EEverett and the Born Rule
Alastair I. M. Rae
School of Physics and Astronomy, University of BirminghamBirmingham B15 2TT
Abstract
During the last ten years or so, derivations of the Born rule based on decisiontheory have been proposed and developed, and it is claimed that these arevalid in the context of the Everett interpretation. This claim is critically as-sessed and it is shown that one of its key assumptions is a natural consequenceof the principles underlying the Copenhagen interpretation, but constitutesa major additional postulate in an Everettian context. It is further arguedthat the Born rule, in common with any interpretation that relates outcomelikelihood to the expansion coefficients connecting the wavefunction with theeigenfunctions of the measurement operator, is incompatible with the purelyunitary evolution assumed in the Everett interpretation.
Key words:
Everett, Born Rule, Quantum Measurement
Preprint submitted to Studies in the History and Philosophy of Science October 25, 2018 a r X i v : . [ qu a n t - ph ] J un verett and the Born Rule Alastair I. M. Rae
School of Physics and Astronomy, University of BirminghamBirmingham B15 2TT
Introduction
The conventional (“Copenhagen”) interpretation of quantum mechanicsstates that the result of a measurement is one (and only one) of the eigen-values belonging to the operator representing the measurement and that,following the measurement, the wavefunction “collapses” to become the cor-responding eigenfunction (ignoring the possibility of degeneracy). Accordingto the “Born rule”, the probability of any particular outcome is proportionalto the squared modulus of the scalar product of this eigenfunction with thepre-measurement wavefunction. This analysis underlies many of the pre-dictions of quantum mechanics that have been invariably confirmed by ex-periment. An alternative approach to quantum measurement is the Everettinterpretation (also known as the “relative states” or the “many worlds” in-terpretation) which was proposed by Everett III (1957). The essence of thisapproach is that it assumes no collapse of the wavefunction associated with ameasurement: instead, the time development of the state is everywhere gov-erned by the time-dependent Schr¨odinger equation. After a “measurement-like” event, this results in a splitting of the wavefunction into a number ofbranches, which are then incapable of reuniting or communicating with eachother in any way. This splitting occurs even when a human observer is partof the measurement chain: the resulting branches then each contain a copyof the observer, who is completely unaware of the existence of the others.Since its inception, the Everett interpretation has been subject to con-siderable criticism—e.g. Kent (1990), Squires (1990)—which has three mainstrands (or branches [ sic ]). First, there is its metaphysical extravagance. Thecontinual evolution of the universe into a “multiverse” containing an immensenumber of branches would mean that the universe we observe should be ac-companied by an immense number of parallel universes, which we do not
Preprint submitted to Studies in the History and Philosophy of Science October 25, 2018 bserve and have no awareness of—surely such a postulate must be a grossbreach of the principle of Occam’s razor! Everett himself was aware of thiscriticism and, in a footnote to his original paper, he compares the conceptualdifficulties of accepting his interpretation with those encountered by Coper-nicus when the latter proposed the (in his time revolutionary) idea that theearth moves around the sun. However, the reason that the Occam’s razorargument has not led to the universal rejection of Everett’s ideas is less to dowith the strength or otherwise of the Copernican analogy and more a resultof the fact that the branching of the universe into the multiverse is claimedto be a direct consequence of the time-dependent Schr¨odinger equation: noadditional postulate, such as the collapse of the wavefunction, is required toexplain the phenomenon of quantum measurement and the extravagance withuniverses may therefore be considered a price worth paying for the economyin postulates.The second strand in the criticism of Everett is known as the “preferredbasis” problem. This is because there is an apparent ambiguity in the waythe branches are defined. Thus, if the wavefunction of a system has theform ψ = Aψ + Bψ , then Everett suggests that a measurement should leadto two sets of branches, one associated with each of the states representedby ψ and ψ . However, the original state could just as well be written as ψ = Cφ + Dφ where φ = 2 − / ( ψ + ψ ), φ = 2 − / ( ψ − ψ ), C =2 − / ( A + B ) and D = 2 − / ( A − B ), so why should the branches not bejust as well defined by φ and φ —or indeed any other orthogonal pair oflinear combinations of ψ and ψ ? This problem has been largely resolvedby the appreciation of the importance of the effect of the environment on aquantum system and the associated “decoherence”—Zurek (2008), Wallace(2002), Wallace (2003a). A quantum measurement is inevitably accompaniedby complex, chaotic processes which act to pick out the particular basisdefined by the eigenstates of the measurement operator. This basis is thenthe one “preferred” by the Everett interpretation and this supervenes on theSchr¨odinger wavefunction. This result is now generally accepted, althoughBaker (2006) argues that its derivation uses the Born rule so that there is adanger of circularity if it is then assumed as part of its proof.The third criticism leveled at Everett is the problem of probabilities. Theconventional (Copenhagen) interpretation states that, if the wavefunction be-fore a measurement is ψ = Aψ + Bψ , and if ψ and ψ are eigenstates ofthe measurement operator with eigenvalues q and q respectively, then theoutcome will be either q with