Statistical Interpretation

"The Heisenberg-Bohr tranquilizing philosophy - or religion? - is so delicately contrived that, for the time being, it provides a gentle pillow for the true believer from which he cannot very easily be aroused. So let him lie there."
Albert Einstein (1879-1955)

To interpret QT as a statistical theory is essentially the same as denying the central (predictive) ’completeness dogma’ of the CI. The CI is based on some metaphysical constructs which have been added to the formalisms of QT in order to relieve the shock, brought about by the breakdown of classical physics ⁹⁷ . In sections 2, 3, 4 the most important of these additions have been shown to be unjustified. In section 5 we discussed some of the consequences of the CI’s mixing of physics and metaphysics. Thus, the most important arguments in favor of the statistical interpretation of QT have already been discussed. There are, however, some remaining points which will be briefly dealt with in this section.

"All events, even those which on account of their insignificance do not seem to follow the great laws of nature, are a result just as necessary as the revolution of the sun."
Marquis de Laplace (1749-1827),
"A philosophical essay on probabilities", Dover Publications, p.3

"Nothing in Nature is random.... A thing appears random only through the incompleteness of our knowledge."
Baruch Spinoza (1632 - 1677), Dutch philosopher

An apparent shortcoming of the SI is its fundamental negative position (we use the term SI whenever no distinction between SIS, SII, SID is required; these three variants of the SI take all the same position with regard to predictive completeness). The CI is ’optimistic’ - it tells us first of all that we have a complete, if not ’final’ theory. The SI on the other hand tells us that QT is (predictive) incomplete and leaves a question mark as regards all remaining questions. This refusal of answers, as regards individual events, of the QT is very often considered as a shortcoming of the SI - and not of QT. In fact, this refusal is hard to accept. Science should give definite answers, fuzzy predictions are allowed in psychology, sociology, poetry and medicine (among others) but not in physics. Deterministic predictions form the conceptual basis for exact science and have proven extremely successful in the past. While all this is true it is irrelevant because it is wishful thinking. We should not add metaphysical constructs to a physical formalism in order to satisfy our desires. Since the deterministic dogma is in disagreement with the experimentally confirmed predictions of QT, it should be abandoned.

"Behind it all is surely an idea so simple, so beautiful, that when we grasp it - in a decade, a century, or a millennium - we will all say to each other, how could it have been otherwise?"
John Archibald Wheeler
Annals of the New York Academy of Sciences, 480 (1986)

A second philosophical dogma, closely related to the deterministic (completeness) dogma, is the idea that QT is the physical ’final theory’ from which everything else can be derived. According to the reductionistic way of thinking, which was extremely successful in the past and should certainly not abandoned completely, physical disciplines are arranged in a hierarchical order. This requires e.g. that special relativity reduces to Newtonian mechanics in the limit $c → ∞$ and that nonrelativistic QT reduces to Newtonian mechanics in the limit $ℏ → 0$ . The former is true but the latter is not, as shown in the detailed discussion of section 4. The ’correct’ limiting behavior of QT was an important cornerstone in Bohr’s philosophical constructions and he created the ’correspondence principle’ in order to to establish the orthodox way of thinking. But we know today that there are so many exceptions from this ’principle’ that it should best be abandoned as a whole. Without the ’correct’ classical limiting behavior the dream of QT as a complete or even ’final’ theory remains a dream. Altogether, the SI expresses a much more modest position as regards the limits of our understanding and it seems that this more modest position is also more realistic. The reductionistic dogma provides a basis for string theory and implies that the gravitational field should be quantized. The SI is incompatible with the reductionistic point of view. An alternative paradigm ruling the relation of physical theories exists, which has been coined ’emergence’ [6, 91, 106].

The SI could also be called the minimalist interpretation of QT. The ’quantum revolution’ becomes much more dramatic if QT is interpreted in a purely statistical sense - without the ’tranquilizing’ additions of the CI. We are faced with limitations which had never before been conceivable, and which affect the very basis of all previous science.

