1. A.S. Kotanjyan, A.A. Saharian, V.M. Bardeghyan Department of Physics, Yerevan State University Yerevan, Armenia Casimir-Polder potential in the geometry of the cosmic string with a cylindrical shell

2. Content Motivation Electromagnetic field Green tensor in the geometry of a cosmic string Casimir-Polder forces Casimir-Polder forces in the geometry of cosmic string with a conducting cylindrical shell

3. Motivation Cosmic strings generically arise within the framework of grand unified theories and could be produced in the early Universe as a result of symmetry breaking phase transitions Cosmic strings are still candidates for the generation of a number of interesting physical effects: Generation of Gravitational waves High-energy cosmic rays Gamma ray bursts Effective cosmic string geometry arises in a number of condensed matter systems (for example vortex lines superconductors or in liquid helium)

4. Geometry of the problem: Cylindrical waveguide with a cosmic string along the axis Topological defect (cosmic string) conical (δ-like) singularity angle deficit Cosmic string Conducting cylindrical shell Line element The angle deficit is related to the linear mass density:

5. In quantum field theory the non-trivial topology induced by cosmic strings leads to non-zero vacuum expectation values for physical observables (vacuum polarization) Another type of vacuum polarization arises when boundaries are present (Casimir effect) We consider combined effects of the topology and boundaries on the Casimir-Polder (CP) force acting on a polarizable microparticle Quantum effects from topology and boundaries

6. Boundary-free cosmic string geometry Nontrivial topology due to the cosmic string changes the structure of the vacuum electromagnetic field Neutral polarizable microparticle placed close to the string experiences CP force polarizability tensor Retarded Green tensor for the electromagnetic field in the geometry of a cosmic string Retarded Green tensor in Minkowski spacetime

7. Eigenfrequencies of the electromagnetic field TM wavesTE waves is the th zero of the Bessel function ( ) or its derivative ( ): By using the mode summation method explicit expressions are given for all components of the tensor Off-diagonal components vanish in the coincidence limit

8. CP potential CP potential for general case of the polarizability tensor Integral term vanishes for integer values of q

9. Asymptotic: Large distances At large distances (compared with wavelengths corresponding to oscillator frequencies )from the string For isotropic polarizability the force is repulsive The components of the polarizability tensor in cylindrical coordinates associated with the cosmic string are related to the corresponding eigenvalues by Coefficients depend on the orientation of the polarizability tensor principal axes with respect to the string Dependence of the CP potential on the orientation of the principal axes leads to the moment of force

10. Asymptotic: Small distances Eigenvalues of the polarizability tensor In the leading order To discuss the asypmtotic at small distances we consider the oscillator model In dependence of the eigenvalues for the polarizability tensor and of the orientation of the principal axes, the CP force can be either repulsive or attractive

11. CP force in the single oscillator model

12. CP potential induced by the conducting cylindrical shell Green tensor on imaginary frequency axis is evaluated in a way similar to that used in V.B. Bezerra E.R. Bezerra de Mello, G.L. Klimchitskaya, V.M. Mostepanenko, A.A. Saharian, Eur. Phys. J. C, 71, 1614 (2011) for a cylindrical boundary in Minkowski spacetime by using the Abel-Plana-type summation formula for the series over A is the radius of cylindrical shell

13. CP potential induced by the conducting cylindrical shell CP potential is presented in the decomposed form Potential in the cosmic string geometry without boundaries (first part of this talk) Part in the potential induced by the cylindrical boundary Expression for is obtained for general case of anisotropic polarizability

14. Interior region: Isotropic case In the case of isotropic polarizability: Modified Bessel functions Notation: In the oscillator model:

15. Interior region: Isotropic case Boundary induced part of CP force is attractive with respect to the cylindrical boundary By taking into account that the pure string part is repulsive with respect to the string we conclude that Total force is directed along the radial direction to the cylindrical boundary

16. Exterior region: Isotropic case Boundary induced part in CP potential for the oscillator model: Boundary induced part is attractive In the exterior region the pure string and boundary induced parts in CP force have opposite signs Notation:

17. CP force: Exterior region At large distances from the cylindrical boundary: Near the cylinder the boundary induced part dominates and the total CP force is attractive with respect to the boundary At large distances pure string part dominates and CP force is repulsive

18. CP potential

19. Conclusions Explicit formulae are derived for the CP potential inside and outside of a conducting cylindrical shell in the geometry of a cosmic string In the geometry of boundary-free cosmic string and for isotropic polarizability CP force is repulsive CP force is decomposed into purely string and cylinder induced parts Boundary induced part is attractive with respect to cylindrical boundary for both exterior and interior regions and it dominates near the cylinder At large distances from the cylindrical shell the string part dominates and the effective force is repulsive