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Complementarity and the Uncertainty Principle

The third section of the Como paper deals with how complementarity resolves the apparent paradox of the uncertainty relationship. Since these relations express the crux of the non-classical aspect of quantum theory, understanding them through complementarity will improve our appreciation of Bohr's point of view.
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Bohr's analysis was essentially coeval with Heisenberg's discovery, but he never intended complementarity as merely an explanation of the uncertainty principle. Einstein's criticisms forced Bohr repeatedly into an analysis of experimental situations designed to overcome the limits of the principle. Thus, in the years following the Como paper, Bohr's original approach from an analysis of the application of the classical modes of description was largely lost from sight.
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The belief that Bohr designed complementarity simply as his analysis of the uncertainty principle is historically unfounded, but it is a natural mistake, for he was anxious to show how complementarity explains the uncertainty principle. For physicists raised in the framework of classical physics, the discovery of the uncertainty principle was a stunning surprise. Thus it is hardly remarkable that in speaking to a group of physicists at Como, Italy, Bohr was very eager to show how his new framework of complementarity was able to interpret this principle.
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As Bohr understood it, both his complementarity and Heisenberg's uncertainty principle were the consequences of the quantum postulate. Complementarity was the consequence for a conceptual framework in which phenomena are described, while Heisenberg's discovery was its formal mathematical consequence. In quantum mechanical formalism, these parameters are not independent of each other, but are linked reciprocally by the measure of discontinuity in change of state symbolized by Planck's constant.
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It follows mathematically from uncertainty relations that the classical formalism can only be defined within limits fixed by the element of discontinuity involved in the change of state of a quantum system. Since the initial conditions cannot be defined beyond this point, theoretical predictions of the future state of the system are "indeterministic" or probabilistic in nature. However, Bohr believed that even though the uncertainty principle was logically entailed by the quantum postulate, the formalism itself could not tell us whether this limitation was merely a limit on knowledge or if it reflects a deeper ontological "indeterminacy".
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Nevertheless, classical formalism must be regarded as a special case of statistical determinism in which the statistical spread of determinism is negligible relative to the precision of the measurement. Since these are in fact the sort of interactions which are described classically (in line with Bohr's correspondence principle) the predictions quantum mechanics make will approach those of classical mechanics as the exchange of energy or momentum involved in the exchange increases in size relative to that expressed by Planck's constant. In other words, as we deal with larger and larger objects, statistical determinacy will converge to strict determinism.
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Statistical predictions in quantum mechanics might be interpreted as predictions describing behavior between large collections of atomic systems, each of which can be described by as yet undiscovered deterministic laws. In fact, the statistical spread of the formalism can only be confirmed by measurements on many atomic systems. Bohr categorically rejected this interpretation, however. He writes:

One might perhaps believe that the properties of the elements do not inform us directly of the behavior of individual atoms but, rather, that we are always concerned only with statistical regularities holding for the average conditions of a large number of atoms. ...The elements have, however, other properties which permit of more direct conclusions being drawn with respect to the states of motion of the atomic constituents. Above all, we must assume that the quality of the light which the elements in certain circumstances emit and which is characteristic of each element is essentially determined by what occurs in a single atom. Niels Bohr, The Atomic Theory
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Bohr proposed the quantum postulate to develop a mechanics which would represent atomic systems as mechanically stable. This property of stability must be attributed to each individual atom, thus the quantum postulate expresses a fact about the behavior of individual atomic systems. Since the postulate entails the statistical nature of the predictions of the formalism, the loss of determinism must be a consequence of the behavior of individual atoms. Thus Bohr declares as "vain the...
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...repeated, expressed hopes of avoiding the essentially statistical character of quantum mechanical description by the assumption of some causal mechanism underlying the atomic phenomena and hitherto inaccessible to observation... Above all such hopes would seem to rest upon an underestimate of the fundamental differences between the laws with which we are concerned in atomic physics and the everyday experiences which are comprehended so completely by the ideas of classical physics...the peculiar stability properties of atomic structures which are in obvious contrast with the properties of any mechanical model, but which are so intrinsically connected with the existence of the quantum of action, form the very condition for the existence of the objects and measuring instruments, with the behavior of which classical physics is concerned. Letter From Bohr to John Slater

