Open-string mode expansion

In conformal gauge, a free open-string coordinate with Neumann boundary conditions has a cosine expansion. Canonical quantization turns its nonzero Fourier coefficients into string oscillators.

On $0\le \sigma \le \pi$, a coordinate with Neumann boundary conditions at both ends has a constant position, linear time zero mode and cosine oscillators. A coordinate with Dirichlet boundary conditions at both ends has a linear spatial interpolation between endpoint positions and sine oscillators, with no momentum zero mode. Both have integer frequencies. This differs from an ND boundary condition, which applies different conditions at the two ends of the same coordinate and gives half-integer frequencies.

The Fourier kinetic term ${\sum }_{k>0}i{\alpha }_{-k}\cdot {\dot{\alpha }}_k/k$ supplies conjugate pairs, with ${\alpha }_{-k}={\alpha }_k^{\ast }$. Inverting its symplectic form yields the displayed Poisson brackets. Nonzero modes are independent of the center-of-mass pair, while ${\alpha }_0=\sqrt{2{\alpha }^{\prime }}p$ is the momentum zero mode. The imaginary factors are compatible with a real action up to a total derivative.

A transverse open string coordinate has an ND boundary condition when one endpoint obeys a Neumann boundary condition and the other a Dirichlet boundary condition. On a strip of width $\pi$, its nonconstant spatial eigenfunctions are $\cos ((n+1/2)\sigma )$ when the free endpoint is at zero. They solve the endpoint equations because their derivatives vanish at zero and their values vanish at $\pi$. Reversing the endpoints gives sine eigenfunctions with the same frequencies. The fixed endpoint removes the dynamical worldsheet zero mode.

For one free boson, the mixed ND open-string sector has strip vacuum energy $E_0=1/48$. The strip-to-plane transformation relates this energy to the plane Virasoro algebra zero mode by $E_0=h-c/24$, with central charge $c=1$. Hence its ground conformal weight is $h=1/48+1/24=1/16$. For $d$ independent ND coordinates the weights add to $d/16$. At $d=24$, the plane matter ground weight is $3/2$, while the strip oscillator vacuum energy is $1/2$. The plane physical condition $L_0^{\mathrm{plane}}-1=0$ is therefore equivalent to ${\alpha }^{\prime }{p}_{\parallel }^2+N+1/2=0$. The apparent different constants are coordinate/constraint conventions, not different mass spectra.