Introduction tostrong-field QED
Hidetoshi Taya(an SPDR in iTHEMS, studying high-energy particle & nuclear physics)

Plan(QED = quantum electrodynamics)

Brief introduction to Strong-field QED= an area to study what happens by a super strong light(= electromagnetic field)

PART I: Basics of strong-field QED・ Why interesting ⇒ Unexplored non-perturbative regime of physics
Nontrivial phenomena that we’ve never seen
・ Why timely ⇒ Recent availability of strong fields

e.g. Vacuum decay, Birefringence,
New phase of matter, ...

e.g. High-power laser, Magnetar,heavy-ion collisions, …

PART II: Sauter-Schwinger effect and its connection to other areas
・ An interdisciplinary topic⇒ Exchange of ideas/techniques among physics is quite useful for better understandings⇐ An example from my experience: High-harmonic generation from the vacuum⇐ Math. & cond-mat. ideas helped me a lot[HT, Fujimori, Misumi, Nitta, Sakai, (2021)] [HT, Hongo, Ikeda, (2021)]

PART I: Basics of strong-field physics
PART II: Sauter-Schwinger effect and its connection to other areas

What if field becomes strong ?
final

initialtime evol.

No fieldWhat if field becomes strong ?
final

initialtime evol.

No field

Weak field

Strong fieldWhat if field becomes strong ?
final

initialtime evol.

𝑭𝑭No field

Weak field

Strong field

Only minor changes⇒ Perturbative⇒ Very well understoodin both exp.& theor.

g−2ex.) Electron (anomalous) magnetic moment 𝑎𝑎 ≔2≈ Electron energy shift in a weak magnetic field𝑎𝑎 theor. = 1159652182.03 … × 10−12
𝑎𝑎 exp.
= 1159652180.73 … × 10−12[Aoyama, Kinoshita, Nio (2017)]

What if field becomes strong ?
finalinitialtime evol.

No field𝑭𝑭𝑭𝑭

Weak field𝑭𝑭𝑭𝑭𝑭𝑭…

Strong field

Only minor changes

Big change !

⇒ Perturbative⇒ Non-Perturbative⇒ Very well understoodin both exp.& theor.

⇒ Not understood wellg−2ex.) Electron (anomalous) magnetic moment 𝑎𝑎 ≔2≈ Electron energy shift in a weak magnetic field𝑎𝑎 theor. = 1159652182.03 … × 10−12
𝑎𝑎 exp.
= 1159652180.73 … × 10−12[Aoyama, Kinoshita, Nio (2017)]

If field becomes strong, physics becomes totally different & nontrivialWhen is field “strong” ?

Strong-field condition:

To significantly modify the original system with typical energy Δ,
the field must be more energetic than Δ⇒ Strong-field condition: 𝚫𝚫 < (𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐨𝐨𝐨𝐨 𝐭𝐭𝐭𝐭𝐭𝐭 𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟)

When is field “strong” ?

Strong-field condition:

To significantly modify the original system with typical energy Δ,
the field must be more energetic than Δ⇒ Strong-field condition: 𝚫𝚫 < (𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐨𝐨𝐨𝐨 𝐭𝐭𝐭𝐭𝐭𝐭 𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟)

Estimate of the minimum field strength:Standard Model

The (matter) particle having the minimum energy・ Charged ⇒ Couples toelectromagnetic (EM) field・ 𝒆𝒆𝒆𝒆 𝐌𝐌𝐌𝐌𝐕𝐕 𝟐𝟐 > 𝒎𝒎𝟐𝟐𝒆𝒆 = 𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓 𝐌𝐌𝐌𝐌𝐌𝐌 𝟐𝟐≈ 𝑶𝑶(𝟏𝟏𝟎𝟎𝟐𝟐𝟐𝟐 𝐖𝐖/𝐜𝐜𝐦𝐦𝟐𝟐 )∴ EXTREMELY strong ⇒ Impossible to realize within the current technology

LED ≈ 10−5 W/cm2

Glasses ≈ 103 W/cm2

Laser beam machine < 1010 W/cm2

Recent availability of strong EM fields
The situation is changing:
Becoming able to create/observe strong EM fields⇒ NOW is the best time to study physics of strong EM field(=: strong-field QED)
High-power laser

Systems with extreme conditions

Adopted from [Mourou, Tajima, Bulanov (2006)]

