


Perturbative string theory 






Glossary
 Glossary of string theory


The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate stringlike objects (for example, cosmic strings). It is the starting point of the analysis of zerothickness (infinitely thin) string behavior, using the principles of Lagrangian mechanics. Just as the action for a free point particle is proportional to its proper time—i.e., the "length" of its worldline—a relativistic string's action is proportional to the area of the sheet which the string traces as it travels through spacetime.
It is named after Japanese physicists Yoichiro Nambu and Tetsuo Goto.
Background
Relativistic Lagrangian mechanics
The basic principle of Lagrangian mechanics is that an object subjected to outside influences will "choose" a path which makes a certain quantity, the action, an extremum. The action is a functional, a mathematical relationship which takes an entire path and produces a single number. The physical path, that which the object actually follows, is the path for which the action is "stationary" (or extremal): any small variation of the path from the physical one does not significantly change the action. (Often, this is equivalent to saying the physical path is the one for which the action is a minimum.) Actions are typically written using Lagrangians, formulas which depend upon the object's state at a particular point in space and/or time. In nonrelativistic mechanics, for example, a point particle's Lagrangian is the difference between kinetic and potential energy: L = K − U. The action, often written S, is then the integral of this quantity from a starting time to an ending time:
 $S\; =\; \backslash int\_\{t\_i\}^\{t\_f\}\; L\; \backslash ,\; dt.$
(Typically, when using Lagrangians, we assume we know the particle's starting and ending positions, and we concern ourselves with the path which the particle travels between those positions.)
This approach to mechanics has the advantage that it is easily extended and generalized. For example, we can write a Lagrangian for a relativistic particle, which will be valid even if the particle is traveling close to the speed of light. To preserve Lorentz invariance, the action should only depend upon quantities that are the same for all (Lorentz) observers. The simplest such quantity is the proper time, the time measured by a clock carried by the particle. According to special relativity, all Lorentz observers watching a particle move will compute the same value for the quantity
 $ds^2\; =\; (c\; \backslash ,\; dt)^2\; +\; dx^2\; +\; dy^2\; +\; dz^2,\; \backslash $
and ds/c is then an infinitesimal proper time. For a point particle not subject to external forces (i.e., one undergoing inertial motion), the relativistic action is
 $S\; =\; mc\; \backslash int\; ds.$
Worldsheets
Just as a zerodimensional point traces out a worldline on a spacetime diagram, a onedimensional string is represented by a worldsheet. All worldsheets are twodimensional surfaces, hence we need two parameters to specify a point on a worldsheet. String theorists use the symbols τ and σ for these parameters. As it turns out, string theories involve higherdimensional spaces than the 3D world with which we are familiar; bosonic string theory requires 25 spatial dimensions and one time axis. If d is the number of spatial dimensions, we can represent a point by the vector
 $x\; =\; (x^0,\; x^1,\; x^2,\; \backslash ldots,\; x^d).$
We describe a string using functions which map a position in the parameter space (τ, σ) to a point in spacetime. For each value of τ and σ, these functions specify a unique spacetime vector:
 $X\; (\backslash tau,\; \backslash sigma)\; =\; (X^0(\backslash tau,\backslash sigma),\; X^1(\backslash tau,\backslash sigma),\; X^2(\backslash tau,\backslash sigma),\; \backslash ldots,\; X^d(\backslash tau,\backslash sigma)).$
The functions $X^\backslash mu\; (\backslash tau,\backslash sigma)$ determine the shape which the worldsheet takes. Different Lorentz observers will disagree on the coordinates they assign to particular points on the worldsheet, but they must all agree on the total area which the worldsheet has. The Nambu–Goto action is chosen to be proportional to this total area.
Let $\backslash eta\_\{\backslash mu\; \backslash nu\}$ be the metric on the (d+1)dimensional spacetime. Then,
 $g\_\{ab\}\; =\; \backslash eta\_\{\backslash mu\; \backslash nu\}\; \backslash frac\{\backslash partial\; X^\backslash mu\}\{\backslash partial\; y^a\}\; \backslash frac\{\backslash partial\; X^\backslash nu\}\{\backslash partial\; y^b\}\; \backslash $
is the induced metric on the worldsheet, where $a,b\; =\; 0,1$ and $y^0\; =\; \backslash tau\; ,\; y^1\; =\; \backslash sigma$.
For the area $\backslash mathcal\{A\}$ of the worldsheet the following holds:
 $\backslash mathrm\{d\}\; \backslash mathcal\{A\}\; =\; \backslash mathrm\{d\}^2\; \backslash Sigma\; \backslash sqrt\{g\}$
where $\backslash mathrm\{d\}^2\backslash Sigma\; =\; \backslash mathrm\{d\}\backslash sigma\; \backslash ,\; \backslash mathrm\{d\}\backslash tau$ and $g\; =\; \backslash mathrm\{det\}\; \backslash left(\; g\_\{ab\}\; \backslash right)\; \backslash $
Using the notation that:
 $\backslash dot\{X\}\; =\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; \backslash tau\}$
and
 $X\text{'}\; =\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; \backslash sigma\},$
one can rewrite the metric $g\_\{ab\}$:
 $g\_\{ab\}\; =\; \backslash left(\; \backslash begin\{array\}\{cc\}\; \backslash dot\{X\}^2\; \&\; \backslash dot\{X\}\; \backslash cdot\; X\text{'}\; \backslash \backslash \; X\text{'}\; \backslash cdot\; \backslash dot\{X\}\; \&\; X\text{'}^2\; \backslash end\{array\}\; \backslash right)\; \backslash $
 $g\; =\; \backslash dot\{X\}^2\; X\text{'}^2\; \; (\backslash dot\{X\}\; \backslash cdot\; X\text{'})^2$
the Nambu–Goto action is defined as,
$\backslash mathcal\{S\}\; \backslash $

