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Independent metric and Hamiltonian parametrization. The Lorentzian Polyakov action isThe worldsheet metric is independent of the embedding. Its equation sets the traceless worldsheet stress tensor to zero, making the metric locally conformal to the induced worldsheet metric; substitution recovers the Nambu–Goto action. Eliminating momentum in the Nambu-Goto phase-space action instead gives $(\dot{X}-u{X}^{\prime }{)}^2/(2e)-e{T}^2{X}^{\prime 2}/2$. On the positive-lapse branch, choose the independent metric, up to a positive Weyl transformation, asIts determinant is $-e^2{T}^2$. Multiplication by $-T/2$ recovers the reduced density, including the mixed term $-u\dot{X}\cdot X^{\prime }/e$. This proves the phase-space identification of the Polyakov metric.

Conformal gauge and residual coordinate freedom. In the Hamiltonian formulation, conformal gauge is $e=1/T,u=0$. The equations become $\ddot{X}-X^{\prime \prime }=0$, while ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$ remain as constraints. In the metric formulation, conformal gauge is ${\gamma }_{ab}=e^{2\omega }{\eta }_{ab}$, or simply ${\eta }_{ab}$ after fixing Weyl invariance. The metric parametrization proves these choices equivalent. Gauge-fixing the metric does not discard the Virasoro constraints.

With ${\sigma }^{\pm }=t\pm \sigma$, independent reparameterizations ${\sigma }^{\pm }\mapsto f_{\pm }({\sigma }^{\pm })$ multiply the flat metric by a conformal factor. A compensating Weyl transformation restores its chosen representative. Infinitesimally the Conformal Killing equation is ${\partial }_{-}{c}^{+}=0,{\partial }_{+}{c}^{-}=0$. These are the residual conformal transformations of two-dimensional Minkowski spacetime, with the periodicity conditions appropriate to a closed string.

Ghost determinant and anomaly. For gauge conditions $G_A(\varphi )=0$, the Faddeev-Popov determinant is the determinant of the linearized variationInsert this determinant in the gauge-fixed path integral and exponentiate it with anticommuting ghosts and antighosts: the additional action is proportional to $\int b^A{M}_{AB}{c}^B$. In metric gauge fixing, the trace is removed by Weyl invariance, and a worldsheet diffeomorphism changes the trace-free metric throughConsequently $c^a$ is a vector ghost and $b^{ab}$ a symmetric trace-free antighost. A conventional normalization of the worldsheet ghost action isA rescaling of the antighost changes only this overall kinetic normalization. In conformal gauge, after normalizing the chiral fields, the kinetic terms areThe closed-string ghosts are periodic, with independent left and right systems. Their conformal weights are $2$ for $b$ and $-1$ for $c$, so the supplied bc ghost system expression gives $c_{\mathrm{gh}}=-2(6\cdot 2^2-6\cdot 2+1)=-26$. Each free embedding boson has central charge one per chirality. HenceThis central charge of reparameterization ghosts cancels the string conformal anomaly in each chirality separately. Having two sectors does not double the required target dimension.

Channel variables, poles and the vector state. With particles 1,2 incoming and 3,4 outgoing, the Mandelstam variables are $s=-(p_1+p_2{)}^2$ and $t=-(p_1-p_3{)}^2$. They measure squared center-of-mass energy and momentum transfer. In the specified units ${\alpha }^{\prime }=1$, the external tachyon mass squared is $-1$, so $s+t+u=-4$.

The Gamma function residue at $-n$ is $(-1{)}^n/n!$. At $s=n-1$, its argument $-1-s$ varies with the opposite sign to $s$. The Gamma function recurrence givesThus the Veneziano amplitude pole residues areFor generic fixed $t$ the simple poles are $-1,0,1,\dots$. A zero of the residue product at an exceptional negative integer $t$ makes the corresponding pole removable. Fixed $t$ must be away from its own channel poles; fixing it at a channel singularity does not give an ordinary finite meromorphic function of $s$.

At $s=0$, the residue is $-(t+2)=-(t-u)/2$. This is proportional to $(p_1-p_2)\cdot (p_3-p_4)$, the contraction of conserved scalar-pair vector currents. Each current is orthogonal to the exchanged momentum because the external masses in its pair are equal. Alternatively, with scattering-angle cosine $z$, one has $t=-2(s/4+1)(1-z)$; at this pole $t+2=2z$, a pure spin-one angular dependence. ThereforeThe massless vector pole of the Veneziano amplitude is the massless open-string vector seen in the two-tachyon exchange channel.

