• Preamble to the `Copenhagen interpretation'
  • Impossibility of an unambiguous definition
    The `Copenhagen (or orthodox) interpretation' is not a single coherent view of the meaning of quantum mechanics. According to Feyerabend it is ``not a single idea but a mixed bag of interesting conjectures, dogmatic declarations, and philosophical absurdities.'' In order to set myself a limit I will not discuss all absurdities. Thus, I will ignore as an active factor in quantum measurement the `(human) observer as far as transcending his role as an experimenter setting up the measurement arrangement used to perform a measurement, or reading the data obtained by his measuring instruments' (compare). Even so, contributions to the `Copenhagen interpretation' by the `founding fathers of quantum mechanics' are not mutually compatible in all respects. In particular, Bohr and Heisenberg, who may be seen as the parents of the Copenhagen interpretation, might seem to be as different as the father is from the mother, the father being responsible for the philosophical genes and the mother for the physical ones. Heisenberg once said that Bohr was primarily a philosopher, not a physicist. With respect to Heisenberg the converse statement seems justified. Indeed, `Bohr's cautious attitude with respect to ontological assertions' testifies to a considerably more careful philosophical attitude than `Heisenberg's flight into philosophical vagueness by relying on the Aristotelian notions of potentiality and actuality' in order to arrive at an `ontological account of quantum measurement'.
  • Between `instrumentalism/empiricism'and `scientific realism'
    In view of Feyerabend's remark it will be no surprise that it is impossible to give an unambiguous definition of what the `Copenhagen interpretation' is. A list of properties the `Copenhagen interpretation' in my view can best be characterized by is given here (column to the left). Nevertheless there are ambiguities. Thus, whereas the `Copenhagen interpretation' has started out from an `instrumentalist/empiricist point of view', has the state vector gradually obtained a more `realist meaning', be it in a probabilistic sense. It is a matter of taste whether this `realism' is considered to be consistent with a `Copenhagen understanding', since this development may rather have been stimulated by the rising influence of `scientific realism' at the expense of `instrumentalist/empiricist influences' (compare). During a long time there has been a practice of maintaining the `idea of completeness of quantum mechanics' by considering the `state vector to be a (complete) description of an individual object'. This is amalgamating `Copenhagen instrumentalist/empiricist' and `scientific realist' ideas.
    It is possible to draw the dividing line between `Copenhagen' and `non-Copenhagen' interpretations where the objectivistic-realist tendencies of `scientific realism' threaten to be extended to measurement. This is how I will implement this terminological issue (compare).
  • Heisenberg versus Bohr: physics versus philosophy
    Heisenberg's main concern was with the mathematical formalism of quantum mechanics, dealing with it in a pragmatic physicist's `no nonsense' way, which, notwithstanding its `professed empiricism', is probably best understood as entertaining a realist individual-particle interpretation endorsing ontological probability.
    On the contrary, Bohr's interest was directed more towards the issue of `what we can know about microscopic reality', avoiding as much as possible `ontological statements' in dealing with the mathematical formalism of quantum mechanics, and even denying an ontological meaning to the wave function. This would seem to place Heisenberg and Bohr at opposite extremes of the `ontic versus epistemic dichotomy of interpretations of quantum mechanics', thus explaining their intense disagreements on the meaning of the uncertainty principle (interpreted by Heisenberg as a `result of a physical disturbance of the object by a measurement', whereas for Bohr it was a restriction on the `latitudes observables are defined with within the context of a measurement').
  • That Bohr and Heisenberg, notwithstanding differences, have been accepted by the scientific community as parents of a common interpretation may indicate that these differences were not experienced as particularly telling (compare my qualification of the `ontic versus epistemic' dichotomy as potentially misleading). Indeed, it is hard to believe that the physicist Bohr would not have wanted to deal with `physical reality itself' rather than just with `knowledge': with respect to `quantum mechanical observables' (referred to by Bohr as `physical quantities') his attitude may be seen as a classically realist one -be it `modified in a contextualistic sense'-, the natural interpretation of classical mechanics being a realist one. Nor is Heisenberg's `reference to measurement' free of `epistemological implications', the latter being connected with Heisenberg's professed empiricism.
    Bohr's reluctance to make `ontological statements regarding microscopic reality itself' earned him Einstein's characterization of `Talmudic philosopher'.
  • Bohr, Heisenberg, and `interpretations'
    Bohr and Heisenberg cannot easily be understood using the present-day definitions of `interpretation of the quantum mechanical formalism' (like the realist -either objectivistic or contextualistic- and empiricist interpretations). On the other hand, it may be useful to try to apply these concepts also here. For instance, Bohr's warning against Jordan's assertion that `measurement results are created during measurement' would be futile in the `empiricist interpretation'. On the other hand, Jordan's assertion is in an ontologically-realist way corroborating Bohr's epistemological way of thinking about quantum mechanical observables (having had important implications with respect to the nonlocality issue arising from the latter's discussion with Einstein).
    We should not forget that the quantum mechanical formalism was being developed together with its interpretation. `Classical mechanics' was the starting point for both Bohr and Heisenberg¹⁴. Thus, `physical quantities', in microphysics to be represented by Hermitian operators, were thought to behave classically after having been amplified to the `macroscopic dimensions necessary to render them directly accessible to human observation'. For Bohr this implied that `what we can tell about a physical quantity of a microscopic object' should be expressed in the language of classical mechanics (compare), and does not directly refer to a `microscopic world existing independent of the measurement arrangement'. For Heisenberg it meant that by the measurement an `ontic potentiality' was turned into an `actuality', the microscopic object being assumed after the measurement to actually have the property it prior to the measurement only had potentially. The similarity of Heisenberg's view with Jordan's one¹⁷ may explain the occasional grave disagreements between Heisenberg and Bohr.
  • Essential role of measurement
    The new insight, common to Bohr and Heisenberg, that `measurement plays an essential role in assessing the meaning of quantum mechanics' has been sufficiently important to overlook their differences, and to unite them into one single Copenhagen interpretation, of which the essential role of measurement is the main characteristic. Notwithstanding Bohr's cautiousness, inducing him to avoid as much as possible ontological assertions, most physicists have interpreted quantum mechanical observables in Heisenberg's more ontological sense, thus subscribing to ontological indeterminism of measurement. Bohr's influence is felt if the Copenhagen interpretation is thought to entertain a contextualistic-realist interpretation of quantum mechanical observables, and an instrumentalist interpretation with respect to the wave function or state vector.
  • `Copenhagen interpretation' versus `empiricist interpretation'
    It should be recalled that, notwithstanding Heisenberg's empiricism, the `Copenhagen interpretation' is very different from the empiricist interpretation. Thus, the `Copenhagen interpretation' does not transcend the `classical idea of empiricism' in which the role of a `measuring instrument as a means to transmit information from the microscopic object to the human observer' is not (or, at least, not sufficiently) taken into account. Indeed, when reading the Copenhagen literature on quantum measurement the reader is plagued by the irritating feeling that a measuring instrument is there `just to disturb the microscopic object' (Heisenberg), or even 'just to define the measured observable' (Bohr).
    In the `empiricist interpretation' the role of `measurement' in an unambiguous way is `to obtain information on the microscopic object by means of interaction between microscopic object and measuring instrument, transmitting information from the first to the second' (compare).
  • The `Copenhagen interpretation' and `experiment'
    In assessing the importance of the Copenhagen interpretation it is not unimportant to realize that this interpretation is largely based on the experiments being performed during the time of its inception. These were mainly scattering experiments like the Compton-Simon and Stern-Gerlach experiments in which the measurement result was established by ascertaining the direction in which a particle or a photon had been scattered. Therefore Heisenberg's phenomenon could be identified with the `position of the microscopic object, actualized by the measurement' rather than by the `position of the pointer of a measuring instrument' (as is the case in the empiricist interpretation).
    Nowadays more sophisticated experiments are being performed not allowing a simple analysis like the one sufficient for such scattering experiments. In particular it is necessary to consider the `interaction between microscopic object and measuring instrument' to be able to draw conclusions about the object by reading the final position of a pointer of the instrument. Unfortunately, as a consequence of the `contextualistic-realist interpretation of observables' the role of the measuring instrument is only partly implemented in the Copenhagen interpretation.
  • Negative and positive features of the `Copenhagen interpretation'
    In the following analysis of the Copenhagen interpretation a number of its weak points will be discussed (compare the list of negative features presented here), often stemming from too rashly accepting certain features observed in thought-provoking measurements like the Compton-Simon and the Stern-Gerlach ones as generally valid, or even as being normative for all measurements within the microscopic domain.
    Nevertheless, the Copenhagen interpretation must be honoured for its insight that measurement plays an ineradicable role in assessing the meaning of quantum mechanics (compare the list of positive features presented here). This positive feature seems to me to be of such a fundamental importance that it is worthwhile to try to update the Copenhagen interpretation so as to yield a consistent view circumventing its negative features, and to be able to also cope with the more sophisticated `present-day quantum mechanical measurements not readily describable by the standard formalism of quantum mechanics' (compare).






• The quantum postulate
  • According to Bohr⁴⁵ each quantum phenomenon (representing a `quantum mechanical measurement event' like a `flash on a scintillation screen' or a `track in a Wilson chamber') possesses a feature of `individuality' or `wholeness', in the sense that allegedly no distinction can be drawn between `microscopic object' and `measuring instrument': object and measuring instrument are thought to constitute an `indivisible whole'. This is the quantum postulate. It marks the essential difference between quantum mechanics and classical mechanics, in the latter theory `measurement' being assumed not to play any essential role, whereas in quantum mechanics physical quantities are thought to be well-defined `only within the context of a measurement of that quantity' (compare).
    The `dependence on the measurement arrangement' going with the `quantum postulate' is at the basis of the Copenhagen interpretation. It is applied (although in a rather ambiguous way) by Bohr in his answer to the EPR challenge of the `completeness of quantum mechanics', juxtaposing `Bohr's contextualistic-realist interpretation of quantum mechanical observables' to `Einstein's objectivistic-realist one'.
  • Remarks on the quantum postulate:
    • The `quantum postulate' has its physical basis in the idea of `unavoidable interaction between object and measuring instrument, disturbing the object when a quantum mechanical measurement is performed'. The non-vanishing of the "quantum of (inter)action" (Planck's constant h) allegedly makes it impossible either to neglect this interaction or to compensate for it. According to the `Copenhagen interpretation', due to the non-vanishing of h measurement is inducing a fundamental indeterminism or acausality, thus allegedly making unanalyzable the process of `obtaining knowledge by means of measurement'.
    • Due to the essential role played by the measurement interaction the `Copenhagen interpretation' is sometimes referred to as an `interactional interpretation'. In my view the insight that the measurement interaction plays an essential role in the process of obtaining knowledge on a microscopic object is the most important aspect of the `Copenhagen interpretation' (although it is questionable whether this judgment should be extended to the Copenhagen thesis of the unanalyzability of the quantum phenomenon).
    • The `quantum postulate' is at the basis of many features characterizing the `Copenhagen interpretation', such as ontological indeterminism during measurement, completeness in the restricted sense, correspondence, and complementarity.
      It should be noted that, notwithstanding the empiricist terminology, in the `Copenhagen interpretation' a `quantum phenomenon' is not understood in the sense of the empiricist interpretation. A `flash on a scintillation screen' or a `track in a Wilson chamber' is not interpreted as a `property of the measuring instrument' (viz. a `chain of water drops, the formation of which being induced by ionization of water molecules triggered by a passing charged microscopic object'), but as a `property of the microscopic object made macroscopically observable by the measurement arrangement'. By Bohr quantum mechanical observables are interpreted in the contextualistic-realist sense corresponding to the latter interpretation.
  • Criticisms and appraisal of the `quantum postulate'
    
