Analysis of Wallace’s Proof of the Born Rule in Everettian Quantum Mechanics: Formal Aspects

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D. Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D. Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace’s proof, reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections.

[1]  Edi Karni,et al.  Axiomatic Foundations of Expected Utility and Subjective Probability , 2014 .

[2]  D. Wallace.  The Emergent Multiverse: Quantum Theory according to the Everett Interpretation , 2012 .

[3]  David Wallace,et al.  How to Prove the Born Rule , 2010 .

[4]  David Wallace.  Quantum Probability from Subjective Likelihood: improving on Deutsch's proof of the probability rule , 2003 .

[5]  James B. Hartle,et al.  Quantum Mechanics in the Light of Quantum Cosmology , 2018, 1803.04605.

[6]  Gell-Mann,et al.  Classical equations for quantum systems. , 1992, Physical review. D, Particles and fields.

[7]  H. Everett.  "Relative State" Formulation of Quantum Mechanics , 1957 .

[8]  Meir Hemmo,et al.  Quantum probability and many worlds , 2007 .

[9]  W. Zurek.  Decoherence and the Transition from Quantum to Classical—Revisited , 2003, quant-ph/0306072.

[10]  James B. Hartle,et al.  Quantum Mechanics in the Light of Quantum Cosmology , 1989 .

[11]  Erkki J. Brändas,et al.  Decoherence and the Appearance of a Classical World in Quantum Theory : E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch and I.-O Stamatescu, Springer-Verlag, New York, 2003 , 2004 .

[12]  David Wallace.  Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation , 2003 .

[13]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[14]  H. Price.  Decisions, Decisions, Decisions: Can Savage Salvage Everettian Probability? , 2008, 0802.1390.

[15]  J. Neumann,et al.  The Logic of Quantum Mechanics , 1936 .

[16]  L. Vaidman.  On schizophrenic experiences of the neutron or why we should believe in the many‐worlds interpretation of quantum theory , 1996, quant-ph/9609006.

[18]  J. Wheeler,et al.  Quantum theory and measurement , 1983 .

[19]  Alastair Wilson.  Objective Probability in Everettian Quantum Mechanics , 2013, The British Journal for the Philosophy of Science.

[20]  Bryce S. DeWitt,et al.  The Many-worlds interpretation of quantum mechanics , 2015 .

[21]  David M. Kreps.  Notes On The Theory Of Choice , 1988 .

[23]  Nicolaas P. Landsman.  Decoherence and the quantum-to-classical transition , 2009 .

[24]  David Z. Albert.  8. Probability in the Everett Picture , 2010 .

[25]  W. Zurek.  Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse? , 1981 .

[26]  A. Assis.  On the nature of a*kak and the emergence of the Born rule , 2010, 1009.1532.

[27]  A. Kent.  One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientific confirmation , 2009, 0905.0624.

[28]  Richard D. Gill.  On an argument of David Deutsch , 2003 .

[29]  H. Price.  Probability in the Everett World: Comments on Wallace and Greaves , 2006, quant-ph/0604191.

[30]  David Deutsch.  Quantum theory of probability and decisions , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[31]  Barry Loewer,et al.  Interpreting the many worlds interpretation , 1988, Synthese.

[32]  G. Auletta.  Foundations and Interpretation of Quantum Mechanics: In the Light of a Critical-Historical Analysis of the Problems and of a Synthesis of the Results , 2000 .

[33]  Lurdes Y. T. Inoue,et al.  Decision Theory: Principles and Approaches , 2009 .

[34]  R. Schack,et al.  Quantum probability from decision theory? , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[35]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[36]  T. Maudlin.  Critical Study David Wallace, The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Oxford University Press, 2012, 530 + xv pp. , 2014 .

[37]  S. Carroll,et al.  Many Worlds, the Born Rule, and Self-Locating Uncertainty , 2014, 1405.7907.

[38]  I. Stamatescu,et al.  Decoherence and the Appearance of a Classical World in Quantum Theory , 1996 .

[39]  D. Bernoulli.  Exposition of a New Theory on the Measurement of Risk , 1954 .

[40]  Lina Jansson,et al.  Everettian quantum mechanics and physical probability: Against the principle of “State Supervenience” , 2016 .

[41]  A. Zee,et al.  Discreteness and the origin of probability in quantum mechanics , 2006, hep-th/0606062.

[42]  H. Greaves.  Probability in the Everett interpretation , 2007 .

[43]  POSITION EIGENSTATES AND THE STATISTICAL AXIOM OF QUANTUM MECHANICS , 2001, quant-ph/0102113.

[44]  Simon Saunders,et al.  Chance in the Everett Interpretation , 2016, 1609.04720.

[45]  A. Gleason.  Measures on the Closed Subspaces of a Hilbert Space , 1957 .

[46]  David John Baker,et al.  Measurement outcomes and probability in Everettian quantum mechanics , 2007 .

[47]  James B. Hartle,et al.  Quantum Mechanics of Individual Systems , 1968, 1907.02953.

[48]  Wojciech H. Zurek,et al.  Probabilities from entanglement, Born's rule p{sub k}= vertical bar {psi}{sub k} vertical bar{sup 2} from envariance , 2005 .

[49]  J. Schreiber.  Foundations Of Statistics , 2016 .

[50]  Simon Saunders.  Derivation of the Born rule from operational assumptions , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[51]  Dov M. Gabbay,et al.  Handbook of Quantum Logic and Quantum Structures: Quantum Structures , 2007 .

[52]  When Worlds Collide: Quantum Probability from Observer Selection? , 2001, quant-ph/0108070.

[53]  Hilary Greaves,et al.  Understanding Deutsch's probability in a deterministic multiverse , 2003, quant-ph/0312136.

[54]  Nicolaas P. Landsman,et al.  Essay Review of: Maximilian Schlosshauer, Decoherence and the Quantum-To-Classical Transition (Springer, Berlin, 2007) , 2009 .