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A Bogoliubov vacuum is annihilated by the quasiparticle linear operators that diagonalize a stable quadratic bosonic Hamiltonian. If $a_k=u_k{b}_k-v_k{b}_{-k}^{†}$, its original-mode occupation number is $|v_k{|}^2$. It is generally a multimode squeezed vacuum rather than the original Fock vacuum. In infinitely many modes, existence in the original Fock space requires a separate convergence analysis.

This is the symmetric-particle construction of Fock space.

For one-particle Hilbert space ${\mathcal{H}}_1$, the bosonic Fock space isCreation operators add one symmetrized particle and annihilation operators remove one.

A bounded invertible bosonic Bogoliubov transformation is unitarily implementable in the usual bosonic Fock space precisely when its creation-mixing part is a Hilbert-Schmidt operator. Without this condition, the transformation still preserves the algebraic canonical commutation relations, but its two vacua need not be vectors in a common Fock space. Equal nonzero squeezing in countably many independent modes is a simple counterexample.

For $[a,a^{†}]=1$, the occupation number operator $a^{†}a$ has eigenvalues $0,1,2,\dots$ on the bosonic oscillator Fock space.

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.

The historical interpretation fills the negative-energy electron states of the Dirac equation and uses the Pauli exclusion principle to prevent additional electrons from falling into them. Removing an electron of energy $-E$ and charge $-e$ gives a hole of excitation energy $E$ and charge $+e$, interpreted as a positron. A quantum field theory instead treats negative-frequency field coefficients as independent antiparticle creation operators in a Fock space, with positive excitation energies after normal ordering. No literal infinite material sea is needed. Bosonic antiparticles also exist, although a filled-sea exclusion argument cannot stabilize bosonic negative-energy levels.

For a one-particle Hilbert space $H$, the bosonic or fermionic Fock space is ${\mathcal{F}}_{\pm }(H)={⨁}_{n=0}^{\infty }{H}^{{\otimes }_{\pm }n}$, using symmetric or antisymmetric tensor powers. Its zeroth summand is the one-dimensional vacuum sector. Creation operators and annihilation operators change particle number. The bosonic Fock space and fermionic Fock space implement the corresponding exchange statistics.

For a complex Klein-Gordon field, $j^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })$ is conserved. A mode $\varphi =N{e}^{-iEt+i\mathbf{p}\cdot \mathbf{x}}$ has $j^0=2E|N{|}^2$, so unrestricted positive- and negative-frequency solutions do not have a positive probability norm under this current. In the quantized complex field, the associated conserved quantity is charge: particles and antiparticles contribute with opposite signs while both have positive excitation energy and positive Fock space norm.

The canonical commutation relations imply $[N,{\alpha }_{-k}^i]=k{\alpha }_{-k}^i$ for $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$. With zero level assigned to the oscillator vacuum, a Fock space basis state containing $r_{ki}$ creation operators of each mode has level ${\sum }_{k,i}k{r}_{ki}$. The level is therefore a nonnegative integer; it counts oscillator excitation weighted by frequency, not simply the number of creation operators.

Time-coordinate oscillators produce negative norms in the covariant indefinite Fock space. A physical negative-norm state would violate probabilistic unitarity. Such ghosts differ from the anticommuting worldsheet ghost fields introduced by gauge fixing.

A one-particle state belongs to the one-excitation sector of a Fock space. In a momentum basis with $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, the wavepacket $\int d^{3}pf(\mathbf{p})a_{\mathbf{p}}^{†}|0⟩/(2\pi {)}^3$ has norm squared $\int d^{3}p|f(\mathbf{p}){|}^2/(2\pi {)}^3$. Sharp-momentum states have distributional rather than unit normalization; factors of $\sqrt{2E_{\mathbf{p}}}$ convert to relativistic normalization of a one-particle state.

