Towards Quantum Theory

Science is basically an endeavor for seeking some knowledge about the world. In the previous chapters I have mostly concentrated on statistical inference as a tool for obtaining such a knowledge. I will now turn to the second area which I intend to cover in this book: Quantum theory, which I will also see as a tool for obtaining knowledge of the world. Most of this chapter does not presuppose any familiarity with the quantum formalism; my ambition is to derive this foundation from reasonable assumptions. The basis is taken to be conceptual variables, physical variables that may be accessible or inaccessible, i.e., it may be possible or impossible to assign numerical values to them. In an epistemic process, the accessible variables are just ideal potential observations connected to an actor or to some communicating actors, what I earlier have called epistemic conceptual variables or e-variables. Group actions are defined on these variables, and using group representation theory, this is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible conceptual variables are derived as a result of the approach, and in the discrete case it is argued from the same approach that the possible values taken by the accessible conceptual variables are the eigenvalues of these operators. The interpretation of quantum states (or eigenvector spaces) implied by this approach is as focused questions to nature together with sharp answers to those questions. Orthogonal resolutions of the identity are then connected to the relevant questions; these may be complementary in the sense defined by Bohr. Instead of entering the general derivation of these results at once, I will first give some background, and I will then develop the main ideas in the case of a spin 1/2 particle, where less mathematics is required, before I turn to the general theory.