Has anyone got any idea as to just what precisely the deal is with certain wave functions' spin addition/subtraction, such as the sigma/lambdas? I can't quite make out just why so many other particles don't suffer from this "schizoid" behavior, for example: the Xi+c (c(sd+-ds)/sqrt2)... it appears to have no reason to be recognized as a single particle while being mathematically described in exactly the manner in which sig+c/lam+c (c(ud+-du)/sqrt2) are described, and they are recognized as being quite individually different particles under SU(3).
Yeah, Plato and a couple of others would be good bets for this. Have you seen them around here lately?
Wow, such confidence in my abilities... Rock, I'm afraid I can't really make sense of your notation scheme. Could you perhaps use the Latex notation ? Then again it has been four years since I was into this stuff so I don't know if my rusty memory will serve you any good. Let's see, I think you mean with the sigma's the fermi matrices, or their corresponding operators, right ? Do you mean the wave function with Xi ? That would be \Psi or \Xi I believe in Latex notation. What is the deal with the plus signs, do you mean actual addition or something lie : \Xi^{c} or \Xi_{c} ? Then I don't know why you use the lightspeed constant as an index of any kind. What is the +- deal a short hand for addition and subtraction ? About the SU(3) symmetry, I believe the fermi matrices are SU(2) so are you talking about Dirac matrices then ? But I thought they were SU(4) ? You aren't talking about the color symmetry SU(3) are you ? So I guess what I'm asking is to be a bit more precise and clear. ------------------ I err, therefore I exist ! [This message has been edited by Plato (edited September 07, 2000).]
Sorry about that. Umm, I'm not familiar with the Latex notation, so I'll have to muddle through as best I can. Ok, here we go: Xi+c (c(sd+-ds)/sqrt2)... Xi particle, positive electrical charge, charmed (hence dsc quark composition). The +- is for sd+ds or sd-ds. sig+c/lam+c (c(ud+-du)/sqrt2) would be sigma pos (charmed) and lamda pos (charmed), both composed of udc. ud+du or ud-du. Key point though: the u and d spins may be aligned (ud+du), thus resulting in a larger mass particle (lam) or not (ud-du), thus resulting in a lighter particle (sig). They have the exact same mathematical description, save for the spin addition/subtraction. The Xi (pos., charmed) has an identical description (suitably rewritten for a differing qqq composition), but is identified as only being a single particle rather than two rather different ones. Hmm, SU(3) vs. SU(4)? Yeah, I dropped that ball: SU(4). I have no idea why I wrote SU(3), it wasn't a typo... though I'd like to claim that excuse. Please Register or Log in to view the hidden image!
Oh! I forgot to give you a different rendering of the wave functions: Xi+c can be written as 1/sqrt2((sd+-ds)/c) sig+c as 1/sqrt2((ud+du)c) lam+c as 1/sqrt2((ud-du)c) [This message has been edited by Rock (edited September 08, 2000).]
Hi Rock, I think I'm following know you are talking about baryons and their quark content : hence the SU(3) because there are always three quarks in a baryon. So what you are saying is : while there are two different symbols for the sigma and lamba particle there is only one for the Xi particle, right ? I'm afraid you got me there : I would assume that for different spin additions the mass of the Xi particle would also be different. I can't think of a reason why this would still have the same mass... Are you certain of this ? Perhaps there are two different Xi particles having different masses but people simply use the same symbol ? It looks very odd to me. .... I just did some searching on the net and found this site : http://www.starlight-pub.com/Matter/PartIV/IV2/IV2ChemistryHyperons.html It seems there are neutral and negative Xi particles, a triplet of pos, neg and neutral lamba particles and a singlet sigma paticle. This last symmetry is due to isospin symmetry I guess but I don't really see how the Xi is just a duplet... ------------------ I err, therefore I exist !
"...SU(3) because there are always three quarks in a baryon." Yep, that would probably be it. "So what you are saying is : while there are two different symbols for the sigma and lamba particle there is only one for the Xi particle, right ?" Precisely the problem. As best as I can see it, the subtraction version should lie around 0.2 MeV/c^2 lighter than the addition one, with appropriate property modifications (no, I haven't dealved into what property differences there are in particular). "...two different Xi particles having different masses but people simply use the same symbol ?" An extremely unsettling prospect indeed. I've never read that possibility, but I certainly can't discount it... especially in light of the J/psi, or the difficulties in standardizing excited states' names. I've gone to the site, and it's fascinating, but before I can respond here to any of their info, I'll need time to root around (it seems quite large), and to digest the data. "It seems there are neutral and negative Xi particles, a triplet of pos, neg and neutral lamba particles and a singlet sigma paticle. This last symmetry is due to isospin symmetry I guess but I don't really see how the Xi is just a duplet..." That one I can answer: you're looking at the decuplet, a symmetry tool from c. 1975. It was superceded by the cubedecahedron (triangle, hexagon, triangle, but with a zigzag face along the lower stratum and a simple alternating face along the upper) and the 20-plet (a pyramid with the old decuplet on the bottom, and a singlet CCC at the top). The original decuplet doesn't take into account charmed quarks as they hadn't been hypothesized by '75 (or so). The decuplet also has either the nucleons uud (p) and udd (n) on the fourth line at the top, or the quadruplet uuu (delta ++), uud (delta +), udd (delta o), and ddd (delta -). The deltas are a quad while the nucleons aren't for reasons of symmetry which I've seen alluded to but never spelled out in plain English. The deltas are obviously more massive (spin addition: binding energy). The Xi (NOT charmed versions) are Xi o (ssu) and Xi - (ssd). The reason that there's not a third Xi (+?) here is that there's not a third quark between u/d and s. Other Xi's exist: Xi c+ (scu), Xi co (csd), Xi bo (bsu), Xi b- (bsd). I think it's safe to theorize that Xi t+ (tsu), and Xi to (tsd) aren't too far away... presumably through the LHC or another of it's ilk. You mentioned isospin symmetry... you're quite right in that, it's just that I've got a personal pet peeve about isospin. u and d are treated as if they were special, meanwhile the s/c and b/t symmetries get thrown by the wayside. Does this sound like the action of rational symmetry loving physicists? Yet hypercharge is relegated to near nonexistance, b/t sym hasn't even got a name that I know of, and isospin is glorified as if it were any different from what it is: the simple binary yes/no axis of a partcicle possessing any first generation qaurks.
