real world interpretations of quantum theory 0708.3710

7/27/2019 Real World Interpretations of Quantum Theory 0708.3710

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invariant description of reality compatible with quantum theory and with experiment and observation, which we needand do not currently have. But it would leave our understanding incomplete, in the sense that some choices are leftundetermined.

INTERPRETING A BRANCHING STRUCTURE

Many worlds or one?

As Bell [9] and (probably many) others [10] have noted, if the branching of worlds were precisely and objectivelydened, a many worlds interpretation would seem unnecessarily extravagant. Given a precisely dened branchingstructure, we can just as easily dene a one world interpretation of quantum theory.

To esh this out, let us suppose that we indeed have an algorithm for dening a branching structure for the universalwave function, given a formulation of quantum theory, an initial state (0) at t = 0, and a Hamiltonian H . For thepurposes of this illustration, we follow a traditional view of a branching structure, taking it to be a process by whichthe unitarily evolving wave function is divided into a sum of components at some time t1 , each component then evolvesunitarily but may further sub-divide at some later time, from which point the sub-components then unitarily evolveuntil such time as they sub-divide into sub-sub-components, and so on. On this view, by denition, branches remainindependent: they are not allowed to merge again after dividing. Suppose too that the branching structure becomesasympotically constant in other words that the number of branches becomes constant as t , and that we canapply the algorithm in the asymptotic limit to obtain a complete list of branches B j .

Each branch is then uniquely identied by a nal component, that is, one which will undergo no further sub-division:the branch can be identied throughout time by following the branching diagram backwards from the nal states.Each branch B j thus denes an unnormalised pure quantum state j (t) at each time t, which can be identied fromthe nal component by following the branch backwards and taking the component corresponding to the branch at anygiven time. The branch states need not be, and generally will not be, orthogonal at every time t: in particular, beforeany given sub-division, all branches whose nal components ultimately arise from that sub-division will have the samebranch state. However, we suppose that asymptotically the branch states dene an orthogonal decomposition of (t),so that

limt

((t) j

j (t)) = 0

and

limt

(j (t), j (t)) = pj jj ,

where the pj are constants.We can then dene the probability p(B j ) of each branch B j by the Born rule. If the relevant states also have

asymptotic limits, so that we can dene

= limt

exp( iHt/ )(0) ,

and

j = limt j (t) ,

we have(j ,

j ) = p(B j ) = (

, j ) = pj .

More generally, so long as the relevant inner product has an asymptotic limit, we can dene

p(B j ) = limt ((t), j (t)) , (1)

where (t) = exp( iHt/ )(0) is the unitarily evolved state at time t.If these assumptions the existence of a precisely dened branching structure and the existence of the relevant

asymptotic limits held, then a perfectly sensible interpretation of the mathematics would be to postulate that


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precisely one branch is randomly chosen, via the Born rule probability distribution, and that this branch alone isrealised in Nature.

Clearly, this would be more economical than a many-worlds interpretation of quantum theory, even if (which ishotly contested [11, 12]) we could make recover all the predictions of standard quantum theory from many-worldsquantum theory. Here, just one branch would be realised in Nature, rather than all. It would also make clear what theBorn probabilities are probabilities of . In contrast, it is (at the very least) less immediately clear that many-worldsinterpretations can nd a sensible role for Born probabilities: whatever p(B j ) might possibly be [ 17] in a many-worldsinterpretation, it clearly cannot be the probability of reality being described by the branch B j , since all the branchesare equally real.

That said, the aim here is not to argue that, given our starting assumptions, the one-world view is demonstrablycorrect and the many-worlds view completely indefensible. These questions are considered in detail elsewhere [ 11, 12].The point here is simply to observe that the one-world view appears natural. The one-world view is not actuallylogically necessary to motivate all the interpretational ideas that we set out below. However, it will be adopted fromhere on, on the grounds that it is both pedagogically helpful and fundamentally appealing.

Next we examine some standard intuitions about the form of the branching structure.

Do measurement theory and decoherence alone allow us to dene a satisfactory one-world interpretation?