probability | A | or q with probability | B | ,3here these probabilities reflect the frequencies of the corresponding out-comes after a large number of similar measurements. However, according tothe Everett approach there is no “either-or” because both outcomes are man-ifest, albeit in different branches. Instead of a disjunction to which we canapply standard probability theory, we have a conjunction , where it is hard tosee how probabilities can make any sense—Squires (1990), Graham (1973),Lewis (2004). There have been several attempts to resolve this conundrumand to show how probability (or something else that is in practice equivalentto it) can be used in an Everettian context. David Wallace has proposeda principle that he calls “subjective uncertainty” in which he claims that arational observer should expect to emerge in one branch after a measure-ment, even although she is also reproduced in the other branches—Wallace(2003b), Wallace (2007) . Greaves (2004) has criticized this approach andsuggested an alternative in which we have to take into account the observer’s“descendants” in all the branches, but we should “care” more about somethan others; the extent to which we should care is quantified by a “caringmeasure” that is proportional to the corresponding Born-rule weight. Boththese approaches are designed to explain why some branches appear to befavoured over others, but both attempt to do this without altering Everett’smain principle that the quantum state evolves under the influence of thetime-dependent Schr¨odinger equation with nothing else added, so that theBorn rule supervenes on this. An alternative approach, which I shall notdiscuss any further in this paper, is to maintain most of the fundamentalideas of the Everettian interpretation, but add a further layer of “reality” tojustify the use of probabilities; an example of this can be found in Lockwood(1989).Interest in the Everett interpretation has been on the increase recently—particularly during 2007, which was the 50th anniversary of the publication ofEverett’s original paper, Everett III (1957). Much of the renewed interest hasdeveloped from work by Deutsch (1999) some eight years earlier, which wasthen developed by, Wallace (2003b), Saunders (2004) and Wallace (2007).This programme (which I refer to below by the initials DSW) aims to derivethe Born rule from minimal postulates that are claimed to be consistent withthe Everett interpretation, as well as with other approaches to the measure-ment problem. In fact, Deutsch (1999) makes little reference to the Everettinterpretation in his derivation of the Born rule and Saunders (2004) empha-sizes that he believes that his derivation is independent of any assumptionsabout the measurement process. However, Wallace (2007) assumes the Ev-4rett interpretation and claims that his derivation shows that the Born ruleis completely consistent with it. Gill (2005) examined Deutsch’s derivationand sought to clarify the assumptions underlying it, again without referringto the Everett interpretation as such. A similar approach, but using slightlydifferent assumptions has been developed Zurek and is set out in a recentreview paper, Zurek (2008).The present paper aims to show that some of the postulates underlyingthe above derivations arguments do not follow naturally from the Everettinterpretation and may well not be consistent with it. The DSW Proof of the Born Rule
This section sets out the DSW derivation of the Born rule by applyingit to a particular example. The argument is deliberately kept as simple aspossible and more general treatments can be found in the cited references.Consider the case of a spin-half particle, initially in an eigenstate of an op-erator representing a component of spin in a direction in the xz plane atan angle θ to the z axis, passing along the y axis through a Stern-Gerlachapparatus oriented to measure a spin component in the z direction.Standard quantum mechanics tells us that the initial state α θ can bewritten as a linear combination of the eigenstates of ˆ S z : α with eigenvalue+1 (in units of (cid:126) /
2) and β with eigenvalue -1. We have α θ = cα + sβ (1)where c = cos( θ/
2) and s = sin( θ/ S z = +1 and − associated with the measurement, thetotal wavefunction of the system is ψ = cαχ + + sβχ − (2) At a number of points in this paper, I compare the predictions of the Everett modelwith those produced by the “Copenhagen interpretation”, by which I mean a model inwhich the wavefunction collapses into one of the eigenstates of the measurement operator.This is assumed to occur early enough in the process for the outcomes to be the same aswould be observed if particles were emerge randomly from one or other output channel,with the relative probabilities of the two outcomes determined by the Born rule. χ + ( χ − ) is the wavefunction representing the detectors, including theirenvironment, when a particle is detected in the positive (negative) channel.According to the Copenhagen interpretation, the corresponding probabilitiesfor a positive or negative outcome are given by the Born rule as c and s respectively. From the Everettian point of view, on the other hand, thereis no collapse and the system is always in a state of the form ψ . However,because of the effects of the environment and decoherence, phase coherencebetween the two terms on the right-hand side of (2) is lost, so they cannever in practice interfere. The wavefunction has therefore evolved into two“branches” which then develop independently.The principle of the DSW approach is to describe the process being stud-ied as a game, or series of games, where we receive rewards, or pay penalties(i.e. receive negative rewards) depending on the outcomes. The derivationproposed by Zurek (2008) is quite similar to this, although it does not usegame theory.Imagine a game where the