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On Quantum Mechanics

Carlo Rovelli

Department of Physics and Astronomy,University of Pittsburgh, Pittsburgh, Pa 15260, USA.

Pittsburgh February 1, 1995.

A b s t r a c tI suggest that the common unease with taking quantum mechanicsas a fundamental description of nature (the "measurement problem")could derive from the use of an incorrect notion, as the unease withthe Lorentz transformations before Einstein derived from the notionof observer-independent time. I suggest that the incorrect notionthat generates the unease with quantum mechanics is the notion ofobserver-independent state of a system -or: observer-independentvalues of physical quantities.I reformulate the problem of the "interpretation of quantummechanics" as the problem of deriving the formalism from a set ofsimple physical postulates. I consider a reformulation of quantummechanics in terms of information theory. All systems are assumedto be equivalent, there is no observer-observed distinction, and thetheory describes only the information that systems have about eachother; nevertheless, the theory is complete.

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1. A reformulation of the problem of the "Interpretation ofQuantum Mechanics"

In this paper, I discuss a novel view of quantum mechanics. Thispoint of view is not antagonistic to current ones, as the "Copenhagen"[Heisenberg 1927, Bohr 1935], consistent histories [Griffiths 1984,Omnes 1988], decoherent histories [Gell-Mann and Hartle 1990],many-worlds [Everett 1957, Wheeler 1957, DeWitt 1970], quantumevent [Huges 1989], many minds [Albert and Lower 1988, 1989,Lockwood 1986, Donald 1990] interpretations, but rather combinesand complements them. The discussion is based on a critique of anotion generally assumed uncritically. As such, it bears a vagueresemblance with Einstein's discussion of special relativity, which isbased on the critique of the notion of absolute simultaneity. Thenotion rejected here is the notion of absolute, or observer-independent, state of a system; equivalently, the notion of observer-independent values of physical quantities. The main thesis of thepresent work is that by abandoning such a notion (in favour of theweaker notion of state -and values of physical quantities- relative tosomething), quantum mechanics makes much more sense. Thisconclusion derives from the observation that the experimentalevidence at the basis of quantum mechanics forces us to accept thatdistinct observers give different descriptions of the same events.From this, I argue that the notion of observer-independent state of asystem is inadequate to describe the physical world beyond the ¡‘ 0limit, in the same sense in which the notion of observer-independenttime is inadequate to describe the physical world beyond the c‘ ∞limit. I then consider the possibility of replacing the notion ofabsolute state with a notion that refers to the relation betweenphysical systems.

The motivation for the present work is the commonplace observationthat in spite of the 67 years-lapse from the discovery of quantummechanics, and in spite of the variety of approaches developed withthe aim of clarifying its content and improving the originalformulation, quantum mechanics still maintains a remarkable levelof obscurity. It is even accused of being unreasonable andunacceptable, even inconsistent, by world-class physicists (Forexample [Newman 1993]). My point of view in this regard is thatquantum mechanics synthesizes most of what we have learned so farabout the physical world: The issue is not to replace or fix it, but to

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understand what it actually says about the world; or, equivalently:what precisely we have learned from experimental micro-physics.

Still, it is difficult to overcome the sense of unease that quantummechanics communicates. The troubling aspect of the theory assumesdifferent faces within different interpretations, and therefore acomplete statement of the problem can only be based on a survey ofits solutions already proposed. Here, I do not attempt such a survey;for a classical review see [d'Espagnat 1971], a more modern survey isin the first chapters of [Albert 1992], or, in compact form, the recentterse paper [Butterfield 1995]. I refer to these work, and inparticular to the last two, for a discussion on what one would expectfrom a "satisfactory" interpretation of quantum mechanics. Theunease is expressed, for instance, in the objections the supporters ofeach interpretation raise against other interpretations. For example:Is there something "physical" happening when the wave function"collapses"? Is it really possible that observer and measurement,including wave function reduction, cannot not be described inSchrödinger evolution-terms? How can a classical world emergefrom a quantum reality? If somebody would prepare me in aquantum superposition of two macroscopic states, how would I feel?If the Planck constant were 25 orders of magnitude larger, howwould the world look like? Who chooses the family of consistenthistories that describe a set of events, and can this "chooser" bedescribed quantum mechanically? And so on. Some of thesequestions are perhaps naive or ill-posed, but the fact that they areregularly asked indicates that the problem of the interpretation ofquantum mechanics has not been fully disentangled yet. This unease,and the variety of interpretations of quantum mechanics that it hasgenerated is sometimes denoted the as the "measurement problem".In this paper, I address this sense of unease, and propose a way out.

The strategy of this paper is based by two ideas:1. That this unease may derive from the use of a concept which is

inappropriate to describe the physical world a the quantum level. Iargue that this concept is the concept of observer-independentstate of a system, or, equivalently, the concept of observer-independent values of physical quantities.

2. That quantum mechanics will cease look puzzling only when wewill be able to der ive the formalism of the theory from a set ofsimple physical assertions ("postulates", "principles") about theworld. Therefore, we should not try to a p p e n d a reasonableinterpretation to the quantum mechanics formal ism, but rather to

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de r i ve the formalism from a set of experimentally motivatedpostulates.

The reasons for exploring such a strategy are illuminated by anobvious historical precedent: special relativity. I shall make use ofthis analogy for explanatory reasons, in spite of the evident limits ofthe simile.

Special relativity is a well understood physical theory, appropriatelyaccredited to Einstein's 1905 celebrated paper. It is interesting inthis context to recall that the well known fact that the formal contentof special relativity is entirely coded in the Lorentz transformations,which were written by Lorentz, not by Einstein, and several yearsbefore 1905. So, what was Einstein's contribution? It was tounderstand the physical meaning of the Lorentz transformations.(And more, but this is what is of interest here). We could say,admittedly in a provocative manner, that Einstein's contribution tospecial relativity has been the interpretation of the theory, not itsformalism : the formalism already existed.

Lorentz transformations were amply discussed at the beginning ofthe century, and their in terpretat ion subjected to much debate. Inspite of the recognized fact that they represent an extension of theGalilean group compatible with Maxwell theory, the Lorentztransformation were perceived as rather unreasonable andunacceptable as a fundamental spacetime symmetry transformation,even inconsistent, before 1905. The physical interpretation proposedby Lorentz himself -a physical contraction of moving bodies, causedby complex and unknown electromagnetic interaction between theatoms of the bodies and the ether- was quite unattractive (andremarkably similar to certain interpretations of the wave functioncollapse presently investigated! ). Einstein 1905 paper suddenlyclarified the matter by pointing out the reason for the unease intaking the Lorentz transformations "seriously": the implicit use of aconcept (observer-independent time) inappropriate to describereality when velocities are high. Equivalently: a common deepassumption about reality (simultaneity is observer-independent)which is physically incorrect. The unease with the Lorentztransformations derived from a conceptual scheme in which anincorrect notion -absolute simultaneity- was assumed, yielding anysorts of paradoxical situations. Once removed this notion, the physicalinterpretation of the Lorentz transformations stood clear and specialrelativity is now universally considered rather uncontroversial.

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Here I consider the hypothesis that all "paradoxical" situationsassociated with quantum mechanics -as the famous and unfortunatehalf-dead Schrödinger cat [Schrödinger 1935]- may derive fromsome analogous incorrect notion that we use in thinking aboutquantum mechanics. (Not in using quantum mechanics, since weseem to have learned to use it in a remarkably effective way.) Theaim of this paper, therefore, is to hunt for this incorrect notion, withthe hope that by digging it out, and exposing it clearly to publiccontempt, we could free ourselves from the present unease with ourbest present theory of motion, and fully understand what the theoryis asserting about the world.

Furthermore, Einstein was so persuasive with his interpretation ofthe Lorentz equations because he did not append an interpretationto them: rather, he re-derived them, starting from two "postulates"with terse physical meaning -equivalence of inertial observers anduniversality of the speed of light- taken as facts of experience. Thisre-derivation unraveled the physical content of the Lorentztransformations and provided them with a solid in terpreta t ion. Iwould like to suggest that in order to grasp the full physical meaningof quantum mechanics, a similar result should be achieved: Find asmall number of simple statements about nature -which mayperhaps be apparently contradictory, as the two postulates of specialrelativity do- with clear physical meaning, from which the formalismof quantum mechanics could be derived. To my knowledge, such aderivation has not yet been achieved. In this paper, I do not achievesuch a result in a satisfactory manner, but I discuss a possiblereconstruction scheme.

The program outlined is thus to do for the formalism of quantummechanics what Einstein did for the Lorentz transformations: i. Finda set of simple assertions about the world, with clear physicalmeaning, that we know are experimentally true (postulates); ii.Analyze these postulates, and show that from their conjunction itfollows that certain common assumptions about the world areincorrect; iii. Derive the full formalism of quantum mechanics fromthese postulates. I expect that if this program could be completed, wewould at long last begin to agree that we have "understood" quantummechanics.

In section 2 I analyze the measurement process as described by twodistinct observers. This analysis leads to the main idea: the observerdependence of state and physical quantities, and to recognize a few

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key concepts in terms of which, I would like to suggest, the quantummechanical description of reality "makes sense". Prominent amongthese is the concept of information [Shannon 1949, Wheeler 1988,1989, 1992]. In section 3 I switch from an inductive to a (verymildly) deductive mode, and I put forward a set of notions, and a setof simple physical statements, from which the formalism of quantummechanics can be reconstructed. I denote these statements as"postulates", at the risk of misunderstanding: I do not claim anymathematical nor philosophical rigor, nor completeness, in thederivation -supplementary assumptions are made along the way. Iam not interested here in a formalization of the subject, but only ingrasping its "physics". In particular, ideas and techniques for thereconstruction are borrowed from quantum logic research, butmotivations and spirit are completely different. Finally, in section 4I discuss the picture of the physical world that has emerged, andattempt an evaluation. In particular, I compare the approach I havedeveloped with some currently popular interpretations of quantummechanics, and argue that the differences between those disappear,if the results presented here are taken into account.

In order to prevent the reader from channeling his thoughts in thewrong direction, let me anticipate a remark. By using the word"observer" I do not make any reference to conscious, animate, orcomputing, or in any other manner special, systems. I use the word"observer" in the sense in which it is conventionally used in Galileanrelativity when we say that an object has a velocity "with respect toa certain observer". The observer can be any physical object havinga definite state of motion. For instance, I say that my hand moves ata velocity v with respect to the lamp on my table. Velocity is arelational notion (in Galilean as well as in special relativistic physics),and thus it is always (explicitly or implicitly) referred to "something";it is traditional to denote this something as the "observer", but it isimportant in the following discussion to keep in mind that the"observer" can be a table lamp. Similarly, I use information theoryin its original (Shannon) form, in which information is a measure ofthe number of states in which a system can be -or in which severalsystems whose states are physically constrained (correlated) can be.Thus, a pen on my table has information because it points in this orthat direction. We do not need a human being, a cat, or a computer,to make use of this notion of information.

