9.1: The Program of Renormalization
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A statistical mechanical system is defined by a set of degrees of freedom and by a set of coupling constants \{K\ns_\alpha\}. The degrees of freedom can be discrete, such as Ising spins \sigma\ns_i, or continuous, such as a field \phi(\Br). Additionally, each such system possesses a microscopic length scale \ell. For discrete, lattice-based systems, this length scale is simply the lattice spacing: \ell=a. For continuous systems, we can define a microscopic length scale by imposing a cutoff \RLambda on the wavevectors we integrate over in all Fourier transforms. That is, we replace \int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk) \longrightarrow \int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk)\,\gla(\Bk)\ , where F(\Bk) is any function and \gla(\Bk) is the cutoff function. The simplest such case to imagine is a sharp cutoff which is isotropic in wavevector, \gla(\Bk)=\RTheta(\RLambda-|\Bk|). Other cutoff schemes, however, are possible, including ‘soft cutoffs’ where \gla(\Bk) is smooth. The microscopic length scale is then \ell\sim\RLambda^{-1}, which is the smallest distance in real space over which the system can independently fluctuate.
The idea behind renormalization is that we can successively winnow degrees of freedom from a system in some exact or approximate way, and in so doing we generate a new version of the system, at a different length scale \ell'>\ell, and with different couplings \{K'_\alpha\}. We then iterate this procedure. The result is a set of equations which tell us how the couplings behave under a change of the microscopic length scale. As we shall see, the fixed points of this procedure – where couplings do not change under a change of length scale – are critical points. Such a fixed point is defined by a set of couplings \{K^*_\alpha\}.
If we denote by \CR\ns_b the renormalization procedure \CR\ns_b\big(\ell\,,\{K\ns_\alpha\}\big) = \big(\ell'\,,\{K'_\alpha\}\big)\ , where \ell'=b\,\ell, then we have the composition law \CR\ns_b\,\CR\ns_{b'}=\CR\ns_{b+b'}. The set of transformations \{\CR\ns_b\} is collectively referred to as the renormalization group (RG) because of this mathematical structure. It is somewhat of a misnomer, however, since the transformations are only defined for b\ge 1, which means that there is no inverse operation, and hence no true group structure1. Nevertheless we shall use the RG terminology because it has become universally accepted in the literature.