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Critical closed bosonic strings have independent left- and right-moving Virasoro constraints. Positive modes in both sectors annihilate physical kets, while both zero-mode conditions impose the mass shell and closed-string level matching. Quotient by physical null string states, equivalently use the appropriate BRST cohomology. With $L_0={\alpha }^{\prime }{k}^2/4+N$, the result is $N=\tilde{N}$ and $M^2=4(N-1)/{\alpha }^{\prime }$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.

For a compact string coordinate of radius $R$, choose $p_L=n/R+wR/{\alpha }^{\prime }$ and $p_R=n/R-wR/{\alpha }^{\prime }$, where the left-moving oscillators are denoted by tildes. The two classical zero-mode Virasoro constraints give $M^2=p_L^2+4\tilde{N}/{\alpha }^{\prime }=p_R^2+4N/{\alpha }^{\prime }$. Their difference yields the displayed shifted closed-string level matching condition. Their average is $M^2=(n/R{)}^2+(wR/{\alpha }^{\prime }{)}^2+2(N+\tilde{N})/{\alpha }^{\prime }$. Classically the center-of-mass momentum can be continuous; the integer momentum number belongs to compact-coordinate quantization. Quantum intercepts shift the mass formula but cancel out of level matching.

Covariant quantization keeps target-spacetime Lorentz covariance manifest, imposes canonical commutation relations on every embedding coordinate and removes unphysical states with the Virasoro constraints.

DDF operators are transverse physical string oscillators constructed as contour integrals of weight-one string vertex operators. Choose a null reference momentum $q$ and transverse polarizations ${ϵ}_i$ orthogonal to it and to a reference on-shell momentum. The integrand is proportional to ${ϵ}_i\cdot \partial X{e}^{inq\cdot X}$; its normalization and $q\cdot p$ are fixed so the operators obey the oscillator algebra in the title and commute with the positive-mode Virasoro constraints. For an open-string tachyon seed, one can use $2{\alpha }^{\prime }q\cdot p=1$. Acting with negative modes shifts momentum and raises oscillator level while preserving the physical mass-shell condition. Their transverse Fock space has positive norm and makes the physical content of the no-ghost theorem for the critical bosonic string explicit.

With string tension $T=1/(2\pi {\alpha }^{\prime })$, the free-ended open string $X^0=a\tau$, $X^1+i{X}^2=a\cos \sigma e^{i\tau }$, $0\le \sigma \le \pi$, is straight at every target time. Its Nambu–Goto equations of motion and Virasoro constraints hold, and its ends move at light speed. Integrating the target-space Noether charges of a string gives $M=\pi Ta$ and $J=\pi T{a}^2/2$. Thus $J={\alpha }^{\prime }{M}^2$. Extending the same map to $0\le \sigma \le 2\pi$ describes a double-covered folded closed string, with twice the mass and twice the angular momentum, hence $J={\alpha }^{\prime }{M}^2/2$. The different slopes record the different boundary topology.

For $d=26$, $q^2=-2$ and the physical string state $[{\zeta }_{\mu \nu }{\alpha }_{-1}^{\mu }{\alpha }_{-1}^{\nu }+b_{\mu }{\alpha }_{-2}^{\mu }]|p⟩$, the Virasoro constraints give $b_{\nu }=-q^{\mu }{\zeta }_{\mu \nu }$ and $t={\zeta }^{\mu }{}_{\mu }=-2q\cdot b$. Every solution has the displayed decomposition, where $q\cdot A=0$ and $c$ is symmetric, traceless and transverse to $q$. The vector piece is $L_{-1}(2A\cdot {\alpha }_{-1}|p⟩)$; the scalar piece is $t(L_{-2}+3L_{-1}^2/2)|p⟩/10$. These are null string states, leaving the transverse traceless massive tensor as the physical polarization.