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2. Quantum mechanics is a theory about information

In this section, a preliminary analysis the process of measurement ispresented, and the main ideas and arguments are introduced.Throughout this section, standard quantum mechanics and standardinterpretation -by which I mean for instance: formalism andinterpretation in [Dirac 1930] or [Messiah 1958]- are assumed.

2.1 The third person problem

Consider an observer O (Observer) that makes a measurement on asystem S (System). For the moment we may think of O as a classicalmacroscopic measuring apparatus, including or not including ahuman being - this being irrelevant for what follows. Assume thatthe quantity being measured, say q, takes two values, 1 and 2; andlet the states of the system S be described by vectors (rays) in a two(complex) dimensional Hilbert space HS. Let the two eigenstates ofthe operator corresponding to the measurement of q be |1> and |2>.As it is well known: if S is in a generic normalized state |ψ> = α |1> + β|2>, where α and β are complex numbers and |α |2 + |β |2 = 1, then O canmeasure either one of the two values 1 and 2 - respectiveprobabilities being |α |2 and |β |2.

Let us assume that in a given specific measurement the outcome ofthe measurement is 1. From now on, we concentrate on describingth is specific experiment, which we denote as E . The system S isaffected by the measurement, and at a time t=t2 after themeasurement, the state of the system is |1>. In the physical sequenceof events E , the states of the system at t1 and t2 are thus

t1 ‘ t2

α |1> + β |2> ‘ |1> . (1)

This is the standard account of a measurement, according to quantummechanics.

Let us now consider this same sequence of events E , as described bya s e c o n d observer, which we refer to as P. P is an observerdifferent from O. I shall refer to O as "he" and to P as "she". Pdescribes a system formed by S and O. Therefore she views both Sand O as sub-systems of the larger S-O system she is considering.Again, we assume P uses conventional quantum mechanics. We also

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assume that P does not perform any measurement on the S-O systemduring the t1-t2 interval, but that she knows the initial states of bothS and O, and is thus able to give a quantum mechanical description ofthe set of events E . She describes the system S by means of theHilbert space HS considered above, and O by means of a Hilbert spaceH O. The S-O system is then described by the tensor product HS O =HSˆHO. As it has become conventional, let us denote the vector in HO

that describes the state of the observer O at t=t1 (prior to themeasurement) as |init>. The physical process during which Omeasures the quantity q of the system S implies a physicalinteraction between O and S. In the process of this interaction, thestate of O changes. If the initial state of S is |1> (resp |2>) (and theinitial state of O is |init>), then |init> evolves to a state that we denoteas |O1> (resp |O2>). We think of |O1> (resp |O2>)as a state in which"the position of the hand of a measuring apparatus points towardsthe mark '1' (resp '2')". It is not difficult to construct modelHamiltonians that produce evolutions of this kind, and that can betaken as models for the physical interactions that produce ameasurement. Let us consider the actual case of the experiment E , inwhich the initial state of S is |ψ> = α |1>+β |2>. The initial full state ofthe S-O system is then |ψ>ˆ |init> = (α |1>+β |2>)ˆ |init>. As well known,the linearity of quantum mechanics implies

t1 ‘ t2(α |1> + β |2>)ˆ|init> ‘ α |1> |O1> + β |2> |O2>. (2)

Thus, at t=t2 the system S-O is in the state (α |1>ˆ |O1>+β |2>ˆ |O2>).This is the conventional description of a measurement as a physicalprocess [von Neumann 1932].

I have described an actual physical process E taking place in a reallaboratory. Standard quantum mechanics requires us to distinguishsystem from observer, but it allows us freedom in drawing the linethat distinguishes the two. The peculiarity of the above analysis isjust the fact that this freedom has been exploited in order to describethe same sequence of physical events in terms of two differentdescriptions. In the first description, equation (1), the line thatdistinguishes system from observer is set between S and O. In thesecond, equation (2), between S-O and P. I recall that we haveassumed that P is not making a measurement on the S-O system;there is no physical interaction between S-O and P during the t1-t2

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interval. P may make measurements at a later time t3 : if shemeasures the value of q on S and the position of the hand on O, shefinds that the two agree, because the first measurement collapses thestate into one of the two factors of (2), leaving the secondmeasurement fully determined to the consistent value. Thus, wehave two descriptions of the physical sequence of events that wehave denoted E : The description (1) given by the observer O and thedescription (2) given by the observer P. The point I would like toemphasize here is that these are two distinct correct descriptions ofthe same sequence of events E . In the O description, the system S isin the state |1> and the quantity q has value 1. According to the Pdescription, S is not in the state |1> and the hand of the measuringapparatus does not indicate "1".

Thus, I come to the observation on which the rest of the paper relies.

Main observation: In quantum mechanics different observersmay give different accounts of the same sequence of events.

For a very similar conclusion, see [Zurek 1982]. In the rest of thework I explore the consequences of taking this observation fully intoaccount. Since this observation is crucial I now pause to discuss andreject various potential objections to the main observation. Thereader who finds the above observation plausible may skip thisrather long list of objections and jump to section 2.3.

2.2 Objections to the main observation, and replies

Objection 1. Whether the account (1) or the account (2) is correctdepends on which kind of system O happens to be. There are systemsthat induce the collapse of the wave function. For instance, if O ismacroscopic (1) is correct, if O is microscopic (2) is correct.This implies that O cannot be described as a genuine quantumsystem. Namely there are "special systems" that do not obeyconventional quantum mechanics, but are "intrinsically classical" inthat they produce collapse of the wave functions -or actualization ofquantities' values. This idea underlies a variety of old and recentattempts to unravel the quantum puzzle. The special systems beingfor instance gravity [Penrose 1989], or minds [Albert and Loewer1988], or macroscopic systems [Bohr 1949]. If we accept this idea,we have to separate reality into two kinds of systems: quantummechanical systems on the one hand, and special systems on theother. Bohr claims explicitly that we have to renounce giving a full

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quantum mechanical description of the classical world [Bohr 1949].This is echoed in texts as [Landau and Lifschit 1977]. Wigner pushesthis view to the extreme consequences and distinguishes materialsystems (observed) from consciousness (observer) [Wigner 1961].Here, on the contrary, I wish to assume

Hypothesis 1: All systems are equivalent: Nothing a prioridistinguishes macroscopical systems from quantum systems. Ifthe observer O can give a quantum description of the system S,then it is also legitimate for an observer P to give a quantumdescription of the system formed by the observer O.

Of course, I have no proof of Hypothesis 1. However, let me illustratemy motivations for holding it. I am suspicious toward attempts tointroduce special non-quantum and not-yet-understood systems orspecial new physics, to alleviate the strangeness of quantummechanics: they look very much like Lorentz's attempt to postulate amysterious interaction that Lorentz-contracts physical bodies "forreal" -something that we now perceive as ridiculous, in the light ofEinstein's clarity. Virtually all those views modify quantummechanical predictions, in spite of common statements of thecontrary: if at t2 , the state is as in (1), then P can never detectinterference terms between the two branches in (2), contrary toquantum theory predictions. Admittedly, these discrepancies arelikely to be minute, as shown by the beautiful discovery of thephysical mechanism of decoherence [Zurek 1981, Joos and Zeh 1985],which saves the phenomena. But they are nevertheless differentfrom zero, and thus observable (more on this later). I am inclined totrust that a sophisticated experiment able to detect those minutediscrepancies will fully vindicate quantum mechanics against anydistortion due to postulated intrinsic classicality of specific systems.The question is experimentally decidable; so we shall see. Second, Ido not like the idea that the present over-successful theory of motioncan only be understood in terms of its failures yet-to-be-detected.Finally and most importantly, I maintain it is reasonable to remaincommitted, up to compelling disproof, to the golden rule that allphysical systems are equivalent in their respect to mechanics: thisgolden rule has proven so overwhelmingly successful, that I am notready to dismiss as far as there is another way out.

Objection 2: What the discussion indicates is that the quantum stateis different in the two accounts, but the quantum state is a fictitious

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non-physical mental construction; the physical content of the theoryis given by the outcomes of the measurements.Indeed, one can take the view that outcomes of measurements arethe only physical content of the theory, and the quantum state is asecondary theoretical construction. This is the way I read[Heisenberg 1927] and [van Fraassen 1991]. According to this view,anything in between two measurements outcomes is like the "non-existing" trajectory of the electron, to use Heisenberg's vividexpression, of which there is nothing to say. I am very sympatheticwith this view, which plays an important role in section 3. This view,however, does not circumvent the main observation for the followingreason. The account (2) states that there is nothing to be said aboutthe value of the quantity q of S at time t2. P claims that at t=t2 thequantity q does not have a determined value. On the other hand, Oclaims that, at t=t2 , q has the value 1. From which the mainobservation follows again.

Objection 3: As before (only outcomes of measurements are physical),but the truth of the matter is that P is right and O is wrong.This is undefendable. Since all physical experiments of which weknow can be seen as instances of the S-O measurement, this wouldimply that not a single outcome of measurement has ever beingobtained yet. If so, how could we have learned quantum theory?

Objection 4: As before (only outcomes of measurements are physical),but the truth of the matter is simply that O is right and P is wrong.If P is wrong, quantum mechanics cannot be applied to the S-Osystem (because her account is a straightforward implementation oftextbook quantum mechanics). Thus this objection predictsdiscrepancies, so far never observed, with quantum mechanicalpredictions, which include observable interference effects betweenthe two terms of (2) .

Object ion 5: As before, but under the assumption that O ismacroscopic. Then the interference terms mentioned becomeextremely small because of decoherence effects. If they are smallenough, they are unobservable, and thus q=1 becomes an absoluteproperty of S, which is true and absolutely determined, albeitunknown to P, who could measure it anytime, and would not seeinterference effects.Again, strictly speaking this is wrong, because decoherence dependon which subsequent observation P does. Therefore, the propertyq=1 of S would become an absolute property at time t2, or not,

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according to which subsequent properties of S the observer (P)considers. This is the reason for which the idea of exploiting physicaldecoherence for interpreting quantum mechanics has evolved intothe consistent and decoherent histories interpretations, whereprobabilities are (consistently) assigned to histories, and not to singleoutcomes of measurements within a history. (See, however, thediscussion on the no-histories slogan in [Butterfield 1995].)

Objection 6. There is no collapse. The description (1) is not correct,because "the wave function never really collapses". The account (2 )is the correct one. There are no values assigned to classicalproperties of system, but only quantum states.If so, then the observer P cannot measure the value of the propertyq either, since (because of assumption) there are no values assignedto classical properties but only quantum states; thus the quantity qdoesn't ever have a value. But we do describe the world in terms of"properties" that the systems have and values assumed by variousquantities, not in terms of states in the Hilbert space vector. In adescription of the world purely in terms of quantum states, thesystems never have definite properties and I do not see how tomatch the description with any observation. For a detailedelaboration of this argument, which I view as very strong, see[Albert 1992].