    • Ambiguity of the notion of `quantum phenomenon'
      The `quantum phenomenon' is a rather vague and intuitive notion, being given different meanings by different adherents of the `Copenhagen interpretation'. Identifying a `quantum phenomenon' with the `coming into being of a measurement result' these differences are reflected by the differences between Bohr and Heisenberg, observed by them with respect to the notion of `quantum mechanical measurement result a_m'.
      Moreover, with Bohr the notion of `quantum phenomenon' has been subject to a certain evolution during the development of the `Copenhagen interpretation'. As a consequence of the idea that the nonvanishing value of Planck's constant h should be responsible for quantum jumps during a measurement, `discontinuity'<sup>45</sup> was initially assumed by Bohr to be a determining feature. However, `discontinuity' is absent in his later formulations, being replaced by individuality, representing a generalization of the notion of a `classical particle', in which the influence of the measurement interaction during measurement is acknowledged as an important factor. Accompanying Bohr's growing awareness of the contextual meaning of quantum mechanics while developing his complementarity principle, the emphasis has changed from an `unspecified influence of measurement' to an `influence as occurring in a specified measurement arrangement'.<sup>46</sup>
    • Criticisms of `unanalyzability of the quantum phenomenon'
      • Influence of the classical paradigm
        The idea of `discontinuity' has been responsible for the idea that a quantum mechanical measurement is accompanied by an irreducibly indeterministic disturbance of the microscopic object, and that therefore the `quantum phenomenon' is an `unanalyzable whole' not allowing a clear distinction between microscopic object and measuring instrument (compare e.g. a track in a Wilson chamber). It is remarkable that Bohr, notwithstanding this lack of distinction, has chosen to interpret the `quantum phenomenon' as a `property of the microscopic object' rather than as a `property of the measuring instrument'.
        Probably Bohr's classical way of thinking prevented him from going all the way towards the empiricist interpretation (in which a `quantum phenomenon' corresponds to a final pointer position, obviously being a `macroscopically observable event allowing for the classical account that according to Bohr was necessary'), but instead sending him into the direction of a `realist interpretation of quantum mechanical observables, be it modified in a contextualistic sense'.
        Bohr's `unanalyzability of the quantum phenomenon' is largely a consequence of ignoring the pre-measurement phase which is allowing a `quantum mechanical analysis of quantum measurement' in which `microscopic object' and `measuring instrument' are duly distinguished. It seems that the `classical paradigm' stood in the way of a (quantum mechanical) analysis that later on would be helpful in obtaining a better understanding of the role of measurement in quantum mechanics.
      • Einstein's challenge of the `unanalyzability of the quantum phenomenon'
        Note that in his challenge of the Copenhagen completeness thesis the `quantum phenomenon' was construed by Einstein `differently from the quantum postulate' (in particular by trying to circumvent the `influence of measurement'), viz. in the sense of the `objectivistic-realist interpretation of quantum mechanical observables' satisfying the possessed values principle. The (alleged) unanalyzability of the quantum phenomenon has been criticized by Einstein to the extent of qualifying the Copenhagen interpretation as a ``tranquilizing philosophy''. By accepting `indeterminism (or, more appropriately, acausality) of measurement', and by consequently relinquishing any possibility of `explaining measurement results obtained in a measurement', Bohr, according to Einstein, too hastily accepted `completeness of quantum mechanics'.
        Unfortunately, like Bohr's approach also Einstein's challenge did not transcend the `classical paradigm', thus precluding the analysis from going any deeper than the `wholeness bestowed on the quantum mechanical description by its reliance on Planck's constant'. A deeper-going analysis had to await developments based on a quantum mechanical theory of measurement, allowing a `quantum mechanical description of the interaction of object and measuring instrument', or even based on subquantum or hidden variables theories allowing contemplation of possibilities to reduce `quantum mechanical wholeness' to `subquantum properties of object and measuring instrument'.
        It is fortunate that, notwithstanding the limited scope of Einstein's challenge, it yet did have some effect, viz. it induced Bohr to stress the contextuality of his `quantum phenomenon' (by pointing at an ambiguity of Einstein's `element of physical reality'). By this development the ideas of `discontinuity' and `irreducible indeterminism' have lost much of the importance they initially had (compare).
    • Reliance of the `Copenhagen interpretation' on the notion of `measurement'
      Nowadays an often-heard criticism of the `Copenhagen interpretation' is its reliance on the notion of `measurement' induced by the `quantum postulate'. This criticism has gained momentum after the breakdown of logical positivism/empiricism had decreased the importance attributed to the `phenomena' as opposed to `microscopic reality itself' (this criticism actually goes back to Einstein's idea that a physical theory should describe an `objective observer-independent reality'). Such a criticism is challenging the notion of `quantum phenomenon in which the disturbing influence of the measurement is thought to be so essential that a certain inseparability or wholeness of the system object+measuring instrument had to be assumed'. At the basis of this criticism are the ideas
      i) that an observer-independent⁹ microscopic reality exists;
      ii) that an objective description of this microscopic reality is possible;
      iii) that quantum mechanics itself can provide such an objective description.
      I do not share this criticism. In my view awareness of the `crucial role measurement plays in microscopic physics' is an asset of the `Copenhagen interpretation' rather than a weakness. Within the microscopic domain no information can be obtained without interaction with measuring instruments that are sensitive to the microscopic information, and capable of amplifying it to macroscopically observable dimensions. The `Copenhagen interpretation' should be recognized as having been the first to emphasize the crucial importance of this feature of microscopic physics, and to subsume it under the physics and philosophy of that domain.
      In my view (which is based on an empiricist interpretation of physical theories) a `quantum mechanical description of microscopic reality' may be analogous to a `(phenomenological) description of a billiard ball by the classical theory of rigid bodies', suggesting a `wholeness of the billiard ball' comparable to the wholeness attributed by Bohr to a `quantum phenomenon'. `Unanalyzability of a quantum phenomenon by quantum mechanics' may be compared with `unanalyzability of the rigidity of a billiard ball if one is restricting oneself to the classical theory of rigid bodies'. Then an analysis of rigidity simply is impossible, not so much because the phenomenon would be unanalyzable in an ontological sense (for instance, because it would have no parts), but as a consequence of `insufficient resources provided by the theory of rigid bodies'. Analogously, `quantum phenomena' probably will resist complete analysis as long as Planck's constant h is an axiomatically defined term of quantum mechanics, and no physical explanation has been found for its value (compare).
    • The `quantum phenomenon' in the `EPR experiment'
      According to Bohr the EPR experiment can be understood in terms of the `quantum phenomenon' in a way completely analogous to any other quantum measurement. This unhappy analogy marks the birth of the nonlocality conundrum, the `quantum phenomenon' allegedly comprising both particle 1 and 2. It is unfortunate that the discussion between Bohr and Einstein has been restricted to EPR experiments, thus tending to emphasize `wholeness' rather than `separation' (as might obtain in `correlation measurements of the EPR-Bell type' in which two different `quantum phenomena' are to be observed -viz. Alice's one and Bob's one- which should be considered as `separate' rather than `constituting a single whole', the experiment determining the `correlation between these separate phenomena').
      Even in the `EPR experiment' it is possible to distinguish between two different events; however, in that case Bob's event is a `preparation event' rather than a `measurement event' (compare), the former not easily to be equated with a `phenomenon', and hence being subject to the `Copenhagen fear of the metaphysical' (also).
      By considering EPR-Bell measurements it would have been evident that, in order to test experimentally any prediction on particle 2 it would be necessary to perform a measurement on that very particle, giving rise to its own measurement arrangement, and generating its own `quantum phenomenon' by probing the `preparation event at Bob's position'.
      It is even not impossible that Bohr had in mind something like this, as might be guessed from a famous quotation<sup>47</sup> in his `answer to the EPR challenge'. However, he refrained from any explicit reference to `Bob's measurement arrangement', thus summoning Einstein's conclusion of `nonlocality' (which could be ignored by Bohr on the basis of his epistemological attitude, but which was taken seriously by `more ontology-minded physicists' like Bell).
    • Bohr versus Einstein on the `quantum postulate'
      Summarizing, we must come to the conclusion that on the subject of the `quantum postulate' neither Bohr nor Einstein was completely right, nor was each of them completely wrong. Moreover, if it is duly taken into account which of the ideas of each of the adversaries should be abandoned in the face of new insights, and which ones should be kept, then by hindsight we can say that their controversy over the `quantum postulate' and the related completeness thesis has been a fruitful one (rather than the metaphysical exchange it has often been considered to be) because it directed the attention to the role of measurement in studying microscopic objects.
      On the other hand, it seems to me that neither Bohr nor Einstein was prepared to sufficiently transcend classical thinking to give the `quantum phenomenon' its proper position as a `property of the measuring instrument' rather than as a `property of the microscopic object'. Both kept thinking in terms of a realist interpretation of quantum mechanics (either contextualistic or objectivistic), neither of them contemplated the possibility of an empiricist interpretation. However, Bohr's awareness of the `fundamental role measurement plays in microscopic physics' may earn him the alleged victory over Einstein that has been unjustifiedly attributed to him on the basis of an alleged `aversion of metaphysics'.






