Casimir interaction between spheres in (D + 1)-dimensional Minkowski spacetime

• Published in: The journal of high energy physics Vol. 2014; no. 5; pp. 1 - 31
• Main Author: Teo, L.P.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2014; Springer Nature B.V
• Subjects: Asymptotic expansions; Classical and Quantum Gravitation; Computation; Dirichlet problem; Elementary Particles; Energy use; High energy physics; Integrals; Mathematical analysis; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; Separation; String Theory; Translations; Field Theories in Higher Dimensions; Nonperturbative Effects
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP05(2014)016

A bstract We consider the Casimir interaction between two spheres in ( D + 1)-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar fields. We consider combinations of Dirichlet and Neumann boundary conditions. The TGTG formula of the Casimir interaction energy is derived. The computations of the T matrices of the two spheres are straightforward. To compute the two G matrices, known as translation matrices, which relate the hyper-spherical waves in two spherical coordinate frames differ by a translation, we generalize the operator approach employed in [ 39 ]. The result is expressed in terms of an integral over Gegenbauer polynomials. In contrast to the D =3 case, we do not re-express the integral in terms of 3 j -symbols and hyper-spherical waves, which in principle, can be done but does not simplify the formula. Using our expression for the Casimir interaction energy, we derive the large separation and small separation asymptotic expansions of the Casimir interaction energy. In the large separation regime, we find that the Casimir interaction energy is of order L −2 D +3 , L −2 D +1 and L −2 D −1 respectively for Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions, where L is the center-to-center distance of the two spheres. In the small separation regime, we confirm that the leading term of the Casimir interaction agrees with the proximity force approximation, which is of order , where d is the distance between the two spheres. Another main result of this work is the analytic computations of the next-to-leading order term in the small separation asymptotic expansion. This term is computed using careful order analysis as well as perturbation method. In the case the radius of one of the sphere goes to infinity, we find that the results agree with the one we derive for sphere-plate configuration. When D =3, we also recover previously known results. We find that when D is large, the ratio of the next-to-leading order term to the leading order term is linear in D , indicating a larger correction at higher dimensions. The methodologies employed in this work and the results obtained can be used to study the one-loop effective action of the system of two spherical objects in the universe.