No-ghost Theorem Research Articles

Remarks on mathematical structure of BRST theories

A number of rigorous results concerning the mathematical structure of quantum systems with the BRST symmetry are presented. For studying model-independent properties of such systems, a generalization of the Strocchi-Wightman theory is suggested, which is a Wightman type theory in the Krein space with an OpJ*-algebra of physical and ghost fields. It is proved that the BRST charge operatorQ is either bounded with σ(Q)={0} or unbounded with σ(Q)=¢. A number of conditions equivalent to the “no-ghost theorem” (which states that the kernel ofQ is non-negative) are derived.

Consequence of local U(1)V x U(1)A symmetry in two dimensions.

We study the two-dimensional U(1){sub {ital V}}{times}U(1){sub {ital A}} model in detail. This model has both local vector U(1) and local axial-vector U(1) gauge symmetries. The original Lagrangian is expressed in terms of some fermion fields and a gauge field. We bosonize fermion fields by the same method as in the Thirring model. The U(1){sub {ital V}}{times}U(1){sub {ital A}} symmetry requires the vanishing of the Schwinger term (or, equivalently, the nilpotency of the Becchi-Rouet-Stora-Tyutin charge). The absence of the Schwinger term leads to some negative-norm states (ghost fields). However, the physical states are positive semidefinite if and only if the number of ghost fields is only one. We show the no-ghost theorem in detail.

BRST symmetry and a general ghost decoupling theorem

A no-ghost theorem is proven in the BRST approach to quantization of a gauge theory. The proof is given in a general and model independent formulation. Two distinct classes of quantum matter states are identified and it is shown that only for one of these classes does the BRST symmetry always ensure ghost decoupling.

No-ghost theorem and Osp(1, 1|2) in gauge theories

We explain how generators of the Osp (1, 1|2) extended BRST algebra can be constructed in a general gauge theory and in an arbitrary gauge, order by order in a systematic expansion. As an example we consider both axial and Coulomb gauge Yang-Mills theories. We relate the construction of physical states to the representation theory of quantum Osp(1, 1|2) and explain why the identification of physical states with Osp(1, 1|2) singlet states emerges. This yields a particularly transparent demonstration of the no-ghost theorem.

Nonexistence of a ground state and breaking of the no-ghost theorem on a squeezed string.

For the squeezed string (closed and open) based on the oscillator formalism, we show (i) that all real numbers are allowed for eigenvalues of the Hamiltonian and (ii) neither the Virasoro conditions nor the physical condition in the Becchi-Rouet-Stora formalism eliminate negative-norm states.

A detailed study of the physical state conditions in covariantly quantized string theories

We offer a detailed account of the Goddard-Thorn proof of the no-ghost theorem which incorporates simplifications presented earlier. We extend these methods to obtain a new calculation of the cohomology of the nilpotent operator Q occuring in BRST quantized strings. We further explain why the apparent doubling of transverse states in the cohomology of Q has no physical significance.

Semi-infinite cohomology and string theory

We develop the theory of semi-infinite cohomology of graded Lie algebras first introduced by Feigin. We show that the relative semi-infinite cohomology has a structure analogous to that of the de Rham cohomology in Kähler geometry. We prove a vanishing theorem for a special class of modules, and we apply our results to the case of the Virasoro algebra and the Fock module. In this case the zero cohomology is identified as the physical subspace of the Fock module and the no-ghost theorem follows. We reveal the profound relation of semi-infinite cohomology theory to the gauge-invariant free string theory constructed by Banks and Peskin. We then indicate the connection between gauge-invariant interacting string theories and the geometric realizations of the infinite-dimensional Lie algebras.

BRS cohomology in string theory and the no-ghost theorem

It is shown that the operator counting the number of non-transverse modes of the bosonic string in 26 dimensions can be expressed as the anticommutator of the BRS charge Q with another operator. As a result it is easy to exhibit the cohomology of Q and express the transverse state projection operator of Brink and Olive in terms of Q.

