Picture (string theory)
In string theory, the concept of "picture" refers to a choice of gauge fixing within the operator formalism, specifically when dealing with superstring theory and its quantization. It arises because of the presence of superconformal symmetry and the need to deal with picture-changing operators to ensure gauge invariance of scattering amplitudes.
Superstring theory possesses worldsheet supersymmetry. When quantizing the superstring, one encounters constraints from this supersymmetry in the form of superconformal symmetry. These constraints lead to conserved currents, namely the supercurrent TF.
The picture formalism is introduced to handle the zero modes of a related operator b, which is part of the superghost system (b, c, β, γ) arising from gauge fixing. The superghosts are associated with the superconformal transformations. These zero modes can lead to obstructions when calculating scattering amplitudes.
A state in the string Hilbert space is assigned a "picture number." Physical states must have a total picture number that satisfies certain constraints depending on the specific string theory (e.g., -1 for superstrings in the RNS formalism).
Picture-Changing Operators (PCOs):
Picture-changing operators are vertex operators that can be inserted into correlation functions to change the picture number of a vertex operator or state. They essentially act as a means to redistribute picture number charge to satisfy the overall picture number constraint. The canonical picture-changing operator is generally derived from the supercurrent TF and the superghost fields, and often takes the form X = {Q, ξ} where Q is the BRST charge and ξ is the superghost field associated with the superconformal gauge fixing.
Picture Independence:
The physical S-matrix should be independent of the choice of picture used in the calculation, meaning that different arrangements of picture-changing operators leading to the correct total picture number should give the same result. This is a consistency condition that must be satisfied to ensure the theory is well-defined. Proving picture independence is often a complicated process, involving the use of various superconformal identities and the properties of the superghosts.
Relevance:
The picture formalism is essential for calculating scattering amplitudes in superstring theory. Without it, one encounters inconsistencies and ambiguities in the computations. It allows for a consistent treatment of the superconformal symmetry and the ghost sector, leading to physically meaningful results. The careful insertion and manipulation of picture-changing operators is a crucial aspect of string theory calculations.