In string theory, as the one dimensional string propagates in time, it sweeps out a two-dimensional surface known as the string worldsheet. The spacetime coordinates are taken to be functions $X = X(x, \tau)$ where $\tau$ is like time and $x$ is spacelike. The $X$ obeys the wave equation: $$\frac{\partial^2 X}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 X}{\partial \tau^2} $$ where $c$ is the propagation speed. The initial conditions are $X(x,0) = a(x)$ and $\partial X(x,0) / \partial \tau = b(x)$.

a) Take the Fourier transform with respect to $x$ to get $\tilde X (k,\tau)$. You should now have a first-order ODE in time. The solution for the harmonic equation: $\tilde X (k, \tau) = A(k) \cos (kc\tau)+ B(k) \sin(kc\tau)$.

b) Use the initial conditions to relate $A(k)$ and $B(k)$ to the Fourier transform of $a(k)$ and $b(k)$ and rewrite the solution $\tilde X(k,r)$.

c) Invert the transform to give the solution $X(x,\tau) (use the convolution theorem to do the integrals for the inverse transform).

d) In string theory, the so-called D-branes are surfaces on which open strings can end. The relevant boundary conditions are known as the Dirichlet boundary conditions $X(0,\tau) = 0$ and $X(l,\tau) = y$ for all $\tau$, where $y$ is a constant. Impose this boundary condition on the solution.

Ok, first of all... what? I have that the Fourier transform of $\partial^2 X / \partial x^2$ is $-k^2 \tilde X$, but not much else. I also heard that the integrals in part c diverge?

  • $\begingroup$ It is not clear what you question is and this might get closed for that reason. One correction, after FFT in x you do not have a first order ODE in time, you get a second order ODE in time. $\endgroup$ – ggcg May 3 at 20:33
  • $\begingroup$ Hello OP! There are a few suggestions to improve your question: 1. What do you mean by ''what?'' Is there something that you expected here that is counter intuitive to you? 2. Probably you can ellaborate on what more do you want to do with your strings and then we can suggest whether Fourier transform is enough to solve your problem or if you need more techniques. 3. Could you please provide an example where you find the integral in (c) to diverge? $\endgroup$ – nGlacTOwnS May 4 at 1:33

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