The operator formalism describes a quantum system using states in a Hilbert space and observables represented by self-adjoint operators with suitable domains on that Hilbert space. Evolution and correlation functions are computed from Hamiltonian operators and their time-ordered products. For free quantized fields, the oscillator construction gives a Fock space. The operator-path-integral equivalence relates these matrix elements to integrals over field histories.

A relativistic theory must accommodate the energy-momentum relation $E^2={\mathbf{p}}^2+m^2$, causal interactions and the possibility of particle creation. Promoting the squared relation to a wave equation gives the Klein-Gordon field equationA plane wave $\varphi =N{e}^{-iEt+i\mathbf{p}\cdot \mathbf{x}}$ hasThe positive and negative branches both solve the local differential equation. Its being second order in time means that specifying only a wavefunction, as in the nonrelativistic Schrödinger equation, is not the complete initial datum.

For a complex scalar, the conserved Klein-Gordon inner product comes from the currentFor the plane wave, $j^0=2E|N{|}^2$, which is negative on the negative-frequency branch. Thus this conserved density cannot be a positive single-particle probability density for arbitrary Klein-Gordon solutions. Merely replacing it by $|\varphi {|}^2$ does not supply the missing conserved relativistic probability law. The indefinite Klein-Gordon charge density is an obstacle to that naive one-particle interpretation, not evidence that a consistent scalar field theory has negative probabilities.

The Dirac equation is first order,The gamma-matrix anticommutation relations imply $H_D^2=({\mathbf{p}}^2+m^2)I$, so its spectrum again contains $\pm E_{\mathbf{p}}$. But its conserved Dirac current has density $j^0={\psi }^{†}\psi \ge 0$. The Dirac equation solves the density-sign difficulty while leaving both energy branches. Treating every negative-energy solution as an ordinary accessible one-electron state would leave the energy unbounded below and permit radiative descent without a stable ground state.

The historical Dirac sea filled the negative-energy electron states and interpreted holes as positive-energy antiparticles. It motivates the Positron but is not the general modern explanation: bosons do not obey an exclusion principle and cannot be stabilized by filling a sea. Quantum field theory instead promotes the fields to operators and interprets negative-frequency modes using antiparticle creation operators.

For a complex scalar field, schematically,Bosonic commutation relations give a normal-ordered Hamiltonian and conserved charge of the formBoth particle species have positive excitation energies. The sign-changing scalar current now represents charge, rather than particle probability; opposite charges are physically allowed. The ordinary positive Fock space norm gives nonnegative probabilities. A real scalar field is its own antiparticle and has no corresponding complex-field U(1) charge.

For the Dirac field, the analogous expansion contains electron annihilation operators and positron creation operators. Fermionic anticommutation relations turn the negative-frequency contribution into a positive antiparticle excitation energy after the vacuum constant is removed. The Pauli exclusion principle and relativistic locality are thereby incorporated in a field description, rather than a postulated infinite population of physical negative-energy electrons. The normal-ordering step removes an additive vacuum constant; it does not by itself settle the separate gravitational vacuum-energy problem.

Interactions can change particle number while conserving total energy, momentum and charge. Trying to localize a particle on its Compton wavelength scale requires momenta of order its rest mass, where accessible interactions can involve pair creation. A fixed-particle wavefunction is consequently not a general description of such processes. Creation and annihilation operators act between different particle-number sectors, and field interactions describe scattering and production within one framework.

Finally, relativistic causality concerns observables: spacelike separated local bosonic observables commute, while fermionic fields satisfy the appropriate anticommutation conditions. This microcausality prevents a controllable influence through spacelike separated local operations. Free relativistic one-particle theories remain useful, and low-energy fixed-number approximations can be excellent. What fails is the naive insistence that unrestricted relativistic interactions, positive probabilities, stability and arbitrary particle production all fit an unchanged one-particle Schrödinger picture.^[ref]

