Zero-norm States Research Articles

Reconstructing confined particles with complex singularities

Complex singularities have been suggested in propagators of confined particles, e.g., the Landau-gauge gluon propagator. We rigorously reconstruct Minkowski propagators from Euclidean propagators with complex singularities. As a result, the analytically continued Wightman function is holomorphic in the tube, and the Lorentz symmetry and locality are kept intact, whereas the reconstructed Wightman function violates the temperedness and the positivity condition. Moreover, we argue that complex singularities correspond to confined zero-norm states in an indefinite metric state space.

Recent Developments of the Lauricella String Scattering Amplitudes and Their Exact SL(K + 3,C) Symmetry

In this review, we propose a new perspective to demonstrate the Gross conjecture regarding the high-energy symmetry of string theory. We review the construction of the exact string scattering amplitudes (SSAs) of three tachyons and one arbitrary string state, or the Lauricella SSA (LSSA), in the 26D open bosonic string theory. These LSSAs form an infinite dimensional representation of the SL(K+3,C) group. Moreover, we show that the SL(K+3,C) group can be used to solve all the LSSAs and express them in terms of one amplitude. As an application in the hard scattering limit, the LSSA can be used to directly prove the Gross conjecture, which was previously corrected and proved by the method of the decoupling of zero norm states (ZNS). Finally, the exact LSSA can be used to rederive the recurrence relations of SSA in the Regge scattering limit with associated SL(5,C) symmetry and the extended recurrence relations (including the mass and spin dependent string BCJ relations) in the nonrelativistic scattering limit with the associated SL(4,C) symmetry discovered recently.

Revisiting the Ramond sector of the N=1 superconformal minimal models

Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight $c/24$ in the even series of $\mathcal{N}\!=\!1$ superconformal minimal models with central charge $c$. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible $c/24$ module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

In recent work Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as $PT$. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre et al. our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. In their paper Alexandre et al. presented a variational procedure for the action in which the surface term played an explicit role, to thus suggest that one has to use such a procedure in order to establish the Goldstone theorem in the non-Hermitian case. However, by taking certain fields that they took to be Hermitian to actually either be anti-Hermitian or be made so by a similarity transformation, we show that we are then able to obtain a Goldstone boson using a completely standard variational procedure. Since we use a standard variational procedure we can readily extend our analysis to a continuous local symmetry by introducing a gauge boson. We show that when we do this the gauge boson acquires a non-zero mass by the Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

Recurrence relations of Kummer functions and Regge string scattering amplitudes

We discover an infinite number of recurrence relations among Regge string scattering amplitudes [11, 30] of different string states at arbitrary mass levels in the open bosonic string theory. As a result, all Regge string scattering amplitudes can be algebraically solved up to multiplicative factors. Instead of decoupling zero-norm states in the fixed angle regime, the calculation is based on recurrence relations and addition theorem of Kummer functions of the second kind. These recurrence relations among Regge string scattering amplitudes are dual to linear relations or symmetries among high-energy fixed angle string scattering amplitudes discovered previously.

Higher Spin String States Scattered from D-Particle in the Regge Regime and Factorized Ratios of Fixed Angle Scatterings

We study scattering of higher spin closed string states at arbitrary mass levels from D-particle in the Regge regime. We extract the infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. In this calculation, we have used an identity proved recently based on a signless Stirling number identity in combinatorial theory. The complete ratios calculated by this indirect method include a subset of ratios calculated previously by direct fixed angle calculation. Moreover, we discover that in spite of the non-factorizability of the closed string D-particle scattering amplitudes, the complete ratios derived for the fixed angle regime are found to be factorized. These ratios are consistent with the decoupling of high-energy zero norm states calculated previously.

Massive superstring scatterings in the Regge regime

We calculate four classes of high-energy massive string scattering amplitudes of fermionic string theory at arbitrary mass levels in the Regge regime (RR). We show that all four leading order amplitudes in the RR can be expressed in terms of the Kummer function of the second kind. Based on the summation algorithm of a set of extended signed Stirling number identities, we show that all four ratios calculated previously by the method of decoupling of zero-norm states among scattering amplitudes in the Gross regime can be extracted from this Kummer function in the RR. Finally, we conjecture and give evidence that the existence of these four Gross regime ratios in the RR persists to subleading orders in the Regge expansion of all high-energy fermionic string scattering amplitudes.

Observables and correlators in non-relativistic ABJM theory

Non-relativistic ABJM theory is defined by Galilean limit of mass-deformed = 6 Chern-Simons theory. Holographic string theory dual to the theory is not known yet. To understand features candidate gravity dual might exhibit, we examine local and nonlocal physical observables and their correlations in the non-relativistic ABJM theory. We show that gauge invariant local observables correspond to zero-norm states and that correlation functions among them are trivial. We also show that a particular class of nonlocal observables, Wilson loops, are topological in the sense that their correlation functions coincide with those of pure Chern-Simons theory. We argue that the theory is nevertheless physical and illustrate several physical observables whose correlation functions are nontrivial. We also study quantum aspects. We show that Chern-Simons level is finitely renormalized and that dilatation operator is trivial at planar limit. These results all point to string scale geometry of gravity dual and to intriguing topological and tensionless nature of dual string or M theory defined on it.

