# Paths of least action and loops in time

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

$$S=\int_0^\tau (T-V)dt$$

Questions:

• 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?

• Feynman diagram is not showing particles going backwards in time. The Feynman diagram is showing paths in 2D space with the time variable not shown. The other figure is showing 1D paths with time on the horizontal axis. – Luke Pritchett Feb 21 '19 at 14:59
• The path integral doesn't consider loops in spacetime. Those loops are just loops in space. – Javier Feb 21 '19 at 15:09

1. Concretely, the 2nd figure shows 2 spatial directions, say $$x$$ and $$y$$, not $$x$$ and $$t$$, as already pointed out by Quantumwhisp, Luke Pritchett & Javier. Time $$t$$ is instead the curve parameter, which is monotonically increasing.
$$[a,b]~\ni~\lambda ~~\mapsto~~ (x^0(\lambda),x^1(\lambda),x^2(\lambda),x^3(\lambda))~\in~ \mathbb{R}^4,$$ which might be backward-in-time $$x^0 \equiv ct$$.