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Different Interpretations of the Uncertainty Principle

The difficulty many philosophers and physicists have with indeterminism makes it easy to misinterpret what Bohr was trying to say. To him, indeterminism was a theoretical reflection of a fundamental fact about the behavior of individual atoms. Heisenberg's discovery of the uncertainty principle in 1927 was no surprise for Bohr, as it was for most other physicists of the time. His whole orientation had prepared him for just such a revolution.
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Before going into Bohr's analysis of the uncertainty principle, two points should be made clear. First, the mathematical formalism expressing relationships encompassed by the "uncertainty principle" are straight forward deductive consequences of the quantum postulate. All too often, discussions of the principle begin with a series of thought experiments intended to demonstrate that observations which determine the value of one observable within a certain range require physical conditions which preclude determining its canonically conjugate observable within a range that would contradict the uncertainty principle.
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It is then easy to get the mistaken impression that the principle expresses an empirical generalization derived from analyzing physical situations and that such a presumed empirical discovery is then injected as a postulate into the theoretical formalism. However, this is not the case. Heisenberg first developed his formalism for theoretical representation of the atomic system processes, and then showed that the consequence of his formalism was that the system could not be characterized by a state which was defined in terms of precise values for both canonically conjugate parameters. Since this analysis of complementarity doesn't require studying the mathematical formalism, there is no need to consider the derivation of the principle itself. The concern here is to give Bohr's theoretical conclusions some physical significance.
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Heisenberg's Interpretation

Heisenberg was led to his discovery by proposing that "nature allowed only experimental situations to occur which could be described within the framework of the formalism of quantum mechanics". In considering various thought experiments, he, and later Bohr, tried to show it was impossible to determine with arbitrary precision the values of both of the canonically conjugate observables. The reason for this is notoriously ambiguous in Heisenberg's own presentations, however.
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In some cases, he seems to imply that uncertainty arises because the observation "disturbs" the experiment. In this case, observation would require interaction, and the uncertainty arises there. However, interactions do not arise continually in atomic systems, therefore this interpretation must accept the classical presumption that all physical systems exist in well-defined states theoretically represented by the concept of the classical state. Here it is meaningful to talk about the classical state of the quantum system, even though we cannot precisely know the parameters that define any particular state.
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The Verificationist Viewpoint

However, in other cases, Heisenberg seems to imply a more "verificationist" view that since it's impossible to verify empirical statements for atomic systems, assertions making use of this concept are meaningless. He comments: Any use of the words "position" and "velocity" with an accuracy exceeding that given by [the uncertainty principle] is just as meaningless as the use of words whose sense is not defined...one should remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all - in spite of the fact that these sentences produce some kind of picture in our imagination... one should be especially careful in using the words "reality", "actuality", etc., since these words very often lead to statements of the type just mentioned. From Heisenberg interviews
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The verificationist view is associated with the logical positivist view, which we have already seen was pervasive during that time. It is worth noting a rather obvious inconsistency in the same work from which the above passage was quoted, Heisenberg seems to make a claim about what the electron "really" is. Looking at how Bohr approached this same problem, it's easy to see how different his thinking was.
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How Bohr Saw Heisenberg's Discovery as Complementarity