"I think I can safely say that nobody understands quantum mechanics"
Richard Feynman: The Character of Physical Law, Ch. 6

"The difference between science and religion is that the former wishes to get rid of mysteries whereas the latter worships them."
Sidney Hook, Philosopher (1902-1989)

How can the ’correctness’ of the SI (or any other interpretation) be proven? To qualify an interpretation as ’true’ or ’wrong’ is a somewhat subtle point. An interpretation cannot be tested by direct comparison of predictions with experimental data (see however section 6.3). Unfortunately, an interpretation cannot be tested by mathematical-logical reasoning either; such reasoning transforms as a rule only interpreted mathematical input into interpreted mathematical output⁹⁸ . A possible test is, however, given by the fact that an interpretation defines a direction for future research. Generally speaking, it can be tested in an indirect way by studying its consequences. An indication for the ’truth’ of the SI is that the completeness doctrine of the CI leads to inconsistencies. If the SI is ’true’ then essentially all applications of QT applied to genuine single events should lead to absurd consequences. This is exactly what happens. The long list of absurdities and paradoxes resulting from the CI single-event point of view includes, as prominent members, the measurement problem, the Aharonov-Bohm effect [3], and cause-effect velocities exceeding that of light [100] in EPR-like situations. Many other examples may be found but an exhaustive listing will not be given here; I just mention, as a slightly less prominent example, a recent analysis of tunneling data where signal velocities up to five times the velocity of light were found [108].

According to the SI, QT is predictive incomplete, i.e. unable to determine what will be observed. If the term ’indeterministic theory’ is used as synonym for predictive incompleteness, QT may be called a indeterministic theory while Newtonian mechanics is a deterministic theory. Nevertheless, QT makes definite predictions, but these do not concern individual events but statistical ensembles (QT could be called a single-event indeterministic theory or a many-event deterministic theory)⁹⁹ . The universe described by QT allows (according to the SI) only limited access for its visitors. Nature does not behave completely irregular, but single (microscopic) events are out of our reach. Many-body systems behave (fortunately) effectively like statistical ensembles and this explains the enormous success of QT and its relevance for the macroscopic world.

6.1 ’No-go’ for hidden variables is not ’no-go’ for the statistical interpretation

We know that Einstein believed in a final deterministic description of nature. Interestingly, this position is not completely different from that of the CI. The CI’s belief in a deterministic description reveals in its claim that QT is predictive complete (i. e. deterministic in our nomenclature). There is, of course, a fundamental logical gap between this claim and the probabilistic nature (the unpredictability of single-event observables) of QT. This gap is not denied by the QT. It is closed by postulating that the observed numbers are created by the interaction between the object under study and the measurement device and that the question how this actually takes place - the measurement problem - will be solved sooner or later. By means of this sophistic construction the CI maintains predictive completeness of QT. Thus, fundamentally, the belief in a deterministic description of nature was shared by Einstein and Bohr, despite their totally different opinion with regard to the predictive power of QT.

"The paradox is only a conflict between reality and your feeling what reality ought to be."
Richard Feynman (1918-1988)

This belief in a deterministic description of all observable things continues to be extremely widespread. Still, almost all discussions on the interpretation of QT are centered about the Einstein-Bohr controversy. The possibility of alternative interpretations - which differ from either position - is neglected. This leads very often to the wrong idea that incompleteness with regard to single events implies existence of a ’hidden variable theory’. This belief is frequently expressed in a way similar to the following quotation (from a review article by Pearle [115]):

”Suppose one believes, with Einstein, that quantum theory does not describe the behavior of an individual system. Then, since individual systems do exist in nature, it is reasonable to conclude that there ought to be a theory which does describe individual systems and that this theory (often called a ’hidden variable theory’) ought to supercede quantum theory.”

The problematic nature of the central conclusion (...since individual systems do exist in nature, it is reasonable to conclude that there ought to be a theory.. ) of this statement, has already be discussed (see subsection 5.1). This conclusion is not unreasonable, but it is by no means necessary. It corresponds to the SID and the CI. Why should the fact that individual systems exist, necessarily imply that a theory describing these systems exists. Various other ’reasonable’, but less arrogant, conclusions may be drawn from the existence of individual systems. We may for instance say, that neither the nonexistence nor the existence of a corresponding theory can be concluded from the very fact that something exists (because philosophically speaking epistemology and ontology are two different things). This is the position of the SIS. There may also be philosophical reasons to postulate that the ultimate constituents of matter cannot be described by a deterministic theory. This is the position of the SII (The idea of indeterminism has been analyzed from a philosophical and physical point of view by Karl Popper [120]). Therefore, statements found in the literature [104], claiming that the experimentally confirmed non-existence of Bell-type-hidden-variable-theories presents an argument against the SI, are unjustified.

