Boostlet Frames and Causal Sparsity in Two-Dimensional Quantum Field Theory

A sampled Lorentz-adapted Parseval frame is constructed from smooth boostlet windows on the timelike and spacelike branches of the two-dimensional Fourier plane. For the massive Klein--Gordon field, the real field is extended complex linearly and the Pauli-Jordan form is kept bilinear. This convention yields an antisymmetric boostlet commutator matrix and an exact rapidity-integral representation pairing opposite energy branches. Spacelike-separated centres give entrywise decay of every algebraic order in invariant distance. Timelike-separated centres select the rapidity of the connecting four-vector, with a uniform stationary-phase expansion away from rapidity-tile boundaries. Causal sparsity is formulated as a quantitative entrywise threshold property, supplemented by exact scale and branch zeros. A Sobolev coefficient-summability criterion gives mass-shell convergence of finite frame truncations for a restricted class whose Fourier transforms have vanishing trace on the null lines, and hence gives controlled finite-matrix approximation of the commutator form on that class. The boostlet estimates are consistent with the established causal wavefront geometry of the state-independent propagator.