Typically the two parameters involved in the uncertainty principle are of position and momentum, or of energy and time. Bohr saw Heisenberg's discovery from the point of view of the complementarity of the two classical modes of description. Bohr saw the determination of position or time as the goal of the mode of space-time coordination, while seeing the determination of momentum or energy as a goal of applying the "claim of causality" through the conservation principles. Just as the uncertainty principle allows one to determine position and time or momentum and energy with an arbitrary degree of precision, so the two classical modes could be used together, but only in a complementary way.
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From Bohr's point of view, the uncertainty principle does not express any consequence of "disturbing" the system by observation. But at the same time, his analysis does depend on recognizing that observation involves an interaction that is "uncontrollable" once the quantum postulate is accepted. Bohr tried to show that the classical state of the system is only attainable by assuming it is possible to apply both spatial-temporal and causal modes of description to the system at the same time. Since the quantum postulate expressly says this cannot be done, the classical concept of the atomic system is no longer defined. For this reason, the disturbance principle is at odds with the analysis of the uncertainty principle from the view of complementarity.
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In fact, Bohr already arrived at the need for a complementary point of view before he knew of the uncertainty principle, but quite naturally saw Heisenberg's discovery as confirmation of his own analysis. He even considered using the word "reciprocal" (derived from the mathematical way of expressing the uncertainty principle between the canonically conjugate parameters) instead of "complementary" to refer to the two modes of description.

The Disturbance Principle
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Though it forms no part of complementarity, the disturbance principle was frequently defended as part of the Copenhagen Interpretation and often identified with Bohr's view in the years following Heisenberg's discovery. From the perspective of Heisenberg, it appeared that the basis of the disturbance principle lay in the fact that the instruments doing the observing "disturbed" the observed system such that its state after observation is no longer what was determined in the measurement. This interpretation compares observation of atomic systems to measuring, for example, the inner workings of a wrist watch using a yardstick.
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However, the disturbance interpretation plays havoc with the facts behind the genesis of the uncertainty principle and its status within the mathematical formalism of quantum mechanics. The principle is a straight forward deductive consequence of the quantum theoretical formalism which provides a highly confirmed means of predicting the outcome of interaction between radiation and matter. There is no mention of disturbance in the derivation of the principle itself, nor of how to go about determining the relevant parameters.
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The design of experiments is relevant only to interpreting the physical significance of the principle. The assumption that the classical system really exists in a classical mechanical state supposes the question of whether an experiment could be designed which will yield greater knowledge about the state of the atomic system than the uncertainty principle allows. If this could be done, the theory would be properly judged incomplete.

The disturbance interpretation mistake becomes apparent when we realize, that according to it, we could only approach classical ideals of strict determinism if our measuring instruments were the size of atoms. However, it is only an immense difference between the dimensions of ordinary human experience and those involved in atomic processes that made strict determinism a nearly obtainable goal. If our instruments were the same size as atoms, then the role of the quantum in an interaction would be ever increasing rather than decreasing, as the disturbance interpretation suggest.
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In classical mechanics, the observation also "disturbs" the observed, but the disturbance is either negligible or "controllable" and so can be accounted for in defining the state of an isolated system after the observation interaction. In quantum theory, ordinarily the effect of the interaction cannot be considered negligible nor "controllable". Since the disturbance interpretation makes it appear that the uncertainty principle is a empirical generalization, it's unable to explain why this alleged disturbance cannot be determined in the quantum framework, and allows a return to classical deterministic formalism.
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The Epistemic Interpretation

A second possible interpretation, the "epistemic interpretation", holds that the uncertainty principle is a consequence that it was developed to describe not the properties of atomic systems, but rather what we can know about nature as we experience it. If we accept that the adequacy of a theory rests on its success in predicting observable phenomena, then the uncertainty principle gives no reason to hold that quantum theory is inadequate. The theory is in no way incomplete because it allows only either a measurement on position and time, or momentum and energy, because in fact no phenomena allows us to determine such information. The uncertainty principle merely expresses this fact and warrants accepting it as a mathematical instrument for predicting relevant phenomena.

