The Euler-Lagrange equations follow from minimizing the action. Usually this is done with fixed (e.g. vanishing) boundary conditions such that we do not have to worry about any boundary terms. However, it's also possible to obtain boundary conditions which follow from the action itself, the so-called natural boundary conditions. In terms of some Lagrangian $L$ they take on the form $$ \frac{\delta L}{\delta x'(t)}\Bigg|_{x=a} = 0 $$ or for fields $$ \frac{\delta \mathcal{L}}{\delta \partial_x\phi}\Bigg|_{x=a} = 0~. $$
These boundary conditions minimize the action for a system with variable end-points.
But what happen with these boundary conditions when we quantize the theory? Specifically, do states in the Hilbert space satisfy the natural boundary conditions automatically or is this something that needs to be imposed "by hand"?
