An Introduction to Quantum Mechanics: From Facts to Formalism

GORDON J. TROUP

R. Bonventre

Mark Linne

Emmanuel E. Haven, Andrei Yu. Khrennikov

Mathematical formalism of quantum mechanics: brief introduction In this chapter, we present basic notions of quantum mechanics, and we emphasize “quantum mathematics.” To posit that the mathematical formalism of quantum mechanics can be used outside of quantum physics (as in sociology, economics, or finance) may be of interest. If one can steer away from merely positing and veer instead into convincing the reader that such a position is tenable, then some achievement has been booked. But the price to pay is that the formalism of quantum mechanics needs to be uncovered: how else can one judge whether what we posit is senseful? Thus, a reader can either (i) formally accept our viewpoint about the quantum-like paradigm (see Khrennikov [1]), or (ii) ask for a detailed analysis of the quantum foundations which would justify this paradigm. In the first case, it may well be sufficient to skip this chapter (but we should suggest to attempt to include Chapter 3). In the second case, we invite especially those readers who are concerned about the fact that the formal usage of the mathematical formalism of quantum mechanics may transfer major problems of quantum foundations to social science, economics, and finance, to read the present chapter carefully.

Jānis Ruža

A factual inconsistency of the nonlocality theory is demonstrated via a careful analysis of the conceptual and formal contents of Bell's theorem and the inequalities associated with it. The conceptual inconsistency of the theory follows from the consideration that it is, in principle, impossible to formulate quantum mechanics in terms of local hidden variables. The formal inconsistency of the nonlocality theory follows from the facts that: (i) nonlocality is not a physical property that can be represented by a function or an operator; (ii) it is impossible to derive Bell's inequality in a mathematically consistent way; and (iii) Bell's definition of locality is incorrect. Therefore, Bell's theorem cannot serve as the foundation for the introduction of the nonlocality notion in quantum mechanics.

S. Simons, Herman Erlichson