"Clearly a statement cannot be tested by observation unless it is an assertion about the results of observation. Every item of physical knowledge must therefore be an assertion of what has been or would be the result of carrying out a specified observational procedure."
Sir Arthur Eddington, The Philosophy of Physical Science, 1958, p. 9-10

In the CI there are values of observables which are ’not possessed’ by the corresponding particles but are ’created by the act of measurement’. Terms like ’not pre-determined’ or ’not defined’ or ’not definite’ are sometimes used as synonyms. These unobserved observables do not exist in the CI but play nevertheless a major role because their non-existence is a fundamental assumption in the philosophical construction of the CI. If we really know nothing about a thing how can we know that it does not exist. Such concepts are pure metaphysics. Very often one finds the assumption that the observables in a statistical interpretation of QT, must - in contrast to the CI - have pre-determined values.

”This heresy [meaning the SI] takes the state vector to describe an ensemble of systems and maintains that in each individual member of that ensemble every observable does indeed have a definite value, which the measurement merely reveals when carried out on that particular individual system [emphasis mine].”
N. D. Mermin, Rev.Mod.Phys. vol.65, p.803

This assumption is then combined with Bell’s theorem to obtain an argument against the SI. This way of reasoning is just a variant of the situation considered in the last paragraph [the assumption that the values of the observables must be ’pre-determined’ has apparently been derived from a combination of two closely related metaphysical assumptions of the CI, namely (i) predictive completeness (determinism), and (ii) the unreality of unobserved properties.] But this is, again, pure metaphysics (corresponding to CI, SID) at least two other dogmas - already listed in the last paragraph - are conceivable.

"Nowadays, we have stripped Maxwell of his philosophy and retained only his equations. Perhaps we should do a similar job on quantum mechanics."
H.R. Post, Against Ideology,1977, p. 14.

6.2 Two attacks on the statistical interpretation

As compared to the majority position of the CI, the SI represents the opinion of a qualified minority among physicists. The controversy between both interpretations seems to consist, for the most part, in attacks of the SI against the CI and not the other way round. Most of the more recent attacks have simply been ignored by the supporters of the CI; this is certainly the most efficient way for a majority to react on ”minor” attacks. In particular, no really specific answer to criticism of the kind reported in section 5, has, to the best of my knowledge, ever been published.

"The demand for certainty is one which is natural to man, but is nevertheless an intellectual vice"
Bertrand Russell (1872 - 1970) British philosopher, logician, mathematician, historian, and social critic

An exception to this rule, i.e. an explicit attack of the CI on the SI, may be found in two papers, an older one by Gillespie [57] and a more recent one by Pusey et. al. [125]. In both of these works the authors try to construct a logical contradiction within the mathematical formalism of QT as interpreted by the SI. This claim seems strange because an interpretation can never be disproved by mathematical means¹⁰⁰ . It is the mathematical formalism that needs interpretation and not the other way round.

Indeed, as a first look on these papers shows, it is not the SI which leads to a contradiction but the assumption that a theory of hidden variables exists on top of QT. The authors postulate (assert) that the SI requires the existence of hidden variables (or something equivalent bearing a different name). As shown in detail in subsection 6.1 such a constraint is not required for (all versions of) the SI. But once this postulate is considered to be true, ”QT as interpreted by the SI”, becomes ”QT as supplemented by a theory of hidden variables” - and then it is an easy matter to construct a logical contradiction ”within the SI” because the existence of hidden variables is in conflict with the structure of QT.

"Doubt is not a pleasant condition, but certainty is absurd."
Francois-Marie Arouet (1694 - 1778), known by his nom de plume Voltaire, French writer, historian and philosopher

This common error represents the essence of both papers, but it is useful to study its creation in detail because such a study presents a concrete example for the general classification scheme developed in subsection 5.4.

