This paper begins the study of infinite-dimensional modules defined on bicomplex numbers. It generalizes a number of results obtained with finite-dimensional bicomplex modules. The central concept introduced is the one of a bicomplex Hilbert space. Properties of such spaces are obtained through properties of several of their subsets which have the structure of genuine Hilbert spaces. In particular, we derive the Riesz representation theorem for bicomplex continuous linear functionals and a general version of the bicomplex Schwarz inequality. Applications to concepts relevant to quantum mechanics, specifically the bicomplex analogue of the quantum harmonic oscillator, are pointed out.

The development of quantum information theory has renewed interest in the idea that the state vector does not represent the state of a quantum system, but rather the knowledge or information that we may have on the system. I argue that this epistemic view of states appears to solve foundational problems of quantum mechanics only at the price of being essentially incomplete.

This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including the spectral decomposition theorem. Applications to concepts relevant to quantum mechanics, like the evolution operator, are pointed out.Comment: 21 page

Cramer's transactional interpretation of quantum mechanics is reviewed, and a number of issues related to advanced interactions and state vector collapse are analyzed. Where some have suggested that Cramer's predictions may not be correct or definite, I argue that they are, but I point out that the classical-quantum distinction problem in the Copenhagen interpretation has its parallel in the transactional interpretation. RésuméL'interprétation transactionnelle de la mécanique quantique, proposée par J. G. Cramer, est sommairement revue, et quelques questions liées aux interactions avancées età l'effondrement du vecteur d'état sont analysées. Certains ont suggéré que les prédictions de l'interprétation de Cramer ne sont pas correctes ou ne sont pas bien définies. Je chercheà montrer qu'au contraire elles le sont, mais je signale que le problème de la distinction du classique et du quantique, inhérentà l'interprétation de Copenhague, a son parallèle dans l'interprétation transactionnelle.

We investigate, from a group-theoretical point of view, the possibility of implementing the so-called extended principle of relativity. This consists in postulating that the set of all equivalent reference frames contains frames whose relative velocities are larger than c, in addition to those whose coordinates are related by proper orthochronous Lorentz transformations. We show that implementing the extended principle of relativity by means of either real or complex linear transformations results in strong conflicts with experiment and/or intractable problems of interpretation. We then briefly analyze alternative approaches to four-dimensional superluminal transformations, in which the extended principle of relativity is either weakened or completely abandoned.