Quantum Vacuum on the Worldline

The <strong>Quantum</strong> <strong>Vacuum</strong> 

Worldline Approach 

Worldline applications 

<strong>Quantum</strong> <strong>Vacuum</strong> on the Worldline 

Holger Gies 

Institute for Theoretical Physics 

Heidelberg University 

Holger Gies <strong>Quantum</strong> <strong>Vacuum</strong> on the Worldline

Outline 




1 The <strong>Quantum</strong> <strong>Vacuum</strong> 

Topography of QFT 

A view on the quantum vacuum 

Effective action 

2 Worldline Approach 

Effective Action from Worldline Techniques 

Worldline numerics 

3 Worldline applications 

Generalized Heisenberg-Euler Effective Actions 

Casimir Effect 

Nonperturbative Worldline Dynamics 


Outline 



















Topography of QFT. 







Holger Gies <strong>Quantum</strong> <strong>Vacuum</strong> on the Worldline 

(MERIAN 1620)









(MERIAN 1620)









(MERIAN 1620)









(MERIAN 1620)









(MERIAN 1620)









(MERIAN 1620)

Outline 






















A view on the quantum vacuum. 

⊲ ρ → 0: “pneumatic vacuum” 












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









⊲ Probing the quantum vacuum, e.g., by external fields: 

“modified quantum vacuum” 









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






⊲ Heat bath: quantum & thermal fluctuations 









⊲ Boundary conditions: Casimir effect 









+ 

+ 

+ 

+ 

e 




⊲ electric fields: Schwinger pair production “vacuum decay” 

− 

e 


+ 

−

Outline 






















Universal tool: effective action Γ. 




<strong>Quantum</strong> 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)








<strong>Quantum</strong> vacuum with background A, e.g., scalar QED 

fluctuations → Γ[A] 

 

Γ[A] = − ln 

= 

 

λ 

ln 

Dφ e − R −|D(A)φ| 2 +m 2 |φ| 2 

 

λ 2 + m 2 

⊲ spectrum of quantum fluctuations: −D(A) 2 φ = λ 2 φ 

= 






Γ[A] = 

λ 

 

ln λ 2 + m 2 

Problem solved, “in principle” 




find spectrum λ for a given background A 

sum over spectrum 

= 





Heisenberg-Euler effective action. 

Γ = + 

 

= − F + 1 

 

x 




(EULER, KOCKEL’35; HEISENBERG, EULER’36;WEISSKOPF’36; SCHWINGER’51; RITUS’76) 

8π 2 

x 

1% 

+ + . . . 

 

ds 

s e−im2 

s 

(es) 2 

F 

|G| cot(es 2 +G2 +F) 

 

F 

 

× coth(es 2 +G2 −F) . . . 

Conventions: F = 1 

4 FµνF µν = 1 

2 (B2 − E 2 ), G = 1 

4 Fµν ˜ F µν = −B · E 





Heisenberg-Euler effective action. 




(EULER, KOCKEL’35; HEISENBERG, EULER’36;WEISSKOPF’36; SCHWINGER’51; RITUS’76) 

⊲ weak-field expansion 

Γ = 

 

−F + 8 α 

45 

2 

m4 F 2 + 14 α 

45 

2 

m4 G2 + O(F 6 ) 

= + + . . . 

Conventions: F = 1 

4 FµνF µν = 1 

2 (B2 − E 2 ), G = 1 

4 Fµν ˜ F µν = −B · E 


Outline 






















Remember . . . 

Γ[A] = 

λ 

 

ln λ 2 + m 2 

Problem solved, “in principle” 



find spectrum λ for a given background A 

sum over spectrum 

= 




















Γ[A] = 

BUT: 

λ 

 

ln λ 2 + m 2 



= 

In general practice: 

spectrum {λ} not known 

analytically 

spectrum {λ} not bounded 

 

λ → ∞ (regularization) 

renormalization 





Worldline representation of Γ. 

