don n. page- insufficiency of the quantum state for deducing observational probabilities

8/3/2019 Don N. Page- Insufficiency of the Quantum State for Deducing Observational Probabilities

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arXiv:0808

.0722v3

[hep-th]

17Sep2008

Insufficiency of the Quantum State for Deducing Observational Probabilities

Don N. Page

Theoretical Physics Institute

Department of Physics, University of AlbertaRoom 238 CEB, 11322 89 Avenue

Edmonton, Alberta, Canada T6G 2G7(Dated: 2008 September 17)

It is usually assumed that the quantum state is sufficient for deducing all probabilities for a system.This may be true when there is a single observer, but it is not true in a universe large enough thatthere are many copies of an observer. Then the probability of an observation cannot be deducedsimply from the quantum state (say as the expectation value of the projection operator for theobservation, as in traditional quantum theory). One needs additional rules to get the probabilities.What these rules are is not logically deducible from the quantum state, so the quantum state itselfis insufficient for deducing observational probabilities. This is the measure problem of cosmology.

PACS numbers: PACS 03.65.Ta, 03.65.Ca, 02.50.Cw, 98.80.Qc,

I. INTRODUCTION

All probabilities for a system are believed to be en-

coded in its quantum state. This may be true, but thereis the question of how to decode the quantum state togive these probabilities. In traditional quantum theory,the probabilities are given by the expectation values ofprojection operators. Once a possible observation is spec-ified (including the corresponding projection operator),then its probability is given purely by the quantum stateas the expectation value the state assigns to the projec-tion operator, a mathematization of the Born rule [1].

This prescription works well in ordinary single lab-oratory settings, where there are no copies of the ob-server. Then distinct observations are mutually exclu-sive, so that different ones cannot both be observed. If

one assigns a projection operator to each possible dis-tinct observation in a complete exhaustive set, then theseprojection operators will be orthonormal, and their (non-negative) expectation values will sum to unity, which areconditions necessary for them to be interpreted as theprobabilities of the different possible observations.

However, in cosmology there is the possibility that theuniverse is so large that there are many copies of each ob-server, no matter how precisely the observer is defined.This raises the problem [2, 3, 4, 5] that two observa-tions that are seen as distinct for an observer are notmutually exclusive in a global viewpoint; both can occurfor different copies of the observer (though neither copy

may be aware of that). This would not be a problemfor a putative superobserver who can observe all possi-ble sets of observations by all observers over the entireuniverse, but it is a problem for the assignment of nor-malized probabilities for the possible observations thatare distinct for each copy of the observer. The result [4]is that one cannot get such a set of normalized probabil-

Electronic address: [email protected]

ities as the expectation values of projection operators inthe full quantum state of the universe.

One can still postulate that there are rules for get-

ting the probabilities of all possible observations from thequantum state, but then the question arises as to whatthese rules are. Below we shall give examples of severaldifferent possibilities for these rules, showing that theyare not uniquely determined and thus that they are log-ically independent of the question of what the quantumstate is. Therefore, the quantum state just by itself is in-sufficient to determine the probabilities of observations.

The main application of the logical independence ofthe probability rules is to the measure problem in cosmol-ogy (see [5] for many references), the problem of how tomake statistical predictions for observations in a universethat may be so large that almost all theoretically possi-ble observations actually occur somewhere. The logicalindependence implies that the solution to the measureproblem is not just the quantum state of the universe butalso other independent elements, the rules for getting theprobabilities of the observations from the quantum state.

II. OBSERVATIONAL PROBABILITIES WITH

MANY COPIES OF THE OBSERVER

A goal of science is to come up with theories Ti thatpredict the probabilities of results of observations (obser-vational results). Here for simplicity I shall assume thatthere is a countable set of possible distinct observationsOj out of some exhaustive set of all such observations.This set of possible observations might be, for example,all possible conscious perceptions [6], all possible datasets for one person, all possible contents for an eprintarXiv, or all possible data sets for a human scientific in-formation gathering and utilizing system [2]. If one imag-ines a continuum for the set of observations (which seemsto be logically possible, though not required), in that caseI shall assume that they are binned into a countable num-ber of exclusive and exhaustive subsets that each may be
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considered to form one distinct observation Oj . Then thegoal is to calculate the probability Pj(i) P(Oj |Ti) forthe observation Oj , given the theory Ti.