Define the open-string endpoint momentum flux along the open string by $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Variation of the Nambu-Goto phase-space action givesThe two last equations are the Virasoro constraints. Integration of the momentum equation givesThus the necessary and sufficient condition for total momentum conservation is equality of the two endpoint fluxes, component by component. Vanishing of both fluxes is a sufficient local boundary condition.

The spatial boundary term in the action variation is $-\int dt[J_m\delta X^m{]}_0^{\pi }$. For free ends the endpoint variations are arbitrary, soIn a gauge with $u=0$ and nonzero $e$ this reduces to the usual Neumann boundary condition $X^{\prime m}=0$. The conclusion is conservation of total momentum, not that total momentum must vanish. The PDF contains the derivative in this conclusion; the TeX transcription drops it.

For endpoints confined to $x^1=0$, the Dirichlet boundary condition fixes $\delta X^1=0$. It therefore imposes no requirement that $J_1$ vanish. Preserving the fixed position requires ${\dot{X}}^1=e{P}^1+u{X}^{\prime 1}=0$ at each endpoint, but that is a different condition. For example, in $u=0$ gauge the endpoint momentum $P^1$ is zero while $e{T}^2{X}^{\prime 1}$ can be nonzero. Consequently ${\mathcal{P}}_1$ is generally not conserved: the normal endpoint forces transfer momentum to the hyperplane. A dynamical D-brane recoils and carries the missing momentum. If the hyperplane is idealized as fixed, its external support absorbs the momentum. The combined string-and-support system conserves momentum; freely varying directions tangent to the hyperplane retain their vanishing-flux condition.

For the Monge-gauge Hamiltonian of a three-dimensional string, in Monge gauge, $X^{\prime m}=(0,1,{\Phi }^{\prime })$ and $P_m=(P_0,P_1,\Pi )$. The momentum constraint gives $P_1=-{\Phi }^{\prime }\Pi$. The other Virasoro constraint then readsChoosing the positive-energy branch,Substituting into the phase-space action leaves $I=\int dtd\sigma (\Pi \dot{\Phi }-\mathcal{H})$, soFor an infinite string this is an energy-density expression; even the undeformed string has infinite total energy. One may regulate the length or subtract its constant reference energy when a finite total energy is needed. The positive branch is unambiguous at the density level.

The canonical Hamiltonian field equations areConsequently $\Pi ={\Phi }^{\prime }$ impliesThe second equation also becomes $\dot{\Pi }={\Phi }^{\prime \prime }$, consistent with differentiating the first. These are one-direction traveling profiles along the string.

For the linear profile, the spatial string is the straight line $X^2=k{X}^1+kt$. It is tilted, and its coordinate intercept moves with speed $k$. This is also a uniformly moving straight string: the physical transverse velocity of a tilted string is the velocity normal to the line. With unit tangent $(1,k)/\sqrt{1+k^2}$ and coordinate velocity $(0,k)$, the normal speed isThus $k>1$ does not imply superluminal physical motion. Fixed-$\sigma$ labels have spacelike trajectories when $k>1$, but labels on a string can slide tangentially and are not identifiable material particles. Choosing $\dot{\sigma }=-k^2/(1+k^2)$ removes the tangential velocity and gives the subluminal velocity $(-k^2,k)/(1+k^2)$.

The induced worldsheet metric supplies a coordinate-invariant check:The string worldsheet remains Lorentzian for every finite $k$, including $k=1$ and $k>1$. Its constant energy density is $1+k^2$. This linear profile is a tilted boosted straight string, rather than a localized finite-energy wave.

Set $T=1/(2\pi {\alpha }^{\prime })$ and take the closed-string spatial coordinate periodic with period $2\pi$. One convenient Lorentzian convention for the conformal gauge action isThe two anticommuting bc ghost systems are the Faddeev-Popov ghosts of the worldsheet metric gauge fixing. Their fields have conformal weights $2$ and $-1$. Ghost normalizations can be rescaled without changing the algebra. After a Wick rotation the two terms become the familiar holomorphic and antiholomorphic $b\bar{\partial }c$ and $\tilde{b}\partial \tilde{c}$ actions.

