QED

QED Vacuum

Schwinger Effect for E(x , t) = E(t)

Results

Schwinger Effect for E(x , t)

Schwinger Pair Production in Strong Electric Fields Florian Hebenstreit Advisers: R. Alkofer (KFU Graz), H. Gies (FSU Jena)

Seminar des Graduiertenkollegs FSU Jena 23.06.2009

Summary

QED Vacuum


Results


Summary

Historical overview: Does the nothingness exist? Does the nothingness exist? What is the meaning of the vacuum? • Democritus, Leucippus: ∼ 450BC

Vacuum necessary for the motion of particles

• Aristotle: ∼ 350BC

Nature abhors a vacuum → horror vacui

• Torricelli, Pascal, von Guericke: ∼ 1650AD

Experiments (diluted gases) → vacuum exists

• Quantum Field Theory: ∼ 1950AD

Vacuum fluctuations, virtual particles, vacuum energy...

QED Vacuum


Results

Outline

QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary


Summary

QED Vacuum


Results

Outline



Summary

QED Vacuum



Results

Summary

The Vacuum of QuantumElectroDynamics Vacuum of QFT: State in which no real particles are present BUT Vacuum fluctuations: Virtual particles do exist

Quantum Mechanics

→

↓

Uncertainty Principle: ∆E · ∆t & ~

Quantum Field Theory

←

l

→

Vacuum Fluctuations: Virtual Particles

Virtual particles exist for ∆t ≈

~ mc 2

Special Relativity ↓

←

Mass-Energy Equivalence: E = mc2

≈ 10−21 s

QED Vacuum


Results


Summary

Feynman Diagrams for QED

¯ ∂/ − eA / + m)ψ • QED Lagrangian: L = − 14 F µν Fµν + ψ(i • Amplitudes → Cross sections, Decay rates...

• Feynman Diagrams: Graphical representation

Electron Line

Photon Line

Elementary Vertex

NLO (1-Loop) Diagrams ↔ QED Corrections

Electron Self Energy

Vacuum Polarization

Vertex Function

QED Vacuum


Results


Summary

Effects due to the Non-Trivial Vacuum of QED QED effects in atomic physics: • Anomalous Magnetic Moment: Deviation from g = 2

• Lamb Shift: Lift of degenerate energy levels in the H-atom

QED effects in pure vacuum: • Casimir Effect: Attractive force between two parallel plates

QED effects in perturbed vacuum: • Non-Linear Compton Scattering: → T.Heinzl (21.4.)

• Vacuum Birefringence: Linear → elliptic polarization

• Schwinger Effect: Spontaneous production of e+ e− pairs

QED Vacuum



Results

Summary

QED effects in atomic physics Anomalous Magnetic Moment

• Electron spin ↔ Magnetic moment • Magnetic moment: ~ µ = −gµB~s/~ • Relativistic QM: g = 2

LO (Tree level) ↔ Relativistic QM

NLO ↔ QED Corrections

g − 2 = α/π → QED precision test

QED Vacuum


Results


Summary

QED effects in atomic physics Lamb Shift Lift of degenerate energy levels in the H-atom Non-relativistic QM

Relativistic QM

QED Corrections

2p (n = 2, l = 1) 2s (n = 2, l = 0)

1s (n = 1, l = 0) Most important: Emission and re-absorption of virtual photons

QED Vacuum


Results


Summary


Relativistic QM

QED Corrections

2p (n = 2, l = 1)

H 2p3/2 (j = 3/2, l = 1) 2s (n = 2, l = 0) A H A A 2p (j = 1/2, l = 1) A 1/2

2s1/2 (j = 1/2, l = 0) Most important: Emission and re-absorption of virtual photons

QED Vacuum


Results


Summary


Relativistic QM

QED Corrections

2p (n = 2, l = 1) 2s (n = 2, l = 0) A

2s

(j = 1/2, l = 0)

1/2 A A 2p (j = 1/2, l = 1) A 1/2 2s1/2 (j = 1/2, l = 0)@ 2p1/2 (j = 1/2, l = 1) @

Most important: Emission and re-absorption of virtual photons

QED Vacuum


Results


QED effects in pure vacuum Casimir effect • Outside: Fluctuations with all frequencies/wavelengths • Conducting plates → Boundary condition: Ek = 0 • Inside: Possible frequencies/wavelengths restricted vac ǫvac outside > ǫinside

~cπ 2 FC =− A 240d 4 A. Lambrecht, Physik in unserer Zeit 36 (2005)