Starting soon

Strong EM field (𝒆𝒆𝒆𝒆⁄𝒎𝒎𝟐𝟐𝒆𝒆 > 𝟏𝟏)

𝐼𝐼 > 1024 W/cm2 :~e.g., ELI @ Europe
Yoon et al., (2021)

Current strongest
𝐼𝐼~1023 W/cm2CoReLS @ Korea

CPA technique・ Heavy-ion collisions
𝑒𝑒𝑒𝑒 > 𝑚𝑚𝜋𝜋2 ∼ 140 MeV 2(𝐼𝐼 > 1035 W/cm2 )𝑒𝑒𝑒𝑒 < 1 MeV 2 ∼ 1011 T(𝐼𝐼 ∼ 1029 W/cm2 )Yanovsky et al., (2008)

𝐼𝐼~1022 W/cm2
HERUCLES @ USA

・ Magnetar

At RHIC (2000~), LHC (2012~)

Observation by IXPE, XL-Calibur (2022~)
Press release by Tamagawa lab. (2022)

What can happen with strong EM field ?
When 𝒆𝒆𝒆𝒆 > 𝒎𝒎𝟐𝟐𝒆𝒆 , many non-trivial phenomena have been predicted to occur:
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, Phys. Rept. (2023)]

Sauter (1931), Schwinger (1951)

What I like the most: Production of particles from the vacuum (Sauter-Schwinger effect)

A few examples among many others (don’t explain; ask later if interested)
Birefringence ofphoton in vacuum

Photon splitting～

Photon scattering

Hadron mass change(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[HT (2015)]

QCD phase diagram(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[D’Elia et al. (2022)]

What can happen with strong EM field ?
When 𝒆𝒆𝒆𝒆 > 𝒎𝒎𝟐𝟐𝒆𝒆 , many non-trivial phenomena have been predicted to occur:
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, Phys. Rept. (2023)]

Sauter (1931), Schwinger (1951)

What I like the most: Production of particles from the vacuum (Sauter-Schwinger effect)
Our vacuum is not an empty space,but has a structure similar to semi-conductor(QED) Vacuum

CAN observe

energyDirac (1930)

momentum

gap = 2m

Dirac sea

CANNOT observe

A few examples among many others (don’t explain; ask later if interested)
Birefringence ofphoton in vacuum

Photon splitting～

Photon scattering

Hadron mass change(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[HT (2015)]

QCD phase diagram(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[D’Elia et al. (2022)]

What can happen with strong EM field ?
When 𝒆𝒆𝒆𝒆 > 𝒎𝒎𝟐𝟐𝒆𝒆 , many non-trivial phenomena have been predicted to occur:
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, Phys. Rept. (2023)]

Sauter (1931), Schwinger (1951)

What I like the most: Production of particles from the vacuum (Sauter-Schwinger effect)
Our vacuum is not an empty space,but has a structure similar to semi-conductor(QED) VacuumThen, real particles will be produced out ofthe vacuum by strong enough field

Particle production !

CAN observe

energyDirac (1930)

momentum

gap = 2m

Dirac sea

EM field

CANNOT observe
Sauter (1931)

A few examples among many others (don’t explain; ask later if interested)
Birefringence ofphoton in vacuum

Photon splitting～Photon scattering

Hadron mass change(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[HT (2015)]

QCD phase diagram(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[D’Elia et al. (2022)]

What can happen with strong EM field ?
When 𝒆𝒆𝒆𝒆 > 𝒎𝒎𝟐𝟐𝒆𝒆 , many non-trivial phenomena have been predicted to occur:
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, Phys. Rept. (2023)]

Sauter (1931), Schwinger (1951)

What I like the most: Production of particles from the vacuum (Sauter-Schwinger effect)
Our vacuum is not an empty space,but has a structure similar to semi-conductor(QED) VacuumThen, real particles will be produced out ofthe vacuum by strong enough field

Particle production !

CAN observe

energyDirac (1930)

momentum

gap = 2m

Dirac sea

EM field

CANNOT observe
Sauter (1931)

A few examples among many others (don’t explain; ask later if interested)
Birefringence ofphoton in vacuum

Photon splitting～

Photon scattering

Hadron mass change(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[HT (2015)]

QCD phase diagram(for 𝑒𝑒𝑒𝑒 > Λ2QCD )[D’Elia et al. (2022)]

PART I: Basics of strong-field physicsPART II: Sauter-Schwinger effect and its connection to other areas

Similar particle production mechanisms
appear in many other areas of physics

Similar particle production mechanisms
appear in many other areas of physics

The most prominent example: Landau-Zener transition in solid
Landau (1932), Zener (1932)

Ground state of solids(QED) Vacuum

Conduction
band
gap = 2mv.s.