Objection 7 There is no collapse. The description (1) is not correct,because "the wave function never really collapses". The account (2 )is the correct one. The values assigned to classical properties aredifferent from branch to branch.This is a form of Everett's view [Everett 1957]. Which entails the ideathat when we measure the electron's spin being up, the electron spinis also and simultaneously down "in some other branch" -or "world",hence the Many World denomination of this view. The property ofthe electron of having spin up is then not absolutely true, but justtrue relative to "this" branch, namely we simply have a new"parameter" for expressing contingency: "which branch" is a new"dimension" of indexicality, in addition to the familiar ones as "whichtime" and "which place". Thus, the state of affairs of the example isthat, at t2, q has value 1 in one branch and has value 2 in the other;the two branches being theoretically described by the two terms in(2). This is a fascinating idea that has recently being implemented ina variety of diverse incarnations. Traditionally, the idea has beendiscussed in the context of a notion of apparatus, namely adistinguished set of subsystems of the universe, and a distinguished

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quantity of such an apparatus -the preferred basis. Such a (collectionof) preferred apparata and preferred basis are needed in order todefine branching, and thus in order to have assignment of values[Butterfield 1995]; the view has recently branched (!) into the manymind interpretations, where the distinguished subsystems arerelated to various aspect of the human brain. (See [Butterfield 1995]for a recent discussion). All these versions of Everett idea violateHypothesis 1, and thus I am not concerned with them here.Alternatively, there are versions of Everett idea that reject thespecification of a preferred apparatus and preferred basis, and inwhich the branching itself is indexed by an arbitrarily chosen systemplaying the role of apparatus and an arbitrarily chosen basis. To myknowledge, the only elaborated versions of this view which avoid thedifficulties mentioned in objection 5 have evolved into the historiesformalisms considered below.

Objection 8. What is absolute and observer independent is theprobability of a sequence A 1, ... An of property ascriptions (such thatthe interference terms mentioned above are extremely small -decoherence); this probability is independent from the existence ofany observer measuring these properties.This is certainly correct, and, in fact, this observation is at the root ofthe histories interpretations of quantum mechanics [Griffiths 1984,Omnes 1988, Gell-Mann and Hartle 1990]. However this viewconfirms the observation above that different observers givedifferent accounts of the same sequence of events, for the following-often overlooked- reason. The beauty of the historiesinterpretations is the fact that the probability of a sequence ofoutcomes within a consistent family of sequences does not dependson the observer, precisely as it doesn't in classical mechanics. One canbe content with this powerful aspect of the theory and consistentlystop here. However, probabilities depends on the choice of theconsistent family of histories used to describe a sequence of events.For instance, two contradictory events may both be predicetd withprobability equal to one if considered within two different families.The observer makes a choice in picking up a family of alternativehistories in terms of which he describe the system. Unlike classicalmechanics, this choice is such that alternative properties can both betrue, depending the family chosen [Griffiths 1993, Hartle 1994, Kent1996]. Now, the observer (O) too is a physical system (even MurrayGell-Mann is a physical system). Unless we assume that the observeris an unphysical entity, we then are free to consider how a secondobserver would describe the events, and we are back to the problem

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above: the descriptions of the same sequence of events given by thetwo observers can be different. Classes of observers may agree onsets of outcomes (or interpret the differences as statisticalignorance), but there may always be other observers, perhapsobserving the sequence of events a n d O, who have chosen adifferent family of consistent histories to describe the same sequenceof events. Note not just the probabilities, but even the actualdescription of what has happened in a concrete instance (see note 5,below) can be different. Thus, the histories interpretations do notemphasize but confirm the conclusion that if an observer O gives adescription of a sequence of events, another observer -choosing adifferent family of histories- may give a different description of thesame sequence. I shall return to this point in the last section.

In conclusion, it seems to me that whatever view of quantum theory(consistent with hypothesis 1) one may hold, the main observation isunescapable. I may thus proceed to the main point of this work.

2.3 Main discussion

If different observers give different accounts of the same sequenceof events, then each quantum mechanical description has to beunderstood as relative to a particular observer. Thus, a quantummechanical description of a certain system (state and/or values ofphysical quantities) cannot not be taken as an "absolute" (observerindependent) description of reality, but rather as a formalization, orcodification, of properties of a system relat ive to a given observer.Quantum mechanics can therefore be viewed as a theory about thestates of systems and values of physical quantities relative to othersystems.

A quantum description of the state of a system S exists only if somesystem O (considered as an observer ) is actually "describing" S, or,more precisely, has interacted with S. The quantum state of a systemis always a state of that system with respect to a certain othersystem. More precisely: when we say that a physical quantity takesthe value v, we should always (explicitly or implicitly) qualify thisstatement as: the physical quantity takes the value v with respect tothe so and so observer. Thus, in the example considered in section2.1, q has value 1 with respect to O, but not to P. It seems to me thatthere is no way to escape this conclusion.

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Therefore, I maintain that in quantum mechanics "state" as well as"value of a variable" -or "outcome of a measurement"- are relationalnotions in the same sense in which velocity is relational in classicalmechanics. We say "the object S has velocity v " meaning "withrespect to a reference object O". Similarly, I maintain that "thesystem is in such a quantum state" or "q=1" are always to beunderstood "with respect to the reference O". In quantum mechanicsall physical variables are relational, as velocity is.

If quantum mechanics describes only relative information, one couldconsider the possibility that there is a deeper underlying theory thatdescribes what happens "in reality". This is the thesis of theincompleteness of quantum mechanics (first suggested in [Born1926]!). Examples of hypothetical underlying theories are hiddenvariables theories [Bohm 1951, Belifante 1973]. Alternatively, the"wave-function-collapse-producing" systems can be "special" becauseof some non-yet-understood physics, which becomes relevant due tolarge number of degrees of freedom [Ghirardi Rimini and Weber1986, Bell 1987], complexity [Hughes 1989], quantum gravity[Penrose 1989] or other.

As is well known, there are no indications on physical grounds thatquantum mechanics is incomplete. Indeed, the practice of quantummechanics supports the view that quantum mechanics represents thebest we can say about the world at the present state ofexperimentation, and suggests that the structure of the worldgrasped by quantum mechanics is deeper, and not shallower, thanthe scheme of description of the world of classical mechanics. On theother hand, one could consider motivations on m e t a p h y s i c a lgrounds, in support to the incompleteness of quantum mechanics.One could argue: "Since reality has to be real and universal, and thesame for everybody, then a theory in which the description of realityis observer-dependent is certainly an incomplete theory". If such atheory were complete, our concept of reality would be disturbed.

But the way I have reformulated the problem of the interpretation ofquantum mechanics in sec.1 should make us suspicious and attentiveprecisely to such kinds of arguments. Indeed, what we are lookingfor is precisely some "wrong general assumption" that we suspect tohave, and that could be at the origin of the difficulty inunderstanding quantum mechanics. Thus, I discard here the thesisof the incompleteness of quantum mechanics and tentatively assume

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Hypothesis 2: (Completeness) Quantum mechanics provides acomplete and self-consistent scheme of description of the physicalworld, appropriate to our present level of experimentalobservations.

The conjunction of this hypothesis 2 with the "Main observation" ofsection 2.1 and the discussion above leads to the following idea:

Quantum mechanics is a theory about the physical description ofphysical systems relative to other systems, and this is a completedescription of the world.

The thesis of this paper is that this conclusion is not self-contradictory. If this conclusion is valid, then the "incorrect notion"at the source of our unease with quantum theory has beenuncovered: it is the notion of true, universal, observer-independentdescription of the state of the world. If the notion of observer-independent description of the world is unphysical, a completedescription of the world is exhausted by the relevant informationthat systems have about each other. Namely, there is neither anabsolute state of the system, nor absolute properties that the systemhas at a certain time. Physics is fully relational, not just as far as thenotions of "rest" and "motion" are considered, but with respect to anyphysical quantities. Accounts (1) and (2) of the sequence of events Eare both correct, even if distinct: any time we talk about a "state" or"property" of a system, we have to refer these notions to a specific"observing", or "reference" system. Thus, I propose the idea thatquantum mechanics indicates that the notion of a "universal"description of the state of the world, shared by all observers, is aconcept which is physically untenable, on experimental ground. It isvalid only in the ¡‘0 approximation.1

Thus, the hypothesis on which I base this paper is that accounts (1)and (2) are both fully correct. They refer to different observers: (1)refers to O, while (2) refers to P. I propose to reinterpret everycontingent statement about nature, as for instance: "the electron has

1 To counter objections based on instinct alone, it is perhaps worthwhilerecalling the great resistance that the idea of fully relational notions of "rest"and "motion" encountered at the beginning of the scientific revolution; andthe fact that we should perhaps be ready to accept that quantum mechanics(and general relativity) could well be in the course of triggering of a -not yetdeveloped- revision of world views as far reaching as the seventeenthcentury's one (on this, see [Rovelli 1995]).

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spin up", "the atom is in the so and so excited state", the "spring iscompressed", "the chair is here and not there", as elliptic expressionsfor relational assertions: "the electron has spin up with respect to theStern Gerlac apparatus " ... "the chair is here and not there w i t hrespect to my eyes ", and so on. Quantum states, as well as values ofphysical quantities make sense only when referred to a physicalsystem (which I denote as observer system, or reference system). Ageneral physical theory is a theory about the state that physicalsystems have, relative to each other. I explore and elaborate thispossibility in this paper.

2.4 Relation between descriptions

An immediate issue raised by the multiplication of points of viewinduced by the relational notion of state and physical quantities'values is the problem of the relation between distinct descriptions ofthe same events. What is the relation between the value of a variableq relative to an observer O, and the value of the same variablerelative to a different observer? We do not like a solipsistic world ofun-communicative observers, nor, in any case, this is what quantummechanics describes. Let me reconsider the example of section 2.1.It is clear that there is some relation between the description of theworld illustrated in (1) and in (2). More precisely, we may ask twoquestions: (i.) Does P "know" that S "knows" the value of q? (ii.) DoesP know what is the value of q relative to O? I consider these twoquestions one at the time.

(i.) Does P "know" that O has made a measurement on S at time t2?The answer is a definite yes, for a number of reasons. First, theobserver P has a full account of the events E , so it should be possibleto interpret her description (2) as expressing the fact that O hasmeasured S. Indeed this is possible: The key observation is that inthe final state at t2 in (2) , the variables q (with eigenstates |1> and|2>) and the pointer variable (with eigenstates |O1> and |O2>) arecorrelated. From this fact, the observer P understands that thepointer variable in O has information about q. In fact, the state of S-Ois the quantum superposition of two states: in the first, (|1>ˆ |O1>), Sis in the |1> state and the hand of the observer is correctly on the "1"mark. In the second, (|2>|O2>), S is in the |2> state and the hand ofthe observer is, correctly again, on the "2" mark. In both cases, thehand of O is on the mark that correctly represents the state of thesystem. More precisely, P could consider an observable M that

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checks whether the hand of O indicates the correct state of S. If shemeasured M, then the outcome of this measurement would be "yes"with certainty, when the state of the S-O system is an in (2). Theoperator M is given by

M |1> |O1> = |1> |O1>, M |1> |O2> = 0 ,

M |2> |O2> = |2> |O2> , M |2> |O1> = 0 , (3)

where the eigenvalue 1 means "yes, the hand of O indicates thecorrect state of S" and the eigenvalue 0 means "no, the hand of Odoes not indicates the correct state of S". Thus, it is meaningful tosay that, according to the P description of the events E , O "knows" thequantity q of S, or that he "has measured" the quantity q of S, andthe pointer variable embodies the information.

