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This is the wave equation for relativistic strings using light-cone coordinates.
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Solutions of the Wave Equation
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In the next chapter, we will consider the hamiltonian and stress-energy tensor and write down conserved charges and currents for the string. Right now, let s focus on nding a solution of the wave equation given in Eq. (2.46). From elementary mechanics, we know that the solution of a wave equation can be written in terms of a superposition of waves moving to the left on the string and waves moving to the right on the string. If the motion is in one dimension (call it x), then we can write down a solution of the form f (t , x ) = fL ( x vt ) + fR ( x + vt ) We will write the equations of motion for the relativistic string in the same way. We have a solution which is a superposition of left-moving components X L ( + ) and right-moving components X R ( ):
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X ( , ) = X L ( + ) + X R ( )
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(2.47)
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You should recall from partial differential equations that the most general solution can be written as an expansion of Fourier modes. Here, we denote these modes as k , and write the left-moving and right-moving components as
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2 k ik ( + ) x s s X ( , ) = + p ( + ) + i e 2 2 2 k 0 k
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(2.48)
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X R ( , ) =
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2 x + s p ( ) + i s k e ik ( ) 2 2 2 k 0 k
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(2.49)
String Theory Demysti ed
We have introduced some new terms here. First, we have included the characteristic length of the string which is related to the Regge slope parameter and hence to the tension in the string via T= 1 2 1 2
=
(2.50)
Next, notice the coordinate x and momentum p . These are the center of mass coordinate and the total momentum of the string, respectively. The zeroth -order Fourier mode is de ned in terms of
0 =
0 =
(2.51)
What does this tell us physically The solutions imply that the string can move as a single unit with position and momentum through space-time. In addition, it also has vibrations, which are described by the modes k . When you see modes like this, you should think quantization (think in terms of the harmonic oscillator or elds in quantum eld theory). Remember, we are still in the realm of classical physics, even if it s relativistic classical physics. So the solutions of the wave equation X , X L , and X R must be real functions. This implies that x and p are real (as they must be, given their physical interpretation) and allows us to relate positive and negative modes
k = k
k = k
(2.52)
where * represents the complex conjugate. Now, let s take a look at the solutions of the wave equation with different boundary conditions.
Open Strings with Free Endpoints
Open strings with free endpoints satisfy the Neumann boundary condition that we reproduce here: X =0 when = 0,
CHAPTER 2 Equations of Motion
Now, looking at Eqs. (2.48) and (2.49) we see that
2 X L = s p + 2 2 X R = s p 2 s
k 0 s
ik ( + ) k
2
k 0
ik ( + ) k
Summing these as in Eq. (2.47) and setting = 0, X =0 0 =
( p p ) +
( 2
s k 0
k e ik
This tells us that in the case of an open string with free endpoints, it must be the case that p = p (string cannot wind around itself ) (same modes for left- and right-moving waves)
=
Physically, this means that for an open string with free endpoints the modes combine to form standing waves on the string. Before we move on to our next case, let s consider the other boundary condition, which is imposed at the other end of the string = . 0= = =
X L
=
X R
=
2
k 0
ik ( + ) k
p
2
s k 0
ik ( + ) k
2
k 0 s
ik k
e ik eik ( 2i ) 2i sin( k )
=i 2
k 0
ik k
String Theory Demysti ed
This can only be true if sin k = 0 , which means that k must be an integer. Denoting it by n, a simple exercise shows that Eq. (2.47) can be written as X = x +
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