The Formal Theory 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.

Walter Greiner

Walter Greiner

Michel Le Bellac

We now start what will be the subject of this book: quantum field theory at non-zero temperature and/or non-zero chemical potential. Two competing formalisms have been used at zero temperature in order to study field theory: the operator formalism, which is the older one, and the path integral formalism, which represents the more modern approach. Actually, it may often be illuminating to look at a given problem from both points of view, although in some cases one of the formalisms may prove to be definitely superior to the other: for example, path integrals are much simpler when quantizing gauge theories. At finite temperature, both formalisms are useful, and it is instructive to be able to switch from one approach to the other. Since the simplest case of field theory is field theory with zero space dimension, or, in other words, quantum mechanics, we shall begin with a short description of quantum mechanics at finite temperature, or, equivalently, quantum statistical mechanics. We first deal with the path integral approach, and later on revert to the more conventional operator formalism. We shall be particularly interested in time-ordered products of position operators, which will generalize to time-ordered products of field operators in field theory.

Maciej Błaszak

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.

Walter Greiner