Scaling, Universality and Renormalization

Scaling: In physics we are most often concerned with connecting problems at different length scales. For example, we start from atomic forces and use these to predict the properties of solids. We need some method for describing how to extrapolate effects over many orders of magnitude of different lengths to form connections among different theories. The word for this kind of extrapolation is scaling.

Universality: As one traverses the different length scales, one only retains a few characteristics of the original problem. Thus many different microscopic problems have the same macroscopic manifestation, depending mostly upon the symmetry of the original problem. Thus the Ising model, magnet and liquid-gas phase transition all have the same behaviour near critical point.

Universality Classes: Problems then fall into few different categories depending upon the nature of their solution. These categories are called “universality classes”. Scientists now use this phrase to describe many different kinds of things beyond phase transitions. One of these days I expect to hear a description of different kinds of pop music as different universality classes.

Renormalization:  The renormalization method ties all this together. Wilson described the effect of many renormalizations in which the length scale was changed again and again, as the motion of the interactions toward a “fixed point”, a state of constant and unchanging interaction.

New Calculational Paradigm: Previously one started with a “problem” i.e. a description of the interactions among the particles in the system. The job of the theorist was mostly to calculate an “answer”, that is a detailed description of the behaviour of the particles.

In the new era, one does renormalization calculations in which one starts with one problems (i.e. set of interactions) and constructs another equivalent problem on another length scale. A partial but often sufficient description of the behaviour is encoded in the relation between different scales.

Example:

Old era: One said that quantum electrodynamics was characterized by an interaction strength called α, having the value 1/137.036.

New Era: One says that at smaller distance scale, α gets larger and eventually this electromagnetic interaction gets very strong. In informal language, one says that interaction strength runs.

The practice of physics has changed and moved away from the mode of calculation of Newton, Boltzmann, Einstein, and Dirac, going from solving problems to discussing the relationship among such problems. In addition to the aforementioned people who pushed in this direction, all the workers in the fields determine the subject’s content.

– Leo P. Kadanoff

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