player receives a reward depending on theoutcome of the experiment. Assume that the value of θ is under our controland that, whenever the experimenter observes a particle emerging from thepositive or negative channel of the Stern-Gerlach apparatus, she receives areward equal to x + or x − respectively; these values can be chosen arbitrarilyby the experimenter. In the special cases where θ = 0 or θ = π , the initialspin state is an eigenstate of ˆ S z with eigenvalues +1 and − V ( θ )—of the game as theminimum payment a rational player would accept not to play the game, andlook for an expression for V ( θ ) of the form V ( θ ) = w + ( θ ) x + + w − ( θ ) x − (3)where the w s are non-negative real numbers that we call “weights” and whichare normalized so that their total is unity. We shall find that w + ( θ ) = c and w − ( θ ) = s (4)which are the probabilities predicted by the Born rule for this setup.First consider the effect on the wavefunction of rotating the SG magnetthrough 180 ◦ about the y axis. It follows from the symmetry of the Stern-Gerlach apparatus that spins that were previously directed into the upper6hannel will now be detected in the lower channel and vice versa. Thus V ( θ + π ) = w + ( θ + π ) x + + w − ( θ + π ) x − = w − ( θ ) x + + w + ( θ ) x − (5)From standard quantum mechanics, the effect of this rotation on the wave-function (2) is to transform it to ψ = − sαχ + + cβχ − (6)We now proceed by considering a series of particular values of θ . Case 1
The first case is where θ = 0 so that the initial state, α θ , isidentical with α . As noted above, this state is unaffected by the measurementand the particle is always detected in the positive channel. Thus V (0) = x + , w + (0) = 1 and w − (0) = 0. Similarly, V ( π ) = x − , w + ( π ) = 0 and w − ( π ) = 1. Case 2
In the second case, θ = π/ ψ is as in (2) above, but with c = s = 2 − / . Now consider the effect of rotating the Stern Gerlach appara-tus through an angle π . Using (5) and (6), we get the following expressionsfor V and ψ V (3 π/
2) = w − ( π/ x + + w + ( π/ x − (7) ψ = 2 − / [ − αχ + + βχ − ] (8)The only change in the wavefunction is the change of sign in the term in-volving α . DSW point out that this sign, in common with any other phasefactor, should not affect the value, because it can be removed by performinga unitary transformation on this part of the wavefunction only—e.g. by arotation of the spin through 2 π or by introducing an additional path lengthequal to half a wavelength. Moreover, Zurek (2008) shows that one of theeffects of the interaction of the system with the environment is to removeany physical significance from these phase factors. It follows that the valueshould not be affected by the rotation so that V (3 π/
2) = V ( π/ w + ( π/
2) = w − ( π/
2) = 1 / V ( π/
2) = ( x + + x − ) / θ .7e now extend the result to the case where the number of output channelsis M instead of two and the wavefunction is the sum of M terms, each ofwhich corresponds to a different eigenstate of the measurement operator. Inthe case where the coefficients of this expansion are all equal, any action thathas the effect of exchanging any two output channels (which are numbered 1and 2) must leave the wavefunction unchanged apart from irrelevant changesin phase. The value is then also unchanged, but the roles of w and w arereversed. Hence w x + w x = w x + w x (10)where x i is the reward associated with the i th output channel. It followsthat w = w ; consideration of other permutations immediately extends thisresult to all i and we have w i = N − . Case 3
In the third case, θ = π/ θ/
2) = √ / θ/
2) =1 /
2. We now assume that the system is modified so that, after emerging fromthe Stern-Gerlach magnet and before being detected, the outgoing particlesinteract with a separate quantum system that can exist in one of, or a linearcombination of four eigenstates φ i . Following Zurek (2008), this is referredto as an “ancilla” from now on. The ancilla is designed so that, if θ = 0so that all spins emerge from the positive channel, the ancilla is placed inthe state 3 − / (cid:80) i =1 , φ i ; while, if θ = π and all spins are negative, its statebecomes φ . From linear superposition it follows that if the original spin isin a state of the form (2) with θ = π/
3, the total wavefunction of the spinplus the ancilla isΨ = 3 − / [ φ + φ + φ ] cos( π/ α + φ sin( π/ β = [ φ α + φ α + φ α + φ β ] (11)As the coefficients of each term in the above expansion are equal, it followsfrom the earlier discussion of Case 2 that all four weights are equal to 0.25.If we were to measure on the ancilla a quantity whose eigenstates were oneof the functions φ to φ , we should obtain a result equal to one of thecorresponding eigenvalues. If the result corresponds to one of the first threeeigenfunctions, we can conclude that if, instead, we had measured the spindirectly, we would have got a positive result, while a result corresponding to φ indicates a negative spin. As this is the only such state, it follows thatthe weight corresponding to a negative spin is w − ( π/
3) = 0 .
25 and therefore,8rom normalization, that w + ( π/
3) = 0 .
75. (The last step, which followsZurek (2008), establishes these results without assuming that the weights areadditive.) The value of the game therefore equals 0 . x + + 0 . x − . It canalso be shown quite straight forwardly—Zurek (2008)—that, after a numberof repeats of the experiment, the predicted distribution of the results is asobserved experimentally.Following DSW, the above argument can be extended to the case of a mea-surement made in the absence of the ancilla if we make a further assumption,known as “measurement neutrality”. This states that the outcome of thegame is independent of the details of the measurement process—i.e. the pres-ence or absence of the ancilla—so that w + ( π/
3) = 0 .