It might perhaps (though naively) be argued that we can go on to sketch a satisfactory interpretation of quantumtheory, by dening the branching structure as follows.

First, suppose that each sub-branching each point at which a branch splits into two or more corresponds to aquantum measurement type interaction. Then in principle we can dene the entire branching tree in a cosmologicaltheory by listing all the quantum measurement-type interactions. Each elementary branch can then be dened bylisting a consistent set of possible outcomes of all the measurements encountered on some branching path from theinitial to the nal time. If we apply the Born probability rule as above, then precisely one branch is picked out ascorresponding to reality, and hence precisely one set of consistent measurement outcomes is realised.

Now, standard decoherence arguments tell us that we can calculate the probabilities of quantum measurementoutcomes at any point after they have been classically recorded, and that the answers will be essentially constant afterthat point. In particular, the nal time Born probability rule ( 1) for branches gives essentially the same probabilityfor the set of measurement outcomes as that predicted by Copenhagen quantum mechanics. Thus, it might be argued,we can derive the standard Copenhagen Born rule probabilities for individual measurement events from the nal timeBorn rule for branches, since we have postulated that branchings correspond to quantum measurements. Hence, the

argument runs, we have outlined an interpretation of quantum theory which treats the universe as a closed quantumsystem and which explains both why we see a quasiclassical world and why the Copenhagen interpretation of quantumtheory describes the outcomes of our experiments within that world.

Problems with naive denitions of branching

This is too glib, though. Invoking a quantum measurement type interaction as a fundamental notion reproducesone of the unsatisfactory features of the Copenhagen interpretation. We may think we know roughly what we meanby the term, but we dont have a precise denition. And how can this explanation work without one? Is a cosmiccommittee meant to survey the evolution of the universe and decide case by case whether, and if so precisely when,a measurement took place?

In any case, the notion of quantum measurement isnt sufficiently general to produce a realistic interpretation of a

quantum theory of cosmology. All the quasi-classical objects and structures in our universe, including all measuringdevices, natural and articial, emerged from a purely quantum initial state. [18] The emergence of quasiclassicalstructure from a quantum state doesnt correspond to a quantum measurement interaction in any standard senseof the phrase. Yet, without an account of this process, we cant begin to justify an account of the world as it nowappears or the measurement devices and classical records it now contains.

This is a deeper problem than seems to be generally recognised. The set of possible quasi-classical structures thatcould have emerged from the initial state of our universe is, presumably, a continuous set, not a discrete one, at leastif space-time itself is continuous. The peak of the Moons centre of mass wave function could have been epsilonicallyfurther away from that of the Earth, our galaxys relative momentum to its neighbours could have been slightlydifferent, and so on. All these possible quasi-classical structures are ultimately dened by quantum states. So we


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cannot hope to identify classically distinct possible nal outcomes within a cosmological theory simply by inspection;we need to nd a denition that makes sense within quantum theory.

We could try to solve this problem mathematically by dening a complete set of mutually exclusive quasi-classicalstates, from which precisely one is realised, if we had some natural mathematical prescription; for instance, a naturalprojective decomposition on the subspace spanned by all the quasi-classical states. However, the notion of quasi-classicality doesnt, per se , supply such a prescription.

Another consequence of the continuous multiplicity of quasi-classical structures is that it is hard to dene preciselywhat we mean by a particular experiment in the context of a continuum of possible cosmologies and so labellingthe Everett branches of a cosmology by the outcome of an experiment is much more problematic than it rst appears.Would an instantiation of the experiment in an alternative quasiclassical world with the Moons centre-of-mass wave-function peak very slightly further away from the Earths count as the same experiment or a different one? Whatabout an instantiation in which the experimental apparatus is in a slightly different quasiclassical state? Again,referring to quasiclassicality, measurement, or similarly imprecise intuitions doesnt resolve the ambiguity: we need amathematical criterion.

CHARACTERISING BRANCHES MATHEMATICALLY

We are thus motivated to propose a mathematical characterisation of possible branches. First, we need someassumptions.

