1. July 2010,　Azumino
Thermalization and Unruh Radiation for a Uniformly Accelerated Charged Particle
張　森
Sen Zhang
S. Iso and Y. Yamamoto

2. for accelerating observer
Unruh effect and Unruh radiation
Vacuum: ～ ～
Bogoliubov transformation
Unruh Effect:
Vacuum
for inertial observer
thermal state
for accelerating observer
Hawking Radiation:
Vacuum of free falling observer
Asymptotic observer

3. for accelerating observer
Unruh effect and Unruh radiation
Unruh Effect:
Vacuum
for inertial observer
thermal state
for accelerating observer
Unruh Temperature:
(107K)
How to See?
Unruh Radiation: radiation due to fluctuation of electron
Chen, Tajima ‘99
Schutzhold, Schaller, Habs ‘06

4. (a0～100 for patawatt-class laser)
Previous Results
Chen, Tajima ‘99
Schutzhold, Schaller, Habs ‘06
Radiation from fluctuation
Larmor radiation
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.
The discussion is intuitive and smart …
But more systematic derivation is required
・ 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.

5. Equipartition theorem
Plan
Charged particle
How does it fluctuate actually?
Stochastic equation (general formalism for fluctuation)
Accelerating case
Equipartition theorem
Agrees Chen Tajima’s proporsal
Unruh Radiation
Radiation from fluctuations in transverse directions
Angular distribution
Interference effect
But several problems …

6. Particle

7. Stochastic Equation Real Process Random motion
Focus on Particle Motion
absorption and radiation
Brownian motion

8. Stochastic Equation Scalar for simplicity: Equation of motion:
Solution:
fluctuation
dissipation
Effective equation for a particle interacting with some quantum field

9. Self-force from Larmor radiation (ALD)
Non-local
expansion:
P. R. Johnson and B. L. Hu
Renormalized mass
Self-force from Larmor radiation (ALD)

10. Equation of fluctuations
Fluctuation around uniformly accelerated motion for transverse direction:
Acceleration (1 keV)
Equation of fluctuations
Transverse direction
Longitudinal direction

11. Transverse Fluctuation
Neglecting term:
Relaxation Time:
Including term:
Two point function:
Derivative expansion

12. Equipartition Theorem
thermal

13. Action:
Solution:
Stochastic equation:
Equipartition theorem
Universal

14. Longitudinal Fluctuation
Transform 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

15. Radiation

16. Interference effect
Nonzero
What Chen-Tajima calculated
Depend on

17. Inteference Effect - Unruh Detector
2D: no radiation
Raine, Sciama, Grove 91’s
4D: radiate during thermalization,
but no radiation if the detector
state is thermal state at first
Unruh Detector
Shih-Yuin Lin & B. L. Hu
Eom:

18. Interference term GR
Cancels the radiation from inhomogeneous part

19. Interference effect - charged particle
For transverse fluctuation:

20. Energy momentum tensor:
Larmor Radiation:
Unruh Radiation

21. 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 …

22. UV divergence Four poles
Photon travelling time in Compton wave length
Relaxation time (thermalization time)
: does not contribute for but is dominant for
Cancelled by the interference term, in the calculation
of radiation due to transverse fluctuations
Unruh radiation depends on physics beyond the semi-classical analysis
in our framework. Treatment based on QED will be required.

23. Problem of Radiation Dumping
Abraham-Lorentz-Dirac Force:
Energy momentum conservation
on-shell condition
Runaway Solution
Landau-Lifshitz equation:
Runawayは手で落とせる。
いろんなアプローチはあるが、きちんとした答えがわかっていない。
No back reaction for uniformly accelerated electron !?
What can we say about this problem using QED?