On a patch with ${\alpha }_0^{+}\ne 0$, use worldsheet diffeomorphisms and worldsheet supersymmetry to fix ${\alpha }_{n\ne 0}^{+}=0$ and ${\psi }_r^{+}=0$. The Virasoro constraints and superconformal current constraints then determine the longitudinal modes by the displayed transverse bilinears. In the Ramond sector, reduction of the fermionic zero modes includes the zero-mode Dirac constraint; fixing only nonzero modes does not complete the reduction. Eight transverse bosons and eight transverse fermions remain in the critical ten-dimensional RNS string.

With unit tension and $X^0=t$, $X^1=\sigma$, the Virasoro constraints give $P_1=-\Pi {\Phi }^{\prime }$ and $P_0=-\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$. The reduced energy density is $\mathcal{H}=-P_0$. On the chiral branch $\Pi ={\Phi }^{\prime }$, the Hamiltonian field equation becomes $\dot{\Phi }={\Phi }^{\prime }$.

For a nondegenerate induced worldsheet metric $g_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$, variation gives ${\partial }_a(\sqrt{-g}{g}^{ab}{\partial }_b{X}^m)=0$. In conformal gauge this is the two-dimensional wave equation, supplemented by the Virasoro constraints.

The first-order string action isThe multipliers impose the two Virasoro constraints. The canonical equations give a spatial momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$.

Use flat target spacetime with metric ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,\dots ,1)$, and let $h_{ab}$ be an independent Lorentzian worldsheet metric. The Polyakov action isHere $X$ is the string embedding map and $T$ the string tension. Variation of $X$, followed by integration by parts, givesThus the embedding equation isThe boundary term vanishes by periodicity for a closed string. At an open-string boundary it requires, for example, vanishing normal derivative for a Neumann boundary condition, or $\delta X=0$ for a Dirichlet boundary condition. Mixed choices give corresponding endpoint conditions.

Define the induced worldsheet metric ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. Using $\delta \sqrt{-h}=-\frac{1}{2}\sqrt{-h}{h}_{ab}\delta h^{ab}$, metric variation givesThe second set of equations is therefore the vanishing worldsheet stress tensor:For a nondegenerate induced metric this says $h_{ab}=e^{2\omega }{\gamma }_{ab}$. Substituting into the action gives $S_P=-T\int \sqrt{-\det \gamma }{d}^2\sigma$, proving the classical equivalence of Polyakov and Nambu–Goto actions. The auxiliary metric has not added physical metric degrees of freedom.

There are three important symmetry statements. Worldsheet diffeomorphisms change the parameterization, with $X$ transforming as a scalar and $h$ as a tensor; the integral is invariant. Weyl transformations $h_{ab}\mapsto e^{2\omega (\sigma )}{h}_{ab}$ leave $\sqrt{-h}{h}^{ab}$ invariant specifically in two dimensions. These are gauge redundancies. Locally, the two diffeomorphism functions and one Weyl function fix the three independent components of $h$ to conformal gauge, $h_{ab}=e^{2\omega }{\eta }_{ab}$. Global worldsheet moduli may remain and cannot be gauged away. Separately the flat-target Poincaré group acts globally on $X$ and supplies conserved target momentum and angular momentum, rather than another worldsheet gauge constraint.

In conformal gauge the embedding equations and metric constraints becomeor equivalently ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$. The Virasoro constraints must still be imposed: fixing the metric in the action before varying it would miss them. Residual conformal changes of the two null worldsheet coordinates are generated by these constraints. Classically these statements hold in any target dimension; cancellation of the quantum worldsheet Weyl anomaly is a further condition, not part of the classical variation.

Consider the closed bosonic string on a rectangular product of spatial circles with no background mixing between them. Let $Y^j$ have period $2\pi R_j$, take $0\le \sigma <2\pi$, and choose $Y_L(\tau +\sigma )$ to be the tilded oscillator sector. Closed-string topology permitsSingle-valued center-of-mass wavefunctions give quantized momentum $p^j=n_j/R_j$, $n_j\in \mathbb{Z}$. These are the momentum and winding modes. The oscillator modes remain integer moded; the zero modes are split asThe two intercept-one Virasoro constraints, with noncompact $k^2=-M^2$, giveThus the mass formula and compact-circle closed-string level matching condition areHere $N$ labels right movers and $\tilde{N}$ left movers; exchanging the labels changes the displayed sign convention, not the spectrum. Noncompact oscillator polarizations are still restricted by positive-mode Virasoro conditions and null-state identifications. T-duality exchanges $n_j,w_j$ and $R_j,{\alpha }^{\prime }/R_j$, preserving masses and the physical state conditions.