Suppose we are given a quantum theory of a closed system with Hilbert space H , Hamiltonian H and initial stateI = (0). In non-relativistic quantum mechanics, we can obtain the state at any later time t as (t) = exp( iHt )I .In relativistic quantum eld theory with a suitable xed background space-time, for example Minkowski space, wecan obtain the quantum state on any spacelike hypersurface by the Tomonaga-Schwinger formalism, assuming (as wedo) that we have a eld theory for which the Tomonaga-Schwinger solutions are mathematically well dened on allhypersurfaces. To apply our discussion to a hypothetical quantum theory of gravity, we assume for now withoutworrying about the details that the quantum gravity theory allows something analogous, i.e. that we can evolvethe initial state to produce sequences of states of the matter and gravitational elds that characterise the evolvingphysical system.

It might be objected that we cannot presently justify assuming the existence of a mathematically well denedquantum eld theory that characterises the non-gravitational interactions, let alone a mathematically well denedquantum theory of gravity. Granted, theoretical physics has not produced such theories as yet (or at least notdemonstrably so). But our aim here is to offer a new way of interpreting quantum theories, not to solve all the current

problems of theoretical physics. We need to assume that we have a mathematically well dened theory in order forthere to be any sense in trying to interpret it. We also need to postulate some properties of the theory. We cantbe sure that these postulates will turn out to be valid for the nal quantum theory of everything, if indeed there isone. However, they seem reasonably plausible, are not particularly ad hoc, and do in fact apply to some interestingtheories.

So, we adopt the following assumptions:

Postulate 1 The Hilbert space H of the system has a natural preferred representation as a tensor product H = HA HB , which is dened at every time (or on every spacelike hypersurface, or for every possible evolved state in a quantum gravity theory).

Postulate 2 There is a natural preferred complete set of orthogonal projections {P i }iI dened on the Hilbert space HB at every time. (Or on every spacelike hypersurface, etc.: to save repetition and x our notation we henceforth state the postulates for the case of non-relativistic quantum mechanics and take the (more fundamentally interesting)alternatives of relativistic quantum eld theory and quantum gravity as understood.)

For quantum eld theory in Minkowski space, examples of possible natural tensor products are given by taking HAand HB to be the subspaces corresponding respectively to fermions and bosons, or massive and massless particles.(Or vice versa see the discussion below.) For a quantum gravity theory, possible examples might be for HA and HBto correspond respectively to the gravitational and matter degrees of freedom, or perhaps to the massless particles(including gravitons and photons) and the massive particles. (Or perhaps vice versa again see the discussion below.)

An example of a possible natural projective decomposition (dened via a limit) is the set of projections onto thesimultaneous eigenstates of single-particle position operators. [19]


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Other examples are the sets of projections onto simultaneous eigenstates of momentum operators or of energyoperators. Another example, which depends on the state (t) at any given time (or the state on any given hypersurface,etc.) is determined by the Schmidt decomposition

(t) =j

( pj (t))1 / 2 ej (t) f j (t) ,

where for each time t the sets { ej (t) } and { f j (t) } are subsets of orthonormal bases of HA and HB respectively. Inthis case we can take the projectors P i at time t to be projections onto the subspaces f j (t) : pj (t) = pi (t) spannedby vectors f j (t) whose Schmidt coefficients are equal.

Given the chosen projective decomposition, we can decompose the state at any time as

(t) =k

(I P k ) (t) =k

k (t) , (2)

where k (t) = ( I P k ) (t). Here the range of k may depend on time t: for instance, if the P k are Schmidt projections,the size of their set depends on the number of degeneracies in the Schmidt decomposition. The notation k shouldbe read as implicitly allowing this, and also as allowing integrals over continuous ranges instead of, or combined with,a discrete sum.

To avoid any possible misinterpretation at this point, note that the branching structure we propose below will not be directly dened by this decomposition. The possible branch states at time t will not be dened to be the k (t),but rather certain mixed states depending on them: see equation ( 4) below.

