The purpose of this text is to point out some obscure issues innon-relativistic quantum mechanics and to compare in particular the``orthodox'' view to the many-worlds interpretation [1]. Main attention ispaid to the concepts of reality in both interpretations.
We chose three related questions to be discussed in the framework ofboth viewpoints:
A. How does the Quantum Mechanics predict the behavior ofclassical systems? If a macroscopic body is describedquantum-mechanically, then (in principle) it is capable of being``smeared'' in space or momentum space. Moreover, according to the theory,there are situations when the state vector of a macroscopic body representssuch smeared states (see question B). How does this agree with the usualbehavior of things?
B. (A variant of the Schrödingers cat paradox.) Morespecifically , what happens if the given state vector of a macroscopic bodyrepresents a widely smeared distribution of the body's position? We canprepare such a state by means of a Stern-Gerlach apparatus coupled to amechanism which shifts a cannonball after the spin measurement (the exampleis borrowed from [1]; the cannonball is shifted 1 m up or down according tothe spin direction of a single electron). If the state vector of anincoming electron is a superposition of spin-up and spin-down states, thenthe wavefunction of the cannonball will be a superposition of two peakdistributions. Such a state B.S. DeWitt [2] called a ``schizophrenicstate'' -- a (coherent) superposition of macroscopically distinguishablestates.
C. Is there any objective reality behind the usual classicalworld? Suppose we have an observer in an isolated room performing ameasurement, and an observer outside who has the exact statevector of the whole room (including the observer) before the measurement,along with the complete Hamiltonian of the whole room. Theobserver is capable of calculating the state vector evolutionaccording to the Schrödingers equation. (The example is also from[1].) The measurement consists of recording the final position of thecannonball mentioned in the Question B, using an electron in aspin-to-the-right state. According to the observer, the cannonball wasin fact shifted (e.g.) up after the measurement. But theobserver does not attribute any reality to that event until theroom is unlocked and he observes the actual position of the ball;before that, both the cannonball and the observer were in a ``schizophrenicstate''. We see that the two observers do not seem to agree in what realityreally is.
Now we explain how the two different interpretations of QuantumMechanics deal with those questions.
There are two processes in Nature which are radically different -- thecausal development of a state vector according to Schrödingersequation and ``measurements'' resulting in an instantaneous projection ofthe state vector onto an eigenspace corresponding to a randomly choseneigenvalue. The ``measurement'' always involves a classical ``apparatus''and a (non-classical) ``system''; the ``outcome'' is always amacroscopic change in the ``apparatus'', which is considered toindicate the ``value of an observable'' of a ``system''. If no measurementoccurs due to absence of classical ``apparati'', then the state vectorevolves by the Schrödingers equation. The probabilities of differentvalues of a given observable can be found, but not the actual value whichis unpredictable. Moreover, quantum systems do not ``possess'' propertiesdescribed by the ``observables'' being measured. Therefore, it is not ourlack of knowledge that prevents us from predicting e.g. the position of aparticle, but rather the absence of such a reality element. The statevector is the most complete possible description of real properties of a``system''. Now we briefly describe how the Orthodoxy handles our threequestions.
A. Macroscopic objects are constantly engaging in various``measurement-like'' processes, which result in constant projection oftheir state vectors onto eigenspaces of all sorts of observables and so inkeeping the dispersions of the position, momentum etc. as small aspossible. According to this interpretation, a reader is constantly keepingthe letters in the book in their places by actually recording theirpositions and shapes. This leads to agreement among all observers asregards the behavior of classical systems.
B. A cannonball cannot be observed in more than one position (itis probably its heavy weight which ensures the space and momentum spacelocalization). Its state vector ceases to be smeared after the position ofthe cannonball is observed the first time. Nothing strange is in the factthat before the measurement the position was not definite since theOrthodoxy denies objective existence to anything that is not measurable (weimagine the laboratory to be a completely insulated room).
C. The same applies to the problem of the observer underobservation. What the observer saw was not an element of reality -- infact, he was not really detecting the upward movement of the cannonball;instead, he was in a ``schizophrenic state'' until the observer inhis mercy determined the true state of affairs (the sarcasm is borrowedfrom [1]). Technically, the observer thinks that the statevectors of the observer and the ``system'' became correlated after themeasurement in the sense that the total state vector of the laboratory roomis no longer decomposable into a tensor product of the observer's and thestate vectors of the ``system''; after he measured their state,the total state vector reduces to a tensor product of eigenvectors and thusthe observer regains his (its?) classical status.