6.2.1 Untenability of simple ensemble interpretations?¹⁰¹

"The difficulty lies not so much in developing new ideas as in escaping from old ones"
John Maynard Keynes (1883 - 1946), British economist

Gillespie’s paper [57], entitled ”Untenability of simple ensemble interpretations of quantum measurement probabilities”, is characterized in the abstract as ”a repacking of the Bell argument against hidden variables”. A ”simple ensemble interpretation” is defined in the introduction as follows:

The state vector $|ψ ⟩$ specifies, not an individual system, but rather a statistical ensemble of identically prepared systems, each of which has precise values for all its dynamical variables. The values of these observables are distributed among the ensemble systems in such a way that the probability of selecting an ensemble system that has $A = a i$ is equal to the probability given by the prediction postulate [Born’s rule] for obtaining $ai$ in a measurement of $A$ . We shall call this viewpoint a ”simple ensemble interpretation of quantum mechanics”..

The most interesting part of this definition is ...each of which has precise values for all its dynamical variables.... What does it mean, precisely ? What does it imply if it is meaningful?

This statement is metaphysical in nature in the sense that it cannot be verified in any experiment. In fact, fundamental assumptions about the role of B must be part of every physical theory. Therefore, a theory - free of metaphysical assumptions concerning B - which could be used to test the valididy of B does not exist.

"It is undesirable to believe a proposition when there is no ground whatsoever for supposing it is true"
Bertrand Russell, Sceptical Essays (1928), "On the Value of Scepticism"

One would probably agree that a system has precise values if some values are predicted by a theory which are then verified by a corresponding experiment. This means, if we use the abbreviation $A$ for the statement ”we have theoretically predicted values for dynamical variables which have been verified experimentally” and the abbreviation $B$ for the statement ”a system has values for its dynamical variables”, then everybody would probably agree that the implication

A = ⇒ B,

(14)

is true. The problem is that this logical relation is useless because the conclusion in (14), ”a system has values for its dynamical variables”, plays in the present context the role of an premise; we need something with the arrow in the opposite direction. The ’solution’ of Gillespie for this problem is simply to assume that the implication inverse to (14), namely

B = ⇒ A,

(15)

is also true. This means, that for a system that ”has values for its dynamical variables” necessarily a theory exists which successfully predicts these values. This assumption has not been formulated explicitely by Gillespie, but implicitely postulated to be true, as a consequence of the fact that he introduces functions of these values [see Eq. (6) of [57]], which do not occur in the mathematical formalism of QT. Without assumption (15) this cannot be done. The fact that ’something’ exists (whatever that may mean) does not necessarily imply that a theoretical description for this ’something’ exists. Gillespie’s implicit assumption (15) is, like the statement $B$ , of a metaphysical nature.

"the Talmudic philosopher sniffs at 'reality' as at a frightening creature of the naive mind"
Albert Einstein (1879 - 1955), quoted after A. Fine, "The shaky game" p. 36, footnote 10.

The statement (15) is closely related to the philosophical determinism of the 19th century, namely the believe that everything that can be observed can in principle be described (predicted) by a physical theory. This deterministic metaphysical assumption has already been discussed in a related context (see subsection 5.4) and has been denoted for brevity by D. We will refer to D or to assumption (15) depending on context. Gillespie’s implicit metaphysical assumption (15) may equivalently be written in the form

¬A = ⇒ ¬B,

(16)

This means, if it is not the case that ”we have theoretically predicted values for dynamical variables which have been verified experimentally”, then it is not true that ”a system has values for its dynamical variables”. In this form the statement rather expresses the fundamental idealistic aspect of the CI (discussed in more detail in subsection 5.1). It has been put in a compact form by the philosopher Mark Rowlands:

”if you can’t in principle know something then it does’nt exist”.

Thus Gillespie’s implicit metaphysical assumption D is exactly the same as the fundamental metaphysical postulate of the CI (see subsection 5.1). In view of the statistical nature of the quantum mechanical predictions (which is not denied by the CI) it seems strange that a deterministic postulate should be part of the metaphysical framework of the CI. However, this postulate D is urgently needed for the CI in order to make QT a (complete) theory of single particles. The clash with the statistical nature of the quantum mechanical predictions can be avoided (at least superficially) by means of the well-known assertion of the CI, that unobserved values are not real. This strange assertion, which is of course also metaphysical in nature, is a direct consequence of postulate D.

