  We study 1+1 dimensional $\phi^4$ theory using the recently proposed method
of conformal truncation. Starting in the UV CFT of free field theory, we
construct a complete basis of states with definite conformal Casimir,
$\mathcal{C}$. We use these states to express the Hamiltonian of the full
interacting theory in lightcone quantization. After truncating to states with
$\mathcal{C} \leq \mathcal{C}_{\max}$, we numerically diagonalize the
Hamiltonian at strong coupling and study the resulting IR dynamics. We compute
non-perturbative spectral densities of several local operators, which are
equivalent to real-time, infinite-volume correlation functions. These spectral
densities, which include the Zamolodchikov $C$-function along the full RG flow,
are calculable at any value of the coupling. Near criticality, our numerical
results reproduce correlation functions in the 2D Ising model.
