Mathematical properties of renormalization-group transformations

Renormalization-group transformations of the type introduced into statistical physics by K. G. Wilson have been widely used with great success for a variety of many-body calculations. The basic approach consists in integrating out certain degrees of freedom and using a new “image” Hamiltonian to describe the statistical properties of those that remain. The mathematical foundations of the procedure, however, remain somewhat obscure. P.A. Pearce and the author have pointed out that there are serious questions about the existence and properties of a thermodynamic limit for renormalization-group transformations. In particular, certain real-space transformations show peculiarities in this limit. These suggest that there may be serious mathematical problems with renormalization-group procedures, despite their considerable practical sucess.