*Last updated: Nov 27, 2021.*]

This post is (a seed of) a bibliography covering the primordial research area that goes by some of the following names:

- The “preferred-factorization problem”, aka “quantum mereology”, in the context of decoherence.
- The “set-selection problem” in the context of consistent histories.
^{a } - The “problem of defining wavefunction branches,” especially in the context of a many-body lattice.
- The “quantum reality problem”.

Although the way this problem tends to be formalized varies with context, I don’t think we have confidence in any of the formalizations. The different versions are very tightly related, so that a solution in one context is likely give, or at least strongly point toward, solutions for the others.

As a time-saving device, I will mostly just quote a few paragraphs from existing papers that review the literature, along with the relevant part of their list of references. Currently I am drawing on five papers: Carroll & Singh [arXiv:2005.12938]; Riedel, Zurek, & Zwolak [arXiv:1312.0331]; Weingarten [arXiv:2105.04545]; Kent [arXiv:1311.0249]; and Zampeli, Pavlou, & Wallden [arXiv:2205.15893].

I hope to update this from time to time, and perhaps turn it into a proper review article of its own one day. If you have a recommendation for this bibliography (either a single citation, or a paper I should quote), please do let me know.

Contents

### Carroll & Singh

From “Quantum Mereology: Factorizing Hilbert Space into Subsystems with Quasi-Classical Dynamics” [arXiv:2005.12938]:

While this question has not frequently been addressed in the literature on quantum foundations and emergence of classicality, a few works have highlighted its importance and made attempts to understand it better. Brun and Hartle [2] studied the emergence of preferred coarse-grained classical variables in a chain of quantum harmonic oscillators. Efforts to address the closely related question of identifying classical set of histories (also known as the “Set Selection” problem) in the Decoherent Histories formalism [3–7, 10] have also been undertaken. Tegmark [9] has approached the problem from the perspective of information processing ability of subsystems and Piazza [8] focuses on emergence of spatially local sybsystem structure in a field theoretic context. Hamiltonian induced factorization of Hilbert space which exhibit k-local dynamics has also been studied by Cotler et al [14]). The idea that tensor product structures and virtual subsystems can be identified with algebras of observables was originally introduced by Zanardi et al in [15, 16] and was further extended in Kabernik, Pollack and Singh [17]

^{b }to induce more general structures in Hilbert space. In a series of papers (e.g. [18–21]; see also [22]) Castagnino, Lombardi, and collaborators have developed the self-induced decoherence (SID) program, which conceptualizes decoherence as a dynamical process which identifies the classical variables by inspection of the Hamiltonian, without the need to explicitly identify a set of environment degrees of freedom. Similar physical motivations but different mathematical methods have led Kofler and Brukner [23] to study the emergence of classicality under restriction to coarse-grained measurements.

#### Selected references

[1] S. M. Carroll and A. Singh, “Mad-Dog Everettianism: Quantum Mechanics at Its Most Minimal,” arXiv:1801.08132 [quant-ph].

[2] T. A. Brun and J. B. Hartle, “Classical dynamics of the quantum harmonic chain,” Physical Review D 60 no. 12, (1999) 123503.

[3] M. Gell-Mann and J. Hartle, “Alternative decohering histories in quantum mechanics,” arXiv preprint arXiv:1905.05859 (2019) .

[4] F. Dowker and A. Kent, “On the consistent histories approach to quantum mechanics,” Journal of Statistical Physics 82 no. 5-6, (1996) 1575–1646.

[5] A. Kent, “Quantum histories,” Physica Scripta 1998 no. T76, (1998) 78.

[6] C. Jess Riedel, W. H. Zurek, and M. Zwolak, “The rise and fall of redundancy in decoherence and quantum Darwinism,” New Journal of Physics 14 no. 8, (Aug, 2012) 083010, arXiv:1205.3197[quant-ph].

[7] R. B. Griffiths, “Consistent histories and the interpretation of quantum mechanics,” J. Statist. Phys.

36 (1984) 219.

[8] F. Piazza, “Glimmers of a pre-geometric perspective,” Found. Phys. 40 (2010) 239–266,

arXiv:hep-th/0506124 [hep-th].

[9] M. Tegmark, “Consciousness as a state of matter,” Chaos, Solitons & Fractals 76 (2015) 238–270.

[10] J. P. Paz and W. H. Zurek, “Environment-induced decoherence, classicality, and consistency of quantum histories,” Physical Review D 48 no. 6, (1993) 2728.

[11] N. Bao, S. M. Carroll, and A. Singh, “The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional,” arXiv:1704.00066 [hep-th].

[12] T. Banks, “QuantuMechanics and CosMology.” Talk given at the festschrift for L. Susskind, Stanford University, May 2000, 2000.

[13] W. Fischler, “Taking de Sitter Seriously.” Talk given at Role of Scaling Laws in Physics and Biology (Celebrating the 60th Birthday of Geoffrey West), Santa Fe, Dec., 2000.

[14] J. S. Cotler, G. R. Penington, and D. H. Ranard, “Locality from the spectrum,” Communications in Mathematical Physics 368 no. 3, (2019) 1267–1296.

