If an EM wave only gives us a probability of where a photon may be at a given moment, and the HUP tells us that we can't know the exact location of the photon. Then would it be correct to say that a photon does not travel in a straight line?
If this is true, wouldn't the photon's crooked path mean that it must travel faster than $c$ for us to measure its straight line speed at $c$?
Here's a comparison to explain my question further: If two sprinters run a 100 yard dash in 10 seconds, but one sprinter is required to run in a zigzag manner wouldn't that runner need to run faster than the straight line sprinter for both runners to run the race in 10 seconds. Does the crooked traveling photon need to exceed $c$ for us to measure its straight line speed at $c$?
Answer
First of all, as Countto10 pointed out, in this other question you can find a nice discussion on the meaning of trajectories in quantum mechanics.
Heisenberg's uncertainty principle tells you that when you measure the photon's position or momentum (or any other pair of complementary variables), you will find the results to always affect each other. In other words, a photon cannot have a definite value of position and momentum at the same time.
This does not really come into play if you are trying to measure the speed of light using single photons. Let us imagine for the purpose an experiment in which a single photon is emitted at point $A$ at a known time $t$, and travels towards the point $B$. After the photon is emitted, its wavefunction will expand and evolve according to a variety of factors. For example, the smaller the error in the transverse position when it is emitted, the faster its wavefunction will disperse and so the harder it will be to detect it at $B$. This kind of things can however be taken care of relatively easily, so to make the probability of detecting the photon in $B$ high enough. When this detection event happens, we can measure the time it took and derive the speed of light accordingly.
The photon did not follow a zig zag path going from $A$ to $B$. It didn't simply because that is not how a photon usually travels, and nothing in Heisenberg's uncertainty principle says it should. Its wavefunction did evolve in the process, but it is wrong to think of this in terms of a point particle jiggling around. It means instead that the probabilities of detecting the photon in the various transverse points vary with the longitudinal position.
As a final note, it is worth noting that there should be a lot of buts and ifs in the above argument. For example, I assumed that the emitted photon can be more or less be thought of as a particle-like thing, in the sense of it being relatively well localized. A photon can however also be highly delocalized, or have a complex inner structure. This kind of things can matter: an interesting example is a recent experiment by Bareza and Hermosa in which it was shown that photons carrying an orbital angular momentum can have (in the vacuum) a group velocity smaller than $c$. This is an interesting reminder that the speed of light being $c$ only strictly holds for plane waves in the vacuum, not really for light with finite extent and complex inner structure.
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