Sphere-plate Casimir interaction in (D + 1) -dimensional spacetime

2015

We study the cylinder-plate and the cylinder-cylinder Casimir interaction in the (D + 1)-dimensional Minkowski spacetime due to the vacuum fluctuations of massless scalar fields. Different combinations of Dirichlet (D) and Neumann (N) boundary conditions are imposed on the two interacting objects. For the cylinder-cylinder interaction, we consider the case where one cylinder is inside the other, and the case where the two cylinders are outside each other. By computing the transition matrices of the objects and the translation matrices that relate different coordinate systems, the explicit formulas for the Casimir interaction energies are derived. From these formulas, we compute the large separation and small separation asymptotic behaviors of the Casimir interaction. For the cylinder-plate interaction with R ≪ L, where R is the radius of the cylinder and L is the distance from the center of the cylinder to the plate, the order of decay of the Casimir interaction only depends on the boundary conditions imposed on the cylinder. The orders are L −D+1 / ln(L) and L −D−1 / ln L respectively for Dirichlet and Neumann boundary conditions on the cylinder. For two cylinders with radii R1 and R2 lying parallelly outside each other, the orders of decay of therespectively for DD, DN/ND and NN boundary conditions, where L is the distance between the centers of the cylinders. The more interesting and important characteristic of Casimir interaction appears at small separation. Using perturbation technique, we compute the small separation asymptotic expansions of the Casimir interaction energies up to the next-to-leading order terms. The leading terms coincide with the respective results obtained using proximity force approximation, which is of order d −D+1/2 , where d is the distance between the two objects. The results on the next-toleading order terms are more interesting and important. We find some universal behaviors. It is also noticed that for the case of Dirichlet-Dirichlet cylinder-plate interaction, the next-to-leading order term agrees with that obtained using derivative expansion. Hence, based on our results on other boundary conditions and on the cylinder-cylinder interaction, we postulate a formula for the derivative expansion to expand the Casimir interaction energy up to the next-to-leading order terms for DD, DN, ND and NN boundary conditions, for the interaction between two curved surfaces in (D + 1)-dimensional Minkowski spacetime. It is found that the postulate agrees with our previous results on the sphere-sphere interactions except when D = 4.

Section: Discussionmentioning

confidence: 99%

Section: Small Separation Asymptotic Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Scalar cylinder-plate and cylinder-cylinder Casimir interaction in higher dimensional spacetime

2015

“…Various approaches have been proposed to compute the Casimir interaction between two objects in ð3 þ 1Þ-dimensional Minkowski spacetime which can either be classified as the worldline method [15][16][17][18][19], multiple scattering approach [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] or mode summation approach [35][36][37][38]. Generalizing the prescription in [38] to arbitrary dimensions, we have been able to compute the zero temperature Casimir interaction between a sphere and a plate [39] and between two spheres [40] in ðD þ 1Þ-dimensional Minkowski spacetime.…”

Section: Introductionmentioning

confidence: 99%

Finite temperature Casimir interaction between spheres in(D+1)-dimensional spacetime: Exact computations and asymptotic expansions

2014

We consider the finite temperature Casimir interaction between two Dirichlet spheres in ðD þ 1Þ-dimensional Minkowski spacetime. The Casimir interaction free energy is derived from the zero temperature Casimir interaction energy via the Matsubara formalism. In the high temperature region, the Casimir interaction is dominated by the term with zero Matsubara frequency, and it is known as the classical term since this term is independent of the Planck constant ℏ. Explicit expression of the classical term is derived and it is computed exactly using appropriate similarity transforms of matrices. We then compute the small separation asymptotic expansion of this classical term up to the next-to-leading order term. For the remaining part of the finite temperature Casimir interaction with nonzero Matsubara frequencies, we obtain its small separation asymptotic behavior by applying certain prescriptions to the corresponding asymptotic expansion at zero temperature. This gives us a leading term that is shown to agree precisely with the proximity force approximation at any temperature. The next-to-leading order term at any temperature is also derived and it is expressed as an infinite sum over integrals. To obtain the asymptotic expansion at the low and medium temperature regions, we apply the inverse Mellin transform techniques. In the low temperature region, we obtain results that agree with our previous work on the zero temperature Casimir interaction.

“…Based on the pioneering work [10], a perturbative machinery has been developed in [11,[44][45][46][47][48][49][50][51][52][53]. We use the same method with necessary modification to the present scenario.…”

Section: The Small Separation Asymptotic Behaviormentioning

confidence: 99%

Ferminoic Casimir effect between spheres

2015

We consider the Casimir interaction between two spheres corresponding to massless Dirac fields with MIT-bag boundary conditions. Using operator approach, we derive the TGTG-formula for the Casimir interaction energy between the two spheres. A byproduct is the explicit formula for the translation matrix that relates the fermionic spherical waves in different coordinate systems. In the large separation limit, it is found that the order of the Casimir interaction energy is L −5 , where L is the separation between the centers of the spheres. This order is intermediate between that of two Dirichlet spheres (of order L −3 ) and two Neumann spheres (of order L −7 ). In the small separation limit, we derive analytically the asymptotic expansion of the Casimir interaction energy up to the next-to-leading order term. The leading term agrees with the proximity force approximation. The result for the next-to-leading order term is compared to the corresponding results for scalar fields and electromagnetic fields.