11 - Comments: The quantum field theory of physics and of mathematics

Summary

One recognizes that there has been, and continues to be, a great deal of common ground between statistical mechanics and quantum field theory (QFT). Many of the effects and methods of statistical physics find parallels in QFT, particularly in the application of the latter to particle physics. One encounters spontaneous symmetry breaking, renormalization group, solitons, effective field theories, fractional charge, and many other shared phenomena.

Professor Fisher [1] has given us a wonderful overview of the discovery and role of the renormalization group (RG) in statistical physics. He also touched on some of the similarities and differences in the foundations of the RG in condensed matter and high-energy physics, which were amplified in the discussion. In the latter subject, in addition to the formulation requiring cutoff-independence, we have the very fruitful Callan-Symanzik equations. That is, in the process of renormalizing the divergences of QFT, arbitrary, finite mass-scales appear in the renormalized amplitudes. The Callan-Symanzik equations are the consequence of the requirement that the renormalized amplitudes in fact be independent of these arbitrary masses. This point of view is particularly useful in particle physics, although it does make its appearance in condensed matter physics as well.

The very beautiful subject of conformal field theory spans all three topics we are considering: critical phenomena, quantum field theory, and mathematics. The relationship between conformal field theory and two-dimensional critical phenomena has become particularly fruitful in recent years.