Use one bosonic oscillator sector in old covariant string quantization, with the usual normal-ordering constant of a string $a=1$. Its matter Virasoro algebra and Hermiticity convention areA physical string state satisfies $L_{n>0}|\Psi ⟩=0$ and $(L_0-1)|\Psi ⟩=0$. Let $|k⟩$ be the oscillator vacuum with $L_{n>0}|k⟩=0$ and $L_0|k⟩=h|k⟩$. For a level-two descendant, the zero-mode condition gives $h+2=1$, hence $h=-1$. The vacuum carrying this momentum is not itself a physical level-zero tachyon state; its descendant is the candidate physical state. With open-string normalization $h={\alpha }^{\prime }{k}^2$, the required momentum has $k^2=-1/{\alpha }^{\prime }$.

The positive-mode conditions can be computed without assuming the answer. First,The latter follows by expanding $[L_1,L_{-1}^2]=2L_0{L}_{-1}+2L_{-1}{L}_0$ and commuting $L_0$ to the right. Therefore $L_1(L_{-2}+\mu L_{-1}^2)|k⟩=0$ gives $3-2\mu =0$, or $\mu =3/2$. Similarly,At $h=-1$ and $\mu =3/2$ the second condition is $(d/2-13)|k⟩=0$. Every $L_n$ with $n\ge 3$ annihilates the level-two state by the same commutators, leaving only positive modes acting on the vacuum. ThusThis is the level-two scalar Virasoro null state.

For the norm, take the vacuum norm to be one, suppressing the continuum momentum delta function. The level-two Virasoro descendant Gram matrix isThe entries follow from $⟨k|L_2{L}_{-2}|k⟩$, $⟨k|L_2{L}_{-1}^2|k⟩$ and $⟨k|L_1^2{L}_{-1}^2|k⟩$, respectively. At $h=-1$,Consequently the physical state at $d=26$ has norm squared zero. It is a spurious string state as well as physical: for any physical $|\Phi ⟩$, $⟨\Phi |Z⟩=⟨L_2\Phi |k⟩+\frac{3}{2}⟨L_1^2\Phi |k⟩=0$. The null-state quotient of a string therefore identifies it with zero. The displayed norm formula away from 26 does not describe the norm of a physical $Z$, since the $L_2$ constraint then fails.

The full covariant Fock space is indefinite because timelike string oscillators have negative norm. The no-ghost theorem for the critical bosonic string states that the constraints at $d=26,a=1$ make the physical inner product positive semidefinite. Every physical state can be represented by a positive transverse state plus a physical null state. Modding out the latter leaves a positive physical space, rather than retaining negative-norm states and simply ignoring them.

One outline of the proof uses DDF operators. Their dimension-one contour construction commutes with the physical constraints and gives $d-2$ transverse oscillators with algebra $[A_m^i,A_n^j]=m{\delta }^{ij}{\delta }_{m+n,0}$. Their oscillator states have manifestly positive norm. The covariant constraints and spurious descendants can be organized triangularly in the remaining timelike and longitudinal oscillators. At the critical dimension the completeness step shows that their remaining physical contribution is null, so the 24 transverse DDF oscillators represent the whole quotient.

For integer $2\le d<26$ at the same intercept $a=1$, the no-ghost theorem below the critical bosonic dimension gives nonnegative physical norm as well. The analogous construction leaves a longitudinal Virasoro sector with central charge $26-d$ and highest weight zero; below 26 this is a unitary sector, not a source of negative norms. At 26 it becomes null. Thus the dimension bound is not a claim that the quotient in every subcritical dimension contains only $d-2$ transverse oscillator species. The argument assumes the ordinary nonzero-momentum sector where the reference null direction for the DDF construction can be chosen. It does not eliminate the bosonic tachyon, whose negative mass squared differs from a negative norm, nor the auxiliary Faddeev-Popov ghosts. Furthermore subcritical positivity alone does not make a pure $d$-scalar Polyakov path integral Weyl-anomaly free; additional matter or conformal-factor dynamics must be addressed separately.