Attractive force between the plates

Summary

QED Vacuum


Results


Summary

QED effects in perturbed vacuum Vacuum Birefringence • Strong background field polarizes the vacuum • Vacuum polarization → refractive index n • Different n for different polarization states

α n± = 1 + 45π

E Ecr

2

(11±3)

T. Heinzl and A. Ilderton, arXiv:0809.3348 (2008)

Linear polarized probe beam → eliptically polarized

QED Vacuum


Results


Summary

QED effects in perturbed vacuum Schwinger Effect: The Analogy to Atomic Physics • Sub-barrier Tunnelling: Ionization of H-atom by E-field

J. Oppenheimer, Phys. Rev. 31 (1928) 4

• Ground state: Electron bound with Eb = me = −13.6 eV 2~2 • Perturbation: Constant electric field → ΦE ∼ −E z VHrL @eVD r @a.uD

-13.6

2 m2 e5 P ∼ exp − 3 E ~4

4 ∼ exp − 3

√

3/2

2mEb eE ~

!

QED Vacuum


Results


Summary




-13.6

2 m2 e5 P ∼ exp − 3 E ~4

4 ∼ exp − 3

√

3/2

2mEb eE ~

!

QED Vacuum


Results


Summary




-13.6

2 m2 e5 P ∼ exp − 3 E ~4

4 ∼ exp − 3

√

3/2

2mEb eE ~

!

QED Vacuum



Results

Summary

QED effects in perturbed vacuum Schwinger Effect • Sub-barrier Tunnelling: Production of e+ e− pairs by E-field

F. Sauter, Z. Phys. 69, 742 (1931) W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1935)

• Vacuum state: Dirac sea picture → Eb = 2mc 2 • Perturbation: Constant electric field → ΦE ∼ −E z +mc2 -mc2

pz

m2 c 3 P ∼ exp −π eE ~

z

π ∼ exp − 4

√

3/2

2mEb eE ~

!

QED Vacuum



Results

Summary




pz


z

π ∼ exp − 4

√

3/2

2mEb eE ~

!

QED Vacuum



Results

Summary




pz


z

π ∼ exp − 4

√

3/2

2mEb eE ~

!

QED Vacuum


Results


Summary

QED effects in perturbed vacuum Yet another Analogy: Oscillatory Inhomogeneity L. Keldysh, Sov. Phys. JETP. 20 (1965)

• 1st (laser) time scale ω: E (t) = E cos(ω t) √ • 2nd (tunnelling) time scale ωT : ωT ∼

v L

• Keldysh adiabaticity parameter γ: γ =

∼

ω ωT

Eb /2m Eb /eE

=

ω

√

= √ eE

2mEb

2mEb eE

Non-perturbative regime: ω ≪ ωT ↔ γ ≪ 1 Low frequency/Strong fields ↔ ’Instantaneous’; Tunneling √ 3/2 2mE P ∼ exp − 43 eE~b

Perturbative regime: ω ≫ ωT ↔ γ ≫ 1 High frequency/Weak fields ↔ ’No time to tunnel’; Multi-Photon 2Eb /~ω P ∼ 2ω√eE2mE b

QED Vacuum


Results


Summary

QED effects in perturbed vacuum Schwinger Effect with Oscillatory Inhomogeneity E. Brezin and C. Itzykson, Phys. Rev. D. 2 (1970)

• 1st (laser) time scale ω: E (t) = E cos(ωt) • 2nd (tunnelling) time scale ωT : ωT ∼ vL ∼ mc 2c/eE = • Keldysh adiabaticity parameter γ: γ =

ω ωT

=

eE mc

ωmc eE

Non-perturbative regime: ω ≪ ωT ↔ γ ≪ 1 Low frequency/Strong fields ↔ ’Instantaneous’; Tunneling 2 c3 P ∼ exp −π meE~

Perturbative regime: ω ≫ ωT ↔ γ ≫ 1 High frequency/Weak fields ↔ ’No time to tunnel’; Multi-Photon 2 eE 4mc /~ω P ∼ 2ωmc

QED Vacuum


Results


Summary

QED effects in perturbed vacuum

Effect:

Atomic Physics

Schwinger Effect

Ionization of Atoms

Vacuum Pair Production

Keldysh γ:

γ=

γ ≪ 1:

P ∼ exp

γ ≫ 1:

P∼

√ ω 2mEb eE

− 34

√

√eE 2ω 2mEb

γ= 3/2

2mEb eE~

2Eb /~ω

ωmc eE

2 c3 P ∼ exp −π meE~ P∼

2 eE 4mc /~ω 2ωmc

QED Vacuum


Results

Outline



Summary


QED Vacuum


Results

Schwinger Effect: Various Methods Various methods: • WKB / Scattering Theory

• Effective Action Approach • Quantum Kinetic Theory

• Monte Carlo Simulations • ...

