Feynman Lecture 26 - Summation of Probability Amplitudes

Question

How does Fig. 26-3 (shown below) correspond to the following paragraph from this Feynman lecture?

Finally, we give a very crude view of what actually happens, how the whole thing really works, from what we now believe is the correct, quantum-dynamically accurate viewpoint, but of course only qualitatively described. In following the light from A to B in Fig. 26–3, we find that the light does not seem to be in the form of waves at all. Instead the rays seem to be made up of photons, and they actually produce clicks in a photon counter, if we are using one. The brightness of the light is proportional to the average number of photons that come in per second, and what we calculate is the chance that a photon gets from A to B, say by hitting the mirror. The law for that chance is the following very strange one. Take any path and find the time for that path; then make a complex number, or draw a little complex vector, ρeiθ, whose angle θ is proportional to the time. The number of turns per second is the frequency of the light. Now take another path; it has, for instance, a different time, so the vector for it is turned through a different angle—the angle being always proportional to the time. Take all the available paths and add on a little vector for each one; then the answer is that the chance of arrival of the photon is proportional to the square of the length of the final vector, from the beginning to the end!

Any insight greatly appreciated!! Thanks in advance!!

EDIT: Here is a lecture where Feynman talks about this

Answer 1

The picture is a way to compute the probability of going from an initial to a final state. In other words, we are

following the light from A to B.

The little arrows are the result of following this algorithm:

Take any path and find the time for that path; then make a complex number, or draw a little complex vector, ρeiθ, whose angle θ is proportional to the time. The number of turns per second is the frequency of the light. Now take another path; it has, for instance, a different time, so the vector for it is turned through a different angle—the angle being always proportional to the time. Take all the available paths and add on a little vector for each one

In interpreting the above paragraph, note that the space that the picture is being drawn in is an abstract space (the complex plane), and not real space. Real life space is not present in this picture, so for each arrow in the picture, you have to imagine what the path of the photon looks like in real life in your head.

The big arrow is the "final answer" and is the result of adding the little arrows using vector addition. The big arrow is related to the probability of going from initial state to final state, as Feynman says:

then the answer is that the chance of arrival of the photon is proportional to the square of the length of the final vector, from the beginning to the end!