As we can see, the roots of the ``schizophrenia'' lie in our inabilityto abandon certain classical notions such as position, velocity, etc., whendescribing macroscopic objects. Obviously, the Orthodoxy explains exactlywhy we happen to have such notions -- via the state reduction mechanism-- but with one important flaw: it does not give us a criterion for aprocess to be considered a ``measurement''. Of course, physicists know (inall known cases) whether a system is classical enough to be usedas an ``apparatus''; but such a theory is still not entirely satisfactorysince it does not inequivocally tell us how to handle any given physicalprocess.
It has been always stressed that the Orthodox Quantum Mechanics ``hasits classical limit''. Namely, a macroscopic system is (mostly always) in acoherent state with appropriately small dispersions of its classicalobservables and so obeys the classical laws with great precision. But ingeneral such tightly localized ``wave packets'' tend to get smeared as timepasses (the time needed for a body to really get smeared is astronomicallybig, but we are interested here in conceptual aspects of theory). Againthe Orthodoxy resorts to the state reduction to ensure the continuedlocalization of macroscopic objects (when they are observed or at leastengaged in any interaction with other classical systems). As concerns the Orthodox concept of measurement, little can be changedin it because we are rather forced to obtain macroscopic resultsin any real laboratory. Therefore, any valid theory must predict thatmacroscopic objects have definite positions when observed, and that anelectron beam is split in two after passing through a Stern-Gerlachapparatus showing certain relative beam intensities, etc. The Orthodoxydoes all that with help of some make-do devices, which constitute the maintarget of our attention.
The first and the only sacred cow is the Schrödingers equation(or, as in the relativistic quantum theory, other equations describing thecontinuous evolution of the state vector). The EWG propose that a ``statevector of the Universe'' exists in some extremely infinite-dimensionalspace, and that it always evolves continuously, never ``collapsing'' intoan eigenspace -- there is no discontinuous ``measurement-like'' processes.At that point we don't have to distinguish ``classical apparati'' from``quantum systems''. Instead, every time any two systems interact (the``measurement'' just being a case of such interaction) their state vectorsbecome correlated, as it follows from the usual quantum mechanics formalism(as we said, the EWG approach is just an interpretation of theusually accepted formalism of Orthodoxy).
Strictly speaking, there are no separate state vectors for each systemafter a ``measurement'', or any interaction process. After interaction,the systems have ``correlated states'', i.e. the new state vector of thecomposite system is no longer decomposable into a single tensor product oftwo state vectors of each of the two systems, but is rather a sum of suchproducts. Each term of that sum is the product of states of the first andthe second systems, thus expressing a kind of correspondence between thosestates. This correspondence is formalized in the EWG interpretation bydefining in a natural way the ``relative state'' of the first system withrespect to a certain state of the second system and a certain state of thecomposite system, using the decomposition of the total state vector into asum of the tensor products (the orthogonal complement in the Hilbert statespace is used here to ensure the uniqueness of the relative state).
To explain the phenomena actually observed in nature, EWG suggest that theobserver (or, generally speaking, an ``apparatus'') be described by aHilbert state space as well, representing in particular every relevantmemory state of the (not necessarily human) observer. The correlationbetween the states of the quantum system and the observer enables us toascribe to the quantum system its relative state with respect to any givenstate of the observer. Then, as the observer's ``real'' experience isexpressed by his memory state -- and for any given state of his memory, weby definition regard it to be the reflection of the observer's ``reality''-- we say that for each term of the decomposition we have a separate``macroscopic reality''. John Wheeler introduced the term ``world'' foreach of such ``realities'' [2], so we would say that after ameasurement-like process the world was split into several different worlds.(No interaction is conceivable between those imaginary universes, ofcourse.)
The ``splitting of the Universe'' is done each time when a ``classicalsystem'' treads on the Realm of the Quanta (performs a measurement), andthe exact details of the splitting depend on the considered process, sincethe splitting is only a device of the theory and not an actual event of theworld. (We will therefore avoid saying that the splitting ``occurs''.)The measurement appears to the observer as having certain outcomebecause it is so within his branch of the World Tree.
