1. The problem statement, all variables and given/known data(adsbygoogle = window.adsbygoogle || []).push({});

Question (with the following definitions here):

- Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}##

- Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0##

- where ##d## is the dimension of the space ##d=\eta^{uv}\eta_{uv}##

For the L1 operator I am able to get the correct expression of ##1+A-2B=0##

I am struggling with L0

Any help much appreciated.

2. Relevant equations

##\alpha^u_0={p^u}\sqrt{2 \alpha'}##

##\alpha_{n>0}## annihilate

##\alpha_{n<0}## create

## [\alpha_n^u, \alpha_m^v]=n\delta_{n+m}\eta^{uv}## (*)

where ##\eta^{uv}## is the Minkowski metric

##p^u|k>=k^u|k>##

3. The attempt at a solution

Here is my L0 attempt- Consider ##L_0 |x>=0## to show that ##m^{2}=1/\alpha'##

##L_0=(\alpha_0^2+2\sum\limits_{n=1}\alpha_{-n}\alpha_{n}-1)##

So first of all looking at the first term of ##|x>## I need to consider:

##L_0 \alpha_{-1}\alpha_{-1}|k> =(\alpha_0^2+2\alpha_{-1}\alpha_{1}-1)\alpha_{-1}\alpha_{-1}##

Considering the four product operator and using the commutators in the same way as done for ##L_1## I get from this:

##L_0\alpha_{-1}\alpha_{-1}|k> =(\alpha_0^2+4-1)\alpha_{-1}\alpha_{-1}|k>## (**)

Here's how I got it:(dropped indices in places, but just to give idea, ##\eta^{uv}## the minkowksi metric)

##2\alpha_{-1}\alpha_{1}\alpha_{-1}\alpha_{-1} |k>

= 2(\alpha_{-1}(\alpha_{-1}\alpha_1+\eta)\alpha_{-1})|k>

= 2(\alpha_{-1}\alpha_{-1}\alpha_1\alpha_{-1}+\eta\alpha_{-1}\alpha_{-1})|k>

= 2(\alpha_{-1}\alpha_{-1}(\alpha_{-1}\alpha_{1}+\eta)+\eta\alpha_{-1}\alpha_{-1})|k>

=2(\alpha_{-1}\alpha_{-1}(0+\eta|k>)+\eta\alpha_{-1}\alpha_{-1}|k>)

= 2(2\alpha_{-1}.\alpha_{-1})##

so from (**) I have:

##L_0\alpha_{-1}\alpha_{-1}|k> =(\alpha_0^2+3)\alpha_{-1}\alpha_{-1}|k>=0##

##=(2\alpha'p^2+3)\alpha_{-1}\alpha_{-1}|k>=0##

##\implies 2\alpha'p^2+3=0##

## \implies 2(-m^2)\alpha'=-3##

So I get ## m^{2}=3/\alpha'## and not ##1/\alpha'## :(

Any help much appreciated ( I see the mass is independent of ##A## and ##B## so I thought I'd deal with the first term before confusing my self to see why these terms vanish)

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