Unphysical Combination of Complementary Experiments

Let us take a look at how Bohr in his famous 1927 Como Lecture  describes complementarity as a fundamental aspect of Bohr's Copenhagen Interpretation still dominating textbook presentations of  quantum mechanics:

• The quantum theory is characterised by the acknowledgment of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests essentially upon the classical concepts.
• Notwithstanding the difficulties which hence are involved in the formulation of the quantum theory, it seems, as we shall see, that its essence may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to the classical theories and symbolised by Planck's quantum of action.

OK, we learn that quantum theory is based on a quantum postulate about an essential discontinuity symbolised as Planck's constant $h=6.626\times 10^{-34}\, Js$ as a quantum of action. Next we read about necessary interaction between the phenomena under observation and the observer:   

• Now the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. 
• Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation.
• The circumstance, however, that in interpreting observations use has always to be made of theoretical notions, entails that for every particular case it is a question of convenience at what point the concept of observation involving the quantum postulate with its inherent 'irrationality' is brought in.

Next, Bohr emphasizes the contrast between the quantum of action and classical concepts:

• The fundamental contrast between the quantum of action and the classical concepts is immediately apparent from the simple formulas which form the common foundation of the theory of light quanta and of the wave theory of material particles. If Planck's constant be denoted by $h$, as is well known: $E\tau = I \lambda = h$, where $E$ and $I$ are energy and momentum respectively, $\tau$ and $\lambda$  the corresponding period of vibration and wave-length. 
• In these formulae the two notions of light and also of matter enter in sharp contrast. 
• While energy and momentum are associated with the concept of particles, and hence may be characterised according to the classical point of view by definite space-time co-ordinates, the period of vibration and wave-length refer to a plane harmonic wave train of unlimited extent in space and time.
• 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, inherent in our very idea of observation.

Bohr clearly brings out the unphysical aspects of the basic action formula

• $E\tau = I \lambda = h$,  

where energy $E$ and momentum $I$ related to particle are combined with period $\tau$ and wave-length $\lambda$ related to wave.

Bohr then seeks to resolve the contradiction by naming it complementarity as an effect of interaction between instrument and object:

• Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects. 
• In quantum mechanics, however, evidence about atomic objects obtained by different experimental arrangements exhibits a novel kind of complementary relationship. 
• … the notion of complementarity simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instruments and the objects form an integral part of the phenomena. 

Bohr's complementarity principle has been questioned by many over the years:

• Bohr’s interpretation of quantum mechanics has been criticized as incoherent and opportunistic, and based on doubtful philosophical premises. (Simon Saunders)
• Despite the expenditure of much effort, I have been unable to obtain a clear understanding of Bohr’s principle of complementarity (Einstein).

Of course an object may have complementary qualities such as e.g. color and weight, which can be measured in different experiments, but it is meaningless to form a new concept as color times weight or colorweight and then desperately seek to give it a meaning. 

In the New View presented on Computational Blackbody Radiation the concept of action as e.g position times velocity has a meaning in a threshold condition for dissipation, but is not a measure of a quantity which is carried by a physical object such as mass and energy.

The ruling Copenhagen interpretation was developed by Bohr contributing a complementarity principle and Heisenberg contributing a related uncertainty principle based position times momentum (or velocity) as Bohr's unphysical complementary combination. The uncertainty principle is often expressed as a lower bound on the product of weighted norms of a function and its Fourier transform, and then interpreted as combat between localization in space and frequency or between particle and wave. In this form of the uncertainty principle the unphysical aspect of a product of position and frequency is hidden by mathematics.

The Copenhagen Interpretation was completed by Born's suggestion to view (the square of the modulus of) Schrödinger's wave function as a probability distribution for particle configuration, which in the absence of something better became the accepted way to handle the apparent wave-particle contradiction, by viewing it as a combination of probability wave with particle distribution.