Technically, the EWG interpretation says: take the state vector of thecomposite system after interaction, decompose it into a sum of the tensorproducts of state vectors of the two systems; then each of the producedterms describes a ``world'' containing the observer in a certain memorystate (i.e. remembering certain ``real events'') and the quantum system inthe corresponding relative state. So the observer in that worldthinks that the quantum system is in the relative state -- which is easilyseen to be the result of the state reduction of the initial state vector ofthe quantum system into an eigenspace of the observable in question. Ofcourse, if we chose another observable, the world splitting would bechanged -- it really depends on our choice, not being an objectiveevent. The correspondence between the perceptible world and themathematical formalism (being a necessarily ad hoc part of anyphysical theory) is complemented by one more ``recipe'': when the statevector is a linear combination of tensor products of states, and one of thespaces corresponds to a classical system, then we have to split theUniverse accordingly and interpret the classical observables as beingdefinite within each of the resulting worlds.
In any given world after the splitting, the measurements have certainoutcomes and the ``truly quantum'' systems have their states reducedexactly after the Orthodox predictions. So the EWG interpretation claimsthat the Orthodox state reduction seems to be valid to allobservers, and it is possible to explain this phenomenon by using purelycausal continuous state evolution and considering (human) observers asquantum systems having memory states. The ``reality'' appears toall observers (as inferred from their memory) as showing state reductionand the possibility of several measurement ``outcomes''. Hugh Everett IIIproceeded even to show that the ``probability axiom'' of the Orthodoxy maybe reproduced in the EWG interpretation from some general assumptions likethe ``measure'' on the ``set of all worlds'' being in certain sense``uniform''. In particular, he demonstrated that if a single observer wereto repeat an experiment several times, then the sequence of results as heremembers them (within ``most of the worlds'' created by the end of thelast experiment) would be very much like a random sequence with certainprobabilities of the outcomes [2].
Let us now reconsider our three questions.
A. If the state vector of a system represents a ``smeared'' state(which does not necessarily imply any ``measurement'' in its past), then weimagine several worlds, so the observables we are interested in havedetermined values in each of the worlds. An observation made on such asystem by another classical system does not cause further splitting (unlesswe chose the wrong splitting in the first place) but correlates theobserver's state with the state of the system, so in each world theobserver will perceive a well-defined ``measurement outcome''. Therefore,the classical behavior of things is something we only perceive(though what could be more real?), and the theory describes the process ofperception in a general framework of ``pure wave mechanics''.
B. There is, therefore, nothing wrong in saying that a cannonball isin fact in a widely smeared state, since the observers trying todetermine its position will necessarily correlate their memory states withthe cannonball's, and so split the world and never detect anythingunusual. The paradox seems to be solved by abandoning the idea of havingonly one ``actually observed'' reality and instead introducing a bunch of``shadow'' realities which are observed by ``shadow'' observers.
C. Consider now the situation with two observers. In the EWGframework, each one has the same ``degree of reality'', viz. each is ableto split the world. Therefore, nothing is wrong with theobserver's calculations of the ``room state vector'' because thestate of the quantum system is not in fact reduced and theobserver in the room doesn't ascribe any excessive reality to the``actual'' result of the measurement (of course, if he follows the EWGinterpretation). Let us see how both observers agree in that the realityof the room is unchanged after the observer's intervention. Theobserver has split the world immediately after the measurement wasfinished; for him, a second splitting (due to the observer) ispossible but doesn't really split any of the existing branches any further-- in any given world branch, the observer sees only one outcome as real,and the state of the quantum system is reduced, so the observerwon't see anything else. And for the observer the real splittingis done when he enters the enchanted room; the ``room state vector'' hehad, however, provides an adequate description of the observer's statealong with the possibility of the first splitting. We must remember thatsplittings are not real events, but rather devices of the EWGinterpretation of the quantum-mechanical formalism.
The Orthodoxy clearly holds the simpler view: there is only one reality,the one directly perceived by an observer or ``measured''. Quantum systems,as opposed to classical ones, don't have certain properties we would expectthem to have (position, direction of spin, etc.) but rather exhibitdifferent measurement outcomes which we might stubbornly perceive asmanifestations of their properties, but that would only be a consequence ofour mental inertia. In fact, measurement is a process which serves as theonly bridge between the virtually intangible ``wavicles'' and our roughmacroscopic devices, and therefore may be said to create the``real'' properties of quantum systems. Of course, the quantum reality isindependent of observation, (i.e. there is some reality before theobservation, although the observation may affect it) and the reality iscompletely contained in the (partially unobservable) state vector.
