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Guido Bacciagaluppi

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The Role of Decoherence in QuantumMechanicsFirst published Mon Nov 3, 2003; substantive revision Mon Apr 16, 2012

Interference phenomena are a well-known and crucial aspect of quantummechanics, famously exemplified by the two-slit experiment. There aresituations, however, in which interference effects are artificially orspontaneously suppressed. The theory of decoherence is precisely the

study of (spontaneous) interactions between a system and its environmentthat lead to such suppression of interference. We shall make more precisewhat we mean by this in Section 1, which discusses the concept of suppression of interference and gives a simplified survey of the theory,emphasising features that will be relevant to the following discussion. Infact, the term decoherence refers to two largely overlapping areas of research. The characteristic feature of the first (often called ‘dynamical’

or ‘environmental’ decoherence) is the study of concrete models of (spontaneous) interactions between a system and its environment that leadto suppression of interference effects. That of the second (the theory of ‘decoherent histories’ or ‘consistent histories’) is an abstract (and in factmore general) formalism that captures the essential features of thephenomenon of decoherence. The two are obviously closely related, andwill both be reviewed in turn in Section 1.

Decoherence is relevant (or is claimed to be relevant) to a variety of questions ranging from the measurement problem to the arrow of time,and in particular to the question of whether and how the ‘classical world’may emerge from quantum mechanics. This entry mainly deals with therole of decoherence in relation to the main problems and approaches inthe foundations of quantum mechanics. Specifically, Section 2 analysesthe claim that decoherence solves the measurement problem. It also

discusses the exacerbation of the problem through the inclusion of

1



environmental interactions, the idea of emergence of classicality, and themotivation for discussing decoherence together with approaches to the

foundations of quantum mechanics. Section 3 then reviews the relation of decoherence to some of the main foundational approaches. Finally, inSection 4 we mention suggested applications that would push the role of decoherence even further.

Suppression of interference has of course featured in many papers sincethe beginning of quantum mechanics, such as Mott's (1929) analysis of alpha-particle tracks. The modern foundation of decoherence as a subjectin its own right was laid by H. D. Zeh in the early 1970s (Zeh 1970;1973). Equally influential were the papers by W. Zurek from the early1980s (Zurek 1981; 1982). Some of these earlier examples of decoherence (e.g., suppression of interference between left-handed andright-handed states of a molecule) are mathematically more accessiblethan more recent ones. A concise and readable introduction to the theoryis provided by Zurek in Physics Today (1991). (This article was followedby publication of several letters with Zurek's replies (1993), whichhighlight controversial issues.) More recent surveys are the ones by Zeh(1995), which devotes much space to the interpretation of decoherence,Zurek (2003), and the books on decoherence by Giulini et al. (1996) andSchlosshauer (2007). [1]

1. Essentials of Decoherence

1.1 Dynamical decoherence1.2 Decoherent histories

2. Conceptual Appraisal2.1 Solving the measurement problem?2.2 Exacerbating the measurement problem2.3 Emergence of classicality

3. Decoherence and Approaches to Quantum Mechanics3.1 Collapse approaches

The Role of Decoherence in Quantum Mechanics

2 Stanford Encyclopedia of Philosophy

3.2 Pilot-wave theories3.3 Everett interpretations

3.4 Modal interpretations3.5 Bohr's Copenhagen interpretation

4. Scope of DecoherenceBibliographyAcademic ToolsOther Internet ResourcesRelated Entries

1. Essentials of Decoherence

The two-slit experiment is a paradigm example of an interferenceexperiment. One repeatedly sends electrons or other particles through ascreen with two narrow slits, the electrons impinge upon a second screen,and we ask for the probability distribution of detections over the surface

of the screen. In order to calculate this, one cannot just take theprobabilities of passage through the slits, multiply with the probabilitiesof detection at the screen conditional on passage through either slit, andsum over the contributions of the two slits. [2] There is an additional so-called interference term in the correct expression for the probability, andthis term depends on both wave components that pass through one or theother slit.

There are, however, situations in which this interference term (fordetections at the screen) is not observed, i.e. in which the classicalprobability formula applies. This happens for instance when we perform adetection at the slits, whether or not we believe that measurements arerelated to a ‘true’ collapse of the wave function (i.e. that only one of thecomponents survives the measurement and proceeds to hit the screen).The disappearence of the interference term, however, can happen also

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spontaneously, when no collapse (true or otherwise) is presumed tohappen. Namely, if some other systems (say, sufficiently many stray

cosmic particles scattering off the electron) suitably interact with thewave between the slits and the screen. In this case, the reason why theinterference term is not observed is because the electron has becomeentangled with the stray particles. [3] The phase relation between the twocomponents of the wave function, which is responsible for interference, iswell-defined only at the level of the larger system composed of electronand stray particles, and can produce interference only in a suitable

experiment including the larger system. Probabilities for results of measurements performed only on the electron are calculated as if thewave function had collapsed to one or the other of its two components,but in fact the phase relations have merely been distributed over a largersystem. [4] It is this phenomenon of suppression of interference throughsuitable interaction with the environment that we call ‘dynamical’ or‘environmental’ decoherence.

1.1 Dynamical decoherence

The study of ‘dynamical’ decoherence consists to a large extent in theexploration of concrete spontaneous interactions that lead to suppressionof interference. Several features of interest arise in models of suchinteractions (although by no means are all such features common to allmodels).

One feature of these environmental interactions is that they suppressinterference between states from some preferred set, be it a discrete set of states (e.g. left- and right-handed states in models of chiral molecules, orthe upper and lower component of the wave function in our simpleexample of the two-slit experiment), or some continuous set (e.g. thecoherent states of a harmonic oscillator). The intuitive picture is one inwhich the environment monitors the system of interest by continuously



‘measuring’ some quantity characterised by the set of preferred states(‘eigenstates of the decohering variable’). Formally, this is reflected in the

(at least approximate) diagonalisation of the reduced state of the systemof interest in the basis of privileged states (whether discrete orcontinuous).

These preferred states can be characterised in terms of their robustness orstability with respect to the interaction with the environment. Roughlyspeaking, the system gets entangled with the environment, but the statesbetween which interference is suppressed are the ones that wouldthemselves get least entangled with the environment under furtherinteraction. The robustness of the preferred states is related to the fact thatinformation about them is stored in a redundant way in the environment(say, because a Schrödinger cat has interacted with so many strayparticles: photons, air molecules, dust). This information can later beacquired by an observer without further disturbing the system (we observe—however that may be interpreted—whether the cat is alive or dead by

intercepting on our retina a small fraction of the light that has interactedwith the cat).

In this connection, one also says that decoherence induces ‘effectivesuperselection rules’. The concept of a (strict) superselection rule meansthat there are some observables—called classical in technical terminology—that commute with all observables (for a review, see Wightman(1995)). Intuitively, these observables are infinitely robust, since nopossible interaction can disturb them (at least as long as the interactionHamiltonian is considered to be an observable). By an effectivesuperselection rule one means, analogously, that certain observables (e.g.chirality) will not be disturbed by the interactions that actually takeplace. [5]

Interaction potentials are functions of position, so the preferred states will

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tend to be related to position. In the case of the chiral molecule, the left-and right-handed states are indeed characterised by different spatial

configurations of the atoms in the molecule. In the case of the harmonicoscillator, one should think of the environment coupling to (‘measuring’)approximate eigenstates of position, or rather approximate jointeigenstates of position and momentum (since information about the timeof flight is also recorded in the environment), thus leading to coherentstates being preferred. (Rough intuitions should suffice here; see also theentries on quantum mechanics and measurement in quantum theory.)

The resulting localisation can be on a very short length scale, i.e. thecharacteristic length above which coherence is dispersed (‘coherencelength’) can be very short. A speck of dust of radius a = 10 -5cm floatingin the air will have interference suppressed between (position)components with a width of 10 -13 cm. Even more strikingly, the timescales for this process are minute. This coherence length is reached after amicrosecond of exposure to air, and suppression of interference on a

length scale of 10 -12 cm is achieved already after a nanosecond. [6]

One can thus argue that generically the states privileged by decoherenceat the level of components of the quantum state are localised in positionor both position and momentum, and therefore kinematically classical.(One should be wary of overgeneralisations, as already pointed out, butthis is certainly a feature of many concrete examples that have beeninvestigated.)