The Nambu–Goto action is invariant under worldsheet diffeomorphisms. Its canonical Nambu–Goto phase-space constraints areIn conformal gauge, $\Pi =T\dot{X}$, so these become $(\dot{X}\pm X^{\prime }{)}^2=0$. 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, $0\le \sigma \le \pi$. Its open-string mode expansion isCanonical quantization gives the string oscillator relationsWith ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$, the worldsheet stress-energy tensor obeysSubstituting the open-string mode expansion into this quadratic expression derives the Virasoro generators: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 givesApplying 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 $1\le r\le m-1$ contribute, givingHere $D$ is the target spacetime dimension; the timelike free boson also contributes one to the central charge, since a double contraction contains two metric factors. ThusThis derives both the Virasoro generators and their quantum anomaly.

In old covariant string quantization, the physical-state Virasoro conditions for an open string arewhere $a$ 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,For a closed string, there are two commuting Virasoro algebras. With uncompactified zero modes, $L_0={\alpha }^{\prime }{p}^2/4+N_R$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+N_L$. Imposing both positive-mode conditions and both zero-mode conditions gives the closed-string level matching $N_L=N_R$ and $M^2=4(N_L-a)/{\alpha }^{\prime }$.

Covariance initially retains the timelike string oscillators. For example, ${\alpha }_{-1}^0|0;p⟩$ 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 ${\sum }_{n>0}{L}_{-n}|{\chi }_n⟩$. Its inner product with every physical string state vanishes, by $L_n|\psi ⟩=0$ and $L_n^{†}=L_{-n}$. Whenever such a state is itself physical, it is a null string state and represents a gauge redundancy, not an extra polarization. Consequently 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.

Use $[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m){\delta }_{m+n,0}$ and $L_n^{†}=L_{-n}$. A normalized highest-weight vector obeys $L_0|h⟩=h|h⟩$ and $L_n|h⟩=0$ for $n>0$. Its Virasoro descendants are products of negative modes. Their inner products are determined entirely by $c,h$ and the Virasoro algebra; for example $\Vert L_{-1}|h⟩{\Vert }^2=2h$, so unitarity requires $h\ge 0$. An irreducible unitary highest-weight Virasoro module is the Virasoro Verma module quotient by its null submodules, and its surviving inner product must be positive definite.

The classification for $c<1$ is the Unitary Virasoro discrete series:where $m$ is an integer at least $2$, $1\le r\le m-1$, and $1\le s\le m$. The identification $(r,s)\sim (m-r,m+1-s)$ leaves $m(m-1)/2$ distinct modules. The two generating singular-vector levels are $rs$ and $(m-r)(m+1-s)$; all their Virasoro descendants are removed in the irreducible quotient. For $m=2$ the only module is the trivial $c=h=0$ module; for $m\ge 3$ these are the nontrivial unitary discrete-series modules. There are no other irreducible unitary highest-weight representations with $c<1$. This is the classification theorem, rather than a condition inferred only from finitely many descendant norms.

At $c=1/2$, $m=3$ and the possible weights areConstruct them with a single free chiral Majorana fermion:The free chiral Majorana fermion conformal field theory has $T=-:\psi \partial \psi :/2$. In the Neveu–Schwarz sector, $r\in \mathbb{Z}+1/2$, while in the Ramond sector, $r\in \mathbb{Z}$. With positive modes moved to the right and the zero-mode product antisymmetrized, the corresponding generators areThe oscillator relation gives $[L_n,b_r]=-(r+n/2)b_{n+r}$. The stress-tensor operator product expansion, or the reordering of the quadratic generators, then gives the Virasoro algebra with $c=1/2$. One can check both constants directly in the ground states. In the NS sector,In the R sector, $b_0^2=1/2$ and $L_{-1}|R⟩=\frac{1}{2}{b}_{-1}{b}_0|R⟩$, whose norm squared is $1/8$. The relation $[L_1,L_{-1}]=2L_0$ requires $2h_R=1/8$, so $h_R=a=1/16$. Thus the Ramond sector zero-mode shift is essential.

