In principio erat verbum. It seems appropriate to add the first few words of St. John's gospel to the Latin quotations that were bandied about at the beginning of the recent Workshop on Cosmology, which was held at the Fermi National Accelerator Laboratory in Batavia, Ill. This outburst of Latinity and classical scholarship, which even included a dispute over whether the word apparatus belongs to the second or the fifth declension, seemed an appropriate beginning for a conference on a subject the origins of the universe that has fascinated scholars ever since classical times. In the beginning was the word, en arche hen ho logos. One of the meanings of logos in this context is a word descriptive of the basic structure of the universe and, more than descriptive, a dynamical word that outlines and perhaps even determines the course of the history that follows the beginning. From the philosophers of ancient Hellas to the professors of physics in modern universities, most have expected that this word would be mathematical. It is the last generation or two that has expected it to be quantum mechanical. Quantum mechanics has to describe everything, said Murray Gell-Mann of Caltech in Pasadena in the opening talk of the workshop. And it became clear that by everything he meant the work of historians and crime detectives and the songs of birds as well as the motions of galaxies. But he added: questions are all murky and border on the philosophical. The main reason for the dominance of quantum mechanics was cited by James Hartle of the University of California at Santa Barbara: laws of physics are quantum mechanical; quantum cosmology is the proper framework. Nevertheless, the large objects that dominate the universe as we now see it operate according to classical mechanics. The serious problem for all of physics, and for quantum cosmology in particular, is to find some kind of linkage by which classical physics can be generated out of quantum mechanics. A similar question occurs historically: If we trace the expansion of the universe backwards we eventually come to a point in history when the universe was so small that it had to behave as a whole as a quantum mechanical system. Figuring out how the present came out of the past is the problem.
In principio erat verbum. It seems appropriate to add the first few words of St. John's gospel to the Latin quotations that were bandied about at the beginning of the recent Workshop on Cosmology, which was held at the Fermi National Accelerator Laboratory in Batavia, Ill. This outburst of Latinity and classical scholarship, which even included a dispute over whether the word apparatus belongs to the second or the fifth declension, seemed an appropriate beginning for a conference on a subject the origins of the universe that has fascinated scholars ever since classical times. In the beginning was the word, en arche hen ho logos. One of the meanings of logos in this context is a word descriptive of the basic structure of the universe and, more than descriptive, a dynamical word that outlines and perhaps even determines the course of the history that follows the beginning. From the philosophers of ancient Hellas to the professors of physics in modern universities, most have expected that this word would be mathematical. It is the last generation or two that has expected it to be quantum mechanical. Quantum mechanics has to describe everything, said Murray Gell-Mann of Caltech in Pasadena in the opening talk of the workshop. And it became clear that by everything he meant the work of historians and crime detectives and the songs of birds as well as the motions of galaxies. But he added: questions are all murky and border on the philosophical. The main reason for the dominance of quantum mechanics was cited by James Hartle of the University of California at Santa Barbara: laws of physics are quantum mechanical; quantum cosmology is the proper framework. Nevertheless, the large objects that dominate the universe as we now see it operate according to classical mechanics. The serious problem for all of physics, and for quantum cosmology in particular, is to find some kind of linkage by which classical physics can be generated out of quantum mechanics. A similar question occurs historically: If we trace the expansion of the universe backwards we eventually come to a point in history when the universe was so small that it had to behave as a whole as a quantum mechanical system. Figuring out how the present came out of the past is the problem.
Those defile the purity of mathematical and philosophical truthy who confound real quantities with their relations and sensible measures. —ISAAC NEWTON, Principia Before we say anything about quantum mechanics, let us take a quick look at the Newtonian (or ‘classical’) view of the physical universe. According to that view, nature consists of an enormous number of particles. When Newtonian physics is combined with the theory of the electromagnetic field, it becomes convenient to think of these particles as dimensionless (even if there is a kind of conceptual strain involved in trying to think of something as having a mass but not any size ), and as possessing electrical properties – negative charge, or positive charge, or neutrality. This leads to the well-known ‘solar system’ view of the atom – with the electrons whirling around the nucleons (neutrons and protons) just as the planets whirl around the sun. Out of atoms are built molecules; out of molecules, macroscopic objects, scaling from dust motes to whole planets and stars. These latter also fall into larger groupings – solar systems and galaxies – but these larger structures differ from the ones previously mentioned in being held together exclusively by gravitational forces. At every level, however, one has trajectories (ultimately that means the possibility of continuously tracing the movements of the elementary particles) and one has causality (ultimately that means the possibility of extrapolating from the history of the universe up to a given time to its whole sequence of future states).
