String Theory Demystified

32 String Theory Demystifi ed
From elementary calculus, we know that in a given space described by a metric G αβ ,
an element of surface area is written as
dA = −det G d ξ αβ 2
In our case, the metric we need is the induced metric [Eq. (2.15)]. So we take
dA = −γdτdσ. If we integrate from some initial proper time τ to some fi nal
i
proper time τ f and over the length of the string (which we’ll denote as ) then the
action can be written as
Or explicitly, using Eq. (2.17)
τ
τi
∫ ∫
0
f
S =−T dτ dσ
−γ
τ
τi
∫ ∫
0
(2.18)
f
S =− T dτ dσ X 2 2 2
X′ −( X ⋅ X′
) (2.19)
The actions in Eqs. (2.18) and (2.19) are called the Nambu-Goto action, which
describes the dynamics of a (classical) relativistic string. As the motion of a point
particle in space-time serves to minimize the length of the world-line, the motion of a
classical string in space-time acts to minimize the surface area of the worldsheet.
Before moving ahead to a quantum theory of strings, we need to fi nd the equations
of motion for the string which we can then later quantize.
Equations of Motion for the String
Now that we have the action in place (and hence the lagrangian) we can obtain the
equations of motion for the string. We do this using the action principle, which tells
us to vary the action and set the result to 0
δS = 0
When computing the variation of the action, we will derive the equations of motion that
will be a partial differential equations—meaning that we will need to specify boundary
conditions in order to solve them. There are two different types of strings we need to
consider when looking at boundary conditions: open strings and closed strings.
If a string is open, this means exactly what it says, that the string is a free piece
of string with loose ends moving through space-time. The worldsheet in this case