Every subatomic particle is completely described by a small set of observables such as mass m and quantum numbers, such as spin J and parity P. I want to get a macroscopic analogy for these observables and then try to get a more rigorous mathematical definition.
For example, electric charge is an attribute that seems easiest to get a grip on. We see two objects of opposite charge repel one another. So I can imagine the electric charge on two quarks repelling or attacting. Mass I can see as the quark’s trajectory being affected if two quarks with mass approach near each other. If a quark is massless then its trajectory should not be influenced by another quark in so far as mass is concerned. Then there is energy being quantized. An analogy for String Theory is to think of a quark as a small circle of string. If you break the string, hold it at both ends and create a standing wave you can have 2, 3, or more nodes. With the string unbroken, you must have an even number of nodes or there will be a phase mismatch at one spot. This makes it easier to understand the energy in a wave of a string must be quantized and not continuously variable.
Isospin, for example, is an attribute for which I don’t have a macroscopic analogy. It seems physics is saying that the mathematics that describe macroscopic spin are similar to observations made about the strong interactions between neutrons and protons and so Isospin is an attribute of quarks. But what about spinning objects can I use as an analogy?
Perhaps the words are misleading me. I have come to learn that Strangeness, Charm, Topness, Bottomness, and others were created before being attached to actual phenomena. So while Strangeness has something to do with particle decay, it is not strange.
I am trying to read the Wikipedia articles but my college physics courses are years passed and have not left me with enough of an in depth knowlege to get through the definitions. What I am asking for in this thread is first a macroscopic analogy for various quark attributes and then some assistance in getting a more rigorous mathematical foundation. Also, please correct any of the analogies I have given if they are just plain wrong or too misleading.
Most of these conserved quantum numbers are actually labels for the representations of some group that these fields furnish. For example, electric charge is related to the group U(1) of unitary 1-by-1 complex matrices (complex numbers of norm 1). When you hit the system with an element of U(1) you multiply a given field of charge n by the nth power of the complex number. For this to be a group action, n has to be an integer.
Mass is a little more unusual, but again has an interpretation in terms of the representation theory of the Poincaré group of special relativity.
Isospin is called so because it’s related to representations of SO(3) (rotations in 3-d space) but not the SO(3) that actually rotates space. It actually rotates some “internal” 3-d space of parameters on the fields, but the representation theory is the same.
As for the string theory and quarks… you’re confusing what the original string theorists thought (string theory describes strong force) with the representation-theoretic view of Gell-Mann that gave us the quarks we know now.
One change that may or may not help your thinking is that all quarks have mass. Only the so-called gauge bosons - photons, W and Z bosons, and gluons - are massless. The graviton would also be massless if it is ever found. These particles are elementary in and of themselves, i.e. they are not composed of quarks or any smaller particles.
As for trying to translate properties into macroscopic terms, good luck. To be honest I don’t think most of them - like spin or the color force - have any analogies, but maybe some of the physicists here can try.
In this old thread I expanded on what Mathochist says about isospin in somewhat less technical language.
However, it’s probably more important to realise that you shouldn’t get hung up about isospin. It’s actually only an approximation that’s useful in practice without being really fundamental. Understanding things like gauge symmetry will get you closer to the heart of the matter.
Isospin is called that by analogy with quantum mechanical spin, because it shares the same mathematics. Quantum mechanical spin has some vague similarities to macroscopic spin, but also some fundamental differences. There is almost no comparison at all between isospin and macroscopic spinning objects. That said, much of what isospin is is just a way of keeping track of how many up quarks and down quarks (and anti-up and anti-down) there are in a particular particle, so it’s in much the same category as charm, strangeness, etc. Basically, you just have to accept that there’s some property (never mind what it actually is) that makes the six different kinds of quark different from each other. So some quarks have a “flavor” of “up”, some are “down”, some are “strange”, and so on.
Exapno, you’re right that the photon, gluon, and (presumably) graviton are all massless, but the W and Z are quite massive, heavier than most atoms. And there’s no reason in principle (that we know of) you couldn’t have a massless fermion: Neutrinos were long thought to be an example of such (though we now know that they have a nonzero, though very tiny, mass).
As usual, let me preface by saying that I know just less than nothing about QCD, QED, and the like. So, patience, please.
Is isospin essentially a tendency for a down quark to (transiently) change into an up quark, and vice versa? If that is even close to correct, let me ask, then, is isospin a consequence, or a manifestation, of the weak force (by virtue of its “ability” to change a down into an up quark)?
Okay, let’s roll up our sleeves and dive in. There are a whole lot of ways to rotate something around a fixed origin in three-dimensional space. Luckily, they’re really nice in that you can always rotate something back to where it started (rotations are invertible) and if you do one rotation and then another, the result is a third rotation (rotations are composable). Basically, the fact that you can invert and compose rotations makes them into something mathematicians call a “group”. This group is called SO(3).