The bosonic string in D < 26 with and without Liouville fields

It is shown that the Liouville modes for the bosonic string in D < 26 are equivalent to the old Brower modes. Consistent free quantum strings do therefore exist in D < 26 for all values of the Regge intercept α 0 such that α 0 ⩽ 1 for open strings and α 0 ⩽ 2 for closed strings, i.e. for all values allowed by the no-ghost theorem. The properties of the Brower modes restict the allowed Liouville theories as follows: For open strings only Liouville theories with a singularity line at one (or both) end of the interval 0 ⩽ σ ⩽ π are allowed. For closed strings no allowed Liouville theory has been found. Liouville theories with periodic boundary conditions are not allowed.

Covariant quantization of superstrings based on Becchi-Rouet-Stora invariance.

A manifestly covariant quantization of the superstring is presented in the Neveu-Schwarz-Ramond formalism on the basis of the Becchi-Rouet-Stora (BRS) invariance principle. The critical dimension D=10 and the correct Regge intercept are shown to follow from the requirement of the nilpotency ${Q}_{B}$${\mathrm{}}^{2}$=0 of the BRS charge. A modified form of the subsidiary condition of Kugo and Ojima ${Q}_{B}$\ensuremath{\Vert}phys〉=0 to define the physical subspace is sufficient to demonstrate the no-ghost theorem.

Computing the KAC determinant using dual model techniques and more about the no-ghost theorem

We relate some mathematical results concerning the Virasoro algebra to properties of dual models. Specifically, we show how to compute the Kac determinant using well known techniques from dual models. Then we use the Kac formula to discuss the validity of the no-ghost theorem of dual models in a more general setting than the original proofs of that theorem.

Strings in curved space

Polyakov's quantization is extended to strings embedded in a curved space, and the critical dimension calculated. However the no-ghost theorem fails there unless the space is supersymmetric and Ricci-flat.

Covariant quantization of string based on BRS invariance

Covariant canonical quantization of the bosonic string is performed, based on the BRS invariance principle. Defining a physical state by a modified form of Kugo-Ojima's subsidiary condition Q B|phys〉 = 0 ( Q B = BRS charge), we prove a no-ghost theorem in a simple manner. There an interesting relation between critical dimension and BRS symmetry is noticed; namely, the conditions D = 26 and α 0 = 1 for the space-time dimension D and the zero-intercept α 0 of leading trajectory are required by the nilpotency Q B 2 = 0 of the BRS charge. In addition, ghost number fractionization is observed in this system.

Supergauge Algebra and Interaction between Open and Closed Strings

In the Neveu-Schwarz and Ramond models we construct the vertices which describe the emission of closed-string ground-state particles out of open string states. The modified supergauge algebra associated with the new vertices can be shown to hold in the same form as the usual supergauge algebra in free cases. and is crucial for proving the no-ghost theorem in our system.

No-ghost theorem for the pomeron sector of Neuveu-Schwarz dual resonance model

With certain forms for the pomeron propagator in the non-planar single loops for the N-S model with positive G-parity, no-ghost theorem for the pomeron sector is proved by the method of Olive and Scherk.

Supergauge Algebra and Electromagnetic Interaction of the Dual Pion Model. II

We consider the electromagnetic problems of the dual pion model proposed by Neveu and Schwarz. The photon gauge substitution and the supergauge algebra appearing in this model are shown to determine the interaction uniquely. Consequently we find that (i) the electric charge has only two localization points which are at the ends of the string and that (ii) new gauge conditions hold, which are crucial for proving the no-ghost theorem in our t)leory. We also give a proof of the photon-gauge invariance of our system and find (iii) the conserved dual current. Unfortunately we are forced to stay in the on-shell world (i.e., massless photon).