Let ${\mathcal{L}}_n$ and ${\tilde{\mathcal{L}}}_n$ include both matter and ghost contributions. With the conventional plane normalization of the zero mode, their Virasoro algebra isEach free coordinate boson contributes central charge $1$, including the timelike coordinate after the usual analytic continuation. The weight-$(2,-1)$ bc ghost system contributes $-26$ in each chiral sector. Thus the total central charge is $D-26$.

The Virasoro constraints originate in a gauge symmetry. A nonzero central term is a quantum obstruction to realizing that gauge symmetry without an anomaly; equivalently, the BRST charge fails to be nilpotent. Cancellation of the worldsheet Weyl anomaly therefore requiresThis is the critical dimension of the bosonic string. The accompanying physical-state normal-ordering condition gives the usual matter intercept $1$ in each sector. A shift of the zero-mode convention changes the linear term in the central polynomial, but cannot change the required cancellation of its cubic coefficient.

In conformal gauge, ${\gamma }_{\mu \nu }=e^{2\omega }\mathrm{diag}(-1,1)$. The Weyl invariance of the Polyakov action removes $\omega$, leavingIts complete variation isFix the string embedding map on the initial and final time slices. The interior Euler-Lagrange field equation is the wave equationAt each spatial endpoint the remaining variation vanishes under either Dirichlet boundary conditions, $\delta X^m=0$, or Neumann boundary conditions, $X^{m\prime }=0$, independently for each target-space component. Thus a fixed endpoint or a free endpoint makes the boundary contribution vanish. Fixing a spatial point throughout time imposes Dirichlet boundary conditions on its spatial coordinates, while the time coordinate may retain Neumann boundary conditions.

The independent metric tensor equation must also be retained after gauge fixing. Its Virasoro constraints areTogether with the wave equation, these ensure equivalence to the Nambu–Goto action; varying only the already fixed metric would lose these Virasoro constraints.

Let $q=\sigma /L$, with $L>0$, and let $Z=X^1+i{X}^2$. Direct differentiation gives $\ddot{Z}=Z^{\prime \prime }=-Z/L^2$, while the time coordinate also satisfies the wave equation. Using the Minkowski metric,Thus both Virasoro constraints hold, and the induced worldsheet metric is $g_{\mu \nu }={\cos }^{2}q\mathrm{diag}(-1,1)$. The configuration solves the Nambu–Goto action equations in its interior.

Place the fixed end at $\sigma =0$. The first free endpoint has $Z^{\prime }=e^{it/L}\cos q=0$ at $q=\pi /2$. For the usual single unfolded rotating Nambu–Goto string,This parameter interval need not equal the earlier interval $[0,\pi ]$, since that interval was a coordinate convention. The proper length is measured on a constant-$X^0$ slice; the local motion is perpendicular to the string, so there is no longitudinal Lorentz contraction.

All points have the same phase $t/L$, hence the segment rotates rigidly with angular velocity $\Omega =1/L$. At radius $r=L\sin q$, its speed is $v=r/L$. Therefore the free tip moves at the speed of light. This is consistent with its Neumann boundary condition: $X^{\prime }=0$ and the Virasoro constraints imply ${\dot{X}}^2=0$ at the free tip. The null endpoint is a boundary limit, not a nondegenerate interior point.

In this conformal gauge, $P_m=T{\dot{X}}_m$, so the energy density per $\sigma$ is $-P_0=T$. Equivalently, the energy density per proper length is $T/\sqrt{1-r^2/L^2}$. Thusand the rotational kinetic energy isThe angular momentum about the fixed point isTherefore the Regge trajectory isThe endpoint conditions alone also admit folded extensions with parameter length $(n+\frac{1}{2})\pi L$, $n\ge 0$. Their proper length, energy and angular momentum are $(2n+1)$ times the values above, and $\beta =1/[(2n+1)\pi T]$. The displayed answer uses the implicit unfolded-segment convention.

For covariant quantization of the bosonic string,All other commutators between independent canonical variables vanish. The covariant Fock vacuum obeys ${\alpha }_k^m|0;p⟩=0$ for every $k>0$ and every $m$. Normal ordering moves these annihilating string oscillators to the right and givesThen ${\hat{N}}_{\mathrm{cov}}|0;p⟩=0$. Its commutator $[{\hat{N}}_{\mathrm{cov}},{\alpha }_{-k}^m]=k{\alpha }_{-k}^m$ grades a Fock state basis by $N=\sum k{n}_{km}$, although its inner product is indefinite because of the timelike oscillator.

