Copenhagen Interpretation

We use the term ”Copenhagen interpretation” (CI) for a conglomerate of ideas invented by Bohr, Heisenberg, von Neumann and others. The terms ”orthodox interpretation” and ”standard interpretation” could be used as synonyms for CI. We do not pay attention to historical accuracy as regards individual contributions; for that matters see e.g. Howard (preprint), Faye (web-article).

5.1 Completeness requires metaphysics

Picture on cover of Alter and Yamamoto's book: Schroedinger's realistic and Heisenberg's surrealistic picture of Mona Lisa
O. Alter and Y. Yamamoto, "Quantum Measurement of a Single System", Wiley, 2001 .

Let us start with the following compact description of the CI, found in a review article by N. D. Mermin [104] (pdf):

”It is a fundamental quantum doctrine that a measurement does not, in general, reveal a preexisting value of the measured property. On the contrary, the outcome of a measurement is brought into being by the act of measurement itself, a joint manifestation of the state of the probed system and the probing apparatus. Precisely how the particular result of an individual measurement is brought into being - Heisenberg’s transition from the possible to the actual - is inherently unknowable. Only the statistical distribution of many such encounters is a proper matter for scientific inquiry.”

This means: As far as concrete predictions are concerned QT is a statistical theory. The comment on how results of individual measurements are ’brought into being’ cannot, of course, be verified experimentally (using the mathematical framework of QT); it belongs to metaphysics and could be replaced by any other metaphysical construction without changing any concrete predictions of QT. A problem arises if we contrast this quotation with claims for ’completeness’ of the CI:

"We regard quantum mechanics as a complete theory for which the fundamental physical and mathematical hypotheses are no longer susceptible of modification."
Heisenberg and Max Born, paper delivered to Solvay Congress of 1927

How can a statistical theory be ’complete’ as regards individual atomic events ? Obviously, the above notion of ’completeness’, used by Heisenberg and Born, must be rather different from our macroscopic, ’common sense’ notion of completeness. To mention a trivial example, we definitely know that statistical information about the average lifetime of all persons in a country is not sufficient to predict the lifetime of any individual person in that country.

An assumption implicit in the common sense notion - which is so ’trivial’ that it is hardly ever mentioned in a macroscopic context - is that the objects of our study (the persons in the above example) have properties which can be ascribed to them no matter whether or not they are studied, counted, observed, or whatever experimental action may be required for their quantitative description. In short, the objects - defined by their properties - possess independent ’reality’ or ”objectivity” (the nomenclature is not unique and we neglect here also more subtle philosophical distinctions). This is a very fundamental assumption for science. Note that this assumption does not require that the objects (their properties) cannot be influenced by the act of observation; it concerns the very existence of the properties (the objects). It is this assumption, - sometimes called ’objectivity’ - that is abandoned in the CI.

"Procrustes: a villainous son of Poseidon in Greek mythology who forces travelers to fit into his bed by stretching their bodies or cutting off their legs "
Definition from Merriam-Webster dictionary

Abandoning ’objectivity’ the ’completeness’ of QT may now be achieved as follows: Properties in QT are formally represented by linear operators $Qˆ$ . Possible values of $Qˆ$ are given by the solutions $q$ of the eigenvalue equation $Qˆ |q ⟩ = q |q ⟩$ . The states $|q ⟩$ are the eigenstates of $ˆ Q$ belonging to the eigenvalues $q$ (an abstract representation of states seems now more appropriate). The physical state of an object is generally given, at each instant of time, by a mathematical state (vector) $|ψ ⟩$ . Depending on the value of $|ψ ⟩$ two completely different situations may result:

Accepting these two postulates the common-sense-incompleteness of QT (which seems evident due to the probabilistic nature of its predictions) has, in fact, been formally eliminated. The fact that no predictions about non-existing properties (non-existing objects) can be made is no indication of incompleteness. In this sense the above postulates 1 and 2 ’solve’ the problem of incompleteness.

The question is, however, if we accept postulates 1 and 2 as reasonable. Is it allowed to transform a (epistemic) state of insufficient knowledge in a (ontic) state of non-reality? Does it really make sense to make such an extreme assumption about ’reality’ (the deepest philosophical mystery of all), requiring that microscopic particles travel there and back from ’existence’ to ’nonexistence’, depending on the value of $|ψ (t )⟩$ and the act of observation? Is postulate 2 not rather the result of wishful thinking declaring as irrelevant all observations which we are unable to make in our ’complete’ theory; thus cutting off and axiomatically defining away all problems with logical ’brute force’?. In fact, the simple ’solution’ of the completeness problem in terms of the above axioms (in particular postulate 2) creates a never ending series of quantum theoretical mysteries and paradoxes (which indicate a general ”quantum weirdness”). Strangely enough these situation has not led to a critical reconsideration of the basic postulates but rather to enjoyment in and inflationary expansion of mysteries and paradoxes.

Matters like ’existence’ and ’reality’ do not belong to science but to the part of philosophy called metaphysics. So, let us discuss first the validity of postulates 1,2 from a philosophical point of view.

"...all those bodies which compose the mighty frame of the world have not any subsistence without a mind, that their being is to be perceived or known."
Rene Descartes (1596 - 1650)

The non-realistic aspect of Copenhagen philosophy has been brought in mainly by Heisenberg (alternative terms to characterize this philosophical way of thinking are ”idealism” or ”immaterialism”). It goes back to Aristotle (384BC - 322BC), Descartes (1596 - 1650), and Berkeley (1685-1753). Heisenberg refers explicitly to Berkeley’s philosophy [61]:

”The next step was taken by Berkeley. If actually all our knowledge is derived from perception, there is no meaning in the statement that the things really exist; because if the perception is given it cannot possibly make any difference whether the things exist or do not exist. Therefore, to be perceived is identical with existence.”

Heisenberg used Berkeley’s philosophy to interpret QT two centuries later. How do present-day professional philosophers think about Berkeley’s idealism? There is, of course, no unique position. Some incorporate Heisenberg’s ideas. A new ’verificationist’ theory of truth may be invented and QT may then - using this new theory of truth - shown to be ’complete’ (see Garola and Sozzo [56])⁵⁹ . Others take a rather critical position against the ”Berkeley-Copenhagen Interpretation” (a term coined by Paul Mermet) and its mixing up of ontology and epistemology (see Popper [120], Bunge [30],[31],[32], Fine [51], Held [64], Adler [1]).

"The indeterminacy discovered by physical measurements of subatomic phenomena simply tells us that we cannot know the definite position and velocity of an electron at one instant of time. It does not tell us that the electron, at any instant of time, does not have definite position and velocity. [Physicists]...convert what is not measurable by them into the unreal and nonexistent"
Mortimer Jerome Adler (1902-2001), Philosopher. Quoted after: Mermin, Rev.Mod.Phys. vol.65, p.803 (1993)

Let us also hear the rather explicit answer of the philosopher Mark Rowlands (University of Miami), interviewed by Dan Schneider, on the following question:

DS: What is the principle difference between ontology and epistemology, without simply giving a dictionary definition?