Suppose now that we have some hypothetical fundamental physical theory which stipulates that time runs onlyfrom 0 to T , so that we have initial state I = (0) and nal state F = (T ). We have the nal state decomposition

(T ) =l

(I P l )(T ) =l

l (T ) . (3)

We will dene a set of branches with a one-to-one correspondence between the branches B l and those states l (T ) inequation ( 3) that are nonzero. [20]

Dene

q T l (k, t ) = | ( l (T ) , exp( iH (T t)/ )k (t) ) |2

and

pT l (k, t ) =q T l (k, t )

k q T l (k, t ).

DeneT l (t) =

k

Tr H B (k (t)k (t))( k (t) , k (t) )

pT l (k, t ) , (4)

where the sum is over indices k for which k (t) is nonzero. We call T l (t) the real state of branch B l at each time t.Note that T l (t) is generally a mixed quantum state and is dened on HA alone.

Postulate 3 In a closed quantum system for which a fundamental physical theory stipulates that time runs only from 0 to T , precisely one of the branches B l is realised. The branch B l is realised with probability

pT l = ( l (T ) , l (T ) ) = ( l (T ) , (T ) ) .

If B l is realised, then reality at each time t in the range 0 t T is described by the real state T l (t).Asymptotic Hypothesis In the fundamental theories of closed quantum systems that we consider, the quan-

tities dened above have asymptotic limits as T . That is, the probabilities pT l tend to constants pl as T

and the real state T l (t), for any nite t, tends to a constant state l (t) as T .Postulate 4 In a closed quantum system, described by a fundamental theory satisfying the asymptotic hypoth-

esis, for which time runs from 0 to , precisely one of the branches B l is realised. The branch B l is realised with probability given by the asymptotic value

pl = limT pT l .

If B l is realised, then reality at each time t 0 is described by the real state dened by the asymptotic limit

l (t) = limT T l (t) .


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DISCUSSION

Ontology

The real quantum state plays a crucial role in real world interpretations, directly representing physical reality.Indeed, perhaps the most natural version of a real world interpretation is to take the real quantum state as theprimary physical quantity, and to relegate the standard quantum state vector to the status of an auxiliary mathe-matical quantity, useful for calculating the possible realised branches and their probabilities, but not necessarily itself corresponding to anything in reality. In Bells terminology, on this view, the real state corresponds to a beable, whilethe standard quantum state need not.

While it may perhaps seem unfamiliar to suggest representing reality directly by a time- or hypersurface-dependentdensity matrix, this is precisely what is proposed in realist many-worlds versions of quantum theory. The maindifferences here are that we have (a proposal aimed at) a realist picture of one quasiclassical world rather than manyeffectively independent such worlds, and a density matrix of generally large rank rather than (as could be possible inmany-worlds quantum theory, given pure unitary evolution from a pure initial cosmological state) of rank one. Therst is a positive advantage, and it is not evident why the second should create new problems.

That said, one could, alternatively, take some mathematical quantity derived from the real state as representingreality (i.e. as the beable), and simplify the picture. We leave this interesting possibility for future exploration.

The factorization of H into HA HB is meant to be encoded into the laws of nature; not an arbitrary choice,nor an observer-dependent construction, nor a split into system and environment dened for a particular experiment.

Moreover, the degrees of freedom characterised by HB play a different role to those characterised by HA : the HBdegrees of freedom are auxiliary mathematical objects, which do not necessarily have any direct correspondence toreality, whereas the HA degrees of freedom do directly represent reality.

Correspondence to reality

Why might one hope such a picture can accurately represent the quasiclassical world we perceive?

Role of the fundamental factorization

Firstly, we require that the factorization H = HA HB allows us to characterise quasiclassical physics entirely in

terms of degrees of freedom corresponding to HA .In the case of quantum eld theory in Minkowski space, it seems fairly clear that this can be done. If HA andHB are the subspaces corresponding respectively to fermions and bosons, or massive and massless particles, then ineither case we can represent the local densities of chemical species in terms of HA degrees of freedom. We can thenexpress quasiclassical variables describing, for instance, the orbit of the Moon around the Earth, or the behaviourof apparatus and measuring devices in a two-slit experiment in terms of these local densities.

Though it may seem slightly less intuitive, this is also presumably true if we swap the roles of HA and HB . In thatcase, the photon degrees of freedom in HA allow us to characterise atoms and larger aggregations of matter in termsof their associated electromagnetic elds, and hence to recover a description of the quasiclassical world.

