Worldline Approach to the Casimir Effect

Worldline Approach to the Casimir Effect

Holger Gies

Institute for Theoretical PhysicsHeidelberg University

Holger Gies Worldline Approach to the Casimir Effect

”during my first years at Philips I did some workthat even Pauli regarded as physics ...

H.B.G. Casimir, Autobiography, 1983


A view on the quantum vacuum.


A view on the quantum vacuum.

a


Casimir effect.

B origin:

E =12

∑~ω[a]− 1

2

∑~ω[∞]

B Hendrik B.G. Casimir 1948:

FA

= − π2

240~ ca4

relativistic: cquantum: ~“universal”: no other parameters

kgm · s2 =

kg ·m2

sms

1m4


Casimir’s derivation

B boundary conditions:

Et

∣∣∣plates

= Bn

∣∣∣plates

= 0

a

B “vacuum energy”

12

∑~ω =

~2

A∫

d2kt

(2π)2

∞∑n=−∞

ω~kt ,n, ω~kt ,n

= c

√~k2

t +(πn

a

)2




Et

∣∣∣plates

= Bn

∣∣∣plates

= 0

a

B “vacuum energy”

12

∑~ω =

~2

A∫

d2kt

(2π)2

∞∑n=−∞

ω~kt ,n, ω~kt ,n

= c

√~k2

t +(πn

a

)2

→∞




Et

∣∣∣plates

= Bn

∣∣∣plates

= 0

a

B subtract energy for very large separations: lima→∞nπa = kz

∆E(a) =~cA4π

∫dkt kt

[ ∞∑n=−∞

ω~kt ,n− 2a

π

∫dkz

√k2

t + k2z

]

B difference of two divergent quantities: Regularization required !


Casimir’s regulator

B smooth cutoff function F (k/km)

F (k/km) = 1 for k � km

F (k/km) = 0 for k →∞

“The physical meaning is obvious: for veryshort waves (X-rays, e.g.) our plate ishardly an obstacle at all and therefore thezero-points energy of these waves will notbe influenced by the position of the plate.”


Casimir’s regulator

B smooth cutoff function F (k/km)

F (k/km) = 1 for k � km

F (k/km) = 0 for k →∞

B regularization~2

∑ω → ~

2

∑ω F (ω/km)

B remove the regulator

∆E(a) = limkm→∞

∆E reg(a) = −c~π2A720a3


Another view on the quantum vacuum.

B ρ→ 0: “pneumatic vacuum”



B QFT: quantum fluctuations BUT: . . . just a picture !



B Boundary conditions: Casimir effect



B Probing the quantum vacuum, e.g., by external fields:“modified quantum vacuum”



=⇒ modified light propagation: “QV ' medium” (PVLAS,BMV,Q&A)



B Heat bath: quantum & thermal fluctuations



+++++

−−−−−

ee + −

B electric fields: Schwinger pair production “vacuum decay”


Universal tool: effective action Γ.

B Quantum vacuum with background A∫fluctuations → Γ[A]

Γ[A] =⇒

δΓ[A]δA = 0, quantum Maxwell equations → (light prop.)

EQV = Γ[A]T , FCasimir = −∂EQV

∂A , Casimir force

W = 2Im Γ[A]VT , Schwinger pair production rate

(HEISENBERG&EULER’36; WEISSKOPF’36; SCHWINGER’51)

(CASIMIR’48)



B Quantum vacuum with background A∫fluctuations → Γ[A]

Γ[A] = − ln∫Dφe−

R−|D(A)φ|2+m2|φ|2 =

=∑

λ

ln(λ2 + m2

)

B spectrum of quantum fluctuations:

ScQED: −D(A)2 φ = λ2 φScalar: (−∂2 + A(x))φ = λ2 φ



Γ[A] =∑

λ

ln(λ2 + m2

)=

Problem solved, “in principle”

find spectrum λ for a given background Asum over spectrum



Γ[A] =∑

λ

ln(λ2 + m2

)=

BUT:

In general practice:

spectrum {λ} not knownanalyticallyspectrum {λ} not bounded∑

λ →∞ (regularization)renormalization


Measurements of the Casimir force.

B Experimental milestones

Sparnaay 1958,van Blokland & Overbeek 1978,Lamoreaux 1997,Mohideen & Roy 1998,

∆FF ' 100%

∆FF ' 50%

∆FF ' 5%

∆FF ' 1%

B Sparnaay’s fundamental requirements:

clean plate surfacesprecise measurement of separation acontrol of electrostatic potentials Vres


Casimir meets AFM.