MR: Ontology is concerned with what exists. Epistemology is concerned with what we can know. The history of human thought is riddled with determined attempts to ignore the difference between the two. I mentioned this earlier in connection with the idea of materialism. Lets take an example. The Copenhagen interpretation of quantum theory uncritically slides from ’we cannot know, simultaneously, the precise velocity and location of a lepton’ to ’a lepton has no precise simultaneous location and velocity.’ The founding principle of modern physics seems to be: if you cant in principle know something, then it doesnt exist. And thats called ’verificationism’. Another word for it might be ’crap’. It is a view that has been thoroughly discredited in philosophy for nearly a century. Scientists still cling to it like its gospel. Here’s another claim you see all the time: we cant know anything about what happened before the universe began. Therefore, there was nothing before the universe began. Utter crap.

Summarizing, the philosophical construction provided by the CI allows effectively to ’neglect’ genuine probabilistic predictions - i.e. those with probabilities different from $1$ . If one disagrees with this radical denial of reality by the CI then one has to accept that QT is ’incomplete’. It turns out that several different meanings of the term ’(in)completeness’ exist in QT⁶⁰ . In order to distinguish these different meanings we denote the present one - concerning the prediction of the value of a single observable - by ’single-event (in)completeness’.

Einstein simply didn't accept this, and at one place he said, "If we have the statistical interpretation, then this whole egg-walking about existence and non-existence is completely superfluous." He used the German word "Eiertanz".
Interview with Dr. Alfred Lande, By Thomas S. Kuhn and John Heilbron, In Berkeley, California, March 8, 1962 .

It is well known that the controversial discussions between Bohr and Einstein played a crucial role for the further development. Einstein did not agree with the CI’s denial of reality and considered QT as a - using the present nomenclature - single-event incomplete theory (for him QT was also incomplete in every other sense of the word). As is also well-known, Bohr and Heisenberg succeeded to convince the majority of physicists and the CI became the standard interpretation of QT. During the following years, in particular after publication of the famous EPR paper, the question of single-event (in)completeness was pushed into the background by other questions involving a different meaning of completeness. We shall come back to this point in subsection 5.3.

5.2 An immediate consequence: The ’measurement problem’

"Observations not only disturb what has to be measured, they produce it... We compel [the electron] to assume a definite position... We ourselves produce the results of measurements."
Pascual Jordan, german physicist (1902-1980)

The first question one encounters in the CI is the following: How do properties come back to reality during the process of measurement ? After all we see spots at screens, tracks in cloud chambers, verify scattering predictions by means of specific values of momentum, etc. How do all these properties which are, according to the CI, nonexistent immediately before the interaction between the considered system and the measurement device, ’come into being’ during the measurement process. Much effort has been devoted to this question, which has been called the ’measurement problem’.

It is obviously impossible to solve this ’problem’ by describing the considered (single event) system alone. QT yields only probabilities but the measured values for a (single) microscopic system disagree, in general, with these numbers. What can be predicted by QT is the set of possible outcomes (for a particular observable), which is the set of all eigenvalues (of the corresponding operator), and the corresponding set of probabilities; the latter depends on the preparation procedure. Therefore to solve the measurement problem the measuring device must also be described quantum mechanically⁶¹ . The first version of a measurement theory is due to von Neumann but this theory has since then been replaced several times; a relatively new creation is the theory of ’unsharp observables’. I do not want to discuss the various theories which have been proposed in the past to solve the measurement problem, although some of these are really strange and even potentially amusing intellectual constructs [149].

A second question, closely related to the measurement problem, concerns the ’collapse’, or ’reduction’ (the corresponding rule is called ’projection postulate ) of the wave function during measurement: What happens exactly when the ’collapse of the wave function’ takes place? The measurement problem and the related collapse problem are unsolved since its creation eighty years ago. For a more detailed discussion and refutation see e.g. Margenau [96], Ballentine [11], [13], and Blood (arXiv:0808.3699).

"One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."
E. Schroedinger, "Die gegenwaertige Situation in der Quantenmechanik" (1935). .

A vivid description of the absurd consequences of the basic premise of the CI, in conjunction with von Neumann’s measurement theory, has been mentioned by Schrödinger. In the quoted text [138] about a cat, frequently referred to as ”cat paradox”, he wanted to show that the basic assumption of the CI that a single object can be described by a linear combination of functions, each one belonging to a different numerical value of an observable, must be wrong. This is the very heart of the absurdity; using a living object (cat) as measuring apparatus and the restriction to two values (dead or alive) are ’dramaturgic effects’ intended to increase the interest/amusement of the reader (and author) of this dramatic novel. In order to show that this assumption, and the related idea of a ”reduction of the wave function”, is wrong, Schrödinger used the logical method ”reductio ad absurdum” ⁶². The relation between the ’state’ of the atom and the ’state’ of the cat has been clearly explained (from a point of view similar to the SI) by Medina [102] (quant-ph/0508014).

5.3 Incompatible observables require more metaphysics

"Niels Bohr brainwashed a whole generation of theorists into thinking that the job (interpreting quantum theory) was done 50 years ago. "
Murray Gell-Mann (1929-)

In subsection 5.1 the ’solution’ of the CI of the contradiction between the probabilistic predictions of QT and the common-sense meaning of completeness has been discussed⁶³ . If one accepts the CI then one may restrict now further consideration to the eigenstates $|q ⟩$ of the operator $Qˆ$ , whose values are to be measured, since properties not predictable (with certainty) are not ’possessed’ by the considered particle. But then, a second problem arises if a second non-commuting observable $ˆ P$ - describing a different property of the considered particle - is to be measured⁶⁴ . The question is: given that we found $q$ in a measurement of $Qˆ$ , which result is to be expected if a measurement of $Pˆ$ is performed immediately afterwards ?

It should be pointed out that this question does only arise if QT is believed to be a theory about single particles, i. e. in the framework of the CI. A well-defined question of this form does not exist in the SI, which is a theory about statistical ensembles. In the SI the ’simultaneous measurement’ of more than one observable is an ill-defined concept. To see this recall that a ’measurement’ in the framework of a probabilistic theory - like the QT as interpreted in the sense of the SI - requires a well-defined ensemble of identically prepared individual systems. A completed measurement on a single system does not, of course, provide such a (new) ensemble. It may play a role as part of a (new) preparation procedure, but then it is not really a measurement but rather a small contribution to the total preparation procedure, which requires $N$ repetitions of the single-event measurement. A completed (individual) measurement on a single system is a (tiny, $1∕N$ ) contribution to a statistical measurement, which requires also $N$ individual measurements for its completion.

