1 Motivation

galileo.gif In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.
Galileo Galilei, physicist and astronomer (1564-1642)

Many different interpretations of the quantum-mechanical formalism exist already but none of them is really satisfying and generally accepted. A closer look shows that all these different interpretations are not really different: they all share certain characteristic assumptions belonging to the Copenhagen interpretation. This raises the question if not one or more of these Copenhagen assumptions should be abandoned and a more radical change in our point of view towards quantum theory (QT) should be made.

This question will be studied in this web-review. In a first step, the three main assumptions underlying the CI will be critically examined. This is done in the first three sections (see sections 234). Several results published in the last eighty years and never refuted are recalled, which show that the Copenhagen interpretation is based on misinterpretations of experimental facts and incomplete or even erroneous reasoning.

The second step, to find a ’better’ interpretation is not an easy task because we do not know how to ’verify’ an interpretation. In contrast to physical predictions - which can be compared with experiments or at least tested with regard to internal consistency by comparing with other theories - interpretations are of a more abstract and more general (philosophical, metaphysical) character and cannot be directly compared with observation2 . A statement beyond question is that every interpretation of a physical theory is influenced both by the structure of the theory and by metaphysical considerations. Thus, it may be useful to think at the beginning about the relative weight which one wants to ascribe to metaphysics - as compared to the structure of the physical theory - in the process of constructing an interpretation.

What one finds is interesting, namely “Philosophy beats Physics” in the following sense. If one starts with philosophical considerations physical questions (which one would normally consider as very important) may become of secondary importance. This is in a sense understandable because philosophical questions are of greater generality (but more difficult to verify) in comparison to physical questions. But it is, of course, not at all satisfying that philosophical reasoning may lead to the complete elimination of important physical questions.

I would like to illustrate this last remark by means of the following example (which was rather impressive for me). In his excellent book [40] on the foundations of quantum theory, W.M. de Muynck starts his analysis with philosophical considerations. He decides to see the world, in particular the quantum world, from the empiristic3 point of view (chapter 2 of [40]). According to this view the mathematical formalism of QT does not describe microscopic objects but rather relations between preparation and measurement procedures (p.76 of [40]). The state vector describes the preparation of a statistical ensemble. This might lead to the conclusion that this interpretation is similar to the statistical interpretation (where the state vector describes not individual particles but statistical ensembles). However, this is not the case. In the empiristic philosophical category only relations between processes can be described. This classification leads to to the following conclusion ”The question of whether an individual-particle interpretation is useful at all, can arise only in a realist4 interpretation” (p.278 of [40]).

What happened here ? A physicist constructs an interpretation of a physical theory. He seems to be satisfied with the result but is unable to tell which objects (single particles or statistical ensembles) are described by the theory, i.e. he is unable to tell the range of validity of the theory. The physicist acted in fact here like a philosopher5 . He did not pay attention to the question how to verify the predictions of the theory. Does the theory describe the behavior of single particles or of ensembles ? This is the most important question and it remains unanswered. This was an example (many others exist), how fundamental physical questions can be eliminated by putting philosophical questions in the foreground.

bohm.jpg From the outset, however, this whole controversy has been plagued by tacit assumptions, very often of a philosophical rather than a physical character...
David Bohm, physicist (1917-1992)

Let us now choose the alternative route and start from the mathematical-physical structure of QT. A fact beyond doubt, confirmed by numerous experiments and accepted by supporters of all interpretations, is that the predictions of QT are probabilities6 . As a simple example, let us imagine that the number           ∫              2
PA    =        |ψ  (x ) | dx    =   0.23
            A gives, according to a correct quantum mechanical calculation, the probability of finding an electron within an area A of a screen. Is this a prediction about a single electron ? The answer is Yes/No if this prediction can/cannot be verified by means of an experiment involving a single electron. The answer is obviously NO - at least if we restrict ourselves to the structure of the physical theory without any additional metaphysical constructs: If a single electron hits the screen its position is either inside A or outside and this happens every time an electron hits the screen. The number P     =   0.23
  A , which is a probability, cannot be used to predict the outcome of single events. In order to verify a probabilistic prediction, like PA    =   0.23 , a large (virtually infinite) number N of experiments with single electrons (either simultaneously or in consecutive order, it does not matter) have to be performed, counting the number of times N
   A the electron hits a point inside A . Since QT can certainly be expected to be true the ratio N     ∕N
   A will be close to P     =   0.23
   A if the total number N becomes large.

The fact that probabilities are predictions concerning statistical ensembles and cannot be used to make predictions on single events is not a consequence of the specific structure of QT. Rather it is true on very general grounds - by definition of the term probability.

A most familiar example is given by classical statistical physics, which deals with large aggregates of particles. Here probability densities are given for initial values but the time-development is deterministic. Thus, classical statistical physics is a logical hybrid containing both probabilistic and deterministic elements. The particles move along definite classical trajectories, determined by the canonical equations, if they start at time t0 at initial points x0 with momenta p0 . We know that the positions and momenta of the particles are known at t if they are known at t
  0 , but we do not know the initial values at t
  0 . The probabilistic nature of the initial values is sufficient to make the whole theory probabilistic. In fact, the output of this theory is given by probabilities and no predictions on single events can be made.

A classical probabilistic theory describing single particles exists, which is, apart from the number of particles, very similar to classical statistical physics. We can write down a Liouville equation which determines the time development of a probability density in six-dimensional phase space7 starting from an initial value for the density at t
 0 . The latter describes the probabilities for various possible initial positions and momenta of a single particle. Given that we define true probabilities as initial values8 , the output of the theory is again a probability and no predictions on single events can be made.

