Reductionism versus Emergence

Recently I attended a talk at LMU (Ludwig-Maximilians-Universität) by Gabriel Kotliar of Rutgers Univ on "Strongly correlated electron systems." It was supposed to be on superconductivity, but he widened the topic in order to discuss his approach to solid state physics, namely "Dynamic Mean Field Theory." I know what Mean Field Theory (MFT) is, it's replacing the effect of large number of nearby electrons in a solid by their average (mean) properties, and using that average to determine the effect on one electron. I don't know what the "dynamic" means. In essence, though, he was arguing for the need for a new "standard model" of solids. Band theory, Fermi liquid theory, and density functional theory have all done that job, but now something more was needed.

To make this point, he started his talk off with a bit of philosophy by quoting Dirac, who, in 1929, wrote* "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that the exact application of these laws leades to equations much too complicated to be soluble." This is a nice statement of the reductionist viewpoint, which holds that we currently know all the physical laws, and in principle we could calculate and explain anything that we wish. However, because of the large number of particles that make up macroscopic systems, this plan is, in practice, not possible.

Dirac continues his quote by giving a way out of this conundrum: "It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation." Presumably this is what DMFT is attempting, although Kotliar did not say so explicitly. He muddied the waters a bit by referring to a famous paper of solid-state theorist Phil Anderson, titled More is Different**, in which Anderson argues that as the number of particles increases to macroscopic size, there are features that emerge and new fundamental laws are needed to explain them. It is not possible, he argues, even in principle, to use approximations to the fundamental laws to obtain these new emergent properties. Unfortunately, Kotliar did not say which side of the debate - reductionism or emergence - he was promoting.

*Dirac makes this claim in the opening paragraph of a paper titled "Quantum Mechanics of Many-Electron Systems," published in Proc. Roy. Soc. London, Ser. A, vol. 123, pages 714-733.

**Science, 1972.

To show how difficult both quantum mechanics and the many-body problem are, there is an entire book dedicated to the Quantum mechanics of one- and two-electron atoms, by Hans Bethe and Edwin Salpeter. Two-electron atoms are an example of a "three-body problem," two electrons and a nucleus, and we really don't understand this simplest of systems completely!


Upcoming events: Two talks next week in Munich at LMU  will focus on cosmology.

• The first is a joint "public talk and discussion" by Paul Steinhardt of Princeton and Eiichiro Komatsu of the Max Planck Institute for Astrophysics. The title is "Visions of the early Universe - Sind wir Zeugen des Urknalls?" (Are we witnessing the Big Bang?). If you miss this, you can catch Steinhardt a few days later - Saturday, 31 Jan - giving the Fred Elston Memorial Lecture at the Daytona Beach campus of Embry-Riddle Aeronautical University.
• The second is the Sommerfeld Theory Colloquium at LMU given by Paolo de Bernardis of the University La Sapienza in Rome, titled "Precision Measurements of the Cosmic Microwave Background."