[15] P. Zanardi, “Virtual quantum subsystems,” Phys. Rev. Lett. 87 (2001) 077901, arXiv:quant-ph/0103030 [quant-ph].

[16] P. Zanardi, D. A. Lidar, and S. Lloyd, “Quantum tensor product structures are observable induced,” Phys. Rev. Lett. 92 (2004) 060402, arXiv:quant-ph/0308043 [quant-ph].

[17] O. Kabernik, J. Pollack, and A. Singh, “Quantum State Reduction: Generalized Bipartitions from Algebras of Observables,” Phys. Rev. A 101 no. 3, (2020) 032303, arXiv:1909.12851 [quant-ph].

[18] M. Castagnino and O. Lombardi, “Self-induced decoherence: a new approach,” Studies in the History and Philosophy of Modern Physics 35 no. 1, (Jan, 2004) 73–107.

[19] M. Castagnino, S. Fortin, O. Lombardi, and R. Laura, “A general theoretical framework for decoherence in open and closed systems,” Class. Quant. Grav. 25 (2008) 154002, arXiv:0907.1337 [quant-ph].

[20] O. Lombardi, S. Fortin, and M. Castagnino, “The problem of identifying the system and the environment in the phenomenon of decoherence,” in EPSA Philosophy of Science: Amsterdam 2009, H. W. de Regt, S. Hartmann, and S. Okasha, eds., pp. 161–174. Springer Netherlands, Dordrecht, 2012.

[21] S. Fortin, O. Lombardi, and M. Castagnino, “Decoherence: A Closed-System Approach,” Brazilian Journal of Physics 44 no. 1, (Feb, 2014) 138–153, arXiv:1402.3525 [quant-ph].

[22] M. Schlosshauer, “Self-induced decoherence approach: Strong limitations on its validity in a simple spin bath model and on its general physical relevance,” Phys. Rev. A 72 no. 1, (Jul, 2005) 012109, arXiv:quant-ph/0501138 [quant-ph].

[23] J. Kofler and C. Brukner, “Classical World Arising out of Quantum Physics under the Restriction of Coarse-Grained Measurements,” Phys. Rev. Lett. 99 no. 18, (Nov, 2007) 180403, arXiv:quant-ph/0609079 [quant-ph].

### Riedel, Zurek, & Zwolak

From “The Objective past of a quantum universe: Redundant records of consistent histories” [arXiv:1312.0331]:

“Into what mixture does the wavepacket collapse?” This is the

preferred basis problemin quantum mechanics [1]. It launched the study of decoherence [2, 3], a process central to the modern view of the quantum-classical transition [4–9]. The preferred basis problem has been solved exactly for so-calledpure decoherence[1, 10]. In this case, a well-definedpointer basis[1] emerges whose origins can be traced back to the interaction Hamiltonian between the quantum system and its environment [1, 2, 4]. An approximate pointer basis exists for many other situations (see, e. g., Refs. [11–17]).

The consistent (or decoherent) histories framework [18–21] was originally introduced by Griffiths. It has evolved into a mathematical formalism for applying quantum mechanics to completely closed systems, up to and including the whole universe. It has been argued that quantum mechanics within this framework would be a fully satisfactory physical theory only if it were supplemented with an unambiguous mechanism for identifying a preferred set of histories corresponding, at the least, to the perceptions of observers [22–29] (but see counterarguments [30–35]). This would address the Everettian [36] question: “What are the branches in the wavefunction of the Universe?” This defines the

set selection problem, the global analog to the preferred basis problem.

It is natural to demand that such a set of histories satisfy the mathematical requirement of consistency, i.e., that their probabilities are additive. The set selection problem still looms large, however, as almost all consistent sets bear no resemblance to the classical reality we perceive [37–39]. Classical reasoning can only be done relative to a single consistent set [20, 31, 32]; simultaneous reasoning from different sets leads to contradictions [22–24, 40, 41]. A preferred set would allow one to unambiguously compute probabilities

^{1}for all observations from first principles, that is, from (1) a wavefunction of the Universe and (2) a Hamiltonian describing the interactions.

To agree with our expectations, a preferred set would describe macroscopic systems via coarse-grained variables that approximately obey classical equations of motion, thereby constituting a “quasiclassical domain” [14, 23, 24, 40, 49, 50]. Various principles for its identification have been explored, both within the consistent histories formalism [15, 26, 39, 49, 51–56] and outside it [57–61]. None have gathered broad support.

^{1}We take Born’s rule for granted, putting aside the question of whether it should be derived from other principles [9, 36, 42–48] or simply assumed. That issue is independent of (and cleanly separated from) the topic of this paper.

#### Selected references

[1] W. H. Zurek, Phys. Rev. D 24, 1516 (1981).

[2] W. H. Zurek, Phys. Rev. D 26, 1862 (1982).

[3] E. Joos and H. D. Zeh, Zeitschrift für Physik B Condensed Matter 59, 223 (1985).

[4] H. D. Zeh, Foundations of Physics 3, 109 (1973).

