Equation of motion of open string with Dirchlet b.c

[benbenny]

Fashioned after the derivation of the equation of motion for a string with Neumann b.c in Zwiebach's a first course of string theory, I have derived the very similar equation using Dirchlet b.c. My result, in natural units, is
$$X^{\mu}(\tau,\sigma)=X_{0}^{\mu}-2\alpha' p^{\mu}\sigma +\sum_{n\ne 0}\left(\frac{\sqrt{2\alpha'}}{\sqrt{n}}\sin(n\sigma) a_{n}^{\mu}e^{-in\tau} \right)$$

Im having a hard time understanding the significance of the term
$$2\alpha' p^{\mu}\sigma$$ .

From comparing this result to the Neumann b.c derived string, I understand that this term signifies translational momentum of the center of mass of the string in spacetime. Since this string has fixed endpoints, my intuitive guess would be that it would have zero translational momentum. Further I am baffled by the sigma dependence of this term which indicates that this momentum term is zero at one endpoint of the string, and maximized at the other end. I am lost on this, any clarification would be much appreciated.

Thanks.

Ben

 

[benbenny]

Fashioned after the derivation of the equation of motion for a string with Neumann b.c in Zwiebach's a first course of string theory, I have derived the very similar equation using Dirchlet b.c. My result, in natural units, is
$$X^{\mu}(\tau,\sigma)=X_{0}^{\mu}-2\alpha' p^{\mu}\sigma +\sum_{n\ne 0}\left(\frac{\sqrt{2\alpha'}}{\sqrt{n}}\sin(n\sigma) a_{n}^{\mu}e^{-in\tau} \right)$$

Im having a hard time understanding the significance of the term
$$2\alpha' p^{\mu}\sigma$$ .

From comparing this result to the Neumann b.c derived string, I understand that this term signifies translational momentum of the center of mass of the string in spacetime. Since this string has fixed endpoints, my intuitive guess would be that it would have zero translational momentum. Further I am baffled by the sigma dependence of this term which indicates that this momentum term is zero at one endpoint of the string, and maximized at the other end. I am lost on this, any clarification would be much appreciated.

Thanks.

Ben

Anyone know why I am getting no answer for this one? Should I post it in the beyond the standard model forum?

Cheers.

 

[diazona]

Maybe ask one of the mentors/moderators to move it... you're probably more likely to get an answer there since this is beyond the level of what usually winds up in the HW forum. (And you're asking more of a conceptual question than "how do I do this problem" anyway )

 

[Mindscrape]

Yeah, who wants to study all the string nonsense anyway? :p

 

[benbenny]

Ok. Just thought that since its undergrad level question...but ill ask a moderator. thanks