I n the early 20th century, the groundbreaking theories of quantum mechanics and special relativity emerged at roughly the same time.At first, both theories appeared to be entirely incompatible with each other.While the quantum equations written out by Erwin Schrödinger and Wolfgang Pauli dealt with the behaviours of electrons, atoms, and molecules moving at speeds encountered in classical physics, the rules of special relativity first imagined by Albert Einstein described the behaviour of matter moving at close to the speed of light.Initially, this apparent clash of theories presented a crisis in physics.However, in the late 1920s, further theoretical work carried out by physicists including Paul Dirac, Oskar Klein, and Walter Gordon culminated in a new set of equations, which proved definitively that quantum mechanics and special relativity are compatible with each other after all. An educational challengeYet with this groundbreaking discovery came an entirely new problem.In adding a new layer of complexity to the equations of non-relativistic quantum mechanics, the more complete theories of relativistic
I n the early 20th century, the groundbreaking theories of quantum mechanics and special relativity emerged at roughly the same time.At first, both theories appeared to be entirely incompatible with each other.While the quantum equations written out by Erwin Schrödinger and Wolfgang Pauli dealt with the behaviours of electrons, atoms, and molecules moving at speeds encountered in classical physics, the rules of special relativity first imagined by Albert Einstein described the behaviour of matter moving at close to the speed of light.Initially, this apparent clash of theories presented a crisis in physics.However, in the late 1920s, further theoretical work carried out by physicists including Paul Dirac, Oskar Klein, and Walter Gordon culminated in a new set of equations, which proved definitively that quantum mechanics and special relativity are compatible with each other after all. An educational challengeYet with this groundbreaking discovery came an entirely new problem.In adding a new layer of complexity to the equations of non-relativistic quantum mechanics, the more complete theories of relativistic
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.
Science has developed through a variety of investigations more or less over the time scale of human existence. On this scale, quantum mechanics is an extremely young field, existing essentially for only just over a century. Even our understanding of classical mechanics has existed for a comparatively long period—roughly having been formalized with Newton's equations published in his Principia Mathematica , in April 1686 (1687). Up until the very end of the nineteenth century, mechanics in the form of Newton's laws were comfortable, being understood and accepted by all who dealt with the motion of reasonable bodies. With the turn of the new century, however, mechanics would be shaken to its roots. The change began already on December 14, 1900, when Planck (1900) presented his new theory of black-body radiation to the German Physical Society, and this change would affect mechanics in physically small structures (and lead to quantum mechanics). Mechanics of large bodies would similarly be changed in 1905 with the arrival of Albert Einstein's (1905a) relativity theory. In this book, we will deal with only the first of these two great breakthroughs.
The turn of the XXth century witnessed a revolution in physics comparable to Isaac Newton’s discovery of the universal laws of mechanics and of gravitation three centuries earlier. The world required to be described in novel terms, as the immutable, deterministic view of our familiar universe had given way to a new world picture, one which featured chance, flux, and an incessant upsurge of waves of matter. Such a worldview was so radically new and counterintuitive that it gave rise to strong debates, to the effect that Albert Einstein himself tried to oppose it on the grounds that “God does not play dice”. In spite of the intense debates that accompanied its emergence, quantum mechanics quickly proved an incredibly efficacious new tool to understand and to predict a wide array of new phenomena. It was so successful that in no time it broke free from the environment of research labs to become part of daily life, making it possible, for example, to understand why some materials were conductors, while others were insulators. Along with it, came, too, the discovery of transistors, on which much of modern electronics rests. It also led to understand how novel materials known as superconductors allow the transport of electricity with no loss, thus paving the way for new developments in the fields of medical imagery or energy distribution. Other aspects of the quantum theory led to the development of atomic clocks of astounding accuracy, which would be wrong by no more than fifteen seconds, had they been set at the beginning of the universe. A hundred years later, at the turn of the XXIst century, Quantum mechanics has lost none of its astounding power. Contemporary research has undertaken the task of exploring its less immediately perceptible aspects. Groundbreaking developments have ensued, such as the teleportation of grains of light or the possibility, once predicted by the great physicist Richard Feynman, to build, one day, novel computers which, unlike the ones we are familiar with today, will be able to process innumerable numbers of operations in parallel. The Wright Colloquium will be for us the occasion to explore, in the company of five internationallyknown specialists in the field, some of the fascinating aspects of quantum mechanics. We will appraise how efficiently quantum physics can describe our world, and confront its limitation when it is faced with the infinitely small (in relation to recent experiments carried out at the CERN) as with the incommensurably large scale of sidereal spaces. We will appreciate the extent to which quantum physics already has impacted our everyday lives, and evoke the way in which novel fields such as quantum computers and quantum information will entail profound changes in the future. The quantum adventure has only just begun! Monday 15 November 2010 - 18h30 Jochen Mannhart Center for Electronic Correlations and Magnetism, University of Augsburg, Germany Quantum Physics on the Scale of Daily Life Tuesday 16 November 2010 - 18h30 Wolfgang Ketterle Nobel Laureate 2001 (Physics), Department of Physics, Massachusetts Institute of Technology, Cambridge, U.S.A. When freezing cold is not cold enough New forms of matter close to absolute zero temperature Wednesday 17 November 2010 - 18h30 David Gross Nobel Laureate 2004 (Physics), Kavli Institute for Theoretical Physics, University of California, Santa-Barbara, U.S.A. Quantum Mechanics of the Very, Very Small and the Very, Very Large Thursday 18 November 2010 - 18h30 Alain Aspect CNRS senior scientist and Professor Institut d'Optique and Ecole Polytechnique Palaiseau, France. From Einstein’s intuition to quantum bits: a new quantum age? Vendredi 19 novembre 2010 - 18h30 Rainer Blatt Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences and University of Innsbruck, Austria The Quantum Way of Doing Computations
The quantum revolution in physics of the 20th century has completely transformed the nature not only of physics and science in general, but also of philosophy and our worldview. Quantum revolution began in the early 19th century with the discovery of ‘quantum’ (plural ‘quanta’) principally by Max Planck, and with theories and discoveries in the field of relativity, linked inseparably with the name of Albert Einstein. Max Planck came across the quanta when he was studying what is known as the ‘blackbody radiation’. In classical physics in particular and the classical world in general, there was no serious problems with objects, objectivity and ontology. In classical physics, particles and fields have an existence independent of the observer. In broader view, everything, the world, exists independent of the observer and of the consciousness. The first quantum-mechanical theories on electron in mid-1920s did not take into account the effect of velocity of light, and were therefore non-relativistic.
Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Additionally, relativistic quantum mechanics is the basis for the standard model of elementary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels. However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to require the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent profound work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of recent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.
Abstract One serious objection to the whole of quantum theory as we have developed it so far, is that it is inconsistent with the theory of relativity. In order to see how we may overcome this defect, at least as far as the special theory of relativity is concerned, we shall first recall some of its basic features. Shortly after Einstein’s original 1905 papers on the subject, Minkowski realized that the physical ideas could also be understood geometrically, by combining space and time into a single four-dimensional real vector space, M, whose elements we shall call four-vectors.