Since the first detailed analyses of the systems of the second compartment were done by Poincaré, I shall name this box P in his honor. It is stuffed with the chaotic dynamic systems that are the bread and butter of science. Among these systems are all the fundamental problems of mechanics, starting with three, rather than only two bodies interacting with one another, such as the earth, moon and sun, or the three atoms in the water molecule, or the three quarks in the proton.
Quantum mechanics, as it has been practiced for about 90 years, belongs in the third compartment, called Q. After the pioneering work of Planck, Einstein and Niels Bohr, quantum mechanics was given its definitive form in four short years, starting in 1924. The seminal work of Louis de Broglie, Werner Heisenberg, Erwin Schrödinger, Max Born, Wolfgang Pauli and Paul Dirac has stood the test of the laboratory without the slightest lapse. Miraculously. it provides physics with a mathematical framework that, according to Dirac, has yielded a deep understanding of "most of physics and all of chemistry" Nevertheless, even though most physicists and chemists have learned how to solve special problems in quantum mechanics, they have yet to come to terms with the incredible subtleties of the field. These subtleties are quite separate from the difficult, conceptual issues having to do with the interpretation of quantum mechanics.
The three boxes R (classic, simple systems), P (classic chaotic systems) and Q (quantum systems) are linked by several connections. The connection between R and Q is known as Bohr's correspondence principle. The correspondence principle claims, quite reasonably, that classical mechanics must be contained in quantum mechanics in the limit where objects become much larger than the size of atoms. The main connection between R and P is the Kolmogorov-Arnold-Moser (KAM) theorem. The KAM theorem provides a powerful tool for calculating how much of the structure of a regular system survives when a small perturbation is introduced, and the theorem can thus identify perturbations that cause a regular system to undergo chaotic behavior.
Quantum chaos is concerned with establishing the relation between boxes P (chaotic systems) and Q (quantum systems). In establishing this relation, it is useful to introduce a concept called phase space. Quite amazingly this concept, which is now so widely exploited by experts in the field of dynamic systems, dates back to Newton.
The notion of phase space can be found in Newton's mathematical Principles of Natural Philosophy published in 1687. In the second definition of the first chapter, entitled "Definitions", Newton states (as translated from the original Latin in 1729): "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly." In modern English this means that for every object there is a quantity, called momentum, which is the product of the mass and velocity of the object.
Newton gives his laws of motion in the second chapter, entitled "Axioms, or Laws of motion." The second law says that the change of motion is proportional to the motive force impressed. Newton relates the force to the change of momentum (not to the acceleration as most textbooks do).
Momentum is actually one of two quantities that, taken together, yield the complete information about a dynamic system at any instant. The other quantity is simply position, which determines the strength and direction of the force. Newton's insight into the dual nature of momentum and position was put on firmer ground some 130 years later by two mathematicians, William Rowan Hamilton and Karl Gustav-Jacob Jacobi. The pairing of momentum and position is no longer viewed in the good old Euclidean space or three dimensions; instead it is viewed in phase space, which has six dimensions, three dimensions for position and three for momentum.