Vacuum energy in quantum field theory
status, problems and recent
advances
M. Bordag
(Leipzig University)
• some historical remarks
• what can be calculated and what cannot
• new representation of Casimir force
some historical remarks
Planck introduced his 2nd quantization hypothesis
as a consequence the energy of the quantum oscillator
became a hº instead of zero at T=0
Planck 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 data
Lindemann and Aston (1919) on no isotope separation
of Neon
Hartree and collaborators (1928) on X-ray diffraction
patterns on rock salt
1911
w hat we should lo ok on:
in t erm s of quant um m echanics,
µ
E n = ¹h !
1
n+
2
¶
t he zero-p oint energy is
¹h
E0 =
!
2
it is t he sam e f or all energy levels, hence it can b e m easured
only t hrough it s dep endence on ext ernal param et ers
like t he m ass of isot op es
zero-p oint energy m ust b e asso ciat ed t o any oscillat ory degree of f reedom ( but not , f or exam ple t o t he rigid rot at or) ,
hence t he f orm ula reads in f act
¹h X
E0 =
!J
2 J
T h e c o n c ep t o f zer o - p o in t en er g y w as n o t g en er ally ac c ep t ed :
Pauli in 1933 crit ized t he general validit y of t he concept of
zero p oint energy. He argued, w hence at t ribut ed t o each degree of f reedom , it m ust b e large and because of it s gravit at ional ¯ eld ' t he radius of t he world would not ext end b eyond
t he m oon' .
Status of zero-point energy e®ects by 1943 was discussed in a
review paper by K. Clusius, especially he discussed the known
experimental con¯rmations
(for more details see Rechenberg's contribution in the QFEXT98
proceedings)
The work by Casimir
'preliminary': in 1948, Casimir and Polder published their
paper on The in° uence of retardation on the London-van
der Waals forces
for the interaction of two neutral atoms they found
23 h
¹ ®1 ®2
E = ¡
4¼ R 7
for one atom in front of a conducting plane even simpler,
3 h
¹®
E = ¡
4¼ R 4
Casimir considered 2 metallic plates and the zero-point energy
of the electromagnetic ¯eld con¯ned in between
h
¹c
E0 =
2
Z
2
d k jj
( 2¼) 2
X1
s
µ
k 12 + k 22 +
n= 0
¼n
L
¶
2
he managed to get rid of the divergences and obtained a ¯nite
answer for the force
@
¼3 h
¹c
F = ¡
E0 = ¡
@L
480 L 4
(scalar case, force per unit area)
alternative approach using Lifshitz formula
Z 1
Z 1 n
£
h
¹
2
¡
E ( a) =
k
dk
d»
l
n
1
¡
r
(
i
»;
k
)
e
?
?
?
TM
4¼2 0
0
o
£
2
¡ 2qa ¤
+ ln 1 ¡ r T E ( i »; k ? ) e
2qa ¤
re° ection coe± cients
" ( i») q( i»; k ? ) ¡ k ( i »; k ? )
r T M ( i»; k ? ) =
;
" ( i ») q( i »; k ? ) + k ( i »; k ? )
q( i »; k ? ) ¡ k ( i »; k ? )
r T E ( i »; k ? ) =
;
q( i »; k ? ) + k ( i »; k ? )
with notations
2
2
2
q = q ( i »; k ? ) = k ?
»2
+ 2;
c
2
2
2
k = k ( i »; k ? ) = k ?
»2
+ " ( i ») 2 :
c
sometimes considered as the only approach,
however one may consider the vacuum energy of a quantum
¯eld ' ( x ) in the background of some classical ¯eld Á( x ) ,
for example
Z E = E cl³ass + E 0 ;
1 energy
4
2
with
the
classical
S = ¡
d x Á( x ) ¤ + M +
Z2 Z
h ³
1 1
2
2
4 r ) ¡ r
2 +
E cl ass = ¡
dr dÁ(
+
M
x ' (x) ¤ + m +
2 2
and the vacuum energy
E 0 ( s) =
¹
2s
2
X
!
J
1¡ 2s
J
2
´
¸ Á ( x ) Á( x )
i
´
2
¸¸~ Á
Á2 (( xr )) Á(
' (rx)) :
St at us
what can be calculated depends on the handling of the
ultraviolet divergences
in general, we need to introduce a regularization,
e.g. zetafunctional regularization, s > 32 , s ! 0
E vac ( s) =
¹
2s
2
X
!
1¡ 2s
n
n
or frequency damping function, ± > 0, ± !
1X
E vac ( ±) =
!