⊲ pedestrian approach 

Γ[A] = 

λ 

= − 

 

= − 

∞ 

1/Λ 2 

∞ 

1/Λ 2 

 

ln λ 2 + m 2 



dT 

T e−m2 

T 

Tr exp D(A) 2 

T 

dT 

T e−m2 

T 

N 

x(T )=x(0) 

 

= Tr ln −(D(A)) 2 + m 2 

 

=〈x|e iH(iT ) |x〉 

Dx(τ) e − 

TR “ ” 

˙x 2 

dτ 4 +ie ˙x·A(xτ) 

0 






⊲ pedestrian approach 

Γ[A] = 

λ 

= − 

 

= − 

∞ 

1/Λ 2 

∞ 

1/Λ 2 

 

ln λ 2 + m 2 



dT 

T e−m2 

T 

Tr exp D(A) 2 

T 

dT 

T e−m2 

T 

N 

x(T )=x(0) 

 

= Tr ln −(D(A)) 2 + m 2 

 

=〈x|e iH(iT ) |x〉 

Dx(τ) e − 

TR “ ” 

˙x 2 


0 






x(T ) = 

 

Γ[A] = − 

∞ 

1/Λ 2 

dT 

T e−m2 

T 

N 

x(T )=x(0) 



Dx(τ) e − 

TR “ ” 

˙x 2 


0 


(FEYNMAN’50) 

. 

(HALPERN&SIEGEL’77) 

(POLYAKOV’87) 

. 

. 

(BERN&KOSOWER’92; STRASSLER’92) 

(SCHMIDT&SCHUBERT’93) 

(KLEINERT’94)





 

Γ[A] = − 

Worldline approach: 

∞ 

1/Λ 2 

dT 

T e−m2 

T 

N 

x(T ) = 

x(T )=x(0) 



Dx(τ) e − 

effective action Γ ∼ closed worldlines x(τ) 

worldline ∼ spacetime trajectory of φ fluctuations 

gauge-field interaction ∼ “Wegner-Wilson loop” 

TR “ ” 

˙x 2 


0 

finding {λ} and 

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


Outline 


















Worldline Numerics. 

 

x(1)=x(0) 

Dx(t) −→ 




nL 

l=1 



, nL = # of worldlines 

x(t) −→ x i, i = 1, . . . , N (ppl) 

→ statistical error 

→ systematical error 

−→ → spacetime remains continuous 







Trajectory of a <strong>Quantum</strong> Fluctuation. 

⊲ Feynman diagram (conventionally in momentum space) 








⊲ worldline (artist’s view) 








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


















































Propertime T . 






T ∼ regulator scale of smeared momentum shells 


Propertime T . 






⊲ “Measuring” the Wegner-Wilson loop exp −ie dx · A in a 

background A 


Outline 



















Magnetic Step. 




















=⇒ “<strong>Quantum</strong> diffusion” of B field 






D = 3 

3/2 4π 

eB Leff(x) 

−0.2 

−0.4 

−0.6 




0 

m=0, set A 

m=0, set B 

m=0, set C 

1/2 

m=0.5 B0 , set A 

1/2 

m=0.5 B0 , set B 

1/2 

m=0.5 B0 , set C 




−0.8 

−2 −1 0 1 2 

x √ eB 


(HG,LANGFELD’01) 

HE limit


D = 3 

3/2 4π 

eB Leff(x) 

−0.2 

−0.4 

−0.6 




0 

m=0, set A 

m=0, set B 

m=0, set C 

1/2 

m=0.5 B0 , set A 

1/2 

m=0.5 B0 , set B 

1/2 

m=0.5 B0 , set C 




−0.8 

−2 −1 0 1 2 

x √ eB 

⊲ diffusion law: L1 

eff (x) ∼ exp −3.255 m x − 0.7627 √ 

eB x 



HE limit

Outline 



















Casimir Effect. 




⊲ Hendrik B.G. Casimir 1948: 

F π2 c 

= − 

A 240 a4 ⊲ precision measurements O(1%) 

(LAMOREAUX’97) 

(MOHIDEEN ET AL.’98+) 

(DECCA ET AL.’03+) 

. 

. 










F π2 c 

= − 





. 










F π2 c 

= − 





. 









