Section 26–6 How it works

 (Photons / All possible paths / Accumulated probability)

 

In this section, Feynman discusses the nature of light as photons, explains Fermat’s principle from the perspective of all possible paths of photons, and how photons contribute to the accumulated probability of an optical image.

 

1. Photons:

“Instead the rays seem to be made up of photons, and they actually produce clicks in a photon counter, if we are using one (Feynman et al., 1963, section 26–6 How it works).”

 

Feynman provides a crude view of Fermat’s principle based on his interpretation of quantum electrodynamics. That is, the path of light, for example, from A to B, does not seem to be in the form of waves and light rays seem to be made up of photons. Note that Feynman formulated his theory in terms of paths of particles in space-time to avoid the field concept. In Wilczek’s (1999) words, “uniquely (so far as I know) among physicists of high stature, Feynman hoped to remove field-particle dualism by getting rid of the fields (p. 13).” However, Feynman was disappointed that his theory of quantum electrodynamics is mathematically equivalent to the conventional quantum theory.

 

Feynman says that light rays seem to be made up of photons instead of light rays are really photons. In addition, he explains that light rays actually produce clicks in a photon counter if we are using one. Perhaps Feynman could have clarified that the photon model adopted is just an idealization. Currently, physicists prefer the concept of quantum field instead of particles. For example, Wilczek writes, “[i]n quantum field theory, particles are not the primary reality.” On the other hand, Hobson argues using the two vacuum effects, Unruh effect and single-quantum nonlocality, to abandon the particle concept. We may also question the definition of point-particle in the sense that it cannot be found at a specific location from the viewpoint of special theory of relativity or quantum theory.

 

2. All possible paths:

“Now let us show how this implies the principle of least time for a mirror. We consider all rays, all possible paths ADB, AEB, ACB, etc., in Fig. 26–3 (Feynman et al., 1963, section 26–6 How it works).”

 

To demonstrate the principle of least time for a mirror reflection, Feynman considers all light rays and all possible paths ADB, AEB, ACB, etc., in Fig. 26–3. The path ADB makes a small contribution, but the next path, AEB, takes a quite different time, so its angle θ is quite different. However, the phrase all possible paths could be misleading because there are only thirty-two paths (instead of infinity) shown in Fig. 26-14. Of course, it is impossible to include all possible paths that are infinitely many. It is more practical to consider a minimum number of paths that is sufficient to obtain a reasonably accurate answer.

 

To have a better idea of Feynman’s sum over all paths, it can be illustrated by a thought experiment involving a point source of light, two slits, and a screen (Feynman et al., 2010). Firstly, we imagine a black plate is inserted between the light source and the two slits. Whenever we drill some holes through the plate, it creates alternative routes for the light and each of this route corresponds to a probability amplitude. Assuming it is possible to drill more holes until the plate no longer exists, what does that mean? Alternatively, we can insert more and more plates between the light source and the screen, and then drill all holes such that there is nothing left. The thought experiment shows that we can sum the amplitudes of all possible paths (or holes) from the source to the screen.

 

3. Accumulated probability:

“Almost all of that accumulated probability occurs in the region where all the arrows are in the same direction (or in the same phase) (Feynman et al., 1963, section 26–6 How it works).”

 

Feynman mentions that almost all of that accumulated probability occurs in a region where all the arrows are in the same direction. This is why the extreme parts of the mirror do not contribute much to the image formation, but it still reflects light just like the other parts of the mirror. In his lecture on QED, he uses the phrases stopwatch hand and arrow to represent quantum probability (instead of complex numbers or complex vector). Furthermore, Feynman (1985) adds that “... all the arrow to point in the same direction, and to produce a whopping final arrow - lots of light! (p. 58).” One may clarify that the meaning of arrows in the same direction means that the light rays are in phase and thus, they reinforce each other.

 

Feynman elaborates that all of the contributions from the paths which have very different times cancel themselves out because they point in different directions. In his QED lecture, he provides a clearer explanation of how the light rays from the left-hand part of the mirror cancel themselves out (with Fig. 25). In his words, “we see that some of the arrows point more or less to the right; the others point more or less to the left. If we add all the arrows together, we have a bunch of arrows going around in what is essentially a circle, getting nowhere (Feynman, 1985, p. 46).” However, a complete cancellation of arrows is not always possible because the length of the arrows is inversely proportional to the distance the light traveled (Feynman, 1985, pp 73-74).”

 

Review Questions:

1. Would you assume light rays are made up of photons in your explanation of Fermat’s principle?

2. Did Feynman consider all possible paths ADB, AEB, ACB… in Fig. 26-14?

3. How does the ultimate picture of photons relate to the accumulation of arrows?

 

The moral of the lesson: the ultimate picture (or image formation) of photons is dependent on the probability of arrival of photons or an accumulation of arrows based on the principle of least time.

 

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

2. Feynman, R. P., Hibbs, A. R., & Styer, D. F. (2010). Quantum mechanics and path integrals (Emended ed.). New York: Dover.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Wilczek, F. (1999). The persistence of ether. Physics Today, 52(1), 11-13.

5. Wilczek, F. (2001). Fermi and the Elucidation of Matter. arXiv preprint physics/0112077.