From Time Asymmetry to Quantum Entanglement: The Humean Unification
FFrom Time Asymmetry to Quantum Entanglement:The Humean Unification
Eddy Keming Chen * June 11, 2020
Abstract
Two of the most di ffi cult problems in the philosophical foundations ofphysics are (1) what gives rise to the arrow of time and (2) what the ontologyof quantum mechanics is. The first problem is puzzling since the fundamen-tal dynamical laws of physics do not include an arrow of time. The secondproblem is puzzling since the quantum-mechanical wave function describes anon-separable reality that is remarkably di ff erent from the objects in our ordi-nary experiences.In this paper, we propose a unified “Humean” solution to the two problems.Humeanism allows us to incorporate the Past Hypothesis and the StatisticalPostulate into the best system, which we then use to simplify the quantumstate of the universe. This allows us to confer the nomological status to thequantum state in a way that adds no significant complexity to the best systemand solves the “supervenient-kind problem” facing the original version of thePast Hypothesis. We call this strategy the Humean unification . It brings togetherthe origins of time asymmetry and quantum entanglement. On this theory,what gives rise to the arrow of time is also responsible for the non-separablephenomena in nature. The result is a more unified theory, with a separable mo-saic, a best system that is simple and non-vague, less tension between quantummechanics and special relativity, and more theoretical and dynamical unity. Wethen compare our proposals to those in the literature that focus on only oneof the two problems. Our analysis further suggests that, in order to obtain adeeper understanding about the problems in philosophy of science, it can betremendously illuminating to explore the full resources of Humeanism, even ifone is not a Humean.
Keywords: arrows of time, quantum entanglement, wave function, density matrix,Humean supervenience, laws of nature, Best System Account, the Past Hypothesis, Statisti-cal Postulate, the Mentaculus, objective probabilities, separability, narratability, vagueness,Lorentz invariance * a r X i v : . [ phy s i c s . h i s t - ph ] J un ontents Two of the most puzzling phenomena in nature are time asymmetry and quantumentanglement. They have played important roles in the development of contempo-rary physics. The study of time asymmetry started a rigorous discipline of statisticalmechanics with applications to many domains. The study of quantum entangle-ment produced profound insights about the foundations of quantum mechanics, aswell as potential technological advances in quantum information and cryptography.In philosophy of science, both problems are treated as useful data for evalu-ating leading theories about laws, chances, and ontology. They frequently comeup in debates about Humeanism vs. anti-Humeanism in the metaphysics of sci-ence, serving as important case studies regarding questions such as whether the2undamental ontology is separable, whether laws supervene on the material ontol-ogy, and whether we should allow fundamental laws about initial conditions and“deterministic chances.”So far, however, they have largely been treated as distinct and unrelated problemsin the foundations of physics and philosophy of science. Humeans have o ff eredingenious solutions to them by conferring nomological status to the Past Hypothesis,a promising explanation for the arrow of time in our patch of the universe, and(recently) to the quantum wave function, which is responsible for the phenomena ofquantum entanglement. However, conferring nomological status is not always easyand could lead to tensions with other things Humeans may believe about laws ofnature. For example, can the Past Hypothesis be a fundamental (Humean) law evenif it is stated in a non-fundamental language, as an infinitely long disjunction, or withvague terms? Can the wave function be considered nomological if it is extremelycomplex and perhaps more complex than the mosaic it aims to summarize? Therehave been proposed answers but they seem to require further modifications of theHumean framework, which may not be fully satisfactory.The purpose of this paper is to focus on some interconnections between the twoproblems and show that they are deeply related such as to permit a unified treatmentin the Humean framework. The unification in the Humean framework shows thatwhat is responsible for time’s arrow can also be responsible for the non-separablephenomena in nature. We do this by adopting a new theory of quantum statisticalmechanics and using the nomological status of the Past Hypothesis to select a naturalinitial quantum state of the universe and to argue for its nomological status. We callthe general strategy the Humean unification . We show that it leads to not only novelsolutions to both problems but also new insights about the relationship betweenHumeanism and foundations of physics. Humean unification suggests that, in orderto obtain a deeper understanding about the problems in philosophy of science, itcan be tremendously illuminating to explore the full resources of Humeanism, evenif one is not a Humean.We proceed as follows. In §2, we review the problems of time asymmetry andquantum entanglement in more details and discuss their relevance to the Humeanframework. In §3, we review the Mentaculus theory, a promising and concretetheory of quantum statistical mechanics, and we construct a new theory called the
Wentaculus that makes central use of density matrices and a new law called the
InitialProjection Hypothesis that replaces the Past Hypothesis. In §4, we “Humeanize” theWentaculus by arguing that the initial quantum state of the universe describedby the Initial Projection Hypothesis can be interpreted nomologically rather thanontologically, which leads to a unified treatment of time asymmetry and quantumentanglement. In §5, we discuss the fruits of Humean unification. In §6, we compareand contrast Humean unification to other related proposals that focus on only oneof the two problems. In this paper, we make use of ideas and methods from the metaphysics of science, philosophy ofphysics, and mathematical physics. We do not worry too much about whether the ideas are purelyphilosophical or scientific. A precise disciplinary boundary here may be di ffi cult to draw. Indeed, natural initial quantum state, our theory provides novelinsights about the foundations of quantum statistical mechanics. The new quantumtheories admit a Humean interpretation, but I believe that they are also compatiblewith a non-Humean interpretation. I discuss this possibility in §7. In this section, we discuss the original problems of time asymmetry and quantumentanglement (A and B) as well as further problems they give birth to (A1-2 andB1-2).The first problem can be stated as follows:
A. The Problem of Time Asymmetry:
Why is there temporal asymmetry in theworld when the fundamental dynamical laws are symmetric in time?Time asymmetry is widespread in nature: ice cubes melt in a cup of hot water butdo not spontaneously form in it; gas expands in a box but does not spontaneouslycontract; wine glasses break into pieces but the broken pieces do not spontaneouslyform wine glasses. In the language of thermodynamics, (isolated) physical sys-tems (typically) evolve from states of lower entropy to states of higher entropy;but not the other way around. The phenomena are summarized by the SecondLaw of Thermodynamics: (isolated) physical systems (typically) do not decrease inentropy. However, the fundamental dynamical laws of physics, such as the Newto-nian equation of motion, the Schrödinger equation, the Dirac equation, and Einsteinfield equations are (essentially) symmetric in time. They allow ice cubes to decreasein size and to increase in size, gas molecules to expand and to contract, and wineglasses to break into pieces and the pieces spontaneously form glasses. They allow(isolated) physical systems to increase in entropy as well as to decrease in entropy.It has been argued that the origin of time asymmetry in our universe lies in alow-entropy boundary condition, now called the
Past Hypothesis . According to thePast Hypothesis, the universe “started” in a state of extremely low entropy. Startingfrom that state, most likely the universe will evolve according to the fundamentalphysical laws into higher entropy states, giving rise to the temporal asymmetrywe observe. We add the probabilistic qualifier “most likely” because there exist we welcome the possibility that some ideas in philosophy may lead to new theoretical possibilitiesin foundations of physics and vice versa. This paper is the second part of a project called “Time’s Arrow in a Quantum Universe.” Forother related papers in the project, see Chen (2018a,b, 2019a) and Chen (2020). Albert (2000) coins the term. See Feynman (2017), Goldstein (2001), Lebowitz (2008), Ehrenfestand Ehrenfest (2002), North (2011), and Penrose (1979) for more discussions about the low-entropyinitial condition. See Earman (2006) for worries about the Past Hypothesis. See Goldstein et al.(2016) for a discussion about the possibility, and some recent examples, of explaining the arrow oftime without the Past Hypothesis.
Statistical Postulate.
Loewer (2012) dubs thepackage of postulates—the dynamical laws, the Past Hypothesis, and the StatisticalPostulate—the Mentaculus.However, the Past Hypothesis and the Statistical Postulate give rise to di ffi cultconceptual issues. Since they play a crucial role in explaining time asymmetry andthe Second Law, and since they are incredibly simple, it has been argued that the PastHypothesis is a fundamental law of nature and the Statistical Postulate providesobjective probabilities. But how can the Past Hypothesis be a law of nature if it is a(macroscopic) boundary condition? And how can the initial probability distributionbe objective if the laws are deterministic?A1. The Status of the Past Hypothesis:
How can the Past Hypothesis be a funda-mental law of nature if it is a (macroscopic) boundary condition?A2.
The Status of the Statistical Postulate:
How can the initial probability distri-bution be objective if the laws are deterministic?The second and seemingly unrelated problem is as follows:
B. The Problem of Quantum Entanglement:
What is the nature of quantum entan-glement?Quantum mechanics is one of the most empirically successful theories. But itpresents numerous conceptual puzzles. At the heart of them is the phenomenonof quantum entanglement. Quantum entanglement is a property of the quantumstate, which is standardly represented by a wave function ψ . Two systems A and B are entangled when their joint state ψ AB is not a product of their individual states ψ A and ψ B . We have good reasons to be realist about quantum mechanics andabout the quantum state. So we may have to postulate the quantum state in theworld. If it is fundamental, then the fundamental ontology would be non-separable :the fundamental state of the world is not determined by the states of its parts.Quantum entanglement is a kind of holism . However, this is not the only surprisingconsequence of quantum entanglement. David Albert (2015) has shown that ifquantum entanglement is among the fundamental facts, i.e. in the mosaic, thenLorentz invariance of special relativity would conflict with a very natural principlecalled narratability : the full history of the world can be narrated in a single temporalsequence, and other ways of narrating it will be its geometrical transformations (e.g. Suggestions that the Past Hypothesis is an additional law of nature can be found in Feynman(2017), Albert (2000), Goldstein (2001), Callender (2004), and Loewer (2012). The inference that thePast Hypothesis may be a fundamental law is based on the fact that it does not seem to be derivedfrom anything else. In this paper, we set aside the interesting possibility raised by Carroll and Chen(2004). See Chen (2019b) for a survey of the realist proposals. See Miller (2016) for more discussions about the notion of holism here.
