The Nambu–Goto action is the simplest invariant
action in
bosonic string theory, and is also used in other theories that investigate string-like objects (for example,
cosmic strings). It is the starting point of the analysis of zero-thickness (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 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
Tetsuo Goto Tetsuo may refer to:
*Tetsuo (given name)
*'' Tetsuo: The Iron Man''
*'' Tetsuo II: Body Hammer''
*'' Tetsuo: The Bullet Man''
* Tetsuo, a character in ''Akira (manga)
is a Japanese cyberpunk post-apocalyptic manga series written and illust ...
.
Background
Relativistic Lagrangian mechanics
The basic principle of Lagrangian mechanics, the
principle of stationary action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
, 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
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
, 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 non-relativistic mechanics, for example, a point particle's Lagrangian is the difference between kinetic and potential energy:
. The action, often written
, is then the integral of this quantity from a starting time to an ending time:
:
(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
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ve ...
, the action should only depend upon quantities that are the same for all (Lorentz) observers, i.e. the action should be a
Lorentz scalar
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
. 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
:
and
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
:
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, hence we need two parameters to specify a point on a world-sheet. String theorists use the symbols
and
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
is the number of spatial dimensions, we can represent a point by the vector
:
We describe a string using functions which map a position in the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the ...
(
,
) to a point in spacetime. For each value of
and
, these functions specify a unique spacetime vector:
:
The functions
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 ''proper area'' which the world-sheet has. The Nambu–Goto action is chosen to be proportional to this total proper area.
Let
be the metric on the
-dimensional spacetime. Then,
:
is the
induced metric In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using ...
on the world-sheet, where
and
.
For the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
of the world-sheet the following holds:
:
where
and
Using the notation that:
:
and
:
one can rewrite the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
:
:
:
the Nambu–Goto action is defined as
:
where
.
The factors before the integral give the action the correct units, energy multiplied by time.
is the tension in the string, and
is the speed of light. Typically, string theorists work in "natural units" where
is set to 1 (along with Planck's constant
and Newton's constant
). Also, partly for historical reasons, they use the "slope parameter"
instead of
. With these changes, the Nambu–Goto action becomes
:
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
:
and
:
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.
[See Chapter 19 of
Kleinert's standard textbook on ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 5th edition]
World Scientific (Singapore, 2009)
(also availabl
online
For strings the analog correction is provided by the
Polyakov action
In physics, the Polyakov action is an action (physics), action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by Lars Brin ...
, 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
In theoretical physics, light cone gauge is an approach to remove the ambiguities arising from a gauge symmetry. While the term refers to several situations, a null component of a field ''A'' is set to zero (or a simple function of other variables ...
.
See also
*
Dirac membrane In quantum mechanics, a Dirac membrane is a model of a charged membrane introduced by Paul Dirac in 1962. Dirac's original motivation was to explain the mass of the muon as an excitation of the ground state corresponding to an electron. Anticipatin ...
References
Further reading
* Ortin, Thomas, ''Gravity and Strings'', Cambridge Monographs, Cambridge University Press (2004). .
{{DEFAULTSORT:Nambu-Goto action
String theory