The quantum Virasoro algebra has central charge $D$:The Virasoro central extension comes from commutators needed to order infinite oscillator sums; it is a quantum effect absent from the classical Poisson brackets. Replacing a classical Poisson bracket by a commutator cannot recover that term without a regularized ordering calculation.

If a state were annihilated by every nonzero ${\hat{L}}_j$, the $j=1,k=-1$ commutator would imply ${\hat{L}}_0|\psi ⟩=0$. The $j=2,k=-2$ commutator would then imply $(D/2)|\psi ⟩=0$. For $D>0$, only the zero vector is annihilated by all nonzero Virasoro constraints. This explains why only positive modes annihilate a physical string state.

The vacuum is a physical string state when $p^2=2\pi Ta$. At level one all states have the form $|A;p⟩=A_m{\alpha }_{-1}^m|0;p⟩$. Since $[{\hat{L}}_j,{\alpha }_k^m]=-k{\alpha }_{j+k}^m$, the positive-mode Virasoro constraints giveThus the complete level-one conditions and norm arewith the common vacuum normalization suppressed.

For $a>1$, choose spacelike momentum $p^m=(0,k,0,\dots )$, $k^2=2\pi T(a-1)$. A purely timelike polarization has $A\cdot p=0$ and norm $-1$. Hence a negative-norm physical string state exists when $a>1$.

For $a<1$, the mass-shell condition gives $M^2=2\pi T(1-a)>0$. In a rest frame, $A\cdot p=0$ forces $A_0=0$, leaving $D-1$ positive-norm vector-particle polarizations, those of a massive vector.

For $a=1$ and nonzero null momentum, $A\cdot p=0$ leaves a null direction $A_m\propto p_m$. The corresponding null string state is $p\cdot {\alpha }_{-1}|0;p⟩=\sqrt{\pi T}{\hat{L}}_{-1}|0;p⟩$. It is orthogonal to every physical string state because ${\hat{L}}_1$ annihilates them. Quotienting by this gauge redundancy, $A_m\sim A_m+\zeta p_m$, leaves $D-2$ positive vector-particle polarizations. Thus the level-one spectrum agrees with light-cone gauge in string theory at $a=1$. This level-one argument alone does not establish consistency or absence of negative norms at all higher levels.

The real canonical variables $x^m,p_m$ describe the center-of-mass position and total momentum of the open string. The nonzero string oscillators are complex Fourier modes with ${\alpha }_{-k}^m=({\alpha }_k^m{)}^{\ast }$; the Lagrange multipliers obey ${\lambda }_{-j}={\lambda }_j^{\ast }$. The Virasoro constraints satisfy $L_{-j}=L_j^{\ast }$, so the multiplier term is real.

For each $k>0$, the oscillator kinetic term differs from the manifestly real expressionby $\frac{i}{2k}\frac{d}{dt}({\alpha }_k\cdot {\alpha }_{-k})$. Thus the written action is real up to a boundary term, which does not change its symplectic form or bulk dynamics. Adding the corresponding endpoint term makes reality exact.

The independent nonzero Poisson brackets areAll brackets between these independent center-of-mass and oscillator variables vanish. The dependent zero mode is ${\alpha }_0^m=p^m/\sqrt{\pi T}$, so, if it is used, $\{x^m,{\alpha }_0^n\}={\eta }^{mn}/\sqrt{\pi T}$.

The Fourier coefficients of the Virasoro constraints areTheir Poisson brackets areThey form the classical Witt algebra, with no Virasoro central extension. In particular they are first-class constraints, closing on the constraint surface, and generate the remaining worldsheet diffeomorphisms rather than independent physical excitations.

The closed string already contains the particle required for quantum gravity. For bosonic string theory in its critical dimension of string theory, the mass-shell condition and closed-string level matching areAt $N=\tilde{N}=1$, the states ${ϵ}_{mn}{\alpha }_{-1}^m{\tilde{\alpha }}_{-1}^n|0;k⟩$ are massless. After imposing the Virasoro constraints and quotienting null string states, the transverse polarization tensor decomposes into symmetric trace-free, antisymmetric and scalar pieces under $SO(D-2)$. They describe a graviton, the Kalb–Ramond field and the dilaton. In particular, the symmetric trace-free sector is a massless spin-two graviton. Its number of degrees of freedom is $D(D-3)/2$.

