A.S. Kotanjyan, A.A. Saharian, V.M. Bardeghyan Department of Physics, Yerevan State University...
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Transcript of A.S. Kotanjyan, A.A. Saharian, V.M. Bardeghyan Department of Physics, Yerevan State University...
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