Only one-dimensional inhomogeneities: E (x) or E (t)! • Technical remark: ~ = c = 1

m2 e ≈ 10−21 s

• Critical field strength: Ecr = • Compton time: tc =

1 m

≈

1018 V/m

Summary

QED Vacuum


Results


Summary

Effective Action Approach (Imaginary Time) • Vacuum persistence amplitude: h0|0iA = eiSeff • Pair production probability: P ≃ 1 − e−2Im[Seff ] ≈ 2Im[Seff ] • General expression: Seff = log det[iγ µ (∂µ − ieAµ ) + m]

How to calculate Im[Seff ]?! • Exact result: Constant field E (t) = E :

∞ e2E 2 X 1 nπm2 Im[Seff ] = V T exp − eE 8π 2 n2 n=1

J. Schwinger, Phys. Rev. 82 (1951)

• Exact result: Sauter-type field E (t) = E sech2 (t/τ ):

1 Im[Seff ] = −V 8π 2

Z

d 3 k ln

i h 1 − e−πω+ (k) 1 − e−πω− (k)

N. Narozhnyi and A. Nikishov, Sov. J. Nucl. Phys. 11 (1970)

QED Vacuum



Results

Summary

Effective Action Approach (Imaginary Time) • Vacuum persistence amplitude: h0|0iA = eiSeff • Pair production probability: P ≃ 1 − e−2Im[Seff ] ≈ 2Im[Seff ] • General expression: Seff = log det[iγ µ (∂µ − ieAµ ) + m]

How to calculate Im[Seff ]?! • Exact result: Constant field E (t) = E :

m4 Im[Seff ] = V T 8π 2

E Ecr

2 X ∞ n=1

1 Ecr exp −nπ E n2 J. Schwinger, Phys. Rev. 82 (1951)

• Exact result: Sauter-type field E (t) = E sech2 (t/τ ):

Im[Seff ] = −V

1 8π 2

Z

d 3 k ln

i h 1 − e−πω+ (k) 1 − e−πω− (k)

N. Narozhnyi and A. Nikishov, Sov. J. Nucl. Phys. 11 (1970)

QED Vacuum



Results

Summary

Quantum Kinetic Theory (Real Time) • Single particle distribution function: f (k, t)

d • Boltzmann-type equation: dt f (k, t) = C(k, t) + S(k, t) • C(k, t): Collission term → Negligible for low densities! • S(k, t): Source term for pair production

f (k, t) and S(k, t) from first principles?! S. Schmidt et al., Int. J. Mod. Phys. E 7 (1998)

• Simplification: sQED instead of QED

• Lagrangian: |(∂ + ieA)φ(x, t)|2 − m2 |φ(x, t)|2 − 41 F µν Fµν • Quantization: Classical vector potential Aµ = (0, A(t)e3 )

• Canonical Quantization: Aµ classical ↔ φ(x, t) quantized

φ(x, t) =

Z

d 3k † [gp (t)ak + gp∗ (t)b−k ]eik·x (2π)3

QED Vacuum


Results


Summary

Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable

[∂t2 + m2 + k2⊥ + (k3 − eA(t))2 ]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k

CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:

eE (t)pk (t) 2ωp2 (t)

Z

t

dt ′

d dt f (k, t)

= S(k, t) in sQED

eE (t ′ )pk (t ′ ) ωp2 (t ′ )

Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′

QED Vacuum


Results


Summary


[∂t2 + ǫ2⊥ + (k3 − eA(t))2 ]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k



Z

t

dt ′

d dt f (k, t)

= S(k, t) in sQED

eE (t ′ )pk (t ′ ) ωp2 (t ′ )

Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′

QED Vacuum


Results


Summary


[∂t2 + ǫ2⊥ + pk2 (t)]gp (t) = 0

• Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k



Z

t

dt ′

d dt f (k, t)

= S(k, t) in sQED

eE (t ′ )pk (t ′ ) ωp2 (t ′ )

Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′

QED Vacuum


Results


Summary


[∂t2 + ωp2 (t)]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k



Z

t

dt ′

d dt f (k, t)