The other question P may be ask is: (ii.) What is the outcome of themeasurement performed by O? It is important not to confuse thestatement "P knows that O knows the value of q" with the statement"P knows what O knows about q". Of course this is a consistentdistinction common in everyday life (I know that you know theamount of your salary, but I do not know what you know about theamount of your salary). Now in general the observer P does notknow "what is the value of the observable q that O has measured"(unless α or β in (2) vanish). The relation between the descriptionsthat different observers give of the same event is characterized bythe fact that an observer with sufficient initial information maypredict what the other observer has measured, but not the outcomeof the measurement. Communication of measurements results ishowever possible (and fairly common!). P can measure the outcomeof the measurement performed by O. She can, indeed, measurewhether O is in |O1>, or in |O2>. Notice that there is a consistencycondition to be fulfilled, which is the following: if P knows that O hasmeasured q, and then she measures q, and then she measures what Ohas obtained in measuring q, consistency requires that the resultsobtained by P about the variable q and the pointer are correlated.Indeed, they are; as was first noticed by von Neumann, and is clearfrom (2). Thus, there is a satisfied consistency requirement in thenotion of relative description discussed. This can be expressed interms of standard quantum mechanical language: From the point ofview of the P description:

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The fact that the pointer variable in O has information about S(has measured q) is expressed by the existence of a correlationbetween the q variable of S and the pointer variable of O. T h eexistence of this correlation is a measurable property of the O-Sstate.

Notice that by representing the fact that (for P) "the pointer variableof O has information about the q variable in S" by means of theoperator M resolves the well known and formidable problem ofdefining the "precise moment" in which the measurement isperformed, or the precise "amount of correlation" needed for ameasurement to be established -see for instance [Bacciagaluppi andHemmo 1995]. Such questions are not classical question, butquantum mechanical questions, because whether or not O hasmeasured S is not an absolute property of the O-S state, but aquantum property of the quantum O-S system, that can beinvestigated by P, and whose yes/no answers are, in general,determined only probabilistically. In other words: i m p e r f e c tcorrelation does not imply no measurement performed, but only asmaller than 1 probability that the measurement has beencompleted.

2.5 Information

It is time to introduce the main concept in terms of which I proposeto interpret quantum mechanics: information. In section 2.3 Iemphasized the relational character of any quantum mechanicalassertion: the complete meaning of "q=1" is "q=1 relative to O". Themain hypothesis here is that this relational character of physicalstatements is not due to incompleteness of quantum theory, but it isdue to the physical inapplicability of the notion of "observerindependent value of q" to the natural world. Now, one may ask,what is the nature of the relation between the variable q and thesystem O expressed in the statement "q=1 relative to O"? In otherwords, does this relation has a comprehensible physical meaning?Can we analyse it in physical terms?

Let me begin by a purely lexical move. I will denote the relationbetween a physical quantity q of a system S, and the observersystem O with respect to which q has a certain value, as"information". I will say "O has the information that q=1" to mean"q=1 relative to S". The use of this expression "information" emergesnaturally in considering the example of section 2.1, where the

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observing systems O and P are complex observers actively gatheringinformation about S; but I ask the reader to temporary suspend anyevocative implication of the expression "information" at this stage.For the moment, "information" is just a word to denote the relationalcharacter of every contingent assertion about values of physicalquantities, or states of systems. The question addressed here is thuswhat is the physical meaning of the fact that O has information aboutthe variable q.

If the statement "q has a value relative to O", or "O has informationabout q", has any comprehensible physical meaning at all, thismeaning should be related to the contingent state of the S-O system.According to the main hypothesis here, asking about the observer-independent contingent state of the S-O system has no meaning.Therefore, it seems at first that we cannot understand what is thephysical nature of the S - O relation. But this conclusion is tooprecipitous. Indeed, we can make statements about the state of theS-O system, provided that we interpret these statements as relativeto a third physical system P. Therefore, it should be possible tounderstand what is the physical meaning of "q has a value relative toO" by considering the description that P gives (or could give) of theS-O system. This description is not in terms of classical physics, butin quantum mechanical terms; it is the one given in detail in section2.4. The result is that "S has information about q" means that thereis a correlation between the variable q and the pointer variable in O.

This result provides a motivation for the use of the expression"information", because information is correlation. The notion ofinformation I employ here should not be confused with other notionsof information used in other contexts. I shall use here a notion ofinformation that does not require distinction between human andnon-human observers, systems that understand meaning or don't,very-complicated or simple systems, and so on. As it is well known,the problem of defining such a notion was brilliantly solved byClaude Shannon [Shannon 1949]: in the technical sense ofinformation-theory, the amount of information is the number of theelements of a set of alternatives out of which a configuration ischosen. Information expresses the fact that a system is in a certainconfiguration, which is correlated to the configuration of anothersystem (information source). The relation between this notion ofinformation and more elaborate notions of information is given bythe fact that the information-theoretical information is a minimalcondition for any "elaborate information". In a physical theory it is

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sufficient to deal with this basic information-theoretical notion ofinformation. This is very weak; it does not require us to considerinformation storage, thermodynamics, complex systems, meaning, oranything of the sort. In particular: (i.) information can be lostdynamically (correlated state may become uncorrelated); (ii.) we donot distinguish between "obtained" correlation and "accidental"correlation (if the pointer of the apparatus indicates the correct valueof the spin, we say that the pointer has information about the spin,whether or not this is the outcome of a "well thought" interaction);Most important: (iii.) any physical system may contain informationabout another physical system. For instance if we have two spin-1/2particles that have the same value of the spin in the same direction,we say that one has information about the other one. Thus "observersystem" in this paper is any possible physical system (with morethan one state). If there is any hope of understanding how a systemmay behave as observer without renouncing the postulate that allsystems are equivalent, then the same kind of processes -collapse-that happen between an electron and a CERN machine, may alsohappen between an electron and another electron. Observers are not"physically special systems" in any sense. The relevance ofinformation theory for understanding quantum physics has beenadvocated by John Wheeler [Wheeler 1988, 1989, 1992].

Now, as section 2.4 has shown, the fact that q has a value relative toO means that q is correlated with the pointer variable in O.Therefore, it means that the pointer variable in O has informationabout q, in the information theoretical sense. This is the reason forchoosing the expression "information" to denote the relational aspectof physical values ascriptions. Thus, the physical nature of therelation between S and O expressed in the fact that q has a valuerelative to S is captured by the fact that S has information (in thesense of information theory) about q. By "O has information aboutq", we mean "relative to S, q has a value" and also: "relative to P,there is a certain correlation in the S and O states". Notice that thisis, in a sense, a partial answer to the question formulated at thebeginning of this section. First, it is a quantum mechanical answer,because P's information about the S-O system is probabilistic.Second, it is an answer that only shifts the problem by one step,because the information possessed by O is explained in terms of theinformation possessed by P. Thus, the notion of information I usehas a double valence. On the one hand, I want weaken all physicalstatements that we make: not "the spin is up", but "we haveinformation that the spin is up" -which leaves the possibility open to

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the fact that somebody other has different information. Thus,i n fo rmat ion indicates the usual ascription of values to quantitiesthat founds physics, but emphasizes the relational aspect. On theother hand, this ascription can be described within the theory itself,as information-theoretical information, namely correlation. But sucha description, in turns, is quantum mechanical and observerdependent, because a universal observer independent description ofthe state of affairs of the world is fantasy.

Physics is the theory of the relative information that systems haveabout each other. This information exhaust everything we can sayabout the world.

At this point, all the main ideas and concepts have been formulated.In the next section, I consider a certain number of postulatesexpressed in terms if these concepts, and derive quantum mechanicsfrom these postulates.

3. On the reconstruction of Quantum Mechanics

3.1 Basic concepts

Physics is concerned with relations between physical systems. Inparticular, it is concerned with the description that physical systemsgive about physical systems. Following hypothesis 1, I reject anyfundamental distinctions as: system/observer, quantum/ classicalsystem, physical system/consciousness. I assume that the world canbe decomposed (possibly in a large number of ways) in a collection ofsystems, each of which can be equivalently considered as anobserving system or as an observed system. A system (observ ingsys tem ) may have i n f o rma t i on about another system (o b s e r v e dsys tem ). Information is exchanged via physical interactions. Theactual process through which information is collected and stored isnot of particular interest here, but can be physically described in anyspecific instance.

Information is a discrete quantity: there is a minimum amount ofinformation exchangeable (a single bit, or the information thatdistinguishes between just two alternatives.) The process ofacquisition of information (a measurement) can be described as a"question" that a system (observing system) asks another system(observed system). This anthropological language is not meant to

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suggest any special feature or complexity of the systems considered.Since information is discrete, any process of acquisition ofinformation can be decomposed into acquisitions of elementary bitsof information. We refer to an elementary question that collects asingle bit of information as a "yes/no question", and we denote thesequestions as Q1, Q2, ... .

Any system S, viewed as an observed system, is characterized by afamily of yes/no questions that can be meaningfully asked to it.These corresponds to the physical variables of classical mechanicsand to the observables of conventional quantum mechanics. Wedenote the set of these questions as W (S) = {Qi , i´ I}, where the index ibelongs to a set I characteristic of S. The general kinematical featuresof S are representable as relations between the questions Qi in W (S),that is, as structures over W (S). For instance, meaningful questionsthat can be asked to an electron are whether the particle is in acertain region of space, whether its spin along a certain direction ispositive, and so on.

By asking a sequence of questions (Q1, Q2 , Q3 ... ) to S, an observersystem O may compile a string

(e1, e2, e3, ...... ) (4)where each ei is either 0 or 1 ("no" or "yes") and represents the"answer" of the system to the question Qi . (More precisely, theinformation that O has about S can be represented as a string.) It isof course a basic fact about nature that knowledge of a portion (e1, ..., en ) of this string provides indications about the subsequentoutcomes (en, en+1, ... ). It is in this sense that a string (4) containsthe information that O has about S.

It is useful to distinguish between information contained in anarbitrary string (4) and re levant information. (Repeating the samequestion (experiment) and obtaining always the same outcome doesnot increase the information on S.) The relevant information (orsimply informat ion ) that O has about S is defined as the non-trivialcontent of the (potentially infinite) string (4), that is the part of (4)relevant for predicting future answers of possible future questions.The relevant information is the subset of the string (4), obtaineddiscarding the ei 's that do not affect the outcomes of futurequestions.