"...the subject of probability theory is long sequences of experiments or observations repeated very often and under a set of invariable conditions."
Richard Von Mises (1883-1953)
Mathematical Theory of Probability, New York, Academic Press, 1964 .

The mathematical structure of QT clearly reflects this limitation of a purely probabilistic theory: there is no theoretical element describing consecutive measurements of different observables, no matter whether commuting or not. Given the state vector $|ψ (t ) ⟩$ the probability to measure a value $a$ of the observable $Aˆ$ is given by $|⟨ ψ (t )|a ⟩|2$ , where $|a ⟩$ is the eigenstate of $Aˆ$ belonging to the eigenvalue $a$ ⁶⁵. After completion of a single particle measurement the measurement of a quantum-theoretical prediction (a probability) has not been completed. In order to complete it, the single particle measurement has, in general, to be repeated $N$ times, using $N$ other members of the same ensemble⁶⁶ . Thus, immediately after the (single-event) measurement of an observable $ˆ A$ , a state vector does not exist and a (single-event) measurement of a second observable is impossible. The concept of simultaneous (single-event) measurements of several observables has no counterpart in the structure of QT.

Let us consider a concrete example to illustrate this point. Assume that a first (single-particle) measurement revealed a value $p$ for the particle’s momentum. As a consequence we may say that the particle possesses this value of momentum immediately after the measurement. Is this the same as to say that the particle is in a state $|p ⟩$ ? Today there is general agreement (not depending on the interpretation) that the wave function before a measurement describes an ensemble - in the CI this is a multitude of possible states, whose number is then dramatically reduced to $1$ by the measurement process, which brings a single state ’into being’. In special cases this multitude of possible states may, of course, be given by a single state of (sharp) $p$ . This then means that all particles in an ensemble have sharp $p$ (the multitude of possible states in the CI degenerates to the single state $|p ⟩$ ) . Can we interpret our hypothetical state $|p ⟩$ - which we tentatively ascribe to our electron with momentum $p$ - in this way ? In order for this idea to work, each repetition of the first measurement must yield the same result $p$ . But the repetition of the first measurement will in general yield a result different from $p$ because the state before the first measurement was not an eigenstate of $ˆ P$ ⁶⁷. Thus, the answer to the last question is no. There seems to be a possibility left, namely that we filter the results of the first measurement in such a way that only particles with the same momentum $p$ survive at the time of the second measurement. But this transforms the first measurement into a preparation procedure (or part of it) and gives again no support to the idea that our particle with momentum is in the state $|p ⟩$ .

"...in mathematics you don't understand things, you just get used to them."
John von Neumann (1903-1957) .

Summarizing, the three concepts measurement, preparation and state vector reduction are not compatible with each other. Until the work of Margenau [95], nobody was aware of the distinction between measurement and preparation, not even physicists like Einstein and Wigner. The foundations of the CI were built without taking this distinction into account. The brilliant mathematician von Neumann should be mentioned too; it seems that the idea of mathematical simplicity played a major role in his theory of measurement [157]⁶⁸ .

This situation poses a serious problem for the CI, which claims to be a (complete) theory for single particles. To solve it a new element must be added to QT: the reduction (or collapse) of the wave function, which has been already mentioned briefly in subsection 5.2 and has also been used (refuted) implicitly in the last two paragraphs. According to this idea, which is due to von Neumann, the measurement leads to a sudden change $|ψ (t )⟩ → |a ⟩$ of the state vector during the measurement process; the state vector ’collapses’ into the eigenstate⁶⁹ $|a ⟩$ of $ˆ A$ which belongs to the measured value $a$ . This is an indeterministic process since the actual value of $a$ - and the resulting state vector $|a ⟩$ - cannot be predicted. The beautiful term ’non-unitary time evolution’ has been invented to denote this process, which cannot be described by any known mathematical methods. A more detailed analysis of the projection postulate leads to the conclusion that it is ”either redundant or wrong” [13].

"The discontinuous 'reduction of the wave packets' which cannot be derived from Schroedinger's equation is ... a consequence of the transition from the possible to the actual".
Werner Heisenberg (1901-1976).

"But it is certainly not possible to insist on one hand that the formalism is complete and to insist on the other hand that its application to 'the actual' actually demands a step which cannot be derived from it."
Karl Popper (1902-1994) .

Given this new assumption of the CI, which is called ’projection postulate’, a consecutive (simultaneous) measurement of a second observable $ˆ P$ , after measurement of the first observable $ˆ Q$ , could make sense in the formalism of QT because now a state vector exists (by definition) after completion of the first measurement (the above discussed problems and contradictions associated with this assumption are simply neglected in the CI - or otherwise eliminated by means of sophisticated philosophical reasoning). Two cases must be distinguished:

Only the second of these points poses an challenging problem and will be discussed in the rest of this subsection. The SI and the CI arrive essentially at the same conclusion, as far as predictability of several observables is concerned, namely that it is impossible. For the SI this does not present any problem, since it does not claim that QT describes individual particles; not for single observables and all the less for several observables. On the other hand it presents a (further) serious problem for the CI because of its apodictic statement that QT provides a complete prediction of single events, no matter how many properties are considered.

"Nothing is real."
John Lennon (1949-1980) Beatles Lyrics - "Strawberry Fields Forever" .

Recall how the problem of completeness with regard to a single observable was handled by the CI (see subsection 5.1). The probabilistic nature of the predictions of QT was reconciled with the claim for completeness by denying objectivity or reality of the measured properties (or of the objects itself) prior to the measurement. The latter are believed to have no own reality but only ’come into being’ by the act of measurement. If one accepts this explanation, it leads to the present situation, which is in a sense even worse, since now - for a measurement of $ˆ A$ in an eigenstate of a non-commuting observable $ˆ B$ - not even probabilities for the possible measurement results $a$ are provided by QT. Therefore the question of completeness of QT in this situation (a question which exists only for the CI) is in a sense more serious than the previous single-event completeness problem. Let us denote the present kind of (in)completeness by ’simultaneity (in)completeness’.

But from the point of view of the CI this new problem is handled in a similar manner as before, by ’commanding reality’: If $ˆ B$ is ’sharp’, the reality of the property corresponding to the observable $ˆ A$ is denied, and vice versa. In short, properties corresponding to non-commuting observables have no simultaneous reality. This is the answer of the CI to this new (simultaneity) completeness problem. We shall come back to this problem in subsection 5.5.

Prediction is very difficult, especially about the future.
Niels Bohr (1885-1962)

Very often, the considered state is not an eigenstate of $Bˆ$ but a wave packet of finite width $△B$ . Then an uncertainty relation similar to Eq. (4) for $x$ and $p$ exists as a statistical relation for the measurement results of $ˆ B$ and a second non-commuting observable $ˆ A$ . As discussed in subsection 2.1, relation (4) has nothing to do with the simultaneous measurement of $xˆ$ and $pˆ$ on individual particles (we shall identify $ˆ A$ and $ˆ B$ with this important pair of operators in what follows). But it is often used to ’prove’ the validity of the abstract uncertainty principle (1) which is supposed to describe the simultaneous measurement of $ˆx$ and $ˆp$ on single particles.