= S(k, t) in sQED

eE (t ′ )pk (t ′ ) ωp2 (t ′ )

Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′

QED Vacuum


Results


Summary





Z

t

dt ′

d dt f (k, t)

= S(k, t) in sQED

eE (t ′ )pk (t ′ ) ωp2 (t ′ )

Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′

QED Vacuum


Results


Summary




eE (t)ǫ⊥ 2ωp2 (t)

Z

t

dt ′

d dt f (k, t)

= S(k, t) in QED

Z t eE (t ′ )ǫ⊥ ′′ ′′ ′ [1−2f dt ω (t ) (k, t )] cos 2 p ωp2 (t ′ ) t′

QED Vacuum


Results


Summary

Quantum Kinetic Theory (Real Time) Quantum kinetic equation (QED) ↔ Integro-differential equation Z t Z ′ eE(t)ǫ⊥ t d ′′ ′′ ′ ′ eE(t )ǫ⊥ dt ωp (t ) dt [1 − 2f(k, t )] cos 2 f(k, t) = dt 2ωp2 (t) −∞ ωp2 (t′ ) t′ • Non-Markovian equation: Statistical factor & Cosine term • Reformulation: First order differential equation system • Backreaction mechanism: E (t) = Eext (t) + Eint (t)

E˙ int (t) = −4e

Z

d3 k (2π)3

2 ˙ pk (t) ωp (t) d eE(t)ǫ ⊥ f(k, t)− f(k, t) + ωp (t) eE(t) dt 8ωp5 (t)

• Advantage (1): Valid for any time-dependency E (t)

• Advantage (2): Momentum space distribution f (k, t) • Advantage (3): Density nqk [e+ e− ] = 2

R

[dk]f (k, ∞)

!

QED Vacuum


Results

Outline



Summary

QED Vacuum



Results

Electric Field: Pulse-Shaped Time dependent field: E (t) = E sech2 (t/τ ) EHtLE 1.0

0.8

0.6

0.4

0.2

tΤ -4

-2

2

4

Exactly solvable in different approaches!

Summary

QED Vacuum


Results


Summary

Electric Field: Pulse-Shaped REMINDER: Instantaneous approximation should be valid for γ≪1 πm2 e2E 2 exp − ninst [e e ] ≃ V T eE 4π 2 + −

REMINDER: Keldysh adiabaticity parameter γ γ=

m eE τ

For field strengths of the order of E ≃ Ecr : • γ ≪ 1: Long pulse lengths • γ ≫ 1: Short pulse lengths

What happens in a region for which γ ≈ 1?

QED Vacuum


Results


Summary

Electric Field: Pulse-Shaped REMINDER: Instantaneous approximation should be valid for γ≪1 ∞

e2 E (t ′ )2 πm2 exp − ninst [e e ] ≃ V dt eE (t ′ ) 4π 2 −∞ + −

Z

′


m eE τ



QED Vacuum


Results


Summary

Electric Field: Pulse-Shaped REMINDER: Instantaneous approximation should be valid for γ≪1 + −

ninst [e e ] ≃ V

Z

∞

πm2 e2 E (t ′ )2 exp − dt eE (t ′ ) 4π 2 −∞ ′


Ecr tc E τ




QED Vacuum


Results

Summary

Electric Field: Pulse-Shaped FH, R. Alkofer and H. Gies, Phys. Rev. D 78 (2008)

E = 0.1 Ecr

5 ´ 10

τ = 10 tc 10

-5

8

à

à æ

à æ

à æ

à æ

0.9

1

à æ

-5

5 ´ 10

-6

æ

inst.

à

q.k.t.

10

à

1 ´ 10

-6

5 ´ 10

-7

à

à

à æ

à æ

à æ

à æ

à æ

æ à

number density @nm-3 D

number density @nm-3 D

à æ

1 ´ 10

à æ à æ

100

æ

inst.

à

q.k.t.

à æ

0.1

æ æ

5

10

-4

10

-7

à

æ

1 ´ 10

-7

æ

10

20

30

40

50

60 Τ @tc D

70

80

90

100

æ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

EEcr

• Instantaneous approximation: Huge deviation for γ = 1 • γ & 0.2: ’Overlap region’ → Multi-photon effects set in

QED Vacuum



Results

Summary

Electric Field: Oscillation with Gaussian Envelope Time dependent field: E (t) = E cos(ω t) exp(−t 2 /2τ 2 ) EHtLE 1.0 Ω×Τ = 0 Ω×Τ = 5 Ω×Τ = 10

0.5

tΤ -4

2

-2

4

-0.5

-1.0

Single pulse: 1 scale τ ↔ Envelope pulse: 2 scales ω, τ • Time scale τ : Total pulse length • Time scale ω: Laser frequency • Combined scale ωτ : Number of oscillations in the pulse

QED Vacuum


Results


Summary

Electric Field: Oscillation with Gaussian Envelope Field strengths of the order of Ecr via crossed laser beams • XFEL (DESY): E ≃ 0.1Ecr reachable → focusing?!

