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Polyakov action
 
 
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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the Polyakov action is the two-dimensional action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
 of a conformal field theory
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
 describing the worldsheet
Worldsheet

In string theory, the worldsheet is a two-dimensional manifold which describes the embedding of the string in spacetime. It is a direct generalization of the familiar worldline of a particle in special relativity and general relativity....
 of a string in string theory
String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
. It was introduced by S.Deser and B.Zumino and independently by L.Brink, P.Di Vecchia and P.S.Howe (in Physics Letters B65, pgs 369 and 471 respectively), and has become associated with Alexander Polyakov
Alexander Polyakov

Alexander M. Polyakov is a theoretical physicist, formerly at the Landau Institute for Theoretical Physics in Moscow, at Princeton University....
 after he made use of it in quantizing the string. The action reads

where is the string tension
Tension (mechanics)

In physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. Tension is measured newtons or pounds-force and is always parallel to the string on which it applies....
, is the metric of the target manifold, is the worldsheet metric and is the determinant of .






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the Polyakov action is the two-dimensional action
Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
 of a conformal field theory
Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal symmetry. Conformal field theory is often studied in two-dimensional geometry dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions....
 describing the worldsheet
Worldsheet

In string theory, the worldsheet is a two-dimensional manifold which describes the embedding of the string in spacetime. It is a direct generalization of the familiar worldline of a particle in special relativity and general relativity....
 of a string in string theory
String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
. It was introduced by S.Deser and B.Zumino and independently by L.Brink, P.Di Vecchia and P.S.Howe (in Physics Letters B65, pgs 369 and 471 respectively), and has become associated with Alexander Polyakov
Alexander Polyakov

Alexander M. Polyakov is a theoretical physicist, formerly at the Landau Institute for Theoretical Physics in Moscow, at Princeton University....
 after he made use of it in quantizing the string. The action reads

where is the string tension
Tension (mechanics)

In physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. Tension is measured newtons or pounds-force and is always parallel to the string on which it applies....
, is the metric of the target manifold, is the worldsheet metric and is the determinant of . The metric signature
Metric signature

The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalisation, and the diagonal entries of each sign counted....
 is chosen such that timelike directions are + and the spacelike directions are -. The spacelike worldsheet coordinate is called wheareas the timelike worldsheet coordinate is called .

Global symmetries


The action is invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
 under spacetime translation
Translation

Translation is the hermeneutics of the Meaning of a text and the subsequent production of an Dynamic and formal equivalence text, likewise called a "translation," that communicates the same message in another language....
s and infinitesimal
Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. For everyday life, an infinitesimal object is an object which is smaller than any possible measure....
 Lorentz transformation
Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
s: where and is a constant. This forms the Poincaré symmetry
Poincaré group

In physics and mathematics, the Poincar? group, named after Henri Poincar?, is the group of isometry of Minkowski spacetime. It is a 10-dimensional compact space Lie group....
 of the target manifold.

The invariance under (i) follows since the action depends only on the first derivative of . The proof of the invariance under (ii) is as follows:

  
  


Local symmetries


The action is invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
 under worldsheet diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
s (or coordinates transformations) and Weyl transformation
Weyl transformation

In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor:The invariance of a theory or an expression under this transformation is called the Weyl symmetry....
s.

Diffeomorphisms

Assume the following transformation:
It transforms the Metric tensor
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
 in the following way:
One can see that:
One knows that the Jacobian
Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
 of this transformation is given by:
which leads to:
and one sees that:
summing up this transformation leaves the action invariant.

Weyl transformation

Assume the Weyl transformation
Weyl transformation

In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor:The invariance of a theory or an expression under this transformation is called the Weyl symmetry....
:
then:
And finally:
  
And one can see that the action is invariant under Weyl transformation
Weyl transformation

In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor:The invariance of a theory or an expression under this transformation is called the Weyl symmetry....
. If we consider n-dimensional (spatially) extended objects whose action is proportional to their worldsheet area/hyperarea, unless n=1, the corresponding Polyakov action would contain another term breaking Weyl symmetry.

One can define the Stress-energy tensor
Stress-energy tensor

The stress-energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress of Newtonian physics....
:
Let's define:
Because of Weyl symmetry the action does not depend on :


Relation with Nambu-Goto action


Writing the Euler-Lagrange equation
Euler-Lagrange equation

In calculus of variations, the Euler?Lagrange equation, or Lagrange's equation, is a differential equation whose solutions are the function s for which a given functional is stationary point....
 for the metric tensor
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
  one obtains that:
Knowing also that:
One can write the variational derivative of the action:
where which leads to:


If the auxiliary worldsheet
Worldsheet

In string theory, the worldsheet is a two-dimensional manifold which describes the embedding of the string in spacetime. It is a direct generalization of the familiar worldline of a particle in special relativity and general relativity....
 metric tensor
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
  is calculated from the equations of motion:
and substituted back to the action, it becomes the Nambu-Goto action
Nambu-Goto action

The Nambu-Goto action is the simplest invariant action in bosonic string theory. It is the starting point of the analysis of string behavior, using the principles of Lagrangian mechanics....
:


However, the Polyakov action is more easily quantized
Quantization (physics)

In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field . This is a generalization of the procedure for building quantum mechanics from classical mechanics....
 because it is linear
Linear

The word linear comes from the Latin word linearis, which means created by lines.In mathematics, a linear map or function f is a function which satisfies the following two properties......
.

Equations of motion


Using diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
s and Weyl transformation
Weyl transformation

In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor:The invariance of a theory or an expression under this transformation is called the Weyl symmetry....
 one can transform the action into the following form:
where

Keeping in mind that one can derive the constraints:
.


Substituting one obtains:


And consequently:


With the boundary conditions in order to satisfy the second part of the variation of the action.
  • Closed strings
Periodic boundary conditions
Periodic boundary conditions

In mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge....
:
  • Open strings
Neumann boundary conditions: Dirichlet boundary conditions:

See also

  • D-brane
    D-brane

    In string theory, D-branes are a class of extended objects upon which open string s can end with Dirichlet boundary conditions, after which they are named....
  • Einstein-Hilbert action
    Einstein-Hilbert action

    The Einstein-Hilbert action in general relativity is the action that yields the Einstein's field equations when action principle to obtain equations of motion for the spacetime metric....