QUANTUM COSMOLOGY It is an honor, of course, but also a pleasure for me to join in this celebration of Murray Gell-Mann's sixtieth birthday and to address such a distinguished audience. Murray was my teacher and more recently we have worked together in the search for a quantum framework within which to erect a fundamental description of the universe which would encompass all scales – from the microscopic scales of the elementary particle interactions to the most distant reaches of the realm of the galaxies – from the moment of the big bang to the most distant future that one can contemplate. Such a framework is needed if we accept, as we have every reason to, that at a basic level the laws of physics are quantum mechanical. Further, as I shall argue below, there are important features of our observations which require such a framework for their explanation. This application of quantum physics to the universe as a whole has come to be called the subject of quantum cosmology. The assignment of the organizers was to speak on the topic “Where are our efforts leading?” I took this as an invitation to speculate, for I think that it is characteristic of the frontier areas of science that, while we may know what direction we are headed, we seldom know where we will wind up.
1124 NOTICES OF THE AMS VOLUME 45, NUMBER 9 I n the twentieth century, the quest for deeper understanding of the laws of nature has largely revolved around the development of two great theories: namely, general relativity and quantum mechanics. General relativity is, of course, Einstein’s theory according to which gravitation results from the curvature of space and time; the mathematical framework is that of Riemannian geometry. While previously spacetime was understood as a fixed arena, given ab initio, in which physics unfolds, in general relativity spacetime evolves dynamically, according to the Einstein equations. Part of the problem of physics, according to this theory, is to determine, given the initial conditions as input, how spacetime will develop in the future. The influence of general relativity in twentiethcentury mathematics has been clear enough. Learning that Riemannian geometry is so central in physics gave a big boost to its growth as a mathematical subject; it developed into one of the most fruitful branches of mathematics, with applications in many other areas. While in physics general relativity is used to understand the behavior of astronomical bodies and the universe as a whole, quantum mechanics is used primarily to understand atoms, molecules, and subatomic particles. Quantum theory has had a much more complex history than general relativity, and in some sense most of its influence on mathematics belongs to the twenty-first century. The quantum theory of particles—which is more commonly called nonrelativistic quantum mechanics—was put in its modern form by 1925 and has greatly influenced the development of functional analysis, and other areas. But the deeper part of quantum theory is the quantum theory of fields, which arises when one tries to combine quantum mechanics with special relativity (the precursor of general relativity, in which the speed of light is the same in every inertial frame but spacetime is still flat and given ab initio). This much more difficult theory, developed from the late 1920s to the present, encompasses most of what we know of the laws of physics, except gravity. In its seventy years there have been many milestones, ranging from the theory of “antimatter”, which emerged around 1930, to a more precise description of atoms, which quantum field theory provided by 1950, to the “standard model of particle physics” (governing the strong, weak, and electromagnetic interactions), which emerged by the early 1970s, to new predictions in our own time that one hopes to test in present and future accelerators. Quantum field theory is a very rich subject for mathematics as well as physics. But its development in the last seventy years has been mainly by physicists, and it is still largely out of reach as a rigorous mathematical theory despite important efforts in constructive field theory. So most of its impact on mathematics has not yet been felt. Yet in many active areas of mathematics, problems are Edward Witten is professor of physics at the Institute for Advanced Study. His e-mail address is witten@ias.edu.
It will be presented in this chapter a historical account of the consistent histories interpretation of quantum mechanics based on primary and secondary literature. Firstly, the formalism of the consistent histories approach will be outlined. Secondly, the works by Robert Griffiths and Roland Omnès will be discussed. Griffiths' seminal 1984 paper, the first physicist to have proposed a consistent-histories interpretation of quantum mechanics, followed by Omnès' 1990 paper, were instrumental to the consistent-histories model based on Boolean logic. Thirdly, Murray Gell-Mann and James Hartle's steps to their own version of consistent-histories approach, motivated by a cosmological perspective, will then be described and evaluated. Gell-Mann and Hartle understood that spontaneous decoherence could path the way to a concrete physical model to Griffiths' consistent histories. Moreover, the collective biography of these figures will be put in the context of the role played by the Santa Fe Institute, co-founded by Gell-Mann in 1984 in Santa Fe, New Mexico, where Hartle is also a member of the external faculty.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.