THE DEVELOPMENT OF DIFFERENTIAL CALCULUS

After the advent of algebra in the sixteenth century, mathematical discoveries inundated Europe. The most important were differential calculus and integral calculus, bold new methods for attacking a host of problems that had challenged the world's best minds for more than 2000 years. Differential calculus deals with ideas such as speed, rate of growth, tangent lines, and curvature, whereas integral calculus treats topics such as area, volume, arc length, and centroids.

Work begun by Archimedes in the third century B.C. led ultimately to the birth of integral calculus in the seventeenth century. This development has a long and fascinating history to which we shall return in Chapter 7.

Differential calculus has a relatively short history. Its principles were first formulated early in the seventeenth century when a French mathematician, Pierre de Fermat, tried to devise a way of finding the smallest and largest values of a given function. He imagined the graph of a function having, at each of its points, a direction given by a tangent line, as suggested by the points labeled in Fig. 3.1.

THE RISE OF VECTOR ANALYSIS

Throughout their history, mathematics and physics have been intimately related; a discovery in one field led to an improvement in the other. Early natural philosophers grappling with quantities such as distance, speed, and time used geometry inherited from the Greeks to explore physical problems. But in the tumultuous years of the seventeenth century, physics underwent a transformation – a shift in emphasis from numerical quantities, such as distance and speed, to vector quantities, such as displacement and velocity, which have direction as well as magnitude.

The transition was neither abrupt nor confined to that century. It was necessary to invent new mathematical objects – vectors – and new mathematical machinery for manipulating them – vector algebra – to embody the properties of the physical quantities they were to represent.

FORMS OF ENERGY

The concept of energy, as we saw in Chapter 13, is subtle, elegant, and rich. It describes a dynamic property of the universe which is strictly and absolutely conserved; energy can neither be created nor destroyed. Not even in the presence of friction is energy ever lost; it is simply transformed into other forms. Nevertheless, the universe is winding down. Energy tends to be transformed from well-organized forms into more disorganized forms, until it becomes completely useless.

For many years the physical interpretation of quantum theory has been dominated by the “wave-particle duality” attitude of the “Copenhagen school.” This point of view is eloquently described in Bohr's collection of essays on the subject. Despite persistent concerns with apparent paradoxes and limitations to this interpretation (as exemplified by Schrodinger's cat, the paradox of Einstein, Podolsky and Rosen, among others), the Copenhagen view persists de facto in the daily life of the modern physicist. Traditional quantum theory has so successfully explained such a vast amount of data in atomic and molecular physics, solid state physics (and to a lesser extent, elementary particle physics) that little doubt can exist concerning its essential validity.

As a consequence of the overwhelming practical success of the “Copenhagen interpretation, ” the latter has acquired the status of dogma. For many years, therefore, most physicists have found it expedient to relegate the puzzling aspects of the theory to philosophers, and mathematicians. Nevertheless, a persistent interest in this subject has produced a significant and fascinating literature, reviewed by Jammer. In recent years, concerns over the proper meaning to be ascribed to quantum theory have produced an increasingly deep and incisive series of investigations.

These investigations fall generally into two categories: either (1) discussion of the philosophical content of the theory, or (2) analyses of the mathematical variants of the theory and their connection with differing interpretational schemes. Physicists tend to be detached from a commitment to philosophical issues, because of their realization of the transient character of the meanings attached to theories of the day. As a simple illustration of this we can mention the profound differences between nonrelativistic and relativistic quantum mechanics.

’The Logic of Quantum Mechanics” appeared as the title of a scientific work in 1936, with the paper of Garrett Birkhoff and John von Neumann (9). The use of the same title for this volume outlines the fact that a great part of the subject matter we shall deal with pertains to a research field that originated in that historic paper. This title, however, is not to be interpreted as focusing on the propositional calculus that mirrors the structure of quantum mechanics, the so-called “quantum logic” with the word “logic” used in technical sense; rather, the title should be interpreted as focusing on the mathematical foundations of quantum mechanics. The complex edifice of this theory contains simpler substructures that have direct physical bases; each of them can explain some aspects of the behavior of quantum systems. This was also the idea of the classical books by G. W. Mackey (3), J. M. Jauch (3), and V. S. Varadarajan (3), which appeared in the sixties. The present volume includes results of more than a decade of active research which followed these classical works.

The volume is divided into three parts. The first contains an exposition of the basic formalism of quantum mechanics using the theory of Hilbert spaces and of linear operators in these spaces. We shall not follow the old tradition of striving to use concepts of classical mechanics to explain quantum facts (as in the so-called principle of correspondence or the wave-particle dualism)—a tradition that reflects the unusually long delay suffered by quantum mechanics before it acquired autonomy and internal coherence, breaking with the language of the theory to be superseded. The second part follows the program of decomposing quantum theory into its conceptual constituents, singling out the basic mathematical structures, isolating what may be founded on direct empirical evidence, and controlling how single assumptions contribute to shape the theory. In the third part we face the problem of recovering the Hilbert-space formulation of quantum mechanics, starting from the simpler and more general theoretical schemes examined in the second part.

Composition of Different and of Identical Systems

It is a general tendency of physics to interpret physical systems as composed of simpler, more elementary subsystems. The question “What is it made of?” is a very tempting and popular one. Though this attitude does not reflect a logical necessity, it has proved to be successful in significant circumstances: we think, e.g., of macroscopic matter as composed of atoms and molecules, of molecules as composed of atoms, of atoms as composed of nuclei and electrons, and of nuclei as composed of protons and neutrons. Loosely speaking, one might expect that this attitude will be successful whenever the interaction that packs the constituent subsystems together is not strong enough to make them lose their identity.

Here, we are concerned with the problem of constructing the quantum-mechanical formalism of a compound system knowing the formalisms of the constituent subsystems. Thus our first problem is to specify the Hilbert space of the compound system. For the sake of simplicity, let us restrict ourselves to the case of a system composed of only two subsystems.

The prescriptions of quantum theory vary according to whether one is faced with the composition of identical or nonidentical subsystems.

Sentences Associated with a Quantum System and Their Truth Values

In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject.

Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.