In the case of quantum gravity, the question is, as usual, less clear. Taking HA and HB to correspond respectivelyto the gravitational and matter degrees of freedom allows the spacetime to be characterised entirely by HA degreesof freedom. Since any signicant quasiclassical event involving matter eventually has a signicant effect on thegravitational eld, this might perhaps suffice (though the scale of the timelag and its ontological signicance would

need to be considered carefully). If HA corresponds to massless particles, including gravitons and photons, then, asabove, we can characterise matter distributions in terms of their associated electromagnetic elds, and thus one wouldexpect to be able to give a direct description of both matter and metric at classical scales.

As we lack both a well dened quantum gravity theory and a clear understanding of the relevant physics as theuniverse tends towards its nal state, we can offer nothing better than reasoned conjecture. Still, these possibilitiessuggest it is reasonable to hypothesise that an appropriate factorization might exist in quantum gravity theories.

Given an appropriate choice of factorization, we can think of the degrees of freedom in HA as characterising aquasiclassical system, and those in HB an environment interacting with that system. We can also think of ourselvesas, in a certain sense, contained within the system and distinct from the environment. The system and environmenthere are not disjoint and separated physical entities: it is not the case that every physical degree of freedom within


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the region dened by our bodies belongs to HA . Nonetheless, all the quasiclassical variables which we assume deneour experiences can be dened by degrees of freedom in HA and, in that looser but natural sense, one may say webelong to HA .

Role of the nal projective decomposition

Part of the intuition behind dening branches via the nal state decomposition is that, for a realistic model withrealistic dynamics, an appropriate factorization, and an appropriate nal projective decomposition, (i) each possiblenal state of the environment characterises a possible quasiclassical history of the system, (ii) signicantly differentnal states characterize signicantly different quasiclassical histories, (iii) signicantly different quasiclassical historiesare characterized by signicantly different nal states. In particular, the effects of any signicant quasiclassical eventare irreversibly recorded in the environment in such a way that the record can be read by an appropriate abstractlyspecied nal projective decomposition, such as one of those discussed above: incoming photons scatter differentlyoff different positions of a macroscopic pointer, leading to distinct nal states of the photon degrees of freedom, forexample. The idea here is that unitary evolution acting on any two signicantly distinct alternative quasiclassicaloutcomes arising naturally in cosmology, or of a quantum experiment should produce at t = states which are notonly very nearly orthogonal but moreover are distinguished with near certainty by an appropriate natural asymptotic projective decomposition on HB . For example, if the projective decomposition is onto photon momentum states, theintuition is that all the later time quantum states resulting from two different possible outcomes of a measurementare distinguished by photon momenta, because the background electromagnetic eld some time after the experimenteffectively carries a record of the result which will persist to the end of time. [21]

We use the nal projective decomposition to dene an effective post-selection on the ensemble of possible evolutions.One way of picturing this is that Nature rst simulates the evolution of the universe using standard quantum theory,with the initial state I = (0), carries out a measurement at t = , dened by the nal state decomposition, on thissimulated system, and then uses the result of this measurement to determine the branch that is realised in the actualuniverse. A less teleological and less anthropomorphic view is simply to think of the branches as natural structuresthat dene a sample space for a fundamentally probabilistic theory, which has a natural probability distributiondened by the Born rule. This is a probabilistic block universe theory: it simply says that one element of the samplespace is randomly selected, and this random choice determines reality.

Role of the nite time projective decompositions

Suppose now that we adopt the view just mentioned, namely that our universe is effectively post-selected by theoutcome of a nal projective measurement at t = , dened by the asymptotic projective decomposition. One wouldthen expect, if we aim for an interpretation that produces a description of reality at times between 0 and , thatthis description should be determined by the post-selected nal state as well as the initial state.