If misinterpreted in this way the IUP (1) can be used to support the CI’s second suppression of reality explained above. The well-known line of argument goes as follows. Consider a wave packet which has a width $△x$ in ordinary space. According to (1) the width $△p$ of this wave packet in momentum space is proportional to $1 ∕ △x$ . The important point (crucial error of the CI) is that these widths are - according to the IUP - identified with uncertainties in simultaneous measurements of $x$ and $p$ . As a consequence, the higher the accuracy for $x$ the lower that of $p$ and vice versa. In the limiting case considered above one of these quantities is sharp and then the other is completely undetermined. In this way the IUP is considered as confirming the CI’s claim that non-commuting observables have no simultaneous existence⁷³ . This variant of the unreality requirement saves for the CI what we have called simultaneity completeness of QT⁷⁴ .

"No ..It [...the principle of complementarity] is a very fascinating idea, but according to the statistical interpretation, it was an absolutely illogical idea. How can a particle be complementary to one of its own properties, its statistical tendency of distribution in space."
Alfred Lande (1888-1976)

Relation (1) is also used to justify the idea that a particle can be sometimes a particle and sometimes a wave (naive wave-particle, or wave-particle dualism of a single object; see beginning of section 3) as well as Bohr’s concept of complementarity. The latter means that it depends on the measurement process which role (wave or particle) of the particle becomes the preferred one.

In section 2 it has been shown that the IUP, relation (1), is a philosophical postulate, which cannot be justified - neither in the framework of the mathematical formalism of QT nor by experiment. In section 3 it has been shown that a single particle never behaves like a wave; wave-like behavior is always a consequence of the joint action of many (correlated) single particles. The ideas of particle-wave dualism and of complementarity are important elements of the CI. Both vanish into thin air on closer inspection because both are based on the validity of the IUP. Likewise, the IUP cannot be used to confirm the CI’s claim of simultaneous ’nonexistence’ of non-commuting observables on simultaneous measurements.

There is a point - an apparent contradiction on first sight in the above reasoning - still to be clarified: On the one hand I pointed out that the concept of simultaneous measurements is not well-defined in the SI. On the other hand I criticized the CI for claiming that certain (non-commuting) observables cannot be simultaneously measured - or more precisely its conclusion that the corresponding properties are not simultaneously ’possessed’. Is not what I criticize in the CI essentially identical with what I agree with in the SI ? In other words: are not the violations of the IUP discussed in section 2 in contradiction with the SI ?

"In the light of quantum theory these elementary particles are no longer real in the same sense as objects of daily life, trees or stones, but appear as abstractions derived from the real material of observation in the true sense."
Werner Heisenberg (1901-1976)

The answer to the last question is no. From the point of the SI the simultaneous existence of properties corresponding to non-commuting observables is possible, in special situations. Such situations cannot, however, be handled in a standard way by applying the formalism of QT but require additional considerations - typically conservation laws for momentum or energy. While a simultaneous measurement of position and momentum at one and the same time $t$ seems indeed impossible, it seems nevertheless possible to ascribe well-defined values of position and momentum to a particle at a particular time $t$ (and to show in this way a violation of the IUP). The measurements must be done consecutively. If, say, momentum $p$ is determined in a first measurement followed by a position measurement a short time later, then there is no reason to assume that the value of $p$ found in the first measurement may not be ’possessed’ by the particle a short time later, at the time $t$ of the position measurement⁷⁵ . Several violations of the IUP of this kind have been reported in the literature (see subsection 2.1) . These are fully compatible with the SI’s claim that simultaneous measurements of non-commuting observables are not part of QT and invalidate - or make at least superfluous - the complicated non-existence postulate of the CI. An assumption which is of course always inherent in these examples is the reality of the particles (or its properties) and it is mainly this assumption which is criticized by supporters of the CI; the essential source of disagreement between different interpretations being a consequence of incompatible (meta)physical postulates.

5.4 Even more metaphysics and several meanings of completeness

"It is a widely known secret that quantum theory is incomplete: there are facts which can be expressed by propositions formulable in the language of the theory but for which the theory provides no truth conditions. Thus these facts can be neither predicted nor explained by the theory"
Arthur Fine (1963-) "On the Completeness of Quantum Theory", Synthese, vol.27 (1974) p.257 .

In order to describe the further development of QT, and the role of the CI, the metaphysical framework should be widened and alternative metaphysical postulates (in particular those ’keeping reality intact’) should be investigated. The only alternative to the CI considered so far, was the SI. It may be useful, before introducing alternative metaphysical postulates, to characterize the SI, or possible versions of it, more precisely.

In the simplest conceivable version of the SI quantum theory is interpreted as a theory of statistical ensembles and all statements are avoided that cannot be tested experimentally. This means that essentially only the mathematical structure of QT and the experimentally confirmed nature of its predictions, namely probabilities⁷⁶ enter this simplest version of the SI. Properties of single particles (more generally single events) can be measured but cannot be predicted. Also, no statement can be made on the existence or reality of single particles as long they are not observed. The question whether or not a particle exists if it is not observed is irrelevant because such a question can, of course (by its very definition), not be verified by observation; it does not belong to physics but to philosophy. On the other hand, we know that single particles exist, at least, during the interaction process (measurement) which allows to extract numerical values which can be assigned a certain physical meaning in the framework of QT. This assignment of properties to particles at least during a small time interval is all that is needed to compare the predictions of QT with experimental data. It is the acceptance of incompleteness of QT (with regard to single events) which releases this version of the SI - in contrast to the CI - from all metaphysical considerations concerning unobserved single events.

"In this century the professional philosophers have let the physicists get away with murder. It is a safe bet that no other group of scientists could have passed off and gained acceptance for such an extraordinary principle as complementarity, nor succeeded in elevating indeterminacy to a universal law."
James R. Newman (1907-1966), in Scientific American, January 1958 .

Besides the point just discussed, this simplest version of the SI has nothing to say about single particles⁷⁷ . According to our decision (see section 1) to use Occam’s razor as a scientific guideline, we accept this simplest version of the SI as the best one - simply because it does not make use of any metaphysical assumptions. The ’statistical interpretation, simplest version’ will be denoted by SIS for later reference.

According to the SIS quantum theory is an incomplete theory (both single-event incomplete and simultaneity incomplete) with regard to single events because it is no theory about single events. This interpretation will be preferred by those who believe that a physical theory should only make statements which can be tested experimentally. Note also that it is open for later ’extensions’ or ’refinements’ by means of additional postulates. Unfortunately, it does not satisfy our deep desire for final answers - more on that in section 6. However, the fact that philosophical questions are left for the professionals, could also be seen as an advantage of this interpretation.

