Blocking kernel (${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$)

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.