Atomic-Force-Microscope Measurements(MOHIDEEN ET AL.’98)

B Sphere-Plate configuration (DERJAGUIN ET AL.’56)

Force sensitivity: 10−17 N


Casimir meets AFM.


200 µm polystyrene sphere with gold coating 85.6± 0.6 nm


Casimir meets AFM.


B Sphere-Plate configuration (DERJAGUIN ET AL.’56)

B Results

B Corrections:

material

{surface roughness

finite conductivity

finite temperature

geometry

}QFT


Casimir Morphology.(BINNS’05)


Casimir Morphology.

a

R

F = − π2

240~ ca4 A F = ? F = ?


Casimir vs. Newton.

B gravity:

F12 = − Gm1m2

|r1 − r2|2r12

B quantum force:

F‖ = − π2

240~ ca4 A

?Holger Gies Worldline Approach to the Casimir Effect

Casimir Edge Effects.

F‖ = − π2

240~ca4 · A

F1si = ?

(CF. BRESSI,CARUGNO,ONOFRIO,RUOSO’02)



F‖ = − π2

240~ca4 · A

B parallel-plate

energy density



F‖ = − π2

240~ca4 · A

F1si = ?


Worldline representation of Γ.

B pedestrian approach

Γ[A] =∑

λ

ln(λ2 + m2

)= Tr ln

[−(D(A))2 + m2

]

= −∞∫

1/Λ2

dTT

e−m2T Tr exp[D(A)2 T

]︸︷︷︸

=〈x|eiH(iT )|x〉

= −∞∫

1/Λ2

dTT

e−m2T N∫

x(T )=x(0)

Dx(τ) e−

TR0

dτ“

x24 +ie x·A(x(τ)

”



Γ[A] = −∞∫

1/Λ2

dTT

e−m2T N∫

x(T )=x(0)

Dx(τ) e−

TR0

dτ“

x24 +ie x·A(xτ)

”

x(T ) =

(FEYNMAN’50)

...(HALPERN&SIEGEL’77)

(POLYAKOV’87)

(FERNANDEZ,FRÖHLICH,SOKAL’92)

...(BERN&KOSOWER’92; STRASSLER’92)

(SCHMIDT&SCHUBERT’93)

(KLEINERT’94)



Γ[A] = −∞∫

1/Λ2

dTT

e−m2T N∫

x(T )=x(0)

Dx(τ) e−

TR0

dτ“

x24 +ie x·A(xτ)

”

x(T ) =

Worldline approach:

effective action Γ ∼∫

closed worldlines x(τ)

worldline ∼ spacetime trajectory of φ fluctuationsgauge-field interaction ∼ “Wegner-Wilson loop”finding {λ} and

∑λ done in one finite (numerical) step (HG&LANGFELD’01)


Worldline Numerics.

∫x(1)=x(0)

Dx(t) −→nL∑

l=1

, nL = # of worldlines

→ statistical error

x(t) −→ x i , i = 1, . . . ,N (ppl)→ systematical error

−→ → spacetime remains continuous


Trajectory of a Quantum Fluctuation.

B Feynman diagram (conventionally in momentum space)



B worldline (artist’s view)



B worldline numerics: N = 4 points per loop (ppl)


Propertime T .

T ∼ regulator scale of smeared momentum shells


Propertime T .

B “Measuring” the Wegner-Wilson loop exp(−ie

∮dx · A

)


Casimir Effect.

B Casimir effect = “strong-field QFT”

S =12

(∂φ)2 +m2

2φ2 + Aφ2

A(x) = λ

∫S

dσ[δ(x − xσ) + δ(x − xσ)

]

xσ

xσ

(BORDAG,HENNIG,ROBASCHIK’92; GRAHAM ET AL.’03)

B Casimir energy on the worldline:

E [A] = −12

∞∫0

dTT

e−m2T N∫

x(T )=x(0)

Dx(τ) e−

TR0

dτ“

x24 +A(x(τ))

”


Benchmark test: parallel plates


Benchmark test: parallel plates

B for finite m, λ, a

(BORDAG,HENNIG, ROBASCHIK ’92) (HG,LANGFELD,MOYAERTS ’03)


Casimir Effect: curvature effects on the worldline

S2

(a) (b) (c)

S1



−εR

4

0.012

0.01

0.008

0.006

0.004

0.002

0



(THANKS TO K. KLINGMULLER)


Casimir Effect: sphere above plate.