The state–operator correspondence associates each physical string state with a string vertex operator. For example, the matter part of a massless closed-string vertex operator isIts conformal weights are $(1,1)$, so its integrated form $\int d^{2}z{V}_{ϵ,k}$ is invariant under changes of string worldsheet coordinates. At fixed insertion positions it is accompanied by $c\bar{c}$ from the worldsheet ghost fields. Physical vertices represent BRST cohomology classes; longitudinal changes of the polarization tensor are BRST-exact operators and decouple from physical scattering amplitudes. For the graviton, this string-state gauge redundancy becomes the linearized target-space diffeomorphism $h_{mn}\mapsto h_{mn}+{\partial }_m{\xi }_n+{\partial }_n{\xi }_m$.

The Polyakov path integral computes a closed-string scattering amplitude by inserting the external string vertex operators, integrating their unfixed positions and the inequivalent worldsheet moduli, and including the worldsheet ghost fields. On a Riemann sphere three positions are fixed by the residual conformal group. For example, contractions of tachyon vertex operators produce the Koba-Nielsen factor, whose position integral gives the Virasoro–Shapiro amplitude.

The string dual resonance property means that one crossing-symmetric scattering amplitude has equivalent expansions in the different channels: its poles exhibit intermediate string states in each channel, rather than separate channel contributions being added again. The scattering-amplitude factorization at a pole identifies the intermediate particle and its couplings. Set $8\pi T=1$, so ${\alpha }^{\prime }=4$ and the closed-string tachyon has $M^2=-1$. For four such external states the Mandelstam variables obey $s+t+u=-4$. The Virasoro–Shapiro amplitude has generic $s$-channel poles at $s=-1,0,1,\dots$, exactly the bosonic string mass spectrum in these units.

The massless pole gives an especially direct check. With overall normalization $C$ and $u=-4-s-t$, the Gamma function recurrence givesThusThe quadratic residue in $t$ contains a spin-two exchange. A scalar exchange alone could not give this angular dependence. Combined with the known massless spectrum and scattering-amplitude factorization, it identifies exchange of the graviton, with possible scalar contributions from the dilaton; the antisymmetric Kalb–Ramond field does not couple to two identical scalar tachyons here. Consequently the graviton participates in interactions, rather than being an isolated free state.

Decoupling longitudinal graviton polarizations forces a universal coupling to the conserved stress-energy tensor. Consistency of this massless spin-two gauge invariance extends the linearized coupling to the nonlinear dynamics of general relativity. At distances large compared with $\sqrt{{\alpha }^{\prime }}$, the gravitational sector of the effective action begins with the Einstein-Hilbert action, alongside dilaton and Kalb–Ramond field terms and higher-derivative string corrections. The infinite tower of string states supplies the short-distance completion of this quantum gravity expansion.

Interactions are organized by string worldsheet topology. A constant dilaton gives the string coupling $g_s=e^{{\Phi }_0}$, and a connected oriented surface of genus $g$ has Euler characteristic $\chi =2-2g$. Its weight is $g_s^{-\chi }$; with $n$ normalized external vertices,The sphere is tree order, the torus is one loop, and each additional handle adds a factor $g_s^2$. Each coefficient is an integral over worldsheet moduli, so this is string perturbation theory in $g_s$, with an independent low-energy expansion in ${\alpha }^{\prime }$.

For the one-loop vacuum contribution, torus modular invariance identifies $\tau$ with $(a\tau +b)/(c\tau +d)$, $ad-bc=1$. Integration is over the standard fundamental domain of the modular group,The potential short-proper-time region ${\tau }_2\to 0$, responsible for a point-particle ultraviolet divergence, is absent. It would count metrics already represented elsewhere in $\mathcal{F}$. Torus modular invariance is therefore the geometric reason for one-loop ultraviolet finiteness. More explicitly, in $D=26$ the vacuum integrand is proportional to $d^2\tau {\tau }_2^{-14}|\eta (\tau ){|}^{-48}$, with $\eta$ the Dedekind eta function; the complete expression is invariant under the modular group.