At generic radius, neutral $(N,\tilde{N})=(1,1)$ states with one noncompact oscillator and one compact oscillator give two massless vectors per circle. They are the two Abelian gauge fields associated with the compact metric and two-form components. Additional charged vectors can occur at oscillator level sum one. For one circle, set all other compact charges to zero. Take $n=w=\pm 1$ and $(N,\tilde{N})=(1,0)$, with a right-moving noncompact oscillator ${\alpha }_{-1}^{\mu }$; or take $n=-w=\pm 1$ and $(N,\tilde{N})=(0,1)$, with a left-moving noncompact oscillator ${\tilde{\alpha }}_{-1}^{\mu }$. These obey level matching. Their common squared mass isTherefore four extra charged vector particles become massless at $R=\sqrt{{\alpha }^{\prime }}$, the self-dual circle. Away from that radius they are massive. They are genuine vector states: the noncompact level-one oscillator carries a vector polarization, and the Virasoro constraint plus its null identification impose transversality and remove the gauge polarization at zero mass.

The resulting self-dual circle current algebra makes the gauge enhancement explicit. At the special radius, the left-moving charges above have $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$, $p_R=0$. With $Y_L(z)Y_L(w)\sim -{\alpha }^{\prime }\log (z-w)/2$, the currentshave conformal weight one. The operator products give $J^3{J}^3\sim 1/[2(z-w{)}^2]$, $J^3{J}^{\pm }\sim \pm J^{\pm }/(z-w)$ and $J^{+}{J}^{-}\sim 1/(z-w{)}^2+2J^3/(z-w)$. They form a level-one SU(2) current algebra; the right-moving copy supplies another SU(2). Pairing each chiral current with the opposite-sector noncompact derivative yields a weight-$(1,1)$ vector vertex. Thus the generic $U(1{)}_L\times U(1{)}_R$ becomes $SU(2{)}_L\times SU(2{)}_R$, with four charged generators in addition to the two Cartan vectors. Several self-dual circles give the corresponding product enhancement.

Compactification also creates scalar states and moduli; not every additional massless state is a vector. The mechanism just shown uses the intercept-one bosonic spectrum. A superstring projection changes the allowed oscillator states, so the same mass formula cannot simply be applied to a GSO-projected superstring without modification.

Use Minkowski spacetime with signature $(-,+,\dots ,+)$, and write the string tension as $T=1/(2\pi {\alpha }^{\prime })$. If $X^{\mu }(\tau ,\sigma )$ is the string embedding map, its induced worldsheet metric is ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. The Nambu–Goto action isThe first variation uses $\delta {\gamma }_{ab}=2{\partial }_{(a}{X}_{\mu }{\partial }_{b)}\delta X^{\mu }$ and therefore givesAn integration by parts yields the bulk Nambu–Goto equations of motion and the endpoint term:For a closed string, periodicity cancels the endpoint term. For a freely moving open string, arbitrary endpoint variations require zero momentum flux, which is the free-end string boundary condition.

Choose orthogonal conformal coordinates on the nondegenerate part of the worldsheet, so that ${\gamma }_{\tau \sigma }=0$ and ${\gamma }_{\tau \tau }+{\gamma }_{\sigma \sigma }=0$. The Nambu–Goto equations of motion then become a wave equation, with the accompanying Virasoro constraints:Here a dot and a prime mean ${\partial }_{\tau }$ and ${\partial }_{\sigma }$. The last two equations are essential: a solution of the wave equation alone need not be a relativistic fundamental string motion.