The three constructions in Ising Virasoro modules from a Majorana fermion are explicit. The NS vacuum is a highest-weight vector of weight $0$. The state $b_{-1/2}|0⟩$ is a highest-weight vector of weight $1/2$: applying any positive $L_n$ gives $-(n/2-1/2)b_{n-1/2}|0⟩=0$, including the zero coefficient for $n=1$. Since $L_n$ is quadratic in fermions, the two cyclic modules lie in the even and odd fermion parity sectors, respectively. For R, choose a normalized ground state annihilated by all $b_n$ with $n>0$ and with $b_0|R⟩=\pm |R⟩/\sqrt{2}$. All positive $L_n$ annihilate it and its weight is $1/16$. The two zero-mode sign choices give isomorphic Virasoro modules; one choice suffices to construct the third module.

The oscillator inner products are positive, and each cyclic highest-weight representation just constructed is irreducible. Indeed any nonzero proper invariant graded submodule would have a lowest-energy vector $w$ of weight greater than the cyclic ground weight. Every positive $L_n$ annihilates $w$; thus $w$ is orthogonal to every negative-mode descendant of the original ground vector, by moving the modes across the inner product. It is also orthogonal to that ground vector by its different $L_0$ eigenvalue. Since those Virasoro descendants span the cyclic module, $w$ would have zero norm, a contradiction. This proves irreducibility directly inside the positive fermionic Fock space.

In fact the cyclic modules fill the entire even NS, odd NS and chosen R Fock spaces. Otherwise an orthogonal remaining lowest vector would be the ground vector of a further unitary highest-weight Virasoro module; its weight would lie in ${\mathbb{Z}}_{>0}$, $1/2+{\mathbb{Z}}_{>0}$ or $1/16+{\mathbb{Z}}_{>0}$, respectively. None belongs to the classified list $\{0,1/2,1/16\}$. As a check, their Virasoro characters, defined by ${\chi }_h(q)=\mathrm{Tr}{q}^{L_0-c/24}$, areThe NS signs project onto even and odd occupation number; the R product uses the single chosen ground state. Null vectors of the abstract Virasoro Verma modules are automatically zero in this construction. In particular $L_{-1}|0⟩=0$, and the weight-$1/2$ and weight-$1/16$ ground states obey, respectively, $(L_{-2}-\frac{3}{4}{L}_{-1}^2)|1/2⟩=0$ and $(L_{-2}-\frac{4}{3}{L}_{-1}^2)|1/16⟩=0$, as is also verified by their level-two Gram matrices.

Use target signature $(-,+,\dots ,+)$ and a closed-string spatial coordinate $0\le \sigma \le 2\pi$. Write $Y$ for the compact coordinate and $X^A$ for the noncompact coordinates. In conformal gauge, the wave equation factorizes, so every coordinate is a sum of a function of $\tau +\sigma$ and a function of $\tau -\sigma$. The closed-string boundary conditions areThus the derivatives of each chiral part are periodic and have integer Fourier modes; only their worldsheet zero modes can produce the winding displacement. Integrating these Fourier series gives the closed-string mode expansionThe oscillator reality conditions are ${\alpha }_m^{†}={\alpha }_{-m}$ and similarly for the tilded modes. The coefficient of $\tau$ follows by integrating the canonical momentum density $\dot{Y}/(2\pi {\alpha }^{\prime })$. Single-valued wavefunctions of the center coordinate $y\sim y+2\pi R$ have momentum $n/R$, with $n\in \mathbb{Z}$. Equivalently, the compact chiral worldsheet zero modes are ${\alpha }^{\prime }{p}_L(\tau +\sigma )/2$ and ${\alpha }^{\prime }{p}_R(\tau -\sigma )/2$, whereHere tildes denote the left-moving sector. This convention will also fix the sign in compact-circle closed-string level matching.

