Vacuum energy in quantum field theory status, problems and recent

advances

M. Bordag

(Leipzig University)

a aP

s ddgggggg ggga ba f ra s

• some historical remarks• what can be calculated and what cannot• new representation of Casimir force

some historical remarks

1911 Planck introduced his 2nd quantization hypothesisas a consequence the energy of the quantum

oscillator became a instead of zero at T=0Planck called it remaining energy; afterwards it

was renamed simply zero-point energy

1912-1928 consequences of zero-point energy were found by:Nernst and Lindemann (1911) on specific heat

dataLindemann and Aston (1919) on no isotope

separation of Neon Hartree and collaborators (1928) on X-ray

diffractionpatterns on rock salt

hº

what we should look on:in terms of quantum mechanics,

En = ¹h!µn +

12

¶

the zero-point energy is

E0 =¹h2!

it is the same for all energy levels, hence it can be measuredonly through its dependence on external parameterslike the mass of isotopes

zero-point energy must be associated to any oscillatory de-gree of freedom (but not, for example to the rigid rotator),hence the formula reads in fact

E0 =¹h2

X

J! J

T he concept of zero-point energy was not generally ac-cepted:Pauli in 1933 critized the general validity of the concept ofzero point energy. He argued, whence attributed to each de-gree of freedom, it must be large and because of its gravita-tional ¯ eld 'the radius of the world would not extend beyondthe moon'.

Statusof zero-point energye®ectsby1943wasdiscussed in areviewpaper byK. Clusius, especially hediscussed theknownexperimental con rmations

(for more details see Rechenberg's contribution in the QFEXT98proceedings)

Thework by Casimir

'preliminary': in 1948, Casimir and Polder published theirpaper on The in°uence of retardation on the London-vander Waals forcesfor the interaction of two neutral atoms they found

E = ¡23

4¼

¹h®1®2

R 7

for oneatom in front of a conducting plane even simpler,

E = ¡3

4¼

¹h®

R 4

Casimir considered2metallicplatesand thezero-point energyof theelectromagnetic ¯eld con ned in between

E 0 =¹hc

2

Zd2kjj(2¼)2

1X

n=0

s

k21 + k22 +µ¼n

L

¶ 2

hemanaged toget ridof thedivergencesandobtaineda¯niteanswer for the force

F = ¡@

@LE 0 = ¡

¼3

480

¹hc

L 4

(scalar case, forceper unit area)

alternativeapproach using Lifshitz formula

E (a) =¹h

4¼2

Z 1

0k? dk?

Z 1

0d»

nln£1 ¡ r 2TM (i»; k? )e

¡ 2qa¤

+ ln£1 ¡ r 2T E (i»; k? )e

¡ 2qa¤o

re°ection coe±cients

rTM (i»; k? ) ="(i»)q(i»; k? ) ¡ k(i»; k? )

"(i»)q(i»; k? ) + k(i»; k? );

rT E (i»; k? ) =q(i»; k? ) ¡ k(i»; k? )

q(i»; k? ) + k(i»; k? );

with notations

q2 = q2(i»; k? ) = k2? +»2

c2; k2 = k2(i»; k? ) = k2? + "(i»)

»2

c2:

sometimes considered as theonly approach,however onemay consider the vacuumenergy of a quantum¯eld ' (x) in thebackground of someclassical ¯eld Á(x),

E = E class+ E 0;

with theclassical energy

E class =1

2

Zdr Á(r )

h¡ r 2+ M 2+ ¸Á2(r )

iÁ(r )

and thevacuumenergy

E 0(s) =¹ 2s

2

X

J

! 1¡ 2sJ

for example

S = ¡1

2

Zd4x Á(x)

³¤ + M 2+ ¸Á2(x)

´Á(x)

¡1

2

Zd4x ' (x)

³¤ + m2+ ~Á2(x)

´' (x):

Statuswhat can be calculated depends on the handling of theultraviolet divergencesin general, weneed to introducea regularization,e.g. zetafunctional regularization, s > 3

2, s ! 0

E vac(s) =¹ 2s

2

X

n

! 1¡ 2sn

or frequency damping function, ±> 0, ±! 0

E vac(±) =1

2

X

n

! n e¡ ±! n

and express thedivergent part of thevacuumenergy in termsof theheat-kernel coe±cientsin zetafunctional regularization