What about classical dynamical behaviour? Interference is a dynamicalprocess that is distinctively quantum, so, intuitively, lack of interferencemight be thought of as classical-like. To make the intuition more precise,think of the two components of the wave going through the slits. If thereis an interference term in the probability for detection at the screen, itmust be the case that both components are indeed contributing to the



particle manifesting itself on the screen. But if the interference term issuppressed, one can at least formally imagine that each detection at the

screen is a manifestation of only one of the two components of the wavefunction, either the one that went through the upper slit, or the one thatwent through the lower slit. Thus, there is a sense in which one canrecover at least one dynamical aspect of a classical description, atrajectory of sorts: from the source to either slit (with a certainprobability), and from the slit to the screen (also with a certainprobability). That is, one recovers a ‘classical trajectory’ at least in the

sense used in classical stochastic processes.

In the case of continuous models of decoherence based on the analogy of approximate joint measurements of position and momentum, one can doeven better. In this case, the trajectories at the level of the components(the trajectories of the preferred states) will approximate surprisingly wellthe corresponding classical (Newtonian) trajectories. Intuitively, one canexplain this by noting that if the preferred states (which are wave packets

that are narrow in position and remain so because they are also narrow inmomentum) are the states that tend to get least entangled with theenvironment, they will tend to follow the Schrödinger equation more orless undisturbed. But in fact, narrow wave packets follow approximatelyNewtonian trajectories, at least if the external potentials in which theymove are uniform enough along the width of the packets (results of thiskind are known as ‘Ehrenfest theorems’). Thus, the resulting ‘histories’

will be close to Newtonian ones (on the relevant scales).[7]

The most intuitive physical example for this are the observed trajectoriesof alpha particles in a bubble chamber, which are indeed extremely closeto Newtonian ones, except for additional tiny ‘kinks’. As a matter of fact,one should expect slight deviations from Newtonian behaviour. These aredue both to the tendency of the individual components to spread and tothe detection-like nature of the interaction with the environment, which

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further enhances the collective spreading of the components (a narrowingin position corresponds to a widening in momentum). These deviations

appear as noise, i.e. particles being kicked slightly off course.[8]

According to the type of system, and the details of the interaction, thenoise component might actually dominate the motion, and one obtains(classical) Brownian-motion-type behaviour.

Other examples include trajectories of a harmonic oscillator inequilibrium with a thermal bath, and trajectories of particles in a gas(without which the classical derivation of thermodynamics from statisticalmechanics would make no sense; see below Section 4).

None of these features are claimed to obtain in all cases of interactionwith some environment. It is a matter of detailed physical investigation toassess which systems exhibit which features, and how general the lessonsare that we might learn from studying specific models. In particular, oneshould beware of common overgeneralisations. For instance, decoherence

does not affect only and all ‘macroscopic systems’. True, middle-sizedobjects, say, on the Earth's surface will be very effectively decohered bythe air in the atmosphere, and this is an excellent example of decoherenceat work. On the other hand, there are also very good examples of decoherence-like interactions affecting microscopic systems, such as inthe interaction of alpha particles with the gas in a bubble chamber. Andfurther, there are arguably macroscopic systems for which interferenceeffects are not suppressed. For instance, it has been shown to be possibleto sufficiently shield SQUIDS (a type of superconducting devices) fromdecoherence for the purpose of observing superpositions of differentmacroscopic currents—contrary to what one had expected (see e.g.Leggett 1984, and esp. 2002, Section 5.4). Anglin, Paz and Zurek (1997)examine some less well-behaved models of decoherence and provide auseful corrective as to the limits of decoherence.



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1.2 Decoherent histories

As we have just discussed, when interference is suppressed, e.g. in a two-slit experiment, we can also speak (at least formally) about the‘trajectory’ followed by an individual electron. In particular, we canassign probabilities to the alternative trajectories, so that probabilities fordetection at the screen can be calculated by summing over intermediateevents. The decoherent histories formalism (originating with Griffiths1984; Omnès 1988, 1989; and Gell-Mann and Hartle 1990) takes this asthe defining feature of decoherence.

In a nutshell, the formalism is as follows. [9] Take orthogonal families of projections with

! " 1 P" 1

= 1 ,… , ! " n P" n = 1

Given times t 1,… , t n one defines histories as time-ordered sequences of

projections at the given times, choosing one projection from each family,respectively. Such histories form a so-called alternative and exhaustive setof histories.

Take a state #(t ). We wish to define probabilities for the set of histories.If one takes the usual probability formula based on repeated application of the Born rule, one obtains

Tr( P" nU t nt n-1… P" 1U t 1t 0 #(t 0) U *t 1t 0P" 1… U

*t nt n-1P" n)

(where U ts represents the unitary evolution operator from time s to time t ,and its adjoint U *ts the inverse evolution).

We shall take (2) as defining ‘candidate probabilities’. In general theseprobabilities exhibit interference, in the sense that if one sums overintermediate events (if one ‘coarse-grains’ the histories), one does not

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(3)

(4)

obtain probabilities of the same form (2). But we can impose, as aconsistency or (weak) decoherence condition, precisely that interference

terms should vanish for any pair of distinct histories. It is easy to see thatthis condition takes the form

ReTr( P" !

nU t nt n-1… P" !

1U t 1t 0 #(t 0) U *t 1t 0P" 1

… U *t nt n-1P" n) = 0

for any pair of distinct histories. If this is satisfied, we can view (2) asdefining the distribution functions for a stochastic process with thehistories as trajectories. (There are some differences between the variousauthors, but we shall gloss them over.)

Decoherence in the sense of this abstract formalism is thus defined simplyby the condition that (quantum) probabilities for wave components at alater time may be calculated from (quantum) probabilities for wavecomponents at an earlier time and (quantum) conditional probabilitiesaccording to the standard classical formula, i.e. as if the wave had

collapsed.

Models of dynamical decoherence fall under the scope of decoherencethus defined, but the abstract definition is much more general. A strongerform of the decoherence condition, namely the vanishing of both the realand imaginary part of the trace expression in (3) (the ‘decoherencefunctional’), can be used to prove theorems on the existence of (later)‘permanent records’ of (earlier) events in a history, which is ageneralisation of the idea of ‘environmental monitoring’. For instance, if the state # is a pure state | $ ><$ | this strong decoherence condition isequivalent, for all n, to the orthogonality of the vectors

P" nU t nt n-1… P" 1

U t 1t 0 |$ >

and this in turn is equivalent to the existence of a set of orthogonal



projections R" 1..." it i (for any t i%t n) that extend consistently the given set

of histories and are perfectly correlated with the histories of the original

set (Gell-Mann and Hartle 1990). Note, however, that these ‘generalisedrecords’ need not be stored in separate degrees of freedom, such as anenvironment or measuring apparatus. [10]

Various authors have taken the theory of decoherent histories as providingan interpretation of quantum mechanics. For instance, Gell-Mann andHartle sometimes talk of decoherent histories as a neo-Everettianapproach, while Omnès appears to think of histories along neo-Copenhagen lines (perhaps as an experimental context creating a‘quantum phenomenon‘ that can stretch back into the past). [11] Griffiths(2002) has probably developed the most detailed of these interpretationalapproaches (trying to do justice to various earlier criticisms, e.g. byDowker and Kent (1995, 1996)). [12]

In itself, however, the formalism is interpretationally neutral and has the

particular merit of bringing out two crucial conceptual points: that wavecomponents can be reidentified over time, and that if we do so, we canformally identify ‘trajectories’ for the system. As such, it is particularlyuseful as a tool for describing decoherence in connection with attempts tosolve the problem of the classical regime in the context of variousdifferent interpretational approaches to quantum mechanics. In particular,it has become a standard tool in discussions of Everett interpretations,

where ‘worlds’ can be formally described as histories in a consistentfamily (see, e.g., Saunders 1993).

