Collimation processes in quantum mechanics interpreted in quantum real numbers

We reexamine the theory of quantum mechanics using non-standard real numbers, called quantum real numbers (qr-numbers), that are constructed from entities in the standard Hilbert space formulation of the theory. Our motivation is to elucidate certain apparently paradoxical features of the standard theory by enlarging the class of real numbers that physical quantities can take as numerical values. The concept of number is fundamental in the formulation of any physical theory and in this work we use a generalization of this concept that comes from topos theory and which we think provides a fundamental tool for the development of physical theories. Our main result is a reformulation of quantum theory in terms of qr-numbers in which collimation processes clarify the relation between the qr-number value of a physical quantities and the result of its measurement. We recover the probabilities of standard quantum theory as approximations to qr-number formulae. Incidentally we also show that it is possible to give a consistent interpretation of quantum mechanics using functions Tr ρ · A, where A is a fixed operator and the variable state ρ belongs to an open set of state space.