= Solution
\b[Particle gauge fixing and BRST.] Put $C=(p^2+\mu^2)/2$. The classical <constraint> generates $\delta_\varepsilon x^m=\varepsilon p^m$, $\delta_\varepsilon p_m=0$, $\delta_\varepsilon e=\dot\varepsilon$. Consequently the variation of $e-\bar e$ along a gauge orbit is the operator $\partial_t$ on $\varepsilon$. A <path integral> restricted to that gauge must include its <Faddeev-Popov determinant>; anticommuting <FP ghosts> represent the determinant, rather than its inverse. Constant gauge zero modes and any <proper-time modulus> must be treated separately, so the relevant determinant is $\det{}'\partial_t$.
A convenient <point-particle Faddeev–Popov ghost action> at $\bar e=1$ is
$$
\boxed{S=\int dt\left[\dot x\cdot p-C+i b\dot c\right].}
$$
The <FP ghost> normalization has been chosen to give the <graded Poisson bracket> $\{b,c\}=-i$, with the bracket symmetric on two odd variables. The <particle BRST charge and Klein–Gordon constraint> are
$$
Q=cC,\qquad\delta F=\{F,\Lambda Q\},
$$
where the odd constant parameter $\Lambda$ is placed on the left. The product $\Lambda Q$ is even. Accounting for the odd parameter in this convention gives
$$
\boxed{\delta x^m=\Lambda c p^m,\quad\delta p_m=0,\quad
\delta c=0,\quad\delta b=i\Lambda C.}
$$
These are canonical <BRST transformations>. Direct variation checks the sign: the bosonic kinetic term changes by $\Lambda(\dot c\,p^2+c\,p\cdot\dot p)$, while the <FP ghost> term changes by $-\Lambda C\dot c$. Hence
$$
\delta\mathcal L=\frac d{dt}\left[\Lambda c(p^2-C)\right],
$$
and the action is invariant for the corresponding boundary conditions. The <BRST charge> is conserved because $C$ commutes with the gauge-fixed <Hamiltonian> and $c$ is constant on the <FP ghost> equation of motion.
Quantization gives $\{\widehat b,\widehat c\}_+=1$ and $\widehat c^2=0$. Since $C$ commutes with the <FP ghosts>,
$$
\boxed{\widehat Q^2=\widehat c^{\,2}\widehat C^{\,2}=0.}
$$
It generates the same variations by $\delta\widehat F=-i[\widehat F,\Lambda\widehat Q]$, with an ordinary commutator against the even generator. Represent $\widehat c$ by multiplication and $\widehat b$ by differentiation. On a ghost-number-zero wavefunction $\Psi(x)$, $\widehat Q\Psi=0$ is exactly $\widehat C\Psi=0$, namely
$$
\boxed{(-\Box+\mu^2)\Psi=0,}
$$
the <Klein-Gordon equation> for the mostly-plus metric. The specified <FP ghost> sector matters: for an unrestricted wavefunction $\Psi_0+c\Psi_1$, BRST closure only constrains $\Psi_0$, not every component independently. The complete physical prescription uses <BRST cohomology>, not an assertion that every vector in the entire ghost-extended kernel is a new Klein–Gordon particle.
\b[String oscillator algebra and <FP ghost> Virasoro generators.] Use the same bosonic brackets as above and the odd brackets
$$
\{\alpha_k^m,\alpha_l^n\}_{\mathrm{PB}}=-ik\eta^{mn}\delta_{k+l,0},
\qquad\{b_k,c_l\}_{\mathrm{PB}}=-i\delta_{k+l,0}.
$$
Quantization turns them into
$$
\boxed{[\alpha_k^m,\alpha_l^n]=k\eta^{mn}\delta_{k+l,0},\qquad
\{b_k,c_l\}_+=\delta_{k+l,0},\qquad
\{b_k,b_l\}_+=\{c_k,c_l\}_+=0.}
$$
The $b,c$ here are <worldsheet ghost fields>, not the matter <fermion> oscillators used in the NS question.
Contract $b_m$ with the two $c$ factors in the cubic <FP ghost> term of the given BRST operator. The first contraction contributes $-\tfrac12\sum_q(m-q)c_{-q}b_{m+q}$ and the second contributes $+\tfrac12\sum_p(p-m)c_{-p}b_{p+m}$. Their sum is $\sum_q(q-m)c_{-q}b_{m+q}$. Reordering and <normal ordering> therefore give the <ghost oscillator Virasoro generators>
$$
\boxed{L_m=L_m^{(\alpha)}+L_m^{(\mathrm{gh})},\qquad
L_m^{(\alpha)}=\frac12\sum_k:\alpha_k\cdot\alpha_{m-k}:,
\quad L_m^{(\mathrm{gh})}=\sum_k(m-k):b_{m+k}c_{-k}: .}
$$
For $m\ne0$ these formulas have no additive ambiguity. At $m=0$, moving annihilators past creators produces infinite zero-point sums. Their regularized finite constant shifts $L_0$ and corresponds to a term proportional to $c_0$ in $Q$. The convention fixed below has a <ghost oscillator vacuum> weight $-1$, or equivalently the usual intercept-one shift. Whether that shift is written outside the normally ordered sum or incorporated in its definition is a convention; it must not be omitted twice or counted twice. In the chosen oscillator convention this is explicitly
$$
L_0=\frac12\alpha_0^2+\sum_{k>0}\alpha_{-k}\cdot\alpha_k+\sum_{k>0}k(b_{-k}c_k+c_{-k}b_k)-1.
$$
Equivalently, the normally ordered <BRST charge> contains the intercept term $-c_0$; the formal un-ordered operator in the question includes that choice only after its ordering prescription is fixed.
