Anatomy of zero-norm states in string theory

Abstract

We calculate and identify the counterparts of zero-norm states in the old covariant first quantised (OCFQ) spectrum of open bosonic string in two other quantization schemes of string theory, namely the light-cone DDF zero-norm states and the off-shell BRST zero-norm states (with ghost) in the Witten string field theory (WSFT). In particular, special attention is paid to the inter-particle 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 σ-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.Recently it was discovered that [1, 2] the high-energy limit α ′ → ∞ of stringy Ward identities, derived from the decoupling of two types of zero-norm states in the OCFQ spectrum, imply an infinite number of linear relations among high energy scattering amplitudes of different string states with the same momenta. Moreover, these linear relations can be used to fix the proportionality constants between high energy scattering amplitudes of different string states algebraically at each fixed mass level. Thus there is only one independent component of high energy scattering amplitude at each fixed mass level. For the case of string-tree amplitudes, a general formula can even be given to express all high-energy stringy scattering amplitudes for arbitrary mass levels in terms of those of tachyons [1,3].These zero-norm state calculations are independent of the high-energy saddle-point calculations of Gross and Mende [4], Gross [5] and Gross and Manes [6]. In fact, the results of saddle-point calculations by those authors were found [1-3] to be inconsistent with high energy stringy Ward identities of zero-norm state calculations, and thus could threat the validity of unitarity of string perturbation theory. A corrected saddle-point calculation was given in [3], where the missing terms of the calculations in Refs [4,5,6] were identified to recover the stringy Ward identities.The importance of zero-norm states and their implication on stringy symmetries were first pointed out in the context of massive σ-model approach of string theory [7]. Some implications of the corresponding stringy Ward identities on the scattering amplitudes were discussed in [8]. On the other hand, zero-norm states were also shown [9] to carry the spacetime ω ∞ symmetry [10] charges of 2D string theory [11]. This is in parallel with the work of [12] where the gr...

Overview of High Energy String Scattering Amplitudes and Symmetries of String Theory

“…Some implications of the corresponding stringy Ward identities on the string scattering amplitudes were discussed in [32,34]. On the other hand, it was then realized that [31,35] the symmetry in Eq. (2.6) can be reproduced from the off-shell gauge transformations of Witten string field theory (WSFT) [36] by imposing the no ghost conditions.…”

High-energy symmetry of bosonic open string theory in the light-like linear dilaton background

“…In this section, based on the method in [18], we solve for the Virasoro constraints of bosonic open string theory in the linear dilaton background, and derive a general decomposition of physical states in terms of zero and positive norm states at the first massive level.…”

“…It is an approximate symmetry since we are doing a 1 α ′ E 2 expansion for all (tree-level) scattering amplitudes, and it is a global symmetry since we compare scattering amplitudes among independent degrees of freedom. Thus, one might be curious about the connection between the HESS and the infinite target-space gauge symmetry [16,17,18], and wonder that how it is possible to derive HESS from the decoupling of high-energy zero-norm states. For a detailed discussion, see [12,14].…”

High-Energy String Scattering Amplitudes and Signless Stirling Number Identity

“…It is remarkable to see that the identities suggested by string theory calculation can be rigorously proved by a totally different mathematical method in combinatorial theory. It is also very interesting to see that, physically, the identities for arbitrary real values L in equation (8) can only be realized in high-energy compactified string scattering amplitudes considered very recently [18]. This is mainly due to the relation M 2 = K 25 2 +M 2 whereM 2 = 2(N − 1) and K 25 = 2πl−θ l +θ i 2πR (see, e.g., [28]) is the generalized KK internal momentum corresponding to the compactified string coordinate [18].…”