Understanding the Path Integral for Photons - Vince's Q&A

[Pants]

I'm a bit confused about how the path integral for, say, a spin-0 photon is calculated. My understanding of quantum mechanics is somewhere above Feynman's book QED, but somewhere below actually figuring out what every part of the technical definition means. Right now the main sticking point for me is grokking the Hamiltonian, but I don't think I have to figure that out in detail just yet to get the concept.

Anyways, as Feynman describes it in the first chapter of QED, the path integral represents each path that's possible from the source to the detector, and the phase of that path is determined by the energy of the photon and how long it takes to get from source to detector. (Please correct me if I'm misinterpreting this). Here's what I don't get: if the particle is emitted at time 0, and measurement occurs at time T, do we only look at paths that take T seconds to get to the detector traveling at velocity c, or are superluminal paths included in the calculation as well?

Thanks!

-Vince

 

[eljose79]

Dear Vince,

Thank you for your question. Understanding the path integral for a spin-0 photon can be quite tricky, so it's great that you're seeking clarification. Let me break down a few key points that may help you understand the calculation better.

First, you are correct in your understanding that the path integral represents all possible paths that a photon can take from the source to the detector. This includes paths that take T seconds to travel at the speed of light (c), as well as superluminal paths. In fact, the path integral includes all possible paths, regardless of their velocity.

Next, the phase of each path is determined by the energy of the photon and the time it takes to travel along that path. This means that the phase of each path will be different, depending on the energy of the photon and the length of the path. This is where the Hamiltonian comes into play - it is a mathematical operator that represents the total energy of the system, including the energy of the photon.

Now, you may be wondering how we actually calculate the path integral. This is where the technical definition of the Hamiltonian comes into play - it involves a lot of complex mathematical concepts, such as Fourier transforms, Green's functions, and perturbation theory. Without going into too much detail, the basic idea is that we break down the path into small segments and calculate the contribution of each segment to the overall path integral. This is done using the Hamiltonian and other mathematical techniques.

I hope this helps clarify some of your confusion about the path integral for a spin-0 photon. It's a complex topic, so don't be discouraged if you don't fully understand it yet. Keep studying and asking questions, and you'll continue to deepen your understanding. Good luck!