2 n
n
¡ ±! n
e
0
and express the divergent part of the vacuum energy in terms
of the heat-kernel coe± cients
in zetafunctional regularization
µ
¶
a2
1
di v
2
E vac ( s) = ¡
+
l
n
¹
32¼2 s
or with frequency damping function
di v
E vac ( ±)
a 1=2 1
3a 0 1
a1 1
a2
=
+
+
+
ln ±
2
4
3=2
3
2
2
2
2¼ ±
4¼ ±
8¼ ±
16¼
example: hkks for the conducting sphere
½
¾
½ ¾
4¼ 3
¡ 1
2 2¼
3=2 2
a0 =
R ; a1 =
2¼ R ; a 1 = ¨
R;
2
3
1
14
3
½ ¾ 3=2
½ ¾
23 ¼
1
16¼
a3 =
; a2 = ¨
:
2
7
6
7 315R
In general, one needs 5 structure in the classical energy to
accomodate the counterterms
1
E class = pV + ¾S + h 1 R + h 2 + h 3
R
this is not always possible, for example
= 1, outside ² = 1
dielectric ball: inside permittivity ² 6
here we have the known matching conditions across the
surface: ² E n ; E t ; B n ; ¹1 B t
resulting in a hkk
i
h
2656¼ ( c1 ¡ c2 ) 3
4
)
c
¡
c
(
O
+
a2 = ¡
2
1
2
c2
5005R
similar problem in bag model
now consider the limit
of making
the background singulular
¸~ Á( x ) 2 = V ( r ) ! ¸~ ±( r ¡ R )
what happens to the heat kernel coe± cients?
R
a1 =
dx V ( r )
!
a 1 = 4¼®R
a 3=2 = ¼3=2 ®2
a2 =
R
dx V ( r )
2
!
6
a2 =
2¼ ®3
3 R
the classical background becomes singular, too
in some literature, the insertion of a boundary into a quantum
¯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, radius R
this is the hydrodynamic model used by Barton to describe
the ¼-electrons of graphene
quantum ¯eld: elm ° uctuations
interaction via matching conditions ( - - plasma frequency)
li m f l ;m ( k r ) ¡
r ! R+ 0
0
li m ( r f l ;m ( k ; r ) ) ¡
r ! R+ 0
0
li m ( r gl ;m ( k ; r ) ) ¡
r ! R+ 0
=
0;
li m ( r f l ;m ( k ; r ) )
0
=
- R f l ;m ( k R ) ;
li m ( r gl ;m ( k ; r ) )
0
=
0;
r ! R¡ 0
r ! R¡ 0
li m gl ;m ( k r ) ¡
r ! R+ 0
li m f l ;m ( k r )
r ! R¡ 0
li m gl ;m ( k r )
r ! R¡ 0
=
¡
k 2R
( R gl ;m ( k ; R ) )
these are equivalent of a ± resp. ±0 function on the surface
0
We allow for radial vibrations (breathing mode) of the plasma
shell. In C60 these are determined by the elastic forces acting
between the carbon atoms.
we describe these vibrations phenomenologically by a Hamilton
function
H cl ass
p2
m
=
+
!
2m
2
2
b
2
( R ¡ R 0 ) + E r est
with a momentum p = m R_. Here m is the mass of the
shell, ! b is the frequency of the breathing mode, R 0 is the
radius at rest and E r est is the energy which is required to
bring the pieces of the shell apart, i.e., it is some kind of
ionization energy.
the complete energy is
E t ot = E cl ass( R ) + E vac ( R )
and we can perform the renormalization by rede¯ning the
parameters entering ( ! b, R 0 , E r est ) provided all heat kernel
coe± cients have the 'right' structure, i.e., dependence on the
radius
indeed, they do
k
l = 0; 1; : : :
0
0
1=2
0
1
¡ 4¼- R 2
3=2
¼3=2 - 2 R 2
2
¡ 23 ¼- 3 R 2
k
0
1=2
1
3=2
2
l = 1; 2; : : :
0
0
¡ 4¼- R 2
¼3=2 - 2 R 2
¡ 23 ¼- 3 R 2 + 4¼-
l = 0; 1; : : :
0
8¼3=2 R 2
2
¡ 4¼
R
3
14 3=2
3¼
3 2
¡ 8¼- + 2¼
R
15
l = 1; 2; : : :
0
8¼3=2 R 2
2
¡ 4¼
R
3
3=2
¡ 10
¼
3
3 2
¡ 4¼- + 2¼
R
15
with these coe± cients the renormalization can be carried out,
especially contributions growing with - can be removed by a
¯nite renormalization
E( - R )
r en
and a ¯nite vacuum energy can be calculated: E vac =
R
it has for - !
1 the expected limit E( 1 ) = 0:0461766
New represent at ion for t he Casimir force act ing between
separat ed bodies
In general, this force, F = ¡ ( @=@L ) E 0 , is always ¯nite
since it is a measurable quantity
in representations like E 0 =
¹
h
2
P
!
J
we need to calculate
J
the eigenvalues and to subtract out several asymptotic
contributions,
after that a numerical approach is quite hopeless. In fact,
there is no successful attempt in the literature to calculate
the Casimir force for a complicated geometry numerically in
this way.
