# Significance of isospin and hypercharge

by robousy
Tags: hypercharge, isospin, significance
 P: 335 This may sounds like a dumb question but I want to figure out the 'point' of them. I know that isospin was an attempt to describe the proton and nucleon as an isospin doublet and that hypercharge seems to me to be a nifty little relation between the electric charge and the isospin - but what is the point of them now that we know that we cannot describe the nucleon as an isospin doublet and that it is in fact constructed of quarks. Is it that: a) Before we new about quarks it seemed like a good idea. or b) We can indeed build a model based on isospin. I am learning about GUTS at the moment and specifically $$SU(2)_L \times SU(1)_Y$$ What is the significance of the hypercharge Y subscript in the $$SU(1)_Y$$? It seems (to my admittedly ignorant mind) that hypercharge is nothing more than a nifty relation to charge so why does it get elevated to the subscript level $$SU(1)_Y$$ in GUTS??? Please help me understand the significance!
 P: 335 Ok - I know now so I don't want to waste anyone time!! :)
P: 19
 Quote by robousy Ok - I know now so I don't want to waste anyone time!! :)
well tell me what u have understood...i 'll b glad 2 read what u 've interpreted of it.....

P: 21,637

## Significance of isospin and hypercharge

A collection of comments on hypercharge and isospin.
 Conservation of strangeness is not in fact an independent conservation law, but can be viewed as a combination of the conservation of charge, isospin, and baryon number. It is often expressed in terms of hypercharge Y, defined by: Y = S + B = 2(Q-I), where S = Strangeness B = Baryon number Q = charge I = isospin Isospin and either hypercharge or strangeness are the quantum numbers often used to draw particle diagrams for the hadrons.
However - it seems it can more complicated - http://en.wikipedia.org/wiki/Hypercharge
http://en.wikipedia.org/wiki/Isospin
As for strangeness -
 A property of hadrons which is conserved in particle reactions caused by the strong force and which has in a weak interaction.
Eric Weisstein's Scienceworld at Wolfram.com - http://scienceworld.wolfram.com/phys...rangeness.html
http://en.wikipedia.org/wiki/Strangeness - See the definition here, which then states
 The reason for this unintuitive definition is that the concept of strangeness was defined before the existence of quarks was discovered, and for consistency with the original definition the strange quark must have strangeness -1, and the anti-strange quark must have strangeness +1.
Particle physics and supporting theories do appear as strange.

Sometimes one is faced with the need for a theory (understanding) without having all the necessary information. Remember back to the 1800's before quantum physics - the physicists had glimpses of atomic structure, but just didn't have all the information.

A neat bit of trivia -
 Unlike his brother Maurice, who was primarily an experimental physicist, Louis de Broglie had the mind of a theoretician rather than that of an experimenter or engineer. His 1924 doctoral thesis, Recherches sur la théorie des quanta (Research on Quantum Theory), introduced his theory of electron waves.
- from http://en.wikipedia.org/wiki/Louis_de_Broglie

and http://en.wikipedia.org/wiki/Murray_Gell-Mann
P: 335
 Quote by preet0283 well tell me what u have understood...i 'll b glad 2 read what u 've interpreted of it.....

Well, due to the almost identical masses of the proton and neutron we hypothesized the existence of a nucleon isospin doublet. Clearly the p and n have differing electric charges so a formula was created by Gellman and Nishijima linking the isospin with the electric charge: Q=Y+H where H is the hypercharge and Y the isospin. It turns out that hypercharge is actually composed more fundamentally of B-L where B is Baryon and L Letpton number.

Thats the OLD interpretation. I am starting to realize that there is a new interpretation of Isospin which is a bit more complicated and the name is just a carry on from the old days. Something to do with the generators of the groups of the standard model and the Cartan Subalgebra eigenvalues being used to construct the charge generator...(maybe)

If you can expose more of the 'new' isospin idea that would be good.

Thanks!
P: 817
 I am learning about GUTS at the moment and specifically $$SU(2)_L \times SU(1)_Y$$
What is SU(1)_Y?