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The relation between the notions introduced and traditional notionsused in quantum mechanics is rather transparent: A question is ofcourse a version of a measurement. The idea that in quantummechanics measurements can be reduced to yes/no measurements iswell known. A yes/no measurement is represented by a projectionoperator onto a linear subset of the Hilbert space, or by the linearsubset of the Hilbert space itself. Here this idea is not derived fromthe quantum mechanical formalism, but is justified in information-theoretical terms. The notions of observing system and observedsystem reflect traditional notions of observer and system; however,any sort of distinction between classical and quantum systems of thesort advocated by Bohr is rejected here. The set of questions thatcan be asked about a given system S, namely W (S), reflects thenotion of the set of the observables. I recall that in algebraicapproaches a system is characterized by the (algebraic) structure ofthe family of its observables.

On the other hand, there is a notion not mentioned here: the state ofthe system. The absence of this notion is the prime feature of theinterpretation considered here. In place of the notion of state, whichrefers solely to the system, the notion of the information that asystem has about another system has been introduced. I do not viewthe notion of information as "metaphysical", but as a concrete notion:I imagine a piece of paper on which outcomes of measurements on Sare written, or hands of measuring apparata, or memory of scientists,or a two-value variable which is "up" or "down" after an interaction.

For simplicity, in the following I focus only on systems that inconventional quantum mechanics can be described by means of finitedimensional Hilbert spaces. This choice simplifies the mathematicaltreatment of the theory, avoiding continuum spectrum and otherinfinitary issues. I leave to future work the extension to continuumsystems.

3.2 The two main postulates

Postulate 1 (Limited information). There is a maximum amountof relevant information that can be extracted from a system.

The physical meaning of Postulate 1 is that it is possible to exhaust,or "give a complete description of the system", in a finite time. Inother words, any future prediction that can be inferred about the

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system out of an infinite string (6), can also be inferred from a finitesubset

s = [e1, ... , eN] (5)

of (4), where N is a number that characterizes the system S. Thefinite string (5) represents the knowledge that O has about S.2 Onemay say that any system S has a maximal "information capacity" N,where N, being an amount of information, is expressed in bits. Thismeans that N bits of information will exhaust everything we can sayabout the system S. Thus, each system is characterized by a numberN. In terms of traditional notions, we can view N as the smallestinteger such that N ≥ log2 k, where k is the dimension of the Hilbertspace of the system S. Recall that the outcomes of the measurementof a complete set of commuting observables, characterizes the state,and in a system described by a k = 2N dimensional Hilbert space suchmeasurements distinguish one possible outcome out of 2N alternative(the number of orthogonal basis vectors): this means that one gainsinformation N on the system. Postulate 1 is confirmed by ourexperience about the world (within the assumption above, that werestrict to finite dimensional Hilbert space systems. Generalization toinfinite systems should not be difficult).

Notice that Postulate 1 already adds the Planck constant to classicalphysics. Consider a classical system described by a variable q thattakes continuous values; for instance, the position of a particle.Classically, the amount of information we can gather about it isinfinite: we can locate its state in the system's phase space witharbitrary precision. Quantum mechanically, this infinite localization isimpossible because of Postulate 1. Thus, maximum availableinformation can localizes the state only within a finite region of thephase space. Since the dimensions of the classical phase space of anysystem are (L2T -1M ) n, this implies that there is a universal constantwith dimension L2T -1M that determines the minimal localizability ofobjects in phase space. This constant is of course Planck constant.Thus we can view Planck constant just as the transformationcoefficient between physical units (position X momentum) andinformation theoretical units (bits).

2 The string (7) is essentially the state of the system. The novelty here is notthe fact that the state is defined as the response of the system to a set of yes/noexperiments: this is the traditional reading of the state as a preparationprocedure. The novelty is that this notion of state is relative to the observerthat has asked the questions.

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What happens if, after having asked the N question such that themaximal information about S has been gathered, the system O asks afurther question Q = QN+1 ? I introduce the second postulate.

Postulate 2 (Unlimited questions). It is always possible toacquire new information about a system.

If, after having gathered the maximal information about S, thesystem O asks a further question Q, to the observed system S, thereare two extreme possibilities: either the question Q is fullydetermined by previous questions, or not. In the first case, no newinformation is gained. However, the second postulate assures us thatthere is always a way to acquire new information. This postulateimplies therefore that the sequence of responses we obtain fromobserving a system cannot be fully deterministic.

The motivation for the second postulate is fully experimental. Weknow that all quantum systems (and all systems are quantumsystems) have the property that even if we know their quantumstate |ψ > exactly, we can still "learn" something new about them byperforming a measurement of a quantity O such that |ψ > is not aneigenstate of O. This is an exper imen ta l result about the world,coded in quantum mechanics. Postulate 2 expresses this result.Postulate 2 is true to the extent the Planck constant is different fromzero: in other words, for a macroscopic system, getting to questionsthat increase our knowledge of the system after having reached themaximum of our information implies measurements with extremelyhigh sensitivity.

Since the amount of information that O can have about S is limited bypostulate 1, when new information is acquired, part of the oldrelevant information must become irrelevant. In particular, if a newquestion Q (not determined by the previous information gathered), isasked, then O should loose (at least) one bit of the previousinformation. So that, after asking the question Q, new information isavailable, but the total amount of information about the system doesnot exceed N bits.

Rather surprisingly, those two postulates are (almost) sufficient toreconstruct the full formalism of quantum mechanics. Namely, onemay assert that the physical content of the general formalism of

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quantum mechanics is (almost) nothing but a sequence ofconsequences of two physical facts expressed in postulates 1 and 2.This is illustrated in the next section.

3.3 Reconstruction of the formalism, and the third postulate

In this section, I discuss the possibility of deriving the full formalismof quantum mechanics simply from the two simple physicalassertions contained by postulates 1 and 2. This section is rathertechnical, and the uninterested reader may skip it and jump tosection 3.4. The technical machinery to be employed has beendeveloped (in a somewhat different spirit) in quantum logic analyses.See for example [Beltrametti and Cassinelli 1981]. As I mentioned inthe introduction, this reconstruction attempt is not fully successful.In fact I will be forced to introduce a third postulate (besides variousrelative minor assumption). I will speculate on the possibility ofgiving this postulate a simple physical meaning, but I do not haveany clear result. This difficulty reflects the parallel difficulties in thequantum logic reconstruction attempts.

Let me begin by analysing the consequences of the first postulate.The number of questions in W (S) can be much larger than N. Some ofthese questions may not be independent. In particular, one may find(experimentally) that they can be related by implication (Q1 => Q2) ,union (Q3 = Q1 V Q2) and intersection (Q3 = Q1 Λ Q2), that we candefine an always false (Qo) and an always true question (Q∞ ), thenegation of a question (not-Q), and a notion of orthogonality asfollows: If Q1 => not-Q2, Q1 and Q2 are orthogonal (we indicate this asQ1 ⊥ Q2). Equipped with these structures, and under the (non trivial)additional assumption that V and Λ are defined for every pair ofquestions, W (S) is an orthomodular lattice [Beltrametti and Cassinelli1981, Huges 1989].

If there is a maximal amount of information that can be extractedfrom the system, we may assume that one can select in W (S) anensemble of N questions Qi, which we denote as c = {Q i, i=1, N}, thatare independent one from each other. We do not assume there isanything canonical in this choice, so that there may be many distinctfamilies c , b, q, ... of N independent questions in W (S). If a system Oasks the N questions in the family c to a system S, then the answersobtained can be represented as a string that we may denote as

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sc␣ = [e1, ...... , eN]c (6)

The string sc represents the information that O has about S, as aresult of the interaction that allowed it to ask the questions in c . Thestring sc ␣ can take 2N = K values; we denote these values assc(1),␣sc(2), ... ,sc(K) . So that

sc(1) = [0, 0, .... , 0]c, ...., sc(K) = [1, 1, .... , 1]c. (7)

Since the 2N possible outcomes sc (1),␣sc (2), ... ,sc (K) of the N yes/noquestions are (by construction) mutually exclusive, we can define 2N

new questions Qc (1)... Qc (K ) such that the "yes" answer to Qc ( i )

corresponds to the string of answers sc (i) :

Qc(1) = not Q1 Λ not Q2 Λ .... Λ not QN Qc(2) = not Q1 Λ not Q2 Λ .... Λ QN

... Qc(k) = Q1 Λ Q2 Λ .... Λ QN. (8)

We refer to questions of this kind as "complete questions". By takingall possible unions of sets of complete questions Qc (i) (of the samefamily c ), we can construct a Boolean algebra that has Qc (i) as atoms.

Alternatively, the observer O could use a d i f fe rent family of Nindependent yes-no questions, in order to gather information aboutS. Let us denote this other set as b . Then, he will still have amaximal amount of relevant information about S formed by an N-bits string sb ␣ = [e1, ...... , eN]b . Thus, O can give different kinds ofdescriptions of S, by asking different questions. Correspondingly, wedenote as sb (1)... sb (K) the 2N values that sb can take, and we considerthe corresponding complete questions Qb (1)... Qb (K) and the Booleanalgebra they generate. Thus, it follows from the first postulate thatthe set of the questions W (S) that can be asked to a system S has anatural structure of an orthomodular lattice containing subsets thatform Boolean algebras. This is precisely the algebraic structureformed by the family of the linear subsets of a Hilbert space, whichrepresent the yes/no measurements in ordinary quantum mechanics[Jauch 1968, Finkelstein 1969, Piron, 1972, Beltrametti andCassinelli 1981].

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The next question is the extent to which the information (6) aboutthe set of questions c determines the outcome of an additionalquestion Q. There are two extreme possibilities: that Q is fullydetermined by (6), or that it is fully independent, namely that theprobability of getting a "yes" answer is 1/2. In addition, there is arange of intermediate possibilities: The outcome of Q may bedetermined probabilistically by sc . The second postulate statesexplicitly that there are questions that are non determined. We maydefine, in general, as p(Q, Qc (i)) the probability that a "yes" answerto Q will follow the string sc ( i ) . Given two complete families ofinformation sc and sb , we can then consider the probabilities3

pij = p(Qb (i), Qc(j)) (9)

From the way it is defined, the 2N x 2N matrix pi j cannot be fullyarbitrary. First, we must have

0 ≤ pij ≤ 1. (10)Then, if the information sc (i) , is available about the system, one andonly one of the outcomes sb (j) , may result. Therefore

& i pij = 1. (11)We also assume that p(Qb (i), Qc (j)) = p(Qc (j),Qb ( i ) ) (this is a newassumption! There is a relation with time reversal, but I leave it hereas an unjustified assumption at this stage), from which we must have

& j pij = 1. (12)The conditions (10-11-12) are strong constraints on the matrix p i j .They are satisfied if

pij = | Uij |2 (13)where U is a unitary matrix, and that p ij can always be written inthis form for some unitary matrix U (which, however, is not fullydetermined by p ij ) .

In order to take into account questions which are in the Booleanalgebra generated by a family sc , as for instance

3 I do not wish to enter here the debate on the meaning of probability inquantum mechanics. I think that the shift of perspective I am suggesting ismeaningful in the framework of an "objective" definition of probability, tiedto the notion of repeated measurements, as well as in the context of"subjective" probability, or any variant of this, if one does not accepts Jaynescriticisms of the last. The novelty of the proposal regards the notion of state,not the notion of probability.