No-ghost theorem for the Pomeron sector of the dual model

The Pomeron sector of the dual model comprises a spectrum of states obtained by factorizing the residues of poles occurring in a certain channel (with vacuum quantum numbers) of the orientable non-planar single loop graphs of the conventional dual model considered in 26 space-time dimensions. We show that the spectrum of Pomeron states which couple between external physical particle states are transverse and hence positive definite and thus ghost-free. The same states also factorize the Virasoro-Shapiro model which now appears very likely to be the scattering theory of Pomeron ground states.

Spectrum-Generating Algebra and No-Ghost Theorem for the Neveu-Schwarz Model

The no-ghost theorem of Brower for the conventional dual model (intercept ${\ensuremath{\alpha}}_{0}=1$) is extended to the Neveu-Schwarz model, and its spectrum-generating algebra is constructed. These conformally invariant generators (${A}_{n}, {B}_{s}$) are obtained by a systematic use of the gauges ${G}_{r}$. As a consequence it is demonstrated that particular dual models having resonances with positive-definite norms and masses are possible.

Spectrum-Generating Algebra and No-Ghost Theorem for the Dual Model

The dual model is generally factorized using Lorentz oscillators ${a}_{n}^{\ensuremath{\mu}}$ with ghost (or negativenorm) states arising from the indefinite metric ($[{a}_{n}^{0}, {a}_{n}^{0\ifmmode\dagger\else\textdagger\fi{}}]=\ensuremath{-}1$). Here all ghost states are proven to decouple for unit Regge intercept (${\ensuremath{\alpha}}_{0}=1$) as a consequence of the Virasoro gauges (${L}_{n}$). By reformulating vertices in light-cone variables and exploiting the local commutators (for ${Q}^{\ensuremath{\mu}}$, ${P}^{\ensuremath{\mu}}$) on the Koba-Nielson circle, the spectrum-generating algebra (${A}_{n}^{i}$, ${A}_{n}^{(+)}$) is found that commutes with all the gauges ${L}_{n}$. All physical states are explicitly constructed. The noghost theorem follows from the remarkable isomorphism of the transverse generators ${A}_{n}^{i}$ ($i=1, 2$) of Del Giudice, Di Vecchia, and Fubini to the original oscillators $\sqrt{n}{a}_{n}^{i}$, $[{A}_{n}^{i}, {A}_{m}^{j}]=n{\ensuremath{\delta}}_{\mathrm{ij}}{\ensuremath{\delta}}_{n+m,0}$, and the isomorphism (up to $c$ numbers) of the longitudinal generators ${A}_{n}^{(+)}$ with the conformal group generators ${L}_{l}$, $[{A}_{n}^{(+)}, {A}_{m}^{(+)}]=(n\ensuremath{-}m){A}_{n+m}^{(+)}+2{n}^{3}{\ensuremath{\delta}}_{n+m,0}$. Increasing the number of spatial oscillators (${a}_{n}^{i}$, $i=1, \dots{}, D\ensuremath{-}1$), one observes a critical dimension $D=26$. For $Dg26$ ghosts appear, for $Dl26$ there are no ghosts, and ${A}_{1}^{(+)}$ gives the null states postulated by Brower and Thorn. But for $D=26$, all ${A}_{n}^{(+)}$ correspond to null states, so that the second-order Pomeranchukon is precisely a Regge pole (${\ensuremath{\alpha}}_{P}=\frac{1}{2}{\ensuremath{\alpha}}^{\ensuremath{'}}s+2$) as proposed by Lovelace.

Physical states and pomeron poles in the dual pion model

The methods recently developed by Brower for proving the no-ghost theorem for the Veneziano model with α(0)=1 are applied to the dual pion model. We construct operators that generate all the physical states and prove that there are no ghosts for space-time dimensionality d⩽10. In the critical dimension d=10 we find that the pomeron singularities arising from nonplanar loops are simple poles. A new fully symmetrical model (a sort of “pionic” version of the Shapiro-Virasoro model) is formulated and conjectured to correspond to the multi-pomeron amplitudes.