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. Just as the action for a free point particle is proportional to its proper time—"i.e.," the "length" of its world-line—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 T. 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," a minimum. 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": 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 non-relativistic 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:

:$S\; =\; int\_\{t\_i\}^\{t\_f\}\; L\; dt.$

(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

:$-ds^2\; =\; -(c\; dt)^2\; +\; dx^2\; +\; dy^2\; +\; dz^2,$

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

:$S\; =\; -mc\; int\; ds.$

World-sheets

Just as a zero-dimensional point traces out a world-line on a spacetime diagram, a one-dimensional string is represented by a "world-sheet." All world-sheets are two-dimensional surfaces, and we require two parameters to specify a point on the sheet. String theorists use the symbols τ and σ for these parameters. As it turns out, string theories involve higher-dimensional 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

:$x\; =\; (x^0,\; x^1,\; x^2,\; ldots,\; x^d).$

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:

:$X\; (\; au,\; sigma)\; =\; (X^0(\; au,sigma),\; X^1(\; au,sigma),\; X^2(\; au,sigma),\; ldots,\; X^d(\; au,sigma)).$

The functions $X^mu\; (\; au,sigma)$ determine the shape which the world-sheet takes. Different Lorentz observers will disagree on the coordinates they assign to particular points on the world-sheet, but they must all agree on the total "area" which the world-sheet has. The Nambu-Goto action is chosen to be proportional to this total area.

The area $mathcal\{A\}$ of the world-sheet is given by::$mathrm\{d\}\; mathcal\{A\}\; =\; mathrm\{d\}^2\; sigma\; sqrt\{-g\}$where $mathrm\{d\}^2sigma\; =\; mathrm\{d\}sigma\; ,\; mathrm\{d\}\; au$.:Furthermore, $g\_\{ab\}$ is the metric induced on the string: :$g\; =\; mathrm\{det\}\; left(\; g\_\{ab\}\; ight)$:$g\_\{ab\}\; =\; eta\_\{mu\; u\}\; frac\{partial\; X^mu\}\{partial\; sigma^a\}\; frac\{partial\; X^\; u\}\{partial\; sigma^b\}$where is the two-dimensional Minkowski metric.

Using the notation that::$dot\{X\}\; =\; frac\{partial\; X\}\{partial\; au\}$and:$X\text{'}\; =\; frac\{partial\; X\}\{partial\; sigma\},$one can rewrite the metric $g\_\{ab\}$:::$g\; =\; dot\{X\}^2\; X\text{'}^2\; -\; (dot\{X\}\; cdot\; X\text{'})^2$the Nambu-Goto action is defined as,

:

The factors before the integral give the action the correct units, energy multiplied by time. "T₀" 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 $hbar$ and Newton's constant "G"). Also, partly for historical reasons, they use the "slope parameter" $alpha\text{'}$ instead of "T₀." With these changes, the Nambu-Goto action becomes

:$mathcal\{S\}\; =\; -frac\{1\}\{2pialpha\text{'}\}\; int\; mathrm\{d\}^2\; sigma\; sqrt\{(dot\{X\}\; cdot\; X\text{'})^2\; -\; (dot\{X\})^2\; (X\text{'})^2\}.$

These two forms are, of course, entirely equivalent: choosing one over the other is a matter of convention and convenience.

Typically, the Nambu-Goto action is "not" the fundamental action physicists use when they develop quantized versions of string theory. Instead, the quantum theory is developed using the Polyakov action, which is classically equivalent to the Nambu-Goto action, but is more convenient for the quantum formulation. It is, however, possible to develop a quantum theory from the Nambu-Goto standpoint, using the light-cone gauge.

References

* Zwiebach, Barton. "A First Course in String Theory." Cambridge University Press (2004). ISBN 0-521-83143-1. [http://xserver.lns.mit.edu/~zwiebach/firstcourse.html Errata] available online.