Quantum Vacuum on the Worldline

gceWuRCA5A

Quantum Vacuum on the Worldline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

ong>Quantumong> ong>Vacuumong> on the Worldline

Holger Gies

Institute for Theoretical Physics

Heidelberg University

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Topography of QFT.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(MERIAN 1620)


Topography of QFT.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(MERIAN 1620)


Topography of QFT.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(MERIAN 1620)


Topography of QFT.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(MERIAN 1620)


Topography of QFT.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(MERIAN 1620)


Topography of QFT.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(MERIAN 1620)


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

⊲ ρ → 0: “pneumatic vacuum”

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

Topography of QFT

A view on the quantum vacuum

Effective action

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

Topography of QFT

A view on the quantum vacuum

Effective action

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

“modified quantum vacuum”

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

Topography of QFT

A view on the quantum vacuum

Effective action

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

⊲ Heat bath: quantum & thermal fluctuations

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

⊲ Boundary conditions: Casimir effect

Topography of QFT

A view on the quantum vacuum

Effective action

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

A view on the quantum vacuum.

+

+

+

+

e

Topography of QFT

A view on the quantum vacuum

Effective action

⊲ electric fields: Schwinger pair production “vacuum decay”


e

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

+


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Topography of QFT

A view on the quantum vacuum

Effective action

ong>Quantumong> 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)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(CASIMIR’48)


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Topography of QFT

A view on the quantum vacuum

Effective action

ong>Quantumong> 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

=

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Γ[A] =

λ


ln λ 2 + m 2

Problem solved, “in principle”

Topography of QFT

A view on the quantum vacuum

Effective action

find spectrum λ for a given background A

sum over spectrum

=

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Heisenberg-Euler effective action.

Γ = +


= − F + 1


x

Topography of QFT

A view on the quantum vacuum

Effective action

(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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Heisenberg-Euler effective action.

Topography of QFT

A view on the quantum vacuum

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

Effective Action from Worldline Techniques

Worldline numerics

2 Worldline Approach

Effective Action from Worldline Techniques

Worldline numerics

3 Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Remember . . .

Γ[A] =

λ


ln λ 2 + m 2

Problem solved, “in principle”

Effective Action from Worldline Techniques

Worldline numerics

find spectrum λ for a given background A

sum over spectrum

=

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Effective Action from Worldline Techniques

Worldline numerics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Effective Action from Worldline Techniques

Worldline numerics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Universal tool: effective action Γ.

Γ[A] =

BUT:

λ


ln λ 2 + m 2

Effective Action from Worldline Techniques

Worldline numerics

=

In general practice:

spectrum {λ} not known

analytically

spectrum {λ} not bounded


λ → ∞ (regularization)

renormalization

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Worldline representation of Γ.

⊲ pedestrian approach

Γ[A] =

λ

= −


= −


1/Λ 2


1/Λ 2


ln λ 2 + m 2

Effective Action from Worldline Techniques

Worldline numerics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Worldline representation of Γ.

⊲ pedestrian approach

Γ[A] =

λ

= −


= −


1/Λ 2


1/Λ 2


ln λ 2 + m 2

Effective Action from Worldline Techniques

Worldline numerics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Worldline representation of Γ.

x(T ) =


Γ[A] = −


1/Λ 2

dT

T e−m2

T

N

x(T )=x(0)

Effective Action from Worldline Techniques

Worldline numerics

Dx(τ) e −

TR “ ”

˙x 2

dτ 4 +ie ˙x·A(xτ)

0

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(FEYNMAN’50)

.

(HALPERN&SIEGEL’77)

(POLYAKOV’87)

.

.

(BERN&KOSOWER’92; STRASSLER’92)

(SCHMIDT&SCHUBERT’93)

(KLEINERT’94)


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Worldline representation of Γ.


Γ[A] = −

Worldline approach:


1/Λ 2

dT

T e−m2

T

N

x(T ) =

x(T )=x(0)

Effective Action from Worldline Techniques

Worldline numerics

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

dτ 4 +ie ˙x·A(xτ)

0

finding {λ} and

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

Effective Action from Worldline Techniques

Worldline numerics

2 Worldline Approach

Effective Action from Worldline Techniques

Worldline numerics

3 Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Worldline Numerics.


x(1)=x(0)

Dx(t) −→

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

nL

l=1

Effective Action from Worldline Techniques

Worldline numerics

, nL = # of worldlines

x(t) −→ x i, i = 1, . . . , N (ppl)

→ statistical error

→ systematical error

−→ → spacetime remains continuous

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ Feynman diagram (conventionally in momentum space)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline (artist’s view)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline numerics: N = 10 points per loop (ppl)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline numerics: N = 40 points per loop (ppl)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline numerics: N = 100 points per loop (ppl)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline numerics: N = 1000 points per loop (ppl)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline numerics: N = 10000 points per loop (ppl)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

Trajectory of a ong>Quantumong> Fluctuation.

⊲ worldline numerics: N = 100000 points per loop (ppl)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Propertime T .

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

T ∼ regulator scale of smeared momentum shells

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Propertime T .

