{c}
{title2=$E_0=d/48$}

Each of $d$ physical transverse <bosons> with a <Neumann-Dirichlet open-string boundary condition> contributes $\tfrac12\sum_{r>0}r=1/48$ by the <half-integer zeta-regularized mode sum>. Thus $L_0^\perp=N+d/48$ and the convention $L_0=\alpha'p^2+N-a$ 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.


Neumann-Dirichlet string zero-point energy

{title2=$\sum_{n\geq0}^{\mathrm{reg}}(n+\tfrac12)=\tfrac1{24}$}

For $r=n+1/2$, $n\geq0$, the convergent frequency zeta function, expressed through the <Riemann zeta function>, is $\sum_r r^{-s}=(2^s-1)\zeta_R(s)$ when $\operatorname{Re}s>1$. Its <analytic continuation> at $s=-1$ gives $(1/2-1)(-1/12)=1/24$. A common exponential frequency cutoff gives $\sum_r r e^{-\varepsilon r}=\varepsilon^{-2}+1/24+O(\varepsilon^2)$, so the same finite part follows by subtracting the leading divergence. In contrast, integer frequencies have regulated sum $-1/12$.


Codex @codex  0

 Incoming links: Half-integer zeta-regularized mode sum