However, right now they’re just these sort of abstract, flabby “rotations” that we don’t really know how to work or compute with. What we want is to find a matrix (a refresher on how those work is here) for each rotation so that composing rotations corresponds to multiplying matrices, and inverse rotations correspond to inverse matrices. That is, we want to see the group represented by nice simple linear transformations of some n-dimensional space. We call these sorts of things “representations” of the group.
Okay, so for rotations of 3-d space, they’re already such nice linear representatives! That is, the group SO(3) comes with a natural “3-dimensional” representation. But that’s not the only one possible.
Now I need to make a little aside here. Physicists don’t really care about SO(3) for various technical reasons that I’m not entirely sure they all understand themselves. I do, but it’s really annoying to explain without a lot of background. What we’re going to consider is something like two copies of SO(3) sewn together in a certain way.
Grab your favorite mug with a good strong handle and fill it to the brim with the boiling hot beverage of your choice. Now, hold it by the handle and turn it around pivoting at your elbow. If you don’t lift it, you’ll turn the mug over and scald yourself so you have to lift as you go around and make the mug go over your shoulder. Then on the next turn it has to go down under your armpit again. That is, you have to turn it around twice to get back where you started. That’s pretty much how these more general rotations work.
Anyhow, these turn out to be “isomorphic” to (“pretty much the same as, for all we care”) the group SU(2) of 2-by-2 unitary matrices. Just like SO(3), these come with a natural representation, but this one’s 2-dimentional and given by the normal way of multiplying matrices and vectors. However, we can immediately see a 3-d representation: just forget which copy of SO(3) we’re on and use the 3-d representation of SO(3).
And the representation theory of SU(2) goes from there. It’s actually pretty similar to numbers – there are certain basic ones you can “add” and “multiply” together, but now there’s one basic representation (sort of like a prime number) for each dimension. 1-d is the trivial representation sending each group element to “don’t do anything”. 2-d and 3-d we’ve seen, 4-d and higher are a little weirder, but they’re not too hard to understand. Physicists, though, like to label them with a number j so that (2j+1) is the dimension of the representation. That j is the “spin”.
Isospin happens when we aren’t rotating our particles around in space, but through some “internal” space of parameters. We can collect together the proton field and the neutron field into a 2-dimensional vector, for example. Then if you act on this by any isospin rotation (an element of an SU(2) different from (but isomorphic to!) the spatial rotation SU(2) from above) you get another pair of fields. If the first was a possible solution of the equations for protons and neutrons, so will the transformed version. You can gather the three pions together and hit them with the 3-d representation of isospin SU(2) and the same thing happens.
Okay, so the proton is really a pair of fields together that carries the 2-d action of spin-SU(2), and the same for the neutron, so each of them has “spin 1/2”. The two of them can be collected together into a 2-d representation of isospin-SU(2), so they have isospin 1/2. The conserved quantum numbers just tell you what representation you’re using to exhibit the fundamental symmetries of nature!
Just being reminded of the basics again and again is helpful (like all quarks having mass; photons and gluons are massless; W and Z bosons are really heavy). Not dealing with them on a day-to-day basis makes them easy to forget. And just realizing that there may not be macroscopic analogies for some terms is very helpful too. As soon as Wikipedia starts working again, I’ll check “Matrix mathematics”, “Unitary matrices”, and “Guage symmetry” for some background.
Is this the same as the “hold a cup and saucer in one hand and rotate it around without spilling?” You have to rotate it on its axis twice to get your arm back to where you started. If so, your description threw me off because I always start out going under the arm then over the shoulder and I didn’t recognize it at first :). I have to digest alot before I’ll try to ask a halfway decent question. Thank you all.
Quite the opposite. Isospin is only useful when you’re talking about the strong force, precisely because the strong force conserves isospin. So the fact that isospin is conserved tells you, among other things, that in the strong interaction, an up quark doesn’t turn into a down.
At this level of detail, isospin is just a different notation for upness and downness. Really, upness and downness just gives you one component of isospin (commonly called I[sub]3[/sub]) There is actually more to it, but to understand that, you have to start with a good understanding of regular spin. There are a few cases where a particular reaction won’t occur (or at least, won’t occur strongly) because it would violate conservation of isospin, even though it satisfies all the other conservation laws (including conservation of upness and downness).
We don’t care about SO(3) for spin-1/2 systems, because spin-1/2 isn’t SO(3). We do use it occasionally for true vectors (as opposed to spinors) in 3-d space. But usually, if you’re breaking out the group theory for vectors in space, it’s because you’re doing relativity, in which case you want to go to the full Lorentz group, not just SO(3).
Actually, there’s very real sense in which the strong interaction can’t tell the difference between an up and a down quark, so it can’t change one into the other.