Ultraviolet finiteness does not make the bosonic one-loop vacuum energy finite. The remaining long-tube region ${\tau }_2\to \infty$ is an infrared divergence from the tachyon: $|\eta (\tau ){|}^{-48}\sim e^{4\pi {\tau }_2}$. It signals instability of the bosonic vacuum. Tachyon-free consistent backgrounds remove this particular obstruction, though other infrared effects must still be treated. The ultraviolet improvement comes from the extended string and the complete spectrum together with the worldsheet gauge identifications; the genus expansion remains a perturbative description of quantum gravity.

Insert the given closed-string mode expansion into the stress tensor and compare $T(z)={\sum }_n{L}_n{z}^{-n-2}$. Normal ordering gives

The inverse of the worldsheet Laplacian gives the free-field operator-product expansionHenceExtracting the Laurent modes by contour integration yields the string oscillator algebra

The vanishing of the worldsheet stress tensor becomes the Virasoro constraintsSince $L_0={\alpha }^{\prime }{k}^2/4+N$ and the bosonic-string normal-ordering constant is $a=1$, these zero-mode constraints are the target-space string mass-shell conditionThereforeThe positive-mode constraints further requireand null string states identify polarizations that differ by momentum-longitudinal terms. The symmetric trace-free, antisymmetric, and trace sectors of ${ϵ}_{\mu \nu }$ describe the graviton, Kalb–Ramond field, and dilaton, respectively.

Finally, the state–operator correspondence maps ${\alpha }_{-1}^{\mu }{\bar{\alpha }}_{-1}^{\nu }|k⟩$ toAdding its integrated massless closed-string vertex operator to the string nonlinear sigma model changes the background coupling byThus a Fourier mode $\delta G_{\mu \nu }(X)={ϵ}_{(\mu \nu )}{e}^{ik\cdot X}$ is precisely a linearized deformation of the target-space metric. Its transversality, mass-shell condition, and polarization gauge redundancy are the worldsheet statements that the deformation is marginal and is defined up to a linearized spacetime diffeomorphism.

The zero-mode Virasoro constraint iswhere the string level operator isSince the string tension isthe mass formula is

For a string beginning and ending on the same brane, the $N=1$ states are massless. Oscillators polarized tangentially give a gauge field on the $(p+1)$-dimensional worldvolume, while transverse polarizations give scalar fields describing fluctuations of the brane position. Strings joining two separated branes have the additional stretching mass $T|x_2-x_1|$ and are charged under the two endpoint gauge groups. When the two branes coincide, these off-diagonal states also become massless and Coincident-D-brane gauge enhancement enlarges $U(1)\times U(1)$ to $U(2)$. The bosonic open string also retains its $N=0$ tachyon.

A classical string must also satisfy the equations obtained by varying the worldsheet metric before imposing conformal gauge. Vanishing of the worldsheet stress-energy tensor givesIn modes these are the Virasoro constraintsfor every integer $m$. Reality of $X^{\mu }$ also requiresTogether with periodicity and the wave equation, these conditions remove the unphysical longitudinal worldsheet excitations.

Varying the Polyakov action with respect to the worldsheet metric givesIn conformal gauge, variation of $X$ gives the wave equationwhile the Virasoro constraints become $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. The boundary variation is $-T\int d\tau ,X^{\prime }\cdot \delta X$ at each endpoint. It vanishes through a Neumann boundary condition $X^{\prime \mu }=0$, a Dirichlet boundary condition $\delta X^{\mu }=0$, or a direction-by-direction mixture of the two.

The Virasoro constraints determine $X^{-}$ in terms of the $d=D-2$ transverse coordinates $X^i$. Their closed-string expansion introduces independent left- and right-moving string oscillators satisfyingFor $n>0$, ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ annihilate the momentum vacuum, while ${\alpha }_{-n}^i$ and ${\tilde{\alpha }}_{-n}^i$ create excitations. A general state is a product of these creation operators acting on $|p⟩$, subject to closed-string level matching $N=\tilde{N}$.

Normal ordering the quantum constraints introduces the normal-ordering constant of a string $a$. Requiring the Lorentz generators $J^{-i}$ to obey the Lorentz algebra without an anomalous term fixesThus Lorentz invariance resolves both the ordering ambiguity and the critical dimension of string theory.

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.

Varying the independent worldsheet metric gives $T_{ab}=T[{\partial }_{a}X\cdot {\partial }_{b}X-\frac{1}{2}{\gamma }_{ab}{\gamma }^{cd}{\partial }_{c}X\cdot {\partial }_{d}X]$. Its classical vanishing is the metric equation and produces the Virasoro constraints.