## Imaginary Color Charge

Discuss tricky levels, and show off your diabolical creations
19683
Posts: 151
Joined: Wed Jun 05, 2013 12:15 pm

### Imaginary Color Charge

I found the following list of gluon color charges:

1: ((r, anti b)+(b, anti r))/sqrt(2)

2: ((r, anti g)+(g, anti r))/sqrt(2)

3: ((b, anti g)+(g, anti b))/sqrt(2)

4: ((r, anti r)-(b, anti b))/sqrt(2)

5: (-i*((r, anti b)+(b, anti r)))/sqrt(2)

6: (-i*((r, anti g)+(g, anti r)))/sqrt(2)

7: (-i*((b, anti g)+(g, anti b)))/sqrt(2)

8: ((r, anti r)+(b, anti b)-2*(g, anti g)/sqrt(6)

5, 6, and 7 :Have imaginary terms! Gluons can be absorbed by quarks, so does that mean that quarks can :Have imaginary color charge as well?
Binomial Theorem: ((a+b)^n)= sum k=0->k=n((n!(a^(n-k))(b^k))/(k!(n-k)!))

testtubegames
Posts: 1088
Joined: Mon Nov 19, 2012 7:54 pm

### Re: Imaginary Color Charge

That's a really neat question. It seems strange at first to see 'imaginary' charges, and I'll be the first to say I'm not an entire expert on the intricacies of gluons. It seems to me, though, that this is more or less mathematical bookkeeping. Without getting in to too much detail....

You see the same thing all over the place with quantum mechanics. A state (like spin, or the charge value in this case) has some 'phase' associated with it. That basically means a state can be multiplied by any complex number with a magnitude of 1. So you'll most commonly change the phase of a state by multiplying by -1, or with +-i. Since the phase isn't something you can directly observe, this truly is just a math-y use of imaginary numbers. You won't encounter an electron with some 'imaginary spin', as opposed to one with a 'real spin'. You'll always just measure a 'spin'.

It's really interesting how this convention for gluon colors was chosen, surely. But states 5, 6, and 7 are no more special than the others.

19683
Posts: 151
Joined: Wed Jun 05, 2013 12:15 pm

### Re: Imaginary Color Charge

testtubegames wrote:That's a really neat question. It seems strange at first to see 'imaginary' charges, and I'll be the first to say I'm not an entire expert on the intricacies of gluons. It seems to me, though, that this is more or less mathematical bookkeeping. Without getting in to too much detail....

You see the same thing all over the place with quantum mechanics. A state (like spin, or the charge value in this case) has some 'phase' associated with it. That basically means a state can be multiplied by any complex number with a magnitude of 1. So you'll most commonly change the phase of a state by multiplying by -1, or with +-i. Since the phase isn't something you can directly observe, this truly is just a math-y use of imaginary numbers. You won't encounter an electron with some 'imaginary spin', as opposed to one with a 'real spin'. You'll always just measure a 'spin'.

It's really interesting how this convention for gluon colors was chosen, surely. But states 5, 6, and 7 are no more special than the others.
Do quark colors have phase too?
Binomial Theorem: ((a+b)^n)= sum k=0->k=n((n!(a^(n-k))(b^k))/(k!(n-k)!))

testtubegames
Posts: 1088
Joined: Mon Nov 19, 2012 7:54 pm

### Re: Imaginary Color Charge

19683 wrote: Do quark colors have phase too?
Great question. To get into that let me revisit a point I glossed over before...

There is a difference between 'states' and 'charges/color'. The first is a quantum mechanical description of the particle (or probabilities of having different types of particles around). The second is just a property, like my hair color being *brown*.

So when you posted that list of gluons, that was a list of states, not colors. The first state, for instance, is a combination of a red-antiblue gluon and a blue-antired gluon. It's a superposition in the same way as schroedinger's dead and alive cat. It's part one, part the other... and if you we're to measure to see which you had, you'd find just one or just the other.

A state, in quantum mechanics, can have a phase (a complex number that is multiplied in that just has a magnitude of 1). That's important for interference, but in basic calculations wholly unimportant if you just care about the final probabilities. So that's, again - and perhaps slightly clearer now - why the i's and negative values can appear.

For quarks, you can have superpositions in the same way, so you can define a state that's 50% red and 50% blue. Then if you measure it, you'd, just as with the cat, find either a red or blue quark. So states of quarks can have a phase. (Note, colors don't have a phase, just states)

The reason, by the way, you see these strange states for gluons (and not generally when people talk about quarks, say), is that gluons are a bit odd. While you'd naively expect 9 gluons, there are but 8. So if you named them in the simple way (red-antired, red-antiblue, red-antigreen... and so on), you'd have one extra. You could just leave one out... but that's pretty anti symmetrical, and not very elegant. So instead we get those combinations you listed. The states may look a bit confusing, sure, but there are 8 of them, and much more symmetry.