Real time evolution of scalar fields with kernelled Complex Langevin equation

The real time evolution of a scalar field in 0+1 dimensions is investigated on a complex time contour. The path integral formulation of the system has a sign problem, which is circumvented using the Complex Langevin equation. Measurement of the boundary terms allow for the detection of correct results (for contours with small real time extents) or incorrect results (at large real time extents), as confirmed by comparison to exact results calculated using diagonalization of the Hamiltonian. We introduce a constant matrix kernel in the Complex Langevin equation, which is optimized with the requirement that distributions of the fields on the complexified manifold remain close to the real manifold. We observe that reachable real times are roughly twice as large with the optimal kernel. We also investigate field dependent kernels represented by a neural network for a toy model as well as for the scalar field, providing promising first results.