Look, do you mean the electroweak gauge group SU(2)XU(1) or chiral symmetry group SU(2)XSU(2)?
THE 1ST GROUP HAS SUBSCRIPTS (W) OR (L) ON SU(2) TO MEAN WEAK OR LEFTHANDED FIELDS, AND Y ON U(1) TO MEAN THE WEAK HYPERCHARGE(THE GENERATOR OF U(1)).
THE CHIRAL GROUP SU(2)XSU(2) OFTEN WRITTEN AS SU(2)_LXSU(2)_R.
FROM NOW ON, BEFORE I ANSWER ANY OF YOUR QUESTIONS, I NEED TO KNOW WETHER OR NOT YOU ARE LEARNING SOME THING FROM US. TELL ME; ARE WE WASTING OUR TIME?
P: 335
 Quote by samalkhaiat What is SU(1)_Y? Look, do you mean the electroweak gauge group SU(2)XU(1) or chiral symmetry group SU(2)XSU(2)? THE 1ST GROUP HAS SUBSCRIPTS (W) OR (L) ON SU(2) TO MEAN WEAK OR LEFTHANDED FIELDS, AND Y ON U(1) TO MEAN THE WEAK HYPERCHARGE(THE GENERATOR OF U(1)). THE CHIRAL GROUP SU(2)XSU(2) OFTEN WRITTEN AS SU(2)_LXSU(2)_R. FROM NOW ON, BEFORE I ANSWER ANY OF YOUR QUESTIONS, I NEED TO KNOW WETHER OR NOT YOU ARE LEARNING SOME THING FROM US. TELL ME; ARE WE WASTING OUR TIME?
$$SU(3) \times SU(2) \times U(1)_Y$$

Yes, the U(1)_Y weak hypercharge generator.

Yes, I am learning - some aspects faster than others, but please, if you think you are wasting your time then please do not answer. I am sure I would not be offended if noone left an answer.
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P: 1,204
 Quote by robousy If you can expose more of the 'new' isospin idea that would be good.
Perhaps it would be good to remember that the fundamental particles are not actually little indivisible things that cannot be divided. A better description of them is to think of the wave functions one dealt with in quantum mechanics.

With wave functions, you can take the sum of two wave functions and, because the equations are linear, the result is another wave function with properties somehow sort of midway between the other two wave functions. For example:

$$\psi = \psi_A + \psi_B$$

where the two wave functions on the right satisfy an operator equation like:

$$\mathcal{O} \psi_A = A \psi_A$$ and same for B.

The sum of the A and B wave functions is a valid wave function but is not likely to be an eigenfunction for the operator like A and B were. So if you define "particle" as the things that are eigenfunctions for that operator, then the sum is not a "particle".

In the case of the elementary particles, we consider charge, Q, to be one of the operators that define what are the elementary particles. For example, if "e" is the electron, and \psi_e is an electron wave function, we have the operator equation:

$$\mathcal{Q} \psi_e = -e \psi_e$$

since the charge of the electron is -e. Similarly,

$$\mathcal{Q} \psi_\nu = 0$$

since the charge of the neutrino is zero.

So we classify the elementary particles in a way that makes them eigenfunctions of (electric) charge and mass (and parity or whatever).

But just because we classify elementary particles in this way does not mean that we cannot reclasify them in the same we can reclassify wave functions by taking linear combinations of them. And some of the alternative methods of classifying them would make more sense in certain circumstances.

An example is the Cabibbo angle. The up and down quarks (of the "electron family") are eigenstates of the electric charge operator and are eigenstates of mass, but they are not eigenstates of the "weak charge" (when I was in grad school it was called "neutral charge") operator in the sense that when you change an up quark into a down quark by emitting a W-, you don't actually get a pure down quark. Instead, you have to mix in some of the other families of quarks, the "muon family" and "tau family". W+ and W- interactions for quarks involve changing from a +2/3 to a -1/3 (or -2/3 to a +1/3), so you can fix the weak charges by either mixing the (u,c,t), or by mixing the (d,s,b). Nowadays, it is done by mixing the (down,strange,bottom) and it is called the "CKM" matrix.

As an alternative, when you have a pure down quark and arrange for it to emit a W-, it doesn't become a pure up quark but instead ends up with a mixture of top and charm. Thus you could instead define a CKM type matrix as mixing the (u,t,c).

With the leptons, there is also a mixing between the neutral leptons (neutrinos) as compared to the charged leptons. As with the quarks, the mixing appears when you define the particles according to their masses (and therefore into the families or "flavors"). The electron, muon and tau are the mass eigenstates of the charged leptons. When one of these particles emits a W-, the charged lepton changes to a mixture of neutrinos.

Now with the quarks, the effect of the emission of a W+ or W- is a relatively small probability of a change in the family. Therefore we collect the quarks into pairs, (up,down), etc. But with the neutrinos, the mixtures are more democratic so it's hard to say which of the neutrino eigenstates corresponds to the electron and which to the muon and tau. So the neutrinos are generally defined according to what you get when you take a W+/- out of the corresponding lepton mass eigenstate. That is, we talk about an electron neutrino, a muon neutrino and a tau neutrino. This means that the lepton analog of the quark CKM matrix is defined in sort of reverse, an extra opportunity for confusion. It is my belief that the quark mixing is more pure than the lepton mixing arises naturally from the way they are produced from subparticles but that's another story; in the standard model, these are all fairly arbitrary parameters.

