Renormalization group

The renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales. A change in scale is called a “scale transformation”. The renormalization group is intimately related to “scale invariance” and “conformal invariance”, symmetries in which a system appears the same at all scales (so-called self-similarity). (However, note that scale transformations are included in conformal transformations, in general: the latter including additional symmetry generators associated with special conformal transformations.)

As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller scale, with different parameters describing the components of the system. The components, or fundamental variables, may relate to atoms, elementary particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be variable “couplings” which measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances.

There are the:

• Spatial RG
• Momentum-space RG

See also

Scale invariance

Material

• Pedagogical site on Scaling
• K. Huang (2013): A Critical History of Renormalization. arXiv:1310.5533
• Maris, H. J., & Kadanoff, L. P. (1978). Teaching the renormalization group. Am. J. Phys, 46(6), 653-657.

Papers

• Kadanoff, L. P. (1966). SCALING LAWS FOR ISING MODELS NEAR T c. This research was supported in part by the National Science Foundation under grant NSF-GP 4937.
• Callan, C. (1970). Broken Scale Invariance in Scalar Field Theory. Physical Review D 2 (8): 1541.
• Fisher, M. E. (1974). The renormalization group in the theory of critical behavior. Reviews of Modern Physics, 46(4), 597.
• Saleur, H., Sammis, C. G., & Sornette, D. (1996). Renormalization group theory of earthquakes. Nonlinear Processes in Geophysics, 3(2), 102-109.

Books

• Goldenfeld, N. (1992). Lectures on phase transitions and the renormalization group.
• Kadanoff, Leo P. (2000). Statistical Physics: statics, dynamics and renormalization. World Scientific.
• Kopietz, P., Bartosch, L., & Schütz, F. (2010). Introduction to the functional renormalization group. (Vol. 798). Springer.