never favor GRW with small \(\lambda \) over QM; as can be seen by noting that the expression in parentheses in (4.6) is at most one.
If the collapse rate is much smaller, there will be many records only some of which show sequences deemed probable by the Born rule (as in MWI).
Wallace (2014) in considering a similar situation seems to find this—“strictly speaking”—sufficient empirical success as the theory does manage to “explain why the scientific community has so far observed statistical results in accord with quantum mechanics (via the anthropic fact that worlds in which violations were observed are now radioactive deserts [the fate he believes befalls worlds in the tails]). And it explains why it is rational to act as if the predictions of quantum mechanics were true (because in those worlds where they turn out false, we’re all doomed anyway).” Vaidman (2014b, Sect. 8) also seems untroubled by the possibility of death in the tail branches.
Do the copies need to last long enough to have thoughts to cause trouble? I think not. If you survive, you can consider what credences you should have assigned during the short period after splitting when you coexisted with the other copies.
The situation here is like that of the prisoner in Arkham if the period between the splitting event (see Footnote 14) and the deaths were made much shorter.
Anticipating an upcoming branching and subsequent collapse, can one assign a probability to survival before splitting? If so, does the fact that some successor will survive the collapse mean that before splitting survival is certain? These questions need not be answered as the relevant probabilities are those assigned immediately after branching (Footnote 13).
The cutoff \(\tau \) is not a free parameter and not derived from the collapse process. It could be calculated by determining when branching occurs in the absence of collapse (see Sect. 6).
Here it is also assumed that we’re considering familiar experiments, not future ones that probe smaller values of \(\lambda \) (see Footnote 7).
When \(\Delta t\) is large, the probability of the total evidence given by (4.6) is \(|\!\!\left\langle O_i | \Psi \right\rangle \!\!|^4\) because the probability of seeing \(O_i\) is \(|\!\!\left\langle O_i | \Psi \right\rangle \!\!|^2\) and the probability of subsequently surviving is also \(|\!\!\left\langle O_i | \Psi \right\rangle \!\!|^2\). This suggests a crafty maneuver to save GRW with small \(\lambda \). What if the probability of each outcome were one—after all, each outcome will be recorded by someone—so that the probability of total evidence is \(|\!\!\left\langle O_i | \Psi \right\rangle \!\!|^2\) as in QM? This proposal faces a fatal problem: practically every experiment would confirm MWI over single-world theories (see Greaves 2007, Sect. 4).
References
Adler, S. L. (2007). Lower and upper bounds on CSL parameters from latent image formation and IGM heating. Journal of Physics A: Mathematical and Theoretical, 40(12), 2935.
Aicardi, F., Borsellino, A., Ghirardi, G. C., & Grassi, R. (1991). Dynamical models for state-vector reduction: Do they ensure that measurements have outcomes? Foundations of Physics Letters, 4(2), 109–128.
Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of Bohmian mechanics and the Ghirardi–Rimini–Weber theory. The British Journal for the Philosophy of Science, 59(3), 353–389.
Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2011). Many worlds and Schrödinger’s first quantum theory. The British Journal for the Philosophy of Science, 62(1), 1–27.
Arntzenius, F. (2003). Some problems for conditionalization and reflection. The Journal of Philosophy, 100(7), 356–370.
Bacciagaluppi, G. (2012). The role of decoherence in quantum mechanics. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (2012th ed.). Oxford: Blackwell Publishers.
Bassi, A., Deckert, D.-A., & Ferialdi, L. (2010). Breaking quantum linearity: Constraints from human perception and cosmological implications. EPL (Europhysics Letters), 92(5), 50006.
Bassi, A., Lochan, K., Satin, S., Singh, T. P., & Ulbricht, H. (2013). Models of wave-function collapse, underlying theories, and experimental tests. Reviews of Modern Physics, 85(2), 471.
Benatti, F., Ghirardi, G. C., & Grassi, R. (1995). Quantum mechanics with spontaneous localization and experiments. In E. Beltrametti & J.-M. Lévy-Leblond (Eds.), Advances in quantum phenomena (pp. 263–279). Berlin: Springer.
Carroll, S. M., & Sebens, C. T. (2014). Many worlds, the born rule, and self-locating uncertainty. In D. Struppa & J. Tollaksen (Eds.), Quantum theory: A two-time success story: Yakir Aharonov Festschrift. Berlin: Springer.
Feldmann, W., & Tumulka, R. (2012). Parameter diagrams of the GRW and CSL theories of wavefunction collapse. Journal of Physics A: Mathematical and Theoretical, 45(6), 065304.
Ghirardi, G., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34(2), 470.
Gisin, N., & Percival, I. C. (1993). The quantum state diffusion picture of physical processes. Journal of Physics A: Mathematical and General, 26(9), 2245.
Greaves, H. (2007). On the everettian epistemic problem. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 38(1), 120–152.
Leslie, J. (1989). Universes. New York: Routledge.
Lewis, D. (2004). How many lives has Schrödinger’s cat? Australasian Journal of Philosophy, 82(1), 3–22.
Manley, D. (2014). On being a random sample. Manuscript
Maudlin, T. (2010). Can the world be only wavefunction? In S. Saunders, J. Barrett, A. Kent, & D. Wallace (Eds.), Many worlds?: Everett, quantum theory, & reality (pp. 121–143). Oxford: Oxford University Press.
Meacham, C. J. G. (2010). Unravelling the tangled web: Continuity, internalism, uniqueness and self-locating belief. In T. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology. Oxford: Oxford University.
Pearle, P., & Squires, E. (1994). Bound state excitation, nucleon decay experiments and models of wave function collapse. Physical Review Letters, 73, 1–5.
Schlosshauer, M. (2005). Decoherence, the measurement problem, and interpretations of quantum mechanics. Reviews of Modern Physics, 76(4), 1267.
Sebens, C. T., & Carroll, S. M. (2014). Self-locating uncertainty and the origin of probability in Everettian quantum mechanics. arXiv:1405.7577 [quant-ph].
Swinburne, R. (1990). The argument from the fine tuning of the universe. In J. Leslie (Ed.), Physical cosmology and philosophy. New York: MacMillan.
Tegmark, M. (1993). Apparent wave function collapse caused by scattering. Foundations of Physics Letters, 6(6), 571–590.
Vaidman, L. (2014a). Many-worlds interpretation of quantum mechanics. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (2014th ed.). Berlin: Springer.
Vaidman, L. (2014b). Quantum theory and determinism. Quantum Studies: Mathematics and Foundations, 1(1–2), 5–38.
Wallace, D. (2012). The emergent multiverse: Quantum theory according to the everett interpretation. Oxford: Oxford University Press.
Wallace, D. (2014). Life and death in the tails of the GRW wave function. arXiv:1407.4746 [quant-ph].
Zurek, W. H. (2003). Decoherence and the transition from quantum to classical–Revisited. quant-ph/0306072.
Acknowledgments
Thanks to David Albert, David Baker, Gordon Belot, Cian Dorr, J. Dmitri Gallow, Jeremy Lent, David Manley, Laura Ruetsche, Roderich Tumulka, and an anonymous referee. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 0718128.
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Sebens, C.T. Killer collapse: empirically probing the philosophically unsatisfactory region of GRW. Synthese 192, 2599–2615 (2015). https://doi.org/10.1007/s11229-015-0680-x
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DOI: https://doi.org/10.1007/s11229-015-0680-x