"Those are my principles, and if you don't like them... well, I have others."
Julius Henry "Groucho" Marx (1890 - 1977), American comedian

Is postulate D the only metaphysical assumption compatible with Gillespie’s postulate B ? Certainly not. B cannot be verified or falsified, as mentioned already. There is a ”metaphysical degree of freedom” contained in all these discussions. This aspect has been taken into account in early discussions about QT but is very often overlooked in recent years. It seems it should be taken into account in order to obtain all possible interpretations.

There are at least three possible responses, including D, to the metaphysical statement B. These have already been discussed in section 5.4 in a related context, and are given by

For a theory about single particles, which QT is believed to be according to the CI, only the first possibility D is acceptable (at the cost of ”reality”). For a statistical interpretation all three responses can in principle be accepted. This leads to four interpretations (see 5.4):

We see that Gillespie’s argument affects only SID and not SII, or SIS. In particular, the interpretation SIS, which may be seen as the ’most physical’ one - in the sense that it makes no metaphysical assumptions at all - is not affected by Gillespie’s conclusion.

"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions"
Eric T. Bell (1883 - 1960), mathematician and science fiction author

Clearly, the error made by Gillespie (and by the CI in general) is the neglect of the ”metaphysical degree of freedom” contained in his premises¹⁰² . Instead of analysing all possibilities he chooses a particular one, namely the SID, and considers it as self-evident. His conclusion that ’simple ensemble interpretations’ are untenable is, like Bell’s theorem, no final disproof but a valid argument against the SID. It is no argument against the SIS or SII. Both of these variants of the SI are equivalent for all practical purposes. Gillespie’s ”non-simple” ensemble interpretations are from a fundamental point of view identical to his ”simple” ensemble interpretation¹⁰³ and also based on the same deterministic assumption D. Thus, this work does not provide a valid argument against the SI.

Both attacks against the SI discussed in this subsection (see below) may be characterized by a fundamentally deterministic point of view. Both Einstein and Bohr believed in a deterministic description of the world - despite their very different final conclusions. Unfortunately, present day thinking still follows the same deterministic line of reasoning.

Finally, recall that different meanings of the important term ’completeness’ have been introduced in subsection 5.4, as a consequence of the different metaphysical assumptions D,I,S. Distinguishing carefully between these different meanings, an analysis, analogous to the present one, of Bell’s theorem has been performed in subsection 5.6. The conclusion of this analysis is that Bell’s theorem can only be used as an argument against the SID and not against the two other variants of the SI. This conclusion agrees with the present one.

6.2.2 The quantum state cannot (can) be interpreted statistically?¹⁰⁴

"We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it ... You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem."(1982)
Richard Feynman (1918 - 1988), American theoretical physicist

Here some comments will be given on Pusey et al’s paper [125], entitled ”On the reality of the quantum state” (arXiv:1111.3328). The first draft of the published version (essentially the same paper) was entitled ”The quantum state cannot be interpreted statistically”. Some comments will also be given on more recent related work by Lewis et al [92], which was written by two of the authors of [125] together with two other co-authors. The first draft of this latter work was entitled ”The quantum state can be interpreted statistically”.

The implicit metaphysical assumptions which form the basis of [125] are exactly the same as the one found in Gillespie’s paper discussed in subsubsection 6.2.1. Pusey et al. start by assuming that

a system has a ”real physical state”,

which is identical to the above metaphysical assumption B. They then introduce a Bell-like hidden variable description, i.e. they postulate - like Gillespie - implicitly the validity of the metaphysical assumption D (see relation (15)) of the CI. The final conclusions drawn in the analysis of Gillespie’s work can be taken over unchanged to the work of Pusey et al [125]. The present analysis is characterized by the complete absence of any discussions concerning the mathematical models reported in [57] or [125]. We consider these as irrelevant because the ”truth” of interpretations can, as a matter of principle, not be verified by mathematical means.

The work of Pusey et al has been criticised by (at least) two papers. Hofmann (arXiv:1112.2446) discusses several crucial points, including Pusey’s et al’s questionable assumption that all statistical interpretations must be based on hidden variable realism; the present analysis presents a detailed refutation of this assumption. Blood (arXiv:1112.2446) points out that it is unclear wether or not the abstract model designed by Pusey et al. corresponds to something real. Both papers are still unpublished at the time of writing this.