• The Copenhagen completeness thesis
  • The idea of `completeness of quantum mechanics' is probably the best known characteristic of the `Copenhagen interpretation', although in my view it is not the most important one (compare).
  • The discussion on the `(in)completeness of quantum mechanics' is obscured by the fact that two different notions of `(in)completeness' are at stake which are not sufficiently distinguished. These issues are:
    i) (in)completeness in the restricted sense, related to the `quantum postulate', being a strictly quantum mechanical notion, referring to `measurements within the domain of application of quantum mechanics';
    ii) (in)completeness in the wider sense, related to the possibility or impossibility of "hidden variables" implying the possibility of going `beyond quantum mechanics'.
  • Completeness in the restricted sense
    The Copenhagen thesis that `quantum mechanics is a complete theory', originating with Bohr and Heisenberg, refers to the somewhat tautological idea that quantum mechanics is describing `all possible information to be obtained by means of quantum mechanical measurements (being subject to the quantum postulate)'. This will be referred to as completeness in the restricted sense since its validity is restricted to the domain of application of quantum mechanics. It actually is a rather negative notion, its applicability being related to the `impossibility of simultaneously determining sharp values of the quantum mechanical position and momentum observables' (which impossibility actually was a main reason for invoking the `quantum postulate'). `Completeness' should here be taken in the sense of `not completable in the sense of obtaining more information than contained in the probability distributions of the position and momentum observables (satisfying the Heisenberg-Kennard-Robertson inequality for these observables)'. It, in particular, does not refer to the question of the possibility of reconstructing the state vector from the probability distributions obtained by measuring a `quorum' of quantum mechanical (standard) observables', even though quantum mechanics actually is complete in this latter sense (cf. quantum tomography).
    According to Bohr `completeness of quantum mechanics (in the restricted sense)' should be seen as a rational generalization of the `completeness of classical mechanics', where simultaneous knowledge of position (q) and momentum (p) completely determines the state (q,p). `Completeness of quantum mechanics in the restricted sense' refers to the impossibility of surpassing limitations set by the `Heisenberg-Kennard-Robertson inequality' on the joint information to be obtained about an `individual object' from measurements of the quantum mechanical position and momentum observables.
  • The `Copenhagen completeness thesis' has considerably contributed to a custom of interpreting the state vector as a description of an individual particle (object), or -if ensembles are considered at all- to the idea of a von Neumann ensemble in which particles described by identical state vectors are thought to be identical (or identically prepared) even if measurements of the same observable on such particles would yield different values. In the Copenhagen interpretation criticisms of von Neumann's interpretation are circumvented
    i) by the possibility to ignore a possible nonuniqueness of the representation of the density operator ρ_A (as in the EPR problem) by invoking contextuality (symbolized by the index A), to the effect that only the representation corresponding to the actually measured observable A is thought to have physical relevance;
    ii) by restricting attention to strong von Neumann projection (at the expense of weak von Neumann projection)⁶⁵ by ignoring `all possible measurement results not actually realized in an individual measurement', thus leaving undiscussed the problem `that by the measurement an individual particle (represented by a pure state) is transformed into an inhomogeneous ensemble' (compare figure 3).
  • The Copenhagen idea of `completeness in the restricted sense' is the epistemological counterpart of the ontological assumption of irreducibility of quantum probability.
    The "real" properties of a microscopic object are supposed not to be the values of observables (or physical quantities) like position and momentum, but the probabilities p<sub>m</sub> should rather be considered as such. The idea of `completeness in the restricted sense' signifies that the quantum mechanical formalism cannot be completed by specifying for each microscopic object a value the observable possessed prior to measurement.
    Note that the `reasons for irreducibility of quantum probability' were not convincing to Einstein, who pursued a statistical interpretation of the Born rule instead of the Copenhagen `probabilistic interpretation'. Indeed, strictly speaking, the general rejection of Einstein's ideas by the majority of the physics community was far from justified at the time of the Bohr-Einstein controversy on the `(in)completeness of quantum mechanics'. Only much later formal proofs have been given, based on the mathematical formalism of quantum mechanics (compare), demonstrating the impossibility of `incompleteness of quantum mechanics in the restricted sense'.
  • Completeness in the wider sense
    The Copenhagen thesis that `quantum mechanics is a complete theory' is often interpreted in the sense of completeness in the wider sense, signifying the `impossibility of any subquantum or hidden variables theory'. Under the influence of `logical positivist/empiricist abhorrence of metaphysics' `completeness in the wider sense' has become one of the main characteristics of the `Copenhagen interpretation of quantum mechanics', which in Jordan's ontological sense can be characterized as anti-realist. By hindsight, this interpretation is even less justified than the assumption of `completeness in the restricted sense' since the main reasons to reject `subquantum theories' (viz. i) the logical positivist/empiricist philosophy, ii) von Neumann's 1932 "proof" of the impossibility of hidden variables) have turned out to be obsolete as a consequence of insurmountable problems of logical positivism/empiricism, and of Bell's 1960 refutation of von Neumann's "proof", respectively.
    Both Bohr and Heisenberg contemplated the possibility that quantum mechanics could break down in new domains of experience. For instance, Heisenberg admits the possibility of the existence of trajectories of electrons (although he denies their physical relevance on the basis of their `unobservability within the quantum mechanical domain'). However, Bohr saw the `quantum phenomenon' (in particular as it is manifesting itself within complementarity) as a model of the interplay between an observer and the observed object, having a `general significance transcending quantum physics'.
  • Criticisms of the `Copenhagen completeness thesis'
    • Failure to distinguish completeness in the wider sense from completeness in the restricted sense
      Failure to draw a distinction between completeness in the `wider' and `restricted' senses has been a major source of confusion. This holds true not only for the `Copenhagen interpretation itself', but also for Einstein's challenge of the `Copenhagen completeness thesis', in which challenge the `quantum mechanical observables themselves' are `conceived as hidden variables', thus assuming quantum mechanics to be a `very restricted kind of hidden variables theory'. However, it seems to be more appropriate to consider Einstein's approach as an `interpretation of quantum mechanics' (viz. an objectivistic-realist one) rather than a `hidden variables (subquantum) theory', and to see his EPR challenge of the `Copenhagen completeness thesis' as an attempt to refute `completeness in the restricted sense' rather than `completeness in the wider sense'.
      Failure to draw the distinction has had a large influence on the way Bohr's alleged "victory" over Einstein in the (in)completeness debate has been appreciated as a triumph of logical positivism, allegedly refuting all metaphysics by excluding `any subquantum theory'. However, arguments against the existence of Einstein's `restricted kind of hidden variables' based on peculiarities of the mathematical formalism of quantum mechanics like the Kochen-Specker theorem<sup>0</sup>, demonstrating the impossibility of `quantum mechanical elements of physical reality'. They do not prove the non-existence of `elements of physical reality of subquantumtheories', that are not necessarily subject to the quantum mechanical peculiarities (compare).
      Whereas it may be justified to consider `incompleteness in the restricted sense' as falsified (no `simultaneously possessed sharp values' being attributable to the quantum mechanical `position' and `momentum' observables), is the possibility of `incompleteness in the wider sense' still wide open. Indeed, `completeness of quantum mechanics in the restricted sense' may be compatible with `incompleteness of quantum mechanics in the wider sense'. This is particularly evident in the `empiricist interpretation' of the quantum mechanical formalism, in which a clear distinction can be drawn between the `phenomena described by quantum mechanics' and the reality behind the phenomena, to be described by subquantum theories. If de Broglie's idea⁰ is correct that a microscopic object is accompanied by a wave (as seems to be required by interference phenomena), then this leaves room for the assumption that de Broglie's wave should not be equated with the quantum mechanical wave function (or with another solution of the Schrödinger equation as is supposed by de Broglie), but could correspond to an `element of a subquantum theory'.
    • Metaphysical character of `Copenhagen completeness'
      The main weakness of the Copenhagen individual-particle interpretation, associating the quantum mechanical wave function or state vector with an individual object, is that it transcends present-day experimental data by assuming that the quantum mechanical wave function has replaced de Broglie's wave. It is by now evident that the wave function describes an ensemble. It is beyond observation whether perhaps it `also describes an individual element of an ensemble'. It, hence, turns out that the `Copenhagen completeness thesis' is no less metaphysical than is Einstein's idea of the existence of `elements of physical reality'. Even though Einstein's challenge of the `Copenhagen completeness thesis' was not carried out in a proper way (compare) is it evident that Einstein was correct when supposing that quantum mechanics is about `ensembles' rather than about an `individual particle'. Even if the `vacuum fluctuations of quantum field theory' would describe certain empirical aspects of `de Broglie's wavelike phenomenon', would the `quantum field theoretical description' still refer to an ensemble rather than to an individual object.
    • The main reason for the `Copenhagen completeness thesis' is the `quantum postulate', and not the `logical positivist' idea (based on the notion of a von Neumann ensemble) that an ensemble interpretation would not be opportune (because properties distinguishing `different individual objects described by identical state vectors' would not be experimentally verifiable). On the contrary, it could be retorted that by not duly distinguishing `completeness of quantum mechanics in the restricted sense' from `completeness of quantum mechanics in the wider sense' progress of science has been hampered because it implied a belief in the universal validity of quantum mechanics, thus discouraging attempts to experimentally probe the limits of the domain of application of quantum mechanics.
      This also induces the question of whether `measurement disturbance as involved in the quantum phenomenon' is a `fundamental characteristic of all measurement in the microscopic domain', or whether it is just an `artefact of the kind of measurements that are within the domain of quantum mechanics'. If `completeness' is taken in the `restricted sense', then the `Copenhagen interpretation' does not exclude the latter possibility, implying that future experimentation may require theories different from quantum mechanics (i.e. subquantum theories), capable of more fully analyzing (in terms of `subquantum properties') the measurement procedure than is possible using quantum mechanics. If the `quantum postulate' is applicable only to `quantum mechanical measurements', then it is possible that `Einstein's objective elements of physical reality' may exist, but that they are not probed by measurements that are subjected to the`physical restrictions singling out the domain of application of quantum mechanics' (analogous to the way the `atomic constitution of a billiard ball' is not probed by measurements that do not surpass the accuracy sufficient for probing the `classical theory of rigid bodies').

