Final answer:
The Polyakov action is derived by combining the Nambu-Goto action, which describes the area of the worldsheet swept out by the string, with the Polyakov term, which introduces the dynamics of the metric on the worldsheet. The Nambu-Goto action is given by SNG = -T0 ∫dτdσ √(-γ), where T0 is the string tension, τ and σ parameterize the worldsheet, and γ is the determinant of the induced metric on the worldsheet. The Polyakov term is then added to introduce the metric dynamics.
Step-by-step explanation:
Derivation of the Polyakov Action
The Polyakov action is a central concept in string theory and describes the dynamics of a string moving in a curved space-time. It is derived by considering the Nambu-Goto action, which describes the area of a worldsheet swept out by the string, and adding the Polyakov term, which introduces the dynamics of the metric on the worldsheet.
The Nambu-Goto action is given by:
SNG = -T0 ∫dτdσ √(-γ)
where T0 is the string tension, τ and σ parameterize the worldsheet, and γ is the determinant of the induced metric on the worldsheet. The Polyakov term is then added to introduce the metric dynamics:
SP = -rac{1}{4πα'} ∫dτdσ √(-g)gabhαβ
where α' is the Regge slope parameter, g is the determinant of the space-time metric, gab is the inverse space-time metric, and hαβ is the worldsheet metric.
The total action is obtained by combining the Nambu-Goto and Polyakov terms:
S = SNG + SP
This action describes the dynamics of the string in arbitrary curved space-times.