A growing number of commentators have, in recent years, noted the important affinities in the views of Immanuel Kant and Niels Bohr. While these commentators are correct, the picture they present of the connections between Bohr and Kant is painted in broad strokes; it is open to the criticism that these affinities are merely superficial. In this essay, I provide a closer, structural, analysis of both Bohr's and Kant's views that makes these connections more explicit. In particular, I demonstrate the (...) similarities between Bohr's argument, on the one hand, that neither the wave nor the particle description of atomic phenomena pick out an object in the ordinary sense of the word, and Kant's requirement, on the other hand, that both ‘mathematical’ (having to do with magnitude) and ‘dynamical’ (having to do with an object's interaction with other objects) principles must be applicable to appearances in order for us to determine them as objects of experience. I argue that Bohr's ‘complementarity interpretation’ of quantum mechanics, which views atomic objects as idealizations, and which licenses the repeal of the principle of causality for the domain of atomic physics, is perfectly compatible with, and indeed follows naturally from a broadly Kantian epistemological framework. (shrink)

An inequality in quantum mechanics, which does not appear to be well known, is derived by elementary means and shown to be quite useful. The inequality applies to 'all' operators and 'all' pairs of quantum states, including mixed states. It generalizes the rule of the orthogonality of eigenvectors for distinct eigenvalues and is shown to imply all the Robertson generalized uncertainty relations. It severely constrains the difference between probabilities obtained from 'close' quantum states and the different responses they can have (...) to unitary transformations. Thus, it is dubbed a master inequality. With appropriate definitions the inequality also holds throughout general probability theory and appears not to be well known there either. That classical inequality is obtained here in an appendix. The quantum inequality can be obtained from the classical version but a more direct quantum approach is employed here. A similar but weaker classical inequality has been reported by Uffink and van Lith. (shrink)

In part-1, I shall outline the principle details of Bohr’s interpretation. Bohr’s basic interpretive insight is ‘ quantum inseparability’ . Complementarity of phenomena and a “revision to our attitude towards physical explanation” then follow. Together , these can be said to constitute Bohr’s general viewpoint of ‘complementarity’. Bohr does not quite clearly spell out the content of these three ideas; I do.

We show that Bohr's principle of complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the CCRs. In particular, in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa. Equivalently, if there are nonzero projections corresponding to sharp position values, all spectral projections of the momentum operator map onto the zero element.