If we live in such a universe, then we have no direct knowledge of the post-selected nal measurement outcome,of course. But, given a theory which xes the initial state and the Hamiltonian, we could in principle list allthe possible nal measurement outcomes, and use the standard quantum pre- and post-selection rules to infer theprobabilities of our observations (of our own experiments or of other events) conditioned on any given outcome of thisnal measurement. Calculating the probabilities of our observations this way, although unnecessarily complicated,would be consistent with the usual calculations of the probabilities in standard (non-post-selected) quantum theory:postulating a nal measurement does not affect the intervening dynamics or the overall (unconditioned) probabilityof any intervening event.

Let us imagine further that a measurement were carried out at time t on HB , dened by the projective decompositionat time t. If our experience is characterised by HA i.e. if the only degrees of freedom we need to describe it arethose of HA then we learn nothing directly about the outcome of the measurement on HB . However, supposingfor the moment that this is the only measurement intervening between the initial and nal states, we can infer theprobabilities of the various outcomes from the initial state and any possible nal state. We can further calculate adensity matrix describing the state of the HA degrees of freedom at time t, using these probabilities: this densitymatrix depends on the nal state l , and hence (since we dene the branch by the nal state) on the branch B l . Thisis precisely the calculation by which we arrive at the real state ( 4).

Now we take a further step. We calculate density matrices on HA at every time t using the above procedure, and wesuppose that we can give a coherent time-dependent description of evolving physical worlds using the density matrices


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l (t) corresponding to branch B l at each time t.[22] Although it ts quite naturally within the quantum formalism,this postulate does not follow from standard quantum theory. It can ultimately only be justied empirically.

What could constitute empirical justication? Roughly speaking, we would need to show that, given a realisticmodel, the proposed interpretation explains why we experience a persistently quasiclassical world of the type wedo. One good test would be whether it predicts with high probability that the realised world will be persistentlyquasiclassical and that our world is in some sense typical in the set of realised worlds.[ 23] A weaker, but still signicant,and arguably necessary, criterion is whether it predicts, with high probability, that if the realised world is quasiclassical,with large-scale structure, for some reasonably long time interval, then it will continue in that state for a further longinterval.[ 24]

These properties will clearly not hold for worlds dened by a generic choice of time-dependent projective decompo-sition on HB , nor will the asymptotic hypothesis hold true for generic choices. The particular choices suggested aboveare chosen because it seems plausible that one or more of them might indeed dene persistently quasiclassical worldsin realistic models, and that (for essentially the same reason) the asymptotic hypothesis might indeed hold true insuch models. The picture underlying this intuition is one of an ever-expanding universe, starting from a conventionalbig bang and tending towards a big freeze, in which, as matter-matter interactions become less and less frequent, sodo unpredictable quasiclassical events and thus so should branchings of quasiclassical worlds. While of course thereare many unknowns and uncertainties, this picture appears consistent with our current understanding of cosmology.

The idea then is that the split HA HB and the postulated natural projective decompositions at each time (oneach hypersurface, etc) are chosen so that the projective decomposition at any given time characterises quasiclassicalinformation about HA effectively irreversibly recorded in degrees of freedom of HB . Each possible outcome of thenal projective decomposition characterises a possible branch, in which all the information necessary to characterisethe nal quasiclassical state of HA is encoded in the corresponding outcome state in HB . For each possible branch, wecan dene the real state at any given time t, a mixed state on HA dened by the pre- and post-selected measurementprobabilities for the natural projective decomposition on HB at time t. This state approximately characterises thequasiclassical information about HA that is already, at time t, irreversibly encoded in HB .

The real state of any given branch is thus dened via a procedure that is dened by rules based on standard unitaryquantum dynamics (with a specied Hamiltonian) and on the projection postulate though the real state itself neither follows standard unitary quantum dynamics nor standard projection postulate induced state collapse.

Can we, in fact, show that for one of the natural factorizations and natural sets of projective decompositionsdescribed above, it is indeed the case that, to good approximation, in realistic cosmological models, the projectivedecomposition at any given time characterises quasiclassical information about HA irreversibly recorded in degrees of freedom of HB , and the asymptotic hypothesis holds? The question should be, at any rate, decidable, since there arerather few choices that have any serious claim to be regarded as natural. The projective decomposition most studied

to date in other contexts in particular in investigations of modal interpretations [3] of quantum theory is theSchmidt decomposition. Reasons have been identied for doubting that it generally identies quasiclassical degreesof freedom in realistic models [ 16], It seems fair to say, though, that the question is not yet denitively resolved. Inany case, the alternatives of projective decompositions dened by the position and momentum eigenstates of elds inHB look very promising.