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Effective Action from Worldline Techniques

Worldline numerics

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

background A

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

2 Worldline Approach

Effective Action from Worldline Techniques

Worldline numerics

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

3 Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Magnetic Step.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Magnetic Step.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Magnetic Step.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

=⇒ “ong>Quantumong> diffusion” of B field

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Magnetic Step.

D = 3

3/2 4π

eB Leff(x)

−0.2

−0.4

−0.6

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

−0.8

−2 −1 0 1 2

x √ eB

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(HG,LANGFELD’01)

HE limit


Magnetic Step.

D = 3

3/2 4π

eB Leff(x)

−0.2

−0.4

−0.6

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

−0.8

−2 −1 0 1 2

x √ eB

⊲ diffusion law: L1

eff (x) ∼ exp −3.255 m x − 0.7627 √

eB x

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(HG,LANGFELD’01)

HE limit


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

2 Worldline Approach

Effective Action from Worldline Techniques

Worldline numerics

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

3 Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Casimir Effect.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

⊲ 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+)

.

.

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Casimir Effect.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

⊲ 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+)

.

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Casimir Effect.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

⊲ 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+)

.

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Casimir Effect.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

⊲ 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)

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

x σ

x σ

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Benchmark test: parallel plates

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Benchmark test: parallel plates

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Benchmark test: parallel plates

⊲ for finite m, λ, a

-2 (4π) 2 E/m 3

1e+06

10000

100

1

0.01

0.0001

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Casimir Effect: curvature effects on the worldline

S1

S2

(a) (b) (c)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Casimir Effect: curvature effects on the worldline

0.012

0.01

0.008

0.006

0.004

0.002

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

0

−εR 4


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Casimir Effect: curvature effects on the worldline

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Casimir Effect: sphere above plate.

E Casimir /E PFA (a/R


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Casimir Edge Effects.

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

F = −γ

F1si = ?

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

c

· A

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


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Casimir Edge Effects.

Σ1

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

a

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

Σ2


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Casimir Edge Effects.

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

0

-0.002

-0.004

-0.006

-0.008

-0.01

-0.012

-0.014

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

εCasimira 4


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Casimir Edge Effects.

⊲ effective description of a finite plate

area A boundary C

F = −γ

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

c

Aeff,

a4 ⊲ effective area: Aeff A + γ1si

γ aC, γ1si = 5.23(2) × 10 −3

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(HG,KLINGMULLER’06)


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Casimir Edge Effects.

⊲ effective description of a finite plate

area A boundary C

F = −γ

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

c

Aeff,

a4 ⊲ effective area: Aeff A + γ1si

γ aC, γ1si = 5.23(2) × 10 −3

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(HG,KLINGMULLER’06)


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Further Worldline Applications.

+

+

+

+

e −

e+


Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(HG,LANGFELD’02)

(HG,KLINGMÜLLER’05)

(HG,SÁNCHEZ–GUILLÉN,VÁZQUEZ’05)


Outline

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

1 The ong>Quantumong> ong>Vacuumong>

Topography of QFT

A view on the quantum vacuum

Effective action

2 Worldline Approach

Effective Action from Worldline Techniques

Worldline numerics

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

3 Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

⊲ Feynman diagrammar:


Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

D d p1

(2π) D

D d p2

(2π) D

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


∆i(qi)

i


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

Worldline:



T

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics


⊲ 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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

1

|x 1 − x 2| D−2


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

Worldline:



T

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics


e −ie H dx·A(x)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline



dτ1dτ2 ∆(x(τ1), x(τ2))

x


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

⊲ Feynman diagrammar:

+ ∼

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


∆i(qi)

i


∆i(qi)

i


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

Worldline:



T


⎪⎨

⎪⎩


⊲ both diagrams in one expression

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

+


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

x


⎪⎬

⎪⎭

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

Worldline:



T


⎪⎨

⎪⎩


⊲ both diagrams in one expression

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

+


⎪⎬

⎪⎭

2

dτ1dτ2 ∆(x(τ1), x(τ2))

x

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Higher loops per pedes

Worldline, all possible photon insertions:




T



exp − e2


2

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics


dτ1dτ2 ∆(x(τ1), x(τ2))

x

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(FEYNMAN’50)


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Systematics: small-Nf expansion

∼ Nf


T

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

+ + + . . .


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)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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)

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Photon effective action

⊲ quenched approximation

ΓQA[A] =


1

2 (∂µA) 2 −

1

2(4π) 2


x

0

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

x


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Self-interaction potential

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Self-interaction potential

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Quenched effective action

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

⊲ 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]

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

x

=


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Massless Limit?

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

. . . large log’s summable


Conclusions.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Conclusions.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


Conclusions.

The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

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

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

P


Dx Dψ e

A

− R T

0 dτLspin


The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

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

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

T D

2 +1 e−m2 T

σF


⎝ ie

2

σF

σF

σF

T

0

σF

σF


Wspin[A]

dτ σµνF µν

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline


x



The ong>Quantumong> ong>Vacuumong>

Worldline Approach

Worldline applications

Fermions on the worldline III.

Generalized Heisenberg-Euler Effective Actions

Casimir Effect

Nonperturbative Worldline Dynamics

⊲ 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 )

Holger Gies ong>Quantumong> ong>Vacuumong> on the Worldline

(HG&HAMMERLING’05)

More magazines by this user
Similar magazines