Furthermore, it is assumed that any classical system has macroscopicallydefinite values of observables, by virtue of its being macroscopic, andthat those values are real, since all observers agree aboutthem. It is here where we encounter a difficulty: if a macroscopic body isnot being observed, there is no guarantee that it still has definiteobservables, and we need a measurement to find it out. And theinconsistency pointed out above stems from assuming that an observerhimself (especially when human) must always have definiteobservables, i.e. should never be in a ``smeared state'' of memory. Ofcourse, such assumption is natural and psychologically (i.e. in some senseexperimentally) proven; still, it is flawed because the proof is based onthe observer's own observations. EWG show that it is possible to interpretthe same observations in a totally different way and to allow arbitrarilysmeared states of the observer.
Another undesirable feature of the Orthodoxy is the necessity of adistinction between macroscopic and quantum systems. This is necessarybecause we apply different reality concepts to those systems when theformalism of state reduction is used to create the ``macroscopic'' realityout of the ``quantum'' one.
The EWG interpretation at first sight seems to be an ingenious, if slightlyfar-fetched, improvement. The major advantage is the consistency intreating observers as quantum systems with memory states (this isespecially appealing to die-hard materialists, appalled by von Neumann'sconcept of mentality being necessary to ``finalize'' the outcome of themeasurement). But let us think about the reality-related assumptions ofEWG.
The splitting of the world, according to EWG, doesn't occur -- itis just a device of the theory, to be applied each time when a classicalsystem is in danger of falling into a ``schizophrenia''. Once we havechosen the observables to be ``measured'' (the splitting depends on thatchoice), we apply the appropriate splitting to describe macroscopic systemsin terms we understand -- we always want to have definite macroscopicobservables. The reality, therefore, consists of many simultaneous``worlds'' split in all possible ways (if we interpret it in our roughmacroscopic terms); however, the theory predicts only a continuous,deterministic change in the (unobservable!) state vector of theUniverse.
Again, we have to know what is macroscopic and what isn't; this time it isneeded to split the world in the former case and do without it in thelatter. Here two questions arise: why is it necessary to split theUniverse at all? And why don't we apply the splitting to quantum systemsas well? Apparently the splitting explains the observed definiteness ofmacroscopic observables, which is not otherwise predicted by thetheory. But if we apply the splitting to a quantum system, then it wouldalways have reduced states in the corresponding world branches, exactly asa classical system which has only definite values of the observables. Indoing so we would eventually eliminate any difference between ``quantum''and ``classical'' worlds, but we won't get any closer to our intuitivelyperceived ``reality''. The main question asked of the theory -- ``What isobserved?'' -- necessarily contains an idea of some ``real'' event whichis to be described in terms external to the theory, and so an idea of``macroscopic reality'' is introduced in the theory from outside, with allambiguity and qualitativeness so characteristic of our language.
We see that in both interpretations, reality seems to be irreconciliablydivided between quantum and classical realms. Orthodoxy tolerates no shadowto be cast onto the reality of a classical body having in fact certain values of the observables (and so has to admit a certain vaguenessof that reality manifested by situations like that of our question C); andthe EWG interpretation seems to manipulate the ``splitting'' in such a wayas to achieve ``normality'' of each world branch within itself - a``normality'' which, according to the interpretation itself, has nophysical meaning. Neither of the interpretations seems to answer thequestion C by providing a procedure which would have told us that, althoughvery small systems composed of elementary particles behave weirdly, allsufficiently big systems are ``normal''. As we said, there is little reasonto expect such a procedure to be just a natural, direct consequence of theformalism; however, the distinction between classical and quantum systemsshould be quantitative in the first place. As we have seen, inboth interpretations it is not the case.
We see that due to the nature of our everyday macroscopic world, we arebound to ask ``macroscopic'' questions which contain some non-quantitativenotions, inevitably bringing those notions into the theory (or at leastinto its interpretation). Therefore, there is no reason to seek for awholly consistent interpretation until some subtle linguistic orphilosophical points in the notions of reality are clarified. As we know,physicists tend to ignore those conceptual problems as long as the theory``works''; probably, this is the best we can do.
2. Hugh Everett III, Theory of the Universal WaveFunction, in [1].
3. John Archibald Wheeler, Assessment of Everett's``Relative State'' Formulation of Quantum Mechanics, in [1].