0.001 0.01 0.1 1 10 100a/R

0

0.5

1

1.5

2

EC

asim

ir/E

PFA

(a/R

<<

1)

PFA plate-basedPFA sphere-based



0.001 0.01 0.1 1 10 100a/R

0

0.5

1

1.5

2E

Cas

imir/E

PFA

(a/R

<<

1)

PFA plate-basedPFA sphere-basedworldline numerics



0.001 0.01 0.1 1 10 100a/R

0

0.5

1

1.5

2E

Cas

imir/E

PFA

(a/R

<<

1)

PFA plate-basedPFA sphere-basedoptical approximationworldline numerics"KKR" multi-scattering map

(HG,LANGFELD,MOYAERTS’03; JAFFE,SCARDICCHIO’04; BULGAC,MAGIERSKI,WIRZBA’05; HG,KLINGMÜLLER’05)


Future Casimir curvature measurements.

B cylinder-plate geometry (BROWN-HAYES,DALVIT,MAZZITELLI,KIM,ONOFRIO’05)

(EMIG,JAFFE,KARDAR,SCARDICCHIO’06; HG,KLINGMULLER’06; BORDAG’06)



F‖ = −γ‖~ca4 · A

F1si = ?

(CF. BRESSI,CARUGNO,ONOFRIO,RUOSO’02)



Σ2

aΣ1



εC

asim

ira

4

0

-0.002

-0.004

-0.006

-0.008

-0.01

-0.012

-0.014



B effective description of a finite plate

area A boundary C

F = −γ‖~ca4 Aeff,

B effective area: Aeff ' A + γ1siγ‖

aC, γ1si = 5.23(2)× 10−3

(HG,KLINGMULLER’06)


Further Worldline Applications.

Heisenberg-Euler effective actions, spinor QED,flux tubes, quantum-induced vortex interactions

(HG,LANGFELD’01; LANGFELD,MOYAERTS,HG’02)

thermal fluctuations, free energies(HG,LANGFELD’02)

+++++

−−−−−

ee + − “spontaneous vacuum decay”, Schwinger pairproduction in inhomogeneous electric fields

(HG,KLINGMÜLLER’05)

nonperturbative effective actions(HG,SÁNCHEZ–GUILLÉN,VÁZQUEZ’05)


Higher loops per pedes

B Feynman diagrammar:

∼∫

dDp1

(2π)D

∫dDp2

(2π)D

∏i

∆i(qi)



B Worldline:

∼∫

T

⟨ ∫dτ1dτ2 ∆(x(τ1), x(τ2))

⟩x

B photon propagator in coordinate space

∆(x1, x2) =

∫dDp

(2π)D1p2 eip(x1−x2) =

Γ(D−2

2

)4πD/2

1|x1 − x2|D−2



B Worldline:

∼∫

T

⟨e−ie

Hdx·A(x)

∫dτ1dτ2 ∆(x(τ1), x(τ2))

⟩x



B Feynman diagrammar:

+ ∼∫

dDp1

(2π)D

∫dDp2

(2π)D

∫dDp3

(2π)D

∏i

∆i(qi)

+

∫dDp1

(2π)D

∫dDp2

(2π)D

∫dDp3

(2π)D

∏i

∆i(qi)



B Worldline:

+

∼∫

T

⟨ ∫dτ1dτ2dτ3dτ4 ∆(x(τ1), x(τ2))∆(x(τ3), x(τ4))

⟩x

B both diagrams in one expression



B Worldline:

+

∼∫

T

⟨ (∫dτ1dτ2 ∆(x(τ1), x(τ2))

)2 ⟩x

B both diagrams in one expression



B Worldline, all possible photon insertions:

∑∼

∫T

⟨exp

(−e2

2

∫dτ1dτ2 ∆(x(τ1), x(τ2))

) ⟩x

=⇒ “quenched approximation” (further charged loops neglegted)(FEYNMAN’50)


Systematics: small-Nf expansion

+ + + . . .

