  Using the example of the two-dimensional (2D) Ising model, we show that in
contrast to what can be done in configuration space, the tensor renormalization
group (TRG) formulation allows one to write exact, compact, and manifestly
local blocking formulas and exact coarse grained expressions for the partition
function. We argue that similar results should hold for most models studied by
lattice gauge theorists. We provide exact blocking formulas for several 2D spin
models (the O(2) and O(3) sigma models and the SU(2) principal chiral model)
and for the 3D gauge theories with groups Z_2, U(1) and SU(2). We briefly
discuss generalizations to other groups, higher dimensions and practical
implementations.
