String oscillator

The Fourier modes ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ of the transverse closed-string coordinates obey harmonic-oscillator commutation relations. Modes with negative index create string excitations.

An oscillator vacuum is annihilated by all positive-frequency oscillator annihilation operators. For a Ramond sector string this leaves a degenerate space on which the fermionic zero modes act through the Ramond zero-mode Clifford algebra. Vacuum annihilation therefore does not select a unique spinor state.

For a Neumann-Dirichlet open-string boundary condition, the transverse modes have frequencies $r\in \mathbb{Z}+1/2$. The canonical commutation relations give $[{\alpha }_r^i,{\alpha }_s^j]=r{\delta }^{ij}{\delta }_{r+s,0}$ and ${\alpha }_r^{†}={\alpha }_{-r}$. One derivation expands $X-y={\sum }_{r>0}{q}_r\cos (r\sigma )$: spatial orthogonality reduces the Polyakov action to $(8{\alpha }^{\prime }{)}^{-1}\int {\sum }_r({\dot{q}}_r^2-r^2{q}_r^2)d\tau$. These are harmonic oscillators of mass $1/(4{\alpha }^{\prime })$, whose normalized creation operators satisfy ${\alpha }_{-r}=i\sqrt{r}{a}_r^{†}$ in the conventional phased expansion.

The string level operator is the weighted oscillator number $N={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$. A closed string has independent operators $N$ and $\tilde{N}$ for its two chiral sectors.

The canonical commutation relations imply $[N,{\alpha }_{-k}^i]=k{\alpha }_{-k}^i$ for $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. With zero level assigned to the oscillator vacuum, a Fock space basis state containing $r_{ki}$ creation operators of each mode has level ${\sum }_{k,i}k{r}_{ki}$. The level is therefore a nonnegative integer; it counts oscillator excitation weighted by frequency, not simply the number of creation operators.

Closed-string periodicity and the residual spatial constraint require equal left- and right-moving excitation levels, $N=\tilde{N}$.

The normal-ordering constant is the zero-point shift in the quantum Virasoro generator. Closure of the target-space Lorentz algebra fixes $a=1$ and the spacetime dimension $D=26$ for the bosonic string.

In the usual critical Lorentz-invariant bosonic string, level-one transverse vector polarizations fit a massless representation, whereas a massive vector would require an extra polarization. This fixes the intercept to one. Polarization counting alone does not establish that conclusion in every dimension: in three dimensions the one transverse polarization transforms trivially under the transverse rotation group and can be interpreted as a scalar. The quantum consistency assumptions must accompany an argument that the first level must be massless.

Each of $d$ physical transverse bosons with a Neumann-Dirichlet open-string boundary condition contributes $\frac{1}{2}{\sum }_{r>0}r=1/48$ by the half-integer zeta-regularized mode sum. Thus $L_0^{\perp }=N+d/48$ and the convention $L_0={\alpha }^{\prime }{p}^2+N-a$ has $a=-d/48$. For $d=24$, the vacuum shift is $+1/2$. The ordinary integer-moded open-string shift is instead $-d/24$; these endpoint sectors have different normal-ordering constants.

With 24 physical half-integer open-string oscillators, the first three string level operator eigenvalues are $0,1/2,1$. Their states are the vacuum, $a_{1/2}^{i†}|0⟩$, and $a_{1/2}^{i†}{a}_{1/2}^{j†}|0⟩$. The creation operators commute, so the third level is the symmetric square of the 24-dimensional transverse vector, of dimension $24\cdot 25/2=300$. Its trace is a scalar and its trace-free part has dimension 299. Including the vacuum shift gives ${\alpha }^{\prime }{M}^2=1/2,1,3/2$ when the two longitudinal directions are common directions with Neumann boundary conditions. The representation labels refer to the surviving ND rotation group, not full unbroken target Lorentz symmetry.

For a closed bosonic string,

A matter vertex with left and right oscillator levels $N,\tilde{N}$ and momentum $k$ has weights $N+{\alpha }^{\prime }{k}^2/4,\tilde{N}+{\alpha }^{\prime }{k}^2/4$. An integrated string vertex operator must have weights $(1,1)$, giving the displayed mass shell and level matching with $M^2=-k^2$. Physical states are further identified modulo null string states, equivalently through BRST cohomology; merely counting all weight-one primaries would include gauge redundancies.

Each level-two chiral sector of the critical closed string is the symmetric traceless square $S_0^{2}V$ of the massive little group vector space. The tensor product of its two sectors contains the exterior square ${\Lambda }^{2}V$: the equivariant map $(S,T)\mapsto ST-TS$ is onto the antisymmetric matrices. Compact-group complete reducibility realizes this quotient as a subrepresentation. It describes a massive two-form field with $M^2=8\pi T=4/{\alpha }^{\prime }$.

In the critical bosonic string theory, the states ${\alpha }_{-1}^i{\tilde{\alpha }}_{-1}^j|0;p⟩$ have $M^2=0$. Their coefficient matrix decomposes into a symmetric trace-free tensor, an antisymmetric tensor, and a scalar trace, representing the graviton, Kalb–Ramond field, and dilaton. Their transverse particle polarization counts add to $(D-2{)}^2$. The normal-ordering constant of a string is one.

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.