5y Lorentz transformations). The conflict could be a problem for Everettian (andsome GRW-type) theories that aspire to be (fully) Lorentz invariant.B1.
The Problem of Non-Separability:
The state of the world is not determinedby the states of its parts.B2.
The Conflict between Lorentz Invariance and Narratability:
If quantumentanglement is in the mosaic, then Lorentz invariance conflicts with narrata-bility.
The problem of time asymmetry and the problem of quantum entanglement havebeen much discussed in foundations of physics and metaphysics of science. Bothproblems have come up when evaluating Humeanism: the first problem has beenused to support Humeanism and the second one against Humeanism. Each of themhas also inspired much interesting, original, and insightful work about the Humeanframework in the metaphysics of science. These include: Loewer (1996), Cohen andCallender (2009), Callender and Cohen (2010), Miller (2013), Esfeld (2014), Bhogaland Perry (2015), Callender (2015), Albert (2015), Miller (2016), Esfeld and Deckert(2017).So far, the two problems have been treated as distinct problems. We haveseen impressive progress in developing interesting solutions to these two problems.Interestingly, the solutions both have something to do with laws of nature. However,the solutions not fully satisfactory. I will discuss some prima facie problems below.The Humean framework in the metaphysics of science can be roughly character-ized by the following theses: • Humean Mosaic: the fundamental physical ontology is a separable mosaic. Inthe terminology of Lewis (1986), it consists in “local matter of particular fact.” • Best System Account of Lawhood: the fundamental laws are the axioms of thesummary that best balances a host of theoretical virtues such as simplicity andstrength.Humean supervenience is the thesis that all there is is a Humean mosaic consistingof local matter of particular fact and all else supervenes on that. Loewer (2001, 2004)suggests that we should allow the best system to admit objective probabilities evenwhen the laws are deterministic, so long as admitting them makes the system moreinformative without adding too much complexity. This is an important modificationof the original Humean framework, but it is arguably continuous with the Mill-Ramsey-Lewis account. Given the success of statistical mechanics, it also representsan important advancement in our understanding of objective probabilities that playa central role in physics. For the rest of this paper, I will adopt the modified Humeanframework as the starting point to think about Humeanism. For a di ff erent perspective, see Scha ff er (2007).
6n the one hand, in response to the problem of time asymmetry, it has beenargued that the (modified) Humean framework can solve the problem and theworries raised in A1 and A2. It is highly plausible that the Past Hypothesis andthe Statistical Postulate belong to the Humean best summary, as they vastly in-crease the informativeness of the system without adding much complexity. This istrue despite the fact that the Past Hypothesis describes a boundary condition andthe fundamental dynamical laws may be deterministic. Hence, on the (modified)Humean account, the Past Hypothesis (PH) and the Statistical Postulate (SP) are asnomic as the dynamical equations of motion.However, new problems arise as we consider the character of the two newHumean laws. For PH, there is a language problem. Adapting the terminology ofCohen and Callender (2009), we can call it the “supervenient-kind problem”: termssuch as entropy is obviously not fundamental, and as such PH may not be fit to bea fundamental law if the axioms of the best system require the vocabulary to be entirely infundamental physical terms . They note that if we flesh out “low entropy” in termsof the microlanguage, it will be an infinitely long disjunction of microstates whichdoes not seem to be simple at all. (The supervenient-kind problem is part of themotivation for “language-relativization” in the Better Best System Account, whichis arguably a radical departure from the original Humean framework.)In fact, the situation may be even worse: not only is an infinite disjunctionprobably too long to be an axiom of the best system, but also are the macroscopicterms such as entropy unlikely to correspond to exactly one set of disjuncts. Giventhe inherent vagueness in the bridge between the macroscopic and the microscopic,it is plausible that there will be borderline cases of whether some microstates fallunder the allowed range of states dictated by the Past Hypothesis. Macrostateshave vague boundaries. Any precise boundary would seem artificial and arbitrary.To be sure, these problems do not refute the Humean understanding of the PastHypothesis, but they seem to suggest that it may be premature to marry the Humeanframework to something like the Mentaculus. Hence, Humeanism initially seemedfriendly to treating the Past Hypothesis as a law, but upon closer inspection thereare deep and di ffi cult problems about language.For SP, the Humean solution treats it as objective as with the postulates ofquantum-mechanical probabilities. It would be desirable if we can unify the two or See, for example, Callender (2004) and Loewer (2012). Some may respond that we can just replace the PH and SP by specifying a single probabilitydistribution on the state space that does not su ff er from the language problem described here. Butit is plausible that, in the standard framework, the simplest way to specify SP is to first specify theinitial macrostate (using something like PH) which will then serve as the “support” of the probabilitydistribution. Nevertheless, new possibilities will be available in the Wentaculus framework. But weare getting ahead of ourselves. This is essentially Lewis’s insistence that the terms of the best system must refer to “perfectlynatural properties,” or the fundamental properties picked out by fundamental physics. See Sider(2011) for a similar proposal for “structural” and “joint-carving” properties. What these amount toand whether this requirement is tenable is a controversial issue. But it is important to note that suchproperties play an important role in response to the “problem of the (x)Fx” and the new riddle ofinduction. I discuss this point more systematically in Chen (2020). prima facie threat-ening to Humeanism. First, Teller (1986) and Maudlin (2007) suggest that B1 isa problem for Humeanism, as the entanglement relations would make the mo-saic non-separable. Second, if we would like to keep Lorentz invariance (e.g. forEverettian theories) in a non-separable mosaic, we would have to sacrifice narrata-bility. This is undesirable. Since Lorentz transformations does not fully preservethe quantum-mechanical data as argued by Albert, in order to tell the story of themosaic in a temporal sequence, we would have to specify the quantum state not justalong one foliation but along all foliations of space-like hypersurfaces. Describingthe Humean mosaic temporally will become infinitely more complex—a potentiallyundesirable result.A promising response to B1 is also a “nomic” strategy: it recommends thatHumeans allow the quantum state (represented by a wave function) into the bestsystem. In quantum theories with additional ontologies beyond the quantum state(such as Bohmian mechanics, GRW spontaneous collapse theories with a matterdensity ontology or a flash ontology, and Everettian quantum theory with a matterdensity ontology), it may be tempting to think that the quantum state is part of thesummary of the local ontology consisting in particles, matter density, or flashes inphysical space. However, the quantum state may be too complex to be nomological.In fact, the typical quantum wave function is highly complex as a function onconfiguration space. In response to the complexity worry, one may follow Dürret al. (1996), Goldstein and Teufel (2001), Goldstein and Zanghì (2013) to connect thenomological interpretation to the Wheeler-DeWitt equation in quantum gravity. Asa solution to that equation, the universal wave function must be time-independentand thus may be simple. But for Humeanism, it seems premature to tie its tenabilityto a particular idea in quantum gravity, especially when it is not clear whether theWheeler-DeWitt equation will survive future development in quantum gravity. (Ithas yet to play a central role in string theory.)These concerns with the nomological interpretation of the wave function areby no means decisive refutations, but they suggest that we may need to thinkoutside the box and look for other ways to solve the problems that avoid the aboveissues. Nevertheless, the nomological interpretation seems promising and on theright track, especially since a successful nomological interpretation can solve bothproblems—B1 and B2—at the same time. If the entanglement relations are not in themosaic, then it can satisfy separability, narratability, as well as Lorentz invariance(for theories with such an aspiration). This is in contrast to the interesting proposalof the high-dimensional Humean interpretation of the wave function developed byLoewer (1996).The treatments of the two problems are so far largely unrelated to each other. Inthe next three sections, we discuss some important connections between the two.The Humean unification will take advantage of their connections. We use the PastHypothesis to simplify the quantum state (by choosing a natural, simple, unique,objective, but mixed quantum state), so that we can find solutions to both B1 and B28ithout making the law system overly complex . We then use the chosen quantumstate to connect the initial low-entropy macrostate to the microdynamics, providinga solution to the supervenient-kind problem and the vagueness problem.
In this section, I first review the standard account of quantum mechanics in a timeasymmetric universe. For concreteness, we focus on the quantum Mentaculus, aneo-Boltzmannian account of quantum statistical mechanics. Next, I propose analternative account called the
Wentaculus.
It replaces the universal wave functionwith a (mixed-state) universal density matrix, the pure-state dynamics with mixed-state dynamics, and the Past Hypothesis with the Initial Projection Hypothesis. HereI focus on the conceptual ideas as much as possible, leaving most mathematicaldetails to the footnotes.
