Casimir Energy of an Open String with Angle-Dependent Boundary Conditions
- PDF / 377,745 Bytes
- 7 Pages / 612 x 792 pts (letter) Page_size
- 92 Downloads / 226 Views
I, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Casimir Energy of an Open String with Angle-Dependent Boundary Conditions A. Jahana,* and I. Brevikb a Research
Institute for Astronomy and Astrophysics of Maragha (RIAAM), University of Maragheh, P. O. Box 55136-553, Maragheh, Iran b Department of Energy and Process Engineering, Norwegian University of Science and Technology 7491, Trondheim, Norway * e-mail: [email protected] Received April 17, 2019; revised May 9, 2019; accepted May 26, 2019
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. DOI: 10.1134/S1063776119110049
1. INTRODUCTION The Casimir energy is a physical manifestation of vacuum energy [1]. It is purely a quantum phenomenon which, for example, causes two parallel conducting plates to attract each other. The vacuum energy of open and closed strings, as simple cases, has been investigated by several authors. Lüscher et al. were the first ones who calculated the Casimir energy of an open string which is now called the Lüscher potential [2–4]. They obtained this potential by considering a static quark-antiquark with the chromo-electric field between them vibrating string. The Casimir energy of a piece wise string was considered in [5–9]. The vacuum energy of an open string placed between two beads was obtained in [10]. The Lüscher potential is recovered when the masses of the beads become large. The quantum corrections to the Lüscher potential were calculated in [11], where the authors interpreted the corrections as a sort of non-local effect in a bosonic string. The Nambu–Goto model of an open string was used to model the inter-quark potential in [12, 13]. The string was assumed to end on point masses with mass m. It was shown that one recovers the Lüscher potential in the limits m → 0, ∞. The present work is to a large extent a sequel to a previous one where we obtained the Casimir energy as the zero-temperature limit of free energy of an open string in an angle-dependent set-up [14]. We there obtained the finite-temperature free energy using the
path integral method. Here, we make use of three different methods to calculate the Casimir energy for an open string whose ends are located on two s
Data Loading...