The tendency has always been strong to believe that whatever received a name must be an entity or being, having an independent existence of its own. And if no real entity answering to the name could be found, men did not for that reason suppose that none existed, but imagined that it was something peculiarly abstruse and mysterious.
John Stuart Mill (1806-1873), British philosopher

Let us now look for other interpretations of QT, which keep reality (in contrast to the CI) intact but may contain (in contrast to the SIS) metaphysical elements - i.e. statements which cannot be tested experimentally. There are (at least) two possibilities of this kind:

The present classification scheme shows a certain similarity to the one used by Ballentine (see p. 81 of [9]) in a slightly different context: His (A), (C) correspond roughly to CI,D; his (B) corresponds to the union of SIS and I.

"Is the world ruled by strict laws or not? This question I regard as metaphysical. The laws we find are always hypotheses; which means that they may always be superseded, and that they may possibly be deduced from probability estimates. Yet denying causality would be the same as attempting to persuade the theorist give up his search; and that such an attempt cannot be backed by anything like a proof..."
Karl Popper (1902-1994)

Let us compare the interpretations CI, SIS, D, I. Let us mention first some obvious differences. The fundamental difference of SIS, D, I as compared to the CI is the reality of properties (and particles) no matter whether observed or not. The significant feature of SIS, as compared to CI,D,I is the complete absence of metaphysical postulates. Indeterminism is a fundamental and irreducible feature of reality in I, in pronounced contrast to D⁸¹ . In order to proceed to a comparison of ’finer details’ of the above four interpretations, we should first analyze the three meanings of the term ’(in)completeness’ encountered so far.

The first two versions of completeness, introduced in sections 5.1 and 5.3, have been called ’single event completeness’ and ’simultaneity completeness’ respectively⁸² . These two terms have a relatively clear meaning as attributes of a physical theory - provided real properties and particles are considered: They state that experimental results may be predicted successfully - the predictions concerning either a single observable or two observables belonging to a single event at a given instant of time. Thus, these two term may be subsumed under the common heading ’predictive’.

The assertion that a theory is predictive (in)complete is a physical assertion insofar as a theory and experimental results are required to test its truth.

"For the idea instructs us only in regard to a certain unattainable completeness, and so serves rather to limit the understanding than to extend it to new objects."
Immanuel Kant (1724-1804), Critique of Pure Reason .

On the other hand, the term ’ontological completeness’ as defined in I means: ’nothing better exists’. A ontologically complete theory will, in its domain of validity, never be replaced by a better theory and does not even require any modification or extension; it might also be called a ’final theory’ (for its domain of validity). Ontological completeness is obviously fundamentally different from predictive completeness⁸³ . It is a ’metaphysical’ term, insofar as it cannot be verified by experiment. In contrast to ’predictive completeness’ ontological completeness is a fictitious concept: It characterizes QT in a indirect way - by the non-existence of possible ’better’ physical theories. Since an infinite number of possibilities exists, such an assertion cannot be verified definitively⁸⁴ .

Note that, for the present (metaphysical) discussion, the difference between predictive completeness and ontological completeness is not a semantic subtlety since it addresses different metaphysical postulates [namely (ir)reality and (in)determinism]. Thus, the difference is important, despite the fact that it seems to go widely unnoticed; see however Fine [50], Ballentine [15]⁸⁵ , and, in particular, Held [63, 64]).

The fact that D,I are ’realistic’ interpretations implies immediately that they are predictive incomplete; if reality is kept intact the fact that only probabilistic predictions can be made is equivalent to predictive incompleteness. Thus, both D and I are statistical interpretations which cannot be used to describe the behavior of individual events. To stress this aspect, which seems to be the most fundamental of all from a physical (but not philosophical) point of view, we replace the abbreviations D and I by SID (statistical interpretation, deterministic version) and SII (statistical interpretation, indeterministic version).

Using this notation we may now list the values the interpretations CI, SIS, SID, SII take with regard to the two different notions of completeness. Interpretations which claim both predictive (in)completeness and ontological (in)completeness will be denoted ’(in)complete’ for brevity.

Note that only three ”metaphysical” interpretations are allowed; predictive completeness combined with ontological incompleteness does not make sense. Note that the SII (the indeterministic variant of the two metaphysical versions of the SI) is not totally different from the CI, since both declare QT to be ontologically complete. A point, which should be mentioned even if ’trivially’ true, is that a large number of published statements like ”quantum mechanics is (in)complete” are - in absence of more detailed explanations - ambiguous.

"..[the wave function] is correlated one-to-one with the real state of the real system.. If this works, then I speak of a complete description of reality by the theory"
Albert Einstein, Correspondence with Schroedinger

Let us close this subsection with a few historical remarks. During the first big confrontation between Einstein and Bohr (an important part of which took place at the Solvay conferences during the period from 1911 to 1930) ’predictive incompleteness’ was the main point of (concrete) criticism raised by Einstein against QT. It remained his main point of criticism for the rest of his life. He believed in a deterministic world and was, therefore a pronounced advocate of the SID. He did not like QT and thought it would be replaced sooner or later by a deterministic theory. But note that two quite different kinds of attacks against QT are involved here⁸⁶ : His concrete criticism of predictive incompleteness of the real existing QT may be true while his metaphysical belief in a deterministic replacement of QT may be wrong at the same time⁸⁷ . But this subtle difference - as well as the corresponding difference between predictive and ontological completeness - was not taken into account and mixed up in the discussion.

"Bohr's shift of the problem from completeness to soundness (=freedom from contradiction)."
Karl Popper, Quantum Theory and the Schism in Physics, p.39

On top of that QT was applied to a lot of problems during that period of time and it turned out to be very successful⁸⁸ . Consequently, Einstein became increasingly convinced that QT was correct and began to think about a deterministic theory which had to agree, on the average, with the statistical predictions of QT. But in this situation his opponents could not resist to gain advantage from the following simple rhetoric trick: misinterpret Einsteins claim for (predictive) incompleteness of QT as a claim for incorrectness of QT. This strategy was successful and was one of the reasons (among several others) for the proclaimed ’victory’ of the CI in this first battle between Einstein and his opponents.

5.5 The EPR paper - the collapse of the Copenhagen interpretation

"This onslaught came down upon us as a bolt from the blue. Its effect on Bohr was remarkable. ...everything else was abandoned...so it went for a while with growing wonder for the unexpected subtlety of the argument..."
Leon Rosenfeld (1904 - 1974), collaborator of Niels Bohr

In 1935 Einstein, Podolsky, and Rosen (EPR) published a paper [46], entitled ”Can Quantum-Mechanical Description of Physical Reality Be Considered Complete” which had an enormous influence on further activities in this field. The EPR paper consists of two parts. The first part starts with two general definitions and continues with a concrete example, which leads to a general proposition characterizing the Copenhagen interpretation of QT. In the second part a particular situation involving two (entangled) particles is studied. Combining the conclusions of both parts leads to the final conclusion of incompleteness of QT. This ’gedankenexperiment’ is described and discussed, from very different points of view, in a large number of publications, see e.g. Fine [51], Ballentine [8], Redhead [130]. We will first describe the paper and then discuss its meaning, in particular the question: ”which kind of completeness is referred to?”