2. Conceptual Appraisal

2.1 Solving the measurement problem?

The fact that interference is typically very well suppressed between

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localised states of macroscopic objects suggests that it is relevant to whymacroscopic objects in fact appear to us to be in localised states. A

stronger claim is that decoherence is not only relevant to this question butby itself already provides the complete answer. In the special case of measuring apparatuses, it would explain why we never observe anapparatus pointing, say, to two different results, i.e. decoherence wouldprovide a solution to the measurement problem of quantum mechanics.As pointed out by many authors, however (e.g. Adler 2003; Zeh 1995, pp.14–15), this claim is not tenable.

The measurement problem, in a nutshell, runs as follows. Quantummechanical systems are described by wave-like mathematical objects(vectors) of which sums (superpositions) can be formed (see the entry onquantum mechanics). Time evolution (the Schrödinger equation)preserves such sums. Thus, if a quantum mechanical system (say, anelectron) is described by a superposition of two given states, say, spin in x-direction equal +1/2 and spin in x-direction equal -1/2, and we let it

interact with a measuring apparatus that couples to these states, the finalquantum state of the composite will be a sum of two components, one inwhich the apparatus has coupled to (has registered) x-spin = +1/2, and onein which the apparatus has coupled to (has registered) x-spin = -1/2. Theproblem is that, while we may accept the idea of microscopic systemsbeing described by such sums, the meaning of such a sum for the(composite of electron and) apparatus is not immediately obvious.

Now, what happens if we include decoherence in the description?Decoherence tells us, among other things, that plenty of interactions aretaking place all the time in which differently localised states of macroscopic systems couple to different states of their environment. Inparticular, the differently localised states of the macroscopic system couldbe the states of the pointer of the apparatus registering the different x-spinvalues of the electron. By the same argument as above, the composite of



electron, apparatus and environment will be a sum of (i) a statecorresponding to the environment coupling to the apparatus coupling in

turn to the value +1/2 for the spin, and of (ii) a state corresponding to theenvironment coupling to the apparatus coupling in turn to the value -1/2for the spin. Again, the meaning of such a sum for the composite systemis not obvious.

We are left with the following choice whether or not we includedecoherence: either the composite system is not described by such a sum,because the Schrödinger equation actually breaks down and needs to bemodified, or it is described by such a sum, but then we need to understandwhat that means, and this requires giving an appropriate interpretation of quantum mechanics. Thus, decoherence as such does not provide asolution to the measurement problem, at least not unless it is combinedwith an appropriate interpretation of the theory (whether this be one thatattempts to solve the measurement problem, such as Bohm, Everett orGRW; or one that attempts to dissolve it, such as various versions of the

Copenhagen interpretation). Some of the main workers in the field such asZeh (2000) and (perhaps) Zurek (1998) suggest that decoherence is mostnaturally understood in terms of Everett-like interpretations (see belowSection 3.3, and the entries on Everett's relative-state interpretation and onthe many-worlds interpretation).

Unfortunately, naive claims of the kind that decoherence gives a completeanswer to the measurement problem are still somewhat part of the‘folklore’ of decoherence, and deservedly attract the wrath of physicists(e.g. Pearle 1997) and philosophers (e.g. Bub 1997, Chap. 8) alike. (To befair, this ‘folk’ position has at least the merit of attempting to subjectmeasurement interactions to further physical analysis, without assumingthat measurements are a fundamental building block of the theory.)

2.2 Exacerbating the measurement problem

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Decoherence is clearly neither a dynamical evolution contradicting theSchrödinger equation, nor a new interpretation of the theory. As we shall

discuss, however, it both reveals important dynamical effects within theSchrödinger evolution, and may be suggestive of possible interpretationsof the theory.

As such it has much to offer to the philosophy of quantum mechanics. Atfirst, however, it seems that discussion of environmental interactionsshould actually exacerbate the existing problems. Intuitively, if theenvironment is carrying out, without our intervention, lots of approximateposition measurements, then the measurement problem ought to applymore widely, also to these spontaneously occurring measurements.

Indeed, while it is well-known that localised states of macroscopicobjects spread very slowly with time under the free Schrödinger evolution(i.e., if there are no interactions), the situation turns out to be different if they are in interaction with the environment. Although the different

components that couple to the environment will be individually incrediblylocalised, collectively they can have a spread that is many orders of magnitude larger. That is, the state of the object and the environmentcould be a superposition of zillions of very well localised terms, each withslightly different positions, and that are collectively spread over amacroscopic distance, even in the case of everyday objects. [13]

Given that everyday macroscopic objects are particularly subject todecoherence interactions, this raises the question of whether quantummechanics can account for the appearance of the everyday world evenapart from the measurement problem in the strict sense. To put it crudely:if everything is in interaction with everything else, everything isgenerically entangled with everything else, and that is a worse problemthan measuring apparatuses being entangled with the measured systems.And indeed, discussing the measurement problem without taking



decoherence (fully) into account may not be enough, as we shall illustrateby the case of some versions of the modal interpretation in Section 3.4.

2.3 Emergence of classicality

What suggests that decoherence may be relevant to the issue of theclassical appearance of the everyday world is that at the level of components of the wave function the quantum description of decoherencephenomena can display tantalisingly classical aspects. The question isthen whether, if viewed in the context of any of the main foundational

approaches to quantum mechanics, these classical aspects can be taken toexplain corresponding classical aspects of the phenomena. The answer,perhaps unsurprisingly, turns out to depend on the chosen approach, andin the next section we shall discuss in turn the relation betweendecoherence and several of the main approaches to the foundations of quantum mechanics.

Even more generally, one can ask whether the results of decoherencecould thus be used to explain the emergence of the entire classicality of the everyday world , i.e. to explain both kinematical features such asmacroscopic localisation and dynamical features such as approximatelyNewtonian or Brownian trajectories in all cases where such descriptionshappen to be phenomenologically adequate. As we have mentionedalready, there are cases in which a classical description is not a gooddescription of a phenomenon, even if the phenomenon involvesmacroscopic systems. There are also cases, notably quantummeasurements, in which the classical aspects of the everyday world areonly kinematical (definiteness of pointer readings), while the dynamics ishighly non-classical (indeterministic response of the apparatus). In asense, if we follow Bohr in requiring the world of classical concepts inorder to describe in the first place ‘quantum phenomena’ (see the entry onthe Copenhagen interpretation), then, if decoherence gives us indeed the

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everyday classical world, the quantum phenomena themselves wouldbecome a consequence of decoherence (Zeh 1995, p. 33; see also

Bacciagaluppi 2002, Section 6.2). The question of explaining theclassicality of the everyday world becomes the question of whether onecan derive from within quantum mechanics the conditions necessary todiscover and practise quantum mechanics itself, and thus, in Shimony's(1989) words, close the epistemological circle.

In this generality the question is clearly too hard to answer, depending asit does on how far the physical programme of decoherence (Zeh 1995, p.9) can be successfully developed. We shall thus postpone the (partlyspeculative) discussion of how far this programme might go until Section4.

3. Decoherence and Approaches to QuantumMechanics

There is a wide range of approaches to the foundations of quantummechanics. The term ‘approach’ here is more appropriate than the term‘interpretation’, because several of these approaches are in factmodifications of the theory, or at least introduce some prominent newtheoretical aspects. A convenient way of classifying these approaches isin terms of their strategies for dealing with the measurement problem.

Some approaches, so-called collapse approaches, seek to modify theSchrödinger equation, so that superpositions of different ‘everyday’ statesdo not arise or are very unstable. Such approaches may have intuitivelylittle to do with decoherence since they seek to suppress precisely thosesuperpositions that are created by decoherence. Nevertheless their relationto decoherence is interesting. Among collapse approaches (Section 3.1),we shall discuss von Neumann's collapse postulate and theories of spontaneous localisation (for which see also the entry on collapse



theories).

Other approaches, known as ‘hidden variables’ approaches, seek toexplain quantum phenomena as equilibrium statistical effects arising froma deeper-level theory, rather strongly in analogy with attempts atunderstanding thermodynamics in terms of statistical mechanics (see theentry on philosophy of statistical mechanics). Of these, the mostdeveloped are the so-called pilot-wave theories (Section 3.2), in particularthe theory by de Broglie and Bohm (see also the entry on Bohmianmechanics).

