Natural Order and Science

[little_monkey]

Ok, I correct myself, you have lecturing as well as issuing random exams, if the latter can be called a form of communication. You are really one of the most obsessively overbearing people I have ever "discussed" physics with, and not with the best reading comprehension either as I noticed in the other thread. Why do I feel like I'm in 5th grade again arguing whose action figure has the better superpower during recess?

Should I indulge you? Fine.

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So first of all, your question is not worded very clearly, and one could get the impression that you are labouring under the misconception that there are always two states in the massless case. If you think that, you shouldn't go round giving exams.

There are two issues here, one is how to "embed particles in fields" in the sense of quantizing the classical field theory, and the other, how particles of different spin are represented as fields. Now, you seem to be mainly alluding to the latter, although there are related subtleties in the former.
First of all, in the massless case, one is interested in transverse states, so instead of considering the full Poincaré group, one analyses the corresponding representations of the little group and resulting helicity states. Now, in the case of J=0, there is nothing much to analyse, one has a singlet under the little group as well as parity, and there is one state, not two. Now, so far we haven't seen a massless helicity 0 particle in nature, but the picture of course doesn't change in the massive case when going to the full Poincaré group. The representation is still trivial, still no spin. Going to massless higher spins, most famously photons or gravitons, the little group will have a single representation again, but this time with nonvanishing helicity. Applying parity then flips the helicity, and we get the two states one is used to corresponding to circular polarizations in the case of the photon, and the usual left- and righthanded rotating strain solutions of linearized gravity. Once again, going to the massive case, we switch to the full Poincaré group with, accordingly, larger irreducible representations.

All this is a purely group theoretic analysis of spin, and the situation in quantized field theory is a bit more involved because we have to explicitly say how and why the states in the massless case are reduced to representations of the little group. In the case of photons, the connection is as follows - they are massless because everything is in a linear representation of the gauged U(1), and it is precisely the same gauged U(1) which allows us to eliminate the additional longitudinal state, for example via GB or BRST. For spin-2, a similar but more involved construction is possible, as here we have the "gauge transformations" as relics of the diffeomorphism invariance of the full theory.

When masses are forbidden by such linearly realized symmetries, they arise via an explicit or spontaneous breaking of said symmetry. This then leads to extended conditions for the physical states involving the Goldstone or Stückelberg fields if you're doing GB.

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Is there anything you would like to add, oh ye pompous one?  Ok, now I feel dirty. One should talk about the science for science's sake, to improve one's knowledge, and with a bit of humility. You treat science as if you were a baboon and physics your red swollen ass, a display unbecoming to its dignity.

The pompous side of me wants to add that it was Wigner who classified the state for m=0, you get two states, and for m>0, you get 2j+1 states, that the Poincare group has no finite-dimensional representation, but you get the infinite ones by embedding a polarization vector which is momentum dependent, making the basis vector infinite-dimensionally. Yes to your post about removing certain degrees of freedom, yes to your post to point out the connection to gauge theory. You should have mentioned the Ward identities, but I'll forgive you. Ok, I misjudged you, but that was because of how you behaved towards me. My initially post was not addressed to you, but you took the liberty to jump him, and belittled me. I didn't know if you were deliberately obstinate in refusing to see my point about a simple definition. Also, I put into one of my posts a word that anyone in QFT would have reacted but you didn't, so I concluded that you were an impostor. But this post corrects this impression you had given me so far - an impression all on your doings. So, yes I'm a professor who taught this stuff for 25 years, so I know it like the back of my hand. I'm also retired with lots more time on my hand, and I guess I will be spending more time on this forum. If I threw some dirt on you it's because YOU started it. Most likely we will cross path again on this forum. I'll leave it up to you if you want to restart afresh or continue this hostility.