The oscillator brackets directly give
$$
[L_m,b_n]=(m-n)b_{m+n},\qquad
[L_m,c_n]=-(2m+n)c_{m+n}.
$$
If $Q^2=0$, then $[L_n,Q]=[\{b_n,Q\},Q]=[b_n,Q^2]=0$. The graded Jacobi identity now yields <Virasoro closure from BRST nilpotence>:
$$
\begin{aligned}
[L_m,L_n]&=[\{b_m,Q\},L_n]\\
&=\{[b_m,L_n],Q\}+\{b_m,[Q,L_n]\}\\
&=(m-n)\{b_{m+n},Q\}=(m-n)L_{m+n}.
\end{aligned}
$$
In particular a central extension cannot remain in the total generators if the quantum <BRST charge> is nilpotent.
\b[Low-level descendant calculation.] Let $|\Omega;p\rangle$ be the oscillator ground state, annihilated by $\alpha_{n>0}$, $b_{n\ge0}$ and $c_{n>0}$. All $L_{n>0}$ annihilate it. The nonzero-mode <FP ghost> generators act as
$$
L_{-1}^{(\mathrm{gh})}|\Omega\rangle=-b_{-1}c_0|\Omega\rangle,
\qquad
L_{-2}^{(\mathrm{gh})}|\Omega\rangle
=-2b_{-2}c_0|\Omega\rangle-3b_{-1}c_{-1}|\Omega\rangle.
$$
For example $L_1(-b_{-1}c_0|\Omega\rangle)=-2b_0c_0|\Omega\rangle=-2|\Omega\rangle$. Similarly $L_2$ on the two level-two terms gives $-8|\Omega\rangle$ and $-9|\Omega\rangle$, hence $-17|\Omega\rangle$. These signs are essential: <FP ghosts> have an indefinite pairing.
The matter generators create
$$
L_{-1}^{(\alpha)}|\Omega\rangle=\alpha_0\cdot\alpha_{-1}|\Omega\rangle,
\qquad
L_{-2}^{(\alpha)}|\Omega\rangle=
\left(\alpha_0\cdot\alpha_{-2}+\frac12\alpha_{-1}\cdot\alpha_{-1}\right)|\Omega\rangle.
$$
Contracting the level-one term with $L_1$ gives $\alpha_0^2$. At level two the first term gives $2\alpha_0^2$, and the double contraction of the second gives $D/2$; cross terms vanish because oscillator levels differ. Thus the normalized oscillator/Virasoro descendant coefficients, conventionally denoted by the requested norms, are
$$
\boxed{\|L_{-1}|\Omega\rangle\|^2_{\mathrm{osc}}=\alpha_0^2-2,\qquad
\|L_{-2}|\Omega\rangle\|^2_{\mathrm{osc}}=2\alpha_0^2+\frac D2-17.}
$$
Comparing the first with $[L_1,L_{-1}]=2L_0$ gives the oscillator highest weight
$$
\boxed{h_0=\frac12\alpha_0^2-1.}
$$
In this normalization the question's $\langle0|L_0|0\rangle$ is $h_0$. Comparing level two with $[L_2,L_{-2}]=4L_0$ gives
$$
2\alpha_0^2+\frac D2-17=4\left(\frac12\alpha_0^2-1\right),
\qquad\boxed{D=26.}
$$
This is the <bosonic-string critical dimension from ghost descendants>. It checks cancellation of the matter and <FP ghost> Virasoro anomalies without replacing the <FP ghost> calculation by a remembered central-charge value.
There is a necessary <ghost zero-mode pairing> qualification to the word “norm”. If $b_0,c_0$ are Hermitian in the full <FP ghost> space and $b_0|\Omega\rangle=0$, then
$$
\langle\Omega|\Omega\rangle
=\langle\Omega|\{b_0,c_0\}|\Omega\rangle=0.
$$
A bare full-ghost inner product cannot at the same time normalize this ket to one. <FP ghost> zero modes need saturation in actual amplitudes. The above equations are the coefficients of $L_1L_{-1}|\Omega\rangle$ and $L_2L_{-2}|\Omega\rangle$, or the corresponding normalized contravariant Virasoro form. They are not positive-definite Hilbert norms. The algebraic coefficient calculation is sufficient for the requested critical-dimension argument and remains valid without that false normalization.
Finally, impose the stated ground-state conditions $Q|\Omega;p\rangle=b_0|\Omega;p\rangle=0$. They imply $L_0|\Omega;p\rangle=\{b_0,Q\}|\Omega;p\rangle=0$, so $h_0=0$ and
$$
\boxed{\alpha_0^2=2,\qquad p^2=\frac1{\alpha'},\qquad
M_{\mathrm{ground}}^2=-\frac1{\alpha'}.}
$$
This is the bosonic <tachyon>. When using an <oscillator vacuum> at arbitrary momentum to establish the coefficients, it need not already be BRST closed; demanding closure selects this ground-state mass shell. The two uses should not be confused.
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