For a free-ended open string, take $0\le \sigma \le \pi$ and $a>0$, and setwith the other coordinates constant. Every component satisfies the wave equation, andMoreover $X^{\prime }=0$ at both endpoints, giving the Neumann boundary condition. At fixed target time $t=a\tau$, the spatial image is the straight segmentThus its angular velocity is $1/a$. The speed at coordinate $r$ is $|r|/a$, so the endpoints have the null motion of a free string endpoint. The induced worldsheet metric degenerates there; the conserved densities below have finite limits, so the endpoint degeneracy causes no divergent charge.

In these coordinates ${\Pi }^{\tau \mu }=T{\dot{X}}^{\mu }$. The target-space Noether charges of a string give the energy, momentum and angular momentum:Since this is the centre-of-momentum frame, the rest mass is $M=E$. Eliminating $a$ proves the free-ended straight-string Regge relation:A spacetime Lorentz transformation of the solution changes its frame, not this rest-frame relation.

There is also a folded closed string version: use the same formula over $0\le \sigma \le 2\pi$. The spatial segment is then covered twice, with folds at $\sigma =0,\pi$, and the closed-string mode expansion is periodic. Each charge is twice its open string value, soThe folds also have degenerate induced metric and are understood through the finite canonical densities or a limiting smooth motion. The two classical Regge trajectories have different slopes because the closed string has two coincident branches. No quantum string intercept enters this classical calculation.

Let $Y$ be the compact coordinate, identified modulo $2\pi R$. The closed string may wrap the circle:Single-valued momentum eigenfunctions $e^{ipY}$ require $p=n/R$, $n\in \mathbb{Z}$. These are the momentum and winding modes. Write the chiral zero-mode momenta asThe left-moving string oscillators will be tilded: $N_L=\tilde{N}$, $N_R=N$. This fixes the sign in compact-circle closed-string level matching.

Define the lower-dimensional mass by $M^2=-p_{\mathrm{noncompact}}^2$. The two closed-string physical-state Virasoro conditions, with bosonic string intercept one, areTheir average and difference derive the complete free bosonic string mass spectrum on the circle:The allowed states have nonnegative integer oscillator levels and obey the Virasoro constraints within each chiral sector. Compactification has not removed oscillator polarizations; it has added quantized compact momentum and allowed winding while changing the lower-dimensional interpretation of those polarizations.

At large $R$, the momentum and winding modes with $w=0$ have closely spaced momentum energies $|n|/R$, whereas nonzero winding costs $|w|R/{\alpha }^{\prime }$. At small $R$ their roles reverse. The precise equality is T-duality:It leaves $p_L$ unchanged and reverses $p_R$, and leaves the mass formula and closed-string level matching unchanged. On the string embedding map it reverses the right-moving part of $Y$. Therefore the small-radius spectrum is the large-radius spectrum of the dual theory, rather than a spectrum in which all charged states simply become heavy.

At a generic radius, the neutral level $(N_L,N_R)=(1,1)$ contains the lower-dimensional graviton, Kalb–Ramond field and dilaton, two gauge bosons from the mixed components of the metric and two-form, and the radius modulus of a string compactification. The two vector charges are the Cartan charges of $U(1{)}_L\times U(1{)}_R$.

Consider instead the following charged states:Every state satisfies compact-circle closed-string level matching. The single noncompact string oscillator supplies a vector polarization. At the massless point, the level-one Virasoro constraint imposes transverse polarization and its physical null string state identifies polarizations differing by the noncompact momentum. Thus these are genuine lower-dimensional vector states. Their squared mass isThus four additional charged massless spin-one particles appear atFor $(N_L,N_R)=(0,1)$ the charges have $p_R=0$, $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$; for $(1,0)$ they have $p_L=0$, $p_R=\pm 2/\sqrt{{\alpha }^{\prime }}$. This identifies the two pairs of charged roots.

