Divergencies in bulk Casimir energy

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Divergencies in bulk Casimir energy

• Fialkovsky I.V.
• St Petersburg State University

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Contents of the talk

• Motivation and background
• Casimir effect as such
• Implementing boundaries in QFT
• Controvercies over (all different types of)
  divergencies
• Casimir energy of Bulk systems
• generating fucntional within Path Integral
  approach
• exact calculation of the Casimir energy
• week coupling and/or short distance asymptotics
• Results and discussion

This talk is based on results obtained in
collaboration withYuriy Pismak and Vladimir
Markov
for details see I.V. Fialkovsky, V.N. Markov,
Yu.M. Pis'mak, On the Casimir energy for scalar
fields with bulk inhomogeneities,
arXiv0804.3603v2
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Casimir effect
in vacuum neutral, -thin perfectly
conducting plates
related terminology - vacuum polarization-
vacuum energy- Casimir-Polder interaction
Casimir, 1948
Essential modification of the ground (vacuum)
state and its energy in presence of boundaries
(spatial defects)
Experimentally - confirmed with 0.5
error e.g. G. L. Klimchitskaya, et al. 05, U.
Mohideen, A. Roy, 98, 99
Theoretically - well established for forces
between separated objects - still controversial
for self-pressure of isolated objects
Most recent Reviews Mostepanenko,
quant-ph/0702061, Klimchitskaya, Mostepanenko,
quant-ph/0609145
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Implementing boarders into QFT

• Explicit Boundary conditions (BC)

• Quantum fields are subject to BC either imposed
  restricting the measure of Path Integral, or
  while solving the equation of motion

e.g. Bordag, Robaschik, Wieczorek84
Problems confinig all modes with the same
condition is not physical

• matching conditions interaction with
  Delta-potentials

• Quantum fields interacting with extarnal field of
  delta-function profile. Strong coupling limit
  reproduce BC.

Symanczik 81, lots of works by Bordag, Milton,
Jaffe, our group, etc. see also talks by Pismak
and Bordag at RG2008
Should exist a limit of smoother (realistic)
interaction leading to delta-potential

• smoother potentials theta-function, etc.

• Quantum fields interacting with extarnal field
  with theta-function profile. Sharp limit
  reproduces delta-potential. Strong coupling then
  reproduces BC.

Milton02,05, Graham et al. 03
Controvercies all around.
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Divergencies in Casimir Energies (Forces)

• Interaction of Disjoint bodies

• Force is (always) finite - all divergencies in
  energy (contribution of the whole empty space,
  etc.) are independent of the distancce.

• Self-energy of Single objects

• The so calledlater appeared to be simply an
  UV-divergency subject to renormalization and
  introduction of counter-termsBordag,
  Vassilevitch 04, also talk by Bordag at RG2008

• Coordinate divergencies in local densities vs.
  finite total energy

Deutch, Candelas 79, Olum, Graham 02,
Milton'04, 05, etc.

• Sharp-limit divergency of our interest, to be
  discussed now

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Bulk systems the statement of the problem
The model is described by the action
with profile function
i.e. piecewise constant or square-well
potential.
Normalization is equivalent to condition of
matter conservation within the slab.
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Sharp limit
Recalling the normalization condition for
one would expect to have selfconsistent limit
procedure from theta- to delta-potentials.
Basing on perturbation analysis it was claimed
that
thus invalidating phycal grounds for
consideration of -potentials.
Graham, et al. 03
Still, in Milton05 found no divergencies in
sharp limit when analyzing energy
dencities. Never through exact result, which
was merely forgotten.
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Generating functional
To describe the physical properties of the system
we consider
normalization condition of the generating
functional
defines the reference point for the total energy
of the system
To perform the integration, one first represents
the defect action as
- auxiliary fields defined of the support of the
layer only
Surprisiungly, this procedure goes exactly as in
the case of delta-potentials.
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Generating functional
introducing projector operator
we can integrate out first the s and then the
s to obtain
with modified propagator
All understod as linear operations
normalization factor
Thus, the Casimir energy
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Casimir energy
For the case of single layer, in Fourier
representation
with help of we can write
We note that U is proportional to a Green
function of an ordinary differential operator
then
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Casimir energy explicit calculation of
Using symmetry conditions which follow from
definition
we get
yet again from definitions of U and Q it follows
that
which yelds the coefficients
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Casimir energy, after all
Using obtained
with obvious
first derived by Bordag95, recently rederived by
Vassilevitch, Konoplja07 forgotten and
confronted by Fosco et al.07 and yet again
supported by our group
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Modified propagator
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Asymptotical behaviour week coupling
Can be rewritten in dimentionless units
convergent for
Non-trivial behaviour for masless theory
Surprisingly also gives asymptotics for short
separation
The same -behavior is found for massive theory.
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Sharp limit divergency Physical interpretation

• Finite renormalization

we can imagine the matter of the layer having a
classical energy
with constants a,b that can cancel the divergency
in the Casimir energy

• Physical Divergency

who told us we can shrink the layer to zero width?
We have repultion starting with some small
distance. Remeber, the amount of matter is
assumed conserved while shrinking.
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Sharp limit and renormalization
On the other hand with exact result it is
straightforward
Just the vacuum energy of a single
delta-potential plate!!!
Usually this quantaty is merely thrown away.
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Sharp limit and renormalization
self-interacting finite width layer perfectly
observavble
self-interacting delta-potential layer
non-observable
Thus, we dont see any controvercy for planar
case.
Sharp limit for shells of non-trivial geometry
still to be adressed.
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Conclusions

• Scalar field interacting with a finite width
  layer

• considered in the framework of pure QFT (path
  integral approach)
• Casimir energy rederived
• complete modified propagator calculated for the
  first time

• Asymptotic behaviour of Casimir energy

• rigorous derivation of week coupling behaviour
• rigorous derivation of sharp-limit asymptotics
• contradiction of sharp-limit for plane geometry
  resolved

Future task

• Sharp limit for non-trivial shells (sphere, etc.)
  must be considered analyticaly on the grounds of
  EXACT results rather then numerically within
  perturbation theory

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Thank you for your attention.