Choose a noncompact light-cone pair and impose light-cone gauge in string theory $X^{+}=x^{+}+{\alpha }^{\prime }{p}^{+}\tau$, with $p^{+}\ne 0$. The Virasoro constraints $({\partial }_{\pm }X{)}^2=0$ then solve for ${\partial }_{\pm }{X}^{-}$ in terms of the transverse coordinates. For example, with ${\partial }_{\pm }={\partial }_{\tau }\pm {\partial }_{\sigma }$, one has ${\partial }_{\pm }{X}^{-}=({\partial }_{\pm }{X}^i{)}^2/(2{\alpha }^{\prime }{p}^{+})$. Thus there are two independent transverse oscillator Fock spaces, including $Y$ among the transverse coordinates. Quantization gives $[{\alpha }_m^i,{\alpha }_k^j]=m{\delta }^{ij}{\delta }_{m+k,0}$, and the same relation for the tilded oscillators, with the two sectors commuting. Lorentz consistency gives the critical dimension of the bosonic string $D=26$ and string intercept one in each sector. Accordinglycount 24 transverse oscillator species. The two zero-mode constraints give the lower-dimensional bosonic string mass spectrumTaking their difference and average yieldsEvery choice of transverse string oscillators and integer charges satisfying this closed-string level matching is a physical string state in light-cone gauge. The eliminated longitudinal oscillators do not supply additional polarizations.

To classify all massless states, positivity of the two compact terms implies $N+\tilde{N}\le 2$. When the sum is two, masslessness forces $n=w=0$, and closed-string level matching then forces $N=\tilde{N}=1$. The states ${\alpha }_{-1}^i{\tilde{\alpha }}_{-1}^j|0⟩$ have $24^2=576$ polarizations. In 25 noncompact dimensions, their decomposition gives a graviton, a Kalb–Ramond field, a dilaton, two Abelian gauge vectors from the metric and two-form with one compact index, and the radius scalar field. Their physical polarization counts areThe first two numbers are the symmetric traceless second-rank tensor and antisymmetric second-rank tensor of the massless little group $SO(23)$.

When $N+\tilde{N}=1$, closed-string level matching requires $|nw|=1$. The compact contribution obeysand equality occurs precisely at $R=\sqrt{{\alpha }^{\prime }}$. The charges $n=w=\pm 1$ have $(N,\tilde{N})=(1,0)$; the charges $n=-w=\pm 1$ have $(N,\tilde{N})=(0,1)$. Each charge has 24 polarizations supplied by its single excited sector. At a general radius these have

Finally, $N=\tilde{N}=0$ requires $nw=0$. The exceptional massless ground states of a bosonic circle occur atEach charge is a scalar field. These special radii must be included if “general radius” is to mean a complete classification, rather than only a generic radius. At a generic radius the massless spectrum is the neutral 576-state spectrum above; at exceptional radii one adds exactly the states just listed. The neutral ground state remains tachyonic.

The mapsends $p_L$ to $p_L$ and $p_R$ to $-p_R$. Reversing all right-moving compact string oscillators, ${\alpha }_m^Y\mapsto -{\alpha }_m^Y$, preserves their commutators and oscillator number; the left-moving oscillators are unchanged. Both the mass formula and compact-circle closed-string level matching are therefore unchanged, with a one-to-one map of the complete oscillator basis. This proves T-duality of the full spectrum, including winding, massive and tachyonic states.

