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

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

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

(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.

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 …

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

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

Equation of fluctuations 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 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 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:

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 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.

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?