The Mathematical Formalism of Quantum Mechanics

R. Bonventre

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.

Sol Wieder

Y. B. Band, Y. Avishai

Carlo Rovelli C

Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.

Shabnam Siddiqui

This chapter discusses the mathematical formalism of the theory of quantum mechanics and present the differences between the theory of classical mechanics and quantum mechanics. The mathematics that is used to account for the indeterminacy of quantum mechanics renders the mathematical approach of quantum mechanics distinct from that of classical mechanics. In quantum mechanics, any dynamical variable that is an observable that can be measured experimentally is described as a linear Hermitian operator. In quantum mechanics, it is not only necessary to define an observable, but it is also necessary to define the states associated with it. When a system is in a linear superposition of states of an observable, the number of different values of an observable observed in an experiment is equal to the number of states that a quantum mechanical system can jump into upon measurement of an observable.