Understanding quantum and classical complex systems

Physics is the quest for understanding fundamental laws of Nature. An important part of this quest is to reveal universal behaviors of individual objects. The ability to access ultra-high energies and ultra-fine scales allows us to identify elementary particles that are indivisible (e.g. electrons, quarks, etc.). Great successes have been achieved in the construction of effective mathematical models for the dynamics of these elementary particles, exploiting the symmetry and topology of the space-time of our universe.

This reductionism approach has yielded great theories in physics (e.g. the general relativity, the standard model, and even string theories). If we view our universe as a vast chessboard, we now have a quite good understanding of what the chess pieces are, and the rules on how each chess piece moves. Yet within this analogy, we also realize that a winning strategy in chess play is so much more than just knowing what the rules are. It is the interaction between different chess pieces, both spatially and temporally, potentially with very long range correlations, that makes the chess play so much more fascinating.

So is our universe. This emergentism perspective was eloquently promoted by P.W. Anderson's Science article "More is Different" (1972), and R.B. Laughlin's book "A Different Universe: Reinventing Physics from the Bottom Down" (2006). Stephen Wolfram's "A New Kind of Science" (2002) also gives a very comprehensive set of illustrations on how very simple rules can lead to profound complexity - a more computer scientists' perspective that models our universe as a giant computer. The gist of the story is that understanding the "source code" may not tell us much about what will really happen, if you run the program dictating almost countless particles over billions of years.

Thus begins the adventurous journey to uncover fundamental laws arising from not just a small number of individual objects, but from the collective behaviors of many. We call these the many-body systems, which can either be classical or quantum. By definition, such system consists of a large number of components interacting with each other. It becomes non-trivial when the interaction is large enough (as is often the case in the real world) that the constituent components are entwined with each other and lose their identities. This is when new objects or degrees of freedom emerge, that can be completely different from the microscopic ones. Our job is to recognize these new objects, understand the new physical laws governing these objects, and to be able to classify these objects based on universal characteristics independent of certain microscopic details.

The world of physics is so much richer with emergent behaviors from many-body systems, behaviors for which we can construct elegant effective theories at different energy scales. There are also more complex phenomena that depend very much on microscopic details. These phenomena tend to be chaotic with rare events that are hard to predict. New methodologies, mathematical tools and even new physics are needed to understand this largely uncharted territory. One thing is certain: even if we eventually have a grand unified theory for all elementary particles in the universe, it is only the time when we finish laying board for the game to start!