At the self-dual circle the four single-oscillator charge families give $4\cdot 24=96$ new massless states. Each family decomposes into a 25-dimensional gauge vector with 23 polarizations and one scalar field. There are also four massless ground states, with charges $(\pm 2,0)$ and $(0,\pm 2)$. Thus the massless spectrum at the bosonic self-dual circle hasThe gauge interpretation follows directly from the chiral compact-boson operators. With $⟨Y_L(z)Y_L(0)⟩=-{\alpha }^{\prime }\log z/2$, the exponential $e^{ik{Y}_L}$ has conformal weight ${\alpha }^{\prime }{k}^2/4$. At the self-dual radius,have weight one and satisfy $J^3(z)J^3(0)\sim 1/(2z^2)$, $J^3(z)J^{\pm }(0)\sim \pm J^{\pm }(0)/z$, and $J^{+}(z)J^{-}(0)\sim z^{-2}+2J^3(0)/z$. These are the level-one $SU(2)$ current relations; the right sector supplies another copy. Multiplying these currents by a noncompact oscillator produces six gauge vectors in total. The nine scalar string vertex operators $J_L^a{J}_R^b$ form the $(3,3)$ representation, accounting for the radius scalar field and eight new scalar fields. The remaining neutral fields are the graviton, two-form and dilaton.

A radius deformation as a Higgs mechanism is the expectation value of the Cartan–Cartan scalar string vertex operator $J_L^3{J}_R^3\propto \partial Y_L\bar{\partial }{Y}_R$. Its stabilizer is $U(1{)}_L\times U(1{)}_R$, soThe four compact single-oscillator scalar field polarizations become longitudinal polarizations of the four massive vectors, exactly as in the Higgs mechanism. The other four extra scalar fields have masses $4/R^2-4/{\alpha }^{\prime }$ or $4R^2/{\alpha }^{\prime 2}-4/{\alpha }^{\prime }$ and need not remain massless; one pair becomes tachyonic on either side of the self-dual point. Together with the original bosonic tachyon, this means that the gauge-symmetry interpretation is formal and does not assert a stable bosonic-string vacuum.

Distinguish target-space light-cone coordinates from the worldsheet $\pm$ labels. In target space take $X^{\pm }=(X^0\pm X^{25})/\sqrt{2}$, so the Minkowski metric is $d{s}^2=-2d{X}^{+}d{X}^{-}+{\sum }_i(d{X}^i{)}^2$. After conformal gauge has been chosen, the residual transformations of $\tau +\sigma$ and $\tau -\sigma$ may be used, on a patch with $p^{+}\ne 0$, to makeFor an open string on $0\le \sigma \le \pi$, the conventional normalization is $\kappa =2{\alpha }^{\prime }{p}^{+}$. For the closed-string convention of the preceding solution, $\kappa ={\alpha }^{\prime }{p}^{+}$. In either case, the two Virasoro constraints readSince ${\partial }_{\pm }{X}^{+}=\kappa /2$, they determine the longitudinal coordinate:Its oscillator content is fixed by the transverse coordinates, leaving $d-2$ independent fields. This is the light-cone gauge in string theory; it solves the constraints rather than imposing them afresh on each transverse state.

For Neumann boundary conditions at both endpoints, the open-string expansion isThe transverse string oscillators satisfy $[{\alpha }_m^i,{\alpha }_n^j]=m{\delta }^{ij}{\delta }_{m+n,0}$, $i,j=1,\dots ,24$, and ${\alpha }_{n>0}^i|0,p⟩=0$. Define $L_n^{\perp }=\frac{1}{2}{\sum }_m:{\alpha }_{n-m}^i{\alpha }_m^i:$ and ${\alpha }_0^{+}=\sqrt{2{\alpha }^{\prime }}{p}^{+}$. The solved longitudinal modes, including the zero-mode constraint, areThe critical bosonic string theory has 24 transverse bosons. Their regulated oscillator vacuum energy is $24\cdot \frac{1}{2}{\sum }_{n\ge 1}n=-1$, since the finite term of ${\sum }_{n\ge 1}n{e}^{-\epsilon n}={\epsilon }^{-2}-1/12+O({\epsilon }^2)$ is $-1/12$. Thus the normal-ordering constant of a string is $a=1$. A normalized transverse Fock space basis for general excited physical string states iswith finitely many nonzero occupation numbers. Arbitrary physical states are superpositions of this basis, with each momentum on its corresponding mass shell. The string level operator and open bosonic string mass spectrum giveAll spatial directions have free endpoints, so this string ends on a space-filling D25-brane. More generally, a D-brane has Neumann boundary conditions along its worldvolume and Dirichlet boundary conditions in its transverse directions; a D$p$-brane has $p$ spatial worldvolume directions. A stack may additionally supply Chan-Paton factors, without changing the oscillator construction above.