One might think that once one has the quantum state,there would be a standard answer to the question of theprobabilities for the various possible observations. Forexample [4], one might take traditional quantum theory(what I there called standard quantum theory) to give the

probability Pj(i) of the observation as the expectationvalue, in the quantum state given by the theory Ti, of aprojection operator Pj onto the observational result Oj .That is, one might take

Pj(i) = Pji, (2.1)

where i denotes the quantum expectation value ofwhatever is inside the angular brackets in the quantumstate i given by the theory Ti. This traditional approachworks in the case of a single laboratory setting where theprojection operators onto different observational resultsare orthogonal, PjPk = jkPj (no sum over repeatedindices).

However [4, 5], in the case of a sufficiently large uni-verse, one may have observation Oj occurring here andobservation Ok occurring there in a compatible way, sothat Pj and Pk are not orthogonal. Then the traditionalquantum probabilities given by Eq. (2.1) will not be nor-malized to obey

j

Pj(i) = 1. (2.2)

Thus one needs a different formula for normalizable prob-abilities of a mutually exclusive and exhaustive set ofpossible observations, when distinct observations within

the complete set cannot be described by orthogonal projection operators.Although many other rules are also possible, as I shall

illustrate below, the simplest class of modifications ofEq. (2.1) would seem to be to replace the projection op-erators Pj with some other observation operators Qj(i)normalized so that

jQj(i)i = 1, giving

Pj(i) = Qj(i)i. (2.3)

Of course, one also wants Pj(i) 0 for each i and j, soone needs to impose the requirement that the expectationvalue of each observation operator Qj(i) in each theoryTi is nonnegative.

The main point [4, 5] is that in cases with more thanone copy of the observer, such as in a large enough uni-verse, one cannot simply use the expectation values ofprojection operators as the probabilities of observations,so that, if Eq. (2.3) is to apply, each theory must as-sign a set of observation operators Qj(i), correspondingto the set of possible observations Oj , whose expectationvalues are used instead as the probabilities of the obser-vations. Since these observation operators are not givendirectly by the formalism of traditional quantum theory,

they must be added to that formalism by each particularcomplete theory. In other words, a complete theory Ticannot be given merely by the dynamical equations andinitial conditions (the quantum state), but it also requiresthe set of observation operators Qj(i) whose expectationvalues are the probabilities of the observations Oj in thecomplete set of possible observations (or else some otherrule for the probabilities, if they are not to be expecta-

tion values of operators). The probabilities are not givenpurely by the quantum state but have their own logicalindependence in a complete theory.

Let us suppose that we can hypothetically partitionspacetime into a countable set of disjoint regions labeledby the index L, with each region having its own referenceframe and being sufficiently small that for each L sepa-rately there is a set of orthogonal projection operatorsPLj whose expectation values give good approximationsto the probabilities that the observations Oj occur withinthe region L. Each region has its own algebra of quantumoperators, and for simplicity I shall make the somewhatunrealistic assumption that the regions are either space-like separated or are so far apart that each operator inone region, such as PLj , commutes with each observable

in a different region, such as PMk for L = M. (However,I am assuming that each observation Oj can in princi-ple occur within any of the regions, so that the contentof the observation is not sufficient to distinguish what Lis; the observation does not determine where one is inspacetime. One might imagine that an observation de-termines as much as it is possible to know about somelocal region, but it does not determine the properties out-side, which might go into the specification of the index Lthat is only known to a hypothetical superobserver thatmakes the partition.)

Now one might propose that one construct the projec-tion operator

Pj = I

L

(IPLj ) (2.4)

(this being the only place where I need the PLj s to com-mute for different L) and use it in Eq. (2.1) to get aputative probability of the observation Oj in the quan-tum state given by the theory Ti. Indeed, this is essen-tially in quantum language [4] what Hartle and Srednicki[2] propose, that the probability of an observation is theprobability that it occur at least somewhere. However,because the different Pj s defined this way are not or-thogonal, the resulting traditional quantum probabilitiesgiven by Eq. (2.1) will not be normalized to obey Eq.(2.2). This lack of normalization is a consequence of thefact that even though it is assumed that two differentobservations Oj and Ok (with j = k) cannot both oc-cur within the same region L, one can have Oj occurringwithin one region and Ok occurring within another re-gion. Therefore, the existence of the observation Oj atleast somewhere is not incompatible with the existence ofthe distinct observation Ok somewhere else, so the sum ofthe existence probabilities is not constrained to be unity.