• The correspondence principles
  • Different forms of the `correspondence principle'
    We should distinguish four different notions of `correspondence', belonging to different phases in the development of quantum physics, three of which being characteristic of the historical development of the `Copenhagen interpretation'. The fourth is presented here as a tribute to that interpretation.
    • i) correspondence in the Old quantum theory<sup>0</sup>, which is a (heuristic) `methodological principle of theory development', applied during the developing stages of quantum physics preceding the conception of `quantum mechanics' (i.e before 1925)<sup>38</sup>; it implies that in dealing with quantized systems classical relations are preserved as much as possible, satisfaction of the principle being seen as an indication of a successful development of the `quantum theory' as a ``rational generalization of the classical theories'';
    • ii) weak form of the correspondence principle, implying the expectation that if (fully developed) quantum mechanics is applied to larger and larger objects the quantum mechanical description finally will coincide with that of classical mechanics; it coincides with the belief that classical mechanics is the `classical limit of quantum mechanics (often simulated by taking the limit h → 0)'; in most textbooks of quantum mechanics this is the only form of the `correspondence principle' that is referred to;
    • iii) strong form of the correspondence principle<sup>4</sup>, dealing with a characterization of the notion of a `quantum mechanical observable', to the effect that
      • a) a `measurement of a quantum mechanical observable' is supposed to establish a correspondence between that `quantum mechanical observable' and a `classical quantity';
      • b) a `quantum mechanical observable' is exclusively defined (i.e. given empirical content) `within the context of the measurement serving to measure that observable'; thus, it does not make sense to talk about the position of a microscopic object if this quantity is not actually measured (this is the epistemological counterpart of Jordan's ontological assertion);
      • c) the definition of an observable may be less than perfect, the observable possibly being defined up to a certain latitude depending on the measurement arrangement;
      this is the form of the correspondence principle Bohr finally arrived at, applied by him in a number of thought experiments and in his discussion with Einstein on the `completeness of quantum mechanics'.
    • A common feature of these three forms of `correspondence' is their reliance on the connection between the microscopic world (requiring a new theory for its description) and the macroscopic world (described by classical mechanics). For this reason the relation between quantum and classical mechanics has played an important role ever since quantum mechanics had been conceived. In particular, the `classical limit of quantum mechanics' has been playing an important role in form ii), which even today is widely considered as the generic form of the `correspondence principle'.
      It should be noted, however, that the importance of this aspect of the `correspondence principle' has been overestimated. As a methodological principle the `relation of quantum mechanics with reality' should be considered to be more important than its correspondence with `classical mechanics'. Indeed, form iii) already shows some awareness of this by stressing the importance of the `correspondence between quantum mechanics and the physical measurement arrangement' even though sticking to the `requirement of a correspondence with classical mechanics'. In Bohr's correspondence philosophy a gradual change can be observed in which the emphasis is switching from the `relation with classical mechanics' to the `relation with the measurement arrangement'.
  • i) Correspondence in the Old quantum theory:
    • The development of the Bohr model of the atom<sup>0</sup> is an early example of the application of `correspondence' arguments. It is found that frequencies of electromagnetic transitions between highly excited Rydberg states<sup>0</sup> calculated on the basis of Bohr's `quantum conditions' coincide with values found classically.
      Note, however, that for acceptance of `Bohr's model of the atom' as a useful tool for studying atomic spectra its `agreement with experimental data like the Balmer series' has been much more important than the above-mentioned correspondence, even though for less excited states these experimental data were in disagreement with classical theory.
    • Note that the `correspondence approach in the Old quantum theory' has given rise to the Bohr-Kramers-Slater (BKS) theory<sup>0</sup> which turned out to be in disagreement with experiment, and therefore was a blind alley in the development of quantum physics.
      On the other hand, `correspondence' has played an important role in Kramers's attempts at calculating the intensities of transitions between states of the Bohr atom<sup>36</sup>. Also has Heisenberg's collaboration with Kramers on the basis of the `correspondence' approach had a considerable influence on the former's ability to guess the `matrix approach' finally yielding `quantum mechanics'. In this sense `quantum theory on the basis of correspondence' may be seen as a useful step in the `development of quantum mechanics'.
  • ii) Weak form of the correspondence principle:
    • The weak form of the correspondence principle should be distinguished from `correspondence in the Old quantum theory' because the `classical limit of quantum mechanics' is a property of `quantum mechanics', and, hence, could not have had any relevance as long as that theory was not yet sufficiently developed.
    • A relation with the `classical limit of quantum mechanics' is induced by the measurement process necessarily amplifying microscopic information to the macroscopic domain, hence summoning an account of measurement in terms of classical concepts (at least on the macroscopic side of the Heisenberg cut). Therefore the existence could be assumed of a `correspondence between quantum mechanical observables and classical quantities' in the sense that in the limit h → 0 quantum mechanical observable A reduces to a (corresponding) classical quantity
      A(q,p). Inversely, starting from the classical quantity it was sometimes possible to guess which quantum mechanical observable should be taken as its counterpart (so-called quantization procedure). In doing so the assumption, often attributed to Dirac, that in this limit the commutator of two observables A and B reduces to the Poisson bracket<sup>0</sup> of the corresponding classical quantities, −i [A, B]<sub>−</sub> → {A(q,p), B(q,p)}, is not an implausible one (compare the example of position and momentum observables given here). Hence, as a heuristic tool for developing the theory of quantum mechanics the `weak correspondence principle' has been rather fruitful. Note, however, that as a result of its non-uniqueness this quantization procedure cannot be conceived as a `derivation of quantum mechanics'.
    • The existence of a correspondence of this kind is also suggested, for instance, by Ehrenfest's theorem<sup>0</sup> as well as by the practical applicability of a (semi-)classical description to certain objects (like the `electromagnetic field'). However, it turns out to be rather unfruitful to demand a general existence of such a limit, the general structure of the `set of quantum mechanical observables' being fundamentally different from the structure of the `set of classical quantities'. The `classical limit' may be valid as `applicability of a (semi-)classical description in special cases', it does not make sense as a general requirement: for instance, the quantum mechanical `spin observable' does not have a classical limit. For these reasons as a methodological principle the `weak form of the correspondence principle' has drawbacks similar to those of `correspondence in the Old quantum theory'.
      It should also be noted that a comparison of `quantum mechanics' and `classical mechanics' as suggested by the `Ehrenfest theorem' is obsolete since the state vector does not refer to an `individual particle' but to an ensemble. This implies that the `classical limit', too, should refer to an ensemble. Hence, if there is a `classical limit' at all, then not `classical mechanics' but `classical statistical mechanics' should be that limit.
  • iii) Strong form of the correspondence principle:
    • The Copenhagen `correspondence principle' has Bohr as its main architect, who in the revolutionary times he lived in, was forced to rebuild the `correspondence house' several times, the `strong form' being one of his masterpieces of architecture. During the years quantum theory was being developed Bohr's concern was with the strange properties of its quantities. Note that while contemplating microscopic physics Bohr always had in mind a classical ontology (``There is no quantum world''). `Classical mechanics' was Bohr's reference as long as quantum mechanics was not yet a fully developed theory, and even longer than that. Bohr usually talked about `physical quantities' rather than `observables'<sup>68</sup> (the quantum mechanical wave function being interpreted by him in an instrumentalist way). As a result his discussions remained in terms of `classical quantities', assumed to have the same ontological status as in `classical physics', be it restricted by `quantum conditions'. In the `strong correspondence principle' these restrictions are thought to be `imposed by the presence of the measuring instrument'. According to Bohr a `physical quantity' can be well-defined by its `correspondence with a classical quantity' only within the context of the measurement arrangement set up to measure that quantity. Implementing this into the mathematical formalism of quantum mechanics this gives rise to application of the contextualistic-realist interpretation at least to `quantum mechanical observables'. It seems reasonable to attribute to attribute such an interpretation to Bohr notwithstanding his cautious attitude with respect to ontological assertions.
      Compared to earlier versions the notion of `correspondence' changed in several ways. Thus, `strong correspondence'
      • α) does not rely on the `classical limit of quantum mechanics' since it is not required that on the macroscopic side of the Heisenberg cut there is a quantum mechanical description next to the classical one (even though it is widely, though unnecessarily, being assumed that such a description is possible, its classical limit allegedly yielding the classical description required by Bohr);
      • β) is stressing the `importance of the experimental arrangement' in establishing the physical meaning of a `quantum mechanical observable', thus transcending the (alleged) `objectivity of classical quantities' by pointing to the `contextuality of the notion of a quantum mechanical observable'; this has become a hallmark of the Copenhagen interpretation;
      • γ) is pointing to the difference between `quantum mechanics' and `classical mechanics' rather than to their similarity, the difference being established by the notion of `incompatibility of quantum mechanical observables'.
    • Remarks on `strong correspondence':
      • By the quantization conditions of `strong correspondence' a relation is established between a `quantum mechanical observable' and the `measurement arrangement used for its measurement'. Such a relation could only be conceived on the basis of the insight that observables may be defined differently if measurement arrangements are different, as is, for instance, the case of incompatible standard observables that are defined by mutually exclusive measurement arrangements (note that this example explains why `strong correspondence' is often presented as a part of `complementarity', compare footnote 4).
      • Due to its apparent operationalism <sup>0</sup> the `strong correspondence principle' has been one reason (next to Heisenberg's self-professed empiricism) to qualify the Copenhagen interpretation as a logical positivist/empiricist philosophy. However, this certainly does not apply to Bohr. It must have been Bohr's classical way of thinking about `physical quantities/observables' which induced him to deny that he would endorse logical positivism/empiricism. Even when talking about a quantum phenomenon Bohr had in mind an `object having classically-realist properties'. This implies that with Bohr, like in classical mechanics, a physical quantity (observable) of a microscopic object should be considered as a `property of that object', the `restriction on the possibility of simultaneous definition of incompatible quantities', involved in complementarity, thought to be the main difference between quantum mechanics and classical mechanics.
      • With Bohr observables are not interpreted in an empiricist sense, but rather in a realist one, be it in a `contextual sense, determined by the measurement arrangement'. This is particularly evident from Bohr's neglect of the difference between the measurement arrangements of EPR and EPR-Bell experiments, in the EPR experiment pointer readings being available only for one of the particles of an EPR pair. Physical quantities of microscopic objects are not defined in terms of the `directly observable quantities of measuring instruments' but are thought to have an existence of their own. In discussions of the `thought experiments' ample use was made of classical properties like `classical momentum conservation' and `classical wave diffraction', being thought to contextually define properties of the microscopic object, without bothering too much about whether the process under consideration is on the directly observational side of the Heisenberg cut or not.
      • The importance of the development of the notion of `correspondence' from `weak correspondence' to `strong correspondence' can hardly be overestimated since it shifted the attention from a methodology associated with the `classical limit' (having limited applicability) to a methodological principle being generally valid for arbitrary `quantum mechanical states' and `quantum mechanical observables' (both standard and generalized ones), liable to be operationally<sup>39</sup> implemented by devising suitable preparation and measurement procedures (yielding as measurement results the quantum mechanical probabilities p<sub>m</sub> of standard and generalized observables).
        However, an important step had still to be made, in which the relation between a `quantum mechanical observable' and its `measurement arrangement' is implemented into the mathematical formalism. This step will lead to a fourth phase in the development of the `correspondence principle', presented here.
    • Latitude of definition of an observable (physical quantity)
      • With Bohr the definition of an observable (physical quantity) A need not be a sharp one (represented by a number), but may have a certain latitude δA (e.g. represented by an interval between two marks on a ruler)⁸⁹. According to Bohr it in general does not make sense to think about an observable (physical quantity) as being more accurately defined than through such an interval; hence not: `the particle has a sharp although unknown position Q somewhere within the interval δQ', but: `the notion of position Q is itself unsharply defined'. The notion of `latitude' fits into the individual-particle interpretation since it is meant to represent a property of an individual microscopic object.
        Even though standard deviations ΔA (as used in the Heisenberg-Kennard-Robertson inequality) are properties of ensembles (viz. of distributions of sharp measurement results a<sub>m</sub>), they have in discussions of the `thought experiments' often been assumed to represent latitudes δA (see e.g. Publ. 52, section 4.5). In a probabilistic interpretation this does not seem to be an unnatural thing to do. Yet, by merging different notions into a single concept it is easily overlooked that `Bohr's latitudes δA' and `standard deviations ΔA' may play very different roles in quantum mechanics: for instance, the distance between two marks on a ruler has nothing to do with the statistical spreading of measurement results. Such differences become evident within the empiricist interpretation of quantum mechanics.
        The confusion as a result of not sufficiently distinguishing `latitudes δA' and `standard deviations ΔA' has given rise to a criticism of the notion of complementarity, to be illustrated here (for a more extensive discussion, see Publ. 48).
      • A `latitude' may be completely defined by the `calibration of the measuring instrument', and therefore may be a property of the measuring instrument rather than of the microscopic object. As follows from Bohr's reference to `definition of an observable', its meaning is epistemological with respect to the microscopic object (compare).
        Strictly speaking Bohr's notion of `latitude' need not be related to irreducible indeterminism, even though its origin is with the quantum postulate. `Latitude' is also referred to as `uncertainty', which notion, however, contrary to Bohr's intention, is often interpreted as `uncertainty about the sharp value allegedly possessed by the observable'. It is also often referred to as `indeterminacy' or `indeterminateness', which is confusing as a consequence of its suggestion of ontological indeterminism.
    • Criticisms and appraisal of `strong correspondence'
      • Reliance of `strong correspondence' on `classical mechanics'
        • Although, by no longer relying on the `classical limit of quantum mechanics', the `strong form of the correspondence principle' differs from its weak form, it still relies on `classical mechanics' by stressing the macroscopic character of a `quantum phenomenon representing a measurement result of an observable'. For Bohr this was an essential aspect of `measurement within the microscopic domain' because `classical language' was thought to be necessary for `unambiguous communication between observers'.
        • This being acknowledged, at the same time it may be put into question whether the `correspondence between a quantum phenomenon and the human experimenter/observer' is really the crucial factor in determining the physical meaning of a `quantum mechanical observable', the human experimenter/observer being screened off from the microscopic world by the macroscopic interfaces of his instruments, and, hence, being dispensable.
        • By stressing in the notion of `correspondence' the essential role of the `measurement arrangement' rather than that of the `human experimenter/observer' Bohr changed the attention from the `relation between the microscopic object and the human observer' to the `relation between the microscopic object and the measuring instrument', thus taking a step into the direction of the empiricist interpretation in which the observer does not play an explicit role, and the measurement process is supposed to be an ordinary physical process, to be described by quantum mechanics.
        • However, at that time this latter insight seems to have been `one bridge too far'. Probably due to a preoccupation with the `role of classical mechanics', nourished by the applicability of (parts of) that theory to the measurements available at that time (mainly limited to scattering experiments allowing unexpected applicability⁰ of the `classical law of conservation of momentum' also within quantum physics), it could be thought that the measurement interaction might be treated classically.
          Moreover, the `strong correspondence principle' seemed itself to frustrate a quantum mechanical treatment of measurement, because a requirement of an `actual presence of a measurement arrangement for each of the multitudinous microscopic constituents of a macroscopic measuring instrument' would prevent an adequate operation of that instrument.
        • However, we now believe that these arguments are not valid because we are convinced that `quantum measurement is an essentially microscopic process, at least in the pre-measurement phase to be described by quantum mechanics', it being determined during that very phase `which observable is measured by the measuring instrument' (compare the quantum mechanical characterization of a POVM given here).
        • Failure of `Bohr's thesis of classical description of measurement', inherent in the `strong correspondence principle', is the main cause of the present obsoleteness of the `Copenhagen interpretation of quantum mechanics'. A quantum mechanical treatment of `measurement within the quantum domain' has turned out to be indispensable both from a physical point of view as well as from a philosophical one. `Correspondence with classical quantities' is not required, and sometimes even impossible (e.g. spin).
        • Nowadays, as a result of the widespread idea (probably having a classical source, too) that quantum mechanics should yield an objective description of nature, the contextuality going with issue b) of the strong form of correspondence is often not sufficiently appreciated. I think that Bohr was justified in stressing the `influence of measurement' in defining quantum mechanical quantities, because simply `all our knowledge on microscopic reality is mediated by measuring instruments processing information so as to make it fit to be observed on a macroscopic scale'. In agreement with his cautious attitude it was not unreasonable for Bohr to consider quantum mechanics as a description of a contextual reality, co-determined by the measurement arrangement actually present. Unfortunately, Bohr did not sufficiently appreciate the `dynamical role of measuring instruments, being evident from the quantum mechanical description of pre-measurement'.
      • `Strong correspondence' without `reliance on classical mechanics'
        • It is important to note here that Heisenberg's approach does not rely on `classical mechanics'. In agreement with his more mathematical attitude he soon applied quantum mechanics to the measurement process, for the scattering experiments of his days arriving at the notion of measurement as formalized by von Neumann. However, although Heisenberg and von Neumann did not share Bohr's preoccupation with classical mechanics, by substituting the `quantum mechanical wave function' for the `phase space point of classical mechanics' they yet retained within their `interpretation of quantum mechanics' the `realism of the usual interpretation of classical mechanics' (be it in the sense of a realist individual-particle interpretation endorsing ontological probability).
          In agreement with Feyerabend's observation, Heisenberg and von Neumann, even though already in an early stage revolting against Bohr's authority, are yet considered (co-)founders of the `Copenhagen interpretation', their mathematical/physical approach not being felt as opposing Bohr's philosophical/physical one (compare): a `realist individual-particle interpretation' in which a microscopic particle was viewed upon as a `wave packet flying around in space' just seemed to be "equally physical" as a `classical object' (even though the wave packet had to be interpreted as a probability wave). Contextuality of the definition of a standard observable could be taken into account both mathematically and physically by assuming that the measurement arrangement actually present would dictate which mathematical representation `has actual physical reality within the context of a measurement'. Thus, in an `interference experiment' the object would (in a realist interpretation) "be a wave", or (in Bohr's instrumentalist interpretation) "manifest itself as a wave"; in a `which-way experiment' it would "be a particle" c.q. "manifest itself as such".
          During a long time the idea of an `individual microscopic object' as a `wave packet flying around in space' could be upheld due to experimental inability to observe `detection of an individual object'. Only when it became possible to do so it became evident that the quantum mechanical wave function does not describe an individual object, but an ensemble. Unfortunately, even today this has not sufficiently been taken up by textbooks of quantum mechanics, thus perpetuating the influence of classical thinking within the domain of quantum mechanics.
        • Being `more quantum mechanical', the Heisenberg/von Neumann approach of quantum measurement might seem to be superior to Bohr's classical one. This is not necessarily true, though. As a matter of fact, reliance on von Neumann projection tended to ignore what is really going on in a quantum measurement<sup>40</sup>. In particular, `von Neumann projection' does not refer to the measuring instrument as playing any dynamical role in establishing a value of the measured observable. When reading the Copenhagen literature one gets the impression that the measuring instrument is there in the first place to `disturb observables incompatible with the measured one' rather than to `determine the value of the measured observable'. Thus, in measurements like the Stern-Gerlach one it is seldom noted that a measurement is completed only when it is ascertained `which of the outgoing beams the particle is in' (which is realized, for instance, by placing detectors in the beams, one of these to be triggered by the particle, thus realizing a `final pointer position').
          In general, however, in the Copenhagen interpretation there is no reference to a process establishing the transition between initial and final states of a pointer of the measuring instrument. Instead, quantum mechanical measurement result a_m is associated with a transition to the final state |a_m> of the microscopic object. This characterizes the Copenhagen interpretation as a realist interpretation, be it of a contextualistic blend (a contextuality which, however, under the influence of the classical paradigm has disappeared from most textbooks of quantum mechanics in favour of a presentation in terms of an objectivistic-realist interpretation).
  • iv) Empiricist form of the correspondence principle
    • Taking into account these criticisms of the `strong correspondence principle' it is straightforward to arrive at the `empiricist interpretation of quantum mechanics' by dropping the requirement of `correspondence with classical mechanics', and by reinforcing Bohr's requirement of `correspondence with the measurement arrangement' so as to take into account in a dynamical way the `interaction between microscopic object and measuring instrument' (thus actually changing `correspondence with a measurement arrangement' into `correspondence with a measurement procedure').
      Such a form of `correspondence' might be referred to as an empiricist form of the correspondence principle, being part of the `neo-Copenhagen interpretation' developed in Publ. 52 and Publ. 53 (compare the sketch of this interpretation given here).
      It should be noted that we have reached here a crucial point in the development of the notion of `correspondence', viz. a transition from a notion of `correspondence between theories' (viz. `quantum mechanics' and `classical mechanics') to `correspondence between theory (i.c. quantum mechanics) and reality'. The latter notion is in agreement with the notion of `interpretation' as used here<sup>43</sup> as well as with the notion of `correspondence' as applied in the so-called `correspondence rules<sup>0</sup> attributing physical meaning to theoretical terms of a physical theory'. In a way the development towards `empiricist correspondence' completes the development of Bohr's thinking about `correspondence' referred to here.
    • It should also be stressed that in the `empiricist form of the correspondence principle' the `measurement arrangement' is no longer specified by its macroscopic parts (as Bohr used to do), but it is realized that the correspondence between a `quantum mechanical observable' and its `measurement arrangement' is determined by the microscopic phase of the measurement (on the microscopic side of the Heisenberg cut), that is, by the pre-measurement, and that the macroscopic phase, although important by making the measurement result empirically accessible, does not contribute to the essence of the correspondence. Consequently, `empiricist correspondence' is determined by the `quantum mechanical treatment of the interaction between microscopic object and measuring instrument', which, notwithstanding Bohr's objection, has turned out not only to be possible but to be even necessary.
      In particular, the correspondence of measurement arrangements of generalized observables with their POVMs does not seem to be liable to `formulation in classical terms'. In the `empiricist form of the correspondence principle' Bohr's `reliance on classical mechanics', still being observed in `strong correspondence', has been abandoned.
    • Strictly speaking the `empiricist form of the correspondence principle' does not belong to the `Copenhagen interpretation'. It is part of a `neo-Copenhagen interpretation' developed in Publ. 52 and Publ. 53, a sketch of which is given here.

