# What, explicitly, happens as EM radiation travels from A to B?

I'm in my fourth year of a Masters Physics course, and am quite concerned that I don't fully understand this.

Suppose a photon is emitted at point A and absorbed at point B one light-minute away. We 'look at' the system 20s after the photon is emitted. Define point C as the point one-third of the way between A and B.

1. How should the light be described? I want to say "the photon is at point C" but I know Fock states are unphysical and coherent states are how best to describe light. But, is this superposition of photon numbers still at point C, or is there a spatial distribution too? If there is a spatial distribution, does that not give a chance that it will be absorbed at B at eg 59s or 61s?
2. Where are the magnetic and electric fields modified? Is it just at point C ; everywhere along the path from A to B (which would seem to have information travel faster than light) ; everywhere between A and C ; some Gaussian-like volume around C ; something else?
3. How are the E and B fields modified? I "know" that it's by sin(kx-wt), but depending on the answer to Q2 that may mean different things. (Eg, if E and B change only between A and C, do they just "switch off" once it's absorbed at B (again raising locality questions), or does the magnitude dampen with time after it's initially altered? In general, how do they change with time, and at various events (emission, absorption, 'the phtoton passing point C'?)
4. What does it mean to say that E and B are perpendicular? Do they literally vary spatially, ie if I move upwards from C with my detector I read a B but no E field, while if I move sideways then I get an E but no B field? If not, to what does the perpendicularity refer?

The bold is my questions (obviously); after that is my guesses at answers and/or hints at where I'm confused, but feel free to ignore these and talk about anything that's relevant to answering.

To be clear, I'm not asking about if I actually made a measurement at point C, I'm asking what we can deduce from the equations etc we know describing light and the EM field. (Think calculating where a ball I throw in the air will be after 0.2s, as opposed to catching and measuring it after 0.2s and hence changing the rest of the motion.)

Also, I'm looking for a model of what happens, and (as far as possible) the simplest full explanation why. (I tried asking one lecturer but he started talking about Fourier transforms and stuff I didn't really understand/see the relevance of, and didn't give a direct "they're modified like this" answer, hence this note.)

Finally, if you refer to modes of EM waves/fields in your answer (or are feeling particularly generous), please define exactly what you mean by this as well - many times I've Googled and still I don't entirely understand these.

Much much thanks!

• @Xander, we had strikes which caused my formal Fourier transform lessons to be missed, and the only time I'd really worked with them beforehand was briefly in a computational class, where I was thinking about the code more than the physics. I can't find a "Particle Physics" by Griffiths, unless you're referring to his "Introduction to Elementary Particles"? – Angus Buck Nov 28 '18 at 16:04
• I believe I've edited the question to make sense now. – Angus Buck Nov 28 '18 at 16:10
• In (2) and (3) you seem to be disregarding relativity of simultainety. For example, if you have two like charges one light second apart (like the Earth and the Moon), and one charge suddenly moves toward the other, then the moving charge would feel the repulsion increase instantly, but the other charge would feel the repulsion increase only after one second. Now change the reference frame to the rest frame of the first charge and the situation reverses. This doesn't answer what you are asking, but illustrates that light moves with the speed of time. – safesphere Nov 28 '18 at 16:16
• Hi Angus, +1 and I've deleted my comment. I'm not sure that Griffith covers your specific questions, but if you don't get an answer, (as there are far better people than I here), I'll give it a shot. Best of luck with it – user213900 Nov 28 '18 at 16:28
• @safesphere Forgive me, I'm not sure where the simultaneity is relevant for my question? I want to know what the E&B fields looks like before, after and at/around point C when 20s have passed, in the rest frame of A and B. Where does the relativity of simultaneity come into play (other than my reservations about there being any E&B field modification between C and B)? – Angus Buck Nov 28 '18 at 16:29

Suppose a photon is emitted at point A and absorbed at point B one light-minute away. We 'look at' the system 20s after the photon is emitted. Define point C as the point one-third of the way between A and B.