A new approach was found only quite recently. As shown by
Bulgac et al. (2006) and Emig et al. (2006), it is possible to
rewrite a representation of the vacuum energy in terms of a
functional determinant which does not contain any ultraviolet
divergences and which is ¯nite in all intermediate steps.
before that ¯rst we need a
speci¯c represent at ion of t he propagat or wit h boundary
condit ions
(M.B., D.Robaschik, E.Wieczorek, 1985)
functional integral representation of generating functional
Z
Y
iS[' ]
Z [¨ ] =
D'
±( ' ( x ) ) e
:
x2 S
Z
Y
±( ' ( x ) ) = C
Db
R
i
d¹ ( ´ ) b( ´ ) ' ( u ( ´ ) )
e S
;
x2 S
rewrite the exponential
Z
Z
Z
4
d¹ ( ´ ) b( ´ ) ' ( u ( ´ ) ) =
d¹ ( ´ )
d x b( ´ ) H ( ´ ; x ) ' ( x )
S
S
with a newly de¯ned kernel H ( ´ ; x ) = ±4 ( x ¡ u ( ´ ) )
Gaussian path integral, diagonalization
Z [0]
=
C ( det K )
¡ 1=2
³
det K~
´
¡ 1=2
and the vacuum energy (distance dependent part) is
1
E0 =
2¼
Z
1
d» Tr l n K~ »:
Z
0
K~ ( ´ ; ´ )
=
=
Z
4
d x
4 0
0
0
0
d x H (´ ; x )G(x ; x )H (´ ; x )
0
G(u(´ ); u(´ )):
is a new kernel for the b-¯eld
It is the projection of the free space propagator onto the
boundary surface S
now we assume the surface S to consist of two nonintersecting
parts, SA and SB , with
S = SA [ SB ;
SA \ SB = 0:
the kernel takes a block structure
Ã
0
~
K
(
´
;
´
)
»;A
A
(
b)
A
A
K~ » =
K~ »;B A ( ´ B ; ´ 0A )
K~ »;A B ( ´
K~ »;B B ( ´
A; ´
B;
´
0
B)
0
B)
!
the next essential step is a factorization of the parts resulting
from the individual surfaces
!
! Ã
Ã
~ »;A A 0
0
1
K
b)
(
~
K» =
0 K~ »;B B
1
0
!
Ã
1
¡
K~ »;A A K~ »;A B
1
£
1
K~ ¡ 1 K~ »;B A
»;B B
using
Ã
det
!
1
C
B
1
= det ( 1 ¡ B C ) = det ( 1 ¡ C B ) :
and dropping the distance independent parts we come to
Z 1
³
´
1
¡ 1
¡ 1
E =
d» Tr l n 1 ¡ K~ »;A A K~ »;A B K~ »;B B K~ »;B A :
2¼ 0
which is the desired representation
Note: all integrations and summations entering do converge
Example:
Applicat ion t o cylindrical geomet ry
y
RA
b
RB
0
x
b
B
A
calculate all quantities in the corresponding basis
ei l '
;
jl > = p
2¼
³
and
K~ ° ;A B
³
´
l0
ll0
= ( ¡ 1) I l ( ° R A ) K l ¡ l 0( ° L ) I l 0( ° R B ) :
K~ ° ;A A
´
ll0
= ±l l 0 I l ( ° R A ) K l ( ° R A ) :
¯nally one comes to the energy in the form
Z 1
1
E =
d° ° Tr l n ( 1 ¡ M
4¼ 0
°
)
with the matrix elements
1
M ° ;l l 0 =
K l + l 0( ° L ) I l 0( ° R )
K l(° R)
(cylinder in front of a plane)
Limit ing cases
1. large separations
Dirichlet boundary conditions
E
D
1
1
=
+
O
R
4¼L 2 l n L
µ
ln
¡ 2
R
L
¶
:
Neumann boundary conditions
2
E
N
5R
= ¡
+ O
4
2¼L
Ã
4
R
R
ln
4
L
L
!
:
2. small separat ions
proximity force approximation and correction beyond
(M.B. 2006)
3
E
E
N
D
¼
= ¡
1920L 2
¼3
= ¡
1920L 2
r
r
R
2L
"
Ã
R
2L
7 L
1+
+ O
36 R
"
µ
1+
7
40
¡
36
3¼2
¶
2
! #
L
R2
Ã
L
+ O
R
L2
R2
! #
similar results were obtaind also for a sphere in front of a
plane
(M.B., V. Nikolaev 2008)
3
spher e
E Dirichlet
=
spher e
=
E Neumann
( ² = a=R )
¼ R
¡
1440 L 2
½
³
´¾
1
2
1+
² + O ²
3
½
µ
¶
¾
3
³
´
¼ R
1
10
2
¡
1
+
¡
²
+
O
²
1440 L 2
3
¼2
Conclusions
² Basic problems are clari¯ed
² Some problems with renormalization still persist
² Computational tools for the calculation of Casimir forces
are available