But where do spin-1/2 systems come from? Why pass to SU(2) at all? I know even Ph.D.s in physics who just don’t know and take it as a given. There are at least two extremely elegant reasons I can think of off the top of my head (which admittedly are slightly related), but most books just pass to SU(2) and move on.
As for the Lorentz group, it’s effectively just two copies of SO(3) stuck together, so the representation theory for SO(3) (and SU(2)) completely determines its representation theory.
I have the notion that it’s related to the symmetries of the polarization states, but I’m not sure I can express it coherently. It seems to make sense in my own head, but maybe that’s just acclimation, not understanding.
OK, now you’ve got me curious. There’s clearly one copy of SO(3) in there for the purely spatial rotations. But that would mean that the boosts would comprise the other copy, but they don’t have the right commutation relations: There’s no combination of boosts in the X and Y directions that’ll result in a boost in the Z direction. Or are both the copies you’re thinking of comprised of combinations of boosts and rotations?
The problem with SO(3) is that it’s not simply connected. If you draw the curve in SO(3) of rotations around the z-axis from 0 to 2pi radians, that loop can’t be smoothly contracted to a point. Spin(3) is the double cover (as I described heuristically), which is simply connected – all loops can be contracted to points.
In the QFT setting we’re looking for unitary representations of symmetry groups, but our measurements can’t detect phases. That means that when we act by two rotations successively, the matrix product might not line up just right, but only up to a phase. If our group is simply connected we can pick these phases to all be zero so we do get an honest representation. That’s why we use SU(2) rather than SO(3). Half-integer spin fields are representations of SU(2) that can’t be made into honest representations of SO(3).
They’re composed of combinations. Now remember that what QFT really talks about is representing the Lie algebra of the group, not the group itself. When you look at the Lie algebra of the Lorentz group it splits into the direct sum of two copies of su(2) (the Lie algebra of SU(2)), and CPT exchanges these copies. If I recall correctly, J generates rotations and K generates boosts, and we have
A = J + iK B = J - iK
Each of which obeys the commutation relations of su(2).
A general representation consists of a representation of each part. The vector representation is (1/2,1/2). The usual spinor representation is (1/2,0) + (0,1/2), one of which is left-chiral and the other of which is right-chiral.
Not surprisingly, I’m confused. For sure, I don’t understand the group-theoretic approach (and that’s probably why I should probably give up right now). But, one more kick at the cat.
Basically, what I was getting at is, if protons and neutrons are basically “two states of the same thing”, does that mean that, from the perspective of the strong force at least, they are constantly changing from one into another (and thus a down quark to an up quark) OR does the fact they are indistinguishable on some level explain the strong force itself (i.e. the more they are separated physically, the more their identities become separate; so the strong force, then, being a way to keep them indistinguishable - by pulling them together?)
I have this fear that, to someone who knows this stuff, my questions are similar to someone asking me ‘why are diseases caused by evil humours?’ - where would I start? How would I answer without being patronizing?
It’s more the latter, but not in the way you’re thinking.
Looking back I think I (along with others) was making a mistake. Isospin is actually connected to the weak interaction. The strong interaction is dealt with by the “color” theory of quarks, quantum chromodynamics (QCD). That has a group SU(3) governing it.
The Abdus-Salaam-Weinberg (sp?) theory of the electroweak interactions goes like this: You’ve got the group U(2) which is SU(2) from before and U(1) for phases. Every representation is determined by a representation from SU(2), which gives the isospin of the field, and saying how U(1) acts on the field, which tells the “hypercharge”. Let’s consider the up-down doublet, which is the 2-d representation of SU(2) with some value for the hypercharge. If we pick a basis (two perpendicular vectors in this 2-d space) we can say one is the up quark and one is the down quark. Then we can read off the regular charges depending on which is which.
The thing is, our choice is sort of arbitrary. I can make my choice here, and you can make yours there, and they should be independent to the extent that the places where we make our choices can’t send signals to each other (remember special relativity). So how do we compare our choices at different points in spacetime?
We have to set up a new field to “connect” our choices, and that “connection” consists of four complex fields since U(2) has four complex dimensions. These are four massless particles that form an SU(2) singlet (1-d representation) and an SU(2) triplet (3-d representation). At first glance they’re all massless, but when they interact with another field that’s floating around, the triplet fields gain a mass. At energies we can normally access, they’re the W+, Z, and W- bosons of the weak force, and the remaining massless field is our old friend the photon.
In the process, the nice symmetry between up and down quarks is lost. At high enough energies, the symmetry should patch itself up, the Higgs field comes out of hiding, and we again can’t tell up from down quarks. At our energy scales, though, ups are ups and downs are downs. Every so often, though, an up can fire off a gauge boson and turn into a down, or a down can fire off a boson and turn into an up. When this happens inside a proton and the boson then creates other particles that fly away, the proton turns into a neutron and we call it “radioactive decay”.