Fermi surface
gap = Δ
= O(1 eV)

Dirac sea

Valenceband

・ The responses of the vacuum and ground state by an external EM field should be similar・ Indeed, the particle prediction rate and the excitation rate obey the same exponential formula2

ΓSauter−Schwinger = # exp[ # 𝑚𝑚 �𝑒𝑒𝑒𝑒 ]2ΓLandau−Zener = # exp[ # Δ �𝑒𝑒𝑒𝑒 ]Note: Similar exponential formula holds for particle production by other strong fields as well
Strong gravitational field ⇒ Hawking radiation, Strong inflaton field ⇒ (p)reheating of the early Universe, …

Similar particle production mechanisms
appear in many other areas of physics

The most prominent example: Landau-Zener transition in solid
Landau (1932), Zener (1932)

Ground state of solids(QED) Vacuum

Conduction
bandgap = 2mFermi surfacev.s.

gap = Δ= O(1 eV)

Dirac seaValenceband・ The responses of the vacuum and ground state by an external EM field should be similar・ Indeed, the particle prediction rate and the excitation rate obey the same exponential formula2

ΓSauter−Schwinger = # exp[ # 𝑚𝑚 �𝑒𝑒𝑒𝑒 ]2ΓLandau−Zener = # exp[ # Δ �𝑒𝑒𝑒𝑒 ]Note: Similar exponential formula holds for particle production by other strong fields as well
Strong gravitational field ⇒ Hawking radiation, Strong inflaton field ⇒ (p)reheating of the early Universe, …

・ Sauter-Schwinger effect has not been verified yet,but Landau-Zener transition has been observed and utilized, e.g., to design devices.

⇒ Importing (exporting) ideas from (to) other areas of physics is quite useful tobetter understand the Sauter-Schwinger effect (or strong-field QED in general)Example: High-harmonic generation from the vacuum
Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac seaVacuum

Pair production

Photon emission✔ Consider AC field with frequency Ω:・ Very naïve expectation: photon has frequency ω same to AC field ω=ΩExample: High-harmonic generation from the vacuum
Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac seaVacuum

Pair production

Photon emission✔ Consider AC field with frequency Ω:・ Very naïve expectation: photon has frequency ω same to AC field ω=Ω✔ Recent observation in semi-conductors
・ Naïve expectation is wrong(also in many other materials)⇒ Exp. observation: Plateau structure
・ Theoretical formulation is still immature even in cond.-mat⇒ mostly numerics, and analytical understanding is lacking[Ghimire et al., (2011)]

Example: High-harmonic generation from the vacuum
Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac seaVacuum

Pair production

Photon emission✔ Consider AC field with frequency Ω:・ Very naïve expectation: photon has frequency ω same to AC field ω=Ω✔ Recent observation in semi-conductors
・ Naïve expectation is wrong(also in many other materials)⇒ Exp. observation: Plateau structure
・ Theoretical formulation is still immature even in cond.-mat⇒ mostly numerics, and analytical understanding is lacking

Q1: High-harmonic generation in QED from the vacuum ?
Q2: Is there any nice analytical method ?
A: Yes. Mathematics helped me to answer those !

[Ghimire et al., (2011)]

How to formulate ? Typical math of strong-field QEDRoughly speaking, strong-field-QED calculations is reduced to the following:

Step 1: Solve a differential equation (to obtain wavefunction Φ ≈)The simplest example: Klein-Gordon equation with a time-dependent potential

0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙(𝑡𝑡)something small(Planck constant)decodewavefunction
Potential(= info. of strong field)

Step 2: Analyze the wavefunction (to predict sthg measurable O[Φ] ≈・ O depends on what you want to measure
・ Typically, a bi-linear 𝜙𝜙 † 𝑋𝑋𝑋𝑋)How to formulate ? Typical math of strong-field QEDRoughly speaking, strong-field-QED calculations is reduced to the following:

Step 1: Solve a differential equation (to obtain wavefunction Φ ≈)The simplest example: Klein-Gordon equation with a time-dependent potential

0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙(𝑡𝑡)something small(Planck constant)decodewavefunction
Potential(= info. of strong field)

Step 2: Analyze the wavefunction (to predict sthg measurable O[Φ] ≈・ O depends on what you want to measure
・ Typically, a bi-linear 𝜙𝜙 † 𝑋𝑋𝑋𝑋

Solve a differential equation is purely a mathematical problem⇒ Any good idea from mathematics⇒ Exact WKB)Exact WKB method
Exact WKB = a nice method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

Exact WKB method
Exact WKB = a nice method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)]

A perturbation theory w.r.t. ℏ (or adiabatic approx.)