• Complementarity
  • `Complementarity of preparation' versus 'complementarity of measurement'
    The notion of `complementarity' deals with the `incompatibility of standard observables', preventing such observables from being simultaneously or jointly measured in an ideal way (i.e. yielding a sharp and unambiguous value for each of them). Like the notion of `correspondence' also the notion of `complementarity' has changed in the course of time. Neither was it exactly the same for different authors.
    In all forms `complementarity' is a direct consequence of the `strong correspondence principle', combined with `mutual exclusiveness of measurement arrangements corresponding to incompatible standard observables', frustrating the possibility of simultaneously obtaining sharp values (i.e. with non-zero latitudes/uncertainties) of incompatible observables.
    In the standard formalism of quantum mechanics `complementarity' is usually represented by the Heisenberg-Kennard-Robertson inequality ΔAΔB ≥ ½|<[A, B]<sub>−</sub>>|, equating standard deviation ΔA with Bohr's latitude δA.
    However, this identification must be seen as a consequence of an unhappy coincidence of the `mathematical derivation of the Heisenberg-Kennard-Robertson inequality' and the `physical attempts to come to grips with quantum measurement by studying thought experiments'. It caused even the best physicists to jump to the conclusion that the former could only be the mathematical expression of the latter.
    Only as late as 1970 doubts were raised on this issue in an important paper by Ballentine<sup>57</sup>, stressing that the `standard deviations figuring in the Heisenberg-Kennard-Robertson inequality' only depend on the initial state (i.e. the state obtaining before the measurement) of the microscopic object, and do not in any way depend on the `properties of the measurement which they purportedly should represent'. It was realized only recently (cf. Publ. 52) that an explanation of the shortsightedness leading to the view of `the Heisenberg-Kennard-Robertson inequality as a property of joint measurement of incompatible observables' may be found in the circumstance that the `standard formalism of quantum mechanics' is not capable of yielding a satisfactory description of `thought experiments' like the double-slit experiment, but that rather the generalized formalism is necessary for that purpose. By restricting oneself to the `standard formalism' it was overlooked that the notion of `complementarity' actually consists of two parts, viz.
    i) complementarity of preparation (expressed, for instance, by the Heisenberg-Kennard-Robertson inequality, which is derived from the `standard formalism'),
    ii)complementarity of measurement (to be expressed, for instance, by the Martens inequality, which is derived from the `generalized formalism').
    As a consequence of the confusion stemming from a general ignorance of this distinction it was possible that in his answer to EPR Bohr could convince the majority of physicists that Einstein's idea of `incompleteness of quantum mechanics' (based on an assumption of `complementarity of preparation') would be unacceptable because it would be in disagreement with Bohr's idea of `complementarity of measurement'.
  • Bohr's notion of `complementarity'
    Bohr's form of `complementarity', too, has its origin in his `preoccupation with classical mechanics when considering the issues of completeness of quantum mechanics and correspondence'. When dealing with `incompatibility of quantum mechanical observables' Bohr stressed the `restricted applicability of classical mechanics', but at the same time he tried to save as much as possible the ideas of classical mechanics deemed necessary by him to describe `measurement phenomena'. From this point of view it seemed plausible to assume that the standard quantum mechanical observables of position (Q) and momentum (P) would take over the roles of classical position (q) and momentum (p). However, since Q and P are incompatible observables there should be differences.
    Bohr's notion of `complementarity' consists of two ingredients:
    • i) The `uncertainty principle'<sup>52</sup>
      The Copenhagen `uncertainty principle' has its origin in the incompatibility of position and momentum observables, and the ensuing impossibility of assigning to a microscopic object `simultaneous sharp values of position and momentum (i.e. having both latitudes of definition δQ and δP equal to zero)'. By studying so-called thought experiments Bohr found that the latitudes satisfy a relation of the type δQ δP ≥ ch, h Planck's constant, and c some positive constant depending on the specific problem. Such an inequality is referred to as an uncertainty relation.
      Usually the Heisenberg-Kennard-Robertson inequality (with A = Q and B = P) is considered to be an implementation of the `uncertainty principle' into the mathematical formalism of quantum mechanics. However, this is subject to strong doubts (compare).
    • ii) Incompatible pictures supplementing each other⁵⁴
      Even though position Q and momentum P are incompatible and hence not measurable simultaneously, the combined information from separate measurements of these observables might be thought to yield a complete characterization of `what can be known about the state<sup>44</sup> of a microscopic object'. In this sense `complementarity' was seen by Bohr as a `rational generalization of the classical ideal of determinism', according to which the pair (q,p) is yielding a complete description of the state of a classical point particle. Within the atomic domain `determinism' (often referred to as `causality') was deemed to be necessarily replaced by `complementarity', thus providing room for `quantum mechanical indeterminism'. Note that this restricts `complementarity' to canonically conjugate observables satisfying `canonical commutation relations' of the type [Q, P]<sub>−</sub> = iI, having a `weak correspondence relation' with a classical Poisson bracket.
  • Remarks on Bohr's notion of `complementarity'
    • `Complementarity' and `completeness'
      Although initially for Bohr issue ii) may have been the most important one, it was soon realized that there cannot be any `completeness' in the sense that the `quantum mechanical state' (described by the wave function or state vector) would be completely determined by the probability distributions of position and momentum.<sup>48</sup>
      A `dwindling influence of the classical paradigm' has probably been the cause that issue ii) is usually not considered to be a constituent of the notion of `complementarity' any more, the `uncertainty principle' alone being considered as expressing its meaning. As a consequence the notion of `complementarity' has obtained a wider significance, no longer being restricted to `canonically conjugate (maximally incompatible) observables (for which sharp knowledge of one is implying maximal uncertainty of the other)'. In the notion of `complementarity' the requirement of `maximality of the uncertainty' has been relinquished so as to consider as `complementary' any pair of observables that satisfy a nontrivial uncertainty relation (for which the constant c in its righthand side is greater than zero). This actually makes `complementary' any pair of standard observables corresponding to incompatible Hermitian operators.
    • `Particle-wave duality' and `particle-wave complementarity'
      During the early stages of the development of quantum mechanics the idea has been developed (de Broglie) that a microscopic object has both particle-like and wave-like properties (particle-wave duality⁰). It seemed that the object (sometimes called a wavicle) can manifest itself either as a particle or as a wave.
      During some time this `particle-wave duality' has been considered paradigmatic for the notion of `complementarity'. Particle and wave pictures of the microscopic object were considered as mutually exclusive, but also as mutually completing each other (compare). In the context of a `position measurement' position Q was thought to be well-defined, allowing a particle picture of the quantum mechanical object (in which, however, the particle's momentum is completely undefined). Analogously, in a `momentum measurement' momentum P was thought to be well-defined. Then the corresponding wave function should be a plane wave. Hence, in the context of a momentum measurement the microscopic object would allegedly have to be pictured as a wave.
      This application of `particle-wave duality' is referred to as `particle-wave complementarity'. Since `complementarity' is associated with `mutual exclusiveness of measurement arrangements', it was generally believed that in a `double-slit experiment' the interference pattern would be erased as soon as it is attempted to determine through which slit the particle or photon has passed. It was thought that in interference experiments (testing the wave character of the object) the particle picture would be inapplicable.
    • Critique of `particle-wave complementarity'
      However, it was soon realized that `particle-wave duality' is not a sound application of `complementarity', since in an interference experiment both particle and wave aspects can be observed. Thus, whereas the interference pattern is described by the wave function, and hence is a consequence of the `wave aspect of quantum mechanics', is the `particle aspect' obvious within the same experiment when it is observed (see for instance Akira Tonomura, Double-slit experiment with single electrons, Video clip 1) how the interference pattern is built up by localized impacts on a screen. Experimental evidence is consistent with an `ensemble interpretation of the quantum mechanical wave function', the localized impacts suggesting a `particle-like character of the individual objects' in `interference measurements' as well as in `which-way measurements'.
      Unfortunately, over the years `particle-wave complementarity' has been folk wisdom (even with Bohr), thus obscuring the inapplicability of the `Copenhagen individual-particle interpretation of the wave function'.
      The idea of `wave-particle complementarity' has been able to come into being, and survive during such a long time, as a result of a restriction of experimental and theoretical attention to the standard formalism of quantum mechanics, being able to describe only measurements of standard observables like Q and P, interpreted as `which-way' and `interference' measurements, respectively (a `which-way measurement' being associated with `particle-like behaviour').
      However, this latter stereotyped idea cannot be maintained in the face of recently performed generalized measurements that can be interpreted as `joint nonideal measurements of interference and path observables' (e.g. Publ. 27), yielding information on both standard observables (be it, in agreement with the Martens inequality, subjected to a restriction of complementarity quite different from `particle-wave complementarity'). It is found that the pure `interference measurement' and `which-path measurement' are just two extremes of a whole sequence of `intermediate measurements' connecting the extremes by changing a parameter governing the measurement arrangement. Taking into account the `intermediate measurements' makes obsolete the polar view as endorsed in the idea of `particle-wave complementarity'. In particular, there is no experimental evidence of the validity of the often-heard assertion that interference would be completely wiped out as soon as an attempt is made at determining which way the particle is going (although the generalized treatment is revealing a more moderate influence of changing the parameter, entailing a continuous change of the probabilities p_m when the measurement arrangement is gradually changed from one extreme to the other).
  • Heisenberg's notion of `complementarity'
    • Heisenberg's disturbance theory of measurement
      According to Heisenberg the meaning of `complementarity' is that measurements of incompatible observables like position Q and momentum P are mutually disturbing, to the effect that in a measurement `determining Q with uncertainty ΔQ' momentum is `determined with uncertainty ΔP', the standard deviations satisfying the same `uncertainty relation' as do Bohr's latitudes.
      Like in Bohr's notion of `complementarity' also `Heisenberg disturbance' is a consequence of `mutual exclusiveness of measurement arrangements', causing measurement results of incompatible standard observables to be influenced by the measurement arrangement that is actually present. Heisenberg's disturbance theory of measurement is mathematically implemented by von Neumann's projection postulate, prescribing a way the quantum mechanical state is influenced by a measurement. According to this convention a sharp measurement of Q (having ΔQ = 0) must be accompanied by maximal uncertainty with respect to momentum (ΔP = ∞), and vice versa.⁷²
      More generally, the standard deviations ΔQ and ΔP are often considered to be mathematical representations of Bohr's latitudes δQ and δP, both quantities possibly being finite. This identification of standard deviations and latitudes has been promoted by the idea that there must be a a close relation between the `uncertainty principle' and the Heisenberg-Kennard-Robertson inequality.
      Although `Heisenberg's disturbance theory of measurement' has turned out to be a realistic feature of `quantum measurement' (compare) is the above-mentioned relation subject to strong doubts (compare), and therefore is not assumed here.
  • Differences between Bohr and Heisenberg regarding the `uncertainty principle'
    • A crucial difference between Bohr and Heisenberg is that Bohr's approach is epistemological (compare), whereas Heisenberg's is ontological/physical, `measurement disturbance' allegedly yielding a physical explanation of the indeterminacy expressed by Bohr's latitudes.
    • With Bohr the `uncertainty principle' refers to an `unanalyzed measurement phenomenon' representing the knowledge conveyed by a measurement, no distinction being made between different phases of that measurement. A `measurement result' is attributed to the `unanalyzed measurement phenomenon'.
      With Heisenberg different phases of the measurement are distinguished. Consistent with `strong von Neumann projection' a measurement result is assumed to refer to the final state of the microscopic object. `Mutual exclusiveness of measurement arrangements' is manifest as a consequence of the impossibility that the final object state simultaneously be an eigenvector of both Q and P. This is in agreement with Heisenberg's contention that the `uncertainty relation' does not refer to the past (i.e. the initial state) but to the future (i.e. the final state).
      Although Heisenberg's approach is an improvement with respect to Bohr's one, it is, as a result of its `reliance on von Neumann's projection postulate' hardly more realistic than is Bohr's. A fundamental criticism of Heisenberg's approach of quantum measurement is given here.
  • Critique of `Heisenberg's disturbance theory of measurement'
    • Heisenberg's disturbance theory of measurement is based on the restrictive view of measurement symbolized in figure 3, in which a measurement is tantamount to a `preparation of a final state of the microscopic object'. For such measurements Heisenberg's `preparative measurement disturbance' merges with the determinative notion of `disturbance of information on the initial state of the object, obtained by measurement'. In more general measurements preparative (predictive) aspects should be carefully distinguished from determinative (retrodictive) ones, thus giving rise to two different kinds of complementarity, one for `preparation', and one for `simultaneous measurement of incompatible observables' (see Publ. 48).
      Restriction to the `standard formalism' (according to which simultaneous measurement of incompatible observables is not even possible) has caused quite a bit of confusion with respect to the latter interpretation of `complementarity'. However, on the basis of the generalized formalism `Heisenberg disturbance' can be given an unambiguous determinative (retrodictive) meaning without having to rely on the special kind op `preparation of the microscopic object' symbolized in figure 3 (compare the Martens inequality). The impossibility of a `simultaneous sharp definition of incompatible standard observables' can be interpreted as a `limitation on the simultaneous measurability of such observables' in a much more general way than envisaged by Heisenberg.
      As noted by Ballentine<sup>57</sup>, as far as the `preparative feature of Heisenberg's disturbance theory of measurement' is related to `complementarity', it is `complementarity of preparation of the final state of the microscopic object' rather than `complementarity of observables measured in the initial state'. Here the same criticism applies as advanced with respect to `strong von Neumann projection': in general there is no reason to believe that the final state of the microscopic object is related to the measurement result that is obtained. As is clearly exhibited by the empiricist interpretation it is not the `final state of the microscopic object' that is relevant for the success of a measurement procedure, but it is the `final state of the pointer of the measuring instrument'. The microscopic object may be disturbed in any way consistent with the properties of the pre-measurement. Only in very special measurement procedures is it possible to interpret a `preparation of the microscopic object in a final state' as a `preparation of the pointer of a measuring instrument in a final pointer state' (compare).
      This conclusion has important consequences for the interpretation of the Heisenberg-Kennard-Robertson inequality.
  • Critique of `complementarity'
    