75 and w − ( π/
3) = 0 . ( θ/
2) and sin ( θ/
2) re-spectively, so the derived weights are the same as those predicted by theBorn rule. By choosing an appropriate ancilla, the above argument can bedirectly extended to examples where the ratio of the weights is any rationalnumber, and then to the general case by assuming that the weights are con-tinuous functions of θ . Hence, the expression for the value is the same asthat predicted by the Born-rule: V ( θ ) = c x + + s x − (12)Further generalization to experiments with more then two possible outcomesis reasonably straightforward and does not introduce any major new princi-ples. Discussion
There have been a number of criticisms of the DSW proof when appliedto the Everett model in particular—e.g. Baker (2006), Barnum et al. (2000),Lewis (2005), Lewis (2007), Hemmo and Pitowski (2007); some of these evenchallenge the result (9) for the symmetric case. I shall shortly develop argu-ments to show that, although the symmetric results appear to be consistentwith the Everett model, this may not be not so in the asymmetric case.First consider how the above translates into predictions of experimentalresults. The game value is the minimum payment a rational observer wouldaccept in order not to play the game. This means that after playing thegame a number of times, a rational observer should expect to receive a setof rewards whose average is equal to the game value. Thus, if we consider9 sequence of N such observations in which n + and n − (= N − n + ) particlesare detected in the positive and negative channels respectively, the totalreward received will be n + x + + n − x − , and this should equal N ( w + x + + w − x − )implying that w + = n + /N and w − = n − /N . This, of course, is just what isobserved in a typical experiment provided N is large enough for statisticalfluctuations to be negligible. It should be noted that frequencies are not beingused to define probabilities, but the derived weights are used to predict theresults of experimental measurement of the frequencies.The above results are of course consistent with the standard Copen-hagen interpretation, whose fundamental mantra was set out by Bohr (1935):“...there is essentially the question of an influence on the very conditionswhich define the possible types of predictions regarding the future behaviourof the system”. In the present context, this means that, because an experi-ment designed to demonstrate interference would involve a different experi-mental arrangement, the experiment can be modelled as a classical stochasticsystem in which spins emerge from either the positive or the negative chan-nel of the Stern-Gerlach apparatus. (It should be noted that this paper doesnot aim to justify the Copenhagen interpretation, but employs its results asa comparator with the Everettian case.)Why should an Everettian observer have experiences such as those justdescribed? In the Everett interpretation, the quantum state evolves deter-ministically and on first sight, there would appear to be no room for uncer-tainty. However, after a splitting has occurred, observers in different brancheshave the same memories of their state before the split, but undergo differentexperiences after it. Given this, it may be meaningful for an experimenter tohave an opinion about the likelihood of becoming a particular one of her suc-cessors. This introduces a form of subjective uncertainty, and Wallace (2007)claims that this plays a role in the Everett interpretation that is equivalentto that played by objective stochastic uncertainty in the Copenhagen case.However, we should note that such subjective uncertainty can only come intoplay at the point where the experimenter becomes aware of an experimentalresult, in contrast to the Copenhagen model where the splitting is assumedto occur as the particles emerge from the Stern-Gerlach magnet. I shallshortly proceed to compare and contrast the Copenhagen and Everettianinterpretations of the different experiments discussed above. To help focusthe discussion, I shall initially assume that in such experiments each par-ticular result is associated with only one branch of the final wavefunction.This assumption has been strongly criticized by DSW and others and I shall10eturn to the question of how it affects our conclusions at a later stage. Inow analyze our earlier arguments step by step. Case 1: Copenhagen
As the initial spin state is in an eigenstate of S z ,the result is completely determined. The probability of the result equallingthe corresponding eigenvalue is 1 and the probability of the alternative iszero. Case 1: Everett
There is only one branch and this contains the onlycopy of the observer who invariably records the appropriate eigenvalue.There is therefore no difference between the observers’ experiences in case1 under the Copenhagen and Everettian interpretations.
Case 2: Copenhagen
The probabilities of positive and negative resultsare both 0.5. After a large number of repeats of the experiment, the ex-perimenter will have recorded approximately equal numbers of positive andnegative results, so her average reward will be ( x + x ) /
2, which is the sameas the game value.