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The epistemic interpretation may have been motivated by talk that properties of a transphenomenal nature is meaningless and irrelevant to science. Classically, the notion of an independent reality "existing" apart from its phenomenal manifestation could be regarded as that concept of an isolated system. Since classical formalism describes the motions of particles and waves, the reality we experience should be consistent regarding such particles and waves. However, what quantum theory tells us is that if its description of its atomic systems is considered complete, since it cannot define a classical mechanical state for these systems, they cannot be considered particles or waves. It follows that the notion of an independent reality cannot be described as previously thought from the classical viewpoint.
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While the concept of an independently existing reality can no longer be assumed to be describable, it doesn't necessarily follow that this concept of an independent reality bears no relation to the world as experienced or as described in scientific theory. However, if the epistemic interpretation cannot establish that the notion of an independently existing reality is meaningless, how does it justify its claim that our knowledge is limited in the way expressed by the uncertainty principle?
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On one hand, if the meaningfulness of any claims concerning the character of an independent reality is denied, then it cannot be said the reason we cannot know, for example, the simultaneous position and momentum of a particle is because in reality there are no properties to be known. But if there is no such reason for the conclusions of the uncertainty principle, why not instead conclude that an altogether different theoretical formalism will one day be invented to escape these limitations and return to strict determinism?
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On the other hand, if the possibility of describing an independently existing reality is allowed, then the theory is complete in providing everything that can be known about such objects. The fact that there are limitations on how we precisely determine the parameters is a consequence of a deeper limitation on what it is that we can actually know. In this interpretation, the fact that nature has thus far hidden from us any means of simultaneously determining position and momentum of a particle may be revealed as only a limitation on our imagination, with deeper mysteries lurking underneath that.
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Either way, the epistemic interpretation cannot be made consistent with Bohr's claim that quantum theory is complete without admitting that the limitation on knowledge expressed by the uncertainty principle results from a character of a reality behind the phenomena. If it is accepted, as is the classical viewpoint, that parameters do correspond to properties of an independent reality, since quantum theory uses such parameters implies that nature possesses the corresponding properties. The proper conclusion would then be that quantum theory is indeed incomplete and that one day we will find a complete one.
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Einstein and Bohr

Einstein seemed to think Bohr was defending the epistemic interpretation with his framework of complementarity. By this time, Einstein had totally repudiated the view that science is only concerned about developing a formal means for predicting phenomena. This is the reason he repeatedly attempted to devise thought experiments which would reveal more about the system than the quantum formalism would permit, thereby dis-proving Bohr's belief it was complete. In fact, if Bohr had defended the epistemic interpretation of the uncertainty principle, as Einstein believed, Einstein's attitude would have expressed open-minded characteristics of scientific progress while Bohr would be seen as dogmatic.
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Bohr's Interpretation of the Uncertainty Principle