    • The identification of Bohr's `latitude δA', figuring in the strong correspondence principle, with the `standard deviation ΔA of the Heisenberg-Kennard-Robertson inequality' has been a considerable source of confusion.
      First, a latitude may correspond to a parameter of the measurement arrangement like, e.g. the slit width in a double-slit experiment. This parameter is independent of the properties of the incoming wave packet, which in a determinative approach is the interesting quantity. Instead, the parameter determines the standard deviations of the lateral position observable in the outgoing state |ψ_m> of each slit (cf. figure 3), thus playing a preparative rather than a determinative role (compare).
      Second, contrary to the idea of a latitude δA, in a standard deviation ΔA every individual object is yielding a sharp measurement result.
      Third, the `Heisenberg-Kennard-Robertson inequality' is not in any obvious way related to the simultaneous measurement of the observables that are involved. On the contrary, the inequality can be tested by separate measurements of the observables (i.e. performed on different individual objects of an ensemble).
    • It is also important to note once again that Heisenberg and Bohr differed considerably in their understanding of the meaning of the `uncertainty relation'. Whereas with Bohr these are valid `within the context of the measurement (i.e. during the measurement)', with Heisenberg they are `referring to the future (i.e. after the measurement)'.
      Both Bohr and Heisenberg differ from the textbook understanding in which the `Heisenberg-Kennard-Robertson inequality' is taken in the initial state, thus referring to the past (i.e. the state valid immediately before the measurement). In figure 12 the main differences are given between Bohr, Heisenberg and Ballentine<sup>57</sup> with respect to the way `measures of uncertainty/indeterminacy' are used (δA = latitude, ΔA = standard deviation, determined either in the initial or final state of the measurement).
      figure 12
      

      prior to measurement during measurement posterior to measurement Bohr δA Heisenberg ΔAfin Ballentine ΔAin