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If one were the hypothetical superobserver who has ac-cess to what is going on in all the regions, one could makeup a mutually exclusive and exhaustive set of joint ob-servations occurring within all of the regions. However,for us observers who are confined to just one region, theprobabilities that such a superobserver might deduce forthe various combinations of joint observations are inac-cessible for us to test or to use to predict what we might

be expected to see. Instead, we would like probabilitiesfor the observations we ourselves can make. I am as-suming that each Oj is an observational result that inprinciple we could have, but that we do not have accessto knowing which region L we are in. (The only proper-ties of L that we can know are its local properties thatare known in the observation Oj itself, but that is notsufficient to determine L, which might be determined byproperties of the spacetime beyond our local knowledge.)

III. EXAMPLES OF DIFFERENT

OBSERVATIONAL PROBABILITIES FOR THE

SAME QUANTUM STATE

Let us demonstrate the logical freedom in the rules forthe observational probabilities Pj(i) P(Oj |Ti) by ex-hibiting various examples of what they might be. Forsimplicity, let us restrict attention to theories Ti that allgive the same pure quantum state |, which can be writ-ten as a superposition, with fixed complex coefficientsaN, of component states |N that each have differentnumbers N of observational regions:

| =

N=0

aN|N, (3.1)

where M|N = MN. (Different values of N modeldifferent sizes of universes produced by differing amountsof inflation in the cosmological measure problem.) Thedifferent theories will then differ only in the prescrip-tions they give for calculating the observational proba-bilities Pj(i) from the single quantum state |. Thesedifferences will illustrate the logical independence of theobservational probabilities from the quantum state, thefact that the observational probabilities are not uniquelydetermined by the state.

In each component state, the index L can run from 1to N (except for the component state |0, which hasno observational regions at all). As above, let us sup-

pose that PL

j is a complete set of orthogonal projectionoperators for the observation Oj to occur in the regionL. Then if only the region L existed, and the state were|N, then the quantum probability of the observationOj would be

pNLj = N|PLj |N. (3.2)

However, in reality, even just in the component state|N for N > 1, there are other regions where the obser-vation could occur, so the total probability Pj(i) for the

observation Oj in the theory Ti can be some i-dependentfunction of all the pNLj s. The freedom of this functionis part of the independence of the observational proba-bilities from the quantum state itself.

Let us define existence probabilities pj(i) that mightnot be normalized to add up to unity when summed over

j for the different possible observational results Oj , andthen use Pj(i) for normalized observational probabilities

obeying Eq. (2.2). The different indices i will denotedifferent theories, in this case different rules for calculat-ing the probabilities, since for simplicity we are assumingthat all the theories have the same quantum state |.

Next, let us turn to different possible examples.For theory T1, let us suppose that the existence prob-

ability pj(1) = 0 if there is no region L in any nonzerocomponent of the quantum state (|N with aN = 0)that has a positive expectation value for PLj , so that

N,L |aN|2pNLj = 0, but that pj(1) = 1 otherwise, that

is if

N,L |aN|2pNLj > 0. This theory is essentially tak-

ing the Everett many worlds interpretation to imply thatif there is any nonzero amplitude for the observation to

occur, it definitely exists somewhere in the many worlds(and hence has existence probability unity).

Now of course these existence probabilities pj(1) arenot necessarily normalized, since M > 1 of them can beunity, with the rest zero, so the sum of these existenceprobabilities is M. However, one could say that so far asthe probability goes of making a particular one of the Mactually existing observations, that could be consideredequally divided between the M actually existing possibil-ities (ifM > 0), so that one has normalized observationalprobabilities Pj(1) = pj(1)/M. This would be the theorythat every observation that actually does exist is equallyprobable.

For theory T2, define the existence probabilities to bepj(2) = |Pj |, the expectation value in the full quan-tum state | of the projection operator Pj defined byEq. (2.4) for the existence of the observation Oj in atleast one region L. This is not the full many-worlds exis-tence probability, which is unity if the observation doesoccur somewhere, but it might be regarded as the quan-tum probability for a superobserver to find that at leastone instance of the observation Oj occurs.

Again, these existence probabilities will not in generalsum to unity, but one can normalize by dividing by thesum M (now generically not an integer) to get normalizedprobabilities Pj(2) = pj(2)/M. For a universe that isvery large (e.g., most of the |aN|

2s concentrated on verylarge Ns), one would expect a large number of pj(2)s tobe very near unity (since it would be almost certain thatthe observation Oj occurs at least somewhere among thehuge number of regions), so that there will be a largenumber of nearly equal but very small Pj(2)s.

