1
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
On the vacuum energy between a sphere and a plane at finite temperature
I. G. Pirozhenko (BLTP, JINR, Dubna, Russia)
QFEXT11, 18-25 September 2011, Benasque
Based on the papers:M. Bordag, I. Pirozhenko, Phys. Rev. D81:085023, 2010; Phys.Rev.D82:125016,2010;arXiv:1007.2741 [quant-ph] ,
2
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
L
dR
This configuration at finite temperature was studied by
Alexej Weber, Holger Gies, Phys.Rev.D82:125019,2010 ; Int.J.Mod.Phys.A25:2279-2292,2010
Antoine Canaguier-Durand, Paulo A. Maia Neto, Astrid Lambrecht, Serge ReynaudQFEXT09 Proceedings; Phys.Rev.Lett.104:040403,2010 ; arXiv:1005.4294 ; arXiv:1006.2959 ; arXiv:1101.5258
At zero temperatureEmig et al, Wirzba, Bulgac et al, Bordag, Canaguier-Durand et al …
3
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
Basic formulas
QFEXT11, 18-25 September 2011, Benasque
,1ln2
n
nTrT MF
nTn 2
M
1ln22
10 TrdE
n
dT2
lll
ll
lllllmll HLK
LRdM
24 21,
0000121212' mm
lllllllllH l
ll
xd l depends on the boundary conditions on the sphere
K
Id
KId N
lDl ,For scalar field
The free energy
turns into the vacuum energy when
are the Matsubara frequencies,
1 Bkc
where nTn 2
4
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
For the electromagnetic field one has to account for polarizations:
Rd
RdHLK
LRdΜ TM
l
TEl
lllll
llllll
ll
ll
llllll
0
0~
~2
4 '
'
with the factors
112~
112111
llllLm
llllllllll
ll
lll
The general formulae for the dielectric ball
zKzzKnznInzInznzIzK
zIzzInznInzInznzIzId TE
l
2222
21,, ln TEl
TMl dd
T.Emig, J.Stat. Mech, 2008
5
In the limit of perfect magnetic, and fixed
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
In the limit of perfect conductor, and fixed
,zKzIzd TE
l
,
.2
2zKzzK
zIzzIzd TMl
,2
2zKzzK
zIzzIzd TEl
.zKzIzd TM
l
Thus the trace of the “polarization” matrix P in the case of a ball with has the opposite sign
TM
l
TElcond
l dd0
0P
,
,
TE
l
TMlmagn
l dd0
0P
In this case we expect the strongest repulsion.
6
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
L
dR
PFA at finite temperature
RTdTRTdT
RTdT
11
1Temperature scale
1 Bkc
Low temperature:
Medium temperature:
High temperature:
In each case holds, Rd 1Rd
The free energy per unit area for two parallel plates
,1ln
2),(
2222
kk
nd
n
edTTd
||F d
k
k is the momentum parallel to the plates
nTn 2
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
,21720
),( 3
2
dTgd
Td
||F
The free energy may be represented in the form
1
23
4
1 0
233
sinhcoth451
1ln90345122
m
n
knx
xmxmxm
xmx
edkkxxxg
The function has several representations:
)(xg
It obeys the inversion symmetry
xgxxg 14
And possesses the asymptotic expansions
.1345
,0345
~3
433
xforx
xforxxxg
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
We apply the idea of the PFA to the free energy per unit area of two parallel plates at finite temperature
,,, TyxhddydxPFA||FF
where
yxhd , is the separation between the plane and the sphere at the point yx,
In polar coordinates with
TRtdtdtR
TRdddrrR
PFA
,12
,cos1
1
0
2
2
00
||
||
F
FF
sinRr
d
R
The corresponding approximation for the force
1
0|||| ,,2 TRtddtTdR
df PFAPFA FFF 0,2 ||0
dRfT
PFA F
(in the limit )0Rd
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
Substituting the free energy for parallel plates we obtain for the free energy
Rd
This expression is meaningful if 0
,21211
720
1
03
22
3
tRTtgtdt
dRPFA
F
Low and medium temperature limits RTdT
,2121720
1
03
22
3
ttRTgtdt
dRPFA F
1dTLow temperature, ,33601720
33
22
3
RT
dRPFA
F
Medium temperature, 1RT ,201720
222
3
RTdRPFA F
High temperature, 1dT ,43
dRTPFA F
10
(exponentially suppressed at high temperature, )
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
Free energy at high temperature
,1ln2
n
nTrT MF
The leading order of high temperature expansion is given by the lowest Matsubara frequency, i.e. the term with 0n
TRTdFRdFT ,10 Fcollects contributions from 0n
01ln21
0 MTrF
For different boundary conditions
llll
llDll H
llll
LRM
2321221
20
1
llll
Dll
TMll M
llM
010 llll
Dll
TEll MM
00
01
0 Dll
Nll M
llM
With these expressions for any finite the function can be calculated numerically. A. Canague-Durand et al, Phys. Rev. Lett.104,040403 (2010)
0 Rd0F
nTdll enM 4~
11
In the limit the convergence of the orbital momentum sum gets lost. One has to find an asymptotic expansion of
I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
Large separations, , only lowest momenta contribute
,2
0)0(00
LRM mD ,
20
3)0(
11
LRM mN
,2
03
)0(11
LRM mTE ,
220
3)0(
11
LRM mTM
,4LTRD F ,
2
3
LRTNF .