In short, there are good intuitive reasons to take the hypothesis seriously. We may, in fact, have a relatively naturalsolution to the measurement problem, which essentially respects unitary quantum dynamics and Lorentz or generalcovariance. The form of the real state also means that it is at least not obvious that this interpretation has the sort of potentially worrisome tails problem that arises in modied dynamical theories in which suppressed components of thestate vector corresponding to unselected quasiclassical worlds acquire exponentially small amplitudes but nonethelessretain (something like) the same mathematical structure and information content of the selected component.

Detailed calculations in realistic models remain to be carried out; they should illuminate these questions furtherand, one might reasonably hope, give persuasive evidence that a well dened real world interpretation can adequately

describe the quasiclassical reality we experience.

ACKNOWLEDGMENTS

Many thanks to Jeremy Buttereld for many helpful comments on an earlier version of the manuscript; thanks alsoto Lucien Hardy, Jonathan Oppenheim, Tony Short, Rafael Sorkin, Rob Spekkens and James Yearsley for helpfulconversations. This research was partially supported by a Leverhulme Research Fellowship, a grant from the JohnTempleton Foundation, an fqxi mini-grant and by Perimeter Institute for Theoretical Physics. Research at Perimeter


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Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario throughthe Ministry of Research and Innovation.

[1] H. Everett III, Reviews of Modern Physics 29 , 454-462 (1957). Reprinted in The many-worlds interpretation of quantum mechanics , DeWitt, B. and N. Graham (Eds.) (Princeton: Princeton University Press, (1973)).

[2] J.S. Bell, Quantum Mechanics for Cosmologists in Quantum Gravity 2 , C. Isham, R. Penrose and D. Sciama (eds.),(Clarendon Press, Oxford, 1981); reprinted in J.S. Bell, Speakable and unspeakable in quantum mechanics , (CambridgeUniversity Press, Cambridge, 1987).

[3] B. van Fraassen, A formal approach to the philosophy of science, in Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain , Colodny, R. (ed.), pp. 303-366 (Pittsburgh: University of Pittsburgh Press, 1972); B.van Fraassen, Synthese 29 , 291-309 (1974); S. Kochen, A new interpretation of quantum mechanics in Symposium on the Foundations of Modern Physics 1985 , Mittelstaedt, P. and Lahti, P. (eds), pp. 151-169 (Singapore: World Scientic,1985); D. Dieks, Annalen der Physik, 7 , 174-190 (1988); D. Dieks, Foundations of Physics, 19 , 1397-1423 (1989); R. Healey,The Philosophy of Quantum Mechanics: An Interactive Interpretation (Cambridge: Cambridge University Press, 1989);G. Bacciagaluppi and M. Dickson, Foundations of Physics 29 , 1165-1201, (1999).

[4] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34 , 470 (1986); G. C. Ghirardi, P. Pearle, and A. Rimini, Phys.Rev. A 42 , 78 (1990).

[5] M. Gell-Mann and J.B. Hartle, gr-qc/9404013; J.B. Hartle, Physica Scripta T76 67 (1998); R. B. Griffiths, Consistent Quantum Theory (Cambridge University Press, Cambridge, 2002).

[6] W.C. Davidon, Il Nuovo Cimento 36 34 (1976); Y. Aharonov and E.Y. Gruss, quant-ph/0507269.

[7] S. Saunders, Synthese 102 , 235-266. (1995); D. Wallace, Studies in the History and Philosophy of Modern Physics 33 ,637-661 (2002); D. Wallace, Studies in the History and Philosophy of Modern Physics 34 , 87-105 (2003).[8] R. Blume-Kohout and W.H. Zurek, Phys. Rev. A 73 , 062310 (2006)[9] J.S. Bell, The measurement theory of Everett and de Broglies pilot wave in Quantum Mechanics, Determinism, Causality

and Particles , M. Flato et al. (eds), (D. Reidel, Dordrecht, Holland, 1976); reprinted in J.S. Bell, Speakable and unspeakable in quantum mechanics , (Cambridge University Press, Cambridge, 1987).

