Open bosonic string mass spectrum (${\alpha }^{\prime }{M}^2=N-a$)

The string level operator for a free-ended open string is $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. The quantum condition is ${\alpha }^{\prime }{M}^2=N-a$, where $a$ is the normal-ordering constant of a string. Lorentz-consistent bosonic string theory has $a=1$.

The first levels are the oscillator vacuum, ${\alpha }_{-1}^I|0⟩$, and ${\alpha }_{-2}^I|0⟩,{\alpha }_{-1}^I{\alpha }_{-1}^J|0⟩$. At level two the transverse vector representation and symmetric square assemble into the massive symmetric traceless square of one extra rotation dimension.