worldline approach to the casimir effect
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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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
→∞
Holger Gies Worldline Approach to the Casimir Effect
Casimir’s derivation
B boundary conditions:
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 !
Holger Gies Worldline Approach to the Casimir Effect
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.”
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
B ρ→ 0: “pneumatic vacuum”
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
B QFT: quantum fluctuations BUT: . . . just a picture !
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
B Boundary conditions: Casimir effect
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
B Probing the quantum vacuum, e.g., by external fields:“modified quantum vacuum”
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
=⇒ modified light propagation: “QV ' medium” (PVLAS,BMV,Q&A)
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
B Heat bath: quantum & thermal fluctuations
Holger Gies Worldline Approach to the Casimir Effect
Another view on the quantum vacuum.
+++++
−−−−−
ee + −
B electric fields: Schwinger pair production “vacuum decay”
Holger Gies Worldline Approach to the Casimir Effect
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)
Holger Gies Worldline Approach to the Casimir Effect
Universal tool: effective action Γ.
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
Holger Gies Worldline Approach to the Casimir Effect
Universal tool: effective action Γ.
Γ[A] =∑
λ
ln(λ2 + m2
)=
Problem solved, “in principle”
find spectrum λ for a given background Asum over spectrum
Holger Gies Worldline Approach to the Casimir Effect
Universal tool: effective action Γ.
Γ[A] =∑
λ
ln(λ2 + m2
)=
BUT:
In general practice:
spectrum {λ} not knownanalyticallyspectrum {λ} not bounded∑
λ →∞ (regularization)renormalization
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
Casimir meets AFM.
Atomic-Force-Microscope Measurements(MOHIDEEN ET AL.’98)
B Sphere-Plate configuration (DERJAGUIN ET AL.’56)
Force sensitivity: 10−17 N
Holger Gies Worldline Approach to the Casimir Effect
Casimir meets AFM.
Atomic-Force-Microscope Measurements(MOHIDEEN ET AL.’98)
200 µm polystyrene sphere with gold coating 85.6± 0.6 nm
Holger Gies Worldline Approach to the Casimir Effect
Casimir meets AFM.
Atomic-Force-Microscope Measurements(MOHIDEEN ET AL.’98)
B Sphere-Plate configuration (DERJAGUIN ET AL.’56)
B Results
B Corrections:
material
{surface roughness
finite conductivity
finite temperature
geometry
}QFT
Holger Gies Worldline Approach to the Casimir Effect
Casimir Morphology.
a
R
F = − π2
240~ ca4 A F = ? F = ?
Holger Gies Worldline Approach to the Casimir Effect
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)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = − π2
240~ca4 · A
B parallel-plate
energy density
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = − π2
240~ca4 · A
F1si = ?
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = − π2
240~ca4 · A
F1si = ?
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = − π2
240~ca4 · A
F1si = ?
Holger Gies Worldline Approach to the Casimir Effect
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)
”
Holger Gies Worldline Approach to the Casimir Effect
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)
”
Holger Gies Worldline Approach to the Casimir Effect
Worldline representation of Γ.
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)
Holger Gies Worldline Approach to the Casimir Effect
Worldline representation of Γ.
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)
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B Feynman diagram (conventionally in momentum space)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline (artist’s view)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 4 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 10 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 40 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 100 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 1000 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 10000 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Trajectory of a Quantum Fluctuation.
B worldline numerics: N = 100000 points per loop (ppl)
Holger Gies Worldline Approach to the Casimir Effect
Propertime T .
T ∼ regulator scale of smeared momentum shells
Holger Gies Worldline Approach to the Casimir Effect
Propertime T .
B “Measuring” the Wegner-Wilson loop exp(−ie
∮dx · A
)
Holger Gies Worldline Approach to the Casimir Effect
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))
”
Holger Gies Worldline Approach to the Casimir Effect
Benchmark test: parallel plates
B for finite m, λ, a
(BORDAG,HENNIG, ROBASCHIK ’92) (HG,LANGFELD,MOYAERTS ’03)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Effect: curvature effects on the worldline
S2
(a) (b) (c)
S1
Holger Gies Worldline Approach to the Casimir Effect
Casimir Effect: curvature effects on the worldline
−εR
4
0.012
0.01
0.008
0.006
0.004
0.002
0
Holger Gies Worldline Approach to the Casimir Effect
Casimir Effect: curvature effects on the worldline
(THANKS TO K. KLINGMULLER)
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
Casimir Effect: sphere above plate.
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
Holger Gies Worldline Approach to the Casimir Effect
Casimir Effect: sphere above plate.
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)
Holger Gies Worldline Approach to the Casimir Effect
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)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = −γ‖~ca4 · A
F1si = ?
(CF. BRESSI,CARUGNO,ONOFRIO,RUOSO’02)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = −γ‖~ca4 · A
F1si = ?
(CF. BRESSI,CARUGNO,ONOFRIO,RUOSO’02)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
F‖ = −γ‖~ca4 · A
F1si = ?
(CF. BRESSI,CARUGNO,ONOFRIO,RUOSO’02)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
εC
asim
ira
4
0
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
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)
Holger Gies Worldline Approach to the Casimir Effect
Casimir Edge Effects.
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)
Holger Gies Worldline Approach to the Casimir Effect
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)
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
B Feynman diagrammar:
∼∫
dDp1
(2π)D
∫dDp2
(2π)D
∏i
∆i(qi)
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
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
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
B Worldline:
∼∫
T
⟨e−ie
Hdx·A(x)
∫dτ1dτ2 ∆(x(τ1), x(τ2))
⟩x
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
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)
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
B Worldline:
+
∼∫
T
⟨ ∫dτ1dτ2dτ3dτ4 ∆(x(τ1), x(τ2))∆(x(τ3), x(τ4))
⟩x
B both diagrams in one expression
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
B Worldline:
+
∼∫
T
⟨ (∫dτ1dτ2 ∆(x(τ1), x(τ2))
)2 ⟩x
B both diagrams in one expression
Holger Gies Worldline Approach to the Casimir Effect
Higher loops per pedes
B Worldline, all possible photon insertions:
∑∼
∫T
⟨exp
(−e2
2
∫dτ1dτ2 ∆(x(τ1), x(τ2))
) ⟩x
=⇒ “quenched approximation” (further charged loops neglegted)(FEYNMAN’50)
Holger Gies Worldline Approach to the Casimir Effect
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)
Holger Gies Worldline Approach to the Casimir Effect
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)
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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<
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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
Holger Gies Worldline Approach to the Casimir Effect
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