23
3
LRTEMF
,22
103
)1(11
LRM mN
,22
103
)1(11
LRM mTE
3)1(
11 20
LRM mTM
In agreement with A.Canague-Durand et al, PRL104,040403 (2010)
Short separations, 0,0
.0),(0 forF
).0(,11
1,00 0
10
ii nlnl
s
is
l
l mlj
s
jMZZdndmdl
sF
By expanding the logarithm and substituting the orbital momentum sums by integrals one obtains
Bordag, Nikolaev, JPA41,2008,PRD 2010
43TF Coincides with the PFA result
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
Free energy at low temperature
iiind TT
MM 1ln1ln21
0
TrF follows from Abel-Plana formula
11
TT e
n Thanks to the Boltzman factor the low temperature expansion emerges from
33
22
110 LLL MMMMM
FF TE 0
Then,
3311ln LNLNTr M
,1 11
01 MMTrN
,1
3111
13
11
021
011
0
31
03
MMTrMMMMTr
MMTrN
LR
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
433
32
1 )(15
)(6
TLNLTNT F
Inserting this expansion into the free energy one gets
(here the limits and were interchanged)
0T ml
4
33
321 )(15
)(6
TL
LNTL
LNdLd
TTFf
and the low temperature correction to the force
The first term in this expansion may vanish, depending on the boundary conditions.
To compare this result with those obtained by A.Weber and H.Gies (Int.JMPA,2010) one should expand it for small separation RLd
423
32 TRdcRcTf
A. Scalar field, Dirichlet-Dirichlet bc
LRN 1
does not depend on the truncation ml
The term does not contribute to the force2T
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
)(61
31
32 432
3 ON At large separations
423 4.28.3 TRdRTfAt short separations
43
454 TLRTf
8mlfor
Weber and Gies have .7.2,96.3 32 cc
B. Dirichlet (sphere)-Neumann bc
The leading contribution to the force is 1N2~ T
)(241
61
),(241
61
9633
9633
ON
ON
NN
ND
The expansion starts from
C. Neumann (sphere)-Dirichlet bc, N-N bc
011 NNND NN
At large separations
)(~ 43 TN
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
C. Electromagnetic field
mllTM
TE
mllTM
TE
mllmllTM
TE
i
iimll
mllTM
TE
mll
MMMM
MM
MM
MM
MMMM
',3213
12333
',2
22
',211
121
',0
0
0,
,21
12
,
00
00
00
MM
From the structure of the expansion it follows that 0101 M1M
)(121ln1ln 53103 OMMiii MM
ml
l
l
m
TMTMTETEEMEM MMMMNN1 1
103
10331 11,0
For the functions defining the low temperature expansion we have
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
C1. Conductor bc
TEN3
TMN3
Contributions growing with l
Sho
rt di
stan
ces
Might be interpreted as non-commutativity of the limits mlT ,0
At short separations one can expect contributions decreasing slower than 4T
At large separations
)(121
31
32
)(1921
121
31
119633
119633
ON
ON
TM
TE
43
6343
3
6015T
LRTRT
FThe low temp correction to the free eneregy
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
C2. Results for dielectric ball in front of conducting plane
Fixed permittivity
.1,
962
23
33 231
2312
ONN TM
Dilute approximation
32333 2
271
92 ON
1
Fixed permeability
962
23
33 231
2312
ONN TE
.1,
Plasma model
.1,1 2
2
p
Large separations
53
23
53
3
coth13132
ON
ON
p
p
p
TE
TM
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I.G. Pirozhenko, On the vacuum energy between a sphere and a plane at finite temperature
QFEXT11, 18-25 September 2011, Benasque
Conclusions
We developed the PFA for a sphere in front of a plane at finite temperature which is valid for a the free energy which behaves like 0,0,~ dd(d,T)F
Using the exact scattering formula for the free energy of we considered high and low temperature corrections to the free energy and the force for scalar and electromagnetic fields and found analytic results in some limiting cases.
42
21 TfTfTF
At low temperature, the corrections have general form
The coefficient is present in DD and DN cases, and absent in all other cases.1f