Editing Many-worlds interpretation (section)

== Probability and the Born rule ==

Since the many-worlds interpretation's inception, physicists have been puzzled about the role of probability in it. As put by Wallace, there are two facets to the question:<ref name="wallace2003"/> the ''incoherence problem'', which asks why we should assign probabilities at all to outcomes that are certain to occur in some worlds, and the ''quantitative problem'', which asks why the probabilities should be given by the [[Born rule]].

Everett tried to answer these questions in the paper that introduced many-worlds. To address the incoherence problem, he argued that an observer who makes a sequence of measurements on a quantum system will in general have an apparently random sequence of results in their memory, which justifies the use of probabilities to describe the measurement process.<ref name=everett56/>{{rp|69–70}} To address the quantitative problem, Everett proposed a derivation of the Born rule based on the properties that a measure on the branches of the wave function should have.<ref name=everett56/>{{rp|70–72}} His derivation has been criticized as relying on unmotivated assumptions.<ref name="ballentine1973">{{cite journal |author1=Ballentine |first=L. E. |date=1973 |title=Can the statistical postulate of quantum theory be derived?—A critique of the many-universes interpretation |journal=Foundations of Physics |volume=3 |issue=2 |pages=229–240 |bibcode=1973FoPh....3..229B |doi=10.1007/BF00708440 |s2cid=121747282}}</ref> Since then several other derivations of the Born rule in the many-worlds framework have been proposed. There is no consensus on whether this has been successful.<ref>{{cite book |first=N. P. |last=Landsman |chapter-url=http://www.math.ru.nl/~landsman/Born.pdf |quote=The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle. |chapter=The Born rule and its interpretation |title=Compendium of Quantum Physics |editor-first=F. |editor-last=Weinert |editor2-first=K. |editor2-last=Hentschel |editor3-first=D. |editor3-last=Greenberger |editor4-first=B. |editor4-last=Falkenburg |publisher=Springer |year=2008 |isbn=978-3-540-70622-9 }}</ref><ref name="kent2009">{{Cite book |last1=Kent |first1=Adrian |title=Many Worlds? Everett, Quantum Theory and Reality |publisher=Oxford University Press |year=2010 |editor=Saunders |editor-first=S. |chapter=One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation |bibcode=2009arXiv0905.0624K |editor2=Barrett |editor-first2=J. |editor3=Kent |editor-first3=A. |editor4=Wallace |editor-first4=D. |arxiv=0905.0624}}</ref><ref>{{cite journal |last1=Kent |first1=Adrian |year=1990 |title=Against Many-Worlds Interpretations |journal=International Journal of Modern Physics A |volume=5 |issue=9 |pages=1745–1762 |arxiv=gr-qc/9703089 |bibcode=1990IJMPA...5.1745K |doi=10.1142/S0217751X90000805 |s2cid=14523184}}</ref>

=== Frequentism ===

[[Bryce DeWitt|DeWitt]] and Graham<ref name="dewitt73"/> and Farhi et al.,<ref>{{cite journal |last1=Farhi |first1=Edward |last2=Goldstone |first2=Jeffrey |last3=Gutmann |first3=Sam |date=1989 |title=How probability arises in quantum mechanics |journal=Annals of Physics |volume=192 |issue=2 |pages=368–382 |bibcode=1989AnPhy.192..368F |doi=10.1016/0003-4916(89)90141-3}}</ref> among others, have proposed derivations of the Born rule based on a [[Frequentist probability|frequentist]] interpretation of probability. They try to show that in the limit of [[Uncountable set|uncountably many]] measurements, no worlds would have relative frequencies that didn't match the probabilities given by the Born rule, but these derivations have been shown to be mathematically incorrect.<ref>{{Cite journal|last=Benioff|first=Paul|author-link=Paul Benioff|date=October 1978|title=A note on the Everett interpretation of quantum mechanics|journal=[[Foundations of Physics]]|language=en|volume=8|issue=9–10|pages=709–720|doi=10.1007/BF00717501|bibcode=1978FoPh....8..709B|s2cid=123279967|issn=0015-9018}}</ref><ref>{{cite journal |last1=Caves |first1=Carlton M. |author-link1=Carlton M. Caves |last2=Schack |first2=Rüdiger |date=2005 |title=Properties of the frequency operator do not imply the quantum probability postulate |journal=[[Annals of Physics]] |volume=315 |issue=1 |pages=123–146 |arxiv=quant-ph/0409144 |bibcode=2005AnPhy.315..123C |doi=10.1016/j.aop.2004.09.009 |s2cid=33263618}}</ref>

