Creation Operators application

[lonewolf219]

Can creation operators be used to find a matrix representation in a larger dimension? Is that maybe how I could find the 3D representation for SU(2) ?

 

[tom.stoer]

Suppose you have generators, i.e. su(n) matrices, T^a_ik in the fundamental representation with the commutation relations

$[T^a,T^b] = if^{abc}T^c$
$a = 1 \ldots n^2 - 1$

Then one can define the operators

$Q^a = \sum_{ik}a^\dagger_i\,T^a_{ik}\,a_k$

using creation and annihilation operators of the n-dim. harmonic oscillator.

These operators form a representation acting on the Hilbert space; the commutation relations are

$[Q^a,Q^b] = if^{abc}Q^c$

Therefore all algebraic relations for the T's can be translated for the Q's; the eigenstates can be classified according to the representations of the su(n) algebra.

A nice side-effect is that the n-dim. harmonic oscillator has a much larger symmetry group, namely not SO(n) but SU(n). One can show that all Q's a conserved, i.e. that

$[Q^a,H] = 0$

holds

 

[lonewolf219]

Thanks, Tom... I still have a lot to learn! Thanks for mentioning Hilbert Space... When does Minkowski space enter the picture? Does QFT use both Hilbert and Minkowski space, depending on the speed of the particles?

 

[tom.stoer]

Hilbert space and Minkowskli space have nothing to do with each other.

Hilbert space is the ∞-dim space of quantum states, e.g. harmonic oscillator states |n> or wave functions ψ.
Minkowski space is the 4-dim. space of spacetime points x.

 

[lonewolf219]

Thanks... by the way, I appreciate your previous post... it was just a little too advanced, and of course, that is my problem to fix!

 

[tom.stoer]

I appreciate your previous post... it was just a little too advanced ...

do you mean #2 or #4? what's the problem? don't hesitate to ask

 

[lonewolf219]

Thanks for the nudge! #2! I have read that the azimuthal quantum number is equal to 1 and therefore the magnetic quantum number (does the magnetic number always define the dimension of the space?) is equal to -1,0,1 for the 3D rep of Pauli matrices, but I don't know exactly how the value of l was found... and I also don't know why we would construct the z component first, then use the ladder operators to find the other two matrices?? I have not taken QM (probably should've mentioned that!)

 

[tom.stoer]

Now it seems that you are asking for the 3-dem. rep. of the su(2) matrices; the 2-dim. rep. are just the Pauli matrices (times 1/2) which give you s(s+1) = 3/4 and s_z = 1/2. For all other n-dim. reps. procedure is always the same: find the matrices satisfying the commutation relations for su(2), calculate the eigenvalues for s_z and calculate the the total spin s.

All this has nothing to do with the harmonic oscillator but is valid on one the algebraic level, i.e. for matrices.

In #2 I did two things: I generalized from su(2) to su(n), and I constructed the operator Q from the matrices T. This works only in the n-dim. rep. of su(n).

Please note that the n-dim. rep. of su(2) and su(n) are two different things. You where asking in #1 for the 3D rep. of su(2) using the 3-dim. harmonic oscillator. My answer was that you don't get the 3-dim. rep. of su(2) but the 3-dim. rep. of su(3) which is not the same.

 

[lonewolf219]

Thanks for pointing out there is a difference between the two representations...

 

[tom.stoer]

There is not only a difference of the reps, we are talking about different groups

 

[lonewolf219]

Yes, I see... Do the 3 Pauli matrices have any relation to the three force carriers W+ W- and Z0?

 

[tom.stoer]

Yes.

In an SU(n) gauge theory you have n²-1 gauge bosons; and there are n²-1 matrices T^a with a=1,2,...,n²-1. For SU(2) you have 2²-1 Pauli matrices, for SU(3) you have 3²-1 Gell-Mann matrices.

The gauge fields are described by n²-1 vector fields A^a_μ where again a=1,2,...,n²-1 is an su(n) index, and where μ=0,1,2,3 is a spacetime index. One can introduce matrix-values fields

$A_\mu = \sum_a A_\mu^a\,T^a$

to describe the gauge fields which are then "rotated" by SU(n) matrices constructed from angles θ^a as

$U(\theta) = \exp[\sum_a \theta^a\,T^a]$

So this how to describe the 3 gauge fields for W and Z in weak interactions, and the 8 gluon fields n QCD.

 

[lonewolf219]

Great! I was wondering where spacetime fit in... and I appreciate the clear distinction that there are the same number of generators as gauge bosons...