Renormalization-group transformation ($g^{\prime }=R_b(g)$)

A renormalization-group transformation integrates out selected short-distance degrees of freedom and restores chosen cutoff and field-normalization conventions. It defines a map $R_b$ on couplings at a length factor $b$. Its discrete Jacobian matrix multipliers differ from eigenvalues of the generator of a continuous renormalization-group flow; the relation for a scaling direction is $\lambda =b^y$.

For dimensionless free energy per site $F(u,C)=f(u)+\beta C$, an exact blocking kernel preserves the partition function and gives $F(u,C)=b^{-D}F(u^{\prime },C^{\prime })$. Thus $f(u)=b^{-D}f(u^{\prime })+g(u)$, where $g(u)=\beta [b^{-D}{C}^{\prime }-C]$ is the additive contribution from the eliminated degrees of freedom. Repeating this equality gives the displayed sum. After subtracting analytic backgrounds, the singular free-energy density obeys the homogeneous scaling hypothesis for critical phenomena, subject to marginal or dangerous-variable qualifications.

A blocking kernel assigns a conditional probability to coarse configurations given microscopic configurations. Multiplying each microscopic Boltzmann weight by the kernel and summing over microscopic configurations defines the coarse Hamiltonian. The normalization preserves the partition function exactly when the coarse operator space is not truncated. In $D$ dimensions a blocking factor $b$ changes the lattice spacing to $ba$ and the number of sites to $N/b^D$. Generated interactions and additive constants must be retained for the transformation to remain exact.