In this monograph, we shall present a new mathematical formulation of quantum theory, clarify a number of discrepancies within the prior formulation of quantum theory, give new applications to experiments in physics, and extend the realm of application of quantum theory well beyond physics. Here, we motivate this new formulation and sketch how it developed. Since the publication of Dirac's famous book on quantum mechanics Dirac, 1930] and von Neumann's classic text on the mathematical foundations of quantum mechanics two years later von Neumann, 1932], there have appeared a number of lines...