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July 2010, Azumino
Thermalization and Unruh Radiation for a
Uniformly Accelerated Charged Particle
張 森
Sen Zhang
S. Iso and Y. Yamamoto
Unruh effect and Unruh radiation
Vacuum:
～
～
Bogoliubov transformation
Vacuum
thermal state
for inertial observer
for accelerating observer
Unruh Effect:
Hawking Radiation:
Vacuum of free falling observer
Asymptotic observer
Unruh effect and Unruh radiation
Unruh Effect:
Vacuum
thermal state
for inertial observer
for accelerating observer
Unruh Temperature:
(107K)
How to See?
Unruh Radiation: radiation due to fluctuation of electron
Chen, Tajima ‘99
Schutzhold, Schaller, Habs ‘06
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.
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 …
Particle
Stochastic Equation
Real Process
Random motion
Focus on Particle Motion
absorption and radiation
Brownian motion
Stochastic Equation
Scalar for simplicity:
Equation of motion:
Solution:
fluctuation
dissipation
Effective equation for a particle interacting with some quantum field
Non-local
expansion:
P. R. Johnson and B. L. Hu
Renormalized mass
Self-force from Larmor radiation (ALD)
Fluctuation around uniformly accelerated motion for transverse direction:
Acceleration (1 keV)
Equation of fluctuations
Transverse direction
Longitudinal direction
Transverse Fluctuation
Neglecting
term:
Relaxation Time:
Including
term:
Two point function:
Derivative expansion
Equipartition Theorem
Equipartition theorem
thermal
Action:
Solution:
Stochastic equation:
Equipartition theorem
Universal
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
Radiation
Interference effect
Nonzero
What Chen-Tajima calculated
Depend on
Inteference Effect - Unruh Detector
2D: no radiation
Raine, Sciama, Grove 91’s
Unruh Detector
4D: radiate during thermalization,
but no radiation if the detector
state is thermal state at first
Shih-Yuin Lin & B. L. Hu
Eom:
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
…
UV divergence
Four poles
Photon travelling time in
Compton wave length
: does not contribute for
Relaxation time
(thermalization time)
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.
Problem of Radiation Dumping
Abraham-Lorentz-Dirac Force:
Energy momentum conservation
on-shell condition
Runaway Solution
Landau-Lifshitz equation:
No back reaction for uniformly accelerated electron !?
What can we say about this problem using QED?