Chapter 9. The Algebraic Formulation of Quantum Mechanics

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.

Ajit Kumar

Antonino Drago

シロウ イシカワ, Shiro Ishikawa, 史郎 石川

9.1 Statistical measurements (W∗-algebraic formulation) 9.2 The relation among systems (Proclaimʷ∗2 in SMTʷ∗) 9.3 Quantum mechanics in B(L²(R)) 9.4 The method of moments 9.5 Principal components analysis in MT

Rocco Duvenhage

Quantum mechanics is essentially a statistical theory. Classical mechanics, however, is usually not viewed as being inherently statistical. Nevertheless, the latter can also be formulated statistically. Furthermore, a statistical formulation of both can be expressed in terms of unital C*-algebras, in which case it becomes clear that they have the same general structure, with quantum mechanics a noncommutative generalization of the classical case. In purely mathematical terms it is seen that quantum mechanics is a noncommutative generalization of probability theory. The most important insight in this respect is that the projection postulate of quantum mechanics is a noncommutative conditional probability. This is the subject of Chapter 1 of the thesis. As ergodic theory (the long term behaviour of a dynamical system) is done in classical probability theory, it is then also done for noncommutative probability theory in Chapter 2. In particular, generalizations of Khintchine's recurrence theorem and a variation thereof for ergodic systems are proved, as well as various characterizations of noncommutative ergodicity. Lastly, in Chapter 3, recurrence and ergodicity is then investigated from a physical perspective in quantum and classical mechanics, by means of a quantum mechanical analogue of Liouville's Theorem in classical mechanics which was suggested in Chapter 1.