"In science there are no paradoxes"
William Thomson - Lord Kelvin - (1824 - 1907), British mathematical physicist and engineer

On the other hand as regards the irreality-camp one can read already in a prestigious physics journal [137] that

”This [Pusey-Barrett-Rudolph] theorem has emerged as a far-reaching no-go result whose implications are cited as possibly even more dramatic than Bells theorem.”

This statement is really strong since Bell’ theorem has already been praised as follows [145]:

”Bell’s theorem is the most important discovery in science.”

The former quote indicates that a theorem even more important than the most important one might exist in science, and that new, even more exciting paradoxical and magical results could possibly be found if experimental data are interpreted in terms of the PBR theorem instead of the Bell theorem. Are we to expect a revival of the Bell-hype ? Or is this ’proof’ of the ’reality’ of the wave function even the beginning of the final take over of irreality in physics ? Pusey et al’s paper created a lot of confusion; some people interpret it even as a denial of the statistical nature of the predictions of QT - and as a kind of restoration of deterministic physics. In fact, Pusey et al.s paper arXiv:1111.3328 is extremely misleading since the real weak points (irreality of unobserved properties) of their own individuality interpretation (reality of the wave function) are not even mentioned.

The authors of the second paper [92] (arXiv:1201.6554) are aware of the fact that interpretations are determined by metaphysical assumptions. Unfortunately, they are using philosophical categories - epistemic versus ontic - to classify physical theories. It seems much simpler to start the analysis by introducing a physical classification scheme - single particle versus ensemble¹⁰⁵ . The authors [92] arrive, after a complicated and somewhat cryptic analysis, at a conclusion opposite to the first [125] ! From the present point of view, this conclusion does not support the SI, because the second paper uses the same Bell-like hidden variable assumption as the first and is therefore based on the same erroneous exclusive use of the metaphysical assumption D.

Summarizing, the papers discussed here do only provide arguments against a particular version of the SI, but are irrelevant for other versions, which do not contain the hidden metaphysical assumption D of the CI.

6.2.3 Quantum theory cannot describe the ’real physical state’, whatever that may be

"What really matters for me is ... the more active role of the observer in quantum physics ... According to quantum physics the observer has indeed a new relation to the physical events around him in comparison with the classical observer, who is merely a spectator."
Wolfgang Pauli (1900 - 1958), Austrian theoretical physicist

Let us quote the following statement by Jaynes [75] which mentions an aspect (scrambling of objective reality and subjective lack of information) which is closely related to the problem of hidden variables:

”But our present QM formalism is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature - all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective and objective aspects of the formalism, we cannot know what we are talking about; it is just that simple.”

We can see by means of a simple consideration that it is impossible to unscramble the part of our knowledge from the objective part of nature [83] (arXiv:0806.4335). The ’scrambling up’ of these two different ingredients is not due to Heisenberg and Bohr, but is in fact a property of the quantum-theoretic formalism itself (of nature - if you prefer).

"...quantum phenomena do not occur in a Hilbert space, they occur in a laboratory."
Asher Peres, "Quantum Theory: Concepts and Methods" p.XI

To illustrate the impossibility of unscrambling it suffices to compare the mathematical structure of Schrödinger’s equation with the corresponding classical equations. We do not use the Hilbert space formalism of QT to discuss this question. This might be considered as a lack of precision but is in fact a gain, because the Hilbert space formalism - no matter how powerful and elegant it might be - has been constructed (by von Neumann) as a generalization of the linear solution space of Schrödinger’s equation. Thus, the abstract formalism contains less information than the fundamental differential equation it is derived from. In particular, Hilbert space formalism says nothing about the concrete form of the Hamiltonian - and this is exactly what is required to answer the question. We will consider the simplest non-trivial problem, a ’single particle’ (ensemble) in an external potential $V (x )$ .

