In the book
Quantum Field Theory for the Gifted Amateur
one is presented with this picture.
Let's postulate the path between $A$ to $B$ is defined as $x=p(t)$. In this case, the action is
$$ S=\int_0^\tau (T-V)dt $$
- Why is the path $p(t)$ restricted to a function of $t$. In other words, why are no loops allowed? To produce a loop in a path the particle would have to be able to backtrack both in space and in time along its path. From this, I infer that an asymmetry between the $x$ and $t$ axis is postulated for paths in the classical case. What is the intuition between this asymmetry? Are we just artificially banish "backward-time-travel" along paths?
- Now, I compare this to the paths of quantum theory using the Feynman path integral. For instance, the following figure (link: https://en.wikipedia.org/wiki/Path_integral_formulation) clearly shows loops. In this case the path is not a function $p(t)$.
But I am confused, because the Feynman path integral is a sum over all classical paths. So, if classically paths are $p(t)$ (i.e. no loops allowed), then why is the second graph showing loops in the paths?
By assuming that loops are not allowed in paths, are we injecting an assumption about the irreversibility of time (e.g. arrow of time) into the principle of least action?