”every element of the physical reality must have a counterpart in the physical theory.”

This is a necessary condition which states that a physical theory is incomplete if for at least one ’element of the physical reality’ a theoretical counterpart does not exist in this theory.

"..what EPR intends to challenge is not quantum theory itself, but rather a particular version of the Copenhagen interpretation."
Arthur Fine
The Shaky Game - Einstein Realism and the Quantum Theory, p.4

The completeness condition requires explanation how to identify an ’element of physical reality’. EPR’s definition of the condition for the existence of an element of physical reality is:

”If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”

(Note that this is not a definition of (an ’element of’) ’reality’ but just a sufficient condition for an ’element of reality’ to exist.) Taken literally this definition is incomprehensible or at least ambiguous. A prediction alone is a mental act and cannot disturb a system (excluding paranormal phenomena). What means ’we can predict with certainty’ ? One would expect reference to the act of experimental verification which seems necessary in order to justify the term ’with certainty’. But this definition does not contain any reference to experimental observation or verification. In fact, this definition defines reality in terms of a theory alone without reference to observation. We may say that it refers to the internal (predicative) structure of a physical theory. But this is exactly the way the term ’reality’ is used (defined) in the CI ! The question whether or not EPR’s definition of reality is comprehensible is (no matter how abstrusely this may sound on first sight) not of primary importance. The important thing is that it reflects the thinking of the supporters of the CI⁸⁹ . In this way the assumptions underlying the EPR paper could not be attacked (by the CI’s). In short, the EPR paper constructs a contradiction within the CI.

This idea helps clarifying a number of unclear points and represents a (my) key to the understanding of the EPR paper. It must, however, be pointed out that Einstein was probably not (fully) aware of the incompatibility of some of the assumptions in the EPR paper with his own statistical interpretation of the quantum mechanical formalism. The reason is that many facts which are known today were unknown at that time. We know today that the projection postulate is incompatible with the statistical interpretation. We (or many of us) know today that a measurement is not the same as a preparation and that an erroneous mixing of these two concepts provides a strong argument in favor of the CI [see the discussion in section 2.1]. At the time of writing of the EPR paper all this was unknown⁹⁰ . For example, the difference between measurement and preparation was recognized by Margenau just in the course of his correspondence with Einstein (as described by Jammer [73]). This fact presents a further complication and potential source of misunderstandings. Nevertheless we know today that all the assumptions concerning questions of interpretation in the EPR paper belong to the CI and differ from Einstein’s own (statistical) interpretation of QT.

"The laws of quantum physics are of a statistical character. This means: they concern not one single system but an aggregation of identical systems; they cannot be verified by measurement of one individual, but only by a series of repeated measurements"
A. Einstein and L. Infeld, The Evolution of Physics, Cambridge University Press. 1938, p.299

a statement diametrically opposed to Einstein’s own, well-documented position. The fact, that any specific discussion of EPR’s reasoning has been deliberately avoided by the CI supporters (see e.g. Bohr’s reply) can also be seen as a confirmation of this idea. A terrible side-effect of the extended EPR discussion is, however, that QT is identified today by a majority of scientists with ’QT as interpreted by CI’.

EPR study the problem of completeness in a situation of two non-commuting observables, using position and momentum as concrete examples. This kind of (in)completeness - involving two non-commuting observables $Aˆ$ , $Bˆ$ - has been referred to as ’simultaneity (in)completeness’ in the discussion of subsection 5.3. As discussed there, a necessary condition for completeness (according to the CI) in such a situation is:

This is just a repetition of the CI’s claim , as discussed in section 5.3, endowed with the more elaborate definition (of ’element of reality’) of the EPR. If the assertion that QT is complete is denoted by $Q$ , the situation may be briefly described by the logical implication $Q ⇒ P$ . The latter is equivalent to the implication $~ P ⇒ ~ Q$ . In other words

SIMULTANEOUS REALITY $⇒$ INCOMPLETENESS

$~$ SIMULTANEOUS REALITY $∨$ INCOMPLETENESS

which concludes the first part of the EPR paper. It means that it is impossible that both of the propositions ”QT is complete” and ” $A$ and $B$ are simultaneously real” are true; at least one of these propositions (or both) must be wrong. In particular, if it can be shown that ” $A$ and $B$ are simultaneously real” it follows that ”QT” (we should put it between quotation marks, because it’s all the time not QT but ’QT as interpreted by the CI’) is incomplete. The above conclusion agrees exactly with the conclusion obtained in section 5.3 using a more intuitive reasoning. This intermediate result of the EPR paper is in perfect agreement with common sense⁹² . The fundamental ’battle’ in the interpretation of QT takes place between the notions of ’completeness’ and ’reality’; there seems to be no other way to make a probabilistic theory ’complete’ than by denying reality.

"It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we say about Nature."
Niels Bohr (1885-1962), Danish Physicist

Let us finally state the last assertion more explicitly, as formulated by EPR near the end of the first part of their paper:

From this follows that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality.

In the second part of the EPR paper, the authors show that the above definitions and postulates imply that (2) is wrong. The final conclusion is that (1) is true, i.e. quantum mechanics is incomplete, given that all of the above definitions and postulates (as well as others introduced in the second part) are true.

To show that (2) is wrong, a two-particle system $S$ consisting of two (entangled) single-particle subsystems $S1$ and $S2$ is studied in a ”gedankenexperiment”. Initially, the two particles interact and are then separated a large distance from each other; much too large to exert any influence on each other (in the considered example this happens in such a way that relative position and total momentum are conserved. The latter are commuting observables of the system $S$ and can be measured simultaneously with arbitrary accuracy).

