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

7

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

8

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

9

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

17

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