    • We should distinguish two different `sources of complementarity' (cf. Publ. 48), viz. preparation and measurement, each being subject to its own restriction with respect to the simultaneous consideration of incompatible observables. Whereas textbooks and Heisenberg refer to preparation (Heisenberg differing from the textbook interpretation by referring to preparation by means of a measurement), is only Bohr's interpretation strictly related to the `simultaneous measurement of incompatible observables'. `Complementarity due to preparation' is a restriction on our ability of preparing a quantum mechanical state. It can be expressed by means of the Heisenberg-Kennard-Robertson inequality in terms of standard deviations, obtained in ideal measurements of the standard observables, or by means of an entropic uncertainty relation.
    • In assessing the meaning of `complementarity' the founding fathers of quantum mechanics have been severely hampered by the fact that only the `standard formalism of quantum mechanics' was available (or even just being developed). This made them believe that the inequality they derived in a rather informal way for the so-called `thought experiments' should be represented within the mathematical formalism by the `Heisenberg-Kennard-Robertson inequality' (being the only theoretical relation available at that time). However, the `standard formalism' is not able to deal with `simultaneous or joint measurement of incompatible observables'. According to this formalism only compatible standard observables (corresponding to commuting Hermitian operators) can be simultaneously measured.
      In order to describe the `simultaneous measurement of incompatible observables' (which actually is the subject discussed in the `thought experiments'), a generalization of the mathematical formalism of quantum mechanics is necessary, allowing to define a concept of joint (nonideal) measurement (see, for instance, Publs 25, 26, 27, 31, 36, 48). Complementarity as discussed in the `thought experiments' actually is about measurement (in)accuracies that can be described by `nonideality measures' like the average row entropies of nonideality matrices figuring in the theory of `joint measurement of generalized observables'. The Martens inequality, rather than the `Heisenberg-Kennard-Robertson inequality', must be seen as in this sense expressing `complementarity'. This is a second kind of `complementarity' different from the `complementarity described by an uncertainty relation like the Heisenberg-Kennard-Robertson inequality, or by the entropic inequality'. The interpretation of these latter inequalities as `properties of joint measurement' has been a formidable source of confusion with respect to the notion of `complementarity'.
  • Heisenberg-Kennard-Robertson inequality and `complementarity'
    Confusion with respect to the role of the `Heisenberg-Kennard-Robertson inequality' regarding `complementarity' has many layers because the inequality allows a number of different interpretations, closely related to the different views on
    (in)determinism discussed here.
    • i) `Quantum mechanical acausality' versus `classical determinism'
      In the `Copenhagen interpretation' the `Heisenberg-Kennard-Robertson inequality' is seen as a restriction on `classical determinism' to the effect that as a result of irreducible indeterminism `no sharp values of position and momentum can simultaneously be attributed to a microscopic object', thus preventing any (classically deterministic) causal account of microscopic processes. With Bohr the `Heisenberg-Kennard-Robertson inequality' is seen as restricting `simultaneous definability of position and momentum' to the effect that one observable is less sharply defined as definition of the other one is sharper. Note that here the quantities ΔQ and ΔP, as well as the `Heisenberg-Kennard-Robertson inequality', are thought to be properties of an individual object.
      On the other hand, for Einstein the meaning of the `Heisenberg-Kennard-Robertson inequality' amounted to a restriction of the possibility of quantum mechanics to yield a deterministic/causal description of an `allegedly deterministic/causal world', assuming ΔQ and ΔP, as well as the `Heisenberg-Kennard-Robertson inequality', to be properties of an ensemble rather than of an `individual object'.
    • ii) `Free evolution' versus `measurement'
      The `Heisenberg-Kennard-Robertson inequality' may be thought to be a property valid either during `free evolution' or during `measurement'. It seems to me that the Copenhagen interpretation's `acknowledgement of the essential role of measurement within quantum mechanics' points into the direction of the second issue. Indeed, within the Copenhagen interpretation the `Heisenberg-Kennard-Robertson inequality' is usually seen as a restriction on the `possibility of jointly measuring incompatible observables like Q and P', formalizing within the quantum mechanical theory the `uncertainty relation found in a more intuitive way by studying thought experiments'. Although Bohr and Heisenberg do not precisely agree on the way the `uncertainty relation' must be interpreted, their contextualistic-realist reliance on `measurement' is sufficient to unite them into a single interpretation. That their identification of the uncertainty relation with the `Heisenberg-Kennard-Robertson inequality for position and momentum' has maintained itself during such a long time before its inadequacy was realized (compare) should perhaps be held against the instrumentalist philosophy ("Shut up and calculate") pervading that era, rather than against the physical insight of the `founding fathers'.
      Due to the fact that completeness (rather than the essential role of measurement) is often considered to be characterizing the `Copenhagen interpretation', the `Heisenberg-Kennard-Robertson inequality' has also been taken as a `property of a freely moving individual microscopic object', representing a certain `indeterminacy independent of any measurement'. In textbooks of quantum mechanics the inequality is usually presented in this (objectivistic-realist) sense. This tendency to get rid of `measurement' as an important interpretational issue (accompanying a development within the philosophy of science⁰ from `logical positivism/empiricism' towards `scientific realism') is at the basis of Ballentine's criticism of the meaning of the `Heisenberg-Kennard-Robertson inequality', realizing that the inequality refers to the initial state, and, hence, cannot reflect any influence of measurement.
      Whereas the influence of the `Copenhagen interpretation' is still being felt as far as an individual-particle interpretation of the wave function or state vector is being entertained, is Einstein's ensemble interpretation considered as anti-Copenhagen since it offers the possibility to interpret the `Heisenberg-Kennard-Robertson inequality' as a true `uncertainty relation' (in the sense of the possessed values principle assuming all quantities to have sharp values), and interpreting standard deviations as `uncertainties with respect to the sharp values<sup>59</sup> the quantities (simultaneously) really have'. In this interpretation the `Heisenberg-Kennard-Robertson inequality' is seen as a property of an ensemble rather than an `individual object'.
      Note that in actual practice the quantum mechanical description can only be applied to ensembles (by performing the same experiment a large number N of times). This holds true also in Heisenberg's disturbance interpretation, the difference with `Einstein's ensemble interpretation' being that Heisenberg refers to relative frequencies obtained in the final state of a `von Neumann ensemble of individual objects', whereas Einstein refers to `statistical uncertainties in the initial ensemble'.
    • iii) Ontological versus epistemological meaning
      The `Heisenberg-Kennard-Robertson inequality' may be taken either in an ontological or in an epistemological sense: within the `Copenhagen interpretation' incompatible observables may either be thought `not to simultaneously have sharp values prior to measurement', or it may be deemed `impossible to determine such sharp values by means of a simultaneous measurement because such measurements are mutually disturbing' (compare).
      The first (ontological) view is consistent with Jordan's ideas⁵⁸ (which may be thought to hold either during `free evolution' or during `measurement'). The experimental standard deviations are then interpreted as `measures of the indeterminacies of the individual objects in their initial states'.
      On the other hand, Heisenberg's empiricism makes him consider as metaphysical and useless the assumption of the `existence of properties possessed by the object independent of measurement', although he does acknowledge the `metaphysical possibility of particle trajectories'. In this respect Heisenberg's position is in agreement with `Bohr's cautious epistemological attitude with respect to ontological assertions', the `directly observable experimental data observed in the final state' expressing `(epistemological) knowledge on the initial state', although having as `phenomena' an ontological meaning of their own.
      Note, however, that the epistemological part of the above analysis would probably not have been appreciated by Heisenberg (who considered such knowledge as metaphysical), `preparation of the final state of the object' being considered by him as the `only physically relevant function of measurement' (a criticism of this position can be found here).
      As an anti-Copenhagen example can be mentioned here Einstein's interpretation of the `Heisenberg-Kennard-Robertson inequality', the inequality being seen as an `ontological property of an ensemble'.
      As far as the `Heisenberg-Kennard-Robertson inequality' is considered in an ontological sense as a `property of the initial state' can it be interpreted as a `restriction on the possibility of preparing an initial state' either in the `Copenhagen probabilistic sense' or in `Einstein's statistical sense'.
    • iv) `Completeness in the wider sense' versus `completeness in the restricted sense'
      Since applicability of the `Heisenberg-Kennard-Robertson inequality' is restricted to quantum mechanical quantities, the inequality should be considered as an expression of completeness in the restricted sense.
      As a consequence of Einstein's dubious identification of `elements of physical reality' with `quantum mechanical observables' the inequality has often been considered a reason to `endorse completeness in the wider sense', denying the existence not only of `hidden variables corresponding to quantum mechanical observables', but of `hidden variables of whichever kind'. However, no `derivation of the Heisenberg-Kennard-Robertson inequality from hidden variables theories' being available, there is no `a priori reason' to deny that inequality an interpretation as a `restriction on the possibility of preparing quantum mechanical states corresponding to arbitrarily concentrated distributions of subquantum elements of physical reality in a series of individual preparations constituting an ensemble' (as could have been done by Einstein in the EPR proposal if he had not set himself to prove `incompleteness in the restricted sense' rather than `incompleteness in the wider sense').
      It seems to me that the confusion of `completeness in the restricted sense' and `completeness in the wider sense' may be caused by the `idea that quantum mechanics is universally valid', thus ignoring its restricted applicability (compare).
    • v) `Realist' versus `empiricist' interpretation
      Perhaps in Heisenberg's empiricism the Copenhagen interpretation approaches the empiricist interpretation: the `Heisenberg-Kennard-Robertson inequality' is thought to refer to the post-measurement state, thus possibly encompassing a certain influence of measurement. However, it is the final state of the microscopic object rather than that of the `measuring instrument' that is intended. Hence, it seems that the empiricism is only apparent, the final state being attributed to the microscopic object in the sense of a `realist interpretation of the quantum mechanical formalism'.
      As was mentioned here, also Bohr's interpretation should be seen as a realist one, although it is not impossible that Bohr's conclusion in his answer to EPR, viz. that there is "no question of a mechanical disturbance" [of particle 2 by the measurement on particle 1, but] "there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system", could be read in an empiricist sense as referring to `pointer readings of future measurements to be performed on particle 2'. This, in any case, would have yielded a `consistent extension of the logical positivist phenomenalism/operationalism of the strong correspondence principle to EPR-Bell measurements'.
      Unfortunately, Bohr's answer is not sufficiently clear on this matter, leaving ample room for the classical paradigm to dominate. Although the Copenhagen interpretation did not fall pray to the possessed values principle, by the older forms of the correspondence principle it was heavily biased towards `classical mechanics', in which the measuring instrument used to stay out of sight. Whenever, in studying the `thought experiments', it turned out that the measuring instrument should be taken into account, its influence was thought to be restricted to `disturbing the microscopic object'.
      A full-blown quantum mechanical description of `quantum measurement in which the measuring instrument is playing a dynamical role' had to await the decline of the influence of the Copenhagen doctrine of `classical description of measurement'. It seems to me that here a close relation can be observed between `developments in quantum mechanics' and a `growing realization within the philosophy of science of theory-ladenness of measurement/observation'.

