**4**polarizations (unlike observable photons, which have only two). At least in the Lorenz gauge. I am truly shocked and traumatized!

But my favorite quote from today's Quantum Field Theory lecture was when the prof said:

"So why is our Hilbert space not positive definite? It's because of that

**damn**gauge invariance again. But if we can convince ourselves that there is some subspace within which the extra polarizations all cancel out, then we can restrict our initial and final states to this subspace, and everything is unitary again, and everything is positive definite, and everything is happy, and it should take up a very warm place in your heart."

Somehow, it's not taking up a warm enough place in my heart. My faith in the electromagnetic 4-vector potential has been shaken!

- Current Mood: geeky

## Comments

firmamentspoonlesscan'tbe the end of the story. The problem is, we haven't seen that clue and so this is the best we have to work with.For decades it's been the case that any number an experimentalist can measure in a lab, no matter what it is, turns out to be either something you could have calculated with QFT without doing the experiment, or one of the 30-something "input parameters" to the theory (for example, the mass of the electron, and the fine structure constant). So the only thing we can do is try to predict those input parameters with new theories, or wait for an experimentalist somewhere to find a discrepency. The former doesn't seem to be happening, although there has been a lot of progress toward "making the theory more pretty" or to some extent justifying it in the minds of the theorists.

fallen_x_ashesfirmamentAnd we needn't be too frightened of the theory's accuracy. Just because we cannot come up with theories as accurate right off, doesn't mean we shouldn't give attention to alternative theories we can come up with. And in fact, it may be the case that we can't even see the was in which the current theory fails empirically without the results of a new theory to point them out to us.

This kind of situation happens all the time in the history of science. Ptolemaic astrophysics was pretty darn accurate; maybe not as accurate as QFT, but one of the best going predictive theories of its day. Copernicus comes along, having read some ancient Greek astronomy, having some funny ideas about mathematical and metaphysical elegance, and he posits a new theory. Now, the old theory did pretty well, and wasn't obviously dissatisfactory. The new theory was, at first, less accurate than the old theory. Nevertheless, Copernicans persevered, improved the theory, used it to find problems in the old theory, and eventually Copernicus won. We may be in the same boat with contemporary physics, and we would be remiss to wait around for the experimentalists to find a crucial problem with it. That would get the process exactly backwards.

spoonlessBut the theory is both metaphysically and explanatorily dissatisfactory, as you've pointed out, and that itself is a pretty deep criticism for any theory.

It quite possibly may be metaphysically and explanatorily dissatisfactory at some level, but I'm not going to throw in the towel just because I don't undestand it at first glance. It's not the first time I've encountered something which seemed bizarre and counter-intuitive only to later realize it makes perfect sense. I consider this my very preliminary "first reaction" and not by any means a final statement on whether I like it. It also may just indicate that I need to sharpen up some of my metaphysical assumptions or tweak them a bit.

Copernicus comes along, having read some ancient Greek astronomy, having some funny ideas about mathematical and metaphysical elegance, and he posits a new theory.

I think the best place to start, though, in looking for a successor to QFT, is by examining it and trying to understand

whythe math works, rather than starting from scratch. Which in some sense, is exactly what Copernicus did. He noticed a bunch of ugly equations for how things moved in the sky... and figured out a much simpler way they could be expressed, and then took that idea seriously metaphysically. In the same way, Einstein came up with his theory by looking at how Maxwell's equations worked, even though the metaphysics (luminiferous aether) Maxwell had tied to it was entirely different. Einsten wouldcertainlynot been able to come up with relativity without carefully examining Maxwell's ideas. In the same way, in the absense of any new experimental data, I suggest we should start from the accuracy of QFT and try to reinterpret its metaphysics in any way that makes more sense. (Perhaps that's exactly what Finkelstein did, I don't know. But I'm just trying to make the point that the accuracy of itisextremely important.)lars_larsenfallen_x_ashesspoonlessThe thing that bothers me the most about this alledged difference in the number of polarizations is that in theory, there shouldn't be any difference between real and virtual photons. In the end, they are both the same thing (or so I thought) and should behave the same--other than what they happen to run into or avoid. All of this makes me want to drop either my rigid interpretation of Lorentz invariance (Einstein's special theory of relativity), or my belief in what's called the "electromagnetic 4-vector potential", an object that can't be Lorentz invariant unless these extra polarizations exist. Either way, I don't think there should be a difference in the # of polarizations!