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Qc(jk) = Qc(j) V Qc(k), (14)we cannot consider probabilities of the form p(Qb (i), Qc (jk)), because a"yes" answer to Qc (jk) is less than the maximum amount of relevantinformation. But we may for instance consider probabilities of theform

pi(jk)i = p(Qb (i), Qc(jk) Qb (i) ) (15)defined as the probability that a "yes" answer to Qb (i) will follow a"yes" answer to Qb (i) (N bits of information) and a subsequent "yes"answer to Qc (jk) (N-1 bits of information). As is well known, we have(experimentally !) that

pi(jk)i ›p(Qb(i)), Qc(j)) p(Qc(j),Qb (i)) + p(Qb (i),Qc(k)) p(Qc(k),Qb (i)) == (pij )2 + (pik)2 (16)

Accordingly, we can determine the missing phases of U in (21) bymeans of

pi(jk)i = | UijU ji + UikUki |2 (17)It would be extremely interesting to study the constraints that theprobabilistic nature of the quantities p implies, and to investigate towhich extent the structure of quantum mechanics can be derived infull from these constraints. One could conjecture that eqs.(13-17)could be derived solely by the properties of conditional probabilities-or find exactly the weakest formulation of the superpositionprinciple directly in terms of probabilities: this would be a strongresult. Even more interesting would be to investigate the extent towhich the already noticed consistency between different observers'descriptions, which I believe characterises quantum mechanics somarvelously, could be taken as the missing input for reconstructingthe full formalism. I have a suspicion this could work, but have nodefinite result. Here, I content myself with the much more modeststep of introducing a third postulate. For strictly related attempts toreconstruct the quantum mechanical formalism from the algebraicstructure of the measurement outcomes, see [Mackey 1963,Maczinski 1967, Finkelstein 1969, Jauch 1968, Piron 1972].

Postulate 3 (Superposition principle). If c and b define twocomplete families of questions, then the unitary matrix Ucb i n

p( Qc(i), Qb (j) ) = |Ucb ij |2 , (18)

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can be chosen in such a way that for every c , b and r , we haveUcr = Ucb U br and the effect of composite questions is given byeq.(17).

It follows that we may consider any question as a vector in acomplex Hilbert space, fix a basis |Qc (i)> in this space and representany other question |Qb (j)> as a linear combination of these:

|Qb(j)> = & i Ubcji |Qc(i)>. (19)The matrices Ub c i j could then be interpreted as generating a unitarychange of basis from the |Qc (i)> to the |Qb (j)> basis. Recall now theconventional quantum mechanical probability rule: if |v(i)> are a setof basis vectors and |w(j)> a second set of basis vectors related to thefirst ones by

|wj> = & i U ji |vi>, (20)then the probability of measuring the state |wj> if the system is inthe state |v i> i s

pij = | <vi | wj> |2 . (21)(28) and (29) yield pij = |Uij |2, which is equation (22). Therefore theconventional formalism of quantum mechanics and its probabilityrules follow. The set W (S) has the structure of a set of linearsubspaces in the Hilbert space. For any yes/no question Qi , let Li bethe corresponding linear subset of H. The relations { =>, V , Λ , not,⊥ } between questions Qi correspond to the relations { inclusion,orthogonal sum, intersection, orthogonal-complement, orthogonality }between the corresponding linear subspaces Li . The kinematics ofquantum mechanics can therefore be reconstructed from the threepostulates given. Here, however, that the Hilbert space on whichstate vectors live is not attached to a system, but to two systems,namely to a system-observer pair.

The inclusion of dynamics in the above scheme is thenstraightforward. We simply notice that two questions can beconsidered as different questions if defined by the same operationsbut asked at different moments of time. Thus, any question Q can belabelled by the time variable t, indicating the time at which it isasked: we denote as t ‘ Q(t) the one-parameter family of questionsdefined by the same procedure performed at different times. As wehave seen, the set W (S) has the structure of a set of linear subspacesin the Hilbert space. Assuming that time evolution is a symmetry inthe theory, then the set of all the questions at time t2 must beisomorphic to the set of all the questions at time t1. Therefore the

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corresponding family of linear subspaces must have the samestructure; therefore there should be a unitary transformation U(t2-t1)such that

Q(t2) = U(t2-t1) Q(t1) U-1(t2-t1). (22)By conventional arguments, we then have that these unitarymatrices form an abelian group and that U(t2-t1) = exp{-i(t2-t1) H } ,where H is a self-adjoint operator on the Hilbert space, theHamiltonian. The Schrödinger equation follows immediately. Thus,the time evolution too can be viewed as a structure on the set of thequestions that can be asked to a system, or, more precisely, astructure on the family of all the questions that can be asked to thesystem at all times.

3.4 The observer observed

We now have the full formal machinery of quantum mechanics, butwith some interpretative novelty: There is no "state of the system"in this framework. At this point, I can return to the issue of therelation between information of observers. The important point inthis regard is that the information possessed by different observerscannot be compared directly. This is a delicate but crucial point ofthe entire construction. A statement about the information possessedby O is a statement about the physical state of O; the observer O is aregular system on the same ground as any other system; thus, wemust discuss his state in physical terms. However, since there is noabsolute meaning to the state of a system, any statement regardingthe state of O is to be referred to some other system observing O.The notion of absolute state of a system, and thus a fortiori absolutestate of an observer, is not defined. Therefore, the fact that anobserver has information about a system is not an absolute fact: it issomething that can be observed by an observer. A second observer Pcan have information about the fact that O has information about S.But any acquisition of information implies a physical interaction. Pcan get new information about the information that O has about Sonly by physically interacting with the O-S system.

I believe that a common mistake in analyzing measurement issues inquantum mechanics is to forget that precisely as an observer canacquire information about a system only by physically interactingwith it, in the same fashion two observers can compare theirinformation only by physically interacting with each other. Thismeans that there is no way to compare "the information possessedby O " with "the information possessed by P", without considering a

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physical interaction between the two. Information, as any otherproperty of a system, is a fully relational notion.

How can a system P have information about the fact that O hasinformation about S? The observer P considers the coupled S-Osystem. Thus, she may ask questions to S, or to O, or to the twotogether. In particular she may ask questions concerning theinformation that O has about S. Information is simply a property ofsome degrees of freedom in O being correlated with some property ofS. A question about the information possessed by O is in no waydifferent from any other physical question. As far as P is concerned,knowing the physics of the S-O system means knowing the set W (S-O) of the meaningful questions that can be asked to the S-O system,and its full structure. In particular, it also means knowing howanswers at time t1 determine answers at time t2.

At the risk of repeating what was presented in section 2.5, let medescribe the information that O has about S, as content of theinformation that P posses. This exercise will also show how thequestion formalism may work. All questions below are posed by P. Ameaningful question to S is whether q=1 is true. Denote this questionas Q1. Notice that the fact that this is a meaningful question to ask Sis not relational (is not contingent), and thus both S and P can askthis same question to S. A meaningful (complete) question about O iswhether his measuring apparatus is ready to (and going to) ask S thequestion wether q=1 is true. Denote this question (that P can ask toO) as QReady. Another meaningful question is whether the hand of hismeasuring apparatus is on "O1". Denote this question as QO1. Similarlyfor QO2. Each of these questions can be asked at any time t.

QReady(t): O is going to measure whether q=1 on SQ1(t): q=1Q2(t): q=2QO1(t): O has the information that q=1QO2(t): O has the information that q=2 (23)

We also consider a complete question QMix (t) to S such that

p( Q1(t), QMix (t)) = p( Q2(t), QMix (t)) = 1/2. (24)

and define

QReady,Mix (t1) = QReady(t) Λ QMix (t). (25)

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Knowing the dynamics of the coupled S-O system, P can work out thepossible outcomes of questions asked at time t2, given a yes answerto QR e a d y,M i x (t1). Assuming standard Hilbert space Hamiltoniandynamics and assuming that the coupling Hamiltonian produces agood measurement, P will compute the probabilities

p( Q1(t2), QReady,Mix (t1) ) = 1/2 (26a)p( Q2(t2), QReady,Mix (t1) ) = 1/2 (26b)p( QO1(t2), QReady,Mix (t1) ) = 1/2 (26c)p( QO2(t2), QReady,Mix (t1) ) = 1/2 (26d)

Thus, she has no information whether q=1 or q=2 at time t2, norinformation about what the hand of the O apparatus indicates.

However, she may consider asking the question wether O knowsabout q or not. By this we mean whether or not the pointer is on theright position. Let us denote this question as Q"O-knows". This is theconjunction of two coupled questions:

Q"O-knows" = [Q1 Λ QO1] V [Q2 Λ QO2], (27)

and corresponds to the operator M described in sec.2.5. Note that Q1,Q O 1, Q2 and QO 2 are compatible questions, so we can considerquestions in the Boolean algebra they generate. The dynamics gives

p( Q"O-knows" (t2), QReady,Mix (t1) ) = 1 (28)

Thus, P has information that O has information about S [equation(28)]. This of course is not in contradiction with the fact that she (P)has no information about S [equation (26a,b)], nor has informationabout which specific information O has about S [equation (26c,d)].Thus the notion "a system O has information about a system S" is aphysical notion that can be studied experimentally (by a thirdobserver) in the same way as any other physical property of asystem. It corresponds to the fact that relevant variables in systemsS and O are correlated.

Now, the question "Do observers O and P get the same answers outof a system S?" is a meaningless question. Because it is a questionabout the absolute state of O and P. What is meaningful is torephrase this question in terms of some observer. For instance, wecould ask this question in terms of the information possessed by a

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further observer, or, alternatively, by P herself. Consider this lastcase. At time t1, O gets information about S. P has informationabout the initial state, and therefore has the information that themeasurement has been performed. The meaning of this is that sheknows that the states of the O-S systems are correlated, or moreprecisely she knows that if at a later time t3 she asks a question to Sconcerning property A, and a question to O concerning his knowledgeabout A (or, equivalently, concerning the position of a pointer), shewill get consistent results.

From the dynamical point of view, notice that knowledge of thestructure of the family of questions W (S) that can be asked to Simplies the knowledge of the dynamics of S (because W (S) includesall Heisenberg observables at all times). In Hilbert space terms, thismeans knowing the Hamiltonian of the evolution of the observedsystem. If P knows the dynamics of the O-S system, she knows thetwo Hamiltonians of O and S and the interaction Hamiltonian. Theinteraction Hamiltonian cannot be vanishing because a measurement(O measuring S) implies an interaction: this is the only way in whicha correlation can be dynamically established. From the point of viewof P, the measurement is therefore a fully unitary evolution, which isdetermined by the interaction Hamiltonian between O and S. Theinteraction is a measurement if it brings the states to a correlatedconfiguration. On the other hand, O gives a dynamical description ofS alone. Therefore he can only use the S Hamiltonian. Since betweentimes t1 and t2 the evolution of S is affected by its interaction with O,the description of the unitary evolution of S given by O breaks down.The unitary evolution does not break down for mysterious physicalquantum jumps, due to unknown effects, but simply because O is notgiving a full dynamical description of the interaction. O cannot havea full description of the interaction of S with himself (O), because hisinformation is correlation, and there is no meaning in beingcorrelated with oneself.

