Polyakov effect

The Polyakov action (English Polyakov action ) is the two-dimensional effect of a conformal field theory , which describes the world surface of a bosonic string . It is named after Alexander Markowitsch Polyakow .

It was introduced in 1976 by Lars Brink , Paolo Di Vecchia and PS Howe and independently of Stanley Deser and Bruno Zumino . Polyakov used it in 1981 to quantize string theory. It is equivalent to the older Nambu-Goto effect .

formulation

Parameterization of the world area of ​​an open string by σ and τ,
X ⁰ and X are the target space time and space coordinates.

The Polyakov effect has the following form

${\displaystyle S={T \over 2}\int _{\Sigma }d\sigma d\tau {\sqrt {-|\gamma |}}\gamma ^{ab}g_{\mu \nu }\partial _{a}X^{\mu }(\sigma ,\tau )\partial _{b}X^{\nu }(\sigma ,\tau )}$.

The symbols in this equation have the following meanings:

• ${\displaystyle \Sigma }$ is the two-dimensional surface of the string.
• ${\displaystyle T}$is the string tension , which indicates how great the tendency of the string to oscillate, analogous to a rubber band, which also has a certain internal tension. This parameter is a free parameter of the theory and determines z. B. the mass of the excited states in a quantized theory. The so-called regge slope parameter is often used instead of , for historical reasons. ${\displaystyle T}$ ${\displaystyle \alpha '=(2\pi T)^{-1}}$
• ${\displaystyle \gamma ^{ab}}$is an independent metric on the surface of the world (the indices take the values ​​0 and 1), which is only introduced as an auxiliary variable, since it does not represent a dynamic field and can be eliminated by using the equations of motion (this leads to the Nambu-Goto effect ).
• ${\displaystyle |\gamma |}$is the determinant of . The signature of the metric is chosen so that time-like directions have a positive sign and space-like directions have a negative sign . The space-like world surface coordinate is designated with , the time- like coordinate with . ${\displaystyle \gamma _{ab}}$ ${\displaystyle \sigma }$ ${\displaystyle \tau }$
• ${\displaystyle g_{\mu \nu }}$is the metric of the target space ( spacetime ), with the indices running from 0 to D-1 when D is the dimension of the target space.
• The target space coordinates are through if they provide pictures of the two-dimensional world space to the tangent of the target area is so . ${\displaystyle X^{\mu }}$ ${\displaystyle X:\Sigma \to T(M)}$

Symmetries

The effect is invariant under the following symmetry transformations :

• Surface diffeomorphisms
• Weyl transformations and Poincaré transformations of the target space.

The Weyl symmetry is characteristic of a two-dimensional theory - if one considers the effect of higher-dimensional objects, one finds that an effect proportional to their world volume contains additional terms that break the Weyl symmetry.

Equivalence to the Nambu-Goto effect

In order to show the equivalence of the Polyakov effect to the Nambu-Goto effect, it is sufficient to use the equations of motion for the induced metric on the world surface: ${\displaystyle h_{ab}=\partial _{a}X^{\mu }\partial _{b}X_{\mu }}$

${\displaystyle {\delta S \over \delta \gamma ^{ab}}=0\quad \to \quad h_{ab}={1 \over 2}\gamma _{ab}\gamma ^{cd}h_{cd}}$.

This can be used to eliminate from the effect and you get exactly the Nambu-Goto effect ${\displaystyle \gamma }$

${\displaystyle S_{NG}=T\int _{\Sigma }d\sigma d\tau {\sqrt {-|h|}}}$.

Individual evidence

1. ↑ Brink, Di Vecchia, Howe: A locally supersymmetric and reparametrization invariant action for the spinning string, Physics Letters B Volume 65, 1976, pp. 471-474
2. ↑ Deser, Zumino, A complete action for the spinning string, Physics Letters B, Volume 65, 1976, p. 369
3. ↑ Polyakov, Quantum geometry of the bosonic string, Physics Letters B, Vol. 103, 1981, p. 207