Understanding the world we live in requires us to understand all the regularitiesin nature. As we discussed in §2, many regularities we are familiar with, such asice melting, smoke dispersing, and face getting more wrinkled with time, are time-asymmetric. A large class of these time-asymmetric phenomena can be understoodas entropic asymmetries in time: the past events have lower entropies than futureevents. To give a full account of entropic asymmetries of time in terms of scientificexplanations, we can postulate some low-entropy boundary conditions and prob-ability distributions beyond the fundamental dynamical equations. The field ofstatistical mechanics has devoted considerable energy in justifying the conjecturethat something like a low-entropy initial condition (together with some probabilitydistributions) will lead to a typical monotonic increase in entropy. For concreteness,we can focus on one particular proposal of Albert (2000), Loewer (2012), and Loewer(2016): 9 he Classical Mentaculus Fundamental Dynamical Laws (FDL): the classical microstate of theuniverse is represented by a point in phase space a (encoding the positionsand momenta of all particles in the universe) that obeys F = ma .2. The Past Hypothesis (PH) : at a temporal boundary of the universe, themicrostate of the universe lies inside M , a low-entropy macrostate that,given a choice of C-parameters, b corresponds to a small-volume set ofpoints on phase space that are macroscopically similar.3. The Statistical Postulate (SP) : given the macrostate M , we postulate auniform probability distribution c over the microstates compatible with M . a The phase space is the 6 N -dimensional state space for a classical system with N pointparticles with precise locations and velocities in physical space. b The C-parameters are certain conventional choices—the coarse-graining variables—thatconnect the macrostates to sets of microstates. c The uniform probability distribution here is with respect to the canonical Lebesgue mea-sure on phase space.
This is the classical-mechanical version of the Mentaculus theory. It is a versionof the neo-Boltzmannian account of classical statistical mechanics. However, it is apretty strong version as it specifies a particular low-entropy macrostate M and aparticular probability distribution (the uniform one). The detailed di ff erences do notmatter here. Most theorems and conjectures in statistical mechanics apply to it just aswell as they apply to weaker versions of the PH and SP. We chose the Mentaculus notto commit ourselves to it but merely to write it down as a (concrete) representativeof a standard way of thinking about time’s arrow in a classical universe.Next, we move to quantum statistical mechanics. Let us consider how to ex-tend the classical Mentaculus to the quantum version. The key will be to replacethe classical state space (phase space) with the quantum state space—the Hilbertspace—and to reformulate Boltzmannian statistical mechanics in terms of resourcesin Hilbert space. Here we can follow the suggestions of Albert (2000)§7 and themathematical framework of Goldstein et al. (2010a).10 he Quantum Mentaculus Fundamental Dynamical Laws (FDL): the quantum microstate of theuniverse is represented by a wave function Ψ that obeys the Schrödingerequation i ̵ h ∂ψ∂ t = ˆ H ψ .2. The Past Hypothesis (PH) : at a temporal boundary of the universe, thewave function Ψ of the universe lies inside a low-entropy macrostatethat, given a choice of C-parameters, a corresponds to H PH , a low-dimensional subspace of the total Hilbert space.3. The Statistical Postulate (SP) : given the subspace H PH , we postulatea uniform probability distribution b over the wave functions compatiblewith H PH . a In addition to the ones mentioned in the classical Mentaculus, the C-parameters herealso include conventional choices about the cut-o ff threshold of quantum state macrostateinclusion. b The uniform probability distribution is with respect to the surface area measure on theunit sphere of H PH . The quantum Mentaculus is a concrete version of a standard way of thinkingabout time’s arrow in a quantum universe. Given this setup, the aim is to showthat typical universal wave functions compatible with these postulates will evolvein such a way that most subsystems increase in entropy. There has been impressiveresults that are highly suggestive along this direction. Let us provide some explanations of the quantum Mentaculus. First, the quan-tum microstate of the universe is represented by a wave function Ψ . It correspondsto a unit-length vector in Hilbert space. The Hilbert space is an infinite dimensionalstate space for quantum theory. But a slightly more perspicuous picture of thewave function is to think of it as a function on the configuration space R N . Theconfiguration space is analogous to the phase space in classical mechanics exceptthat it has only 3 N dimensions instead of 6 N dimensions (where N is the numberof particles in the universe), and each point in the configuration space representsa possible configuration of particles in physical space in terms of their locationsonly. The wave function assigns values to every point in configuration space. Howto interpret the wave function is a central question in the foundations of quantummechanics.But even before we engage in the philosophical questions about the interpreta-tion of the wave function, it is important to realize there is a scientific question atthe heart of quantum theory: what is the dynamics of quantum mechanics? Thewave function changes over time and obeys the Schrödinger equation. Since thewave functions can superpose into other wave functions, and since the Schrödingerequation is linear, we encounter the notorious quantum measurement problem , aboutwhich we will return to shortly. For example, see Goldstein et al. (2010b). H PH of the total Hilbert space. The PastHypothesis subspace H PH has very low entropy. In classical statistical mechanics,Boltzmann defines entropy of a phase point to be proportional to the logarithm ofthe volume of the macrostate that includes the phase point. Analogously, the Boltz-mann entropy of a wave function is proportional to the logarithm of the dimensionof the subspace it (almost entirely) belongs. Hence, H PH is a low-dimensionalsubspace. By comparison, PH for classical statistical mechanics postulates that theinitial phase point lies inside a macrostate of very small volume.Third, to make it overwhelmingly likely that the initial wave function is entropic,i.e. evolves to higher-entropy states, we introduce the quantum version of theStatistical Postulate (SP). It provides a uniform probability distribution over theinitial wave functions in the subspace. Because of time-reversal invariance, it isplausible that there exist an infinity of “bad” wave functions that are anti-entropic(i.e. evolve to lower entropy). But the uniform probability distribution assigns muchlower weight on them than on the entropic wave functions. By comparison, SP inclassical statistical mechanics is a uniform probability distribution on the classicalphase points compatible with the Past Hypothesis.These three postulates make up the quantum version of the Mentaculus. How-ever, the Mentaculus cannot be the entire story of quantum mechanics in a timeasymmetric universe. As we mentioned before, quantum mechanics itself faces themeasurement problem. It seems that the Schrödinger evolution of the wave func-tion is interrupted by sudden collapses. The wave function typically evolves intosuperpositions of macrostates, such as the cat being alive and the cat being dead.This can be represented by wave functions on the configuration space with disjointmacroscopic supports X and Y . During measurements, which are not preciselydefined processes in the standard theory, the wave function undergoes random col-lapses. The probability that it collapses into any particular macrostate X is given bythe Born rule. As such, quantum mechanics is not a candidate for a fundamental physicaltheory. It has two dynamical laws: the deterministic Schrödinger equation and theindeterministic collapse rule. What are the conditions for applying the former, andwhat are the conditions for applying the latter? Measurements and observationsare extremely vague concepts. Take a concrete experimental apparatus for example.When should we treat it as part of the quantum system that evolves linearly andwhen should we treat it as an “observer,” i.e. something that stands outside thequantum system and collapses the wave function? That is, in short, the quantum See Goldstein et al. (2010a) for more rigorous definitions. Since the wave functions have to be normalized, they form a unit sphere in the subspace. So thedistribution is only on the unit sphere S ( H ) . That is, P ( X ) = ∫ X ∣ ψ ( x )∣ dx . Various solutions have been proposed to solve the measurement problem. Bohmianmechanics (BM) solves it by preserving the Schrödinger dynamics, adding particlesto the ontology, and an additional guidance equation for the particles’ motion.Ghirardi-Rimini-Weber (GRW) theories postulate a spontaneous collapse mecha-nism, making wave function collapses independent of the observers. Everettianquantum mechanics (EQM) simply removes the collapse rules from standard quan-tum mechanics and suggest that there are many (emergent) worlds, correspondingto the branches of the wave function, which are all real. My aim here is not toadjudicate among these theories. Su ffi ce it to say that they are all quantum theoriesthat remove the centrality of observations and observers.Both BM and GRW use probabilistic postulates to account for the Born rule. BMpostulates the Quantum Equilibrium Distribution, which dictates that the initialparticle configuration is distributed by the Born rule (see Dürr et al. (1992)). GRWpostulates probabilistic modification of the Schrödinger equation by which the cen-ter of wave function collapses is distributed randomly according to (something closeto) the Born rule. EQM, developed and defended by David Wallace (2012), is theonly one that does not introduce any objective probabilities (but seeks to derivethem from preference axioms). On BM and GRW, SP will postulate a fundamentallydi ff erent kind of probabilities from the quantum mechanical probabilities. It wouldbe desirable if they can be unified. On EQM, the aspiration is to come up with atheory that explains the probabilistic phenomena in nature, for which the objectivestatistical mechanical probabilities of SP will be an obstacle. Recent work in the foundations of quantum mechanics suggests that just as wecan add particles in Bohmian mechanics (BM), we can add additional ontologiesto GRW and Everettian theories: GRW with a flashy ontology (GRWf), GRW witha mass-density ontology (GRWm), and Everettian theory with a mass-density on-tology (Sm). Let us call them quantum theories with additional ontologies . UnlikeBohmian particles, these additional ontologies are not independent variables fromthe wave function.The above quantum theories—BM, GRW, GRWm, GRWf, EQM, Sm—make plau-sible the view that I call
Wave Function Realism : the universal quantum state is (1)ontic and (2) completely represented by the universal wave function. This is incontrast to the epistemic views about the wave function that maintain that thequantum state, represented by a wave function, corresponds to only our epistemicuncertainties over the actual state of the world.In short, the quantum Mentaculus contains the quantum version of the PastHypothesis and that of the Statistical Postulate that support the claim that typicalinitial microstates will be entropic. Such an understanding can be supplementedwith further interpretations about the meaning of the wave function. But the mar- See Bell (1990) and Myrvold (2017) for introductions to the quantum measurement problem. There have been some proposals of how to solve these problems, such as Albert (2000), Ch.7 andWallace (2011). They rely on two plausible conjectures. Sm was introduced in Allori et al. (2010). ff erent approach: • Non-separability and non-narratability problems if we keep the quantum statein the mosaic; • Complexity problems if we move the quantum state from the mosaic to thebest system; • Supervenient-kind and vagueness problems of the Past Hypothesis; • (Not a problem but worth mentioning: dualism of statistical mechanical prob-abilities and quantum mechanical probabilities.) In this section, I construct an alternative framework—the Wentaculus. As we discussin §4 and §5, the Wentaculus solves the problems above and provides additionaltheoretical virtues. I proceed in two steps: (1) Density Matrix Realism and (2)the Initial Projection Hypothesis. I will also explain how Humeanism providesmotivations for this new framework.