[5] W. H. Zurek, Physics Today 44, 36 (1991).

[6] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).

[7] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. (SpringerVerlag, Berlin, 2003).

[8] M. Schlosshauer, Decoherence and the Quantum-toClassical Transition (Springer-Verlag, Berlin, 2008); in Handbook of Quantum Information, edited by M. Aspelmeyer, T. Calarco, and J. Eisert (Springer, Berlin/Heidelberg, 2014).

[9] W. H. Zurek, Physics Today 67, 44 (2014).

[10] M. Zwolak, C. J. Riedel, and W. H. Zurek, Physical Review Letters 112, 140406 (2014).

[11] J. R. Anglin and W. H. Zurek, Physical Review D 53, 7327 (1996); D. A. R. Dalvit, J. Dziarmaga, and W. H. Zurek, Physical Review A 72, 062101 (2005).

[12] O. Kübler and H. D. Zeh, Annals of Physics 76, 405 (1973).

[13] W. H. Zurek, S. Habib, and J. P. Paz, Phys. Rev. Lett. 70, 1187 (1993).

[14] M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993).

[15] M. Gell-Mann and J. B. Hartle, Phys. Rev. A 76, 022104 (2007).

[16] J. J. Halliwell, Phys. Rev. D 58, 105015 (1998).

[17] J. Paz and W. H. Zurek, Phys. Rev. Lett. 82, 5181 (1999).

[18] R. B. Griffiths, Journal of Statistical Physics 36, 219 (1984).

[19] R. Omnès, The Interpretation of Quantum Mechanics (Princeton University Press, Princeton, NJ, 1994).

[20] R. B. Griffiths, Consistent Quantum Theory (Cambridge University Press, Cambridge, UK, 2002).

[21] J. J. Halliwell, in Fundamental Problems in Quantum Theory, Vol. 775, edited by D.Greenberger and A.Zeilinger (Blackwell Publishing Ltd, 1995) arXiv:grqc/9407040.

[22] F. Dowker and A. Kent, Phys. Rev. Lett. 75, 3038 (1995).

[23] F. Dowker and A. Kent, Journal of Statistical Physics. 82, 1575 (1996).

[24] A. Kent, Phys. Rev. A 54, 4670 (1996).

[25] A. Kent, Phys. Rev. Lett. 78, 2874 (1997).

[26] A. Kent and J. McElwaine, Phys. Rev. A 55, 1703 (1997).

[27] A. Kent, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by A. F. J. Cushing and S. Goldstein (Kluwer Academic Press, Dordrecht, 1996); arXiv:quant-ph/9511032.

[28] E. Okon and D. Sudarsky, Stud. Hist. Philos. Sci. B 48, Part A, 7 (2014).

[29] E. Okon and D. Sudarsky, arXiv:1504.03231 (2015).

[30] R. B. Griffiths and J. B. Hartle, Physical Review Letters 81, 1981 (1998).

[31] R. B. Griffiths, Physical Review A 57, 1604 (1998).

[32] R. B. Griffiths, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44, 93 (2013).

…

[39] M. Gell-Mann and J. B. Hartle, in Proceedings of the 4th Drexel Conference on Quantum Non-Integrability: The Quantum-Classical Correspondence, edited by D.-H. Feng and B.-L. Hu, Drexel University (International Press of Boston, Hong Kong, 1998); arXiv:grqc/9509054.

[40] M. Gell-Mann and J. Hartle, arXiv:gr-qc/9404013 (1994).

[41] R. Omnès, Rev. Mod. Phys. 64, 339 (1992).

…

[48] A. V. Nenashev, Physica Scripta T163, 014033 (2014); arXiv: 1601.08205 (2016).

[49] M. Gell-Mann and J. B. Hartle, in Proceedings of the 25th International Conference on High Energy Physics, edited by K. K. Phua and Y. Yamaguchi, South East Asia Theoretical Physics Association and Physical Society of Japan (World Scientific, Singapore, 1990).

[50] M. Tegmark, Chaos, Solitons & Fractals 76, 238 (2015).

[51] R. B. Griffiths, Phys. Rev. A 54, 2759 (1996).

[52] C. Anastopoulos, International Journal of Theoretical Physics 37, 2261 (1998).

[53] A. Kent, Physica Scripta T76, 78 (1998).

[54] T. A. Brun and J. B. Hartle, Phys. Rev. E 59, 6370 (1999).

[55] A. Kent, in Relativistic Quantum Measurement and Decoherence, Lecture Notes in Physics, Vol. 559, edited by H.-P. Breuer and F. Petruccione (Springer Berlin / Heidelberg, 2000) pp. 93–115.

[56] T. Brun, arXiv:quant-ph/0302034 (2003).

[57] L. de Broglie, in Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 Octobre 1927 sous les Auspices de l’Institut International de Physique Solvay (GauthierVillars, Paris, 1928).

[58] D. Bohm, Physical Review 85, 166 (1952).

[59] R. Bousso and L. Susskind, Physical Review D 85, 045007 (2012).

[60] A. Kent, Foundations of Physics 42, 421 (2012).

