Infinite dimensional representation of su(2)

matematikawan

I'm trying to understand this paper which the author claimed that he had constructed an infinite dimensional representation of the su(2) algebra. The hermitian generators are given by

$$J_x=\frac{i}{2}(\sqrt{N+1}a-a^\dagger\sqrt{N+1} )$$
$$J_y=-\frac{i}{2}(\sqrt{N+1}a+a^\dagger\sqrt{N+1} )$$
$$J_z=N+\frac{1}{2}$$

where the creation and the annihilation operators $$a^\dagger$$ and a satisfy the commutator relation $$[a,a^\dagger]=-1$$ and $$N\equiv-a^\dagger a$$.

My question is, what is the meaning of the expression $$\sqrt{N+1}$$? So that I can proceed with the manipulation.
Does it means the binomial expression
$$\sqrt{N+1}= 1 + \frac{1}{2}N + ...$$?

I have been thinking along that line. How do we show that the representation is infinite dimensional?

ps How do you create the superscript operator dagger in tex?

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Ben Niehoff

Gold Member
The dagger symbol is just \dagger. I don't know the answer to any of your other questions. :D

matematikawan

Thanks I have done the necessary changes. They don't have the template for dagger.

Hi,

I think you are a little confused. The N in J_z represents the Number Operator which is defined as a^dagger*a
Where as the N in the square root is just a number and is equal to the eigen value of the Number operator.

Conventionally, that's what I would think of, when I see the equations you have written
But of course, I haven't seen the paper you are referring to and I might be wrong. matematikawan

If N is just a number why the author bother to write the expression to the right of $$a^\dagger$$ .

Yes I agree usually the number operator $$N=a^\dagger a$$. But that is in the case of finite dimensional representation.

For your information, the papers that I trying to understand are:

1. Andre van Tonder, Ghosts as Negative Spinors, Nuc. Phys. B 645(2002) pp 371-386.
2. Andre van Tonder, On the representation theory of negative spin, Nuc. Phys. B 645(2002) pp 387-402.

vkroom

What is attempted in your equations is known in physics as Schwinger's Boson representation of angular momentum algebra ( SU(2) in your case. ) The$$J_i$$ are the generators of this algebra.
Therefore, one would conclude that the 'N' in the first two eqns are actually numbers:eigenvalue of the operator $$\hat{N} = a ^\dagger a$$, whose spectrum is infact the set of all non-negative integers, denoted by $$N$$.
In the third eqn its actually the operator $$\hat{N}$$. As u can see it doesn't matter if it is $$N$$ or, $$\hat{N}$$. Both yields the same behavior. If it were a simple number, that eqn denotes (N+1/2)Id, where Id-> identity operator.

matematikawan

OK I make a mistake. But it is in the expression Jy. There shouldn't be the imaginary number i. Other than that I think I have copied correctly.

$$J_y=-\frac{1}{2}(\sqrt{N+1}a+a^\dagger\sqrt{N+1} )$$

If it true that $$\sqrt{N+1}$$ is just a number then

$$[J_x , J_y]=-\frac{i(N+1)}{4}[a , a^\dagger] = \frac{i(N+1)}{4}.$$

which is not equivalent to the su(2) algebra $$[J_x , J_y]=iJ_z$$.

However if I assume $$\sqrt{N+1}$$ to be an operator I'm able to verify the relation $$[J_x , J_y]=iJ_z$$.

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