• Copenhagen confusion of `preparation' and `measurement'
  • Most probably the main shortcoming of the `Copenhagen interpretation' is its lack of distinction between the notions of preparation and measurement (Publ. 48). This is a consequence of the fact that quantum mechanical observables are thought not to have well-defined values preceding and independent of measurement, which made it impossible to view upon `measurement' as a `determination of the value the observable had prior to measurement'.
    As a second best option it was proposed that the observable should possess (in a realist sense) the measured value after its measurement (so as to allegedly assure that a second measurement of the same observable yield the same measurement result with certainty).
    This is the basis of von Neumann's projection (or reduction) postulate, which has become the characteristic trait of measurement according to the `Copenhagen interpretation'. Hence, in this interpretation a measurement is a `preparation of the object in a certain post-measurement (final) state (measurement of the first kind)'.
    Undoubtedly the most notorious example of the confusion of `preparation' and `measurement' is the EPR experiment, which was presented as a `measurement of particle 2' (by carrying out a `measurement on particle 1'), but which actually is a `preparation of particle 2 conditional on a measurement result obtained for particle 1'.
  • The view of `measurement as a preparation of a post-measurement state' is consistent with Heisenberg's insight that the `uncertainty principle is not relevant to the past (i.e. the initial state) but to the future'. According to this view the meaning of `complementarity' is that, due to the incompatibility of position and momentum, it is impossible to prepare, by means of a simultaneous measurement of the two observables, the object in a state described by a simultaneous eigenvector of Q and P (since such joint eigenvectors do not exist!).
    More generally, a similar impossibility obtains for any pair of incompatible observables having no eigenvectors in common.
  • Criticisms of `measurement as preparation'
    • The widespread use of the notion of measurement of the first kind in the literature on the foundations of quantum mechanics contrasts sharply with the virtual experimental non-existence of such measurements (compare). It is very unfortunate that, by almost exclusively dealing with `measurements of the first kind', the foundational discussion has largely lost touch with experimental practice.
    • The Copenhagen view of `what is a quantum mechanical measurement' is alien to the common-sense view that `measurement should yield information on the object as it was preceding the measurement' (or, in the empiricist interpretation, information on the `preparation procedure that prepared the object preceding the measurement'). Under the influence of the `Copenhagen interpretation' this latter determinative (or retrodictive) view of measurement has been replaced by a preparative (or predictive) one.
      It is fortunate that in recent years interest in the subject of quantum information has revived the common-sense idea that the main goal of quantum mechanical measurement is a `determination of the initial state' (irrespective of whether this state is taken in the realist or in the empiricist sense). In particular the development of the generalized formalism of quantum mechanics has made it possible to determine more accurately the relation between an experimental measurement and the information on the initial state obtained by it (Publ. 47).
      As a general conclusion it must be noted here that we should be appreciably less satisfied with `the precise way the Copenhagen interpretation has implemented measurement in the interpretation of quantum mechanics' than with its insight that such an implementation is necessary.

Sketch of a neo-Copenhagen interpretation

• The `neo-Copenhagen interpretation' proposed here (cf. Publ. 53 and Publ. 56) pays a tribute to the `Copenhagen interpretation' by considering itself a heir to that interpretation, keeping from it what is still useful, but relinquishing what is made obsolete by recent experimental and theoretical developments. It has turned out that many of the Copenhagen ideas cannot be maintained in the face of the experimental and theoretical developments having taken place during the eighty odd years since the twin conception of the `(standard) quantum mechanical formalism' and the `Copenhagen interpretation'. In particular, by the development of a generalized formalism of quantum mechanics, both demonstrating the `limited applicability of the standard formalism' as well as the `fundamental flawedness of the standard interpretation of the Heisenberg-Kennard-Robertson inequality', the necessity was indicated of an interpretation that is `more faithful to actual experimental practice and theoretical insights'.
  Let me start by presenting the following
  Lists of positive and negative features of the `Copenhagen interpretation'
  (judgments as seen from the standpoint of the `neo-Copenhagen interpretation'):

  List of positive features - crucial role of measurement - empiricist attitude

  List of negative features - restriction to the `standard formalism' - von Neumann projection as a `measurement principle' - confusion of `preparation' and `measurement' - realist interpretation of the quantum mechanical formalism - claim of completeness in the wider sense - claim of completeness in the restricted sense - individual-particle interpretation of the state vector - classical account of measurement - ambiguous notion of `correspondence' - confused notion of `complementarity' - boosting the idea of `nonlocality'

• Remarks on `positive and negative features of the Copenhagen interpretation'
  • The common basis of both the Copenhagen and the neo-Copenhagen interpretation is the fundamental insight that a theory of microscopic physics (i.c. quantum mechanics) cannot be understood without duly taking into account the `crucial role played by the measuring instrument and its interaction with the microscopic object'. This insight is implemented into the `Copenhagen interpretation' through Bohr's contextualistic-realist interpretation of quantum mechanical observables, replacing the `objectivistic-realist interpretation of classical mechanics'. Within the `neo-Copenhagen interpretation' the nonobjectivity of the Copenhagen interpretation has been further developed towards the empiricist interpretation of the quantum mechanical formalism.
  • By adopting an `empiricist interpretation of the quantum mechanical formalism' the `neo-Copenhagen interpretation' resumes the empiricist tendencies that were so important during the conception of the `Copenhagen interpretation'. In this way the `scientific realist tendency towards a more realist interpretation of the quantum mechanical formalism' (being operative also within the Copenhagen interpretation) is corrected without maintaining the untenable part of logical positivist/empiricist influences.
  • Notwithstanding that issues of the `Copenhagen interpretation' find a continuation, it is evident that these issues cannot be copied into the `neo-Copenhagen interpretation' without having been duly updated. One reason for such an update is the restricted applicability of the standard formalism of quantum mechanics which has been developed together with the Copenhagen interpretation. Later developments have shown the `necessity to apply a generalized formalism', both in order to be able to `describe new experiments' as well as to `better understand old ones'.
  • As a particularly harmful element of the `standard formalism' should be mentioned `von Neumann's projection postulate' criticized here. During a long time the Heisenberg-Dirac-von Neumann view of `measurement' as `preparation of the final state of the microscopic object' has been maintained as a kind of `(unattainable) ideal model of a quantum measurement', without realizing that its attainability is neither `necessary', nor `practically realizable in even the most paradigmatic measurements'. Probably the postulate would have been abandoned already a long time ago if it would not have been closely related to the next issue.
  • Von Neumann's projection postulate epitomizes the fundamental confusion of `preparation' and `measurement' that is characteristic of the `Copenhagen interpretation'. Indeed, the role of the measurement interaction is restricted to disturbing the microscopic object, rather than being acknowledged as `dynamically important for preparing the final state of the measuring instrument'.
  • Right from the start the classical paradigm has caused the `Copenhagen interpretation' to have insufficient attention for the `dynamical role of measurement' (compare the lack of distinction between EPR and EPR-Bell experiments). Moreover, in the wake of criticisms of logical positivism/empiricism during the 60s and 70s of the 20<sup>th</sup> century, and the rise of scientific realism, the `role of measurement' has been downplayed to the extent that the empiricist lessons of the founding fathers have largely been forgotten, and a `realist interpretation of the quantum mechanical formalism' has become fashionable both in the physical literature and in quantum mechanics textbooks.
  • In order to present an alternative to the overoptimistic conjecture of `scientific realism' that quantum mechanics is describing an `objective quantum reality, it seems necessary to restore `empiricism' in such a way that the formalism genuinely refers to the `phenomena', i.e. to `events that are actually observed', rather than to `certain (ontological) states of the microscopic object, assumed to be prone to faithful measurement'. This is realized by adopting the empiricist interpretation of the quantum mechanical formalism, thus at the same time saying farewell to the (Copenhagen) idea of completeness in the wider sense.
  • As a consequence of restricting itself to the standard formalism of quantum mechanics the `Copenhagen interpretation' is incomplete in the restricted sense, the generalized formalism being necessary to realize `completeness of quantum mechanics on its full domain of application'.
  • Another aspect of the `completeness issue in the restricted sense' is the Copenhagen idea that the wave function or state vector is yielding a (probabilistic) description of an individual particle, which seems to be obsolete by now both on experimental and theoretical grounds. Moreover, apart from the classical paradigm there is no pressing reason to stick to the `Copenhagen individual-particle interpretation' rather than adopting an ensemble interpretation.
  • The Copenhagen idea of the `necessity of a classical account of measurement' should be abandoned as being a `relic of the obsolete idea of theory-independence of observation/measurement'. Instead, within the quantum domain a `fully quantum-mechanical description of pre-measurement' is necessary.
  • With the `Copenhagen interpretation' three different notions of `correspondence' could be distinguished, all of which being burdened by the classical paradigm. In view of the foregoing issue a `fully quantum mechanical correspondence principle' is required, relating `quantum mechanical measurement procedures' to the mathematical formalism. In the `neo-Copenhagen interpretation' for this purpose an empiricist form of the correspondence principle has been proposed to replace the Copenhagen ones.
  • With the `Copenhagen interpretation' there is a fundamental confusion with respect to the `relation between the notion of complementarity and the Heisenberg-Kennard-Robertson inequality'. This confusion is caused by not recognizing the difference between `complementarity of preparation' and `complementarity of measurement'.
    This difference can be seen clearly on the basis of `application of the generalized formalism' to the joint nonideal measurement of incompatible standard observables, yielding the Martens inequality to take over the function unjustifiedly attributed to the `Heisenberg-Kennard-Robertson inequality', thus allowing the latter inequality to take up its rightful place as a description of `complementarity of preparation'.
  • By neglecting the difference between the EPR experiment and EPR-Bell experiments the `Copenhagen interpretation' has boosted the idea of EPR nonlocality. Within the `neo-Copenhagen interpretation' application of the empiricist interpretation avoids any reason to expect such a `nonlocality', von Neumann's projection postulate and a realist individual particle interpretation of the state vector being responsible for such an expectation.
    The `Copenhagen interpretation' cannot be held responsible for Bell's `denial of completeness of quantum mechanics in the wider sense' allowing him to derive the Bell inequality. It may yet have contributed to the remarkable popularity of the `nonlocality idea'as a result of the fact that `nonlocality' seemed to be derivable from `opposite points of view'.
    On this issue we are confronted with the fact that the `Copenhagen interpretation' is not a well-defined set of rules, but has developed in different directions, even contradictory ones. By relying on the empiricism interpretation of the mathematical formalism of quantum mechanics I think the `neo-Copenhagen interpretation' is able to shed light on the `nonlocality conundrum as expressed by violation of the Bell inequality' by taking seriously the empiricist lessons of the the founding fathers of quantum mechanics both within (generalized) quantum mechanics as well as within hidden variables theories.
• Comparing Copenhagen and neo-Copenhagen interpretations
  (in the following table I compare qualifications by `each interpretation's own endorsers'):

  Copenhagen interpretation - standard formalism - completeness in the wider sense - completeness in the restricted sense - indeterminism/acausality - strong correspondence - instrumentalist or realist interpretation - individual-particle interpretation - `measurement' as `preparation of final state of microscopic object' - von Neumann projection - classical account of measurement - uncertainty principle (`Heisenberg-Kennard-Robertson inequality') - EPR nonlocality

  Neo-Copenhagen interpretation - generalized formalism - incompleteness in the wider sense - completeness in the restricted sense - possibility of `determinism/causality' - empiricist correspondence - empiricist interpretation - ensemble interpretation - `measurement' as `obtaining information on initial state' - conditional preparation - quantum mechanical description of (pre-)measurement - `complementarity of preparation' and `complementarity of measurement' (`Heisenberg-Kennard-Robertson inequality' and `Martens inequality', respectively) - EPR-Bell understood locally within quantum mechanics - EPR-Bell understood locally within hidden variables theory