∼ Nf

∫T

⟨e−

e22

R R∆

⟩x

+ N2f

∫T 1,T 2

⟨F2{x1, x2}

⟩x1,x2

+ N3f

∫T 1,T 2,T 3

⟨F3{x1, x2, x3}

⟩x1,x2,x3

+ . . .

=⇒ “particle-~ expansion” (HALPERN&SIEGEL’77)

=⇒ arbitrary g, “small” A (. . . but not perturbative in A)


A scalar model in quenched approximation

φ: “charged” matter field, A: “scalar” photon

L(φ,A) =12

(∂µφ)2 +12

m2φ2 +12

(∂µA)2 − i2

h Aφ2.

well-defined perturbative expansionwell-defined small-Nf expansion∼ h Aφ2 superrenormalizable, [h] = 1, in D = 4imaginary interaction ∼ QED

(. . . imaginary Wick-Cutkosky model)


Photon effective action

B quenched approximation

ΓQA[A] =

∫x

12

(∂µA)2 − 12(4π)2

∫ ∞

0

dTT 3 e−m2T

⟨eih

RdτA e−h2 V [x ]

⟩x

= , (1)

B Worldline self-interaction potential

h2 V [x ] :=h2

8π2

∫ T

0dτ1dτ2

1|x1 − x2|2


Self-interaction potential


Quenched effective action

B soft-photon effective action, A ' const. ( . . . á la Heisenberg-Euler)

ΓQA[A] = − 12(4π)D/2

∞∫0

dTT 1+D/2 e−m2T eihAT

⟨e−h2 V [x ]

⟩x

=

B PDF analysis⟨e−h2 V [x ]

⟩x

=

∫dV Px(V ) e−h2V


Renormalized effective action(HG,SANCHEZ-GUILLEN,VAZQUEZ’05)

ΓQA,R[A] = − 132π2

∫d4x

∞∫0

dTT 3 e−m2

RT(

eihAT − 1− ihAT +(hAT )2

2

)

×

(β

β + h2

8π2 T

)1+α

, α ' 0.79, β ' 13.2

0.5 1 1.5 2A

-0.002

-0.0015

-0.001

-0.0005

Re G 8h=1<

0.5 1 1.5 2A

-0.8

-0.6

-0.4

-0.2

Re G 8h=5<


Massless Limit?

B one-loop small-φ-mass limit: IR divergence

Γ1-loop[A]∣∣

hAm2

R�1 ' −

164π2

∫d4x (hA)2 ln

hAm2

R

B quenched small-φ-mass limit: finite

ΓQA,R[A]|mR=0 = −[−Γ(−2− α)] cos π

2α

25−3απ2(1−α)βα

∫d4x (hA)2

(Ah

)α

[1+O((A/h))]

=⇒ break-down of massless limit ∼ artifact of perturbation theory

. . . large log’s summable


Conclusions.

Probing the quantum vacuum by strongfields, Casimir boundaries, etc . . .

. . . brings QFT to the desktop

“quantum fields meet micro mechanics”

Worldline numerics :

efficient tool

intuitive picture


Fermions on the worldline I.

B Grassmann loops

Γ1spin = ln det

[γµ∂µ + ieγµAµ + m

]= − 1

2(4π)D/2

∫ ∞

1/Λ2

dTT 1+D/2 e−m2T

∫PDx∫

ADψ e−

R T0 dτLspin

Lspin =14

x2 + iexµAµ+12ψµψ

µ − ieψµFµνψν


Fermions on the worldline II.

B spinor QED (parity-even part):

Γ =− 1

2

(4π)D2

∫dDxCM

∞∫1/Λ2

dTT D

2 +1e−m2T

⟨Wspin[A]

⟩x

Wspin[A] = W [A] × PT exp

ie2

T∫0

dτ σµνFµν

Fσ Fσ

FσFσ

FσFσFσ

FσFσ

Fσ

FσFσ


Fermions on the worldline III.

B Spin factor (STROMINGER’80,POLYAKOV’88)

Γ[A] =12

1(4π)D/2

∫ ∞

0

dTT (1+D/2)

e−m2T⟨

e−ieH

dxA(x) Φ[x ]⟩

x

Φ[x ] := trγP : ei2

R T0 dτ σω(τ) :

ωµν(τ) =14

limε→0

∫ ε

−ε

dρρ xµ(τ +ρ

2)xν(τ − ρ

2)

(HG&HAMMERLING’05)


Worldline Approach to the Casimir Effect

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