=== Decision theory ===
A [[decision theory|decision-theoretic]] derivation of the Born rule was produced by [[David Deutsch]] (1999)<ref>{{Cite journal|arxiv=quant-ph/9906015|last1=Deutsch|first1=David|title=Quantum Theory of Probability and Decisions|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume=455 | issue=1988 | pages=3129–3137 | year=1999 | doi=10.1098/rspa.1999.0443 | bibcode=1999RSPSA.455.3129D |s2cid=5217034}}</ref> and refined by Wallace<ref name="wallace2003">{{cite journal | last1 = Wallace | first1 = David | year = 2003 | title = Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation | journal = Stud. Hist. Phil. Mod. Phys. | volume = 34 | issue = 3| pages = 415–438 | bibcode = 2003SHPMP..34..415W | doi = 10.1016/S1355-2198(03)00036-4 | arxiv = quant-ph/0303050 | s2cid = 1921913 }}</ref><ref>{{cite arXiv |eprint=quant-ph/0211104|last1=Wallace|first1=David|title=Quantum Probability and Decision Theory, Revisited|year=2002}}</ref><ref>{{cite arXiv |eprint=quant-ph/0312157 | last1=Wallace | first1=David | title=Quantum Probability from Subjective Likelihood: Improving on Deutsch's proof of the probability rule|year=2003}}</ref><ref>{{cite arXiv |eprint=0906.2718|last1=Wallace|first1=David|title=A formal proof of the Born rule from decision-theoretic assumptions|class=quant-ph|year=2009}}</ref> and Saunders.<ref>{{cite journal | last1 = Saunders | first1 = Simon | year = 2004 | title = Derivation of the Born rule from operational assumptions | journal = Proc. R. Soc. Lond. A | volume = 460 | issue = 2046| pages = 1771–1788 | bibcode = 2004RSPSA.460.1771S | doi = 10.1098/rspa.2003.1230 | arxiv = quant-ph/0211138| s2cid = 1459183 }}</ref><ref>{{cite book|arxiv=quant-ph/0412194|last1=Saunders|first1=Simon|title=Quo Vadis Quantum Mechanics?|pages=209–238|year=2004|doi=10.1007/3-540-26669-0_12|chapter=What is Probability?|series=The Frontiers Collection|isbn=978-3-540-22188-3|s2cid=117218061}}</ref> They consider an agent who takes part in a quantum gamble: the agent makes a measurement on a quantum system, branches as a consequence, and each of the agent's future selves receives a reward that depends on the measurement result. The agent uses decision theory to evaluate the price they would pay to take part in such a gamble, and concludes that the price is given by the utility of the rewards weighted according to the Born rule. Some reviews have been positive, although these arguments remain highly controversial; some theoretical physicists have taken them as supporting the case for parallel universes.<ref name="newsci">{{Cite news | last = Merali | first = Zeeya | title = Parallel universes make quantum sense | magazine = New Scientist | issue = 2622 |date=2007-09-21 | url = https://www.newscientist.com/article/mg19526223.700-parallel-universes-make-quantum-sense.html | access-date = 2013-11-22 }} (Summary only).</ref> For example, a ''[[New Scientist]]'' story on a 2007 conference about Everettian interpretations<ref>{{cite web| url = http://www.perimeterinstitute.ca/conferences/many-worlds-50| title = Perimeter Institute, Many worlds at 50 conference, September 21–24, 2007}}{{cite web| url = http://www.perimeterinstitute.ca/video-library/collection/many-worlds-50-2007| title = Videos}}</ref> quoted physicist Andy Albrecht as saying, "This work will go down as one of the most important developments in the history of science."<ref name="newsci"/> In contrast, the philosopher [[Huw Price]], also attending the conference, found the Deutsch–Wallace–Saunders approach fundamentally flawed.<ref>{{Cite book |last1=Price |first1=Huw |title=Many Worlds? Everett, Quantum Theory and Reality |publisher=Oxford University Press |year=2010 |editor=Saunders |editor-first=S. |chapter=Decisions, Decisions, Decisions: Can Savage Salvage Everettian Probability? |editor2=Barrett |editor-first2=J. |editor3=Kent |editor-first3=A. |editor4=Wallace |editor-first4=D. |arxiv=0802.1390}}</ref>

