The Bicomplex Heisenberg Uncertainty Principle

Abstract

Quantum mechanics is one of the two fundamental pillar of modern physics. The success of the theory can be found everywhere in our everyday life and essentially in every new product that we build. We just have to remember that every semiconductor chip usually uses a quantum behavior in an essential way, for example quantum tunneling, to work. Until now, none of the thousand of experiments realized have succeeded to contradicted or to find a problem with the predictions given by quantum mechanics. However, in spite of this incredible success, many profound questions are still open. For example, we have some problems understanding the measurement, the coherence and the decoherence process, as well as the interpretation of what the theory tell us about the world we live in (Schlosshauer, 2005). Among the possible ways of investigation that we have, we think that stressing the foundations of the theory at the level of the mathematical structure, on which the theory stands, could be a good way to understand why and how the theory works. The mathematical structure of quantum mechanics consists in Hilbert spaces defined over the field of complex numbers (Birkhoff & Von Neumann, 1936). The success of the theory has led a number of investigators, over many decades, to look for general principles or arguments that would lead quite inescapably to the complex Hilbert space structure. It has been argued (Stueckelberg, 1960; Stueckelberg & Guenin, 1961), for instance, that the formulation of an uncertainty principle, heavily motivated by experiment, implies that a real Hilbert space can in fact be endowed with a complex structure. The proof, however, involves a number of additional hypotheses that may not be so directly connected with experiment. In fact Reichenbach (Reichenbach, 1944) has shown that a theory is not straightforwardly deduced from experiments, but rather arrived at by a process involving a good deal of instinctive inferences. This was also pointed out more recently by Penrose (Penrose, 2005, p. 59);