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Use metric signature $(-,+,+,+)$ and units $c=1$. Write ${\dot{x}}^{\mu }=d{x}^{\mu }/d\lambda$ in this part. Varying the worldline einbein, without first fixing it, givesThe canonical four-momentum is $p_{\mu }=m{g}_{\mu \nu }{\dot{x}}^{\nu }/e$, so the same mass shell constraint is the mass shell $g^{\mu \nu }{p}_{\mu }{p}_{\nu }=-m^2$.

The Euler-Lagrange equation for the trajectory isMultiplying by $e$ and then by the inverse metric, and collecting the metric tensor derivatives into the Christoffel symbols, givesThis is a geodesic equation in a possibly nonaffine parameter.

Under a change of parameter, $e^{\prime }({\lambda }^{\prime })=e(\lambda )d\lambda /d{\lambda }^{\prime }$, so $ed\lambda$ is invariant. On the positive branch, the mass shell constraint identifies this invariant with the proper time increment $ds=\sqrt{-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}=ed\lambda$. Choosing ${\lambda }^{\prime }=s$ therefore sets $e^{\prime }=1$. The trajectory equation is then the affinely parametrized geodesic equation and its tangent has norm minus one. This is proper-time gauge for a massive worldline einbein. The mass shell constraint must still be imposed: fixing $e$ in the action before varying would lose it. A fixed parameter interval can retain a proper-time modulus; here changing to the actual proper-time interval is permitted.

Particle gauge fixing and BRST. Put $C=(p^2+{\mu }^2)/2$. The classical constraint generates ${\delta }_{\epsilon }{x}^m=\epsilon p^m$, ${\delta }_{\epsilon }{p}_m=0$, ${\delta }_{\epsilon }e=\dot{\epsilon }$. Consequently the variation of $e-\bar{e}$ along a gauge orbit is the operator ${\partial }_t$ on $\epsilon$. 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 $\det {}^{\prime }{\partial }_t$.

A convenient point-particle Faddeev–Popov ghost action at $\bar{e}=1$ isThe 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 arewhere the odd constant parameter $\Lambda$ is placed on the left. The product $\Lambda Q$ is even. Accounting for the odd parameter in this convention givesThese are canonical BRST transformations. Direct variation checks the sign: the bosonic kinetic term changes by $\Lambda (\dot{c}{p}^2+cp\cdot \dot{p})$, while the FP ghost term changes by $-\Lambda C\dot{c}$. Henceand 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 $\{\hat{b},\hat{c}{\}}_{+}=1$ and ${\hat{c}}^2=0$. Since $C$ commutes with the FP ghosts,It generates the same variations by $\delta \hat{F}=-i[\hat{F},\Lambda \hat{Q}]$, with an ordinary commutator against the even generator. Represent $\hat{c}$ by multiplication and $\hat{b}$ by differentiation. On a ghost-number-zero wavefunction $\Psi (x)$, $\hat{Q}\Psi =0$ is exactly $\hat{C}\Psi =0$, namelythe 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.

String oscillator algebra and FP ghost Virasoro generators. Use the same bosonic brackets as above and the odd bracketsQuantization turns them intoThe $b,c$ 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 $-\frac{1}{2}{\sum }_q(m-q)c_{-q}{b}_{m+q}$ and the second contributes $+\frac{1}{2}{\sum }_p(p-m)c_{-p}{b}_{p+m}$. Their sum is ${\sum }_q(q-m)c_{-q}{b}_{m+q}$. Reordering and normal ordering therefore give the ghost oscillator Virasoro generatorsFor $m\ne 0$ 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 explicitlyEquivalently, 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 giveIf $Q^2=0$, then $[L_n,Q]=[\{b_n,Q\},Q]=[b_n,Q^2]=0$. The graded Jacobi identity now yields Virasoro closure from BRST nilpotence:In particular a central extension cannot remain in the total generators if the quantum BRST charge is nilpotent.

