>>11151592I'm not an undergrad. Dynamical systems applies to classical mechanics and not quantum mechanics in any direct way.
The vacuum being the lowest energy eigenstate very much means that it does not change in time. The vacuum is not a particular field configuration in the path integral, which is how you seem to want to think of it. The entire path integral describes the vacuum.
No need to get defensive. I am just pointing out that you have a misconception about quantum mechanics.