It looks as if you believed that the photon goes from A to B following the straight line between those points. I also have some doubt about your wording "a photon is emitted at point A". You don't say it but it would seem that photon's emission does happen at a precise instant of time.

I would prefer a more elaborate statement, something like this:

"An atom is placed in position A of an inertial frame. Assume the atom has only two (nondegenerate) energy levels, $$E_0$$, $$E_1$$ (or the others may be neglected). Since a long time the atom has stayed in its lowest level $$E_0$$. By a properly chosen laser pulse it is brought to level $$E_1$$, in a very short time. So we may say that at time $$t=0$$ it is in that state. Afterwards it will decay emitting a photon, e.g. via an electric dipole transition whose mean life is short, say $$10\,\mathrm{ns}$$.

At a distance of 1 light-minute is placed, in A's frame, an array of detectors. At time $$t=1\,\mathrm{min}$$ one of them, say B, detects the photon."

Then your questions could follow. We'll see presently.

We 'look at' the system 20s after the photon is emitted. Define point C as the point one-third of the way between A and B.

This makes sense.

I want to say "the photon is at point C"

Surely you understand this isn't a reasonable option. At that time the photon doesn't occupy a precise position in space. Photon emission isn't a "Nadelstrahlung" (needle radiation) as Einstein believed. You could find the photon everywhere at a distance of about 20 light-secs from A. Detector B too has only a (small) probability of detecting the photon. It could be detected instead by any other detector.

but I know Fock states are unphysical and coherent states are how best to describe light.

Wait a moment. Where did you learn that Fock states are unphysical? This comes novel to me. Maybe the exact argument was different. Could you give some reference?

As to coherent states, they are OK to represent a state of e.m. field resembling a macroscopic e.m. wave. But in our case, where we know exactly one photon is present, this is far from a coherent state. In a coherent state the number of photons is undetermined. it is a superposition of states with different photon numbers. Maybe you know there is an uncertainty relation between photon number and field phase?

Going on: I can't understand your question 2. Sorry.

1. How are the E and B fields modified?

Modified wrt what? I'm afraid you have some misunderstandings of the whole matter of quantum fields. Did you take courses on that matter? E and B are operators defined on Fock space (sorry :-) ). It makes no sense to speak of "modifying" them. Your quantum system is expressed mathematically as a Hilbert space (Fock space) and operators defined in it (e.g. E, B, but also the Hamiltonian and so on).

We are reasoning about free fields (with some abuse). In Schrödinger picture the state of e.m. field evolves in time. In my description I vaguely described the state at time near $$t=0$$. The only certain thing is that after - say - $$t=50\,\mathrm{ns}$$ it is an eigenstate of the observable "photon number", at eigenvalue 1. The state changes with $$t$$, exhibiting photon propagation from source outwards.

1. What does it mean to say that E and B are perpendicular?

Well, E and B are vectors. This means they are operators on Fock space, more precisely two triplets of operators to represent components of e.m. field. These operators are so defined that an identity $$\vec E\cdot\vec B=0$$ holds. I can't dwell on it here, but you will find it explained in Ch. from 1 to 3 (according book's organization) of every book on QFT.

• Thank you for your answer! Perhaps I'm mixing up classical and quantum models here, making my question confusing; my apologies. What's bothering me is diagrams like this, which eg make me wonder if the fields are altered everywhere after 20s, make me concerned that Fock states don't give sinusoidal fields etc. Does this clarify my questions? – Angus Buck Dec 6 '18 at 19:41
• It's clear my quantum needs some work, I had no idea that eg "You could find the photon everywhere at a distance of about 20 light-secs from A." I'm also very new to quantum fields as you guessed, so basically an addendum to my above comment/question - am I simply mixing classical/quantum models and hence confusing myself/asking nonsense questions? Thank you for your patience! – Angus Buck Dec 6 '18 at 19:42