⊲ Casimir effect ˆ= “strong-field QFT” 

S = 1 

2 (∂φ)2 + m2 

2 φ2 + V φ 2 

V (x) = λ 

 

S 

 

 

dσ δ(x − xσ) + δ(x − xσ) 




x σ 

x σ 

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





Benchmark test: parallel plates 

















⊲ for finite m, λ, a 

-2 (4π) 2 E/m 3 

1e+06 

10000 

100 

1 

0.01 

0.0001 




exact result, λ=100m 

1500 v loops 2048 ppl 

massless Dirichlet limit 

1e-06 

0.01 0.1 1 10 

am 

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








Casimir Effect: curvature effects on the worldline 

S1 

S2 

(a) (b) (c) 









0.012 

0.01 

0.008 

0.006 

0.004 

0.002 


0 

−εR 4















Casimir Effect: sphere above plate. 

E Casimir /E PFA (a/R




Casimir Edge Effects. 




F = −γ 

F1si = ? 


c 

· A 

a4 (CF. BRESSI,CARUGNO,ONOFRIO,RUOSO’02)





Σ1 




a 


Σ2








0 

-0.002 

-0.004 

-0.006 

-0.008 

-0.01 

-0.012 

-0.014 


εCasimira 4





⊲ effective description of a finite plate 

area A boundary C 

F = −γ 




c 

Aeff, 

a4 ⊲ effective area: Aeff A + γ1si 

γ aC, γ1si = 5.23(2) × 10 −3 


(HG,KLINGMULLER’06)





⊲ effective description of a finite plate 

area A boundary C 

F = −γ 




c 

Aeff, 

a4 ⊲ effective area: Aeff A + γ1si 

γ aC, γ1si = 5.23(2) × 10 −3 


(HG,KLINGMULLER’06)




Further Worldline Applications. 

+ 

+ 

+ 

+ 

e − 

e+ 

− 




Heisenberg-Euler effective actions, spinor QED, 

flux tubes, quantum-induced vortex interactions 

thermal fluctuations, free energies 

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

“spontaneous vacuum decay”, Schwinger pair 

production in inhomogeneous electric fields 

nonperturbative effective actions 



(HG,KLINGMÜLLER’05) 

(HG,SÁNCHEZ–GUILLÉN,VÁZQUEZ’05)

Outline 






















Higher loops per pedes 

⊲ Feynman diagrammar: 

∼ 




D d p1 

(2π) D 

D d p2 

(2π) D 


 

∆i(qi) 

i





⊲ Worldline: 

∼ 

 

T 




 

⊲ photon propagator in coordinate space 

 

∆(x 1, x 2) = 

D d p 

(2π) D 

1 

p2 eip(x Γ 

1−x 2) 

= 

D−2 

2 

4πD/2 

dτ1dτ2 ∆(x(τ1), x(τ2)) 

x 


1 

|x 1 − x 2| D−2






∼ 

 

T 




 

e −ie H dx·A(x) 


 

 

dτ1dτ2 ∆(x(τ1), x(τ2)) 

x





⊲ Feynman diagrammar: 

+ ∼ 




D d p1 

(2π) D 

D d p2 

(2π) D 

D d p3 

(2π) D 

D d p1 

+ 

(2π) D 

D d p2 

(2π) D 

D d p3 

(2π) D 


 

∆i(qi) 

i 

 

∆i(qi) 

i






∼ 

 

T 

⎧ 

⎪⎨ 

⎪⎩ 

 

⊲ both diagrams in one expression 




+ 

 

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

x 

⎫ 

⎪⎬ 

⎪⎭ 







∼ 

 

T 

⎧ 

⎪⎨ 

⎪⎩ 

 

⊲ both diagrams in one expression 




+ 

⎫ 

⎪⎬ 

⎪⎭ 

2 

dτ1dτ2 ∆(x(τ1), x(τ2)) 

x 






⊲ 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 




+ + + . . . 

 

R R 

e2 − 

e 2 ∆ 

+ N 3 f 

 

T 1,T 2,T 3 

x 

+ N 2 f 

 

T 1,T 2 

 

 

F3{x 1, x 2, x 3} 

 

F2{x 1, x 2} 

x 1,x 2,x 3 

x 1,x 2 

+ . . . 

