Paths of least action and loops in time

Question

In the book

Quantum Field Theory for the Gifted Amateur

link: https://books.google.ca/books?hl=en&lr=&id=nIk6AwAAQBAJ&oi=fnd&pg=PP1&ots=JZjwG_qDt5&sig=TdY8z03owRBtHu9x9s9rVQougiM#v=onepage&q&f=false

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?

Answer 1

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.

2. However OP's question about "backward-in-time" is actually a very good and deep question, with a long history in QFT. E.g. in a path integral formulation of RQM we can consider paths
   $$[a,b]~\ni~\lambda ~~\mapsto~~ (x^0(\lambda),x^1(\lambda),x^2(\lambda),x^3(\lambda))~\in~ \mathbb{R}^4, $$ which might be backward-in-time $x^0 \equiv ct $.

Answer 2

The 2nd picture that you show shows a loop. But this is not because the particle in the 2nd picture does travel back in time, but instead because your second picture shows the complete trajectory of the particle in space, which (in your example) is either 2 or 3 dimensional. While doing so, it doen't say anything about about the time. There is no time axxis in your 2nd picture.

Your first graph on the other side just shows a particle trajectory in 1 dimension, with the 2nd axxis being the time axxis. If you would plot a 3 dimensional graph, with one axxis being the time axxis, and the other 2 being for example the x- and the y- coordinate of a 2 dimensional movement, you could observe the particle going in circles there as well.

Long story short: There is no contradiction between the 2 pictures you show, they just show different things. Feynmans Path Integral sums over all classical trajectories. But none of this trajectories include trajectories where particles travell backwards in time.