Lectures on String Theory

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It follows from the Schwarz inequality that
(α ∗ n α m)(α ∗ m α n) = |(α ∗ n α m)| 2 ≤ (α ∗ n α n)(α ∗ m α m) ≤ nm(α ∗ n α n)(α ∗ m α m) .
Therefore,
∞X
J 2 ≤
n,m=1(α ∗ n α n)(α ∗ m α m) = 1 ∞X
X ∞ 1
α n α −n α m α −m =
4
n=−∞
m=−∞
(2πT ) 2 M 4
because in the open string case
∞X
M 2 = πT α n α −n = 2πT X α n α −n .
n=−∞ n>0
Thus, for an open string motion we found inequality
J ≡
pJ 2 ≤ 1
2πT M 2 .
Here the parameter
α ′ = 1
2πT
is called a slope of the Regge trajectory. The function J = α ′ M 2 is a straight line in the (M 2 , J) plane whose slope is α ′ .
Consider a closed (pulsating) string solution
x = R cos σ cos τ , y = R sin σ cos τ , t = Rτ .
We see that
P 0 = 2πRT =⇒ T = P 0
2πR ≡
i.e. tension is energy per unit length.
3.4 Strings in physical gauge
E
2πR ,
As we have seen upon fixing conformal gauge we are still left with the gauge freedom.
It corresponds to reparametrizations of the special type (solutions to the conformal
Killing equation):
σ + → ξ + (σ + ) , σ − → ξ − (σ − ) ,
where ξ ± are two arbitrary functions (periodic in σ). This freedom can be further
fixed leaving only physical excitations. This is achieved by imposing the so-called
light-cone gauge.
3.4.1 First order formalism
Introduce the light-cone coordinates in the d-dimensional Minkowski space
X ± = 1 √
2
(X 0 ± X d−1 ), X i , i = 1, . . . , d − 2 .
Consider the Polyakov action and introduce the light-cone momenta conjugate to
the light-cone coordinates
P ± =
∂L
∂Ẋ± ,
P i = ∂L
∂Ẋi (3.56)