To see T-duality of D-brane boundary conditions directly, decompose the compact coordinate as $X=X_L(\tau +\sigma )+X_R(\tau -\sigma )$ and define its dual by $\tilde{X}=X_L-X_R$. Differentiation givesTherefore the original Neumann boundary condition ${\partial }_{\sigma }X=0$ becomes ${\partial }_{\tau }\tilde{X}=0$ at each endpoint. The dual endpoint coordinate is time-independent: the dual condition is Dirichlet. Conversely a fixed endpoint in $X$ becomes Neumann in the dual coordinate. For a circle, the T-duality radius is $\tilde{R}={\alpha }^{\prime }/R$. A wrapped Neumann brane loses one spatial dimension in the dual picture, for example D25 becomes D24. The two dual endpoint constants need not be identical; their positions encode the original Wilson-line data.

For the separated-brane problem, use two static parallel D$p$-branes and no relative angle or background field. Let their displacement in transverse directions be $d^I$, with ${\sum }_I(d^I{)}^2=d^2$. A Dirichlet coordinate contains the classical termThe spatial gradient contributes to the worldsheet HamiltonianParallel branes have integer modes in their NN and DD directions, so their oscillator vacuum energy is still $-1$. The zero-mode Virasoro constraint is consequentlyWith string tension $T=1/(2\pi {\alpha }^{\prime })$, the Open string stretched between D-branes hasThe classical stretching energy is $Td$, while the quantum ground-state energy, when real, isThe vacuum correction is an additive term in energy squared, not a constant subtraction from the classical stretching energy. It follows that the bosonic stretched-string tachyon threshold isBelow this threshold the lowest state is a tachyon: small-momentum modes have imaginary frequency and the background is unstable, rather than having an ordinary real negative ground-state energy. It is massless at the threshold. The assumption of parallel branes is needed for the integer moding used here; relative angles change the vacuum contribution.

Use a free real scalar field obeying a linear Klein-Gordon equation, for example $(\square -{\mu }^2-\xi R)\varphi =0$. This linearity is the assumption that permits mode evolution by a Bogoliubov transformation. Work with a complete set of normalized modes, or normalizable wave packets when the spectrum is continuous. The stationary asymptotic regions supply time translations relative to which positive-frequency solutions and their complex conjugates can be distinguished.

For solutions define the conserved Klein-Gordon inner productHere $\Sigma$ is a Cauchy surface with future-directed normal. Its conservation follows by applying the wave equation to the divergence of its current, provided boundary flux is included or vanishes. Let $u_j$ be a positive-frequency basis in the past and $v_s$ one in the future, each propagated as a solution through the intermediate geometry. Normalize them by $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and $(u_i,u_j^{\ast })=0$, with the corresponding identities for the future basis. The two field expansions areThe canonical commutation relations give $[a_i,a_j^{†}]={\delta }_{ij}$ and $[b_s,b_t^{†}]={\delta }_{st}$. The in-vacuum satisfies $a_j|0_{-}⟩=0$, while the out-vacuum satisfies $b_s|0_{+}⟩=0$. They are generally different vacua.

Fix the coefficient convention by expanding future modes in past modes:The minus sign is due to the negative norm of the conjugate modes. Taking the Klein-Gordon inner product with the field, which is antilinear in its first entry, givesBoth frequency sectors are needed: if $\beta \ne 0$, a future annihilation operator contains a past creation operator. This is the mechanism of particle number from Bogoliubov coefficients, not a classical nonzero mean field.