Finally, there are approaches that seek to solve (or dissolve) themeasurement problem strictly by providing an appropriate interpretationof the theory. Slightly tongue in cheek, one can group together under thisheading approaches as diverse as Everett interpretations (see the entrieson Everett's relative-state interpretation and on the many-worldsinterpretation), modal interpretations and the Copenhagen interpretation.

We shall be analysing these approaches specifically in their relation todecoherence (we discuss the Everett interpretation in Section 3.3, themodal interpretations in Section 3.4, and the Copenhagen interpretation inSection 3.5).

3.1 Collapse approaches

3.1.1 Von Neumann

It is notorious that von Neumann (1932) proposed that the observer'sconsciousness is somehow related to what he called Process I, otherwiseknown as the collapse postulate or the projection postulate, which in hisbook is treated on a par with the Schrödinger equation (his Process II).There is some ambiguity in how to interpret von Neumann. He may havebeen advocating some sort of special access to our own consciousnessthat makes it appear to us that the wave function has collapsed; this would

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suggest a phenomenological reading of Process I. Alternatively, he mayhave proposed that consciousness plays some causal role in precipitating

the collapse; this would suggest that Process I is a physical process takingplace in the world on a par with Process II. [14]

In either case, von Neumann's interpretation relies on the insensitivity of the final predictions (for what we consciously record) to exactly whereand when Process I is used in modelling the evolution of the quantumsystem. This is often referred to as the movability of the von Neumann cut between the subject and the object, or some similar phrase. Collapsecould occur anywhere along the so-called von Neumann chain: when aparticle impinges on a screen, or when the screen blackens, or when anautomatic printout of the result is made, or in our retina, or along theoptic nerve, or when ultimately consciousness is involved. Von Neumannthus needs to show that all of these models are equivalent, as far as thefinal predictions are concerned, so that he can indeed maintain thatcollapse is related to consciousness, while in practice applying the

projection postulate at a much earlier (and more practical) stage in thedescription.

Von Neumann poses this problem in Section VI.1 of his book. In SectionVI.2, by way of preparation, he discusses the relation between states of systems and subsystems, in particular the partial trace, and thebiorthogonal decomposition theorem, i.e. the theorem stating that anentangled quantum state can always be written in the special form

! k ck &k ' k

for two suitable bases (note the perfect correlations in (5)). Then inSection VI.3, after discussing his insolubility argument (see againfootnote 14), von Neumann shows that there always is a Hamiltonian thatwill lead from a state of the form ! k ck &k ' 0 to a state of the form (5).



This concludes von Neumann's argument.

What von Neumann has shown is that, under suitable modelling of themeasurement interaction, applying the collapse postulate directly to themeasured observable or applying it to the pointer observable of theapparatus (or by extension to the ‘optic nerve signal observable’, etc.)leads to the same statistics of results.

What he has not shown is that the assumption that the collapse occurs atthe level of consciousness is equivalent to the assumption that it happens

at any other earlier stage if one considers also other possiblemeasurements that could be carried out along the von Neumann chain.Indeed, if collapse occurs only at the level of consciousness, it is inprinciple possible, instead of looking at the pointer, to perform a differentmeasurement on the composite of system and apparatus that would detectinterference between the different components of (5).

This is now precisely where decoherence plays a role. Indeed, while suchmeasurements are possible in principle, decoherence will make themimpossible to perform in practice. Therefore, if we assume that Process Iis a real physical process, decoherence makes it in practice impossible todetect where along the measurement chain this process takes place, thusallowing von Neumann to postulate that it happens when consciousnessgets involved. This aspect will be relevant also in the next subsection.

3.1.2 Spontaneous collapse theories

The best known theory of spontaneous collapse is the so-called GRWtheory (Ghirardi Rimini & Weber 1986), in which a material particlespontaneously undergoes localisation in the sense that at random times itexperiences a collapse of the form used to describe approximate positionmeasurements. [15] In the original model, the collapse occursindependently for each particle (a large number of particles thus

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‘triggering’ collapse much more frequently); in later models thefrequency for each particle is weighted by its mass, and the overall

frequency for collapse is thus tied to mass density.[16]

Thus, formally, the effect of spontaneous collapse is the same as in someof the models of decoherence, at least for one particle. [17] Two crucialdifferences on the other hand are that we have ‘true’ collapse instead of suppression of interference (cf. above Section 1), and that spontaneouscollapse occurs without there being any interaction between the systemand anything else, while in the case of decoherence suppression of interference generally arises through interaction with the environment.

Can decoherence be put to use in GRW? The situation may be rathercomplex when the decoherence interaction does not approximatelyprivilege position (e.g. when it selects for currents in a SQUID instead),because collapse and decoherence might actually ‘pull’ in differentdirections. [18] But in those cases in which the decoherence interaction

also takes the form of approximate position measurements, the answerpresumably boils down to a quantitative comparison. If collapse happensfaster than decoherence, then the superposition of components relevant todecoherence will not have time to arise, and insofar as the collapse theoryis successful in recovering classical phenomena, decoherence plays norole in this recovery. Instead, if decoherence takes place faster thancollapse, then (as in von Neumann's case) the collapse mechanism canfind ‘ready-made’ structures onto which to truly collapse the wavefunction. Simple comparison of the relevant rates in models of decoherence and in spontaneous collapse theories (Tegmark 1993, esp.Table 2) suggests that this is generally the case. Thus, it seems thatdecoherence should play a role also in spontaneous collapse theories.

A further aspect of the relation between decoherence and spontaneouscollapse theories relates to the experimental testability of spontaneous



collapse theories. Exactly as we have just discussed in the previoussubsection in the context of von Neumann's Process I, if we assume that

collapse is a real physical process, decoherence will make it extremelydifficult in practice to detect empirically when and where exactlyspontaneous collapse takes place (see the nice discussion of this point inChapter 5 of Albert (1992)).

Even worse, at least with the proviso that decoherence may be put to usealso in no-collapse approaches such as pilot-wave or Everett (possibilitiesthat we discuss in the next sub-sections), then in all cases in whichdecoherence is faster than collapse, what might be interpreted as evidencefor collapse could be reinterpreted as ‘mere’ suppression of interference(for instance in the case of measurements), and only those cases in whichthe collapse theory predicts collapse but the system is shielded fromdecoherence (or perhaps in which the two pull in different directions)could be used to test collapse theories experimentally.

One particularly bad scenario for experimental testability is related to thespeculation (in the context of the ‘mass density’ version) that the cause of spontaneous collapse may be connected with gravitation. Tegmark 1993(Table 2) quotes some admittedly uncertain estimates for the suppressionof interference due to a putative quantum gravity, but they arequantitatively very close to the rate of destruction of interference due tothe GRW collapse (at least outside of the microscopic domain). Similarconclusions are arrived at by Kay (1998). If there is indeed such aquantitative similarity between these possible effects, then it wouldbecome extremely difficult to distinguish between the two. In thepresence of gravitation, any positive effect could be interpreted as supportfor either collapse or decoherence (with the above proviso). And in thosecases in which the system is effectively shielded from decoherence (say,if the experiment is performed in free fall), if the collapse mechanism isindeed triggered by gravitational effects, then no collapse should be

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expected either.

The relation between decoherence and spontaneous collapse theories isthus indeed far from straightforward.

3.2 Pilot-wave theories

3.2.1 De Broglie-Bohm and related theories

Pilot-wave theories are no-collapse formulations of quantum mechanics

that assign to the wave function the role of determining the evolution of (‘piloting’, ‘guiding’) the variables characterising the system, say particleconfigurations, as in de Broglie's (1928) and Bohm's (1952) theory, orfermion number density, as in Bell's (1987, Chap. 19) ‘beable’ quantumfield theory, or again field configurations, as in various proposals forpilot-wave quantum field theories (for a recent survey, see Struyve 2011).

