Effetto Casimir

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Introduction to Casimir effect

F. Gliozzi

DFT & INFN, Universit di Torino

Workshop, 31/10/06

SF

z

F. Gliozzi ( DFT & INFN, Universit di Torino) Casimir effect Workshop, 31/10/06 1 / 31
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Plan of the talk

1 The Casimir Effect and the vacuum energy

2 The Casimir Effect from the Stefan-Boltzmann law

3 Generalisations

4 Main experiments

5 Applications

6 Conclusions

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Plan of the talk



3 Generalisations

4 Main experiments

5 Applications

6 Conclusions

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Plan of the talk



3 Generalisations

4 Main experiments

5 Applications

6 Conclusions

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Plan of the talk



3 Generalisations

4 Main experiments

5 Applications

6 Conclusions



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Plan of the talk



3 Generalisations

4 Main experiments

5 Applications

6 Conclusions

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Plan of the talk



3 Generalisations

4 Main experiments

5 Applications

6 Conclusions


The Casimir Effect and the vacuum energy


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Casimir effect and the vacuum energy


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The set-up

l

L

L>>l

Casimir , 1948:

Two perfectly conducting, parallel, plates

at a distance and at low temperature

are subjected to an attractive forcethe pressure p depends only on

p=

2

240

c

4


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Why? Casimir explanation: zero-point energy

c The empty space is filled with an infinite family of decoupled

harmonic oscillators associated to the normal modes of the

electro-magnetic field

c The energy spectrum is (n+ 12 )

c depend on the boundary conditions

c the number of allowed is infinite

The vacuum zero-point energy E0 =

allowed 12 diverges,

however cannot be measured

Only energy differences are measurable

c QFT puts E0 = 0 in a large box

it may be E0 = 0 in a smaller system, like the region between twoplates


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l

L

d Fixed boundary conditions (conducting

boundaries)

d = c

( m )2 + ( nL )

2 + ( kL )2

n, m, k

0

E0 = 2

n,m,k

/2 diverges !

Regularise by introducing a cut-off, e.g.

n, m, k < N


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L

L

l

System A

EA0 = E0() + 2 E0(L2 2 )

L

L/3

System B

EB0 = 3 E0(L3 )

ECasimir limL

( EA0 EB0 ) p = ECasimir

L2


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gy

A judicious choice of the cut-off yields

ECasimir = c2

720

L2

3

Agrees with other regularisations, but...

P.A.M Dirac, 1980 :... we should no longer have to make use of such illogical

processes as infinite renormalisation. This is quite nonsense

physically,and I have always opposed to it. It is just a rule of thumb

that gives results


The Casimir Effect from the Stefan-Boltzmann law
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Casimir effect as black body radiation in disguise


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An intriguing relation- dimensional analysis

c Black body radiation

c Reflecting cavity in thermalequilibrium

c only one relevant scale: T

Eb/V =[E][L3]

= f(T, , c) =

b T(c/T)3 = b (T)4

(c)3

c b =2

15

c Casimir effect

c Two parallel, conducting platesat T 0

c only one relevant scale :

Ec/V =[E][L3]

= g(, , c) =

c c/3 = cc4

c c = 2720

b = 48 c

Why?


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Heat bath

T

in D dimensions

Quantum system

in D+1 dimensions with aimaginary time, compactified

Classical statistical system

direction with = 1/k T


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Canonical partition function of a quantum system

Z = Tr eH

|eH | , = 1/T

Put = N Trotter formula:

Tr eH = limN

1

1|(1 H)2

|2 2|(1 H)

3 |

3

3

|(1

H)

N |N

N

|(1

H)

|1

new periodic dimension which acts as an imaginary time


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Sum over configurations i

1 2 3 4 5

Any quantum model in D dimensions at a temperature T = 1/ is

equivalent to a classical statistical system in D+ 1 Euclideandimensions. The temperature direction is periodic:

(x, ) = (x, + ) , x = (x1, x2,

, xD)


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Black body radiation

in equilibrium

at a temperature T

in a box L L L

Free electro-magnetic

field in a 4-D box

L L L

periodic in Configurations = stationary waves in a 4-D box

eigenvalues of the 4-D laplacian 2 [ put = c = = 1]

n,mi = 2 n 2

+3i=1

miL

2

n = 0,1,2, , mi = 1, 2, . . . Canonical partition function: Zb = 1/

det 2

No need of explicit calculations: the result is known from the

elementary approach: Eblack body

log Z()

=2V

154

log Zb=

2L3

45 3


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Black body

Casimir

L

L

>>

l

V=BL2

V=L3

log Zb =2

45

V

3=

2

45

L3

3, log Zc =

1

2

2

45

B L2

(2)3=

2

720

B L2

3

Eblack body = log Zb

=2

15

L3

4; ECasimir =

log ZcB

= 2

720

L2

3


Generalisations
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Generalisations


Generalisations
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Variations on a theme: unstable boxes

Generalisations to other space-time dimensions are

straightforward many different shapes and boundary conditions have been studied

the corresponding Casimir effect strongly depends on shapes and

b.c.