• Optical Laser (ELI): Probably ’only’ E ≃ 0.01Ecr reachable t @a.u.D 0

0 x @a.u.D

Crossed laser beams: 2 counter-propagating wave packages

QED Vacuum


Results


Summary

Electric Field: Oscillation with Gaussian Envelope Field strengths of the order of Ecr via crossed laser beams • XFEL (DESY): E ≃ 0.1Ecr reachable → focusing?!

• Optical Laser (ELI): Probably ’only’ E ≃ 0.01Ecr reachable 1.0

0.5

t @a.uD

-0.5

Interaction region x = 0: Oscillation with Gaussian Envelope

QED Vacuum


Results


Summary

Electric Field: Oscillation with Gaussian Envelope V. Popov, JETP Letters 74 (2001)

• WKB / Scattering Theory: Semiclassical treatment

• Gaussian approximation: Production probability for γ ≪ 1

i Ecr 1 2 1 h 2 2 2 γ˜ pk + p⊥ P(p) ∼ exp −π 1 − γ˜ ·exp − E 8 eE with γ˜ =

s

(ωτ )2 + 1 γ (ωτ )2

Accuracy of the Gaussian approximation?

QED Vacuum



Results

Summary

Electric Field: Oscillation with Gaussian Envelope FH, R. Alkofer, G. Dunne and H. Gies, Phys. Rev. Lett. 102 (2009)

E = 0.1Ecr , τ = 100 tc , ω = 25 keV −→ ωτ = 5 and γ = 0.5

5. ´ 10

-14

4. ´ 10

-14

3. ´ 10

-14

2. ´ 10

-14

1. ´ 10

-14

p° @keVD -400

-200

0

200

400

600

• Width: f (k, ∞) NOT Gaussian → Steeper decay! • Structure: Oscillatory behaviour with ∆pkmax = ω

QED Vacuum



Results

Summary

Electric Field: Oscillation with Gaussian Envelope FH, R. Alkofer, G. Dunne and H. Gies, Phys. Rev. Lett. 102 (2009)

E = 0.1Ecr , τ = 100 tc , ω = 25 keV −→ ωτ = 5 and γ = 0.5

5. ´ 10

-14

4. ´ 10

-14

3. ´ 10

-14

2. ´ 10

-14

1. ´ 10

-14

p° @keVD -400

-200

0

200

400

600

• Width: f (k, ∞) NOT Gaussian → Steeper decay! • Structure: Oscillatory behaviour with ∆pkmax = ω

QED Vacuum



Results

Electric Field: Additional Phase Shift Time dependent field: E (t) = E cos(ω t + φ) exp(−t 2 /2τ 2 ) EHtLE 1.0 Gaussian Φ=0 Φ = -Π4 0.5

Φ = -Π2

tΤ -4

2

-2

4

-0.5

-1.0

• For φ = 0: Time symmetric field E (t) = E (−t) • For φ 6= {0, ±π/2}: Mixed time symmetry • For φ = ±π/2: Time antisymmetric field E (t) = −E (−t)

Effect of the phase shift φ?

Summary


QED Vacuum


Results

Summary

Electric Field: Additional Phase Shift E = 0.1Ecr , τ = 100 tc , ω = 25 keV −→ ωτ = 5 and γ = 0.5 φ = −π/4

φ = −0 -14

φ = −π/2 5. ´ 10

-14

4. ´ 10

-14

3. ´ 10

-14

-14

-14

2. ´ 10

-14

-14

1. ´ 10

-14

-14

5. ´ 10

5. ´ 10

-14

-14

4. ´ 10

4. ´ 10

-14

3. ´ 10

3. ´ 10

-14

2. ´ 10

2. ´ 10

-14

1. ´ 10

1. ´ 10

p° @keVD -400

-200

0

200

400

600

p° @keVD

p° @keVD -400

-200

0

102

200

400

600

Huge qualitative difference!