Case 2: Everett
The observer will split into two copies each time a spinis observed and the weights of the two branches are equal for the reasonsdiscussed earlier. After a large number ( N ) of repeats of the experiment thevast majority of observers will have recorded close to N/ N/ Case 3: Copenhagen
As emphasized above, this assumes that the ex-periment is a stochastic process in which a particle emerges from either thepositive and or the negative channel and the relative probabilities of the out-comes are equal to the Born weights. In the presence of the ancilla, a particleis detected in one (and only one) of the equally-weighted states φ to φ , andall four outcomes have equal probability. To have been observed in any ofthe first three states, the spin must have emerged from the Stern-Gerlach ex-periment through the positive channel, while if the final result correspondedto φ , it must have come through the negative channel. It follows directlythat if the ancilla were absent, three times as many spins would be detectedas positive than as negative. Thus, the principle of measurement neutral-ity, assumed in stage 3 of the earlier derivation, follows naturally from theassumptions underlying the Copenhagen interpretation. Case 3: Everett
We first consider the situation where an ancilla ispresent so that the state is described by (11); there are therefore four equally-weighted branches, one corresponding to each of the φ i . The observer splits11nto four equally-weighted copies and should expect her descendants to recordan equal number of each of the four possible results and therefore concludethat there are three times as many positive as negative spins. However, inthe absence of an ancilla, there are only two branches and the observer issplit into two copies each time a result is obtained. To show that a typicalEverettian observer should record results that are consistent with the Bornweights, we again have to apply the principle of measurement neutrality.We saw above that this is a natural, if not inevitable, consequence of theCopenhagen interpretation, but we shall now demonstrate that this is notthe case in an Everettian context.Under the Copenhagen interpretation, particles are assumed to emergefrom either the positive or the negative channel and then into one, and onlyone , of the states φ i . This is not true in the case of the Everett interpreta-tion, where the system evolves deterministically and the state is describedby a linear combination of the wavefunctions associated with a particle beingpresent in each channel. Apparent stochasticity, or subjective uncertainty,only enters the situation at the point where the experimenter observes theresult and splits into a number of descendants—two in the absence of theancilla and four if it is present. There is no requirement for the frequenciesto be the same in both cases—i.e. no a priori reason to apply the principleof measurement neutrality. In the language of decision theory, the valuesof the two games are not necessarily the same, so a decision on whether ornot to accept a payoff may depend on whether the game is being playedwith or without an ancilla. Indeed, as in the absence of an ancilla thereare only two branches, we might expect each observer’s experience to be thesame as in case 2, with equal numbers of positive and negative results andan equal reward for each outcome—i.e. the statistical outcomes would beindependent of the weights. I shall argue later that this is a natural con-sequence of the Everettian interpretation, but at present simply emphasizethat the principle of measurement neutrality is a self-evident consequence ofthe assumptions underlying the Copenhagen interpretation, but constitutesa major additional postulate in the context of Everett.I further illustrate this last point by considering a simple classical examplethat consists of a box with two exit ports from each of which a series of ballsemerges as in figure 1. The apparatus can be operated in one of two modesthat we denote as “C” and “E”. In the C mode, balls emerge one at a timefrom one of two output ports and, on average, three times as many come outof the upper port as from the lower. An experimenter observes the balls as12 Mode E Mode
Figure 1:
In the cop mode a ball is emitted from the first box through either theupper or the lower port and detected either before or after entering the secondbox; the figure shows one possible outcome. In the eve mode, balls emerge fromboth ports and one of them is detected either before or after the second box, whichreleases three balls every time one enters. they emerge and confirms this relative likelihood. Still in the C mode, theexperiment is modified so that when a ball emerges from the upper port, itpasses into a second, “ancillary” box and then emerges at random throughone of three output channels before being detected. The experimenter nowdetects a ball either in one of these three channels or emerging from the lowerport. Clearly the first of these results is three times as likely as the second,so the observed frequencies are independent of the presence or absence ofthe second box. Thus the equivalent of measurement neutrality holds in thiscase.Now consider the game in the E mode, which is also illustrated in figure1. In this case two balls emerge from the box simultaneously: a black ball,from the upper port and a white ball from the lower. The two balls fallinto a receptacle (not shown in the figure) and an experimenter draws oneat random; after repeating the experiment a number of times she sees equalnumbers of black and white balls. The experiment is now modified so thatthe black balls are directed into an ancillary box which now contains a devicethat releases three identical black balls, one through each of the three outputports, whenever one enters. These three balls along with the white one nowfall into the receptacle and the observer again draws one at random; she nowsees a black ball three times as often as a white ball. Thus, the relativelikelihood of a black or white ball depends on the presence or absence ofthe second box, and we can conclude that measurement neutrality is notnecessarily preserved when the game is played in the E mode.A more whimsical analogy follows the precedent set by Schr¨odinger’s catby using animals to illustrate our point. First consider Copenhagen rabbits.13hese come in two colours—black and white; they are all female and capableof giving birth to one (and only one) baby rabbit which is always of thesame colour as its mother. Let us suppose we have four Copenhagen rabbits,three black and one white in a hat and suppose that one, of them, chosenat random, is pregnant. We first play the game of “pick out the pregnantrabbit” by putting our hand in the hat, identifying and then pulling out thepregnant rabbit. We are paid different rewards ( x b and x w ) depending onwhether the extracted rabbit is black or white. After playing the game anumber of times, we find that we have pulled out three times as many blackrabbits as white, so that the game value is (3 x b + x w ) /