The enhanced gauge group can be derived rather than just inferred from counting. At the self-dual circle, the chiral compact boson satisfies $Y_L(z)Y_L(u)\sim -({\alpha }^{\prime }/2)\log (z-u)$. The self-dual circle current algebra has currentsThe exponentials have conformal weight ${\alpha }^{\prime }{p}_L^2/4=1$. Free-boson contractions give the operator product expansionsThey form the level-one SU(2) Lie algebra, and there is an independent right-moving copy. The string vertex operators $J_L^a\bar{\partial }{X}^{\mu }$ and $\partial X^{\mu }{J}_R^a$ give the six massless vectors. ConsequentlyChanging $R$ gives the four charged vectors the mass found above, consistent with the radius deformation as a Higgs mechanism.

For completeness, the enhanced spectrum also has scalar fields. Using a compact rather than noncompact oscillator in the four singly excited charged families gives four scalars. At $(N_L,N_R)=(0,0)$, closed-string level matching requires $nw=0$; the mass equation then gives the four charges $(\pm 2,0),(0,\pm 2)$. These exceptional massless ground states of a bosonic circle give another four scalars. Together with the radius scalar they form nine scalars in the $(3,3)$ representation, in addition to the neutral dilaton. The ever-present neutral bosonic tachyon still has $M^2=-4/{\alpha }^{\prime }$. The enhancement establishes the additional massless vectors, not stability of the entire bosonic vacuum.

The RNS string adds worldsheet Majorana fermions ${\psi }^{\mu }$ to the coordinate fields $X^{\mu }$. The new fields are spinors on the worldsheet but vectors under the target Lorentz group; this distinction matters when interpreting their oscillator states. In a flat target, a convenient Lorentzian convention for the free gauge-fixed action iswhere ${\rho }^a$ are two-dimensional gamma matrices. The bosonic and fermionic kinetic terms are related by worldsheet supersymmetry. Before fixing local worldsheet supersymmetry, one introduces a worldsheet gravitino along with the worldsheet metric. After superconformal gauge fixing, the residual matter constraints are the Virasoro constraints and the constraints generated by the superconformal current $G\propto \psi \cdot \partial X$.

Each real worldsheet Majorana fermion contributes $1/2$ to the chiral central charge, while each coordinate contributes one. The reparameterization bc system contributes $-26$, and fixing local worldsheet supersymmetry also produces commuting superconformal ghosts of weights $(3/2,-1/2)$, with central charge $11$. ThusIn light-cone gauge in string theory there are eight physical transverse coordinate fields and eight transverse worldsheet Majorana fermions.

On the doubled spatial interval of an open string, the two allowed fermionic periodicities areThey arise from the relative signs of the left-right fermion gluing conditions at the two ends. Thus the Ramond sector has integer-mode oscillators $d_n$, including $d_0$, and the Neveu–Schwarz sector has half-integer oscillators $b_r$, $r\in \mathbb{Z}+1/2$, without zero modes. In either sector,A closed string has independent left- and right-moving periodicities, giving four sector choices rather than two.

The Ramond zero-mode Clifford algebra isHence its ground states carry a Spinor representation of the Lorentz group. The zero-mode supercurrent constraint reduces on a ground state to the Dirac equation $p_{\mu }{\Gamma }^{\mu }u=0$. In the Neveu–Schwarz sector the oscillator vacuum instead carries a scalar representation. Consequently the open string NS sector describes spacetime bosons, while the R sector describes spacetime fermions. Exciting a target-vector oscillator does not change this statistics assignment.

The string intercepts follow from the transverse zero-point energy. Per boson the integer-mode contribution is $\frac{1}{2}\zeta (-1)=-1/24$. Per fermion it is the negative of the corresponding half-sum. Since $\zeta (-1,1/2)=1/24$, the eight transverse pairs giveThus $a_{\mathrm{NS}}=1/2$, $a_{\mathrm{R}}=0$, and the open string mass formula is ${\alpha }^{\prime }{M}^2=N-a$. This distinguishes the lowest states explicitly:

