Basic Notions of Quantum Mechanics

Eike Neumann, Martin Pape, Thomas Streicher

The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch's Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category $\mathbf{QCB}_0$ which is equivalent to the category $\mathbf{AdmRep}$ of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann's Spectral Theorem.

William W. Parson

Jiří Souček

The new approach to quaternion quantum mechanics is given. It is shown that such a theory describes tachyons and that the quantum theory of tachyons should be a quaternionic one. This theory needs fundamental changes in basic physical assumptions and in a sense it is complementary to known physics. An important analogy between the basic notions of quaternionic and complex quantum mechanics emerges from the new scheme. The consequences for the theory of quantum and classical tachyons seem to be very important.

Adam M. Brandenburger

Correlation is a basic concept in both game theory and quantum mechanics. We show that there is a formal correspondence between the treatment of correlation in the two domains. We use this correspondence to comment on the notion of a quantum game. 1