= Solution
The <Nambu–Goto action> is invariant under <worldsheet diffeomorphisms>. Its canonical <Nambu–Goto phase-space constraints> are
$$
\Pi\cdot X'=0,\qquad \Pi^2+T^2X'^2=0.
$$
In <conformal gauge>, $\Pi=T\dot X$, so these become $(\dot X\pm X')^2=0$. Their smeared generators implement the residual <conformal transformations> rather than independent propagating degrees of freedom. Classically their Fourier modes form the <classical Virasoro constraint algebra>. Quantization promotes them to the <Virasoro algebra>, and its constraints select the <physical string states> from the indefinite covariant <Fock space>.
For clarity, start with a <Neumann boundary condition> at both ends of an <open string>, $0\leq\sigma\leq\pi$. Its <open-string mode expansion> is
$$
X^\mu=x^\mu+2\alpha'p^\mu\tau+i\sqrt{2\alpha'}\sum_{n\ne0}\frac{\alpha_n^\mu}{n}e^{-in\tau}\cos n\sigma,
\qquad \alpha_0^\mu=\sqrt{2\alpha'}\,p^\mu.
$$
Canonical quantization gives the <string oscillator> relations
$$
[\alpha_m^\mu,\alpha_n^\nu]=m\delta_{m+n,0}\eta^{\mu\nu},\qquad
(\alpha_n^\mu)^\dagger=\alpha_{-n}^\mu.
$$
With $\partial_\pm=(\partial_\tau\pm\partial_\sigma)/2$, the <worldsheet stress-energy tensor> obeys
$$
\frac1{\alpha'}\partial_\pm X\cdot\partial_\pm X
=\sum_{m\in\mathbb Z}L_m e^{-im(\tau\pm\sigma)}.
$$
Substituting the <open-string mode expansion> into this quadratic expression derives the <Virasoro generators>:
$$
\boxed{L_m=\frac12\sum_{r\in\mathbb Z}:\alpha_{m-r}\cdot\alpha_r:},
\qquad
L_0=\alpha'p^2+N,\qquad N=\sum_{r>0}\alpha_{-r}\cdot\alpha_r.
$$
The colons mean <normal ordering>, placing the negative-mode creation operators to the left. The <string level operator> $N$ increases by $r$ when an oscillator $\alpha_{-r}$ is added.
Using the <commutator> once gives
$$
[L_m,\alpha_n^\mu]=-n\alpha_{m+n}^\mu.
$$
Applying this to the quadratic expression for $L_n$ produces $(m-n)L_{m+n}$, together with the double-contraction <Virasoro central extension>. To determine its coefficient, take $m>0$ and evaluate $[L_m,L_{-m}]$ on a zero-momentum <oscillator vacuum>. Only the creation pairs with $1\leq r\leq m-1$ contribute, giving
$$
\frac D2\sum_{r=1}^{m-1}r(m-r)=\frac D{12}(m^3-m).
$$
Here $D$ is the target <spacetime dimension>; the timelike free boson also contributes one to the <central charge>, since a double contraction contains two metric factors. Thus
$$
\boxed{[L_m,L_n]=(m-n)L_{m+n}+\frac D{12}(m^3-m)\delta_{m+n,0}},
\qquad L_m^\dagger=L_{-m}.
$$
This derives both the <Virasoro generators> and their quantum anomaly.
In <old covariant string quantization>, the <physical-state Virasoro conditions for an open string> are
$$
L_{n>0}|\psi\rangle=0,\qquad (L_0-a)|\psi\rangle=0,
$$
where $a$ is the <normal-ordering constant of a string>. Requiring all positive and negative modes to annihilate a state would contradict the nonzero <Virasoro central extension>; the positive-mode prescription is the appropriate covariant subsidiary condition. The zero-mode equation gives the <open bosonic string mass spectrum>,
$$
\alpha'M^2=N-a.
$$
For a <closed string>, there are two commuting <Virasoro algebras>. With uncompactified zero modes, $L_0=\alpha'p^2/4+N_R$ and $\widetilde L_0=\alpha'p^2/4+N_L$. Imposing both positive-mode conditions and both zero-mode conditions gives the <closed-string level matching> $N_L=N_R$ and $M^2=4(N_L-a)/\alpha'$.
Covariance initially retains the timelike <string oscillators>. For example, $\alpha_{-1}^0|0;p\rangle$ has negative norm because $\eta^{00}=-1$. Its existence in the unconstrained space is not an inconsistency if it is absent from the physical quotient. The <no-ghost theorem for the critical bosonic string> states that, at $D=26$ and $a=1$, the <inner product> on <physical string states> at ordinary nonzero momentum is positive semidefinite. After quotienting physical <null string states>, this physical space is represented by the 24 positive-norm transverse <string oscillators>.
A <spurious string state> has the form $\sum_{n>0}L_{-n}|\chi_n\rangle$. Its <inner product> with every <physical string state> vanishes, by $L_n|\psi\rangle=0$ and $L_n^\dagger=L_{-n}$. Whenever such a state is itself physical, it is a <null string state> and represents a gauge redundancy, not an extra polarization. Consequently \b[the physical quotient has no negative-norm propagating states], and agrees with <light-cone gauge in string theory>. This is the significance of the theorem; a proof is not needed here. It does not eliminate the bosonic <tachyon>, whose negative mass squared is distinct from negative norm, nor does it remove the auxiliary <worldsheet ghost fields> used in <gauge fixing>.
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