Finally, let's turn to Bohr's interpretation of the uncertainty principle. He writes According to the quantum theory, just the impossibility of neglecting the interaction with the agency of measurement means that every observation introduces a new uncontrollable element. Indeed, it follows from the above considerations that the measurement of the positional co-ordinates of a particle is accompanied not only by a finite change in the dynamic variables, but also the fixation of its position means a complete rupture in the causal description of its dynamical behavior, while the determination of its momentum always implies a gap in the knowledge of its spatial propagation. Just this situation brings out most strikingly the complementary character of the description of atomic phenomena which appears as an inevitable consequence of the contrast between the quantum postulate and the distinction between object and agency of measurement. Como Lecture
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In his debates with Heisenberg, Bohr would insist that in order for the theoretical representation of the physical system to have any empirical content at all, it must be possible to derive from it a description of the observed phenomena, treated as a causal result of the behavior of the atomic system. To give classical terms the empirical reference they must have requires interacting with the system, however. But the quantum postulate states that the interacting whole has an individuality which prohibits unambiguously defining the state of the system while it is in observation.
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Thus, the physical conditions necessary for observation are complementary to those necessary for defining the state of the system. Bohr regarded the uncertainty principle as directly expressing the formal consequence of this complementarity between the mode of space time coordination and the mode of causality description. Since the quantum postulate denies the classical justification for regarding the parameters used to define the state as pictures of the properties of an independent reality, Bohr repudiated the classical correlation of the parameters with the properties of the object regarded as isolated from any observed interaction. The "pictures" we can form using these concepts of an isolated object refer not to a concrete reality lying behind the phenomena, but to what Bohr called "abstractions":
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On the whole, it would seem scarcely justifiable, in the case of the interaction problem, to demand a visualization by means of space-time pictures. In fact all our knowledge concerning the internal properties of atoms is derived from experiments on their radiation and collision reactions [i.e., on interactions], such that the interpretation of experimental facts ultimately depends on the abstractions of radiation in free space and free material particles [i.e., on systems as isolated]. Hence, our whole space-time view of physical phenomena, as well as the definition of energy and momentum depends ultimately on these abstractions. In judging the application of these auxiliary ideas, we should only demand inner consistency, in which connection special regard has been paid to the possibilities of definition and observation. Como Lectures
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The use of classical terms to describe atomic systems as "particles" or "waves" interacting with the measuring apparatus refers not to a concrete underlying reality, but to an abstraction which is necessary for describing the phenomena as interactions between the measuring agencies and the physical systems decried in quantum theory. Since the disturbance interpretation holds that the objects of the atomic domain really do have properties corresponding to the parameters that define their classical state, Bohr's interpretation is that the limit expressed by the uncertainty principle is merely a consequence of the formalism we must use to describe what it is we are observing.
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Common Misunderstandings About Bohr's Viewpoint

Because we must still describe observations in terms of the classical framework, it is easy to view the uncertainty principle as a limit on knowledge. However, this misunderstanding is a consequence of presupposing that the descriptive concepts of "particle" and "wave" picture the object to which they refer as existing in isolation away from any observation. In other words, we fail to make the quantum postulate a true postulate in our description of nature. Consequently, we imagine somehow that the limitations expressed by the postulate, and the uncertainty principle, are not inherent in nature. And one day they will be circumvented when we gain improved empirical information.
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Bohr, on the other hand, argued against this classical tendency to interpret the concepts as also referring unambiguously to an independent and real object. He concluded that the concepts by which we express what we learn in an observational interaction refer to properties of phenomenal objects which can be described only in the interaction of the system with the observing instruments. By recognizing the complementarity between space-time coordination and causal description, Bohr avoided the dilemma that gives trouble to the epistemic interpretation, since each phenomenal appearance is a unique individual event.
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Wave-particle dualism is not a feature of reality, but rather a feature of our way of representing reality, and of course then the possibility is open for different descriptions which may remove such a dualism. The fact that Bohr emphatically believed such a search was futile reveals his conviction that complementarity was a result of the very way we observe our reality.

As Bohr would say, "we must relearn the presuppositions governing the use of our most elementary concepts". Complementarity tells us that since specific parameters of a physical theory have empirical evidence only if interpreted as referring to specific properties of observed phenomena, when those same parameters are used to define the state of the system isolated from observation, we cannot assume they can be interpreted as referring to properties possessed by that system apart from observational interactions.
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At the same time, the framework of complementarity cannot be coherent if it ignores the question of how to refer to this "independent reality" which in interacting with observational instruments produces the phenomena which are described by the theory. Facing up to the epistemological and ontological consequences concern the next two parts of this review. This ends Part 4 of this review. Thanks for reading! Comments are appreciated!

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The Framework of Complementarity

Part 1 - Overview Early Years Bohr Formulates Complementarity
Part 2 - Argument for Complementarity
Part 3 - Comments on Complementarity
Part 4 - Complementarity and the Uncertainty Principle
Part 5 - Refinement of Complementarity
Part 6 - Extension of Complementarity
Part 7 - The Nature of Empirical Knowledge
Part 8 - Complementarity and the Metaphysics of Quality