4. The second gameis one where we wait until the pregnant rabbit has given birth and then pullout and identify the colour of the baby. Clearly the results and the value arethe same as in the first game.Now consider Everettian rabbits, which are also either black or white. Incontrast to the Copenhagen rabbits, they are capable of carrying and givingbirth to more than one offspring. Suppose we have two pregnant Everettianrabbits: a white rabbit that is pregnant with a single offspring, and a blackrabbit that is expecting triplets. If we draw one of the two pregnant rabbitsfrom the hat at random, the game value will be ( x b + x w ) /
2. However, if,instead, we wait until after the rabbits have given birth and then draw outone of the offspring at random, the game value will now be (3 x b + x w ) / ad hoc ruleswould have to be built into the physics of the set up when it was designed andconstructed. In the quantum case under the Everett interpretation, measure-ment neutrality therefore has to be an additional assumption, rather thanfollowing directly from the structure of the theory as in the Copenhagen case.Gill (2005) shows that measurement neutrality is equivalent to assuming thatmeasures of probability are invariant under functional transformations —i.e.the probability of obtaining a particular result when measuring a variable isthe same as that pertaining when a function of the variable is measured. Heconsiders that functional invariance in the case of one-to-one transformationsis “more or less definitional”, but is much less obvious in the many-to-onecase, which is required for situations such as case 3 above. Gill’s discus-sion relates to probabilities as conventionally defined and his paper makesno reference to the Everett interpretation. I believe that the above argumentshows that many-to-one transformations are also “more or less definitional”under the Copenhagen interpretation, but not in the Everettian context.Measurement neutrality and an associated principle that he calls “equiv-alence” have been argued for by David Wallace in a number of papers—Wallace (2002), Wallace (2003a),Wallace (2003b), Wallace (2007). He con-siders games in which the measurement result is erased after it triggers anassociated reward and before the experimenter has recorded the outcome. Inthe symmetric ( θ = π/
2) case, the final states are independent of the patternof rewards, which reinforces the arguments leading to (9). However, this isnot an issue in the present discussion, which challenges the assumption ofmeasurement neutrality only in the asymmetric case. Another point empha-sized by Wallace (2007) is that the boundary between what is usually takenas preparation and what is part of the “actual” measurement is essentiallyarbitrary, particularly in the context of the Everett interpretation. However,the observation and recording of the result by a conscious observer is part ofthe measurement proper, and it is only at this point that subjective uncer-tainty or the relevance of a caring measure is introduced into the Everettiantreatment of the Born rule.Up to this point I have argued that the assumptions underlying the deriva-tion of the Born rule, in particular measurement neutrality, are not necessaryin an Everettian context, though they may be treated as added postulates.15 now intend to go further and argue that there is an inconsistency betweenthe assumptions underlying the Everett interpretation and the Born rule—or, indeed any rule that relates the likelihood of a measurement outcome tothe amplitudes ( c and s in the above example) associated with the branchingof the wavefunction in a non-trivial way. I shall continue to use the exampleof the measurement of the spin component of a spin-half particle as a focusof the discussion.The scenario I now discuss is one where an observer (“Bob”) records thenumber of positive spins ( M ) in a set of measurements of the state of N identically prepared spins that have passed through a Stern-Gerlach appara-tus. We consider the particular case where Bob does not know the value of θ before he makes any measurements ; that is, he has not seen the apparatusor been told how the magnet is oriented, which means that his initial stateis represented by a wavefunction which is independent of θ . However, if Bobknows the Born rule, he can estimate the value of θ as 2 cos − ( M/N ) / andhis confidence in this value will be the greater, the larger are M and N . As aresult of this experience, Bob’s state has been changed from one of ignoranceto one where he has some knowledge of θ . This change must therefore havebeen reflected in Bob’s quantum state, causing a modification to his wave-function, which now depends on θ . To further emphasize this point, supposethat the value of θ can be changed without Bob’s direct knowledge by anotherexperimenter (“Alice”) who has control of the Stern-Gerlach apparatus. Ifshe does this and the experiment is repeated a number of times at the newsetting, Bob will find that his expectations have been consistently wrong.He may initially attribute this to statistical fluctuation, but eventually hewill amend his state of expectation to bring it into line with his experience.Indeed, Bob may know that Alice is able to do this, in which case he willbe more likely to amend his state of expectation at an earlier stage. Alicecould then send signals to Bob by transmitting sets of N particles using thesame value of θ for each set, but changing it between sets. If the Born ruleapplies, Bob can deduce the values of θ that Alice has used from the relativenumbers of positive and negative results, so Alice has again caused changesin Bob’s state of expectation and therefore of his wavefunction.It is one of the principles of the Everett interpretation that, once branch-ing has occurred and the possibility of interference between branches has beeneliminated, the wavefunction associated with a branch describes the “relativestate” of the system contained in that branch, which cannot be influencedby the state of any other branch. Moreover, the form of the relative state16unctions, which