=== Symmetries and invariance ===
In 2005, Zurek<ref name="zurek2005">{{cite journal |last1=Zurek |first1=Wojciech H. |author-link=Wojciech H. Zurek |year=2005 |title=Probabilities from entanglement, Born's rule from envariance |journal=Physical Review A |volume=71 |issue=5 |page=052105 |arxiv=quant-ph/0405161 |bibcode=2005PhRvA..71e2105Z |doi=10.1103/physreva.71.052105 |s2cid=18210481}}</ref> produced a derivation of the Born rule based on the symmetries of entangled states; Schlosshauer and Fine argue that Zurek's derivation is not rigorous, as it does not define what probability is and has several unstated assumptions about how it should behave.<ref>{{cite journal |last1=Schlosshauer |first1=M. |last2=Fine |first2=A. |year=2005 |title=On Zurek's derivation of the Born rule |journal=Foundations of Physics |volume=35 |issue=2 |pages=197–213 |arxiv=quant-ph/0312058 |bibcode=2005FoPh...35..197S |doi=10.1007/s10701-004-1941-6 |s2cid=119100306}}</ref>

In 2016, Charles Sebens and [[Sean M. Carroll]], building on work by [[Lev Vaidman]],<ref>Vaidman, L. "Probability in the Many-Worlds Interpretation of Quantum Mechanics". In: Ben-Menahem, Y., & Hemmo, M. (eds), The Probable and the Improbable: Understanding Probability in Physics, Essays in Memory of Itamar Pitowsky. Springer.</ref> proposed a similar approach based on self-locating uncertainty.<ref>{{cite journal |last1=Sebens |first1=Charles T. |last2=Carroll |first2=Sean M. |year=2016 |title=Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics |journal=The British Journal for the Philosophy of Science |volume=69 |issue=1 |pages=25–74 |arxiv=1405.7577 |doi=10.1093/bjps/axw004 |s2cid=53648469}}</ref> In this approach, decoherence creates multiple identical copies of observers, who can assign credences to being on different branches using the Born rule. The Sebens–Carroll approach has been criticized by [[Adrian Kent]],<ref>{{Cite journal|last=Kent|first=Adrian|author-link=Adrian Kent|date=February 2015|title=Does it Make Sense to Speak of Self-Locating Uncertainty in the Universal Wave Function? Remarks on Sebens and Carroll|journal=Foundations of Physics|language=en|volume=45|issue=2|pages=211–217 |arxiv=1408.1944|doi=10.1007/s10701-014-9862-5|issn=0015-9018|bibcode=2015FoPh...45..211K|s2cid=118471198}}</ref> and Vaidman does not find it satisfactory.<ref>{{Cite book|last1=Vaidman|first1=Lev|chapter=Derivations of the Born Rule|title=Quantum, Probability, Logic: Itamar Pitowsky's Work and Influence |editor=Meir Hemmo |editor2=Orly Shenker |publisher=Springer Nature Switzerland |year=2020|id=PhilSci: [http://philsci-archive.pitt.edu/15943 15943]}}</ref>

=== Branch counting ===
In 2021, [[Simon Saunders]] produced a branch counting derivation of the Born rule. The crucial feature of this approach is to define the branches so that they all have the same magnitude or [[norm (mathematics)|2-norm]]. The ratios of the numbers of branches thus defined give the probabilities of the various outcomes of a measurement, in accordance with the Born rule.<ref name=BranchCounting>{{Cite journal|last=Saunders|first=Simon|author-link=Simon Saunders|date=24 November 2021|title=Branch-counting in the Everett interpretation of quantum mechanics.|journal=Proceedings of the Royal Society A|language=en|volume=477|issue=2255|pages=1–22 |doi=10.1098/rspa.2021.0600|issn=|arxiv=2201.06087|bibcode=2021RSPSA.47710600S |s2cid=244491576 }}</ref>