E divvac(s) = ¡

a232¼2

µ1

s+ ln ¹ 2

¶

or with frequency damping function

E divvac(±) =

3a02¼2

1

±4+

a1=24¼3=2

1

±3+

a18¼2

1

±2+

a216¼2

ln ±

example: hkks for theconducting sphere

a0 =4¼

3R 3; a1

2=½¡ 1

1

¾2¼3=2R 2; a1 = ¨

½2

14

¾2¼

3R ;

a32=½23

7

¾¼3=2

6; a2 = ¨

½1

7

¾16¼

315R:

In general, one needs 5 structure in the classical energy toaccomodate thecounterterms

E class = pV + ¾S + h1R + h2+ h31

Rthis is not always possible, for exampledielectric ball: insidepermittivity ² 6= 1, outside ² = 1here we have the known matching conditions across thesurface: ²E n; E t; B n; 1

¹ B t

resulting in a hkk

a2 = ¡2656¼

5005R

(c1 ¡ c2)3

c22+ O

h(c1 ¡ c2)

4i

similar problem in bagmodel

now consider the limitof makingthebackground singulular

~Á(x)2 = V (r ) ! ~±(r ¡ R )

what happens to theheat kernel coe±cients?

a1 =Rdx V (r ) ! a1 = 4¼®R

a3=2 = ¼3=2®2

a2 =Rdx V (r )2 6! a2 = 2¼

3®3

R

theclassical background becomes singular, too

in someliterature, theinsertionof aboundary intoaquantum¯eld was considered a 'unnatural act'

There is an example for handling such situation(M.B., N. Khusnutdinov, PRD 77 (2008) 085026)

classical background: spherical plasma shell, radiusRthis is the hydrodynamic model used by Barton to describethe¼-electrons of graphenequantum¯eld: elm°uctuationsinteraction viamatching conditions (­ - plasma frequency)

limr ! R+0

f l;m(kr ) ¡ limr ! R ¡ 0

f l ;m(kr ) = 0;

limr ! R+0

(r f l;m(k; r ))0¡ lim

r ! R ¡ 0(r f l ;m(k; r ))

0 = ­ R f l ;m(kR );

limr ! R+0

(rgl;m(k; r ))0¡ lim

r ! R ¡ 0(rgl ;m(k; r ))

0 = 0;

limr ! R+0

gl ;m(kr ) ¡ limr ! R ¡ 0

gl ;m(kr ) = ¡­

k2R(Rgl ;m(k; R ))

0

theseareequivalent of a±resp. ±0 function on thesurface

Weallow for radial vibrations (breathingmode) of theplasmashell. In C60 thesearedetermined by theelastic forces actingbetween thecarbon atoms.

wedescribethesevibrationsphenomenologicallybyaHamiltonfunction

H class =p2

2m+m

2! 2b (R ¡ R 0)

2+ E rest

with a momentum p = m _R . Herem is the mass of theshell, ! b is the frequency of the breathing mode, R 0 is theradius at rest and E rest is the energy which is required tobring the pieces of the shell apart, i.e., it is some kind ofionization energy.

thecompleteenergy is

E tot = E class(R ) + E vac(R )

and we can perform the renormalization by rede ning theparameters entering (! b, R 0, E rest) provided all heat kernelcoe±cientshavethe 'right' structure, i.e., dependenceon theradius

indeed, they dok l = 0; 1; : : : l = 1; 2; : : :0 0 01=2 0 01 ¡ 4¼­ R 2 ¡ 4¼­ R 2

3=2 ¼3=2­ 2R 2 ¼3=2­ 2R 2

2 ¡ 23¼­

3R 2 ¡ 23¼­

3R 2+ 4¼­

k l = 0; 1; : : : l = 1; 2; : : :0 0 01=2 8¼3=2R 2 8¼3=2R 2

1 ¡ 4¼3 ­ R

2 ¡ 4¼3 ­ R

2

3=2 143¼

3=2 ¡ 103¼

3=2

2 ¡ 8¼­ + 2¼15­

3R 2 ¡ 4¼­ + 2¼15­

3R 2

with thesecoe±cients therenormalization canbecarried out,especially contributions growingwith ­ can be removed by a¯nite renormalizationand a ¯nitevacuumenergy can becalculated: E ren

vac =E(­ R )R

it has for ­ ! 1 theexpected limit E(1 ) = 0:0461766

New representation for the Casimir force acting betweenseparated bodies

In general, this force, F = ¡ (@=@L )E 0, is always ¯nitesince it is ameasurablequantity

in representations like E 0 = ¹h2

P

J! J we need to calculate

the eigenvalues and to subtract out several asymptoticcontributions,after that a numerical approach is quite hopeless. In fact,there is no successful attempt in the literature to calculatethe Casimir force for a complicated geometry numerically inthis way.

