String Theory-Virasoro Generators -- show commutator relation

[binbagsss]

Homework Statement

(I have dropped the hats on the $\alpha_{n}^{u}$ operators and $L_{m}$)

$[\alpha_{n}^u, \alpha_m^v]=n\delta_{n+m}\eta^{uv}$
$L_m=\frac{1}{2}\sum\limits_{n=-\infty}^{\infty} : \alpha_{m-n}^u\alpha_{n}^v: \eta_{uv}-\delta_{m,0}$

where : denotes normal-ordered.

Show that : $[\alpha_{m}^u,L_n]=m\alpha_{m+n}^u$

Homework Equations

see above

The Attempt at a Solution

For a given $n$ we are looking at the following commutator: $[\alpha_m,\alpha_{n-m}\alpha_m]$

to use commutator relation:

$[a,bc]=-a[b,c]-[a,c]b$

$a= \alpha_m$
$b= \alpha_{n-m}$
$c= \alpha_m$

$[a,c]=0$
$[b,c]=(n-m)\delta_{n=0}\eta^{uv}$ using (1)

$\implies [\alpha_{m}^u,L_n]=\alpha_m^u(m)\eta^{uv}$ which is wrong...

thanks in advance

 

[binbagsss]

bump please. many thanks in advance, very grateful.

 

[king vitamin]

The commutator relation you've written down is wrong.

 

[binbagsss]

The commutator relation you've written down is wrong.

Sorry to re-bump an old thread but I thought it would be a waste to start a new one.
I think I understood this at the time but right now I am not getting it.
Why is the commutator relation wrong?

Let me instead use:

$[a,bc]=[a,b]c+b[a,c]$

$a=\alpha_m$
$b=\alpha_{n-m}$
$c=\alpha_m$

$[\alpha_m, \alpha_{n-m}]\alpha_m + \alpha_{m-n}[\alpha_m,\alpha_m] = m\delta_n \eta^{uv}\alpha_m + m\delta_{2m}\eta^{uv}$

which is wrong..

 

[king vitamin]

You shouldn't use the same index $m$ for both the $\alpha_m$ and the summed index in the definition of $L_n$.

So using the (now correct) relation

$$[a,bc] = [a,b]c + b[a,c]$$

with

$$a = \alpha_m^u \\ b = \alpha_{n-k}^{\nu} \\ c = \alpha_k^{\alpha}$$

you should get the correct answer after contracting with $\eta_{\nu \alpha}$.