Thermalization and Unruh Radiation for a Uniformly Accelerated Charged Particle

July 2010, 　 Azumino

張　森Sen Zhang

S. Iso and Y. Yamamoto

Unruh effect and Unruh radiation

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Vacuum:

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Hawking Radiation:

Bogoliubov transformation

Vacuum of free falling observer

Vacuum for inertial observer

thermal state for accelerating observer

Unruh Effect:

Asymptotic observer

Vacuum for inertial observer

(107K)

thermal state for accelerating observer

Unruh Effect:

Unruh Temperature:

How to See?Unruh Radiation: radiation due to fluctuation of electron

Schutzhold, Schaller, Habs ‘06Chen, Tajima ‘99

Unruh effect and Unruh radiation

Radiation from fluctuation

Larmor radiation

Previous Results

The discussion is intuitive and smart …

But more systematic derivation is required

Schutzhold, Schaller, Habs ‘06Chen, Tajima ‘99

Dimensionless laser strength parameter(a0 ～ 100 for patawatt-class laser)

Unruh radiation is very small compare to Larmor radiation. The angular distribution is quite different.

・ Unruh radiation are treated in a complete different way from Larmor radiation.・ How does the path of the uniformly accelerated particle fluctuate?・ The interference effect were not considered.

Plan• Charged particle

• Unruh Radiation

Stochastic equation (general formalism for fluctuation)

Accelerating case

Equipartition theorem Agrees Chen Tajima’s proporsal

How does it fluctuate actually?

Radiation from fluctuations in transverse directions

Angular distribution

Interference effect

But several problems …

Particle

Stochastic Equation

Focus on Particle Motion

Real Process Random motion

absorption and radiation Brownian motion

Stochastic Equation

Equation of motion:

Solution:

fluctuation

dissipation

Effective equation for a particle interacting with some quantum field

Scalar for simplicity:

Self-force from Larmor radiation (ALD)

P. R. Johnson and B. L. Hu

expansion:

Non-local

Renormalized mass

Fluctuation around uniformly accelerated motion for transverse direction:

Acceleration (1 keV)

Equation of fluctuations

Transverse direction

Longitudinal direction

Two point function:

Derivative expansion

Transverse Fluctuation

Relaxation Time:

Neglecting term:

Including term:

Equipartition Theorem

Equipartition theorem

thermal

Action:

Solution:

Stochastic equation:

Equipartition theorem

Universal

Longitudinal FluctuationTransform variables for the accelerated observer :

Problem of coordinates:

The expectation values change, but the Bogoliubov transformation is same

Problem on constant electric field:

Different longitudinal coordinates

means different acceleration

Difficult to say if the longitudinal is same to the transverse

Fluctuation in longitudinal direction for uniformly accelerated obserber:

Very different from transverse direction

Radiation

Interference effect

Depend on

What Chen-Tajima calculated

Nonzero

Unruh Detector

2D: no radiationRaine, Sciama, Grove 91’s

4D: radiate during thermalization, but no radiation if the detector state is thermal state at first

Shih-Yuin Lin & B. L. Hu

Eom:

Inteference Effect - Unruh Detector

Interference term

GR

Cancels the radiation from inhomogeneous part

Interference effect - charged particle

For transverse fluctuation:

Energy momentum tensor:

Larmor Radiation:

Unruh Radiation

Summary and Future Work• An uniformly accelerated particle satisfies a stochastic

equation. The transverse momentum fluctuations satisfy the equipartition theorem for both scalar field and gauge field.

• Longitudinal direction is more complicated.• Radiations due to the fluctuations are calculated partly.• The interference effect are important.• There may be a problem on validity of approximation

which relates to the UV divergence. Treatment based on QED will be required.

• Longitudinal contribution, Angular distribution, QED case …

Four poles

Relaxation time (thermalization time)

Photon travelling time in Compton wave length

: does not contribute for but is dominant for .

Unruh radiation depends on physics beyond the semi-classical analysis

in our framework. Treatment based on QED will be required.

UV divergence

Cancelled by the interference term, in the calculationof radiation due to transverse fluctuations

Problem of Radiation Dumping

Energy momentum conservation

Abraham-Lorentz-Dirac Force:

Runaway Solution

Landau-Lifshitz equation:

No back reaction for uniformly accelerated electron !?

on-shell condition

What can we say about this problem using QED?