Stationary action with maximized action

Question

I would like to ask for an example (a lagrangian) both in classical and quantum level for which the action is maximaized (rather than minimized). What is special in these cases?

Answer 1

At quantum level the action is an operator, so to maximize or minimize is not well defined without further information. At classical level there is a simple example: The length of timelike curves in a $n+1$ dimensional spacetime with (smooth) metric $g$ (with signature $+, -, \ldots, -)$. The action reads: $$S[x] = \int_{a}^b \sqrt{g_{ab}(x(t))\dot{x}^a\dot{x}^b}dt$$ Locally the future-oriented timelike geodesic joining two fixed events, $x_a:= x(a)$ and $x_b=x(b)$, maximize the above functional in the open set of the $C^1$ future-orientes timelike curves $x=x(t)$, with $[a,b]$, joining the said couple of events.