[10] E.g. A. Kent, Int. J. Mod. Phys. A5 1745 (1990).[11] A. Kent, One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientic conr-

mation in Many Worlds? Everett, Quantum Theory and Reality , S. Saunders, J. Barrett, A. Kent and D. Wallace (eds),(Oxford University Press, Oxford, 2010); arXiv:0905.0624.

[12] D. Albert, Probability in the Everett Picture in Many Worlds? Everett, Quantum Theory and Reality , S. Saunders,J. Barrett, A. Kent and D. Wallace (eds), (Oxford University Press, Oxford, 2010). See also other articles in the samevolume for other perspectives, for and against.

[13] D. Deutsch, Quantum Theory of Probability and Decisions , Proceedings of the Royal Society of London A 455, 3129-3137(1999).

[14] D. Wallace, How to prove the Born rule in Many Worlds? Everett, Quantum Theory and Reality , S. Saunders, J. Barrett,A. Kent and D. Wallace (eds), (Oxford University Press, Oxford, 2010);

[15] D. Papineau, A Scandal of Probability Theory, in Many Worlds? Everett, Quantum Theory and Reality , S. Saunders,J. Barrett, A. Kent and D. Wallace (eds), (Oxford University Press, Oxford, 2010);

[16] G. Bacciagaluppi, M. Donald and P. Vermaas, Continuity and discontinuity of denite properties in the modal interpreta-tion , Helvetica Physica Acta, 68 , 679-704, (1995); G. Bacciagaluppi and M. Hemmo, Modal interpretations, decoherence and measurements , Studies in History and Philosophy of Modern Physics 27 , 239-277, (1996); G. Bacciagaluppi, Delocal-ized properties in the modal interpretation of a continuous model of decoherence , Foundations of Physics, 30 , 1431-1444(2000).

[17] See e.g. [1315] for some proposals.[18] And, plausibly, there is a natural sense in which the universe continues to become more quasi-classical: in other words,

new quasi-classical structure continues to emerge, and perhaps always will.[19] That these eigenstates are not dened in the original Hilbert space H does not imply that the objects we are ultimately

interested in, the real states ( 4), are not well-dened density matrices on H A . In fact, physical intuition suggests they

should be well-dened, since they represent intermediate states of theH

A subsystem given a nal position measurement(of unknown outcome) on H B .[20] Here, to simplify the notation, we consider the case where the sum is discrete. If we have a continuous set of branches,

then to obtain nite quantities we need to integrate over small intervals ( l, l + l ) in branch parameter space, and thenconsider the limit as l 0.

[21] The intuition is that this is normally true, not that it is absolutely necessarily true of every possible experiment. Forinstance, with sufficiently advanced technology, the measurement itself and the electromagnetic elds it generates mightbe made part of a macroscopic interference experiment. In such a case, a real world interpretations description of realitymight not necessarily imply that the measurement must have an essentially denite outcome in the realised branch. Wepresently have no experiential data to suggest otherwise, of course.

[22] Recall that we do not suppose that the measurements at each time t actually take place. If they did, they would produce
http://arxiv.org/abs/gr-qc/9404013http://arxiv.org/abs/quant-ph/0507269http://arxiv.org/abs/0905.0624http://arxiv.org/abs/0905.0624http://arxiv.org/abs/quant-ph/0507269http://arxiv.org/abs/gr-qc/9404013


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a non-unitary evolution of the quantum state, and any picture of evolving quasiclassical worlds which emerged would bevery different from that proposed here.

[23] In other words, we see the sort of world we do because almost all worlds are of that sort.[24] In other words, if the realised world contains the sort of structures that we see for some time interval, then it will very

probably persist in that state. If so, we see a persistently quasiclassical world, plausibly, because almost all worlds in whichlife could emerge in the rst place are of that sort.

real world interpretations of quantum theory 0708.3710

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