Casimir Energy of an Open String with Angle-Dependent Boundary Conditions

Abstract

We consider an open string with ends laying on the two different solid beams (rods). This setup is equivalent to two scalar fields with a set of constraints at their end-points. We calculate the zero-point energy and the Casimir energy in three different ways: (1) by use of the Hurwitz zeta function, (2) by employing the contour integration method in the complex frequency plane, and (3) by constructing the Green’s function for the system. In the case of contour integration we also present a finite temperature expression for the Casimir energy, along with a convenient analytic approximation for high temperatures. The Casimir energy at zero temperature is found to be a sum of the Luscher potential energy and a term depending on the angle between the beams. The relationship of this model to an analogous open string model with charges fixed at its ends, moving in an electromagnetic field, is discussed.

APPENDIX A

To evaluate the infinite sum in the first line of (50) we write

$$\begin{gathered} \sum\limits_{n = - \infty }^\infty {\ln (n_{r}^{2} + {{\kappa }^{2}}) = \ln \prod\limits_{n = - \infty }^\infty {(n_{r}^{2} + {{\kappa }^{2}})} } \\ = \ln \left[ {\prod\limits_{n = - \infty }^\infty {({{n}_{r}} + i\kappa )} \prod\limits_{n = - \infty }^\infty {({{n}_{r}} - i\kappa )} } \right], \\ \end{gathered} $$

((A.1))

where n_r = n + r. Then by virtue of the formula

$$\prod\limits_{n = - \infty }^\infty {(nx + y) = \sin \left( {\frac{{\pi x}}{y}} \right)} $$

((A.2))

we get

$$\begin{gathered} \sum\limits_{n = - \infty }^\infty {\ln (n_{r}^{2} + {{\kappa }^{2}})} \\ = \ln \sinh \pi (\kappa + ir) + \ln \sinh \pi (\kappa - ir). \\ \end{gathered} $$

((A.3))