Automatic Regularization by Quantization in Reducible Representations of CCR: Point-Form Quantum Optics with Classical Sources

Electromagnetic fields are quantized in a manifestly covariant way by means of a class of reducible “center-of-mass N-representations” of the algebra of canonical commutation relations (CCR). The four-potential Aa(x) transforms in these representations as a Hermitian four-vector field in Minkowski four-position space (without change of gauge), but in momentum space it splits into spin-1 massless photons and two massless scalars. What we call quantum optics is the spin-1 sector of the theory. The scalar fields have physical status similar to that of dark matter (spin-1 and spin-0 particle numbers are separately conserved). There are no negative-norm or zero-norm states. Unitary dynamics is given by the point-form interaction picture, with minimal-coupling Hamiltonian constructed from fields that are free on the null-cone boundary of the Milne universe. SL(2,C) transformations as well as the dynamics are represented unitarily in the Hilbert space corresponding to N four-dimensional oscillators. Vacuum is a Bose-Einstein condensate of the N-oscillator gas and is given by any N-oscillator product state annihilated by all annihilation operators. The form of Aa(x) is determined by an analogue of the twistor equation. The same equation guarantees that the set of vacuum states is Poincare invariant. The formalism is tested on quantum fields produced by pointlike classical sources. Photon statistics is well defined even for pointlike charges, with ultraviolet and infrared regularizations occurring automatically as a consequence of the formalism. The probabilities are not Poissonian but of a Renyi type with α=1−1/N; the Shannon limit N→∞ is an ultraviolet/infrared-regularized Poisson distribution. The average number of photons occurring in Bremsstrahlung splits into two parts: The one due to acceleration, and the one that remains nonvanishing even for inertially moving charges. Classical Maxwell electrodynamics is reconstructed from coherent-state averaged solutions of Heisenberg equations. We show in particular that static pointlike charges polarize vacuum and produce effective charge densities and fields whose form is sensitive to both the choice of representation of CCR and the corresponding vacuum state.

Scatterings of massive string states from D-brane and their linear relations at high energies

We study scatterings of bosonic massive closed string states at arbitrary mass levels from D-brane. We discover that all the scattering amplitudes can be expressed in terms of the generalized hypergeometric function F 2 3 with special arguments, which terminates to a finite sum and, as a result, the whole scattering amplitudes consistently reduce to the usual beta function. For the simple case of D-particle, we explicitly calculate high-energy limits of a series of the above scattering amplitudes for arbitrary mass levels, and derive infinite linear relations among them for each fixed mass level. The ratios of these high-energy scattering amplitudes are found to be consistent with the decoupling of high-energy zero-norm states of our previous works.

Linear relations among four-point functions in the high energy limit of string theory

The decoupling of zero-norm states leads to linear relations among 4-point functions in the high energy limit of string theory. Recently it was shown that the linear relations uniquely determine ratios among 4-point functions at the leading order. The purpose of this paper is to extend the validity of the same approach to the next-to-leading order and higher orders.

Comments on the high energy limit of bosonic open string theory

In previous works, ratios among four-point scattering amplitudes at the leading order in the high-energy limit were derived for the bosonic open string theory. The derivation was based on Ward identities derived from the decoupling of zero-norm states and was purely algebraic. The only assumption of the derivation was that the momentum polarization can be approximated by the longitudinal polarization at high energies. In this paper, using the decoupling of spurious states, we reduce this assumption to a much weaker one which can be easily verified by simple power counting in most cases. For the special cases which are less obvious, we verify the new assumption for an example by saddle-point approximation. We also provide a new perspective to our previous results in terms of DDF states. In particular, we show that, by using DDF states, one can easily see that there is only one independent high energy scattering amplitude for each fixed mass level.

Notes on high-energy limit of bosonic closed string scattering amplitudes

We study bosonic closed string scattering amplitudes in the high-energy limit. We find that the methods of decoupling of high-energy zero-norm states and the high-energy Virasoro constraints, which were adopted in the previous works to calculate the ratios among high-energy open string scattering amplitudes of different string states, persist for the case of closed string. However, we clarify the previous saddle-point calculation for high-energy open string scattering amplitudes and claim that only ( t , u ) channel of the amplitudes is suitable for saddle-point calculation. We then discuss three evidences to show that saddle-point calculation for high-energy closed string scattering amplitudes is not reliable. By using the relation of tree-level closed and open string scattering amplitudes of Kawai, Lewellen and Tye (KLT), we calculate the high-energy closed string scattering amplitudes for arbitrary mass levels. For the case of high-energy closed string four-tachyon amplitude, our result differs from the previous one of Gross and Mende, which is NOT consistent with KLT formula, by an oscillating factor.