De Broglie's idea was to modify classical Hamiltonian mechanics in such

a way as to make it analogous to classical wave optics, by substituting forHamilton and Jacobi's action function the phase S of a physical wave.Such a ‘wave mechanics’ of course yields non-classical motions, but inorder to understand how de Broglie's dynamics relates to typical quantumphenomena, we must include Bohm's (1952, Part II) analysis of theappearance of collapse. In the case of measurements, Bohm argued thatthe wave function evolves into a superposition of components that are and

remain separated in the total configuration space of measured system andapparatus, so that the total configuration is ‘trapped’ inside a singlecomponent of the wave function, which will guide its further evolution, asif the wave had collapsed (‘effective’ wave function). This analysisallows one to recover qualitatively the measurement collapse and byextension such typical quantum features as the uncertainty principle andthe perfect correlations in an Einstein-Podolsky-Rosen experiment. (Thequantitative aspects of the theory are also very well developed, but we



shall not describe them here.)

It is natural to extend this analysis from the case of measurementsinduced by an apparatus to that of ‘spontaneous measurements’ asperformed by the environment in the theory of decoherence, thus applyingthe same strategy to recover both quantum and classical phenomena. Theresulting picture is one in which de Broglie-Bohm theory, in cases of decoherence, describes the motion of particles that are trapped inside oneof the extremely well localised components selected by the decoherenceinteraction. Thus, de Broglie-Bohm trajectories will partake of theclassical motions on the level defined by decoherence (the width of thecomponents).

This use of decoherence would arguably resolve the puzzles discussed,e.g., by Holland (1996) with regard to the possibility of a ‘classical limit’of de Broglie's theory. One baffling problem, for instance, is thattrajectories with different initial conditions cannot cross in de Broglie-

Bohm theory, because the wave guides the particles by way of a first-order equation, while, as is well known, Newton's equations are second-order and possible trajectories in Newton's theory do cross. Now,however, the non-interfering components produced by decoherence canindeed cross, and so will the trajectories of particles trapped inside them.

The above picture is natural, but it is not obvious. De Broglie-Bohmtheory and decoherence contemplate two a priori distinct mechanisms

connected to apparent collapse: respectively, separation of components inconfiguration space and suppression of interference. While the formerobviously implies the latter, it is equally obvious that decoherence neednot imply separation in configuration space. One can expect, however,that decoherence interactions of the form of approximate positionmeasurements will.

If the main instances of decoherence are indeed coextensive with

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instances of separation in configuration, de Broglie-Bohm theory can thususe the results of decoherence relating to the formation of classical

structures, while providing an interpretation of quantum mechanics thatexplains why these structures are indeed observationally relevant. In thatcase, the question that arises for de Broglie-Bohm theory is not only thestandard question of whether all apparent measurement collapses can beassociated with separation in configuration (by arguing that at some stageall measurement results are recorded in macroscopically differentconfigurations), but also whether all appearance of classicality can beassociated with separation in configuration space. [19]

A discussion of the role of decoherence in pilot-wave theory in the formsuggested above is still largely outstanding. An informal discussion isgiven in Bohm and Hiley (1993, Chap. 8), partial results are given byAppleby (1999), some simulations have been realised by Sanz and co-workers (e.g. Sanz and Borondo 2009); and a different approach issuggested by Allori (2001; see also Allori & Zanghì 2009). Appleby

discusses Bohmian trajectories in a model of decoherence and obtainsapproximately classical trajectories, but under a special assumption. [20]

The simulations currently published by Sanz and co-workers are based onsimplified models, but fuller results have been announced. [21] Alloriinvestigates in the first place the ‘short wavelength’ limit of de Broglie-Bohm theory (suggested by the analogy to the geometric limit in waveoptics). The role of decoherence in her analysis is crucial but limited to

maintaining the classical behaviour obtained under the appropriate shortwavelength conditions, because the behaviour would otherwise breakdown after a certain time.

While, as argued above, it appears plausible that decoherence might beinstrumental in recovering the classicality of pilot-wave trajectories in thecase of the non-relativistic particle theory, it is less clear whether thisstrategy might work equally well in the case of field theory. Doubts to



this effect have been raised, e.g., by Saunders (1999) and by Wallace(2008). Essentially, these authors doubt whether the configuration-space

variables, or some coarse-grainings thereof, are, indeed, decoheringvariables. [22] At least in the opinion of the present author, further detailedinvestigation is needed.

3.2.2 Nelson's stochastic mechanics

Nelson's (1966, 1985) stochastic mechanics is strictly speaking not apilot-wave theory. It is a proposal to recover the wave function and the

Schrödinger equation as effective elements in the description of afundamental diffusion process in configuration space. Insofar as theproposal is successful, however, it then shares many features with deBroglie-Bohm theory. In particular, the current velocity for the particlesin Nelson's theory turns out to be equal to the de Broglie-Bohm velocity,and the particle distribution in Nelson's theory is equal to that in deBroglie-Bohm theory (in equilibrium).

It follows that many results from pilot-wave theories can be imported intoNelson's stochastic mechanics. However, decoherence has been very littlediscussed in the literature on stochastic mechanics, if at all, and thestrategies used in pilot-wave theories to recover the appearance of collapse and the emergence of a classical regime still need to be appliedspecifically in the case of stochastic mechanics. This would presumablyalso resolve some conceptual puzzles specific to Nelson's theory, such asthe problem of two-time correlations raised in Nelson (2006).

3.3 Everett interpretations

Over the years, since the original paper by Everett (1957), some verydiverse ‘Everett interpretations’ have been proposed, which possibly onlyshare the core intuition that a single wave function of the universe shouldbe interpreted in terms of a multiplicity of ‘realities’ at some level or

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other. This multiplicity, however understood, is formally associated withcomponents of the wave function in some decomposition. [23]

Various such Everett interpretations, roughly speaking, differ as to how toidentify the relevant components of the universal wave function, and howto justify such an identification (the so-called problem of the ‘preferredbasis’ — although this may be a misnomer), and differ as to how tointerpret the resulting multiplicity (various ‘many-worlds’ or various‘many-minds’ interpretations), in particular with regard to theinterpretation of the (emerging?) probabilities at the level of thecomponents (problem of the ‘meaning of probabilities’).

The last problem is perhaps the most hotly debated aspect of Everett.Clearly, decoherence enables reidentification over time of both observersand of results of repeated measurement (and thus definition of empiricalfrequencies). In recent years progress has been made especially along thelines of interpreting the probabilities in decision-theoretic terms for a

‘splitting’ agent (see in particular Deutsch (1999) and Wallace (2003b,2007)). [24]

The most useful application of decoherence to Everett, however, seems tobe in the context of the problem of the preferred basis. Decoherenceyields a natural solution to the problem, in that it identifies a class of ‘preferred’ states (not necessarily an orthonormal basis!), and allows oneto reidentify them over time, so that one can identify ‘worlds’ with the

trajectories defined by decoherence (or more abstractly with decoherenthistories). [25] If part of the aim of Everett is to interpret quantummechanics without introducing extra structure, in particular withoutostulating the existence of some preferred basis, then one will try to

look for potentially relevant structures that are already present in the wavefunction. In this sense, decoherence is the ideal candidate for identifying‘worlds’ (see e.g. Wallace 2003a).



A justification for this identification can be variously given by suggestingthat a ‘world’ should be a temporally extended structure and thus

reidentification over time will be a necessary condition for definingworlds; or similarly by suggesting that in order for observers to haveevolved there must be stable records of past events (Saunders 1993, andthe unpublished Gell-Mann & Hartle 1994) (see the Other InternetResources section below); or that observers must be able to access robust states, preferably through the existence of redundant information in theenvironment (Zurek's ‘existential interpretation’, 1998).