The supersymmetric GSO projection keeps odd oscillator fermion number in the NS sector. Defining $F_{\mathrm{osc}}$ so that the NS vacuum has $F_{\mathrm{osc}}=0$, its projector isIt removes the scalar tachyon and keeps the massless vector. In the R sector, fermion parity contains the zero-mode chirality operator, so one may writeHere $F_{\mathrm{osc}}$ counts nonzero-mode fermionic excitations. On the ground states, this keeps one Majorana-Weyl spinor chirality: 16 real covariant components, or eight on-shell polarizations after the Dirac equation. The projected massless open-string states are an eight-polarization gauge vector and an eight-polarization gaugino. The compatible projected tower exhibits spacetime supersymmetry; unprojected worldsheet supersymmetry by itself did not eliminate the tachyon or equalize these massless counts.

For a closed string, both chiral zero-mode constraints must hold:This shifted closed-string level matching is particularly important in the mixed sectors. The lowest states and their spacetime nature are as follows.

For type IIA superstring theory, the two R sector ground-state chiralities are opposite. The Ramond–Ramond potentials can be represented by $C_1,C_3$, with transverse counts $8+56=64$. The two mixed-sector gravitinos have opposite chiralities, and the ten-dimensional theory is nonchiral. For type IIB superstring theory, the two R sector chiralities agree; the Ramond–Ramond potentials are $C_0,C_2,C_4$, with self-dual five-form field strength, giving $1+28+35=64$. Its two gravitinos have the same chirality. Thus both projected type II theories have 128 massless bosonic and 128 massless fermionic polarizations and no tachyon, while their chiralities and form spectra differ.

If unoriented open strings are included in the complete theory, the supersymmetric example is type I string theory. Worldsheet orientation reversal projects the closed type IIB superstring theory spectrum and identifies the two mixed sectors; the surviving closed fields include the graviton, dilaton, a Ramond–Ramond potential $C_2$, and one gravitino and dilatino. The open states form vector and gaugino multiplets with Chan-Paton factors; tadpole and anomaly cancellation select $SO(32)$. This is an additional orientation projection, not part of the GSO projection itself. The heterotic string instead pairs one supersymmetric chiral side with a bosonic chiral side carrying an internal gauge sector; it does not have a Ramond choice on both sides.

Use a mostly-plus target metric tensor and the standard open-string mode expansion convention ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{p}^{\mu }$. The integer string oscillator level is $N={\sum }_{n\ge 1}{\alpha }_{-n}\cdot {\alpha }_n$, whose eigenvalues are nonnegative integers. Then $L_0=N+{\alpha }^{\prime }{p}^2$, so the string mass-shell condition givesThe ground state is a tachyon, and the level-one state is massless. The physical string state conditions in old covariant string quantization areOnly the positive-mode Virasoro constraints annihilate a ket; their negative-mode partners annihilate the corresponding bra. Imposing both signs on every ket would contradict the nonzero Virasoro central extension.

The central charge comes from normal ordering the quadratic oscillator expressions. Classically their Poisson brackets have no central term. Quantum mechanically, moving annihilation operators through creation operators produces double contractions. From the oscillator commutator one obtains $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$, which gives the noncentral part $(m-n)L_{m+n}$.

The coefficient is fixed by the free-boson Virasoro central term. On a formal zero-momentum vacuum, for $m>0$,The two contraction pairings giveHere ${\eta }_{\mu \nu }{\eta }^{\mu \nu }=D$; the time coordinate contributes one, just as each spatial coordinate does. Comparing with the Virasoro algebra provesThis counts covariant target coordinates. It is distinct from the $D-2$ physical transverse polarizations. Including the central charge of reparameterization ghosts gives $c_{\mathrm{total}}=D-26$; cancellation of the worldsheet Weyl anomaly gives the usual critical dimension of the bosonic string $D=26$.

A general level-one state isThe zero-mode constraint requires $k^2=0$. Since $[L_1,{\alpha }_{-1}^{\mu }]={\alpha }_0^{\mu }$, the remaining nontrivial positive-mode constraint is $k\cdot ϵ=0$; all $L_m$ with $m\ge 2$ act trivially at this level. Thus the polarization vector is transverse to a nonzero null momentum.