L

L

L3

c 0.408 < L3L < 3.48 ECasimir > 0c in particular in a conducting cube

(L3 =

L)

Ecube 0.0916 cL


Generalisations
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A B

sphere: outward pressure Boyer,1968

two hemispheres attract

More generally, two bodies A,B related by a reflection attract

Kenneth and Klich, 2006


Generalisations
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Dynamical Casimir Effect

d If one of the plates is not fixed, but moves with a speed there areperturbative corrections in /c [Bordag, Dittes, Robaschik (1986)]

p = 2

240

c4

1 102 23

2

c2 + O

2

c4

d If the mutual distance of the plates vibrates the system emits

photons with a calculable rate [Dodonov and Klimov (1996)]


Generalisations
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D=1 case: strings

3 normal modes (fixed or free b.c.) = n (n= 1, 2, . . . )

3 Spectrum: (m+ 12 )

3 Zero point energy

Eo = (d 2)2

n=1 n diverges !

l

3 function regularisation [F.G.(1976), Hawking (1977)]:(s) =

n=11

ns holomorphic function with a simple pole at s = 1

(1) = n is finite, namely (1) = 112

Eo = (d 2) 2 (1) = (d 2) 124h This has an important role in the string theory


Generalisations
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Toward realistic boundary conditions

z

()

()


Generalisations
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Toward realistic boundary conditions

Finite conductivity [ < ] (Lifshitz 1956)

ECasimir() = E0

1 + 1

2()

Free electron plasma in the metal (Bezerra, Klimchitskaya,

Mostepanenko 2000)

ECasimir() = E0

1 16

3 + 24 2 640

7

1

2

210

3 +

= o/ , o = c/p , p =4Ne2

m


Main experiments
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Experiments


Main experiments
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The Casimir force on a pair of plates of 1 cm2 at a distance of 1

m is 1.3 107

N a macroscopic effect, but extremelydifficult to observe because of many parasitic effects Van der Waals forces roughness of surfaces edge effects hard to configure parallel plates at = 1m finite conductivity of the plates thermal fluctuations residual electric charges on the surfaces

First experimental observation : Tabor and Winterton, 1968

truly convincing evidence with a laser interferometer: Lamoreaux,1997


Main experiments
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A precision experiment (Harris,Chen,Mohideen,2000)

cantilever

sphere

laser

photodiodes

piezo

plate

l

AFM

c Plate-sphere config.c Casimir force

F = 3 R360 c3c polystyrene sphere

coated with goldR = 191m

c plate = optically

polished sapphire disk

(diameter 1 cm)

c is varied with thepiezo-electric tube


Main experiments
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- 1 2 0

- 1 0 0

-8 0

-6 0

-4 0

-2 0

0

2 0

2 0 0 3 0 0 4 0 0 5 0 0

nm

Casim

irForce(10-12

N)


Applications
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Applications


Applications

C i i f d t
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Casimir forces and nano- systems

At separations below a few ten nm the Casimir force dominates

over other forces

Movable components in nano-scales devices often stick together

due to Casimir force (stiction process) This leads to poor yield in the fabrication of micro- and

nano-mechanical systems

It would be of much promise to develop systems with zero or

suppressed Casimir force.


Applications

C i i f t t f h i
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Casimir force as a test for new physics

Many extensions of the standard model assume that the

dimensionality of the space-time is larger than four The additional spatial dimensions are compactified at a small

length scale

some models suggest a length scale of the order of the fraction of

millimeter As a result, the Newtonian gravitational potential is modified at

short distances V(r) = Gm1 m2r + er/ with 10 m

At r gravity is no longer the dominant force between neutralbodies. Casimir force is much stronger

A precise determination of the Casimir force with the torsion

pendulum is the best way to test the predictions of

extra-dimensional physics


Conclusions


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Conclusions


Conclusions


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The Casimir effect is a quantum phenomenon which is strictlyrelated to the nature of the vacuum

It strongly depends on the shape of the bodies involved and on the

boundary conditions

It has been checked by precision experiments between metal

surfaces

Its role in nano technology is rapidly increasing

It could be useful to test fundamental forces and new physics at

the m scale

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