A.S. Kotanjyan, A.A. Saharian, V.M. BardeghyanDepartment of Physics, Yerevan State University

Yerevan, Armenia

Casimir-Polder potential in the geometry of the cosmic string with a cylindrical shell

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

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)

Geometry of the problem: Cylindrical waveguide with a cosmic string along the axis

a

Topological defect (cosmic string)

conical (δ-like) singularityangle deficit

02 Cosmic string

Conducting cylindrical shell

Line element

The angle deficit is related to the linear mass density:

G80

2

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

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

Eigenfrequencies of the electromagnetic field

1 ,0 , 22)(,

)(,

znmnm kj

TM waves TE waves

zknm ,...,2,1 ,...,2,1,0)(

,nmj is the th zero of the Bessel function ( ) or its

derivative ( ):

0)(

,)( /2 ,0)( qjJ nmqm

n 01

By using the mode summation method explicit expressions are given for all components of the tensor

Off-diagonal components vanish in the coincidence limit

CP potential

CP potential for general case of the polarizability tensor

Integral term vanishes for integer values of q

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

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

CP force in the single oscillator model

30g

Fr

r0 q

q

0

2

q

10 r

CP potential induced by the conducting cylindrical shell

a

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 )(

,nmj

A is the radius of cylindrical shella

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

Interior region: Isotropic case

In the case of isotropic polarizability:

Modified Bessel functions

Notation:

In the oscillator model:

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

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:

CP force: Exterior region

At large distances from the cylindrical boundary:)0(

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

CP potential

)(rU

ra

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