= Solution
In the displayed version of <light-cone gauge in string theory>, the oscillators have $D-2$ transverse components and
$$
N=\sum_{k=1}^{\infty}\boldsymbol\alpha_{-k}\cdot\boldsymbol\alpha_k.
$$
The center-of-mass <Poisson brackets> remain $\{x^m,p_n\}=\delta^m_n$, and the transverse oscillator <Poisson brackets> are $\{\alpha_j^a,\alpha_k^b\}=-ij\delta^{ab}\delta_{j+k,0}$. Other independent brackets vanish. With $\hbar=1$, <canonical quantization> gives the <canonical commutation relations>
$$
[x^m,p_n]=i\delta^m_n,\qquad
[\alpha_j^a,\alpha_k^b]=j\delta^{ab}\delta_{j+k,0},\qquad
(\alpha_k^a)^\dagger=\alpha_{-k}^a.
$$
The oscillator <Fock vacuum> at <momentum> $p$ is defined by $\alpha_k^a|0;p\rangle=0$ for $k>0$. It is the ground state of one string, rather than the empty spacetime vacuum. Set $a_k^a=\alpha_k^a/\sqrt{k}$ for $k>0$. The <normal ordering> prescription gives the <string level operator>
$$
\widehat N=\sum_{k>0,a}\alpha_{-k}^a\alpha_k^a
=\sum_{k>0,a}k\,(a_k^a)^\dagger a_k^a,\qquad \widehat N|0;p\rangle=0.
$$
A <Fock state> basis is obtained by applying $\prod_{k,a}((a_k^a)^\dagger)^{n_{ka}}/\sqrt{n_{ka}!}$ to $|0;p\rangle$, with <string level operator> eigenvalue $N=\sum_{k,a}k n_{ka}$. In particular, its level-one states are
$$
\boxed{\alpha_{-1}^a|0;p\rangle,\qquad a=1,\ldots,D-2.}
$$
They transform as the transverse vector of the <little group> rotation subgroup $SO(D-2)$, precisely the <vector-particle polarizations> of a massless vector. A massive vector would instead require $D-1$ <vector-particle polarizations>. This conclusion uses a quantization compatible with the <Lorentz group> of the <bosonic string theory>.
The classical mass constraint alone has no quantum <zero-point energy> shift. Its quantum version includes the <normal-ordering constant of a string> $a$:
$$
(p^2+2\pi T(\widehat N-a))|\mathrm{phys}\rangle=0,\qquad
M^2=2\pi T(N-a).
$$
Masslessness at level one fixes $a=1$, so
$$
\boxed{M_1^2=0,\qquad M_0^2=-2\pi T.}
$$
Thus \b[the ground state is a <tachyon>]. Without the quantum ordering shift, the displayed classical constraint would give $M_1^2=2\pi T$ and would not support the stated massless interpretation. In the usual transverse vacuum regularization, $a=(D-2)/24$; consistency with $a=1$ also gives the <critical dimension of string theory> $D=26$.
Back to article page