If you want to always define "particle" as things that are eigenstates of electric charge and mass, then you will have the weak interactions mixing particle types. But you could instead define "particle" as things that are eigenstates of weak charge and mass, and that would leave you with electric interactions that mixed particle types.

It is also possible to define particles as the things that are eigenstates of both electric and weak interactions. If you do this, then you will have particles that are of mixed mass eigenstates. I suspect that things will be simplest in this basis. In any of these three cases, it is important to note that the mixing is over corresponding particles in the different families, and that the families differ only in their masses.

Now what does this have to do with weak isospin and weak hypercharge? Weak isospin and hypercharge, together, give electric charge and weak charge. Weak isospin is the SU(2) type symmetry between the two objects that a weak force W+/- converts between. Once you've defined weak isospin and electric charge, the difference between these is also defined and (twice the difference) is called weak hypercharge.

When the electric and weak interactions are combined into an electroweak interaction, the appropriate currents (i.e. moving charges) are weak isospin (which is a vector) and weak hypercharge (a scalar). These two are mixed by the Weinberg angle so that instead of interacting directly with gauge bosons according to weak isospin and weak hypercharge, they instead use weak charge and electric charge. Thus the photon is a mixture of a weak isospin and weak hypercharge interaction.

It's late. I hope I haven't made too many serious mistakes.

Carl
 P: 4,008 CarlB, Just wanted to let you know that i really like your last post. You have the ability to explain (conceptually) difficult stuff in an easy language. I like your style, man. I wanna ask you to contribute to the "elementary particles presented thread" if you want. For example, you could make a reference to this post into that thread. Or, if you feel that certain aspects/topics of theoretical physics need to be explained, i invite you to post them there and thus expand our elementary particles-library. I would be very greatful regards marlon
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P: 1,204
 Quote by marlon CarlB, Just wanted to let you know that i really like your last post. You have the ability to explain (conceptually) difficult stuff in an easy language. I like your style, man. I wanna ask you to contribute to the "elementary particles presented thread" if you want. For example, you could make a reference to this post into that thread. Or, if you feel that certain aspects/topics of theoretical physics need to be explained, i invite you to post them there and thus expand our elementary particles-library. I would be very greatful regards marlon
Whoa! Before you conclude that I know what I'm talking about, you should be aware that (a) I am not associated with any physics (or math) departments and in fact make a living by doing stuff like driving forklifts, using a nail gun, and soldering; (b) I started grad school back when Carter was president and never finished a PhD; (c) I don't believe in Einstein's relativity; (d) I believe that using symmetry to define elementary particles is fundamentally misguided; (e) I don't believe in the quantum mechanical vacuum (along with Schwinger as it turns out); (f) I believe that the symmetry breaking seen in the standard model is actually a part of spacetime itself rather than an attribute of the particles per se; (g) I believe that standard matter is condensed from tachyons; and (h) I have a nasty habit of going to physics conferences and being ignored for supporting points (c-g). In short, I really can't think of anyone you'd want less to be telling you about the standard model.

On the other hand, I do believe that what I wrote above is a fairly accurate portrayal of what both I and the standard model believe about the elementary particles and you're certainly welcome to copy it where you wish.

The mixing matrix for the weak force as applied to the quarks is called the "CKM" matrix. The same thing applied to the leptons is called the "MNS"
matrix. Naturally I'm busily attempting to unify all this with my own version of particle theory, but if you value your sanity (or your standing in the standard physics community, I advise you to stay far away).

Carl
 P: 4,008 CarlB at what university did you study ? Besides, you can have whatever personal opinion that you want, i do not care. I read your "personal caracterization" very thouroughly and i admit i would be the exact opposite of you. However, this does not take away the fact that lot's of your post are of high quality and that is my honest opinion. So, YES, i would like to have you on board when it comes to explaining difficult concepts to laymen, which one of the primary intentions of this great forum. For the rest, you are forgiven :) marlon ps : why is it that only the really good and talented people are so very honest and direct ? :)
 P: 335 CarlB. I've been meaning to thank you for ages now since you kindly posted a long and no doubt lucid explanation to my question. TBH I haven't had a chance to go through it yet - but I plan to next week (when a 'long lost friend' I am entertaining returns home) - but thanks again!

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