In string theory, the Ramond–Neveu–Schwarz (RNS) formalism is an approach to formulating superstrings in which the worldsheet has explicit superconformal invariance but spacetime supersymmetry is hidden, in contrast to the Green–Schwarz formalism where the latter is explicit. It was originally developed by Pierre Ramond, André Neveu and John Schwarz in the RNS model in 1971, which gives rise to type II string theories and can also give type I string theory. Heterotic string theories can also be acquired through this formalism by using a different worldsheet action. There are various ways to quantize the string within this framework including light-cone quantization, old canonical quantization, and BRST quantization. A consistent string theory is only acquired if the spectrum of states is restricted through a procedure known as a GSO projection, with this projection being automatically incorporated in the Green–Schwarz formalism.
History
The discovery of the Veneziano amplitude describing the scattering of four mesons in 1968 launched the study of dual resonance models which generalized these scattering amplitudes to the scattering with any number of mesons. While these are S-matrix theories rather than quantum field theories, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind gave them a string interpretation, whereby mesons behave as strings of finite length.
In 1970 Pierre Ramond was working at Yale trying to extend the dual resonance models to include fermionic degrees of freedom through a generalization of the Dirac equation. This led him to constructing the first superalgebra, the Ramond superalgebra. At the same time, Andre Neveu and John Schwarz were working at Princeton to extend existing dual resonance models by adding to them anticommutating creation and annihilation operators. This originally gave rise to a model containing only bosons. Shortly after their second paper on this topic, they realized that their model can be combined with Ramond's fermionic model, which they successfully did to give rise to the Ramond–Neveu–Schwarz (RNS) model, referred to at the time as the dual pion model.
This work was done with only hadronic physics in mind with no reference to strings, until in 1974 Stanley Mandelstam reinterpreted the RNS model as a model for spinning strings. Joël Scherk and John Schwartz were the first to suggest that it may describe elementary particles rather than just hadrons when they showed that the spin-2 particle of the model behaves as a graviton.
At the time, the main issue with the RNS model was that it contained a tachyon as the lowest energy state. It was only in 1976 with the introduction of GSO projection by Ferdinando Gliozzi, Joël Scherk, and David Olive that the first consistent tachyon-free string theories were constructed.
Overview
The RNS formalism is an approach to quantizing a string by working with the string worldsheet embedded in spacetime with both bosonic and fermionic fields on the worldsheet. There are a number of different approaches for quantizing the string in this formalism. The main ones are old covariant quantization, light-cone quantization, and BRST quantization via the path integral. The last approach starts from the Euclidean partition function
where is the worldsheet action with some gauge symmetry group that represents an overcounting of the physically distinct configurations of the fields that the action depends on. This overcounting is eliminated by dividing by the volume of the gauge group . BRST quantization proceeds by gauge fixing the path integral via the Fadeev–Popov procedure, which gives rise to a ghost action in addition to the now gauge fixed action.
The RNS model originates from using the supergravity action which upon gauge fixing gives the RNS action together with a ghost action describing holomorphic and antiholomorphic ghosts that are necessary to eliminate the unphysical temporal excitations of the fields. The physical states of this theory split up into a number of sectors depending on the periodicity condition of the fermionic fields. The full theory is inconsistent and contains an unphysical tachyon, however projecting out a number of these sectors can give rise to consistent tachyon-free theories. In particular, the RNS model gives rise to type IIA and type IIB string theory for closed strings, while combining the open string with a modified version of the IIB string gives rise to type I string theory. Starting instead from a supergravity action gives rise to heterotic string theories.