In §3.1, we saw that the quantum Mentaculus assigns probabilities over wave func-tions. Now, there is a well-known method of encoding the probabilities in thequantum state itself. Instead of saying that the wave function lies inside some sub-space H ν and that there is a uniform probability distribution over the wave functionsin H ν , quantum theory provides a compact way of putting these two pieces of infor-mation together into one mathematical gadget—a density matrix. The probabilitydistribution over wave functions can be represented by a density matrix ˆ W ν . We should not be misled by the language here. Even though we talk about“constructing a density matrix from a collection of wave functions,” there is a more intrinsic way of understanding the density matrix that is independent of the wavefunctions. A density matrix is a well-defined object in its own right in Hilbert space. More precisely, the density matrix is equal to an integral over wave functions inside the unitsphere of the subspace with respect to the uniform distribution given by the surface area measure:ˆ W ν = ∫ S ( H ν ) µ ( d ψ ) ∣ ψ ⟩ ⟨ ψ ∣ . Here is a more intuitive way of understanding the construction procedure.Start from the subspace H ν . It is compatible with many vectors representing di ff erent initial wavefunctions. Take an arbitrary vector ∣ ψ ⟩ . We can construct a projection operator (projecting to ∣ ψ ⟩ ) as ∣ ψ ⟩ ⟨ ψ ∣ . If we apply ∣ ψ ⟩ ⟨ ψ ∣ to any other vector ∣ φ ⟩ , it will first take the inner product ⟨ ψ ∣ φ ⟩ and outputa scalar c , which measures “how much” of ∣ φ ⟩ overlaps with ∣ ψ ⟩ . Then it multiplies the scalar to ∣ ψ ⟩ , which yields c ∣ ψ ⟩ . Take all the vectors on S ( H ν ) , the unit sphere in the subspace, construct thecorresponding projection operators, and then take the “weighted sum” over the projection operators.Since there is a continuous infinity of objects to sum over, instead of using an infinite sum, we usean integral over them with respect to the surface area measure on the unit sphere µ ( d ψ ) . Thisconstruction produces a density matrix that represents the probability distribution over the initialwave function. H ν . The object contains nomore and no less information than what is contained in the subspace itself. It iscalled the normalized projection, whose mathematical representation can be writtenas follows: ˆ W ν = I ν dim H ν , (1)This is the normalized projection onto the subspace H ν . The normalization is achievedby dividing the subspace identity operator I ν by the dimension of the subspace dim H ν . The identity operator is restricted to the subspace: it does nothing to vectorscontained inside the subspace and projects into the subspace everything that is notcompletely contained within. Given the intrinsic understanding of density matrices in Hilbert space, is therea sense we can provide an intrinsic understanding of it on configuration space thatis independent from wave functions and equally objective? The answer is yes. Wecall this perspective Density Matrix Realism, in contrast to Wave Function Realism.Just as we can think of the wave function as a function that assigns values to theconfiguration space R N , we can think of the density matrix as a function that assignsvalues to the Cartesian product of the configuration space with itself. Moreover, wecan also think of it as a function that assigns values to every ordered pair of pointsin configuration space.Wave Function Realism is motivated by the idea that the wave function is centralto the dynamics and the kinematics of quantum mechanics. In order to motivateDensity Matrix Realism, we need to reformulate quantum mechanics directly interms of a fundamental density matrix. Can we do that? The answer is yes. First, the density matrix has an evolution equation analogous to that of the wavefunction. While the wave function obeys the Schrödinger equation, the densitymatrix obeys its generalization to mixed states—the von Neumann equation: i ̵ h d ˆ W ( t ) dt = [ ˆ H , ˆ W ] , (2)where the commutator bracket represents the linear evolution analogous to thelinear evolution described by the Schrödinger equation.Second, the Born rule distribution can be written in terms of the density matrix: P ( q ) dq = W ( q , q ) dq (3)Third, we can reformulate BM, GRW, and EQM in terms of the density matrix. Since the diagonal entries of ˆ W ν add up to 1, it is a density matrix. Density Matrix Realism has already been suggested but not necessarily endorsed by some in theliterature. For some recent examples, see Dürr et al. (2005), Maroney (2005), Wallace and Timpson(2010) and Wallace (2011, 2012). What is new in this paper is the combination of Density MatrixRealism with the Past Hypothesis in forming the Initial Projection Hypothesis (§2.2.2) and theargument for the Humean Unification based on that. For W-EQM, equation (2) is all there is to govern the fundamental quantum state W .
15e can show that each reformulation of the realist quantum theory in terms of auniversal density matrix W is empirically equivalent to its wave-function counter-part, if on the latter theories the uncertainty over the universal wave function isrepresented by a statistical density matrix W . Therefore, these are also empiricallyadequate quantum theories without observers. We call these theories
W-Bohmianmechanics, W-GRW theory , and
W-Everettian quantum mechanics . Thus, we can thinkof W as the central dynamical object in quantum mechanics that produces quan- For W-EQM with a mass-density ontology, we can define the mass density function in terms of thedensity matrix: m ( x , t ) = tr ( M ( x ) W ( t )) , (4)where M ( x ) = ∑ i m i δ ( Q i − x ) is the mass-density operator, which is defined via the position operator Q i ψ ( q , q , ... q n ) = q i ψ ( q , q , ... q n ) . This allows us to determine the mass-density ontology at time t via W ( t ) .For W-BM, we can postulate the guidance equation as follows: dQ i dt = ̵ hm i Im ∇ q i W ( q , q ′ , t ) W ( q , q ′ , t ) ( q = q ′ = Q ) , (5)Finally, we can impose a boundary condition similar to that of the Quantum Equilibrium Hypothesis: P ( Q ( t ) ∈ dq ) = W ( q , q , t ) dq . (6)Since the system is also equivariant, if the probability distribution holds at t , it holds at all times.Equivariance holds because of the following continuity equation: ∂ W ( q , q , t ) ∂ t = − div ( W ( q , q , t ) v ) , where v denotes the velocity field generated via (5.) This theory is first described in Dürr et al. (2005)and Maroney (2005). Dürr et al. (2005) call this theory W-Bohmian mechanics.For W-GRW (first suggested in Allori et al. (2013)), between collapses, the density matrix willevolve unitarily according to the von Neumann equation. It collapses randomly, where the randomtime for an N -particle system is distributed with rate N λ , where λ is of order 10 − s − . At a randomtime when a collapse occur at “particle” k at time T − , the post-collapse density matrix at time T + isthe following: W T + = Λ k ( X ) / W T − Λ k ( X ) / tr ( W T − Λ k ( X )) , (7)with X distributed by the following probability density: ρ ( x ) = tr ( W T − Λ k ( x )) , (8)where W T + is the post-collapse density matrix, W T − is the pre-collapse density matrix, X is the centerof the actual collapse, and Λ k ( x ) is the collapse rate operator defined as follows: Λ k ( x ) = ( πσ ) / e − ( Qk − x ) σ , where Q k is the position operator of “particle” k , and σ is a new constant of nature of order 10 − mpostulated in current GRW theories.For W-GRWm, we can let the density matrix determine the mass density function on space-timeby (4). For W-GRWf, we postulate flashes that are the space-time events at the centers ( X ) of theW-GRW collapses. This is because the predictions of quantum theory are probabilistic; it does not matter whetherthe density matrix we use to extract predictions is statistical or fundamental. See Dürr et al. (2005),Wallace (2016), and Chen (2019a) for more detailed arguments.