High-Energy Zero-Norm States and Symmetries of String Theory

High-energy limit of zero-norm states in the old covariant first quantized spectrum of the 26D open bosonic string, together with the assumption of a smooth behavior of string theory in this limit, are used to derive infinitely many linear relations among the leading high-energy, fixed-angle behavior of four-point functions of different string states. As a result, ratios among all high-energy scattering amplitudes of four arbitrary string states can be calculated algebraically and the leading order amplitudes can be expressed in terms of that of four tachyons as conjectured by Gross in 1988. A dual calculation can also be performed and equivalent results are obtained by taking the high-energy limit of Virasoro constraints. Finally, we compute all high-energy scattering amplitudes of three tachyons and one massive state at the leading order by saddle-point approximation to verify our results.

Zero-norm states and stringy symmetries

We identify spacetime symmetry charges of 26D open bosonic string theory from an infinite number of zero-norm states (ZNS) with arbitrary high spin in the old covariant first quantized string spectrum. We give various evidences to support this identification. These include massive sigma-model calculation, Witten string field theory calculation, 2D string theory calculation and, most importantly, three methods of high-energy stringy scattering amplitude calculations. The last calculations explicitly prove Gross's conjectures in 1988 on high energy symmetry of string theory.

High energy scattering amplitudes of superstring theory

We use three different methods to calculate the proportionality constants among high-energy scattering amplitudes of different string states with polarizations on the scattering plane. These are the decoupling of high-energy zero-norm states (HZNS), the Virasoro constraints and the saddle-point calculation. These calculations are performed at arbitrary but fixed mass level for the NS sector of 10D open superstring. All three methods give the consistent results, which generalize the previous works on the high-energy 26D open bosonic string theory. In addition, we discover new leading order high-energy scattering amplitudes, which are still proportional to the previous ones, with polarizations orthogonal to the scattering plane. These scattering amplitudes are of subleading order in energy for the case of 26D open bosonic string theory. The existence of these new high-energy scattering amplitudes is due to the worldsheet fermion exchange in the correlation functions and is, presumably, related to the high-energy massive spacetime fermionic scattering amplitudes in the R-sector of the theory.

One-Loop Massive Scattering Amplitudes and Ward Identities in String Theory

We calculate bosonic open string one-loop massive scattering amplitudes for some low-lying string states. By using the periodicity relations of Jacobi theta functions, we explicitly prove an infinite number of one-loop type I stringy Ward identities derived from type I zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. The subtlety in the proofs of one-loop type II stringy Ward identities is discussed by comparing them with those of string-tree cases. High-energy limit of these stringy Ward identities can be used to fix the proportionality constants between one-loop massive high-energy scattering amplitudes of different string states with the same momenta. These proportionality constants cannot be calculated directly from sample calculation as we did previously in the cases of string-tree scattering amplitudes.

Covariant Operator Formalism for Massive Arbitrary Spin Fields

Quantum theories of massive fields with arbitrary fixed spins are formulated in terms of manifestly covariant operators. Unphysical modes are confined in zero-norm states by virtue of the subsidiary conditions Q BRST |phys> = 0 with BRST charges. This restriction is analogous to that in quantum gauge field theories. Smooth massless limits of the massive field theories are briefly studied.

Solving all 4-point correlation functions for bosonic open string theory in the high-energy limit

We study the implication of decoupling zero-norm states in the high-energy limit, for the 26-dimensional bosonic open string theory. Infinitely many linear relations among 4-point functions are derived algebraically, and their unique solution is found. Equivalent results are also obtained by taking the high-energy limit of Virasoro constraints, and as an independent check, we compute all 4-point functions of 3 tachyons and an arbitrary massive state by saddle-point approximation.

Anatomy of zero-norm states in string theory

We calculate and identify the counterparts of zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string in two other quantization schemes of string theory, namely, the light-cone Del Giudice--Di Vecchia--Fubine zero-norm states and the off-shell Becchi-Rouet-Stora-Tyutin (BRST) zero-norm states (with ghost) in the Witten string field theory (WSFT). In particular, special attention is paid to the interparticle zero-norm states in all quantization schemes. For the case of the off-shell BRST zero-norm states, we impose the no-ghost conditions and recover exactly two types of on-shell zero-norm states in the OCFQ string spectrum for the first few low-lying mass levels. We then show that off-shell gauge transformations of WSFT are identical to the on-shell stringy gauge symmetries generated by two types of zero-norm states in the generalized massive $\ensuremath{\sigma}$-model approach of string theory. The high-energy limit of these stringy gauge symmetries was recently used to calculate the proportionality constants, conjectured by Gross, among high-energy scattering amplitudes of different string states. Based on these zero-norm state calculations, we have thus related gauge symmetry of WSFT to the high-energy stringy symmetry of Gross.