Let us start with a classical single particle ensemble (interpreted in a probabilistic sense not in the sense of a fluid). It is given by an infinite number of trajectories, each one ruled by the same deterministic law, namely Newton’s equation. No predictions concerning the particle’s position at time $t$ can be made since only a probability density $ρ (x, t)$ is available - as a consequence of the fact that the exact initial values for the particle are unknown, and are distributed according to a probability density $ρ (x, 0 )$ . Very often such a system is studied in phase space [using a probability density $ρ (x, p, t )$ instead of $ρ (x, t)$ ]. However, in order to compare with QT, it is much more useful to study the basic equations for such an ensemble in configuration space. These are given by

( ) ∑ 2 ∂--S- --1--- -∂-S--- + + V = 0, ∂ t 2m ∂ xk k

(17)

Eqs. (17) and (18) for the dynamic variables $S (x, t )$ (the classical action) and $ρ (x, t)$ have to be solved for given initial conditions $S (x, 0 ), ρ (x, 0)$ . Our ”incomplete human information” is described by the fact, that $ρ (x, 0 )$ is [we exclude here the delta function limit, which would make our information complete but is incompatible with the structure of QT [85] (html)] a continous function of finite support. In the classical statistical theory (17), (18) our ”incomplete human information” is cleanly separated from the description of ”reality”. This ”unscrambling” is reflected, in the mathematical structure, by the fact that equation (17) is not coupled to (18). The variable $S (x, t )$ , describing the part of ”reality” may be calculated by solving equation (17) alone. Thus, the individuality of the ensemble members, described by $S$ , is not affected by $ρ$ . On the other hand, the time development of $ρ$ depends on $S$ , which plays the role of an ’input parameter’ in (18) .

Let us consider the corresponding problem in QT. If we replace in Schrödinger’s equation (8) the real and imaginary parts of $ψ$ by new variables $S$ and $ρ$ [by means of the Madelung transformation (10)] then we obtain (see subsection 4.2) the two differential equations

( ) ∑ 2 2 ∂-S-- -1---- -∂-S--- -∂-ρ--- -∂--ρ-- + + V + T ( ρ, , ) = 0. ∂ t 2m ∂ xk ∂ xk ∂ x2 k k

(19)

which describe the corresponding quantum-mechanical ensemble. The new equations (19), (20) look similar to (17), (18) but are very different physically. The only formal difference is the appearance of a new term $T$ in (19), which depends on the probability density $ρ$ and its first and second derivatives and is given by

2 2 √ -- ∂ ρ ∂ ρ ℏ △ ρ T ( ρ, ------, -------) = - -------√------. ∂ x ∂ x2 2m ρ k k

(21)

The presence of this coupling term (which is proportional to the square of the new constant $ℏ$ ) means that the time-evolution of $S (x, t)$ is not only determined by the external potential $V (x, t )$ but also by the variable $ρ$ , which describes our subjective incomplete knowledge. A separation (’unscrambling’) of objective reality and incomplete human knowledge is impossible. In other words, objects described by QT, have no identity independent from the observer.

"Have you noticed that Bohm believes (as de Broglie did, 25 years ago) that he is able to interpret the quantum theory in deterministic terms? That way seems too cheap to me"
Albert Einstein, letter of 12 May 1952 to Born

In contrast to the classical ensemble (17), (18), the quantum-mechanical ensemble (19), (20) is not given by an infinite number of independent trajectories but by an infinite (continuous) number of strongly coupled trajectories which cannot be associated in any meaningful way with individual objects. This is another way to express the fact - implied by the presence of the single term (21) - that in QT not only predictions for individual objects are impossible [this holds true also for the corresponding classical statistical theory (17), (18)] but that even an underlying deterministic law cannot be found [in contrast to the corresponding classical statistical theory (17), (18)].

In other words, attempts to interpret QT in the sense of a classical statistical theory by means of the introduction of hidden variables are useless; an intrinsically indeterministic theory cannot be made compatible with deterministic laws. This is the essence of Bell’s theorem and related considerations [see the above subsubsections 6.2.1, 6.2.2]. A simple no-go proof for hidden variables in QT has been reported by Jordan and Sudarshan [76]. The present qualitative consideration is even simpler although less complete in a mathematical sense.

"One does not get an answer to the question, 'What is the state after collision?' but only to the question, 'How probable is a given effect of the collision?' From the standpoint of our quantum mechanics, there is no quantity which causally fixes the effect of a collision in an individual event. Should we hope to discover such properties later ... and determine [them] in individual events? ... I myself am inclined to renounce determinism in the atomic world, but that is a philosophical question for which physical arguments alone do not set standards."
Max Born (1882 - 1970), German-British physicist who was awarded the Nobel Price for his statistical interpretation of the wave function.

