The Question of the Completeness of Quantum Mechanics

When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory.

Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level.

The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.

Yes-No Experiments

The first five sections of this chapter are intended to provide some intuitive physical content to the notions of state and proposition of a quantum system. We deliberately avoid any deductive attitude, and we also avoid formalizations: the ingredients we shall start with (yes-no experiments and preparations) are so primitive and unstructured that the construction of quantum mechanics out of them is certainly not at hand. What we are going to see is simply that there is natural physical motivation for the occurrence at the very basis of every statistical theory—quantum mechanics in particular—of an ordered structure of propositions, of a convex structure of states, and of a relationship between these structures that makes the states behave as probability measures on propositions. From Section 13.6 on, we shall sketch a more formalized model, based on Mackey's approach.

By a yes-no experiment is meant an experiment that makes use of a measuring instrument having just two outcomes, which without loss of generality we can agree to label “yes” and “no”. Of course, when talking of a yes-no experiment we always imagine that we have specified the physical system to which it pertains. We also imagine that the interaction between

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

The present volume is meant to be the first of a series on quantum mechanics and its applications. It deals with the foundations as well as the fascinating logic of quantum mechanics.

This volume is more general and at places less factual than other volumes of the ENCYCLOPEDIA; nevertheless the thorough presentation will guide the reader to gain an overview of a theory that, together with relativity, is regarded as the greatest achievement in physics in this century.

The Hilbert-Space Description

By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given.

In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.

Transition Probabilities between Pure States

In physicists' nomenclature the expression “transition probability” generally refers to some dynamical instability, more specifically to a nonzero probability for the system to make a transition from an initial to a final state. Our use of the term is not directly related to dynamical instabilities; rather we follow von Neumann's terminology, and the transition probability between two states is meant to represent, intuitively, a measure of their overlapping. To visualize this notion in an explicit example consider the states of linear polarization of a photon beam, let a be the polarization state filtered by a Nicol prism N1, and let β be the state filtered by a Nicol prism N2 rotated by an angle ϑ with respect to N1; then the transition probability between a and β is the probability for a photon in state α to pass N2, which is empirically known to be cos2 ϑ (Malus law), and equals the probability for a photon in state β to pass N1.

In the Hilbert-space formulation of quantum mechanics the idea of transition probability we have in mind is simply the modulus squared of the scalar product (see Sections 2.5 and 9.3).