Alternatively to some global notion of ‘world’, one can look at thecomponents of the (mixed) state of a (local) system, either from the pointof view that the different components defined by decoherence willseparately affect (different components of the state of) another system, orfrom the point of view that they will separately underlie the consciousexperience (if any) of the system. The former sits well with Everett's(1957) original notion of relative state, and with the relational

interpretation of Everett preferred by Saunders (e.g. 1993) and, it wouldseem, Zurek (1998) (see the entry on Everett's relative-stateinterpretation). The latter leads directly to the idea of many-mindsinterpretations. [26]

The idea of many minds was suggested early on by Zeh (2000; also 1995,p. 24). As Zeh puts it, von Neumann's motivation for introducing collapsewas to save what he called ‘psycho-physical parallelism’ (arguably to beunderstood as supervenience of the mental on the physical: only onemental state is experienced, so there should be only one correspondingcomponent in the physical state). In a decohering no-collapse universeone can instead introduce a new psycho-physical parallelism, in whichindividual minds supervene on each non-interfering component in thephysical state. Zeh indeed suggests that, given decoherence, this is themost natural interpretation of quantum mechanics. [27]

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3.4 Modal interpretations

Modal interpretations originated with Van Fraassen (1973, 1991) as purereinterpretations of quantum mechanics (other later versions coming moreto resemble pilot-wave theories). Van Fraassen's basic intuition was thatthe quantum state of a system should be understood as describing acollection of possibilities, represented by components in the (mixed)quantum state. His proposal considers only decompositions at singleinstants, and is agnostic about reidentification over time. Thus, it can

directly exploit only the fact that decoherence produces descriptions interms of classical-like states, which will count as possibilities in VanFraassen's interpretation. This ensures ‘empirical adequacy’ of thequantum description (a crucial concept in Van Fraassen's philosophy of science). The dynamical aspects of decoherence can be exploitedindirectly, in that single-time components will exhibit records of the past,which ensure adequacy with respect to observations, but about whoseveridicity Van Fraassen remains agnostic.

A different strand of modal interpretations is loosely associated with the(distinct) views of Kochen (1985), Healey (1989) and Dieks and Vermaas(e.g. 1998). We focus on the last of these to fix ideas. Van Fraassen'spossible decompositions are restricted to one singled out by amathematical criterion (related to the biorthogonal decompositiontheorem mentioned above in Section 3.1), and a dynamical picture is

explicitly sought (and was later developed). In the case of an ideal (non-approximate) quantum measurement, this special decomposition coincideswith that defined by the eigenstates of the measured observable and thecorresponding pointer states, and the interpretation thus appears to solvethe measurement problem (for this case at least).

At least in Dieks's original intentions, however, the approach was meantto provide an attractive interpretation of quantum mechanics also in the



case of decoherence interactions, since at least in simple models of decoherence the same kind of decomposition singles out more or less also

those states between which interference is suppressed (with a provisoabout very degenerate states).

However, this approach fails badly when applied to other models of decoherence, e.g., that in Joos and Zeh (1985, Section III.2). Indeed, itappears that in more general models of decoherence the componentssingled out by this version of the modal interpretation are given bydelocalised states, and are unrelated to the localised components naturally

privileged by decoherence (Donald 1998; Bacciagaluppi 2000). Note thatVan Fraassen's original interpretation is untouched by this problem, andso are possibly some more recent modal or modal-like interpretations bySpekkens and Sipe (2001), Bene and Dieks (2002) and Berkovitz andHemmo (2006).

Finally, some of the views espoused in the decoherent histories literaturecould be considered as cognate to Van Fraassen's views, identifyingpossibilities, however, at the level of possible courses of world history.Such ‘possible worlds’ would be those temporal sequences of (quantum)propositions satisfying the decoherence condition and in this sensesupporting a description in terms of a probabilistic evolution. This viewwould be using decoherence as an essential ingredient, and in fact mayturn out to be the most fruitful way yet of implementing modal ideas; adiscussion in these terms has been outlined by Hemmo (1996).

3.5 Bohr's Copenhagen interpretation

Bohr is often credited with more or less the following view. Everydayconcepts, in fact the concepts of classical physics, are indispensable to thedescription of any physical phenomena (in a way and terminologysomewhat reminiscent of Kant's transcendental arguments). However,experimental evidence from atomic phenomena shows that classical

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concepts have fundamental limitations in their applicability: they can onlygive partial (complementary) pictures of physical objects. While these

limitations are quantitatively negligible for most purposes in dealing withmacroscopic objects, they apply also at that level (as shown by Bohr'swillingness to apply the uncertainty relations to parts of the experimentalapparatus in the Einstein-Bohr debates), and they are of paramountimportance when dealing with microscopic objects. Indeed, they shape thecharacteristic features of quantum phenomena, e.g., indeterminism. Thequantum state is not an ‘intuitive’ ( anschaulich, also translated as‘visualisable’) representation of a quantum object, but only a ‘symbolic’representation, a shorthand for the quantum phenomena that areconstituted by applying the various complementary classical pictures.

While it is difficult to pinpoint exactly what Bohr's views were (theconcept and even the term ‘Copenhagen interpretation’ have been arguedto be a later construct; see Howard 2004), it is clear that according toBohr, classical concepts are autonomous from, and indeed conceptually

prior to, quantum theory. If we understand the theory of decoherence aspointing to how classical concepts might in fact emerge from quantummechanics, this seems to undermine Bohr's basic position. Of course itwould be a mistake to say that decoherence (a part of quantum theory)contradicts the Copenhagen approach (an interpretation of quantumtheory). However, decoherence does suggest that one might want to adoptalternative interpretations, in which it is the quantum concepts that are

prior to the classical ones, or, more precisely, the classical concepts at theeveryday level emerge from quantum mechanics (irrespectively of whether there are even more fundamental concepts, as in pilot-wavetheories). In this sense, if the programme of decoherence is successful inthe sense sketched in Section 2.3, it will indeed be a blow to Bohr'sinterpretation coming from quantum physics itself.

On the other hand, Bohr's intuition that quantum mechanics as practised



requires a classical domain would in fact be confirmed by decoherence, if it turns out that decoherence is indeed the basis for the phenomenology of

quantum mechanics, as the Everettian and possibly the Bohmian analysissuggest. [28] As a matter of fact, Zurek (2003) locates his existentialinterpretation half-way between Bohr and Everett.

4. Scope of Decoherence

We have already mentioned in Section 1.1 that some care has to be takenlest one overgeneralise conclusions based on examining only well-behaved models of decoherence. On the other hand, in order to assess theprogramme of explaining the emergence of classicality using decoherence(together with appropriate foundational approaches), one has to probehow far the applications of decoherence can be pushed. In this finalsection, we survey some of the further applications that have beenproposed for decoherence, beyond the easier examples we have seen suchas chirality or alpha-particle tracks. Whether decoherence can indeed be

successfully applied to all of these fields will be in part a matter forfurther assessment, as more detailed models are proposed andinvestigated.

A straightforward application of the techniques allowing one to deriveNewtonian trajectories at the level of components has been employed byZurek and Paz (1994) to derive chaotic trajectories in quantummechanics. The problem with the quantum description of chaoticbehaviour is that prima facie there should be none. Chaos is characterisedroughly as extreme sensitivity in the behaviour of a system on its initialconditions, in the sense that the distance between the trajectories arisingfrom different initial conditions increases exponentially in time. Since theSchrödinger evolution is unitary, it preserves all scalar products and alldistances between quantum state vectors. Thus, it would seem, closeinitial conditions lead to trajectories that are uniformly close throughout

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all of time, and no chaotic behaviour is possible (‘problem of quantumchaos’). The crucial point that enables Zurek and Paz's analysis is that the

relevant trajectories defined by decoherence are at the level of components of the state of the system. Unitarity is preserved because thevectors in the environment, to which these different components arecoupled, are and remain orthogonal: how the components themselvesmore specifically evolve is immaterial. Explicit modelling yields a pictureof quantum chaos in which different trajectories branch (a feature absentfrom classical chaos, which is deterministic) and then indeed divergeexponentially. As with the crossing of trajectories in de Broglie-Bohmtheory (Section 3.2), one has behaviour at the level of components that isqualitatively different from the behaviour derived for wave functions of an isolated system.

The idea of effective superselection rules was mentioned in Section 1.1.As pointed out by Giulini, Kiefer and Zeh (1995, see also Giulini et al.1996, Section 6.4), the justification for the (strict) superselection rule for

charge in quantum field theory can also be phrased in terms of decoherence. The idea is simple: an electric charge is surrounded by aCoulomb field (which electrostatically is infinitely extended; theargument can also be carried through using the retarded field, though).States of different electric charge of a particle are thus coupled todifferent, presumably orthogonal, states of its electric field. One canconsider the far-field as an effectively uncontrollable environment thatdecoheres the particle (and the near-field), so that superpositions of different charges are indeed never observed.