[61] A. Kent, Physical Review A 90, 012107 (2014).

### Weingarten

From “Macroscopic Reality from Quantum Complexity” [arXiv:2105.04545]:

By linearity of quantum mechanical time evolution, however, it… follows that a measurement with a linear superposition in the initial state will yield a final state also with a superposition…In the measured final state, the meter no longer has a single value but a combination of two values which cannot, by itself, be connected to a recognizable configuration of a macroscopic object. The absence of a recognizable configuration for the macroscopic device is the “problem of measurement”.

The resolution of this problem proposed by the many-worlds interpretation of quantum mechanics [1, 2] is that the states …actually represent two different worlds… Among the problems of the many-worlds interpretation, however, is that in general, for plausible models of a measurement process, the individual worlds given by the Schmidt basis do not have sufficiently narrow coordinate dispersions to count as classical reality [9]. In addition, it is unclear under what circumstances and according to what basis a system larger than just a micro system and a measuring device should be split into separate worlds.

A resolution to the first of these problems, the absence of classical behavior in the split branches, is proposed to occur through environmentally-induced decoherence [3, 4]…Based on these various considerations it is argued that entangled environmental states … behave essentially as permanent classical records of the experimental results. Correspondingly, for many-worlds augmented with decoherence [5], the circumstance under which a system splits into distinct worlds is when a superposition has been produced mixing distinct values of one of these effectively classical degrees of freedom…

A step toward resolving the second problem, the absence of a criterion for branching for the universe as a whole rather than simply for some system-apparatus pair, takes the form of a theorem [6] according to which, for a system as a whole, if a particular spatial pattern of redundant records happens to occur, then there is a unique corresponding decomposition of a state vector into effectively classical branches.

A residual problem of [3–6], however, is that the rules governing their application to the world are intrinsically uncertain. In particular, the record production needed for environmentally-induced decoherence occurs over some nonzero intervals of time and space, and perhaps is entirely completed only asymptotically in long time and large distance limits. What fraction of the initial state in Eq. (3) must become entangled with the environment for splitting into classical branches to occur? When exactly over the time interval of decoherence, does the splitting of the world in parts occur? And since the process extends over space, this timing will differ in different frames related by a Lorentz boost. Which is the correct choice? These various questions may be of no practical consequence in treating the meter readings as nearly classical degrees of freedom after entanglement and using the resulting values to formulate observable predictions. But what seems to me to be clear is that something is missing from the theory. From outside the theory, something additional and arbitrary needs to be supplied by hand to resolve each of these issues. Moreover, no mathematical machinery is present in any of these proposal which allows the process of filling in what is missing to be stated precisely. As a consequence of all of which it appears to me to be implausible that the branches provided by these accounts are by themselves the fundamental substance of reality.

A discussion of issues concerning environmentally induced decoherence and its combination with the many-worlds interpretation of quantum mechanics appears in [10].

#### Selected references

[1] H. Everett, Rev. Mod. Phys. 29, 454 (1957).

[2] B. DeWitt, Physics Today 23, 30 (1970).

[3] H. D. Zeh, Found. Phys. 1, 69 (1970).

[4] W. H. Zurek, Phys. Rev. D24, 1516 (1981); Phys. Rev. D26; 1862 (1982) Mod. Phys. 75, 715 (2003).

[5] D. Wallace, Studies in the History and Philosophy of Modern Physics 34, 87 (2003).

[6] C. Jess Riedel, Phys. Rev. Lett. 118, 120402 (2017).

[7] M. A. Nielsen, arXiv:quant-ph/050207.

[8] A. R. Brown, L. Susskind, arXiv:0701.01107[hep-th] [9] D. N. Page, arXiv:1108.2709v2 [hep-th].

[10] M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2004).

[11] A. Kent, Found. Phys. 43, 421 (2012); Phys. Rev. A 90, 0121027 (2014); Phys. Rev. A 96, 062121 (2017).

[12] D. P. DiVincenzo, Phys. Rev. A 51, 1015(1995), arXiv:cond-mat/9407022.

### Kent

From “Solution to the Lorentzian quantum reality problem” [arXiv:1311.0249]:

Quantum theory is a mathematically beautiful theory that unifies all of known physics with the exception of gravity. Its probabilistic predictions for experimental outcomes have been verified for a very large range of physical phenomena and contradicted by no experiment. Yet, as John Bell so eloquently and persuasively argued [1], we do not know what precisely it is that quantum probabilities are probabilities of. We do not have a mathematically precise description of what Bell called [2,3] the “beables” for quantum theory. That is, we do not have a sample space of events, or histories, or paths, or other mathematical objects, on which the quantum probability distribution is defined. This is the quantum reality problem, sometimes referred to as the measurement problem, rather misleadingly from a modern perspective, since few physicists now believe that the fundamental laws of nature involve measuring devices per se or that progress can be made by analyzing them. As Bell emphasized, the quantum reality problem becomes particularly conceptually problematic when we impose the natural condition that any solution should respect the symmetries of special relativity. We focus here on solutions to the Lorentzian quantum reality problem, i.e., solutions that have this property.