represent the whole branch including the version of Bobassociated with it, are the same whatever are the values of the expansioncoefficients c and s . This implies that the properties of a system representedby such a relative state are not affected by the measuring process. Thus,although these constants enter the expressions, they do so only as expansioncoefficients, which have no effect on the wavefunctions of the relative statesassociated with the component branches. In particular, the observer’s stateof knowledge of the value of θ cannot be altered as a result of this process.This is in direct contradiction to the conclusion reached above, assumingthat the Born rule holds. There is therefore an inconsistency between theprinciples underlying the Everett interpretation and the appearance of a cor-relation between the apparatus setting and the relative frequencies of thepossible outcomes, such as is implied by the Born rule.To develop this point further, consider the state of the whole system after N particles have passed through the apparatus, so that, according to the Ev-erett interpretation, the wavefunction contains 2 N branches that correspondto all possible sequences of the results of the measurements performed so far.That is, using (2), (cid:89) i =1 ,N α θ ( i ) χ −→ (cid:88) P si c m s N − m Ψ( s , s , ...s N ) (13)where α θ ( i ) is the initial state of spin i and χ refers to the initial state ofthe detecting apparatus, including the observer Bob, which is independent of θ , given the assumptions set out earlier. Each parameter s i has two values,+ and − ; Ψ( s , s , ...s N ) represents the state of the whole system (i.e. spins,measuring apparatus and Bob) after the results s i have been recorded ina measurements on spin i for all i from 1 to N ; m equals the number ofpositive spins in this set; (cid:80) P si implies a summation over all 2 N permutationsof s i . Each term in the summation refers to a separate branch in the Everettinterpretation.It follows from (13) that the number of branches in which m positiveresults have been recorded is N ! /m !( N − m )! and the Born weight associatedwith this whole subset equals c m s N − m ) . Under the Copenhagen interpre-tation, the probability of observing m positive results is the product of thesetwo quantities: this has a maximum value when m = M = N c (= 3 N/
4, if θ = π/ | cs | N / (= √ / m . If N is large, the vast majority of observers willobserve approximately equal numbers of positive and negative results and asmall minority will observe results in the vicinity of the ratio predicted bythe Born rule. Repeating the experiment with a different value of θ does notchange the number of observers recording any particular result, so, if thiswere all there were to it, Bob’s experience would not correlate with the ap-paratus setting and he would be unable to deduce a reliable value of θ fromhis observations. However, the Everett interpretation only works if this is notall there is to it. Because of subjective uncertainty, an observer’s successorsin branches that have a high Born-rule weight are somehow favoured over theothers. How this can work is at the heart of the difficulties many critics havewith the Everett interpretation, but let us leave this on one side. The factthat these successors are so preferred, means that they can with confidencededuce the value of θ from their observations of M and N . Acquiring thisinformation must therefore have altered their reduced state, in contradictionto the Everettian assumptions set out above.Several points should be noted about the above. First, the contradictiondoes not arise in the Copenhagen interpretation because, as noted earlier, thisassumes that stochasticity arises at the point where the spin emerges from theStern-Gerlach magnet. The information as to which branch is occupied bythe spin is additional to that contained in the wavefunction and is obtainedby Bob through the collapse process. Hence, no contradiction arises whenthis is used by the experimenter to guide his expectations about subsequentmeasurements.Second, it should be emphasized that the argument applies only to infor-mation about the apparatus setting that is obtained by Bob as a result of themeasurement process . He could of course have been told in advance how theapparatus was set up so, in this case, χ would already be a function of θ .The later argument could probably be extended to show that he should notbe able to obtain further information about θ by the measurement process,but I believe it clarifies the discussion if we focus on the case where Bob hasno prior knowledge of θ : to demonstrate inconsistency, it is only necessaryto establish a contradiction in at least one particular case.Third, although I have focussed on the Born rule, the above argumentswould apply equally well to any model in which the outcome frequencies wereassumed to depend systematically on the expansion coefficients. This is of18ather marginal interest given that the Born rule is the one that is establishedby experiment.If we accept the above, it follows that the only way probability should beable to enter the Everett interpretation is if all branches are assigned equalweight. Might it nevertheless be possible to reconcile this conclusion withexperiment? Up to now, we have assumed one branch per outcome, withoutattempting to justify this. We now turn to the question of “branch count-ing”, which means considering the number of branches associated with anygiven measurement outcome. If we accept the argument that the expansioncoefficients play no role in determining the outcome likelihood in an Ev-erettian context, then an experimenter’s expectation of a particular outcomeshould be proportional to the number of branches associated with it. Suchan assumption is similar to that made in statistical thermodynamics, wherethe ergodic hypothesis states that the result of averaging over an ensembleof systems is the same as the time average for a single system. When appliedto the symmetric case, this is an essential part of the arguments leading to(9) and (10). However, branch counting has been strongly criticized by DSWon a number of grounds. Wallace (2007) considers a scenario in which extrabranching is introduced into one (say the plus) channel by associating with ita device that displays one of, say, a million random numbers. He argues thatthis must be irrelevant to an experimenter who sees only the measurementresult and is indifferent to the outcome of the randomizing apparatus. Thisis because “if we divide one outcome into equally-valued