- 338

- 0

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

[tex]J_x=\frac{i}{2}(\sqrt{N+1}a-a^\dagger\sqrt{N+1} )[/tex]

[tex]J_y=-\frac{i}{2}(\sqrt{N+1}a+a^\dagger\sqrt{N+1} )[/tex]

[tex]J_z=N+\frac{1}{2}[/tex]

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

My question is, what is the meaning of the expression [tex]\sqrt{N+1} [/tex]? So that I can proceed with the manipulation.

Does it means the binomial expression

[tex]\sqrt{N+1}= 1 + \frac{1}{2}N + ... [/tex]?

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?

[tex]J_x=\frac{i}{2}(\sqrt{N+1}a-a^\dagger\sqrt{N+1} )[/tex]

[tex]J_y=-\frac{i}{2}(\sqrt{N+1}a+a^\dagger\sqrt{N+1} )[/tex]

[tex]J_z=N+\frac{1}{2}[/tex]

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

My question is, what is the meaning of the expression [tex]\sqrt{N+1} [/tex]? So that I can proceed with the manipulation.

Does it means the binomial expression

[tex]\sqrt{N+1}= 1 + \frac{1}{2}N + ... [/tex]?

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?

Last edited: