# Presentation: 388542

## Relationship Between Unruh and Sokolov-Ternov Effects*

**Douglas Singleton**
**CSU Fresno**
***APS California Section Meeting***
***LBNL, CA October 26******th******-27******th****** , 2007***
****work in collaboration with E.T. Akhmedov. To be published in IJMPA and JETP Letts.***

## Probability Decay Rate/time

**For a two level system –****Δ****E= ****E****2**** – E****1 **** > 0**** ****–  the transition probability per unit **
**	****w****-**** = excitation and w****+**** = de-excitation **
**G[x(t-****τ****/2), ****x(t+****τ****/2)]**** ****is the Wightman function and x(t) is the spacetime path of the detector. For a scalar field **
**This measures correlations between fluctuations of the scalar field a two vacuum spacetime points. **

## Linear Unruh Effect

**A linearly accelerating observer will see a **_**thermal **_**heat bath with temperature ****T=a****ħ****/2****π****ck****B**** = ****a/2****π****. The path ****x(t)=(1/a)[sinh(at), cosh(at), 0, 0]**
**Semi-circular contour integrations closed in the upper (lower) complex half plane gives**
**	****	****	****	****      **

## Sokolov-Ternov Effect

**For circularly moving charged particles with spin in a magnetic field one naively expects the particles would become completely polarized due to spin flip radiation.  However the spin flips can go both ways both increasing and decreasing the spin energy**** **
- [Sokolov and Ternov, Sov.-Phys. Dokl.,**8 **1203 (1964); J.D. Jackson Rev, Mod. Phys., **48**, 417 (1976**)] **
**The **_**lab frame**_** spin flip calculation yields **[E. Akhmedov and D. Singleton to be published IJMPA]
**Where ****α****=(g-2)/2≈****Δ****E/a<<1 i.e. is not set to zero here**

## Sokolov-Ternov Effect

**With ****α****=0 one finds transition probabilities and polarization**
**To next order in ****α**** one finds **

## Circular Unruh Effect 

**Consider the scalar field detector undergoing circular motion **
**x(****τ****)=(****γτ****, Rcos[****γωτ****],  Rcos[****γωτ****], 0) ; ****γ****=1/√1-(R****ω****)****2****  ; ****τ****=proper time**
**One can not calculate the Wightman function exactly as in the linear case but approximately**

## Circular Unruh Effect

**Using this approximate, circular Wightman function one finds excitation (de-excitation) probabilities**
**And equilibrium population of upper to lower level**

## Circular Unruh = Sokolov-Ternov

**Comparing the probabilities transition rates (w****±****) for the two calculations and the equilibrium population (P****0****) for the two cases we find that they are of the same form since ****Δ****E/a≈(g-2)/2=****α**

## Conclusions

**The detector rest frame (circular) Unruh effect can be said to be verified via the observed, lab frame Sokolov-Ternov effect.**
** ****The characteristic signal of the Unruh effect is the appearance of the characteristic exponential in he probabilities (w****±****). These characteristic exponentials ******** 1 in the case of the Sokolov-Ternov effect since g≈2 or ****α≈****0.**
**The differences between the circular Unruh calculation and Sokolov-Ternov effect have to do with scalar particle coupling versus spinor E&M particle coupling, and rest frame versus lab frame.**
**The difference between our work and previous work **[Bell and Leinaas, NPB **212**, 131 (1983); NPB **284**, 488 (1987)]** lies in the emphasis that exactly Circular Unruh = Sokolov-Ternov. See also **[D. Barber and S. Mane PRA **37**, 456 (1988); DESY-87-049]

## Acknowledgments

**The work was supported in part by grants from the Agency of Atomic Energy of the Russian Federation (INTAS 03-51-5460), Russian Foundation for Basic Research (04-02-16880), and a CSU Fresno International Activities Grant.**