The reader may convince himself that even if we take into accountseveral observers observing each other, there is no way in whichcontradictions may develop, provided that one does not violate thetwo rules:i. there is no meaning to the state of a system or the information that

a system has, except within the information of a further observer.ii. there is no way a system P may get information about a system O

without physically interacting with it, and therefore breakingdown (at the time of the interaction) the unitary evolution

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description from any observer is not including both S and O (a n dtheir interaction Hamiltonian!) in its Hilbert space description ofthe events.

For instance, there is no way two observers P and O can getinformation about a system S independently from each other: one oftwo (say O) will have to obtain the information first. In doing so, hewill interact with S at a certain time t. This interaction implies thatthere is a non vanishing interaction Hamiltonian between S and O. IfP asks a question to O at a later time t', she will either have toconsider the interacting correlated O-S system, or to realize that theunitary evolution of the O dynamics has broken down, due to somephysical interaction she was not taking into account.

Finally, one can quantitatively study the relation between correlationof quantum states and amount of information. I will not pursue thisdirection here, but I present two comments in this regard. We cansay in general that the property "O1" of the system O contains one bitof information about the property "1" of the system S anytime the S-O system has answered a question Q such that

p( Q, Q"O-knows-1" ) = 1 (29)whe re Q"O-knows-1" = [Q1 Λ QO1] V [not Q1 Λ not QO1]. (30)

and so on. There is an alternative (independent) characterization ofamount of information, as the amount of entanglement between Sand O. Using the Hilbert space formalism, we may consider the state|Ψ > in the tensor product Hilbert space, and the corresponding purestate density matrix ρ = |Ψ ><Ψ |. If we denote the traces in the twofactor Hilbert spaces by TrS and TrO, then it is easy to see that for anon entangled state |Ψ> = |ψ>Sˆ |φ>O, we have

TrO (TrS ρ )2 = 1; (31)

while for an entangled state of the form

|Ψ> = 1

√n &i=1,n |ψi>S |φi>O, (32)

we haveTrO (TrS ρ )2 = 1/n. (33)

Thus, we may consider the amount of "entangling information"defined as

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N = - ln TrO (TrS ρ )2. (34)

On the possibility of defining mutual information from quantumstates, see [Halliwell 1994].

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4. Critique of the concept of state

4.1 "Any observation requires an observer": summary of the ideaspresented

Let me summarize the path covered. I started from the distinctionbetween observer and observed-system. I assumed (hypothesis 1)that all systems are equivalent, so that any observer can bedescribed by the same physics as any other system. In particular, Iassumed that an observer that measures a system can be describedby quantum mechanics. I have analyzed a fixed physical sequence ofevents E , from two different points of observations, the one of theobserver and the one of a third system, external to the measurement.I have concluded that two observers give different accounts of thesame physical set of events (main observation).

Rather than backtracking in front of this observation, and giving upthe commitment to the belief that all observers are equivalent, Ihave decided to take this experimental fact at its face value, andconsider it as a starting point for understanding the world. Ifdifferent observers give different descriptions of the state of thesame system, this means that the notion of state is observerdependent. I have taken this deduction seriously, and haveconsidered a conceptual scheme in which the notion of absoluteobserver-independent state of a system is replaced by the notion ofinformation about a system that an physical system may possess.

I have considered three postulates that this information must satisfy,which summarize present experimental evidence about the world.The first limits the amount of relevant information that a system canhave; the second summarizes the novelty revealed by theexperiments from which quantum mechanics derives by assertingthat whatever the information we have about a system we canalways get new information. The third limits the structure of the setof questions; this third postulate can probably be sharpened. Out ofthese postulates the conventional Hilbert space formalism ofquantum mechanics and the corresponding rules for calculatingprobabilities (and therefore any other equivalent formalism) can berederived.

A physical system is characterized by the structure on the set W ( S )of questions that can be asked to the system. This set has thestructure of the non-Boolean algebra of a family of linear subspaces

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of a complex k-dimensional Hilbert space. The information about Sthat any observer O can possess can be represented as a string s,containing an amount of information N.

I have investigated the meaning of this "information" out of whichthe theory is constructed. I have shown that the fact that a variablein a system O has information about a variable in a system S meansthat the variables of S and O are correlated, meaning that a thirdobserver P has information about the coupled S-O system that allowsher to predict correlated outcomes between questions to S andquestions to O. Thus correlation has no absolute meaning, becausestates have no absolute meaning, and must be interpreted as thecontent of the information that a third system has about the S-Ocouple.

Finally, since we take quantum mechanical as a complete descriptionof the world at the present level of experimental knowledge(hypothesis 2), we are forced to accept the result that there is no"objective", or more precisely "observer independent" meaning to theascription of a property to a system. Thus, the properties of thesystems are to be described by an interrelated net of observationsand information collected from observations. Any complex situationcan be described "in toto" by a further additional observer, and theinterrelation is consistent. However, such a "in toto" description isdeficient in two directions: upward, because an even more generalobserver is needed to describe the global observer itself, and, moreimportantly, downward, because the "in toto" observer knows thecontent of the information that the single component systems possesabout each other only probabilistically.

There is no way to "exit" from the observer-observed global system.Any observation requires an observer (The expression is takenfrom [Maturana and Varela 1920]). In other words, I suggest that itis a matter of natural science whether or not the descriptions thatdifferent observers give of the same ensemble of events is universalor not:

Quantum mechanics is the theoretical formalization of theexperimental discovery that the descriptions that differentobservers give of the same events are not universal.

The concept that quantum mechanics forces us to give up is theconcept of a description of a system independent from the observer

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providing such a description; that is, the concept of absolute state ofa system. The structure of the classical scientific description of theworld in terms of s y s t e m s that are in certain s ta tes is perhapsincorrect, and inappropriate to describe the the world beyond the¡‘0 limit.

It is perhaps worthwhile to emphasize that those considerations donot follow from the theory of quantum mechanics: they follow from acollection of experiments on the atomic world.

There are several aspects of the point of view discussed here thatrequire further development. Among these: i. The reconstruction ofsec.3 can certainly be sharpened, and postulate 3 should be betterunderstood. ii. The quantitative relation between amount ofinformation and correlation between states, which was hinted at, atthe end of sec.3.4, can be studied. iii. I believe it would beinteresting to reconsider EPR-like issues in the light of theconsiderations made here; notice that there is no meaning incomparing the outcome of two spatially separated measurements,unless there is a physical interaction between the observers. Iconclude with a brief discussion of the relation between the viewpresented here with some of the popular views of quantummechanics.

4.2 Relation with other interpretations

I follow [Butterfield 1995] to organize current strategies on thequantum puzzle. The first strategy (Dynamics) is to rejects thequantum postulate that an isolated system evolves according to thelinear Schrödinger equation, and consider additional mechanismsthat modify this evolution -in a sense physically producing the wavefunction collapse. Examples are the interpretations in which themeasurement process is replaced by some hypothetical process thatviolates the linear Schrödinger equation [Ghirardi Rimini and Weber1986, Penrose 1989]. Those interpretation are radically differentfrom the present approach, since they violate hypothesis 2. Myeffort here is not to modify quantum mechanics to make it consistentwith my view of the world, but to modify my view of the world tomake it consistent with quantum mechanics.

The second and third strategies maintain the idea that probabilisticexpectations of values of any isolated physical system are given bythe linear Schrödinger evolution. They must face the problem of

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reconciling probabilities expressed by the state at time t2 in equation(2), - q=1 with probability 1/2 and q=2 with probability 1/2 - withthe assertion that the the observer O assigned the value q=1 to thevariable q at the same time t2. As Butterfield emphasizes, if thisvalues assignment coexists with the probability distributionexpressed by (2), then the eigenstate-eigenvalue link must be insome sense weakened, and the possibility of assigning values tovariables in addition to the eigenstate case (extra values) be allowed.The second and the third strategy in Butterfield's classification differabout whether these extra values are "wholly a matter of physics"(Physics Values), or are "somehow mental or perspectival"(Perspectival Values). In the first case, the assignment is (in everysense) observer-independent. In the second case, it is (in somesense) observer-dependent.

A prime example of the second (Physics Values) strategy is Bohr's, orCopenhagen, interpretation -at least in one possible reading. Bohrassumes a classical world. In Bohr's view, this classical world isphysically distinct from the microsystems described by quantummechanics, and it is precisely the classical nature of the apparatusthat gives measurement interactions a special status [Bohr 1949; fora clear discussion of this point, see Landau Lifshitz 1977].

From the point of view developed here, we can fix once and for all aprivileged system So as "The Observer" (capitalized). This system Socan be formed for instance by all the macroscopical objects aroundus. In this way we recover Bohr's view entirely. The quantummechanical "state" of a system S is then the information that theprivileged system So has about S. Bohr's choice is simply theassumption of a large (consistent) set of systems (the classicalsystems) as privileged observers. This is fully consistent with theview proposed here4. By taking Bohr's step, one becomes blind tothe net of interrelations that are at the foundation of the theory, and,more importantly, puzzled about the fact that the physical theory

4 A separate problem is why the observing system choosen -S0 themacroscopic world- admits, in turn, a description in which expectationsprobabilities evolve classically, namely are virtually always concentrated onvalues 0 and 1, and interference terms are invisible. It is to th is question thatthe physical decoherence mechanism [Joos Zeh 1985, Zurek 1981] provides ananswer. Namely, a f te r having an answer on what determines extra-valuesascriptions (the observer-observed structure, in the view proposed here), thephysical decoherence mechanism helps explaining why those ascriptions areconsistent with classical physics in macroscopic systems.

4 2

treats one system, So -the classical world- in a way which isphysically different from the other systems. The disturbing aspect ofBohr's view is the inapplicability of quantum theory to macrophysics.This disturbing aspect vanishes, I believe, at the light of thediscussion in this paper.

Therefore, the considerations in this paper do not suggest anymodification to the conventional use of quantum mechanics: there isnothing incorrect in fixing the preferred observer So once and for all.Thus, acceptance of the point of view suggested here impliescontinuing to use quantum mechanics precisely as is it is currentlyused. On the other hand, this point of view (I hope) could bring someclarity about the physical significance of the strange theoreticalprocedure adopted in Bohr's quantum mechanics: treating a portionof the world in a different way than the rest of the world. Thisdifferent treatment is, I believe, the origin of the widespread uneasewith quantum mechanics.

