Wouldn't it be lovely if we could use the same variational approaches familiar from quantum mechanics to study quantum field theories? We can, of course; the real question is how do we do this efficiently? In this talk I will discuss the recent revival of approximate Hamiltonian diagonalization as a means to numerically study field theories, both of the strong and weak variety, and of Euclidean and Lorentzian signature. Underlying present successes compared to similar attempts last century is a judicious choice of a basis for the unperturbed Hilbert space. One such basis, that has demonstrated promise, is organized according to the conformal symmetry of the UV fixed point.

Motivated by developing ingredients necessary to implement Hamiltonian truncation, we present a new method using spinors in momentum space to explicitly construct conformal representations in d=2, 3, and 4 dimensions. This turns out to be interesting in its own right; for example, intriguing structures emerge, notably a U(N) action which generalizes the N-particle U(1)^N action of the little group is found to completely characterize primary operators.