For theory T3, refrain from defining the existence prob-abilities pj(3) (since I am regarding it sufficient for atheory to prescribe only the observational probabilitiesof observations in regions within the universe, not forobservations by some putative superobserver). Instead,


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define unnormalized observational measures

j(3) =

L

|PLj | =

N=1

N

L=1

|aN|2pNLj . (3.3)

Then normalize these to let the observational probabili-ties be defined as

Pj(3) = j(3)k k(3)

. (3.4)

For theory T4, define unnormalized observational mea-sures

j(4) =

N=1

1

N

N

L=1

|aN|2pNLj (3.5)

and then normalized observational probabilities

Pj(4) =j(4)k k(4)

. (3.6)

Theory T3 in its sum over L effectively weights eachcomponent state |N by the number of observationalregions where the observation Oj can potentially occur.On the other hand, theory T4 has an average over L foreach total number N of observational regions, so thatcomponent states |N do not tend to dominate the prob-abilities for observations just because of the greater num-ber of observational opportunities within them. TheoryT3 is analogous to volume weighting in the cosmologicalmeasure, and theory T4 is analogous to volume averaging[5].

Of these four rules, T1 does not give the probabilities asthe expectation values of natural observation operators

Qj(i), but the other three do, with the correspondingobservation operators being

Qj(2) =Pj

kPki,

Qj(3) =

LP

Lj

k

LP

Lk i

,

Qj(4) =

N=11

N

NL=1PNP

Lj PN

k

N=11

N

NL=1PNP

LkPNi

, (3.7)

where PN = |NN| is the projection operator ontothe component state with N observation regions.

These examples show that there is not just one uniquerule for getting observational probabilities from the quan-tum state. It remains to be seen what the correct rule is.Of the four examples given above, I suspect that with asuitable quantum state, theory T4 would have the highest

likelihood Pj(i), given our actual observations, since the-ories T1 and T2 would have the normalized probabilitiesnearly evenly distributed over a huge number of possibleobservations, and theory T3 seems to be plagued by theBoltzmann brain problem [5]. One might conjecture thattheory T4 can be implemented in quantum cosmology tofit observations better than other alternatives [5].

Thus we see that in a universe with the possibility ofmultiple copies of an observer, observational probabili-ties are not given purely by the quantum state, but alsoby a rule to get them from the state. There is logicalfreedom in what this rule is (or in what the observationoperators Q

j(i) are if the rule is that the probabilities are

the expectation values of these operators). In cosmology,finding the correct rule is the measure problem.

Acknowledgments

I am grateful for discussions with Tom Banks, RaphaelBousso, Sean Carroll, Brandon Carter, Alan Guth, JamesHartle, Andrei Linde, Seth Lloyd, Juan Maldacena, MarkSrednicki, Alex Vilenkin, an anonymous referee, and oth-ers, and especially for a long email debate with Hartle andSrednicki over typicality that led me to become convincedthat there is logical freedom in the rules for getting ob-servational probabilities from the quantum state. Thisresearch was supported in part by the Natural Sciencesand Engineering Research Council of Canada.

[1] M. Born, Z. Phys. 37, 863-867 (1926).[2] J. B. Hartle and M. Srednicki, Phys. Rev. D 75, 123523

(2007) [arXiv:0704.2630].[3] D. N. Page, Typicality Defended, arXiv:0707.4169.[4] D. N. Page, Phys. Rev. D 78, 023514 (2008)

[arXiv:0804.3592].[5] D. N. Page, D. N. Page, Cosmological Measures without

Volume Weighting, arXiv:0808.0351.[6] D. N. Page, Sensible Quantum Mechanics: Are Only

Perceptions Probabilistic? arXiv:quant-ph/9506010; Int.J. Mod. Phys. D5, 583 (1996) [arXiv:gr-qc/9507024]; inConsciousness: New Philosophical Perspectives, editedby Q. Smith and A. Jokic (Oxford, Oxford Univer-sity Press, 2003), pp. 468-506 [arXiv:quant-ph/0108039];in Universe or Multiverse?, edited by B. Carr (Cam-bridge, Cambridge University Press, 2007), pp. 411-429[arXiv:hep-th/0610101].
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don n. page- insufficiency of the quantum state for deducing observational probabilities

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