Exact Foldy-Wouthuysen transformation for gravitational waves and magnetic field background

1. Exact Foldy-Wouthuysen transformation for gravitational waves and magnetic field background UNIVERSIDADE FEDERAL DE JUIZ DE FORA DEPARTAMENTO DE FÍSICA Bruno Gonçalves This talk is based on the paper B. GONCALVES, Yu. N. Obukhov, I. L. Shapiro, Exact Foldy-Wouthuysen transformation for gravitational waves and magnetic field background. PRD 75 (2007) 124023.

2. This talk is divided in the following topics: • Introduction • Brief Review of the Foldy-Wouthuysen transformation • The exact Foldy-Wouthuysen transformation • Gravitational waves and magnetic field • Equations of motion • Conclusions

3. Introduction • The main purpose of this work is to derive non-relativistic equations of motion for a particle which satisfies Dirac equation in the presence of gravitational waves and magnetic field. • We considered the Dirac Hamiltonian for this case. In order to extract physical information, it is necessary to perform the Foldy-Wouthuysen (FW) transformation. • We developed a method to perform the exact FW transformation for many kind of fields, including the case of gravitational waves and magnetic field. • Within this approach we are able to treat the FW transformation in many different theories as particular cases of a single general transformation.

4. Foldy-Wouthuysen transformation • Let us take the Dirac equation in the form • Now we suppose the solution has two components • And get the equations

5. Foldy-Wouthuysen transformation • The problem of the last equations is that there is a mixture of fields φ e χ. The physical interpretation of the equations in this case is difficult. • In order to solve this problem, one can perform the well known Foldy-Wouthuysen transformation. This transformation is described in • The first step is to define even and odd operators. An odd operator anticommutes with β, it is mixing the two components of the wave function. Even op. commutes with β and does not produce mixing.

6. In order to perform the FW transformation, we write the Hamiltonian in the form • where ε are even operators andO are oddones. • The usual perturbative FW transformation is an expansion in powers of (1/m). The transformed (purely even) Hamiltonian in the order O(1/m3) has the form

7. Exact Foldy-Wouthuysen transformation (EFW) • The EFW transformation has been described in the paper • The authors applied the EFW transformation to many cases such as • Free particle • Particle in the presence of magnetic field • Particle with anomalous magnetic moment in a magnetostatic field • We used these results as references to our work and moreover could obtained all these cases from a unique calculation. These results can be used as a test for our original calculation. • Many others papers on EFW: Oliveira & Tiomno. Nouvo Cimento 24 (1962) 672 ; A. G. Nikitin. J. Phys. A: Math. Gen.A31 (1998) 3297 ; ......

8. Exact Foldy-Wouthuysen transformation • To perform the EFW, one has to verify if • where ηis the involution operator • The next step is to obtain H2 • and the final step is use the formula • In order to extract the square root of operator H, one has to expand this in power series of some parameter of the theory.

9. Exact Foldy-Wouthuysen transformation (EFW) • The advantage to use the EFW instead of using FW is that by the end one has an exact solution for the Hamiltonian. • The transformed Hamiltonian can be analyzed directly or expanded in some parameter. • In the second case, there is the possibility of getting some unexpected terms. • In the particular case of gravitation, we have an example when the EFW result has shown a relevant difference with the one obtained within the perturbative approach Yu.N. Obukhov, Spin, Gravity and Inertia, PRL 86 (2001)

10. Gravitational waves and magnetic field • We take the Dirac equation • Then we write the in the following way • in order to work with the gamma matrices of flat space, we introduce now the verbein defined by the relations

11. Gravitational waves and magnetic field • The next steps are the minimal generalization • And the introduction of spinor connection • Where and

12. Gravitational waves and magnetic field • Then we perform the following calculations

13. Gravitational waves and magnetic field • We used the gravitational wave metric in the form • After the calculations we get the Hamiltonian

14. Gravitational waves and magnetic field • Let us rewrite the Hamiltionian in a more convenient form • where the new terms are • and

15. The operator H2 has the form • where

16. Particular cases • The last result needs to be tested in some form. It includes some particular cases: • Free particle: • This is the simplest case where reduces to and there are no interactions terms because • Particle in presence of magnetic field: • In this case we also havebut now there is the interaction term because • Particle with anomalous magnetic moment in a magnetostatic field: • This is the cases where and

17. Equations of motion • Let`s study the case where the wave has only one polarization (u=0). If we do this, the expressions became much more simple and can be written in the form • where • We also can see that there is no need anymore of notations and . And we have the identities and • We are going to use

18. Expand the Hamiltonian in parameter 1/mc2 and suppose • Let us find the final Hamiltonian for the field φ • In order to find the Equations of Motion for the Spinning Particle we introduce the commutation relations • After calculating commutators, omit the terms that vanish when • The equations obtained in this way are interpreted as semi-classical equations of motion for a spinning particle .

19. Finally, • If we combine the first two equations we get • This is an analog of the Lorentz force in the presence of a gravitational wave.

20. Conclusions • We obtained the operator H2 for a general case, using our notations with Kim and gm. This result includes many particular cases. Every Hamiltonian that can be written using our notations admits the EFW transformation. • Another important equation is the Lorentz force in the presence of gravitational wave. The first term of this equation shows an interesting effect: there is a mixture between the amplitude of the gravitational wave and the external magnetic field • This fact gives the idea to study the motion of a Dirac particle in a region of the space where there is a GW plus a very strong magnetic field. The last can increase the effect of the gravitational wave on the motion of the particle.