fallen_x_ashesThe thing that bothers me the most about this alledged difference in the number of polarizations is that in theory, there shouldn't be any difference between real and virtual photons. In the end, they are both the same thing (or so I thought) and should behave the same--other than what they happen to run into or avoid.That is what I pretty much figured as well, so even though I don't understand much of this, I am fairly disconcerted. Could you take the time to explain this subject to me in laymen's terms? I don't exactly understand what you mean when you talk of polorizations, Lorentz invariance or 4-vector potential.

spoonlessSo photons are individual quantized packets of light waves, they each have a combination of these two polarizations associated with them. If you wear polarized sunglasses, for instance, they will reflect the photons with horizontal polarization and let the photons with vertical polarization pass. This cuts down on the glare, because the ground tends to polarize light in the horizontal direction when it reflects it.

Virtual photons apparently... can have 4 polarizations. How is this possible? Well, the third direction they can be polarized in is along the direction the light is travelling. So we live in 3-dimensions and you'd think that would exhaust all the possibilities right? Well, there's one more dimension we forgot about... the other direction they can be polarized in is... (drum roll)... time. So the fourth polarization of a virtual photon is waving in the direction of time rather than space! (Don't ask me what that means, exactly, cause I don't know!)

I don't know if I want to try and explain what the 4-vector potential is. Suffice to say it's a more abstract but simpler object which the electric and magnetic fields can be derived from (and possibly, arise from). There is a bit of freedom in how to define this object. But the only way of defining it in which it doesn't end up affecting itself at different locations faster than the speed of light (and equivalently, backwards in time) is to choose the so-called "Lorenz gauge". But if you do this, you get these extra polarizations. So you're left with a tough choice between not believing in it at all, believing it doesn't obey causality, or believing that virtual photons are different from real photons in the way I've described. So this is why I say "my faith in it is a bit shaken".

onhavaIn the end, they are both the same thing (or so I thought) and should behave the same--other than what they happen to run into or avoid. All of this makes me want to drop either my rigid interpretation of Lorentz invariance (Einstein's special theory of relativity), or my belief in what's called the "electromagnetic 4-vector potential", an object that can't be Lorentz invariant unless these extra polarizations exist. Either way, I don't think there should be a difference in the # of polarizations!Briefly: in some sense every photon you ever want to talk about is virtual, more or less -- after all, if it doesn't eventually interact with something you don't observe it. How "real" a photon is is just determined by how "on-shell" it is, i.e. how close its mass squared is to the pole in the propagator. The closer you get to this pole, the less you should see any effect from the two "unphysical" polarizations.

Sorry, that's a bit vague, but does it help?

spoonlessFirst off, just to make sure I understand you correctly... would the pole for a photon be at zero, since perfectly "real" photons have zero mass?

Second, I've read that there are two different interpretations of this mass issue... one way is treating virtual particles as if they are off their mass shell... and the other way is by saying that energy is only conserved overall, not internally. I haven't gotten far enough to make a choice as to which sounds better, but I'm guessing their might be problems with the latter interpretation as to explaining how the information of how much energy is to be conserved propagates across without being stored somewhere (like in the mass of the virtual particle). Do you see this as a compelling reason to treat it this way, or are there other better reasons? Or do you see the two explanations equivalent?

Next... I have one idea about how the issue of the polarizations might be solved, philosophically. So tell me if this sounds right, cause it's the only way I can make (some) sense of it for the moment: all photons have some of the extra polarizations (and I suppose some "mass" that they carry), but when we measure them we only have the ability to measure the transverse polarizations, and therefore our measurement collapses the other polarizations and forces the photon into an eigenstate of purely transverse. Equivalently, perhaps measuring them forces them to have zero mass...? Even if my guesses are correct here, I'm still not fully satisfied. Does the mass being closer to the pole make it easier in some way to observe it? I'm not sure how to fit these two things together.

If you're too busy to bother with this, I understand. I'm sure I'll figure it out eventually.