A new approach was found only quite recently. As shown byBulgac et al. (2006) and Emig et al. (2006), it is possible torewrite a representation of the vacuumenergy in terms of afunctional determinant whichdoesnot contain anyultravioletdivergences and which is ¯nite in all intermediatesteps.

before that ¯rst weneed aspeci¯c representation of the propagator with boundaryconditions(M.B., D.Robaschik, E.Wieczorek, 1985)

functional integral representation of generating functional

Z [ ] =Z

D 'Y

x2S

±(' (x)) eiS[' ]:

Y

x2S

±(' (x)) = CZ

Db eiRS d¹ (´) b(´)' (u(´));

rewrite theexponentialZ

S

d¹ (´) b(´)' (u(´)) =Z

S

d¹ (´)Z

d4x b(´)H (´; x)' (x)

with a newly de ned kernel H (´; x) = ±4(x ¡ u(´))Gaussian path integral, diagonalization

Z [0] = C (det K )¡ 1=2³det ~K

´ ¡ 1=2

and thevacuumenergy (distancedependent part) is

E 0 =1

2¼

Z 1

0d»Tr ln ~K »:

~K (´; ´0) =Z

d4xZ

d4x0H (´; x)G(x; x0)H (´0; x0)

= G(u(´); u(´0)):

is a new kernel for theb-¯eldIt is the projection of the free space propagator onto theboundary surfaceS

nowweassumethesurfaceS toconsist of twononintersectingparts, SA and SB , with

S = SA [ SB ; SA \ SB = 0:

thekernel takes a block structure

~K (b)» =

Ã~K »;AA (´ A ; ´

0A ) ~K »;AB (´ A ; ´

0B )

~K »;B A (´ B ; ´0A ) ~K »;B B (´ B ; ´

0B )

!

thenext essential step is a factorization of theparts resultingfromthe individual surfaces

~K (b)» =

Ã~K »;AA 0

0 1

! Ã1 0

0 ~K »;B B

!

£

Ã1 ~K ¡ 1

»;AA~K »;AB

~K ¡ 1»;B B

~K »;B A 1

!

using

det

Ã1 B

C 1

!

= det(1 ¡ B C ) = det(1 ¡ CB ):

and dropping thedistance independent parts wecometo

E =1

2¼

Z 1

0d»Tr ln

³1 ¡ ~K ¡ 1

»;AA~K »;AB ~K ¡ 1

»;B B~K »;B A

´:

which is thedesired representation

Note: all integrations and summations entering do converge

Example: Application to cylindrical geometry

RA RBx

A

B0b b

y

calculateall quantities in thecorresponding basis

jl >=ei l 'p2¼

;

³~K ° ;AB

´

l l0= (¡ 1) l

0I l(°R A )K l¡ l0(° L )I l0(°R B ):

and ³~K ° ;AA

´

l l0= ±l l0I l(°R A )K l(°R A ):

¯nally onecomes to theenergy in the form

E =1

4¼

Z 1

0d° ° Tr ln (1 ¡ M ° )

with thematrix elements

M ° ;ll0=1

K l(°R )K l+l0(° L ) I l0(°R )

(cylinder in front of a plane)

Limiting cases1. large separationsDirichlet boundary conditions

E D =1

4¼L 2

1

ln RL

+ Oµln¡ 2

R

L

¶:

Neumann boundary conditions

E N = ¡5R 2

2¼L 4+ O

ÃR 4

L 4lnR

L

!

:

2. small separationsproximity forceapproximation and correction beyond(M.B. 2006)

E D = ¡¼3

1920L 2

rR

2L

"

1+7

36

L

R+ O

ÃL 2

R 2

! #

E N = ¡¼3

1920L 2

rR

2L

"

1+µ7

36¡

40

3¼2

¶L

R+ O

ÃL 2

R 2

! #

similar results were obtaind also for a sphere in front of aplane(M.B., V. Nikolaev 2008)

E sphereDirichlet = ¡

¼3

1440

R

L 2

½1+

1

3² + O

³²2´¾

E sphereNeumann = ¡

¼3

1440

R

L 2

½1+

µ1

3¡10

¼2

¶² + O

³²2´¾

(² = a=R )

Conclusions

² Basic problems areclari¯ed

² Someproblemswith renormalization still persist

² Computational tools for the calculation of Casimir forcesareavailable