As Bell also stressed [1], mathematical aesthetics are not the main motivation for solving the quantum reality problem. The motivation is the following. On the one hand, the impressive successes of quantum theory and the lack of compelling alternatives make it natural to try to treat quantum theory as fundamental and so to derive everything else in physics from quantum theory. On the other hand, it appears to us that we live in a quasiclassical world, in which macroscopic variables are most of the time approximately governed by deterministic equations of motion, but are also affected by random events of quantum origin. Moreover, it appears as though this quasiclassical world emerged from an initial quantum state with no initial quasiclassical properties. Given a well-defined probabilistic version of the quantum theory of closed systems, we can hope to explain these features from within quantum theory and indeed to sketch a coherent and unified account of cosmology, classical and quasiclassical dynamics, and quantum theory. Without one, we cannot rigorously derive classical or quasiclassical physics from quantum theory nor give a coherent treatment of cosmology from within quantum theory….

The first well-known attempt to address the quantum reality problem directly was the pilot wave theory of de Broglie and Bohm [4,5], in which the beables are particle trajectories whose evolution is defined by the quantum wave function by a guidance equation. However, de Broglie and Bohm’s models apply to nonrelativistic quantum mechanics and are inconsistent with special relativity. No fundamentally relativistic generalization of the models has been found, nor is there a convincing extension to quantum field theory. Many (though not all) physicists also find de Broglie and Bohm’s trajectories and guidance equations rather mathematically unnatural and inelegant additions to quantum theory.

Nonrelativistic dynamical collapse models [6,7] attempt to give another story about physical reality that is consistent with experiment to date at the price of changing the dynamics and hence the experimental predictions of quantum theory. (For some attempts in the direction of relativistic collapse models, see [8–11].) While scientifically interesting, these and other generalizations of quantum theory do not address the main question we focus on here, namely, whether we can find a mathematically precise description of reality consistent with standard quantum theory.

Another line of thought, initiated by Everett, suggests that quantum theory is deterministic and that pure unitary quantum evolution holds at all times. The problems with this idea and with the many incompatible proposals for some form of “many worlds” quantum theory that it has inspired continue to be debated [12]. Still, two relatively uncontroversial points can be made. First, since, according to most of those who advocate some version of many-worlds quantum theory, quantum theory is fundamentally deterministic and the appearance of quasiclassical physics is supposed to arise as an approximation via decoherence, no mathematically precise sample space and probability distribution emerges. Second, many-worlds theories are radically different types of scientific theory from standard “one-world” versions of quantum theory (or indeed from anything previously considered in science) and give a qualitatively different (and fantastically weird) description of reality.

#### Selected references

[1] J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy (Cambridge University Press, Cambridge, U.K., 2004).

[2] J. S. Bell, Epistemol. Lett. 9, 11 (1976).

[3] J. S. Bell, in Quantum Implications: Essays in Honour of David Bohm, edited by B. Hiley and D.Peat (Routledge, Oxon, UK, 1987), pp. 227–234.

[4] D. Bohm, Phys. Rev. 85, 166 (1952).

[5] L. de Broglie, in Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 Octobre 1927 sous les Auspices de l’Institut International de Physique Solvay (Gauthier-Villars, Paris, 1928).

[6] G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42, 78 (1990).

[7] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986).

[8] D. J. Bedingham, Found. Phys. 41, 686 (2011).

[9] Philip Pearle, Phys. Rev. A 59, 80 (1999).

[10] P. Pearle, Phys. Rev. A 71, 032101 (2005).

[11] R. Tumulka, J. Stat. Phys. 125, 821 (2006).

### Zampeli, Pavlou, & Wallden

From “Contrary Inferences for Classical Histories within the Consistent Histories Formulation of Quantum Theory” [arXiv:2205.15893]:

The interpretation of consistent histories has been a heated topic (indicatively, examples of this dialogue can be found on the critical accounts of [18–20] that were followed by responses of proponents [21, 22]). Our work intends to add one more dimension in this scientific exchange strengthening the argument for supplementing the consistency condition with a “set-selection” criterion. This issue was first raised by Dowker and Kent [17] and later by Kent [1] who noted that the problem of contrary inferences is the result of a contextuality particular only to the histories formalism. He later gave a set- selection criterion to restrict “ordered consistent sets” [1, 23, 24]. In [9] it was argued that contrary inferences originate from the existence of zero covers on the histories space and brought forward another set-selection criterion that restricts attention to preclusive con- sistent sets. In [8], Anastopoulos connected the predictability of a consistent set with the persistence in time; while recently, Halliwell proposed to classify multiple consistent sets on whether or not there exists any unifying probability for combinations of incompatible sets which replicates the consistent histories result when restricted to a single consistent set [10]. Gell-Mann and Hartle [25] and Riedel, Zurek and Zwolak [26] on the other hand gave set-selection criteria that were motivated by some physical principle (e.g. thermodynamic considerations) or induced by the environment respectively.