Another claim about the significance of decoherence relates to timeasymmetry (see e.g. the entries on time asymmetry in thermodynamicsand philosophy of statistical mechanics), in particular to whetherdecoherence can explain the apparent time-directedness in our (classical)world. The issue is again one of time-directedness at the level of



components emerging from a time-symmetric evolution at the level of theuniversal wave function (presumably with special initial conditions).

Insofar as (apparent) collapse is indeed a time-directed process,decoherence will have direct relevance to the emergence of this ‘quantummechanical arrow of time’ (for a spectrum of discussions, see Zeh 2001,Chap. 4; Hartle 1998, and references therein; Bacciagaluppi 2002, Section6.1, and Bacciagaluppi 2007). Whether decoherence is connected to theother familiar arrows of time is a more specific question, variousdiscussions of which are given, e.g., by Zurek and Paz (1994), Hemmoand Shenker (2001) and the unpublished Wallace (2001) (see the OtherInternet Resources below).

Zeh (2003) argues from the notion that decoherence can explain ‘quantumphenomena’ such as particle detections that the concept of a particle inquantum field theory is itself a consequence of decoherence. That is, onlyfields need to be included in the fundamental concepts, and ‘particles’ area derived concept, unlike what might be suggested by the customary

introduction of fields through a process of ‘second quantisation’. Thusdecoherence seems to provide a further powerful argument for theconceptual primacy of fields over particles in the question of theinterpretation of quantum field theory.

Finally, it has been suggested that decoherence could be a usefulingredient in a theory of quantum gravity, for two reasons. First, becausea suitable generalisation of decoherence theory to a full theory of quantum gravity should yield suppression of interference betweendifferent classical spacetimes (Giulini et al. 1996, Section 4.2). Second, itis speculated that decoherence might solve the so-called problem of time,which arises as a prominent puzzle in (the ‘canonical’ approach to)quantum gravity. This is the problem that the candidate fundamentalequation (in this approach)—the Wheeler-DeWitt equation—is ananalogue of a time- independent Schrödinger equation, and does not

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contain time at all. The problem is thus in a sense simply: where doestime come from? In the context of decoherence theory, one can construct

toy models in which the analogue of the Wheeler-DeWitt wave functiondecomposes into non-interfering components (for a suitable sub-system)each satisfying a time- dependent Schrödinger equation, so thatdecoherence appears in fact as the source of time. [29] An accessibleintroduction to and philosophical discussion of these models is given byRidderbos (1999), with references to the original papers.

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Other Internet Resources

Crull, E. (University of Aberdeen), and Bacciagaluppi, G.(University of Aberdeen), 2011, ‘Translation of W. Heisenberg: “Isteine deterministische Ergänzung der Quantenmechanik möglich?”’,available online in the Pittsburgh Phil-Sci Archive.Felline, L. (Universidad Autóonoma de Barcelona) andBacciagaluppi, G. (University of Aberdeen), 2011, ‘Locality andMentality in Everett Interpretations: Albert and Loewer's ManyMinds’, available online in the Pittsburgh Phil-Sci Archive.Gell-Mann, M. (Santa Fe Institute), and Hartle, J. B. (UC/SantaBarbara), 1994, ‘Equivalent Sets of Histories and MultipleQuasiclassical Realms’, available online in the arXiv.org e-Printarchive.Wallace, D. (Oxford University), 2000, ‘Implications of QuantumTheory in the Foundations of Statistical Mechanics’, available onlinein the Pittsburgh Phil-Sci Archive.

Wallace, D. (Oxford University), 2002, ‘Quantum Probability andDecision Theory, Revisited’, available online in the arXiv.org e-Printarchive. This is a longer version of Wallace (2003b).The arXiv.org e-Print archive, formerly the Los Alamos archive.This is the main physics preprint archive.The Pittsburgh Phil-Sci Archive. This is the main philosophy of science preprint archive.

A Many-Minds Interpretation Of Quantum Theory, maintained byMatthew Donald (Cavendish Lab, Physics, University of Cambridge). This page contains details of his many-mindsinterpretation, as well as discussions of some of the books andpapers quoted above (and others of interest). Follow also the link tothe ‘Frequently Asked Questions’, some of which (and the ensuingdialogue) contain useful discussion of decoherence.

Guido Bacciagaluppi




Quantum Mechanics on the Large Scale, maintained by Philip Stamp(Physics, University of British Columbia). This page has links to theavailable talks from the Vancouver workshop mentioned in footnote1; see especially the papers by Tony Leggett and by Philip Stamp.Decoherence Website, maintained by Erich Joos. This is a site withinformation, references and further links to people and institutesworking on decoherence, especially in Germany and the rest of Europe.

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quantum mechanics | quantum mechanics: Bohmian mechanics | quantummechanics: collapse theories | quantum mechanics: Copenhageninterpretation of | quantum mechanics: Everett's relative-state formulationof | quantum mechanics: many-worlds interpretation of | quantum theory:measurement in | quantum theory: quantum entanglement and information| quantum theory: quantum field theory | quantum theory: quantum gravity

| quantum theory: the Einstein-Podolsky-Rosen argument in | statisticalphysics: philosophy of statistical mechanics | time: thermodynamicasymmetry in | Uncertainty Principle

Acknowledgments

I wish to think many people in discussion with whom I have shaped myunderstanding of decoherence over the years, in particular Marcus

Appleby, Matthew Donald, Beatrice Filkin, Meir Hemmo, SimonSaunders, Max Schlosshauer, David Wallace and Wojtek Zurek. For morerecent discussions and correspondence relating to this article I wish tothank Valia Allori, Bob Griffiths, Peter Holland, Martin Jones, TonyLeggett, Hans Primas, Alberto Rimini, Philip Stamp and Bill Unruh. Ialso gratefully acknowledge my debt to Steve Savitt and Philip Stamp foran invitation to the University of British Columbia, to Claudius Gros for



an invitation to the University of the Saarland, and for the opportunitiesfor discussion arising from these talks. Finally I wish to thank thefollowing: the referee for this entry, again David Wallace, for his clearand constructive commentary; my subject editor, John Norton, whocorresponded with me extensively over a previous version of part of thematerial and whose suggestions I have taken to heart; our editor-in-chief,Ed Zalta, for his saintly patience; and my late friend, Rob Clifton, whoinvited me to write on this topic in the first place.

Notes to The Role of Decoherence in QuantumMechanics

1. The first version of this entry was based on a talk given at theExploratory Workshop on Quantum Mechanics on the Large Scale, ThePeter Wall Institute for Advanced Studies, The University of BritishColumbia, 17–27 April 2003, on whose website are linked electronicversions of this and several of the other talks (see the Other Internet

Resources).

2. Note that these probabilities are well-defined in quantum mechanics,but in the context of a separate experiment (with detection at the slits).Heisenberg (in the uncertainty paper) notes that for this reason he doesnot like the phrase ‘interference of probabilities’.

3. Realistically, in each single scattering the electron will couple to non-orthogonal states of the environment, thus experiencing only a partialsuppression of interference. However, repeated scatterings will suppressinterference very effectively.

4. Unfortunately, the distinction between ‘true’ collapse (whether or not itis a process that in fact happens in nature) and ‘as if’ collapse issometimes overlooked, muddling conceptual discussions: further on this

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point, see e.g. Pearle (1997) and Zeh (1995, pp. 28–29).

5. As long as decoherence yields only effective superselection rules, onedoes not leave the framework of standard non-relativistic quantummechanics. The discussion of charge in Section 4 below, however,suggests that decoherence might yield strict superselection rules, which ingeneral require the framework of so-called algebraic quantum mechanics.More systematic discussion is needed to fully assess the interpretationalimplications of the latter. (My thanks to Hans Primas for discussion of this point.)

6. These values are calculated based on the classic model by Joos and Zeh(1985). Length and time scales for more massive objects are furtherreduced. For a not too technical partial summary of Joos and Zeh'sresults, see also Bacciagaluppi (2000).