=⇒ “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) = 1 

2 (∂µφ) 2 + 1 

2 m2 φ 2 + 1 

2 (∂µA) 2 − i 

2 h A φ2 . 

well-defined perturbative expansion 

well-defined small-Nf expansion 

∼ h A φ 2 superrenormalizable, [h] = 1, in D = 4 

imaginary interaction ∼ QED 

(. . . imaginary Wick-Cutkosky model) 





Photon effective action 

⊲ quenched approximation 

ΓQA[A] = 

 

1 

2 (∂µA) 2 − 

1 

2(4π) 2 

∞ 

x 

0 




dT 

T 3 e−m2 

T 

e ih R dτA −h 

e 2 

V [x] 

= , (1) 

⊲ Worldline self-interaction potential 

h 2 V [x] := h2 

8π 2 

T 

0 

dτ1dτ2 

1 

|x 1 − x 2| 2 


x




Self-interaction potential 








Self-interaction potential 








Quenched effective action 




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

1 

ΓQA[A] = − 

2(4π) D/2 

∞ 

dT 

T 1+D/2 e−m2 T ihAT 

e 

⊲ PDF analysis 

 

e −h2 

V [x] 

x 

 

= dV Px(V ) e −h2V 0 

 

e −h2 

V [x] 


x 

=




Renormalized effective action 

ΓQA,R[A] = − 1 

32π2 

d 4 x 

∞ 

0 

× 

 

0.5 1 1.5 2 A 

Re h1 

0.0005 

0.001 

0.0015 

0.002 




dT 

e−m2 R 

T 3 T 

β 

β + h2 

8π 2 T 

(HG,SANCHEZ-GUILLEN,VAZQUEZ’05) 

 

e ihAT − 1 − ihAT + 

1+α 

 

(hAT )2 

2 

, α 0.79, β 13.2 

0.5 1 1.5 2 A 

Re 

0.2 

h5 

0.4 

0.6 

0.8 


Massless Limit? 







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

Γ1-loop[A] hA 

m2 1 

− 

≫1 64π R 

2 

 

d 4 x (hA) 2 ln hA 

m2 R 

⊲ quenched small-φ-mass limit: finite 

π 

[−Γ(−2 − α)] cos 2 ΓQA,R[A]|mR=0 = − α 

25−3απ2(1−α) βα 

d 4 x (hA) 2 

α A 

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

h 

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


. . . large log’s summable

Conclusions. 




Probing the quantum vacuum by strong 

fields, Casimir boundaries, etc . . . 

. . . brings QFT to the desktop 

“quantum fields meet micro mechanics” 

⎧ 

⎨ efficient tool 

Worldline numerics : 

⎩ 

intuitive picture 













⎧ 



⎩ 














⎧ 



⎩ 
















Fermions on the worldline I. 

⊲ Grassmann loops 

Γ 1 

spin = ln det γ µ ∂µ + ieγ µ 

Aµ + m 

1 

= − 

2(4π) D/2 

∞ 

1/Λ 2 




dT 

T 1+D/2 e−m2 

T 

Lspin = 1 

4 ˙x 2 + ie ˙x µ Aµ+ 1 

2 ψµ ˙ψ µ − ieψ µ F µνψ ν 


P 

 

Dx Dψ e 

A 

− R T 

0 dτLspin




Fermions on the worldline II. 

⊲ spinor QED (parity-even part): 

Γ = 

1 − 2 

(4π) D 

 

d 

2 

D xCM 

∞ 

1/Λ 2 

dT 

Wspin[A] = W [A] × PT exp 

σF 

σF 

σF 

σF 

σF 

σF 




T D 

2 +1 e−m2 T 

σF 

⎛ 

⎝ ie 

2 

σF 

σF 

σF 

T 

0 

σF 

σF 

 

Wspin[A] 

dτ σµνF µν 


 

x 

⎞ 

⎠




Fermions on the worldline III. 




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

Γ[A] = 1 1 

2 (4π) D/2 

∞ 

0 

Φ[x] := trγP : e i 

R T 

2 0 dτ σω(τ) : 

ωµν(τ) = 1 

4 lim 

ɛ→0 

dT 

T (1+D/2) e−m2 T 

e −ie H 

dxA(x) 

Φ[x] 

x 

ɛ 

−ɛ 

dρρ ¨x µ(τ + ρ 

2 )¨x ν(τ − ρ 

2 ) 


(HG&HAMMERLING’05)

Quantum Vacuum on the Worldline

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