suboutcomes, thatdivision is not decision-theoretically relevant”. However, this argument doesnot fully take into account the Everettian context. Referring again to theclassical game discussed earlier and illustrated in figure 1, we can considerthe additional branching on the right of both setups as due to the presenceof a randomizer with three possible outputs. In the case of the C game, theseare indeed irrelevant to the expectation of the player, because a ball emergesfrom only one of the three channels and must therefore have passed throughthe upper channel at the previous stage. However, in the case of the E game,the chances of observing a black ball are enhanced (tripled) by the splittingand this would have to be taken into account by any rational player, even ifthe only result she sees is the colour of the ball. Similarly, if we introduce arandom number machine as Wallace suggests, then its state will be a linearcombination of its million possible outcomes and all these will be associatedwith a positive value of spin. Given that there is only one branch associatedwith the alternative outcome, we could well expect the subjective likelihood19f a positive result to be one million times greater than that for a negativeoutcome.A second argument deployed to criticize branch counting is based on thefact that the interaction of a quantum system with its environment leads toan immensely complex branching structure. Indeed it is claimed by DSWthat the number of branches is not only very large (possibly infinite), but isalso subject to very large and rapid fluctuations before, during and after theobservation of a result; which may mean that it is not meaningful to talkabout even the approximate number of branches that exist at any time. Thisis adduced as a reason why a rational player should ignore the complexityof the branching structure and instead expect to observe results consistentwith the Born rule. However, if the likelihood of observing a particular resultis proportional to the number of associated branches, the complexity intro-duced by decoherence should actually result in the outcome of a measurementbeing completely unpredictable. The situation is similar to chaos in classicalmechanics or to turbulence in hydrodynamics, whose onset certainly does notlead to increased predictability. In the arguments above, we assumed thateach outcome was associated with a single branch, so what would be thelikely consequences of a complex branch structure in an Everettian context?First, there may well be situations in which we could expect the numberof branches associated with different outcomes to be equal, at least whenaveraged over a number of measurements, and in this case our earlier discus-sion would not be affected. However, we might be able to devise a situation(e.g. one in which a detector was placed in the positive output channel only)where the numbers of branches in the two channels would be expected todiffer greatly. We should then expect to detect a larger number of (say)positive than negative results This would be true even if the Stern-Gerlachapparatus were oriented symmetrically—i.e. with θ = π/
2, so the symmetryon which we based some of our earlier arguments would not hold. The com-plexity and fluctuations of the branch structure in the Everett case wouldrender even the statistical results of a quantum measurement unpredictable.Such a situation is sometimes described as being “incoherent” and it hasbeen argued that this would mean that the universe would be nothing likethe one we experience. However, the obvious conclusion to draw from thisis that the Everett assumptions are falsified, rather than that the Everettmodel is correct and the arguments based on it that lead to this incoherencemust be wrong.It might be thought that branch counting could restore the Born rule if the20umber of branches associated with a particular outcome were proportionalto the Born weight. However, not only is there no obvious mechanism toachieve this, it is also inconsistent with the Everett model for the samereasons as were set out earlier. The quantum description of the branchstructure is contained within Ψ in (13) and therefore cannot depend on theexpansion coefficients for the reasons argued above.
Conclusions
I have argued that attempts to prove the Born rule make assumptions thatare essentially self-evident in the context of the Copenhagen interpretation,but not with the Everett model of measurement. I have further arguedthat probabilities which are functions of the expansion coefficients are notconsistent with the Everett interpretation, because these quantities are notthen accessible to an observer in the reduced state associated with a branch.An alternative scheme that could be consistent with Everett is one whereeach branch has the same probability and the probability of a given outcomedepends on the number of branches associated with it. However, this alsocannot be made consistent with the Born rule and it leads to predictions ofchaotic, unpredictable behaviour, in contrast to the relatively well-orderedbehaviour, invariably demonstrated in experiments. I conclude that the Bornrule is a vitally important principle in determining quantum behaviour, butthat it depends on wavefunction collapse, or something very like it, thatdoes not supervene upon the time-dependent Schr¨odinger equation. It wouldbe possible to retain the many-worlds ontology of the Everett model whileallowing information to be transferred through the measurement, but thestate evolution would no longer be governed by the Schr¨odinger equationalone and the economy of postulates would no longer obviously outweigh themetaphysical extravagance associated with the Everett picture.The debate between the different interpretations of quantum mechanicshas often been metaphysical in the sense that they often make the samepredictions and cannot therefore be be distinguished experimentally. Thepresent paper has argued that this is not so in the case of the Everett in-terpretation, which predicts results different from those that follow from theCopenhagen interpretation, which in turn are supported by experiment. Ifthis is accepted, the Everett model will have been falsified and the search fora consensual resolution of the quantum measurement problem will have tobe focussed elsewhere. 21 am grateful to Peter Lewis for comments on an earlier draft of this paperand to Simon Saunders and David Wallace for useful discussions. I shouldalso like to thank the referees for comments that have helped me clarify someof the points made in the discussion.