$=\; \backslash frac\{T\_0\}\{c\}\; \backslash int\; d\backslash mathcal\{A\}$

$=\; \backslash frac\{T\_0\}\{c\}\; \backslash int\; \backslash mathrm\{d\}^2\; \backslash Sigma\; \backslash sqrt\{g\}$

$=\; \backslash frac\{T\_0\}\{c\}\; \backslash int\; \backslash mathrm\{d\}^2\; \backslash Sigma\; \backslash sqrt\{(\backslash dot\{X\}\; \backslash cdot\; X\text{'})^2\; \; (\backslash dot\{X\})^2\; (X\text{'})^2\}\; \backslash $

where $X\; \backslash cdot\; Y:=\; \backslash eta\_\{\backslash mu\; \backslash nu\}X^\backslash mu\; Y^\backslash nu$.
The factors before the integral give the action the correct units, energy multiplied by time. T_{0} is the tension in the string, and c is the speed of light. Typically, string theorists work in "natural units" where c is set to 1 (along with Planck's constant $\backslash hbar$ and Newton's constant G). Also, partly for historical reasons, they use the "slope parameter" $\backslash alpha\text{'}$ instead of T_{0}. With these changes, the Nambu–Goto action becomes
 $\backslash mathcal\{S\}\; =\; \backslash frac\{1\}\{2\backslash pi\backslash alpha\text{'}\}\; \backslash int\; \backslash mathrm\{d\}^2\; \backslash Sigma$
\sqrt{(\dot{X} \cdot X')^2  (\dot{X})^2 (X')^2}.
These two forms are, of course, entirely equivalent: choosing one over the other is a matter of convention and convenience.
Two further equivalent forms are
 $\backslash mathcal\{S\}\; =\; \backslash frac\{1\}\{2\backslash pi\backslash alpha\text{'}\}\; \backslash int\; \backslash mathrm\{d\}^2\; \backslash Sigma\; \backslash sqrt\; ^2\; \; \{X\text{'}\}^2\},$
and
 $\backslash mathcal\{S\}\; =\; \backslash frac\{1\}\{4\backslash pi\backslash alpha\text{'}\}\; \backslash int\; \backslash mathrm\{d\}^2\; \backslash Sigma\; (\{\backslash dot\{X\}\}^2\; \; \{X\text{'}\; \}^2).$
Typically, the Nambu–Goto action does not yet have the form appropriate for studying the quantum physics of strings. For this it must be modified
in a similar way as the action of a point particle. That is classically equal to minus mass times the invariant length in spacetime,
but must be replaced by a quadratic expression with the same classical value.^{[1]}
For strings the analog correction is provided by the Polyakov action, which is classically equivalent to the Nambu–Goto action, but gives the 'correct'
quantum theory. It is, however, possible to develop a quantum theory from the Nambu–Goto action in the light cone gauge.
References
Literature
 Zwiebach, Barton, A First Course in String Theory. Cambridge University Press (2004). Errata available online.
 Ortin, Thomas, Gravity and Strings, Cambridge Monographs, Cambridge University Press (2004). ISBN 9780521035460.
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