= Physical-state Virasoro conditions for an open string
{title2=$L_{m>0}|\Phi\rangle=0,\ (L_0-a)|\Phi\rangle=0$}
The covariant physical-state conditions impose positive-mode <Virasoro constraints> on kets and the adjoint conditions on bras, with intercept $a$ in the zero mode. They are a string version of <Gupta-Bleuler quantization>. Imposing every positive and negative matter mode on the same state would conflict with the <Virasoro central extension>. In critical <bosonic string theory>, $a=1$ and the matter <central charge> is $26$, canceled by the reparameterization ghosts.
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