In the preclusive consistent sets criterion of [9], histories that are precluded (have

measure zero) have a special role, since such histories and their subsets are not allowed to happen in any “legitimate” (i.e. preclusive) consistent set. Precluded histories play crucial role in another histories formulation, called co-events formulation or anhomomorphic logic [27, 28], that interprets the decoherence functional (or quantum measure [29]) differently. Our results may have consequences in that approach too, but analysing this is left for a future work.

#### Selected references

[1] Adrian Kent. Consistent sets yield contrary inferences in quantum theory. Physical Review Letters, 78(15):2874, 1997. DOI: 10.1103/physrevlett.78.2874.

…

[9] P. Wallden. Contrary inferences in consistent histories and a set selection criterion. Foundations of Physics, 44:1195, October 2014. DOI: 10.1007/s10701-014-9837-6. URL https://ui.adsabs.harvard.edu/abs/2014FoPh…44.1195W.

[10] J. J. Halliwell. Incompatible multiple consistent sets of histories and measures of quantumness. Physical Review A, 96(1), jul 2017. DOI: 10.1103/physreva.96.012123. URL https://doi.org/10.1103%2Fphysreva.96.012123

…

[17] Fay Dowker and Adrian Kent. On the consistent histories approach to quan- tum mechanics. Journal of Statistical Physics, 82(5-6):1575–1646, 1996. DOI: 10.1007/bf02183396.

[18] Elias Okon and Daniel Sudarsky. On the Consistency of the Consistent Histories Approach to Quantum Mechanics. Foundations of Physics, 44(1):19–33, January 2014. ISSN 1572-9516. DOI: 10.1007/s10701-013-9760-2. URL https://doi.org/10.1007/ s10701-013-9760-2.

[19] Elias Okon and Daniel Sudarsky. Measurements according to Consistent Histo- ries. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 48:7–12, November 2014. ISSN 13552198. DOI: 10.1016/j.shpsb.2014.08.011. URL https://linkinghub.elsevier.com/retrieve/ pii/S1355219814000951.

[20] Angelo Bassi and GianCarlo Ghirardi. Decoherent Histories and Realism. Jour- nal of Statistical Physics, 98(1):457–494, January 2000. ISSN 1572-9613. DOI: 10.1023/A:1018647510799. URL https://doi.org/10.1023/A:1018647510799.

[21] Robert B. Griffiths. Consistent quantum measurements. Studies in History and Phi- losophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52: 188–197, November 2015. ISSN 13552198. DOI: 10.1016/j.shpsb.2015.07.002. URL https://linkinghub.elsevier.com/retrieve/pii/S1355219815000489.

[22] Robert B. Griffiths. Consistent Quantum Realism: A Reply to Bassi and Ghirardi. Journal of Statistical Physics, 99(5):1409–1425, June 2000. ISSN 1572-9613. DOI: 10.1023/A:1018601225280. URL https://doi.org/10.1023/A:1018601225280.

[23] Adrian Kent. Reply to: “Comment on: ‘Consistent sets yield contrary inferences in quantum theory’ ” [Phys. Rev. Lett. 81 (1998), no. 9, 1981; 1644140] by R. B. Griffiths and J. B. Hartle. Physical Review Letters, 81(9):1982, 1998. ISSN 0031-9007. DOI:

10.1103/PhysRevLett.81.1982.

[24] Adrian Kent. Quantum histories and their implications. Lect. Notes Phys., 559:93–

115, 2000.

[25] Murray Gell-Mann and James B. Hartle. Quasiclassical coarse graining and thermo-

dynamic entropy. Physical Review A, 76(2):022104, August 2007. DOI: 10.1103/Phys- RevA.76.022104. URL https://link.aps.org/doi/10.1103/PhysRevA.76.022104. Publisher: American Physical Society.

[26] C. Jess Riedel, Wojciech H. Zurek, and Michael Zwolak. Objective past of a quantum universe: Redundant records of consistent histories. Physical Review A, 93(3):032126, March 2016. DOI: 10.1103/PhysRevA.93.032126. URL https://link.aps.org/doi/ 10.1103/PhysRevA.93.032126. Publisher: American Physical Society.

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### Footnotes

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- Relatedly, I have another blog posts that reviews the consistency conditions in consistent histories.↵
- In Sec. 7 and 8 of Kabernik et al. (“Quantum State Reduction: Generalized Bipartitions from Algebras of Observables”, Phys. Rev. A 101, 032303 (2020), arXiv:1909.12851), the authors propose a criteria for a preferred quasiclassical observable, which implicitly defines wavefunction branches as the instantaneous eigenstates thereof. Although very interesting for extending the predictability sieve outside the context of a preferred system-environment tensor structure to (my preferred) context of a lattice (i.e., a microscopic tensor structure induced by spatial locality), I would be surprised if the single (system-dependent) observable they propose could capture all the things we associate with macroscopic outcomes. I also do not understand the motivation, beyond convenience, for (1) the assumption that this preferred observable is strictly a linear combination of local (single-site) observables or (2) the decision to use the Hilbert-Schmidt norm for the sieve. (Almost certainly this definition would not be “covariant under RG flow”, i.e., would depend importantly on the spatial scale at which you choose your microscopic subsystems.) Still, a compelling approach!↵