There is also a null string state $L_{-1}|0;k⟩=\sqrt{2{\alpha }^{\prime }}k\cdot {\alpha }_{-1}|0;k⟩$. It is physical when $k^2=0$ and orthogonal to every physical polarization. The null-state quotient of a string therefore identifiesChoose $k=(E,E,0,\dots ,0)$. Transversality sets ${ϵ}^1={ϵ}^0$, and the null-state identification removes this common component. The remaining $D-2$ spatial components have positive norm. Hence the physical level-one states form a massless vector with $D-2$ polarizations, or 24 at the critical dimension. The removed component is gauge redundancy. States violating the constraints, including independent timelike negative-norm polarizations, are unphysical; physical null states are removed by the quotient rather than counted as additional particles.

The string sigma model in a spacetime metric is the Polyakov actionTake conformal gauge $h_{ab}=\mathrm{diag}(-1,1)$, and choose spatial period $2\pi$ for the closed string. Substitution of the plane-wave metric givesThe stipulated light-cone gauge in string theory $X^{+}=\tau$ sets ${\dot{X}}^{+}=1$ and $X^{\prime +}=0$. Its longitudinal term $-T\int {\dot{X}}^{-}$ is a total time derivative. The transverse string sigma model in an isotropic plane wave is thereforeThe resulting Euler-Lagrange equations are the massive wave equationsThese conventions fix the mass term directly from $X^{+}=\tau$. More generally $X^{+}=\kappa \tau$ would replace $m$ by $|m\kappa |$, with $p^{+}=2\pi T\kappa$ for this spatial period.

Use the real orthonormal Fourier basisFor the plane-wave string mode frequencies, let ${\omega }_0=|m|$ and ${\omega }_{n,c}={\omega }_{n,s}=\sqrt{n^2+m^2}$. The complete real mode expansion for nonzero frequencies isEvery term obeys the wave equation and is periodic. Orthonormality and completeness let its coefficients match arbitrary appropriate initial position and velocity data. For $m=0$, replace the zero-mode bracket by $q_0^I(0)+\tau p_0^I/T$; the zero mode becomes a free particle rather than an oscillator.

The canonical momentum density is ${\Pi }_I=T{\dot{X}}^I$. The transverse Hamiltonian isWrite $X^I={\sum }_a{q}_a^I{f}_a$ and $p_a^I=T{\dot{q}}_a^I$. Since $\int f_a{f}_b={\delta }_{ab}$ and $\int f_a^{\prime }{f}_b^{\prime }=n_a^2{\delta }_{ab}$, substitution gives the plane-wave string mode HamiltonianThis is a collection of independent harmonic oscillators. The last equality follows by substituting the sine-and-cosine time dependence; each oscillator energy is constant.

For canonical quantization of each nonzero-frequency mode, putThen $H_{\perp }={\sum }_{I,a}{\omega }_a(a_a^{I†}{a}_a^I+1/2)$. Opposite travelling-wave oscillator combinations give the equivalent formFor $m\ne 0$, the formal zero-point energy is $E_0=\frac{24}{2}[|m|+2{\sum }_{n\ge 1}\sqrt{n^2+m^2}]$. At $m=0$, replace the zero-mode occupation term by ${\sum }_I(p_0^I{)}^2/(2T)$. The zero-point expression is divergent and needs a regulator or a specified normal ordering convention; no mass-independent intercept has been assumed.

The remaining Virasoro constraints determine the longitudinal coordinate:Periodicity of $X^{-}$ imposes ${\int }_0^{2\pi }{\Pi }_I{X}^{\prime I}d\sigma =0$, the closed-string level matching condition. Labelling the travelling oscillators by their opposite worldsheet momenta gives ${\sum }_{n\ge 1,I}n(N_n^I-{\tilde{N}}_n^I)=0$. The light-cone energy $p^{-}=-p_{+}$ equals $T\int ({\dot{X}}^{-}+m^2{X}^I{X}^I)d\sigma =H_{\perp }$, consistent with the positive mass term in the Hamiltonian.