The strident aspect of Bohr's quantum mechanics is cleanlycharacterized by von Neumann's introduction of the "projectionpostulate", according to which systems have two different kinds ofevolutions: the unitary and deterministic Schrödinger evolution, andthe instantaneous, probabilistic measurement collapse [von Neumann1932]. According to the point of view described here, the Schrödingerunitary evolution of the system S breaks down simply because thesystem interacts with something which is not taken into account bythe evolution equations. Unitary evolution requires the system to beisolated, which is exactly what ceases to be true during themeasurement, because of the interaction with the observer. If weinclude the observer into the system, then the evolution is stillunitary, but we are now dealing with the description of a differentobserver. As suggested by Abhay Ashtekar, the point of viewpresented here can then be characterized by a fundamentalassumption prohibiting an observer to be able to give a fulldescription of "itself" [Ashtekar 1993]. In this respect, these ideasare related to earlier suggestions that quantum mechanics is a theorythat necessarily excludes the observer [Peres and Zurek 1982,Roessler 1987, Finkelstein 1988, Primas 1990]. A recent result inthis regard is a general theorem proven by T Breuer [Breuer 1994],according to which no system (quantum nor classical) can perform acomplete self-measurement. Breuer has shown on general groundsthat no system O can posses internal records capable ofdistinguishing between quantum mechanical states in which the

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system itself has different correlations with another system S. Ifmeasuring the state of S implies the creation of correlations betweenS and O, then it follows that there is a residual irreducibleindeterminacy in any information about a system. We believe thatthe relation between the point of view presented here and Breuer'sresult deserves to be explored.

Other views within Butterfield's second strategy (Physical Values)are Bohm's hidden variables theory, which violates hypothesis 2(completeness), and modal interpretations, which deny the collapsebut assumes the existence of physical quantities' values. Of these, Iam familiar with [van Fraassen 1991], or the idea of actualization ofpotentialities in [Shimony 1969, Fleming 1992]. The assumed valuesmust be consistent with the standard theory's predictions, beprobabilistically determined by a unitary evolving wave function,but are not constrained by the eigenstate-eigenvalue link. One maydoubt these acrobatics could work [See Bacciagaluppi 1995,Bacciagaluppi and Hemmo 1995]. I am very sympathetic with thekey idea that the object of quantum mechanics is a set of quantities'values and their distribution. Here, I have assumed value assignmentas in these interpretations, but with two crucial differences. First thisvalue assignment is observer-dependent. Second, it need not beconsistent with a fully unitary Schrödinger evolution, because theevolution is not unitary when the observed system interacts with theobserver. Namely, there is collapse in each observer dependentevolution of expected probabilities. Clearly, these two differencesallow me to assign values to physical quantities without any of theconsistency worries that plague modal interpretations. The point isthat the break of the eigenstate-eigenvalue link is bypassed by thatfact that the eigenvalue refer to one observer, and the state to adifferent observer. For a fixed observer, the eigenstate-eigenvaluelink is maintained. Consistency should only be recovered betweendifferent observers, but -this is a key point- consistency is onlyquantum mechanical -probabilistic- as discussed in detail in section3.4. Actuality is observer dependent. The fact that the values ofphysical quantities are relational and their consistency is onlyprobabilistically required circumvents the potential difficulties of themodal interpretations.

A class of interpretations of quantum mechanics that Butterfield doesnot include in his classification, but which are presently very popularamong physicists, is given by the consistent histories interpretation[Griffiths1984, Omnes 1988] and the, related, coherent -or

4 4

decoherent- histories interpretation [Gell-Mann and Hartle 1990].These interpretations reduce the description of a system to theprediction of temporal sequences of values of physical variables. Thekey novelties are three: (i) probabilities are assigned to sequences ofvalues, as opposite to single values; (ii) only certain sequences can beconsidered; (iii) probability is interpreted as probability of the givensequence of values within a chosen family of sequences. Therestriction (ii) incorporates the quantum mechanical prohibition ofgiving value, say, to position and momentum at the same time. Moreprecisely, in combination with (ii) it excludes all the instances inwhich observable interference effects make probability assignmentsinconsistent. In a sense, the histories formulations represent asophisticated implementation of the program of discovering aminimum consistent value attribution scheme. The price paid forconsistency is that a single value attribution is meaningless: indeed,whether or not a variable has a value may very well depend onwhether we are asking or not if at a later time another variable has avalue! In section 2.2, I argued that within the history interpretationsdifferent observers d o make different statements about the sameevents if they choose to work with different consistent fami l ies ofhistories. The question relevant to compare this approach with thepresent work is then: What is the relation between probabilitiescomputed by using one family or another? The situation is evenmore serious if we consider factual statements about nature, asopposed to probabilistic predictions. In the histories approaches twopeople may chose to work with two different families of consistenthistories, and therefore give quite unrelated (and possiblycontradictory) accounts of the same events.5 See also the objection 8

5 Consider the following example: suppose we want to make statements aboutthe past evolution of a system (perhaps the Earth), within a historyformulation. I choose a certain consistent set of histories S, appropriate to thesystem, and compute their probabilities. Suppose for simplicity that I obtainthat all the alternative histories have negligible probability, except thehystory A, which has probability close to 1 (perhaps in this history dinosaurswere killed by a meteorite). Then, I would like to say that A is exactly what hashappened. You, the reader, choose a di f fe rent consistent set of histories S'.Suppose that you compute that all the alternative histories have negligibleprobability, except the history B (incompatible with A), which has probabilityclose to 1 (perhaps in B all dinosaurs committed suicide). You would like to say,I presume, that B is exactly what has happened. So what has happened ? (Howdid the dinosaurs become extinct?) These curious circumstances, allowed bythe history framework, do not invalidate the consistency of the formulation,but put me in the condition of wishing to understand what quantummechanics is exactly saying about the world, such that these funnycircumstances may happen. This paper is a tentative answer.

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in section 2.2, and [Kent Dowker 1994, Kent 1996]. Here, I maintainthat the observer does not chooses the family of histories in terms ofwhich she describes the world out of pure thought. The choice isdictated by the physical interaction between the observer and theobserved world (I realize that this is taboo in the histories world).The histories interpretations are not inconsistent with the analysisdeveloped here. What I tried to add here is increased attention tothe process through which the observer - independent, but f am i l y -dependent probabi l i t ies attached to histories, may be related toactual observer-dependent descriptions of the state of the world.

Finally, let me come to the third strategy (Perspectival Values),whose prime example is the many worlds interpretation [Everett1957, Wheeler 1957, DeWitt 1970], and its variants. If the"branching" of the wave function in the many worlds interpretationis considered as a physical process, it raises, I believe, the very samesort of difficulties as the von Neumann "collapse" does. When does ithappen? Which systems are measuring systems that make the worldbranch? These difficulties of the many-world interpretation havebeen discussed in the literature [See Earman 1986]. Alternatively,we may forget branching as a physical process, and keep evolvingthe wave function under unitary evolution. The problem is then tointerpret the observation of the "internal" observers. As discussed in[Butterfield 1985] and [Albert 1992], this can be done by givingpreferred status to special observers (apparata) whose valuesdetermine a (perspectival) branching. See Objection 7 in section 2.2for more details. A natural variant is taking brains -"Minds"- as thepreferred systems that determines this perspectival branching, andthus whose state determines the new "dimension" of indexicality.Preferred apparata, or bringing Minds into the game, violateshypothesis 1.

However, there is a way of having (perspectival) branching keepingall systems on the same footing: the way followed in this paper,namely to assume that all values assignments are completelyrelational, not just relational with respect to apparata or Minds.Notice, however, that from this perspective Everet's wave function isa very misleading notion, because not only it represents theperspective of a non existent observer, but it even fails to containany relevant information about the values observed by any singleobserver! There is no description of the universe "in toto", only aquantum-interrelated net of partial descriptions.

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With respect to Butterfield's classification, the interpretationproposed here is thus in the second, as well as in the third, groups: Imaintain that the extra values assigned are "somehow perspectival"(but definitely not mental!), in that they are observer-dependent,but at the same time "wholly a matter of physics", in the sense inwhich the "perspectival" aspect of simultaneity is "wholly a matter ofphysics" in relativity. In one word: value assignment in ameasurement is not inconsistent with unitary evolution of theapparatus+system ensemble, because value assignment refers to theproperties of the system with respect to the apparatus, while theunitary evolution refer to properties with respect to an externalsys tem.

From the point of view discussed here, Bohr's interpretation,consistent and coherent histories interpretations, as well as the manyworlds interpretation, are all quite l iterally correct, albeitincomplete. The point of view closest to the one presented here isperhaps Heisenberg's. Heisenberg's insistence on the fact that thelesson to be taken from the atomic experiments is that we shouldstop thinking of the "state of the system", has been obscured by thesubsequent terse definition of the theory in terms of states given byDirac. Here, I have taken Heisenberg's lesson to some extremeconsequences.6 Finally, it was recently brought to my attention that

6 With a large number of exceptions, most physicists hold some version ofnaive realism, or some version of naive empiricism. I am aware of the"philosophical qualm" that the ideas presented here may then generate. Theconventional reply, which I reiterate here, is that Galileo's relational notionof velocity used to produce analogous qualms, and that physics seems to havethe remarkable capacity of challenging even its own conceptual premises, inthe course of its evolution. Historically, the discovery of quantum mechanicshas had a strong impact on the philosophical credo of many physicists, as wellas on part of contemporary philosophy. It is possible that this process is notconcluded. But I certainly do not want to venture into philosophical terrainshere, and I leave this aspect of the discussion to more competent thinkers. Onthe other side, the following few observations, may perhaps be relevant. Therelational aspect of knowledge is of course one of the themes around whichpart of western philosophy developed (in Kantian terms, only to mention acharacteristic voice, any phenomenal substance which may be object ofpossible experience is "entirely made up of mere relations" [Kant 1787]). Inrecent years, the idea that the notion of "observer-independent description ofa system" is meaningless has become almost a commonplace in disparate areasof the contemporary culture, from anthropology to certain biology and neuro-physiology, from the post-neopositivist tradition to (much more radically)continental philosophy [Gadamer 1989], all the way to theoretical physicaleducation [Bragagnolo Cesari and Facci 1993]. I find the fact that quantum

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WH Zurek ends his paper [Zureck 1982] with conclusions that areidentical to the ones developed here: "Properties of quantum systemshave no absolute meaning. Rather, they must be alwayscharacterized with respect to other physical systems" and"correlations between the properties of quantum systems are morebasic that the properties themselves" [Zureck 1982].

A c k n o w l e d g e m e n t sThe ideas in this work emerged from: i. Conversations with AbhayAshtekar, Jose Balduz, Julian Barbour, Alain Connes, Jürgen Ehlers,Brigitte Faulknburg, Gordon Flemming, Jonathan Halliwell, Jim Hartle,Chris Isham, Al Janis, Ted Newman, Roger Penrose, Lee Smolin, JohnWheeler and HD Zeh; ii. A seminar run by Bob Griffiths at CarnegieMellon University (1993); iii. A seminar run by John Earman atPittsburgh University (1992); iv. Louis Crane's ideas on quantumcosmology; v. A comment made by Bob Geroch; vi. The teachings ofPaola Cesari on the importance of taking the observer into account.It is a pleasure to thank them all. I also thank Jeremy Butterfield forhis extensive criticisms, and Al Janis and Richard Sisca for theircareful reading of the manuscript.

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