The quantum Mentaculus is most plausible in the framework of Wave FunctionRealism. On that view, the wave function represents the quantum state of the world,the Past Hypothesis is a constraint on the initial wave function, and the StatisticalPostulate is a uniform probability distribution over possible wave functions.In the framework of Density Matrix Realism, the wave function no longer repre-sents the quantum state. Instead, a density matrix completely represents the initialstate. Hence, we can consider reformulating the low-entropy boundary conditionas the constraint on the density matrix. However, just as there are many wavefunctions compatible with H PH , there are many density matrices compatible with H PH , the Past Hypothesis subspace in the total Hilbert space. One could perhapsconstruct a probability distribution over the possible initial density matrices.Interestingly, Density Matrix Realism provides a much simpler constraint thatcombines the Past Hypothesis and the Statistical Postulate that is unavailable in thewave function framework. We postulate that the initial density matrix is the simplestand most natural density matrix associated with H PH : its normalized projection. Itcan be expressed as follows: ˆ W IPH ( t ) = I PH dim H PH , (9)It is the identity operator on H PH divided by the dimension of H PH . I label its Hilbertspace representation as ˆ W IPH ( t ) . In the position representation, it is W IPH ( q , q ′ , t ) .This constraint is motivated by Humeanism. The goal of a Humean theorist isto come up with the simplest and most informative summary of the history of theworld. If we can avoid the postulation of a probability distribution by making theinitial state unique, then the Humean theorist would be motivated to do so. As weshall see in the next section, the postulate leads to further benefits to Humeanism.In contrast, there is no obvious candidate for the simplest or most natural wavefunction compatible with the Past Hypothesis.Therefore, I propose that we add the following postulate to any quantum theoryin the framework of Density Matrix Realism: Initial Projection Hypothesis:
The initial quantum state of the universe is ˆ W IPH ( t ) .The Initial Projection Hypothesis (IPH) plays a similar role as that of the Past Hy-pothesis (PH). They both rule out many available initial states on the state space toexplain the time asymmetry in our universe. They carry the same information aboutinitial entropy. PH selects the initial wave function to be one of the wave functions17igure 1: The expected growth of entropy under the Initial Projection Hypothesis(IPH) and the density-matrix dynamics.inside H PH , and IPH selects the initial density matrix to be the (unique) normal-ized projection on H PH . Both have exactly the same amount of entropy—that of H PH . However, there are some important di ff erences between IPH and PH. First,IPH picks out a unique initial quantum state of the universe while PH does not.In so far as the Past Hypothesis subspace can be unambiguously specified givensome coarse-graining variables, the normalized projection can be unambiguouslyspecified, and IPH also unambiguously specifies the initial state as ˆ W IPH ( t ) . Incontrast, PH narrows down the initial wave function to be inside the subspace H PH ,which is still compatible with an infinite number of di ff erent wave functions.Second, IPH requires no further statistical mechanical probability distributionwhile PH needs to be supplemented with SP. Since IPH chooses a unique initialstate, there is no need to add a probability weighting on the initial states compatiblewith IPH. However, PH is compatible with many wave functions, some of whichwill evolve to lower-entropy states. Hence, PH needs to be supplemented with astatistical mechanical probability distribution (SP) that assigns high weight to the“good” wave functions and low weight to the “bad” ones.When we add IPH to Density Matrix Realism, we arrive at an alternative accountof time’s arrow in a quantum universe: S B ( Ψ PH ( t )) = S B ( W IPH ( t )) = k B log ( dim H PH ) , where S B is the Boltzmann entropy function, k B is the Boltzmann constant, and “dim” counts the dimensionality of the subspace. There are additional subtleties about vagueness, which we explore in §5.2. he Wentaculus Fundamental Dynamical Laws (FDL): the quantum state of the universeis represented by a density matrix ˆ W ( t ) that obeys the von Neumannequation (2). a The Initial Projection Hypothesis (IPH) : at a temporal boundary of theuniverse, the density matrix is the normalized projection onto H PH , alow-dimensional subspace of the total Hilbert space. (That is, the initialquantum state of the universe is ˆ W IPH ( t ) as described in (9).) a For GRW-type theories, the density matrix obeys the stochastic modification of the vonNeumann equation described in footnote
This is the W-version of the Mentaculus. Let us call it the
Wentaculus . To solve thequantum measurement problem, we can construct Bohmian, Everettian, and GRWversions of the Wentaculus. Let us call these theories W
IPH -quantum theories. In§4, we show that the Wentaculus naturally leads to a Humean unification of theorigins of quantum entanglement and time asymmetry. Such a unification will bearmany fruits (§5).
We have seen that the density matrix formalism opens up a new possibility for a time-asymmetric quantum-mechanical world: it can be described by the W
IPH -quantumtheories of the Wentaculus. In this section, I show that Humeanism allows us tofurther simplify the theoretical structure, by unifying the sources of time asymmetryand quantum entanglement and removing the quantum state from the mosaic. First,I argue for the Nomological Thesis. Second, I show that Humeanism allows us toobtain a unified explanation of time asymmetry and quantum entanglement. Third,I discuss two new wrinkles brought up by the Humean unification.
As we discussed in §2, the classical Mentaculus consist in three postulates—the fun-damental dynamical equations, the Past Hypothesis, and the Statistical Postulate—all of which can be admitted, by the best-system account, as Humean laws. The PastHypothesis and the Statistical Postulate are not the usual dynamical laws. In partic-ular, the Past Hypothesis is regarded as a Humean law even though it may look likejust another contingent initial condition. Even before we get into Humeanism, thereare pre-theoretical reasons (reasons that are conceptually prior to a systematic viewabout laws) that support its nomological status. For example, plausibly it playsa starring role in deriving the Second Law of Thermodynamics; and perhaps alsoin deriving the counterfactual asymmetries, the records asymmetry, the epistemic The Wentaculus as it is will be su ffi cient for W IPH -EQM. See §4.2 for the the Bohmian version. plausibly it is a law of natureand not a contingent initial condition.The Humean best-system account provides another argument for the nomologi-cal status of the Past Hypothesis. Take for example the quantum Mentaculus. If wesubtract the Past Hypothesis and the Statistical Postulate from the Mentaculus, thetheory is much weaker. Let us call it the quantum Mentaculus − : The Quantum Mentaculus − Fundamental Dynamical Laws (FDL): the quantum microstate of the uni-verse is represented by a wave function Ψ that obeys the Schrödingerequation.This is the Mentaculus without PH and SP. Since it is time symmetric, it doesnot ground lawful generalizations such as the Second Law of Thermodynamics andmany other temporal asymmetries. As it is, it is much less informative than theMentaculus. The Mentaculus − would be much more informative if we add to it PHand SP.Moreover, PH and SP are not very complex. The uniform surface area measure,specified by SP, is a simple probability measure on the subspace. PH is simple in themacro-language specified in terms of the macro-variables such as temperature, vol-ume, densities, and entropy. In fact, PH could also be simple in the micro-language.A version of PH in the general relativistic cosmological context is the Weyl Curva-ture Hypothesis (WCH), which is a simple postulate about the initial geometry. To have a complete quantum generalization of WCH would require a theory ofquantum gravity, which is still work in progress. However, there are reasons to behopeful. For example, the generalization of WCH in the Loop Quantum Cosmologyprogram has yielded the Quantum Homogeneity and Isotropy Hypothesis (QHIH) That is, we set aside in this paper the possibility explored by Carroll and Chen (2004). In the context of thinking about the origin of the Second Law of Thermodynamics in the earlyuniverse with high homogeneity and isotropy, and the relationship between space-time geometryand entropy, Penrose proposes a hypothesis:I propose, then, that there should be complete lack of chaos in the initial geometry .We need, in any case, some kind of low-entropy constraint on the initial state. Butthermal equilibrium apparently held (at least very closely so) for the matter (includingradiation) in the early stages. So the ‘lowness’ of the initial entropy was not a resultof some special matter distribution, but, instead, of some very special initial spacetimegeometry. The indications of [previous sections], in particular, are that this restrictionon the early geometry should be something like: the Weyl curvature C abcd vanishes at anyinitial singularity . (Penrose (1979), p.630, emphasis original)The Weyl curvature tensor C abcd is the traceless part of the Riemann curvature tensor R abcd . It isnot fixed completely by the stress-energy tensor and thus has independent degrees of freedom inEinstein’s general theory of relativity. Since the entropy of matter distribution is quite high, the originof thermodynamic asymmetry should be due to the low entropy in geometry, which correspondsvery roughly to the vanishing of the Weyl curvature tensor. The Weyl Curvature Hypothesis issimple to state in the language of general relativity. We have good reasons to think that the quantum Mentaculus could be the bestsystem. Thus, we have good reasons to think that PH and SP are parts of the bestsystem. On the modified Humean theory of laws and objective probabilities, itfollows that PH is a Humean law of nature and SP specifies objective probabilitiesin the world.Similarly, the best system from the point of view of Density Matrix Realism is theWentaculus. Given the crucial role IPH plays in the Wentaculus, it is plausible thatIPH should be regarded as a Humean law if the Wentaculus is the best system. Afterall, IPH has the same informational content as PH. They both specify a low-entropyinitial condition. Moreover, IPH is as simple as PH + SP. They pick out the samedensity matrix in the low-entropy subspace.Hence, in so far as we have good reasons to take PH to be a Humean law ifthe quantum Mentaculus is true, we have equally good reasons to take IPH to be aHumean law if the Wentaculus is true. That is, if Wentaculus is the right theory of theactual world, then we have good reasons to confer (Humean) nomological status toIPH that are on a par with our reasons for conferring (Humean) nomological statusto PH. But how do we know which is true: the Mentaculus or the Wentaculus?Here we encounter a case of underdetermination by evidence. The two theoriesare empirically equivalent: no amount of empirical evidence can settle the questionwhich one is correct. However, we can use super-empirical virtues, some of whichwill be discussed in §5.Both PH and IPH are Humean laws that are about the initial quantum state. Asdiscussed before, it is controversial what the quantum states represent. But what isthe nature of the initial quantum state? One promising answer suggests that it isnomological.