Other classical statistical theories do exist, however, the examples given so far should be sufficient to illustrate the essential point. All kinds of statistical theories - no matter whether classical or quantum - make predictions about ensembles and not about individual events. In classical theories the uncertainty (indeterminism) is formulated in terms of probability densities for initial values9 , while in quantum mechanics a more fundamental type of indeterminism exists10 . But the fundamental fact, that predictions can only be made about ensembles is the very same in both theories11 .

I quote the following comment from J. L. Park’s penetrating analysis [114] of the relation between states of ensembles and states of individual particles being elements of ensembles:

The linguistic extension of ψ from its role in describing ensembles to its further function as the state of a single system has given birth to monumental barriers to the understanding of quantum theory as a rational branch of natural philosophy. Problems connected with the general theory of measurement - the nature of quantum measurement, wave-packet reduction, concepts of compatibility and simultaneous measurement - are especially aggravated by this popular convention that the state of an individual system is represented by ψ .

This was written half a century ago; today the list of problems caused by the ’linguistic extension’ would be even longer.

Thus, starting from a simple analysis of the physical structure of QT we arrive immediately at a very important conclusion concerning the range of validity of the theory, namely that QT is not a theory about single particles12 . If, on the other hand, we postulate - as in the CI - for metaphysical reasons13 that QT is a theory about single particles, we are faced with the difficult task to understand how a probabilistic theory can be a ’complete’ theory for single particles14 . This ’problem’ has been called ’the measurement problem’ and had to be invented eighty years ago, simultaneously with the completeness postulate of the CI15 . It is, not unexpectedly for many physicists, still unsolved and will remain unsolved forever because it represents a contradiction to the formalism of the QT. Proofs of its insolubility have also been published [49][140]. The incompatibility, between the very notion of probability and the claim of the CI for single-particle completeness, is the source of all the strange interpretations (psycho-physical parallelism, many worlds, etc.) which have been created so far in order to understand the QT. It is also the reason for the inflationary use of philosophical notions (or extremely abstract mathematics) by physicists16 , as a possible way out of the contradictory assumptions of the CI.

razor_blade.jpg One should not increase, beyond what is necessary, the number of entities required to explain anything.
William of Ockham, ca. 1285-1349.

At the beginning of our search for the ’best interpretation’ of QT we found that priority of metaphysical reasoning may lead to the elimination of important physical questions. If we start our search by studying the physical structure of QT first, it’s the other way round: we arrive at results, which make certain philosophical questions obsolete17 . Now, how can the choice of a proper starting point be justified ? As a matter of fact, if we give priority to physics, this does not mean that we are really free of ’philosophical’ assumptions. However, these philosophical assumptions (such as ”logical realism”) are already part of the general structure, the ’definition’ of the science of physics.

A general idea, which has proven successful in science, is that a minimal number of assumptions should be made in order to obtain an explanation of a scientific problem. This is the principle of simplicity, which is sometimes called ”Occam’s razor”: In science we should use an as small as possible number of (as simple as possible) assumptions and metaphysical principles (dogmas) to explain what we have to explain. Galilei, the father of modern science, who himself eliminated several really severe dogmas of his time, would probably be satisfied with this criterion. This principle could also be expressed as the following requirement for the design of interpretations of physical theories: The number of concepts and notions, which cannot be subject to experimental test, should be as small as possible.

This principle of simplicity is one of the most important principles of physics. It is frequently used to find the mathematical structure of physical theories, sometimes explicitly18 and very often implicitly. It seems reasonable to use it also as a criterion for interpretations. If applied to our present problem it says: do not make philosophical assumptions which are not already part of the mathematical structure of QT19 . The statistical interpretation (SI) of QT fulfills obviously Ockham’s criterion while the CI does (in all of its versions) not. This has been shown already and will be discussed in more detail in sections 2345.

Of course, it is still a matter of intellectual taste, to accept or not accept the principle of simplicity as a valid criterion; it is not possible to prove that a certain interpretation is wrong or right. However, it is a compelling methodical requirement of natural science that scientists are open to all possible interpretations. Interestingly, the SI is not only simpler but also ’more radical’ than the CI. As will be discussed in section 6 the SI is in conflict with the metaphysical principle of reductionism20 . This principle, as applied to physics, was very successful in the past and is very deeply rooted in our thinking. Its breakdown is, however, not in conflict with any observations. As a consequence, the paradigmatic change in natural philosophy21 implied by the SI is more radical than the change implied by the CI; a more detailed discussion will be given in sections 56. It will be shown in section 5 that one of the main motivations of the CI is to establish the compatibility of the QT with the principle of reductionism. In other words, the CI attempts to preserve, in the process of the transition to the QT, an as large part as possible of the logical structure of deterministic physics, i.e. of classical mechanics.

The relation between classical physics and quantum physics (the transition in both directions) is a subject, which can be studied by mathematical means and should allow important conclusions about the meaning of QT. The transition from QT to classical physics is discussed in section 4. The inverse transition, from classical physics to QT, is dealt with in section 7, where several of my papers, which establish a new quantization procedure, based on purely statistical assumptions, are listed. Summarizing sections 4 and 7, one may say that the results of a careful investigation of the relation between classical physics and quantum physics are only compatible with the statistical point of view; individualistic interpretations like the CI lead to inconsistencies and contradictions.

A summary of the essential points dealt with in this web-review may be found in section 8. A general conclusion is drawn in the final section 9.