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.
- 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 four papers: Carroll & Singh [arXiv:2005.12938]; Riedel, Zurek, & Zwolak [arXiv:1312.0331]; Weingarten [arXiv:2105.04545]; and Kent [arXiv:1311.0249].
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 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!b 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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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 . 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 , 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  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 .
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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 , 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 , 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 . 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.
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(↵ returns to text)
- 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!↵