-400

-200

0

137

200

400

600

QED Vacuum


Results


Electric Field: Additional Phase Shift Explanation: Scattering picture • REMINDER: [∂t2 + ωp2 (t)]gp (t) = 0 • 1-dimensional scattering problem

~2 2 Hψ(x) = − ∂ + V (x) ψ(x) = E ψ(x) 2m x

• Formal similarity → ’Scattering potential’: V (t) ∼ −ωp2 (t) • Reflection coefficient ↔ Produced pairs

Schwinger effect ↔ Over-barrier-scattering! • Asymmetric electric field: E (t) = −E (−t) • Symmetric vector potential: A(t) = A(−t) • Symmetric ’scattering potential’: ωp2 (t) = ωp2 (−t)

Resonances ↔ perfect transmission ↔ No pairs produced!

Summary


QED Vacuum


Results

Summary

Electric Field: Additional Phase Shift

sQED : QED :

5. ´ 10

REMINDER: Source term for pair production R ′ ′ eE(t)pk (t) R t ′ eE(t )pk (t ) [1+2f(k, t′ )] cos 2 t dt′′ ω (t′′ ) dt ′ p 2 2 −∞ t 2ωp (t) ωp (t′ ) R R ′ eE(t)ǫ⊥ t ′ eE(t )ǫ⊥ [1−2f(k, t′ )] cos 2 t dt′′ ω (t′′ ) dt ′ p t 2ω 2 (t) −∞ ω 2 (t′ ) p

p

-14

5. ´ 10

-14

4. ´ 10

-14

QED

QED

sQED

sQED

4. ´ 10

-14

3. ´ 10

-14

3. ´ 10

-14

2. ´ 10

-14

2. ´ 10

-14

1. ´ 10

-14

1. ´ 10

-14

p° @keVD -400

-200

0

200

400

600

p° @keVD -100

0

100

Effect of particle statistics becomes obvious!

200

300

QED Vacuum


Results

Outline



Summary

QED Vacuum


Results


Summary

Generalization of Quantum Kinetic Theory Phase-Space formulation of Schwinger effect: {~x , ~p, t} • Quantum Kinetic Theory so far: E (x, t) = E (t)

• ~ k conjugate variable of ~x → No direct generalization!

Approach: Dirac-Heisenberg-Wigner (DHW) function I. Bialynicki-Birula, P. Gornicki and J. Rafelski, Phys. Rev. D 44 (1991)

+ 0 • Cαβ = h0| ψα (x~1 , t), ψ¯β (x~2 , t) |0i = δ3 (x~1 − x~2 )γαβ

− • Cαβ = h0| ψα (x~1 , t), ψ¯β (x~2 , t) |0i

• Wigner transform: Fourier transform w.r.t. ~s = x~1 − x~2 − • DHW function Wαβ (~ x , ~p, t): Wigner transform of Cαβ

QED Vacuum


Results


Generalization of Quantum Kinetic Theory Equation of motion for DHW function • Hartree approximation: Mean electric field ~ =0 • Vanishing magnetic field: B

i i o h n 1 h Dt W αβ = − ∇ γ 0~γ , W − i mγ 0 , W − i γ 0~γ~p, W 2 αβ αβ αβ with

Dt = ∂t + e

Z

1/2

~ ~x + iλ∂p , t)∂p d λE(

−1/2

• Basis set for DHW function: {1, γ5 , γ µ , γ5 γ µ , σ µν } • PDE for 16 generalized phase space functions ci (~ x , ~p , t)

Dt ~c (~x , ~p , t) = M(m, ~p , ∇)~c (~x , ~p, t)

For E (x, t) = E (t) → Equivalent to Quantum Kinetic Theory!

Summary

QED Vacuum


Results

Outline



Summary

QED Vacuum


Results


Summary

Summary QED vacuum is not empty ↔ New physics with new lasers?! • Schwinger effect: Spectacular effect in perturbed vacuum • E (t) = E sech2 (t/τ ): • Instantaneous approach: Breakdown at short time scales • E (t) = E cos(ωt)exp(−t 2 /2τ 2 ): • Crossed laser beams: Realistic model at interaction region • Momentum space: Oscillatory structure; non-Gaussian • E (t) = E cos(ωt + φ)exp(−t 2 /2τ 2 ): • Phase shift: Strong dependence on φ • Particle statistics: sQED ↓↑ ←→ QED ↑↓ • Formalism for general E (x, t): • For E(x , t) = E(t): Identical to quantum kinetic approach