  First of all we shortly illustrate how the symplectic quantization scheme
[Gradenigo and Livi, 2021] can be applied to a relativistic field theory with
self-interaction. Taking inspiration from the stochastic quantization method by
Parisi and Wu, this procedure is based on considering explicitly the role of an
intrinsic time variable, associated with quantum fluctuations. The major part
of this paper is devoted to showing how the symplectic quantization scheme can
be extended to the non-relativistic limit for a Schr\"odinger-like field. Then
we also discuss how one can obtain from this non-relativistic theory a linear
Schr\"odinger equation for the single-particle wavefunction. This further
passage is based on a suitable coarse-graining procedure, when self-interaction
terms can be neglected, with respect to interactions with any external field.
In the Appendix we complete our survey on symplectic quantization by discussing
how this scheme applies to a non-relativistic particle under the action of a
generic external potential.
