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.Relatedly, I have another blog posts that reviews the consistency conditions in consistent histories.a
- The “problem of defining wavefunction branches” especially in the context of a many-body lattice.
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 just quote a few paragraphs from existing papers that review the literature, along with the relevant part of their list of references. 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.
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  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  has approached the problem from the perspective of information processing ability of subsystems and Piazza  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 ). 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  to induce more general structures in Hilbert space. In a series of papers (e.g. [18–21]; see also ) 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  to study the emergence of classicality under restriction to coarse-grained measurements.
 S. M. Carroll and A. Singh, “Mad-Dog Everettianism: Quantum Mechanics at Its Most Minimal,” arXiv:1801.08132 [quant-ph].
 T. A. Brun and J. B. Hartle, “Classical dynamics of the quantum harmonic chain,” Physical Review D 60 no. 12, (1999) 123503.
 M. Gell-Mann and J. Hartle, “Alternative decohering histories in quantum mechanics,” arXiv preprint arXiv:1905.05859 (2019) .
 F. Dowker and A. Kent, “On the consistent histories approach to quantum mechanics,” Journal of Statistical Physics 82 no. 5-6, (1996) 1575–1646.
 A. Kent, “Quantum histories,” Physica Scripta 1998 no. T76, (1998) 78.
 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].
 R. B. Griffiths, “Consistent histories and the interpretation of quantum mechanics,” J. Statist. Phys.
36 (1984) 219.
 F. Piazza, “Glimmers of a pre-geometric perspective,” Found. Phys. 40 (2010) 239–266,
 M. Tegmark, “Consciousness as a state of matter,” Chaos, Solitons & Fractals 76 (2015) 238–270.
 J. P. Paz and W. H. Zurek, “Environment-induced decoherence, classicality, and consistency of quantum histories,” Physical Review D 48 no. 6, (1993) 2728.
 N. Bao, S. M. Carroll, and A. Singh, “The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional,” arXiv:1704.00066 [hep-th].
 T. Banks, “QuantuMechanics and CosMology.” Talk given at the festschrift for L. Susskind, Stanford University, May 2000, 2000.
 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.
 J. S. Cotler, G. R. Penington, and D. H. Ranard, “Locality from the spectrum,” Communications in Mathematical Physics 368 no. 3, (2019) 1267–1296.
 P. Zanardi, “Virtual quantum subsystems,” Phys. Rev. Lett. 87 (2001) 077901, arXiv:quant-ph/0103030 [quant-ph].
 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].
 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].
 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.
 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].
 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.
 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].
 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].
 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 problem in quantum mechanics . 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-called pure decoherence [1, 10]. In this case, a well-defined pointer basis  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  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 probabilities1 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.
1We 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.
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