・ Consider 0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙 𝑡𝑡i𝑡𝑡 ≡ ℏ𝜏𝜏𝑡𝑡⇔𝜕𝜕𝜏𝜏2 + 𝑄𝑄 ℏ𝜏𝜏 𝜙𝜙 𝜏𝜏

⇒ 𝜙𝜙± 𝑡𝑡; ℏ ≔ exp ∓ ∫𝑡𝑡 d𝑡𝑡 ′ 𝑄𝑄 𝑡𝑡 ′ℏ00th order = plane wave
i~ exp[∓ℏ𝑄𝑄𝑡𝑡]

𝑛𝑛× ∑∞
𝑛𝑛=0 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ
Perturbation w.r.t. ℏ[Ecalle (1981)]

Exact WKB method
Exact WKB = a nice method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

A perturbation theory w.r.t. ℏ (or adiabatic approx.)

・ Consider 0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙 𝑡𝑡
i𝑡𝑡 ≡ ℏ𝜏𝜏𝑡𝑡⇔𝜕𝜕𝜏𝜏2 + 𝑄𝑄 ℏ𝜏𝜏 𝜙𝜙 𝜏𝜏

⇒ 𝜙𝜙± 𝑡𝑡; ℏ ≔ exp ∓ ∫𝑡𝑡 d𝑡𝑡 ′ 𝑄𝑄 𝑡𝑡 ′ℏ00th order = plane wave
i~ exp[∓ℏ𝑄𝑄𝑡𝑡]

𝑛𝑛× ∑∞
𝑛𝑛=0 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ
Perturbation w.r.t. ℏ・ WKB expansion makes sense if the perturbative part is convergent
・ However, 𝜓𝜓±,𝑛𝑛 ~𝑛𝑛! in general (e.g., Airy function 𝑄𝑄 𝑡𝑡 ∝ 𝑡𝑡)⇒ WKB expansion has zero radius of convergence ⇒ ill-defined !!!

Exact WKB method
Exact WKB = a nice method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

A resummation scheme for factorially divergent ~n! series
𝑛𝑛・ Consider the div. part of WKB expansion 𝜓𝜓± 𝑡𝑡; ℏ ≔ ∑∞𝑛𝑛=0 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ∞① Construct “Borel transformation”: 𝐵𝐵 𝜓𝜓± 𝑡𝑡; 𝜂𝜂 ≔ �② Laplace trans. gives “Borel sum”:∞𝑛𝑛

𝜓𝜓±,𝑛𝑛 (𝑡𝑡) 𝑛𝑛𝜂𝜂𝑛𝑛!

d𝜂𝜂 −𝜂𝜂/ℏΨ± 𝑡𝑡; ℏ ≔ �𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂)eℏ
0

Exact WKB method
Exact WKB = a nice method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

A resummation scheme for factorially divergent ~n! series
𝑛𝑛・ Consider the div. part of WKB expansion 𝜓𝜓± 𝑡𝑡; ℏ ≔ ∑∞𝑛𝑛=0 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ∞① Construct “Borel transformation”: 𝐵𝐵 𝜓𝜓± 𝑡𝑡; 𝜂𝜂 ≔ �② Laplace trans. gives “Borel sum”:∞

Ψ± 𝑡𝑡; ℏ ≔ �0𝑛𝑛

𝜓𝜓±,𝑛𝑛 (𝑡𝑡) 𝑛𝑛𝜂𝜂𝑛𝑛!

d𝜂𝜂 −𝜂𝜂/ℏ𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂)eℏ・ Ψ± is well-defined and is a natural analytic continuation of 𝜓𝜓±⇐ Indeed, reduces to the original result if you adimit ∑∫ = ∫ ∑∞

∞

∞

∞

𝑛𝑛𝑛𝑛𝑛𝑛=0d𝜂𝜂 −𝜂𝜂/ℏ𝜓𝜓±,𝑛𝑛 (𝑡𝑡) 𝑛𝑛
𝜓𝜓±,𝑛𝑛 (𝑡𝑡) ∞ d𝜂𝜂 −𝜂𝜂/ℏ 𝑛𝑛�e𝜂𝜂 ∼ �e
Ψ± = ��
𝜂𝜂 = � 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ𝑛𝑛ℏ
𝑛𝑛!
𝑛𝑛!
ℏ00⇒ 𝚿𝚿± gives a well-defined version of the WKB solution !