The time-development of the wave function $ψ (x1, x2 )$ of the total system $S$ can be determined with the help of Schrödinger’s equation. It may be expanded in various ways in terms of eigenfunctions of observables $A, B, ..$ belonging to subsystem $S1$ ,

∑∞ ∑ ∞ A A B B ψ (x1, x2 ) = ψ n (x2 ) ϕ n (x1 ) = ψ n (x2 ) ϕ n (x1 ) = .... n=1 n=1

Here, $A ϕ n$ , $B ϕ n , ....$ are eigenfunctions of $A, B, ..$ and $A ψ n$ , $B ψ n , ....$ are corresponding expansion coefficients. Schrödinger’s equation does not (directly) allow to make predictions about the single particles $S1$ and $S2$ . To make progress in that direction, EPR adopt, as a further assumption of the CI, the projection postulate. If valid for subsystems, it says that a measurement on $S1$ of a quantity $A$ with result, say, $ak$ leads to a wave-function collapse or reduction, which may be written as

ψ (x , x ) ⇒ ψA (x ) ϕA (x ). 1 2 k 2 k 1

Only a single term survives the collapse. Alternatively, a measurement on $S1$ of a quantity $B$ with result $bj$ leads to a wave-function collapse of the form

B B ψ (x1, x2 ) = ⇒ ψ (x2 )ϕ (x1 ). j j

If the r.h.s of these equations are really believed to represent the quantum mechanical state of the System $S$ , then $A ψ k (x2 )$ and $B ψ j (x2 )$ are to be interpreted as wave functions of the system $S2$ after the measurement. Thus, two different wave-functions may be assigned to the system $S2$ as a consequence of different measurements on the distant (imagine a astronomical distance) system $S1$

"Reality is that which, when you stop believing in it, doesn't go away."
P. K. Dick (1928-1982), science-fiction author

Given a conservation law for $A$ for the total system it may be possible to ’predict with certainty’ the value of the observable $A$ for system $S2$ by means of a measurement of $A$ on $S1$ . The same may be true for a second observable $B$ : Given a conservation law for $B$ for the total system it may be possible to ’predict with certainty’ the value of the observable $B$ for system $S2$ by means of a measurement of $B$ on $S 1$ . Then two ’elements of reality’ for system $S 1$ exist. If $A$ and $B$ are two non-commuting observables, considered as properties of the individual subsystems, then (2), the second of the above EPR alternatives, is wrong because two elements of reality corresponding to non-commuting observables exist in this situation.

EPR then give a rather special example of a wave function where this situation is realized, with the non-commuting observables given by momentum $p$ and position $x$ (later, a different version, where non-commuting spin components take the role of $p$ and $x$ has been proposed by Bohm and Aharonov). The position $x2$ of the second particle can by definition be considered as an element of reality because it can be predicted with certainty without disturbing particle 2 (by means of a measurement of the position $x1$ of particle 1). The same is true for the momentum $p2$ (it can be determined by means of a measurement of the momentum $p 1$ ). Due to the strange (Copenhagen) definition of ’element of reality’ which has been adopted by EPR (in order to construct a contradiction within the CI), it is not required (and even forbidden in order for the EPR construction to work) that the two measurements on particle 1 are actually performed. But we have now (according to the CI) two elements of reality corresponding to non-commuting observables. This means (2) is wrong and (1) must be true, i.e. QT (as interpreted by the CI) cannot be complete. Summarizing, the assumptions of the CI are incompatible with the CI’s claim that ”QT” provides a complete description of reality. The CI is internally inconsistent.

"However I cannot seriously believe in it because the theory is incompatible with the principle that physics is to represent reality in space and time, without spookish long-distance effects."
Albert Einstein (1879-1955)

This conclusion can only be avoided if a kind of mysterious, ’nonlocal’ influence between measurements taking place at arbitrary large distances is postulated. Of course, even simple and clear facts can be eliminated by invoking sufficiently powerful black magic. In the EPR paper this ’spooky action at a distance’ was excluded. Later, Einstein included this alternative for completeness.

Which kind of ’(in)completeness’ is referred to in the EPR paper ? The whole paper deals with the question whether or not values of observables for single particles can be predicted by QT. Thus, the kind of ’(in)completeness’ referred to is obviously ’predictive (in)completeness’⁹³ . What is the final conclusion to be drawn from the EPR Gedankenexperiment? Recall the behavior of the various interpretations with regard to the two fundamentally different notions of ’completeness’ (predictive and ontological) discussed in subsection 5.4

The lesson to be learned from EPR is that the CI is not acceptable because it is the only interpretation (at least in the present classification) that claims that QT is ’predictive complete’. The EPR gedankenexperiment says nothing about the other three interpretations SIS, SID, SII. However, let us quote the last paragraph of the EPR paper:

”While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.”

which is remarkable in two respects. First, it expresses the well-known believe of Einstein and his co-workers in a fundamentally deterministic world; using the present nomenclature we may say the the authors preferred the SID.

Secondly, it shows that EPR were well aware of the two fundamentally (completely) different meanings of the word ’complete’, even if they did not invent an own notation to distinguish these. What they say in this paragraph is: ”We showed that QT is predictive incomplete but did not treat the question of ontological incompleteness (we did not show that necessarily a better theory exists). However we believe that QT is ontologically incomplete.” At the time of publication of the EPR paper it had been known for years what the authors believed. The important new result is not what they believed but what they proved, namely predictive incompleteness of QT. Of course, predictive completeness implies ontological completeness. But predictive incompleteness does not imply ontological incompleteness; an indeterministic world is perfectly compatible with the conclusions of EPR. But these two different notions of completeness - or more precisely the failure to distinguish these two different notions of completeness - offered a way for the CI to survive the EPR attack.

5.6 EPR, Bell, ”completeness”, and how the collapse became a paradox

"By denying scientific principles, one may maintain any paradox."
Galileo Galilei (1564-1642)

The EPR paper has not changed the predominance of the CI. There is an enormous secondary literature and also an enormous diversity of opinions, see e.g. [8, 161]. A reason, among others, for this enormous confusion seems to be that EPR nowhere explicitly point out that the most important of their postulates disagree with their own positions. Strictly speaking, the difference should not matter, since the conclusions of EPR take the form of logical implications, but actually it does. This misunderstanding is related to the almost uniform (erroneous) identification of QT with ’QT as interpreted by the CI’. One finds, of course, several comments in the literature [8, 51, 64] pointing out that EPR attack (in the ’concrete’ part of their paper, that means in the whole text except the last paragraph), not QT itself but only a particular interpretation of QT. But the central importance of this fact has never been recognized and was completely ignored by a majority of physicists.

"All things are subject to interpretation whichever interpretation prevails at a given time is a function of power and not truth."
Friedrich Nietzsche, German philosopher (1844-1900)

How could it happen that the EPR paper has not changed the predominance of the CI despite the fact that nobody has ever found any concrete shortcomings in the logical or physical reasoning of the authors?⁹⁴ . This is an interesting question but it cannot be answered within the realm of strictly scientific (logical, mathematical, physical) reasoning, so we exclude this question and restrict ourselves essentially to a description of what happened.