The Nomological Thesis:
The initial quantum state of the world is nomological,i.e. it is on a par with laws of nature.The Nomological Thesis, on the one hand, is in tension with the complexity issue inthe quantum Mentaculus. Even though PH is simple, the wave function compatiblewith PH is unlikely to be simple enough to be nomological. The Humeans couldfollow Dürr et al. (1996) and claim that the Wheeler-DeWitt equation in quantumgravity would produce a time-independent wave function that may also be simpleenough. Even though the Wheeler-DeWitt equation leads to fascinating scientificand interpretational questions, the Humeans who endorse this strategy faces severalchallenges. First, a technical challenge: given the time-independence of the wavefunction, the time asymmetry cannot be described in terms of the entropy of theuniversal wave function, and that would require significant changes to the Mentac-ulus program. Moreover, it does not follow that a time-independent wave function For more details, see Ashtekar and Gupt (2016a) and Ashtekar and Gupt (2016b). W IPH ( t ) is nomore complex than IPH itself. So if IPH is simple enough to be nomological, then W IPH ( t ) is simple enough to be nomological. (In contrast, even though PH is simpleenough, the initial wave function in the Mentaculus may not be simple.) Hence, inthe Wentaculus (but not in the Mentaculus), we can easily remove the complexityobstacle by regarding the initial quantum state to be on a par with laws of nature.We propose that the Humeans remove the initial quantum state from the mosaicand move it to the best system. Its values can be completely and in a simple wayspecified by the best system. Hence, moving W IPH ( t ) to the best system is not goingto overburden the best system or make it more complex, since IPH already containsthat information. This is to be contrasted with the situation in the Mentaculus: thePH does not contain all the information to pin down the initial microstate (wavefunction) while the IPH in the Wentaculus does contain all the information to pindown the initial microstate (density matrix). However, after we remove the quantumstates from the mosaic, we need something to be still present in the mosaic. We canuse the additional ontologies in quantum theories such as BM, Sm, GRWm, andGRWf. The fundamental ontology, in each of these theories, will be the particles,matter densities, or flashes, which are separable.What about later quantum states W IPH ( t ) , W IPH ( t ) ,..., and so on? Do we needto postulate them in the mosaic? That is not necessary. For unitary quantumtheories such as BM and Sm, their information can be directly derived from thevon Neumann equation, which is also in the best system. For stochastic theories,the initial quantum state (in the best system) can specify a complete probabilitydistribution and conditional probabilities over possible mosaic histories. If the initial quantum state is nomological, then the Humean best system containsall of its information: without adding any contingent fact from the mosaic, we can The GRW route is di ff erent from the way we think about making predictions on GRW theories.That might mean that the GRW route of Humean unification is less natural than the route on unitarytheories. Hence, Humean unification may be sensitive to empirical questions about whether GRWis correct, and whether quantum theory is exact. Even if one is bothered by this sensitivity, one neednot give it too much weight all things considered. So far, all experimental tests to find violations ofunitary dynamics and the deviations from exact Born rule have only but confirmed exact quantumtheories such as BM and EQM; we have not found any confirmation of GRW over its rivals. For areview, see Feldmann and Tumulka (2012). IPH -quantum theories with additional ontologies. For these theories, we can remove thequantum state from the mosaic without losing any information about entanglementand correlations, for such information is already contained in the best system. Andthe mosaic will not be empty—it will still contain the local beables such as particles,matter densities, and flashes, which make up pointers, tables, and chairs. From aHumean point of view, the best system (now the Wentaculus plus the values of theinitial quantum state) supervenes on the mosaic.Take W
IPH -BM for example. Let us write down the mosaic + best system packagewithout the Humean unification: The W
IPH -BM mosaic: particle trajectories Q ( t ) on physical space-time andthe quantum state W IPH ( t ) . The W
IPH -BM best system: four equations—the simplest and strongest ax-ioms summarizing the mosaic:(A) The von Neumann equation: i ̵ h ∂ ˆ W ∂ t = [ ˆ H , ˆ W ] , (B) The Initial Projection Hypothesis: ˆ W IPH ( t ) = I PH dim H PH (C) The W-Quantum Equilibrium Hypothesis: P ( Q ( t ) ∈ dq ) = W IPH ( q , q , t ) dq , (D) The W-guidance equation: dQ i dt = ̵ hm i Im ∇ qi W IPH ( q , q ′ , t ) W IPH ( q , q ′ , t ) ( q = q ′ = Q ) . What would it look like after Humean unification? It will have fewer things inthe mosaic and fewer equations in the best system.
The W
IPH -BM mosaic: particle trajectories Q ( t ) on physical space-time. The W
IPH -BM best system: three equations—the simplest and strongest ax-ioms summarizing the mosaic:(A) The Initial Projection Hypothesis: ˆ W IPH ( t ) = I PH dim H PH (B) The W-Quantum Equilibrium Hypothesis: P ( Q ( t ) ∈ dq ) = W IPH ( q , q , t ) dq , (C) The combined equation: dQ i dt = ̵ hm i Im ∇ qi ⟨ q ∣ e − i ˆ Ht /̵ h ˆ W IPH ( t ) e i ˆ Ht /̵ h ∣ q ′ ⟩⟨ q ∣ e − i ˆ Ht /̵ h ˆ W IPH ( t ) e i ˆ Ht /̵ h ∣ q ′ ⟩ ( q = q ′ = Q ) In this theory, the mosaic no longer contains the quantum state. IPH still pos-tulates the values of the initial quantum state ˆ W IPH ( t ) . But it is dispensable. Therole it plays in the best system above is to specify the values of the initial probabilitydistribution and the velocity field for particles. We can rewrite any occurrence ofˆ W IPH ( t ) in terms of its explicit functional form. We can construct similar Humeaninterpretations of W IPH -GRWm, W
IPH -GRWf, and W
IPH -Sm.23he nomological role of the quantum state here is similar to that of the Hamil-tonian in classical mechanics. The Hamiltonian specifies the interactions or the“forces” among the component systems. The Hamiltonian is on par with the clas-sical laws of motion as it is a simple part of the Hamiltonian equations. That is, ifwe expand the Hamiltonian function as a function of the variables for things in themosaic (positions and velocities of particles), the equation is still simple. Similarly,the quantum state is a simple part of the von Neumann equation. However, animportant di ff erence is that equation (C) is time-dependent, while the Hamiltonianequations of motion are time-independent.I call such an interpretive strategy the Humean Unification . Humean Unifica-tion recommends that we remove the quantum state from the mosaic. How, then,does one explain the phenomena of quantum entanglement? How can systemsat space-like separation be perfectly correlated, if there is no fact about quantumentanglement in the mosaic? The Humean Unification provides purely nomic expla-nations. There is a law that specifies the quantum entanglement of all the systemsat t , from which we can use the von Neumann equation to derive quantum entan-glement at a later time t . The equation for local beables (such as (4) and (5)) willthen determine the behavior of objects in space-time: e.g. if Alice were to observe“Spin Up” then Bob would observe “Spin Down.” (Both the observers and the ob-served systems will be made out of the local beables and not of the quantum state.)Moreover, the law that specifies the quantum initial condition—IPH—is the samelaw that specifies the low-entropy initial condition. Hence, IPH is the origin of bothquantum entanglement and time asymmetry.The Humean unification provides a unified view on the sources of quantumentanglement and time asymmetry. In the Mentaculus, they have distinct sources—one macroscopic Past Hypothesis and one microscopic wave function. But in theWentaculus, it is one and the same density matrix.The time-dependence of the equation of motion (such as the combined equation(C)) in the unified best system further suggests that there is intertwining betweenthe two. In the Mentaculus picture, the theory as a whole is not time-translation-invariant, because the Past Hypothesis applies only at a particular time. However,we can still understand the sense in which the Mentaculus picture is still time-translation-invariant: we can separate the dynamics from the initial condition, evenwhen both are Humean laws; the dynamics is invariant even when the lawful initialcondition is not. But in the Wentaculus picture, after Humean unification, there The Hamiltonian equations are: ∂ q i ∂ t = ∂ H ∂ p i , ∂ p i ∂ t = − ∂ H ∂ q i . (10)The Hamiltonian function is specified as follows: H ( q , p ) = N ∑ i = p i m i + ∑ ∑ ≤ k < l ≤ N V k , l (∣ q k − q l ∣) , (11)where V k , l is a simple formula for the pair-wise interactions.