・ Note: in practice, some approximations shall be used in Borel resum…

How to formulate ? Typical math of strong-field QED
Step 1: Solve a differential equation (to obtain wavefunction Φ ≈)The simplest example: Klein-Gordon equation with a time-dependent potential

0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙(𝑡𝑡)something small(Planck constant)potential

decodewavefunction

Step 2: Analyze the wavefunction (to predict sthg measurable O[Φ] ≈・ O depends on what you want to measure�̂ or 𝐽𝐽 ∼ 𝜓𝜓𝛾𝛾𝛾𝛾(for fermions)・ Typically, a bi-linear 𝜙𝜙 † 𝑋𝑋𝑋𝑋 ⇐ For HHG, current 𝐽𝐽 ~ 𝑒𝑒𝜙𝜙 † 𝑝𝑝𝜙𝜙)

Application to high-harmonic generation (1/2)
Setup: A monochromatic E-field 𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 cos(Ω𝑡𝑡) applied onto QED vacuum

What I did: Compute harmonic spectrum numerically and analytically with WKBApplication to high-harmonic generation (1/2)
Setup: A monochromatic E-field 𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 cos(Ω𝑡𝑡) applied onto QED vacuum

Harmonic spectrum

What I did: Compute harmonic spectrum numerically and analytically with WKB𝑚𝑚Ω
= 1/4𝑒𝑒𝐸𝐸0・ High harmonics appears also in QED when field becomes strong !

Application to high-harmonic generation (1/2)
Setup: A monochromatic E-field 𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 cos(Ω𝑡𝑡) applied onto QED vacuum

Harmonic spectrum

What I did: Compute harmonic spectrum numerically and analytically with WKB𝑚𝑚Ω
= 1/4𝑒𝑒𝐸𝐸0・ High harmonics appears also in QED when field becomes strong !
・ WKB is good in the plateau regime (i.e., before the spectrum drops)

Application to high-harmonic generation (2/2)
・ WKB works more in the deep non-perturbative regime 𝑬𝑬𝟎𝟎 → 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥, 𝛀𝛀 → 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬

・ Demonstration: magnitude of the harmonic peak at 𝝎𝝎/𝛀𝛀 = 𝟗𝟗
Dependence on strength 𝐸𝐸0

Dependence on frequency Ωanalytical with WKBnumerical

・ Lessons:(1) WKB makes it easier to analyze the non-perturbative regime(2) Saturation & oscillation of the harmonic intensity⇒ consistent with recent semi-conductor exp.

[Xia et al., (2020)](but only 𝐸𝐸0 -dep. is measured and Ω-dep. is our prediction)

PART I: Basics of strong-field physics
PART II: Sauter-Schwinger effect and its connection to other areas

Summary(QED = quantum electrodynamics)

Brief introduction to Strong-field QED= an area to study what happens by a super strong light(= electromagnetic field)

PART I: Basics of strong-field QED・ Why interesting ⇒ Unexplored non-perturbative regime of physics
Nontrivial phenomena that we’ve never seen
・ Why timely ⇒ Recent availability of strong fields

e.g. Vacuum decay, Birefringence,
New phase of matter, ...

e.g. High-power laser, Magnetar,heavy-ion collisions, …

PART II: Sauter-Schwinger effect and its connection to other areas
・ An interdisciplinary topic: Analogues in many other areas of physics⇒ Exchange of ideas/techniques among physics is quite useful for better understandings
e.g. Landau-Zener (cond.-mat), Hawing radiation (gravity), reheating in the early Universe (cosmology), …

・ Mathematically, the problem is essentially solving a differential equation⇒ Mathematical techniques to solve differential equations are quite useful・ As such an example: High-harmonic generation from the vacuum⇐ Based on (exact) WKB in mathematics and an idea in cond.-mat[HT, Fujimori, Misumi, Nitta, Sakai, (2021)] [HT, Hongo, Ikeda, (2021)]

Intuitive picture