I was reading about electron-positron annihilation. Typically it results in two photons, each with an energy of 511 keV, that go shooting out in opposite directions. But I read that in some instances three photons can result. Electrons have an intrinsic spin of ½, while photons have a spin of 1. So if the electron and positron have spins of +½ and -½, you will get two photons with spins +1 and -1. If, on the other hand, the electron and positron have the same spin orientation, say +½ and +½, then three photons will result, with spins +1, +1, and -1.
Well what I’m wondering is, is it possible for only one photon to result from an electron-positron annihilation? This outcome could still abide by the laws of spin conservation. Suppose the positron was travelling with a kinetic energy of 600 keV, and then a faster moving electron was shot from directly behind, say with a kinetic energy of 1.2 MeV, on an intercept course to annihilate with it. Because the kinetic energy of the system outweighs the rest masses of the two particles, any resulting photons from the annihilation would be directed in the forward vector.
You might also have to magnetically polarize the particles before shooting them out, to make sure they had the proper spin orientation for an odd-number multiple photon creation.
I thought about this and I am thinking what would tend to happen is you would have a cone of radiation with a specifically defined angle of emission. But as the kinetic energy was increased, the angle would become narrower and narrower until coherence effects started to predominate. Obviously at some extreme narrow angle you can’t have multiple photons streaming out from the same annihilation point, or at least that’s not statistically favorable.
No, it’s not possible, because it would violate conservation of mass. I know what you’re thinking: Doesn’t annihilation violate that anyway? And the answer is no it doesn’t. An electron-positron pair has a mass of at least 1.022 MeV, and the pair of photons coming out of it also has that mass. But a single photon has zero mass. Mass is not additive.
This is probably easiest to see in the center-of-mass reference frame, because there the problem shows up as lack of conservation of (linear) momentum. In the center-of-mass frame, the total momentum of the electron-positron pair is zero, and so the total momentum of any resulting photons must also be zero. But a single photon always has a momentum, because it’s proportional to its energy, which is fixed by the initial energy.
My first inclination in reading the OP at first was what about conservation of momentum? The OP sort of covered that in the 2nd paragraph. But wrong frame of reference.
You can have two photons go off in ~opposite directions. 3 in 3 different directions. But just one photon?
Force a reference frame? Maybe if you were accelerating the particles during the collision …
The inability for two things to become one thing comes up in less esoteric settings as well. For instance, you can’t have two atoms or molecules combine into a single larger molecule without the participation of another object to balance momentum. The other object might be another atom or molecule, an emitted photon, or a piece of the original system that dissociates away. Electron-ion recombination is another example where two things become one but only if something else happens, too. These considerations can be important in rarefied systems.
I think the following argument, derived from this, is valid as well, but I’m not sure:
You can see this from the Feynman diagram, in that if you draw this interaction in the center-of-mass frame, you have to end up with the zero-momentum photon going straight up, or just sitting at one location on the x-axis as time keeps on slippin’, slippin’, slippin’ into the future… Which is absurd, because photons never sit still, they always go at 45-degree angles to define the edge of the light cone.
This is similarly weird if you rotate the diagram: In a normal electron-positron annihilation diagram, you can rotate it to show an electron being deflected by a photon which it re-emits, a positron being deflected by a photon which it re-emits, or two photons coming together to create an electron and a positron, which can happen if you have a really good day in the lab or a really bad day in a nuclear war.
If you rotate this misbegotten diagram, you get an electron being deflected by a photon that was just sitting at one moment in time (and, apparently, at all locations in space), a positron ditto, or one stationary photon spontaneously decaying into an electron and a positron. Two of those things make absolutely no sense and the third is merely physically impossible.
I’m sure I’m misinterpreting what I can do with these diagrams, but hopefully I’ll be wrong in an interesting way.
I’ve never seen any interpretation of Feynman diagrams which includes information about spatial position, but other than that, the arguments look sound.
You’re ascribing more spatial and temporal meaning to the lines of a Feynman diagram than is warranted. A line can be drawn at any ol’ angle. One can come up with diagrammatic schemes to represent momentum and momentum conservation, but that’s not part of a Feynman diagram. Note too that if the annihilation product were something massive, not massless, the process would still be forbidden for the same reasons, yet your 45-degree argument wouldn’t be relevant.
The rotation would turn the outgoing photon into an incoming photon. If it were drawn as “straight up” originally, one wouldn’t draw it as “straighth sideways” in the rotated version, as that would be a confusing way to draw an incoming particle. It’s not just a literal rotation of the page.
It is indeed impossible, but the form of the diagram isn’t telling you that. Certainly a massive particle could have such a diagram. It’s your separate knowledge about the kinematic issue with a massless particle decay that is pertinent.
Well, would it be possible for the electron-positron pair to emit a pair of photons from their frame of reference, while in actual reality one of the photons would be highly redshifted to insignificance?
What happens if an electron-positron pair are travelling with an immense degree of kinetic energy and then during the annihilation one of the photons is emitted directly in the forward direction? What happens to the other photon?
I know the wavelengths of the photons involved here are extremely small but might it, at least theoretically, be possible to build some sort of laser so coherence effects would direct all the electromagnetic emission in the forward direction?
Every inertial frame of reference has just as much claim of being “actual reality” as any other. And a redshifted photon will still exist.
It will be emitted in the other direction, so as to conserve momentum. (Also mass, as Chronos points out.) Yes, it will be redshifted in the “stationary” frame of reference, but no amount of redshifting will cause the photon to no longer exist.