Low-level descendant calculation. Let $|\Omega ;p⟩$ be the oscillator ground state, annihilated by ${\alpha }_{n>0}$, $b_{n\ge 0}$ and $c_{n>0}$. All $L_{n>0}$ annihilate it. The nonzero-mode FP ghost generators act asFor example $L_1(-b_{-1}{c}_0|\Omega ⟩)=-2b_0{c}_0|\Omega ⟩=-2|\Omega ⟩$. Similarly $L_2$ on the two level-two terms gives $-8|\Omega ⟩$ and $-9|\Omega ⟩$, hence $-17|\Omega ⟩$. These signs are essential: FP ghosts have an indefinite pairing.

The matter generators createContracting 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, areComparing the first with $[L_1,L_{-1}]=2L_0$ gives the oscillator highest weightIn this normalization the question's $⟨0|L_0|0⟩$ is $h_0$. Comparing level two with $[L_2,L_{-2}]=4L_0$ givesThis 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 $b_0|\Omega ⟩=0$, thenA 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 $L_1{L}_{-1}|\Omega ⟩$ and $L_2{L}_{-2}|\Omega ⟩$, 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 $Q|\Omega ;p⟩=b_0|\Omega ;p⟩=0$. They imply $L_0|\Omega ;p⟩=\{b_0,Q\}|\Omega ;p⟩=0$, so $h_0=0$ andThis 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.

For the relativistic particle phase-space action, the first-class constraint $\phi =(p^2+m^2)/2$ generatesIndeed the integrand varies by $\frac{1}{2}d[ϵ(p^2-m^2)]/dt$. The canonical gauge transformation is an invariance when the gauge parameter vanishes at fixed temporal endpoints, or when all fields and the parameter are periodic. The boundary restriction matters for the proper-time modulus.

Normalize the worldline interval to $[0,1]$. Thenis invariant because $\delta s=ϵ(1)-ϵ(0)=0$. Every allowed $e$ in its orbit can be written $e(t)=s+\dot{ϵ}(t)$: set $ϵ(t)={\int }_0^t[e(u)-s]du$. Thus $s$ remains a gauge-invariant integration variable, not another removable nonconstant mode. For a worldline circle the constant gauge parameter is a residual zero mode. Without the endpoint restriction, the assertion that $s$ is invariant would not hold.

The worldline gauge-orbit determinant is the Jacobian from gauge-orbit coordinates $ϵ$ to the nonconstant part of $e$ is the Faddeev-Popov determinant of ${\partial }_t$. Equivalently, the gauge-fixing identity has the formThe determinant is taken between the appropriate boundary-condition spaces, with the modulus removed; on a circle the prime also removes the constant parameter. Gauge fixing therefore leaves a factor $\det {}^{\prime }{\partial }_t$ and a modulus measure, after division by any residual gauge volume. Even though this determinant is field independent in the present Abelian example, it is the required change-of-variables Jacobian. A Grassmann integral over the Faddeev-Popov ghosts exponentiates it:The overall determinant phase depends on the integration convention and can be absorbed into normalization. Zero modes and the same endpoint restrictions must be treated separately rather than included in an invertible determinant.

For the free-ended open string, take $0\le \sigma \le \pi$. A canonical cosine expansion at a fixed time isIt implements the Neumann boundary conditions and has ${\alpha }_n^{†}={\alpha }_{-n}$. With ${\alpha }_0^m=p^m/\sqrt{\pi T}$, its Nambu-Goto phase-space action, up to a total time derivative, isReality requires ${\lambda }_{-n}={\lambda }_n^{\ast }$. Numerical factors can be absorbed into these Lagrange multipliers. In this covariant quantization of the bosonic string the oscillators retain all $D$ spacetime components, in contrast to the transverse oscillators in the preceding solution. The canonical commutation relations areThe oscillator vacuum $|0;p⟩$ is annihilated by ${\alpha }_n^m$ for $n>0$. Its momentum label will sometimes be suppressed.

Define the matter Virasoro algebra generators using normal ordering:No additive intercept is included in this definition of $L_0$. For $n>0$ the indices of the two factors in each term add to $n$; they cannot both be negative. After normal ordering there is a positive-mode annihilation operator on the right, possibly accompanied by the zero mode. Hence $L_n|0;p⟩=0$ for every $n>0$. In $L_0$, commuting positive modes past negative modes formally adds $\frac{D}{2}{\sum }_{k>0}k$. This divergent constant needs a prescription, and a finite shift is an ordering ambiguity. Our convention instead puts the physical string intercept into the constraint $L_0-a$.