24s no such clean separation. The two, as it were, are genuinely unified into onething, so there is no sense in which the theory is fundamentally invariant undertime translation.The violation of time-translation invariance should be viewed not as a bug butas a feature of Humean unification. After all, is it reasonable to insist on savingthe symmetry at the cost of everything else? Of course not. Does this particularviolation make the theory less simple? No; in fact the theory becomes simplerbecause of it. Can we still make sense of the appearance of this symmetry? Yes.At the emergent level of e ff ective dynamics and subsystem dynamics, we can stillmake sense of a time-translation-invariant non-fundamental dynamics. For manysubsystems, they will have (non-fundamental) subsystem density matrices that stillobey time-translation-invariant (e ff ective) laws. The Humean unification is based on a new theoretical possibility opened up bythe conjunction of Density Matrix Realism and the Initial Projection Hypothesis.However, it also leads to some new wrinkles that should be addressed. We focuson two here: the complexity worry and the classical maneuver.(1) The complexity worry. The Initial Projection Hypothesis selects a densitymatrix that is mathematically equivalent to a “disjunction” of wave functions with auniform measure over them. If any wave function in the disjunct is overly complex,shouldn’t the whole disjunction, and therefore the initial density matrix, be overlycomplex? How is that compatible with the earlier claim that the Initial ProjectionHypothesis as well as the initial density matrix are simple enough to be nomological?The intuition behind this worry is that the disjunction inherits whatever complexitythat is in the disjuncts, and the initial density matrix ˆ W IPH ( t ) will be highly complex,and perhaps even more so than typical wave functions. So our criticism about thestandard wave function nomological view comes back to haunt us.That is not quite right. We o ff er a counterexample to the intuition and a positiveargument for the simplicity of the initial density matrix. Let us consider classicalmechanics governed by F = ma . We can think of the content of F = ma as givenby the disjunction of all the solutions to that equation, namely the disjunction ofall complete trajectories of any number of point particles that classical mechanicsallows. Most of those trajectories will be highly complex. However, F = ma is asimple law, even though it is informationally equivalent to the complete disjunctionof its possible solutions. Positive argument: there are ways of understanding thedensity matrix that is independent of the collection of wave functions; the quantumstate space (Hilbert space) permits a straightforward, intrinsic, and geometricalunderstanding of the initial density matrix selected by IPH. In fact, it is (modulo thenormalization constant) equivalent to the subspace itself. While an individual vector Even if one is worried about the loss of (fundamental) time-translation invariance, the theoreticalcost should be viewed in the context of and in balance with the numerous theoretical benefits thatwe discuss in §5.
25n the subspace (the wave function) may require many coordinate numbers to pickout, the subspace requires much less information to specify. The availability of theintrinsic understanding of the fundamental density matrix ˆ W IPH ( t ) also respondsto the supervenient-kind problem raised in §2.(2) The classical maneuver. One might worry that the same “trick” can be playedin the classical context, which somehow makes the Humean unification look tooeasy and perhaps trivial. On first glance, the suggested maneuver is to take the“probability distribution” ( ρ ) as “ontic.” The same thing can presumably be takenin the classical context, where the probability distribution on phase space can begiven a similar ontic interpretation, thus avoiding the problems in the classicaldomain as well. If that is possible, moreover, it seems to show that either we haveproven too much, or that it does not depend on the details of quantum theory.However, that is a mistake. First, it is much less natural to give an ontic in-terpretation of the probability distribution in classical statistical mechanics. If weuse the same idea in the classical domain, we will likely get a many-worlds versionof classical mechanics or lose determinism. The classical probability distribution ρ plays no dynamical role (unlike the density matrix in the W-quantum theories).Moreover, since ρ follows the Hamiltonian dynamics, it will in general be supportedon many macroscopically distinct regions on phase space. If we reify ρ as ontic anddo not modify the dynamics, then we arrive at a many-worlds theory. If we modifythe dynamics to introduce objective “collapses” of ρ into one of the “branches,” itwill look much more artificial and complex than the original Hamiltonian theory. Incontrast, on each of the three interpretations of QM, the artificial e ff ects do not ariseon the Wentaculus. The Bohmian version remains deterministic (and single-world),the GRW version remains stochastic (and single-world), and the Everettian / many-worlds version is still deterministic. On the other hand, even if a classical extensionof our maneuver is possible, it is unclear how it makes the quantum case trivial,since presumably both require di ff erent choices of the ontology and the dynamics. The Humean Unification has consequences for both the Humean mosaic and thebest system. In this section, we list some fruits of this project.
Under Humean Unification, the Humean mosaic becomes simpler, separable, andnarratable. In standard quantum theories, it is plausible that the quantum staterepresents something in the mosaic. This leads to a non-separable entity or relationthat violates one of the tenets of Humean supervenience: that the mosaic must beseparable. The Humean Unification lifts the quantum state from the mosaic into thebest system (without adding too much complexity to the best system). It simplifiesthe mosaic by removing the quantum state ontology and postulating local matters26n spacetime—particles, matter density, or flashes. The separability of the mosaic isnow restored.This result provides a response to Teller (1986)’s and Maudlin (2007)’s influentialargument that Humean supervenience is incompatible with quantum mechanics. Itdoes so without incurring new costs (see contrast with rival proposals in §6) andwith many new benefits (as we see in this section).An under-appreciated consequence of this result is that it also helps with Al-bert (2015)’s recent argument that quantum entanglement is incompatible with fullLorentz invariance. This is best seen in W
IPH -Sm, which aspires to be a fullyLorentz-invariant theory. Albert (2015) shows that the conjunction of Lorentz in-variance and entanglement is inconsistent with narratability, the idea that the fullhistory of the world can be narrated in a single temporal sequence, and other waysof narrating it will be its geometrical transformations. To describe a narratablemosaic temporally, it su ffi ces to list facts along one foliation (with the rest being ge-ometrical transformations); to describe a non-narratable mosaic such as the one thatcontains entanglement relations, we need to list facts along every foliation, whichis infinitely more complex. Thus, one can understand narratability as somethingakin to descriptive parsimony, which is a desirable but defeasible virtue to be bal-anced with other considerations. (After all, one can consistently insist on specifyingstates only directly on the mosaic and not through any temporal sequences pickedout along some foliation.) In so far as one finds narratability a plausible principle,there is tension between Lorentz invariance and quantum entanglement (which is apurely kinematic notion).However, Albert’s argument presupposes that the entanglement relations arein the mosaic. However, if we remove the quantum state from the mosaic, thetrouble-maker is gone, and the mosaic can be both Lorentz invariant and narratable.This avoids the narratability failure mentioned earlier. (A similar result holds forW IPH -GRWm and W
IPH -GRWf.) By allowing the mosaic to be fully narratable andallowing the laws to be fully Lorentz invariant, Humean unification could lead tofurther simplification of the mosaic and the best system. By removing the conflictbetween Lorentz invariance and narratability, the Humean Unification removessome tension between quantum mechanics and special relativity. In summary, the Humean unification strategy simplifies the mosaic by removingthe quantum entanglement relations, leaving with a fundamental ontology that isseparable and narratable that has a better chance of reconciling quantum theorywith fully Lorentz invariance (for those theories that have such aspirations). Thequantum entanglement relations and the quantum state would be absorbed into thebest system, which then supervenes on the new mosaic. I am indebted to discussions with Sheldon Goldstein and Ezra Rubenstein on this point. To be sure, there is still the issue of quantum non-locality. .2 The Best System: Simpler, Less Vague, and More Unified The Humean unification leads to some important modifications of the Humean bestsystem, making it simpler, less vague, and more unified. By lifting the quantumstate from the mosaic to the best system, prima facie we increase the complexity ofthe best system by exactly the amount of complexity of the quantum state. If thequantum state is highly complex, then the resultant best system will be complexas well. Even if the mosaic becomes simpler and separable, the costs may still betoo high for the Humeans. (That may be the case for some of our rivals. See §6.)However, given the analysis in §4 about the simplicity of the Past Hypothesis andthe Initial Projection Hypothesis, the initial quantum state is simple enough to benomological. Making ˆ W IPH ( t ) on a par with Humean laws does not overburdenthe best system. In fact, since the Wentaculus best system already contains IPH,and IPH completely specifies the values of ˆ W IPH ( t ) , there is no added complexitywhen we lift ˆ W IPH ( t ) to the best system. That is a significant advantage over otherversions of quantum Humeanism where the focus is on the quantum wave functionthat is likely much more complex than the initial density matrix.The Humean unification simplifies the best system in another way: it elimi-nates fundamental statistical mechanical probabilities in the best system. In thequantum Mentaculus, there are two kinds of probabilities: the quantum probabil-ities prescribed by the Born rule (or the quantum equilibrium distribution in BM,the collapse probabilities in GRW, and the non-physical decision-theoretic or de se probabilities in EQM) and the statistical mechanical probabilities prescribed by theStatistical Postulate (SP). It would be desirable to unify the two sources of probabili-ties in the theory.Albert (2000)§7 attempts to do it in the GRW framework, relying ona plausible conjecture about Gaussian width during GRW collapses. Wallace (2011)proposes we replace the SP by something like a simplicity constraint, relying on aconjecture about simplicity and reversibility. On Humean unification, however, wehave a completely general way of getting rid of SP. By choosing a unique and naturalinitial density matrix, we no longer have an infinity of possible initial microstates.There is just one state to choose from and we no longer need any probability distri-bution over possible initial microstates. The Humean unification provides a simpleand general way to avoid the dualism of probabilities. This is achieved by makingSP simply unnecessary. In contrast to the quantum Mentaculus (and the classical Mentaculus), Humeanunification contains less vagueness. The Past Hypothesis and the Statistical Pos-tulate are exact only when we choose some arbitrary coarse-graining variables innature. On the Mentaculus, which exact set of microstates counts as the low-entropysubspace, after a certain level of precision, will be entirely arbitrary. There is noth-ing in nature that pins it down. Beyond a level of precision, the exact boundaryof the subspace makes no di ff erence to how things are behaving in physical spaceand what their probabilities are. The same is true for the Statistical Postulate: the For W