The incompleteness problem raised by EPR was not taken seriously and simply denied. Instead, their conclusions were interpreted in the sense that something really strange, like ’spooky action at a distance’ (breakdown of ’local realism’) had been discovered. Thus, the conclusions of the EPR paper were reinterpreted as revealing a ’paradox’. In a second metamorphosis the EPR paper became a ’hidden variable’ theory. Stimulated by papers by John Bell [20, 18], an increasing number of physicists became convinced that what Einstein had in mind when talking about a ’complete’ theory was a theory of ’hidden variables’. Let us quote from an essay-review (2005) on Bell’s book ”Speakable and Unspeakable in Quantum Mechanics” [19], written by M. Zukowski:

”In modern terminology the [EPR-] paper suggests that perfect quantum correlations, which are possible for two subsystems in an entangled state can be used to define a missing element in quantum theory, namely ’elements of reality’”

"The main driving force toward a believe that hidden variables should exist, therefore, is in the religious belief [here the meaning of 'religious' includes e.g. dogmatic atheism,..] that 'nature must be deterministic' and that 'everything happening in nature must be predetermined by previous happenings in the physical world' even where our own knowledge is too limited for grasping the physical causes of what is happening"
F. J. Belinfante, "A survey of hidden-variables theories", p.18

But something like that cannot be found in the EPR paper, thus a rather radical ’rewriting’ has been performed here. There are hardly any facts supporting the modern terminology of what happened in Einstein’s mind (cf. Jammer’s comments on this point [73] and Einstein’s well-known saying ”too cheap” about a different ’hidden variable’ theory), but may be more important is to point out that it is not really of primary importance what he had in mind. We can read what he claimed to be true in the EPR paper and should not automatically replace this claim by what we believe he had in mind. Both things are not necessarily the same. Let us analyze this point in detail.

Einstein believed in a deterministic world and dreamed of a deterministic replacement of QT (we really do not know if he had accepted something like the present hidden variable theories as a realization of his dream). Using the notation introduced in section 5.4 we may say that he believed in the SID.

”Assuming the success of efforts to accomplish a complete physical description, the statistical quantum theory would, within the framework of future physics, take an approximately analogous position to the statistical mechanics within the framework of classical mechanics. I am rather firmly convinced that the development of theoretical physics will be of this type; but the path will be lengthy and difficult.”
A. Einstein, Reply to criticism [44]

On the other hand, what had been proven by EPR is the (predictive) incompleteness of the ”QT” (we use this notation sometimes to indicate that we are talking about ’QT as interpreted by the CI’). and this is not the same as Einstein’s dream (the assertion that QT must be replaced by a deterministic theory). It can be taken for granted that Einstein himself considered this result as an argument in favor of his dream. But this connection breaks down if we remove the metaphysical dogma of determinism which dominated Einstein’s thinking. Two alternative conclusions may be drawn from the EPR paper which are physically equivalent (differ only in a metaphysical sense).

"In doubtful cases the more liberal interpretation must always be preferred."
Marcus Tullius Cicero, Roman Statesman (106 BC - 43 BC)

The three interpretations SIS,SID,SII differ with regard to metaphysical postulates but agree with regard to the following most important point: QT is not a theory about individual particles. This is a concrete assertion which can be tested by observation and in this sense SIS,SID,SII are equivalent from a physical point of view.

A clear separation, in the above discussion, of physical and metaphysical aspects has led us to the conclusion that what Einstein believed is different from what has been shown in the EPR paper. The latter is ’predictive incompleteness’ of QT (breakdown of the CI) while the former is ’ontological incompleteness’ of QT. The difference is irrelevant for followers of a deterministic dogma (like Einstein) but plays an important role for people excluding either all dogmas from physics or replacing the deterministic dogma by an indeterministic dogma.

Very remarkably, exactly the same failure to distinguish ’predictive’ from ’ontological’ (in)completeness has led Einstein’s opponents to the erroneous conclusion that the breakdown of Bell’s hidden variable theory implies the breakdown of the claim of the EPR paper. Let us describe first what happened and then analyze the conclusion. Bell’s attempt to construct a hidden variable theory [16] leads to an inequality which disagrees with QT and is therefore almost certainly (considering the vast amount of experimental evidence in favor of the validity of QT in the atomic range) ’wrong’, i.e. not realized in nature. This fact and the (not really unexpected) results of pertaining experiments were interpreted as a proof for ”completeness” of QT. From this it was concluded that EPR’s claim for ”incompleteness” of QT must be wrong.

"The paradox of Einstein, Podolsky, and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables...In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predicions of quantum mechanics"
John Stewart Bell (1928 - 1990), "On the Einstein Podolsky Rosen paradox"

This conclusion is not justified⁹⁶ . The breakdown of Bell’s hidden variable theory can be seen as an argument in favor of ’ontological completeness’ of QT (no better theory than QT exists) but not as a breakdown of the ’predictive incompleteness’ shown by EPR. These two meanings of the term ’completeness’ are fundamentally different from each other, as discussed already, but the difference was neglected (Interestingly, the difference was neglected for dogmatic reasons both by Einstein and his opponents). Likewise, the difference between the three interpretations SID, SIS, SII was neglected and they were all identified with Einstein’s SID. Using this oversimplified reasoning Bell could claim that his hidden variable theory (or, more precisely, its breakdown) solves ”the Einstein Podolsky Rosen Paradox” [20]. Bell who found the ”silly error” [104] in von Neumann’s proof about the non-existence of hidden variable theories, made a silly error too !

"Your theory is crazy, but it's not crazy enough to be true."
Niels Bohr

"The EPR experiment is as close to magic as any physical phenomenon I know of, and magic should be enjoyed."
Mermin, N. D., "Is the Moon There when Nobody Looks? ", Physics Today, April 1985, p. 47.

The historical development of the Einstein-Bohr controversy, including the development of the mainstream opinion about Bell’s theorem, has been brilliantly described by Wick [161]. After so many years of troubles with the unsolved ’measurement problem’, Bell’s inequality was emphatically welcomed by Copenhagen believers, and its importance stressed almost to breaking point. The triumph continues and leads today to the inflationary use of terms like paradox, magic, mystery... Typically, the magic never happens in the laboratory but always in the brain of the interpreter. The magic is nothing but a consequence of an unjustified extension of the domain of validity of a physical theory. Phenomena not explainable by the theory are declared as magic. In some cases it has been explicitly shown, see Ballentine [10], and Kirkpatrick [82] (arXiv:quant-ph/0106072), how correct application of probability theory eliminates quantum mechanical paradoxes.

Let us illustrate what happened by considering the following hypothetical situation. Today, most people will probably agree, that non-equilibrium statistical mechanics is not an appropriate theory to explain the behavior of living creatures. But imagine that some powerful person, let us call him A, succeeds in convincing the scientific community that non-equilibrium statistical mechanics is a ”complete” theory for living beings. From that point on, scientific activities will produce a lot of mysteries - for all facts not explainable by the ”complete” theory are by definition paradoxes of nature. All A has to do is to convince a first generation of excellent scientists and teachers. After a few generations, it will be a well-established fact, true beyond any doubt, that non-equilibrium statistical mechanics is a ”complete” theory for the description of living beings. This situation then, in our short science-fiction story, corresponds to the present situation in QT.

5 Copenhagen Interpretation