7. For a review of more rigorous arguments see e.g. Zurek (2003, pp. 28–30). In particular they can be obtained from the Wigner function

formalism, see e.g. Zurek (1991) and in more detail Zurek and Paz(1994), who then apply these results to derive chaotic trajectories inquantum mechanics (see below Section 4).

8. For a very accessible discussion of alpha-particle tracks roughly alongthese lines, see Barbour (1999, Chap. 20).

9. For more details of the decoherent histories approach, see the overviewarticle by Halliwell (1995), and for a short discussion of some of itsconceptual aspects, see Section 7 in the entry on Everett's relative-stateformulation of quantum mechanics.

10. Similar results involving imperfectly correlated records can be derivedin the case of mixed states (Halliwell 1999).

11. If so, the idea does not seem quite new. Wheeler's famous ‘delayed



choice’ experiments are presented by him as affecting the past in thisway. And a related discussion of causality was given in detail by Grete

Hermann (1935) (probably the most important philosophical commentatorof the emerging Copenhagen school), specifically in the context of Weizsäcker's analysis of the Heisenberg microscope.

12. In the context of dynamical decoherence, Zurek (1998) has proposedan ‘existential interpretation’, which brings together certain aspects of Bohr and of Everett.

13. As a numerical example, take a macroscopic particle of radius 1cm(mass 10g) interacting with air under normal conditions. After an hour theoverall spread of its state is of the order of 1m. (This estimate usesequations [3.107] and [3.73] in Joos and Zeh (1985).)

14. Von Neumann's justification for espousing a collapse approach in thefirst place arguably relies: (a) on his ‘insolubility theorem’, showing thatthe phenomenological indeterminism in the measurement cannot beexplained in terms of ignorance of the exact state of the apparatus (latergeneralised by several authors; see the discussion in Section 3 of the entryon collapse theories, and references therein); (b) on his ‘no-go’ theoremfor hidden variables, which in his opinion excluded this other alternativeapproach (this theorem was criticised by Grete Hermann, 1935, Sect. 7,and much more famously by Bell, 1987, Chap. 2); (c) on his wish touphold a one-to-one correspondence between mental states and physical

states of the observer (if one gives this up, one obtains some version of the Everett interpretation; see also the discussion in Section 3.3 below).

15. The collapse consists in multiplying the wave function $ (r) by aGaussian of fixed width a, call it a x(r), with a probability distribution forthe centre x of the Gaussian given by ( |a x(r)$ (r)|2dr. In other words, if we denote by A x the operator corresponding to multiplication by the

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Gaussian a x(r), the state | $ > goes over to one of the continuously manypossible states (1/< $ | A x*A x|$ >) A x|$ >, with probability density<$ | A

x* A

x|$ >. In technical language, this is a measurement associated

with a POVM (positive operator valued measure). In the original model,a=10 -5 cm, and collapse occurs with a probability per second 1/ ) , with) =10 16s.

16. This modification was introduced because the original model wouldhave had consequences (that were already ruled out by existingexperiments) for the predicted lifetime of the proton (Pearle and Squires

1994), due to the production of energy associated with the collapse. Latermodels also generally use the formalism of continuous spontaneouslocalisation (CSL), phrased in terms of stochastic differential equations(Pearle 1989, also sketched in the entry on collapse theories), but forpresent purposes we shall stick to the more elementary GRW theory.

17. For N particles, in the case in which the frequencies are independentof mass, it is easy to contrive examples in which the theory gives resultsvery different from decoherence. A state of a macroscopic pointerlocalised in a region A superposed with a state of the pointer localised inB will almost instantaneously trigger a collapse onto one of the localisedstates, which is analogous to what we also expect from decoherence.However, a (contrived) state of the pointer in which all its protons arelocalised in region A and all its neutrons in region B, superposed with astate in which the protons are in B and the neutrons are in A, would also

trigger a collapse, but onto one of these very non-classical states. In the‘mass density’ version this difference will disappear.

18. My thanks to Bill Unruh for raising this issue.

19. By the same token one can dismiss proposed variants of de Broglie-Bohm theory that are not based on the position representation, e.g.Epstein's (1953) momentum-based theory, which would utterly fail to



exhibit the correct ‘collapse’ behaviour and classical regime, preciselybecause decoherence interactions are clearly not momentum-based.

20. I suggest to reinterpret Appleby's assumption as an assumption aboutthe effective wavefunction of the heat bath with which the systeminteracts. That is, I suggest that one should try to justify it from a pilot-wave treatment of the bath.

21. Sanz and Borondo (2009) simulate trajectories for the two-slitexperiment. One model is based on the reduced density matrix of the

system (and thus based on simple de-phasing), and still maintains the no-crossing feature that troubles Holland (1996). The other model, despitealso being simplified, recovers the classical crossing of trajectories. Sanzand co-workers (personal communication, November 2011) have nowcarried out simulations including explicit modelling of the environment,and in situations more complex than the two-slit experiment. Thesesimulations seem to confirm that classical-like trajectories can indeed berecovered in pilot-wave theory using decoherence.

22. Should this be the case, provided the results of observations arerecorded in the configuration-space variables, pilot-wave field theories(as in the ‘minimalist’ model by Struyve and Westman (2007)) could stillrecover at least the appearance of classical trajectories (somewhat inanalogy to the discussion of ‘fooled detectors’ (see e.g. Barrett 2000)).Decoherence would arguably still play a role at this level, similar to the

role it plays in some neo-Copenhagen views (see e.g. the comments onOmnès in Section 1.2 above).

23. For the state of the art on the Everett interpretation(s), see the recentcollection by Saunders et al. (2010), which contains the papers from thetwo international conferences that celebrated the 50 years of Everett'soriginal paper.

Guido Bacciagaluppi




24. This approach to probabilities is captivating, but has also attractedsome serious criticism (see e.g. P. Lewis 2010). Furthermore, part of theproblem of the meaning of the probabilities relates to whether Everettianprobabilities make sense also in confirmation theory, in particular towhether observational data confirms (Everettian) quantum mechanics. Forthe latter ‘epistemic problem’ see in particular Greaves (2007) andGreaves and Myrvold (2010).

25. Such a solution to the preferred basis problem might be only partial inthe sense that there are many inequivalent ways of selecting sets of

decoherent histories (see e.g. Dowker and Kent 1995, 1996). On the otherhand, this does not make ‘our’ set of histories any less decoherent.

26. If one assumes that mentality can be associated only with certaindecohering structures of great complexity, this might have the advantageof further reducing the remaining ambiguity about the preferred ‘basis’(see Matthew Donald's website on ‘A Many-Minds Interpretation of Quantum Theory’, referenced in the Other Internet Resources).

27. It is tempting to see the difference between (Saunders-Wallace)decoherence-based many-worlds and (Zeh) decoherence-based many-minds as merely conventional, since they arguably both share an ontologyof global wave functions and local (perhaps emergent) minds, and onlydiffer in the emphasis they place on different objectively presentstructures in the same decohering wave function. Note also that in Albert

and Loewer's (1988) version of the many-minds interpretation, the mentaldoes not supervene on the physical, because individual minds have trans-temporal identity of their own. This is postulated in order to define astochastic dynamics for the minds and not have to introduce a novelconcept of probability. Even in this case, however, decoherence is of crucial importance, since the dynamical evolution of the minds will havea physical correlate only if the corresponding physical components are



decohered (Felline and Bacciagaluppi 2011). (Thanks to Martin Jones fordiscussion of this last point.)

28. Note that insofar as the movability of the cut between observer andobserved is crucial for the internal coherence of the Copenhageninterpretation (a point stressed by Heisenberg, see e.g. Crull andBacciagaluppi (2011) in the Other Internet Resources), then decoherencecan be argued to be relevant also to this aspect of the Copenhageninterpretation.

29. An analogy from standard quantum mechanics may be helpful here.Take a harmonic oscillator in equilibrium with its environment. Anequilibrium state is by definition a stationary state under the dynamics,i.e. it is itself time-independent. However, one can decompose theequilibrium state of the oscillator as a mixture of localised componentseach carrying out one of the oscillator's possible classical motions (time-dependent!). Such a decomposition can be found e.g. in Donald (1998,Section 2). For a state-of-the-art model, see Halliwell and Thorwart(2002).

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