The above simple consideration indicates also that all attempts to generalize Bell’s assumptions¹⁰⁶ and to ”save QT” over and over again are a waste of time. It may be more useful to accept the single-event indeterministic character of QT and start thinking about the origin of indeterminism; as a matter of fact even classical physics is indeterministic if we take radiation- and self-interaction effects seriously into account.

An interesting contextual explanation for the probability distributions of QT has been given by de Raedt, Michielsen, and coworkers [for a review see [128](arXiv:1208.2365)]. These authors were able to reproduce the probabilistic predictions of QT in a non-trivial way by means of computer simulations. Individual events are not described by a conventional (non-contextual) physical theory but by a set of rules which account for the essential physical aspects of the specific experiment. Probability enters in a ’classical’ way, e.g. in the form of random initial positions of the photons leaving the source in an interference experiment. This theory, which has certainly not much in common with standard hidden variable theories, represents an unconventional and successful attempt ”outside QT” to understand (eliminate) some of the miraculous features of QT.

6.3 Can the wave function of a single system be determined by measurement ?

The question whether or not the wave function of a single particle can be determined (up to gauge freedom) experimentally provides a very fundamental test of the ’validity’ of the CI (as well as the SI). The CI claims that the wave function provides a complete description of a single particle. If this is true it should be possible to verify it and to characterize the quantity ”wave function of a single particle” by means of experimental data. Thus, for proponents of the CI the answer expressed in the title of this subsection is ’Yes’. On the other hand, for proponents of the SI the answer is clearly ’No’; recall that the concept of the ’wave function of a single system’ does not exist in the SI.

The impossibility to measure the wave function of a single object has been predicted for purely theoretical reasons by Blokhintsev [24] and others:

"If [the wave function were a characteristic of a single particle] it would be of interest to perform such a measurement on a single particle (say an electron) which would allow us to determine its own individual wave function. No such measurement is possible"
Blokhintsev, D. I., The Philosophy of Quantum Mechanics, Reidel, Dordrecht, Holland, 1968 .

Of course, an experimental investigation of this question would be useful. At Schrödingers time such experiments were out of reach (see the quotation at the end of this subsection). Today a variety of modern experimental techniques are available which may be used to study the behavior of individual quantum-mechanical systems. These techniques could allow, in principle, the determination of the ’wave function of a single system’ (if it exists) by means of a series of ’nondestructive’ measurements on single systems.

O. Alter and Y. Yamamoto studied this problem. In their book [5], which summarizes experimental work of several years on various systems, the authors arrive after a detailed analysis at the conclusion that it is impossible to determine the wave function of an individual system. Thus, their answer to the question expressed in the title of this subsection is ’No’ and this is in agreement with what proponents of the SI would say.

Unfortunately, matters are slightly more complicated. It has been pointed out (in a book review by Ballentine [14]) that the authors use the projection postulate uncritically, without discussing the question of its validity. As a consequence ”..their arguments are compromised, and their conclusions cast into doubt ..”.

I do not completely agree with this negative assessment and think that Alter and Yamamoto’s result is, nevertheless, very important. The analysis of such questions can only be done in the framework of a given (fixed) interpretation of QT, assuming the validity of all assumptions belonging to this interpretation. The authors took the interpretation which is (unfortunately) accepted by a majority of physicists, namely the CI - and so they had to assume the validity of the projection postulate, which is part of this interpretation. A second essential postulate of the CI is that the wave function describes a single system. Alter and Yamamoto have shown that not both of these postulates can be true (assuming of course that all other assumptions, not specified here, are true), i.e. they have shown in a new way - by comparing theoretical measurement concepts with specific experimental data - that the CI is contradictory. This result supports the SI where both assumptions are rejected.

"We never experiment with just one electron or atom ... any more than we can raise Ichthyosauria in the zoo"
Erwin Schroedinger (1887-1961) .

6.4 A list of publications related to the statistical interpretation

Finally, I provide an incomplete list of publications and articles related to the SI.

6 Statistical Interpretation