I know I’m a broken record on this, but “An algebraic approach to Koopman classical mechanics” (arXiv:1901.00526, DOI there for Annals of Physics 2020) is relevant to this because Section 7.1 shows how to reconcile the Heisenberg picture of measurement —that at the time of measurement wavepacket collapse is necessary— with the Bohr picture of measurement —that a measurement constrains future measurements— by deriving an elementary identity, , where presents collapse as linear projection of an operator to the commutant of (which is known as the Lüders transformer in the measurement theory literature,) whereas requires —indeed, it enforces— that subsequent joint measurements must commute. What I like a lot about this is that it makes (mathematical) sense of Bohr’s Delphic comments. It’s also not prescriptive, in that it allows us our own choice of whether to work in the Heisenberg or the Bohr picture of quantum measurement, just as we can work either in the Heisenberg or the Schrödinger picture of Quantum evolution.

“Classical states, quantum field measurement” (arXiv:1709.06711, DOI there for Physica Scripta 2019) is relevant to the question of coarse-graining because it shows that for the free quantized complex KG field and for the free quantized EM field there is a free random field of commuting observables that is arbitrarily fine-grained over all of Minkowski space-time (that is, the maximal commuting subalgebra of such free fields is a random field everywhere, not just over only a space-like hyperplane.)

Some people find these two papers quite promising for our understanding of QM/QFT, however others don’t.

I’m pretty confident your first paragraph does not address the problems discussed in the post. The problem from the post is about an account of a measuring process that derives, rather than presupposing, a measuring apparatus (e.g., as defined by a tensor decomposition), a preferred set of observables, or a preferred basis. It appears you are just leaving the measured observable $\hat{A}$ unexplained.

Likewise, your second paragraph is unlikely to be addressing the problem because we know there are multiple possible choices of sets of commuting observables that are arbitrarily fine-grained, but do not commute with each other, and we are left with the problem of which to choose. Indeed, in finite-dimensions such a set of commuting observables is equivalent to a choice of basis, and there are many possible choices of basis that are incompatible with each other.

If you disagree, you should be able to explain your proposed solution in the non-relativistic (and, ideally, finite-dimensional) context, where I think it will be more clear where the issue lies.

> The set selection problem still looms large, however, as almost all consistent sets bear no resemblance to the classical reality we perceive

What about an average over all consistent sets?

How would one define an average of a consistent set?

(Remember that, at any fixed time t, each consistent set defines a orthonormal basis, possibly coarse-grained into a set of orthogonal projectors. So even if we put aside the problem of the time structure of the consistent set, defining an average of sets should have all the problems with defining an average of bases.)

I’m not sure how but it seems like it could be necessary. If we assume that each consistent set is a valid framework for looking at the world, then it would be arbitrary to just pick one and ignore the others.

Taking the classical approximation as a model for decision making, at least one could compute a single value for each set as the expected payout of one decision. I don’t know how you could weigh the payouts for all sets; perhaps sets with lower Kolmogorov complexity are rare and naturally easier to reason with.

> If we assume that each consistent set is a valid framework for looking at the world, then it would be arbitrary to just pick one and ignore the others.

I agree we don’t yet have a precise principle for picking only one set of consistent histories, but we surely have good and non-arbitrary (albeit imprecise) reasons for focusing on a very small subset of them: they describe the actual reality we observe! (Likewise, we could imagine all sorts of other weird but internally self-consistent modifications of physics, but we reject these because they do not, in fact, describe the real world.)

> at least one could compute a single value for each set as the expected payout of one decision

Well, first, personally I do not think it makes sense to treat bets/decisions as more fundamental than probabilities. Adrian Kent articulates well why this doesn’t make sense in his contribution to the book “Many Worlds?”.

But even putting that aside, you’d face many problems doing what you’re suggesting. Most consistent sets would not feature a projector corresponding to any given decision, and the ones that do would all give the same answer. (So the average wouldn’t accomplish anything.) And you’d also still not have explained from first principles what the real decisions are, as opposed to the decisions you didn’t take. If you declared a preferred set of decisions without justification, you’d just be smuggling in a set of projectors to act as your preferred basis, begging the question.

> we surely have good and non-arbitrary (albeit imprecise) reasons for focusing on a very small subset of them: they describe the actual reality we observe

If there is one set which fully describes our reality that would be interesting. Since it isn’t a physical division of the universe, but more of a mental model, it would suggest either that it’s not possible for one agent to consider multiple sets or that set has some unique survival advantage for us, which is why we don’t consider others.

> you’d face many problems doing what you’re suggesting. Most consistent sets would not feature a projector corresponding to any given decision, and the ones that do would all give the same answer. (So the average wouldn’t accomplish anything.)

Interesting – I didn’t know that decisions could be represented by projectors or that all sets would either agree or give no preference.

I don’t see how these are problems; it actually makes it easier. For any decision, an agent could ignore all irrelevant sets and if they found any relevant one, they could stop looking, knowing all others would give equal payouts.