IPH -Everettian theories, since the Born rule is not supposed to be fundamental, the elim-ination of SP means that there are no objective probabilities in the world. The actual world isnomologically necessary. Hence, every physical fact would follow from the law statements. The situation is di ff erent under the Wen-taculus picture. The exact values of the initial density matrix makes a di ff erence tothe world just as importantly as constants of nature—di ff erent values will typicallylead to di ff erent microscopic behaviors and probabilities. The exact values of theinitial quantum state is by no means arbitrary— ˆ W IPH ( t ) plays a central role in thefundamental micro-dynamics; it makes a di ff erence to the exact Bohmian velocityfield, GRW collapse probabilities, and configuration of matter densities, etc. That iswhy we can eliminate the vagueness of IPH without objectionable arbitrariness.Another advantage of Humean unification is that it leads to more unity in thebest system. First, it provides mathematical unity between quantum mechanicsand quantum statistical mechanics. Quantum statistical mechanics makes extensiveuse of density matrices. Quantum mechanics, on the other hand, has often beenformulated in terms of wave functions. From the perspective of Humean unification,density matrix is the central object in both theories. A wave function only arisesin special circumstances when the density matrix is pure. Second, there is anincrease in the dynamical unity in some theories. In BM with spin, there doesnot exist a conditional wave function since the particles have only positions butno spin degrees of freedom. Hence, in general, for the subsystems in a W-BMuniverse, there is only a conditional density matrix instead of a conditional wavefunction. As a result, the guidance equation for many subsystems (that are suitablyisolated from the environment) will be the W-guidance equation (5) that refers toa conditional density matrix even in standard BM, while the guidance equationfor the whole universe will be the usual guidance equation that refers to a wavefunction. Therefore, the dynamics for the universe will be importantly di ff erentfrom the dynamics for the subsystems for standard BM. In contrast, in W IPH -BM,the guidance equation for the whole universe and that for the subsystems (that aresuitably isolated from the environment) will be the same—(5). Whether dynamicalunity holds for W
IPH -GRWm and W
IPH -Sm will require a further analysis of thesubsystem dynamics in those theories. But in any case, those theories also witnessan increase in kinematic unity: both the universe and typical subsystems will be inmixed states described by density matrices. This is in contrast to the Mentaculuspicture, where most subsystems are in mixed states (due to entanglement) while theuniverse is in a pure state. I discuss the notion of “nomic vagueness” in more details in Chen (2020). See Dürr et al. (2005) for more details. The standard guidance equation in BM under a universalwave function is: dQ i dt = ̵ hm i Im ψ ∗ ∇ i ψψ ∗ ψ (12) Comparisons
In this section, we discuss two other versions of quantum Humeanism that aremotivated by the problem of quantum entanglement but not the problem of timeasymmetry. We compare and contrast them with the Humean unification we intro-duced in this paper.
One of the earliest proposals of reconciling Humeanism with quantum entanglementis that of Loewer (1996). Loewer argues that if we adopt David Albert’s high-dimensional space fundamentalism, the idea that the fundamental space of theworld is the “configuration space,” then the quantum state (represented by a wavefunction) is entirely separable in that fundamental space. We can reify that as theHumean mosaic. Entanglement and non-locality are merely manifestations of ourperception in the low-dimensional space, which is not fundamental.The move from a low-dimensional space to a high-dimensional space is a rad-ical move. It is revisionary from the Humean perspective. On Lewis’s originalformulation, the Humean mosaic are facts about the physical space-time (or somelow-dimensional manifold). But it is also revisionary from the ordinary scien-tific perspective. There are important reasons to take something like the physi-cal space-time to be fundamental, as it underlies many important symmetries inphysics, including Lorentz invariance. They will be di ffi cult to recover from thehigh-dimensional point of view. In contrast, the Humean Unification does notrequire such radical revisions and even o ff ers additional theoretical benefits. The second class of proposals of reconciling Humeanism with quantum entangle-ment is inspired by Hall (2015), and discussed in Miller (2013), Esfeld (2014), Bhogaland Perry (2015), Callender (2015), Esfeld and Deckert (2017). On this proposal, thequantum state of the universe represented by a wave function is part of the Humeanbest system. There are three ways to interpret this proposal.First interpretation: the wave function itself represents a simple and informativeHumean law of nature (like the classical Hamiltonian). Miller and Callender areclose to endorse this view. However, it requires that the universal wave function tobe extremely simple, or at least simpler than the complete facts about local beablesthrough all time. Otherwise the system containing the wave function would notwin the competition for being the best system. It faces the prima facie problem thatthe universal wave function may not be simple enough to be nomological. See Allori (2013) for related reasons. Chen (2017) argues that the low-dimensional view providesbetter explanation of the Symmetrization Postulate than the high-dimensional view does. Emery(2017) provides reasons to think that other things being equal we should prefer the more common-sense view of physical reality. aspires to be realist. Humeanism requires delicate balances betweenobjectivism and pragmatism, but at least the original Humean proposal has princi-pled constraints on what goes into the best system and what goes into the mosaic,and such constraints are generally compatible with scientific practice.In short, there are prima facie problems facing these versions of wave functionHumeanism. In contrast, the Humean Unification avoids these problems, as weknow that the initial quantum state is simple and unique, and the proposal is fullyrealist. Moreover, the Humean Unification has many other theoretical virtues listedin §5.We do not think of our view as completely opposed to the views discussed above.In fact, there is much common ground; our proposal can be seen as friendly extensionto some of the views above. For example, towards the end of her important essay,Miller (2013) is rightly worried about the potential slippery slope in the strategy she31roposes, and she suggests we find principled limits to curb over-Humeanizationor the“narcissism” if one tries to “Humeanize” away everything one does not like inthe ontology, including all particles outside one’s brain. Our framework can be seenas implicitly suggesting such a limit: carry out Humean unification only when youcan show that the best system will not be over-burdened by extra complexity and thatthere will be additional fruits of unification beyond solving the original problems.Moreover, our framework can be combined with Bhogal and Perry (2015)’s proposalto simplify the supervenient L-state, and with Esfeld and Deckert (2017)’s proposalto further simplify their “super-Humean” best-system. We leave that to future work.
It has been argued that the origin of time asymmetry in our universe lies in itsboundary condition—a low-entropy state now called the
Past Hypothesis . In thispaper, we have used the Past Hypothesis to construct a new class of quantumtheories—W
IPH -theories. We then showed that they allow the Humeans to usethe best system to specify a simple and unique initial quantum state. This ledto the Humean Unification that combined the origins of quantum entanglementand time asymmetry. The data in the Initial Projection Hypothesis, with the helpof the density-matrix dynamics, gives rise to both time asymmetry and quantumphenomena. The result is a new theory with a separable and narratable mosaic aswell as a simpler, less vague, and more unified law system.Can a non-Humean appreciate our new theory? We think so. The Humeanstrategy for regarding the low-entropy initial condition (PH) as nomological doesnot sit well with a governing or dynamical conception of laws (although somenon-Humeans would disagree). However, non-Humeans may be less opposed toour theory. The low-entropy initial condition plays a dynamical role via the initialprojection density matrix and its dynamics. The initial density matrix directlydetermines the Bohmian velocity field, the GRW collapse probabilities, and theEverettian branching structure. Hence, it plays an analogous role as the classicalHamiltonian function, which can be given a non-Humean interpretation. Is our new theory the best theory? Given its empirical equivalence to manyother theories including the quantum Mentaculus, we have to appeal to super-empirical virtues to settle the question. It is unlikely, however, we will ever beable to conclusively settle it. What we can do, at this stage, is to build and refine,to the best of our abilities, di ff erent models, theories, and frameworks. Only afterthat can we meaningfully compare them side by side as complete packages. In themeantime, however, we hope the Humean unification outlined above will provokenew ideas in solving the two problems that we began with. Can a quantum state monist appreciate our new theory? We think so. She can at least appreciatethe W
IPH -quantum theories without local beables: W
IPH -GRW and W
IPH -EQM. The crucial step inthe Humean Unification—§4—would not be possible. The quantum state will have to be in themosaic. Such theories still retain some novel virtues: they make statistical mechanical probabilitiesunnecessary and they are more unified and less vague than their wave-function counterparts. cknowledgement I am grateful for helpful discussions about related ideas with David Albert, CraigCallender, Jonathan Cohen, Michael Esfeld, Sheldon Goldstein, Veronica Gomez,Matthias Lienert, Barry Loewer, Tim Maudlin, Elizabeth Miller, Jill North, EzraRubenstein, Jonathan Scha ff er, Ted Sider, Roderich Tumulka, and David Wallace. References
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