ZeroPoint Energy in Spheroidal Geometries

1
Zero-Point Energy in Spheroidal Geometries

• Adrian Kitson
• Institute of Fundamental Sciences
• Massey University
• New Zealand

2
Outline

• Motivation
• Spheroidal geometry
• Scalar field
• Vector field
• QCD flux tubes

3
Motivation

• Lattice QCD simulations

4
Spheroidal geometry

• Cartesian equation
• Prolate
• Oblate
• Ellipticity
• Prolate
• Oblate

z
b
y
a
x
5
Spheroidal geometry

• Prolate spheroidal coordinates
• where

z
b
y
a
x
6
Scalar field

• Consider a massless scalar field satisfying
  Dirichlet boundary conditions on the surface of a
  prolate spheroid
• Assuming harmonic time dependence, , the
  field equation becomes Helmholtzs equation
• This can be solved by separation of variables in
  prolate spheroidal coordinates

7
Radial function

• The radial function satisfies the following
  differential equation
• In the spherical limit this reduces
  to
• which is the spherical Bessel differential
  equation with

8
Radial function

• For general c
• where
• The coefficients satisfy a
  three-term recursion relation
• The radial functions are normalized to have the
  same asymptotic behaviour as the spherical Bessel
  functions

9
Dirichlet boundary condition

• The Dirichlet boundary condition becomes
• In terms of dimensionless variables
• where e is the prolate ellipticity
• The eigen-energy modes are

10
Zeta-function approach

• The ZPE is defined as
• (though from now on let )
• Spectral zeta function
• The zeta-function approach defines the ZPE as

11
Zeta-function approach

• In terms of dimensionless variables
• This can be written as a contour integral
• This can be analytically continued to give
• which is valid for

12
Zeta-function approach

• For small ellipticity
• The first term is simply the zeta function for a
  spherical boundary

13
Zeta-function approach

• Rescaling , and writing in terms of
  Ricatti-Bessel functions
• where and the Ricatti-Bessel
  functions are

14
Bessel functions

• Uniform asymptotic expansion
• where and

15
Zeta-function approach

• Rescaling , and writing in terms of
  Ricatti-Bessel functions
• where and the Ricatti-Bessel
  functions are

16
Zeta-function approach

• The coefficient of
• Using the uniform asymptotic expansion for the
  Bessel functions the integrand is expanded to

17
Zeta-function approach

• Laurent expansion of the zeta function about
• The contribution to the ZPE from the coefficient
  of
• Including exterior modes gives

18
Greens function approach

• The Greens function for the scalar field
  satisfies
• If all space is considered, the zero-point energy
  is given by
• Working to , the contribution to the
  ZPE from the coefficient of

19
Zero-point energy

• The zero-point energy of a massless scalar field
  subject to Dirichlet boundary conditions on a
  prolate spheroidal shell
• The spherical part is taken from Romeos work on
  the zeta-function approach in spherical
  geometries 1
• 1 A. Romeo, Bessel ?-function approach to the
  Casimir effect of a scalar field in a spherical
  bag, Phys Rev D52 7308 (1995)

20
Results

• Prolate spheroid (with a 0.1)
• For fixed a, the ZPE of a scalar field subject
  to prolate spheroidal boundary conditions
  decreases as b increases

• Oblate spheroid (with b 0.1)
• For fixed b and increasing a the ZPE again
  decreases
• For large oblate ellipticity the ZPE is negative

a
b
21
Results

• ZPE of a scalar field subject to Dirichlet
  boundary conditions on the surface of a spheroid

prolate
b
a
oblate
22
Vector field

• The vector Helmholtzs equation cannot be solved
  by separation of variables in prolate spheroidal
  coordinates
• If axial symmetry is assumed, then the vector
  field can be separated into transverse electric
  (TE) and transverse magnetic (TM) modes
• The boundary conditions are

23
Results

• ZPE of an axially symmetric electromagnetic field
  inside a perfectly conducting prolate spheroidal
  cavity

a 0.1
b
24
QCD flux tubes

• Oxman et al. found for an infrared modified gluon
  bag model the ZPE of a spherical bag is
  attractive 2
• For an infrared modified propagator the
  zero-point energy is
• Assuming the above relation, and accounting for
  the 8 gluons, the QCD ZPE for a prolate
  spheroidal cavity is
• 2 L. E. Oxman, N. F. Svaiter R. L. P.G.
  Amaral, Attractive Casimir effect in an infrared
  modified gluon bag model, hep-th/0507195 (2005)

25
Results

• ZPE of QCD flux tube?

a 0.1
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Summary

• The zero-point energy of scalar fields in
  spheroidal geometries can be calculated for small
  ellipticity
• Vector fields in spheroidal geometries are more
  of a challenge
• Assuming the infrared modified result, the ZPE of
  a QCD flux tube gives rise to an attractive
  inter-quark force

27
Acknowledgments

• Prof. Tony Signal
• Massey University
• Institute of Fundamental Sciences and the Royal
  Society of New Zealand
• QFEXT05 Organizing committee