> And you’d also still not have explained from first principles what the real decisions are, as opposed to the decisions you didn’t take.

True; a complete theory would explain that too. I was assuming free will: the ability to at least choose the question.

> If there is one set which fully describes our reality that would be interesting. Since it isn’t a physical division of the universe, but more of a mental model,

If you’re not a wavefunction realists/physicalist, the branch structure describes the difference between things that did vs. did not occur. So although it’s not a division into multiple physical pieces, it identifies what is physical in the first place.

If you are wavefunction realists/physicalist, then it’s a real physical division.

> it would suggest either that it’s not possible for one agent to consider multiple sets or that set has some unique survival advantage for us, which is why we don’t consider others.

You can’t even define things like “survival” without choosing a preferred consistent set. Asking “did this agent survive?” in most consistent sets is as unanswerable as asking which slit an electron went through in a two-slit experiment.

> I didn’t know that decisions could be represented by projectors

Any physical thing we could measure even in principle (the location of an electron, whether a nerve fired, whether a button was pressed) corresponds to a projector.

> I don’t see how these are problems; it actually makes it easier. For any decision, an agent could ignore all irrelevant sets and if they found any relevant one, they could stop looking, knowing all others would give equal payouts

If you’ve already picked your decision, corresponding to a set of projectors, then you’ve implicitly picked a consistent set. And then of course you can compute a probability. This is just the statement that quantum mechanics is *prior to* (i.e., more fundamental than) decision theory on the ladder of reductionism. But the hard part is to identify the decisions/set from first principles without just sticking them in by hand.

> > And you’d also still not have explained from first principles what the real decisions are, as opposed to the decisions you didn’t take.

> True; a complete theory would explain that too. I was assuming free will: the ability to at least choose the question.

No no, I’m not saying that the actual choice made (yes vs. no) would or should be identified by a set-selection principle. The outcome is still probabilistic (and potentially a place for something like free will to hide if one is sympathetic to that sort of approach). I’m staying that the choice of set determines the

questionthat the decision answers: “Should I press this button?” vs. “Should I exists as a spatially delocalized superposition in both the Milky Way and Andromeda?”.I think I see what you’re saying now: that there is one classical framework imposed by nature and that’s why it doesn’t make sense to consider some sort of average of consistent sets.

I agree that identifying the principles that lead to the selection of the consistent set we are using is the hard part. It has a lot of nice properties, like as you mention, the ability to express our survival.

My view is different though. I see that agents develop their own framework and can choose the questions they’ll ask or the decisions they’ll consider; the free-will assumption. Like when an experimenter measures spin he can choose any direction; it’s constrained only by logic, not a set of predetermined values.

An example of how this might work at a higher level is how one can view the world as lines of competing genes using individuals or as individuals using genes – the primary objects are different.

> If you are wavefunction realists/physicalist, then it’s a real physical division.

I would say what’s physical is the state of the universe and it’s evolution. There are many ways one can describe it and divide it; all the consistent ways are valid but some are less useful then others. Whatever choice is made won’t effect it’s evolution as a whole (except in our little part if the choice isn’t random).

> I see that agents develop their own framework and can choose the questions they’ll ask or the decisions they’ll consider; the free-will assumption. Like when an experimenter measures spin he can choose any direction; it’s constrained only by logic, not a set of predetermined values.

But consider how this actually works in real life: when the experimenter selects a direction to measure a spin, he physically moves a macroscopic object through a series of *classical* states, e.g., the orientation of a big magnet, the turn of a dial, or certain button presses on a keyboard. These classical states already form a preferred basis, as selected by decoherence. Within that classical basis, the actual choice (e.g., 37° vs. 119° angle spin measurement) can be traced back, if you want, to individual nerves firing or not firing, but these also take place in a preferred basis selected by decoherence.

The choice of angle can at least clearly be chosen by a quantum source. I’m not convinced the other details, like the timing and orientation of the magnet are not also quantum in origin, just by accident. Take for example this analysis of coin flips:

https://arxiv.org/abs/1212.0953

It seems like classical states and are always just an approximation and decoherence is almost never perfect.

I’m very familiar with the ideas in Andy’s paper and it doesn’t conflict with my claims. That the choice of angle can ultimately be traced back to a quantum event does not mean the experimentalists selected the angle by putting a macroscopic dial in a coherent quantum superposition, i.e., that he could have avoided the classically preferred angle basis (rather than an incompatible basis formed of superpositions of angles). Yes, of course if you look at the wavefunction of the universe it will contain two branches, one where the experimentalist made one choice and one where he made the other, but there is still a preferred classical basis for that branch structure.

> It seems like classical states and are always just an approximation and decoherence is almost never perfect.

If you chose projectors with a simple mathematical description, this is true. But we don’t actually expect the projectors reflecting the real, physical experiments we perform to be perfectly representable in this way. We can take the actual projectors to give perfect decoherence and, indeed, this is necessary if we want them to describe probabilities that actually sum to 1 exactly. The difference between these projectors and the ones with a simple mathematical description is exponentially small.