  We chart a path toward solving for the nonlinear gravitational dynamics of
cold dark matter by relying on a semiclassical description using the
propagator. The evolution of the propagator is given by a Schr\"odinger
equation, where the small parameter $\hbar$ acts as a softening scale that
regulates singularities at shell-crossing. The leading-order propagator, called
free propagator, is the semiclassical equivalent of the Zel'dovich
approximation (ZA), that describes inertial particle motion along straight
trajectories. At next-to-leading order, we solve for the propagator
perturbatively and obtain, in the classical limit the displacement field from
second-order Lagrangian perturbation theory (LPT). The associated velocity
naturally includes an additional term that would be considered as third order
in LPT. We show that this term is actually needed to preserve the underlying
Hamiltonian structure, and ignoring it could lead to the spurious excitation of
vorticity in certain implementations of second-order LPT. We show that for
sufficiently small $\hbar$ the corresponding propagator solutions closely
resemble LPT, with the additions that spurious vorticity is avoided and the
dynamics at shell-crossing is regularised. Our analytical results possess a
symplectic structure that allows us to advance numerical schemes for the
large-scale structure. For times shortly after shell-crossing, we explore the
generation of vorticity, which in our method does not involve any explicit
multi-stream averaging, but instead arises naturally as a conserved topological
charge.