With this convention the matter Virasoro algebra isA different additive constant in $L_0$ would change the linear-in-$m$ central term, so stating the convention is essential.

For the worldsheet ghost fields, useand choose a ghost oscillator vacuum withThen $c_0|0{⟩}_{\mathrm{gh}}\ne 0$ since $\{b_0,c_0\}=1$. This choice specifies the ghost zero-mode doublet; it is not a claim that both zero modes annihilate one state. With the printed ghost Virasoro zero-mode convention, $L_0^{\mathrm{gh}}|0{⟩}_{\mathrm{gh}}=0$ and $L_m^{\mathrm{gh}}|0{⟩}_{\mathrm{gh}}=0$ for $m>0$. The latter follows by putting positive ghost modes on the right; a possible contraction requires $m=0$ and is absent here.

Apply the supplied BRST charge to the matter state times this ghost oscillator vacuum. Terms with a rightmost $c_m$, $m>0$, vanish, as do the positive-mode ghost generators. ThusThese one-ghost states are independent, as can also be seen by applying $b_0,b_m$. Therefore the BRST physical-state constraints of an open string areFor the matter oscillator vacuum, $L_0={\alpha }^{\prime }{p}^2$, so $p^2=1/{\alpha }^{\prime }$ and $M^2=-1/{\alpha }^{\prime }=-2\pi T$: the physical ground state is a tachyon. The momentum must satisfy this equation; the zero-momentum oscillator vacuum by itself would not be BRST-closed.

Matter and ghost generators commute with one another. Add their two algebras and write ${\mathcal{L}}_m=L_m+L_m^{\mathrm{gh}}$. The given ghost constant must be retained:Define the shifted generators ${\hat{\mathcal{L}}}_m={\mathcal{L}}_m-{\delta }_{m,0}$. ThenBRST nilpotence requires cancellation of the anomalous central term in this shifted constraint algebra, together with the intercept one already present in the charge. At $m=2,n=-2$ the remaining anomalous coefficient is $(D-26)/2$, so $D=26$. At this value the shifted total generators obey the Witt algebra. The unshifted ${\mathcal{L}}_m$ still have the displayed linear zero-mode shift; it must not be silently discarded.

Use $\delta F=ϵ\{F,\phi \}$ with the canonical Poisson bracket. The first-class constraint generatesIndeed the variation of the relativistic particle phase-space action isThe transformation is therefore a gauge invariance when its parameter vanishes at the time endpoints, or suitable boundary conditions remove the total derivative.

On a fixed interval of parameter length $\Delta t$, the proper-time modulusis unchanged by these gauge transformations. Choosing $ϵ(t)={\int }_{t_i}^t(s-e(u))du$ sets $e+\delta e=s$, with $ϵ(t_i)=ϵ(t_f)=0$. Thus the nonconstant part of the worldline einbein can be fixed, but its constant modulus must still be integrated over. Fixing $e=1$ as well would remove inequivalent values of the proper-time modulus; it is legitimate only if the parameter interval is allowed to vary instead. The name proper-time modulus refers to the Schwinger proper-time parameter; after eliminating $p$, the geometric proper length for a massive on-shell trajectory is $M\int edt$ in this normalization.

For the gauge fixing functional $F=e-s$, its variation is $\delta F=\dot{ϵ}$. The Faddeev-Popov determinant is consequentlyThe Grassmann Gaussian integral represents this determinant using anticommuting Faddeev-Popov ghost fields $b,c$:The phase and normalization of the determinant depend on the measure convention. Its domain carries the chosen boundary conditions: on an interval the constant modulus is excluded from the gauge-fixed directions, and on a periodic worldline the constant ghost zero mode in field theory is removed with the residual gauge volume treated separately. A bare determinant with such zero modes left in would vanish.

For a relativistic particle, the Hamiltonian gauge parameter changes the worldline einbein by its time derivative. Fixing the einbein to a constant therefore gives a Faddeev-Popov determinant of ${\partial }_t$, represented by anticommuting FP ghosts. Gauge zero modes and the proper-time modulus are treated separately. The displayed phase convention gives the odd canonical bracket $\{b,c\}=-i$.