Exact Foldy-Wouthuysen transformation for gravitational waves and magnetic field

backgroundBruno Gonçalves

UNIVERSIDADE FEDERAL DE JUIZ DE FORADEPARTAMENTO DE FÍSICA

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.

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

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.

Foldy-Wouthuysen transformation

Let us take the Dirac equation in the form

Now we suppose the solution has two components

And get the equations

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.

In order to perform the FW transformation, we write the Hamiltonian in the form

where ε are even operators and O are odd ones.

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

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 ; ......

Exact Foldy-Wouthuysen transformation To perform the EFW, one has to verify if

where η is 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.

,0},{ H

5i

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)

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

.0)( m

,aae

.,,, abbaa

aaaabba eegeegeeee

Gravitational waves and magnetic field

The next steps are the minimal generalization

And the introduction of spinor connection

Where and

,2

ababi

).(2 abbaabi

.)(41)(

41)(

41 c

cabbaabbabaabab eeeeeeeeeeeeee

Gravitational waves and magnetic field Then we perform the following calculations

Gravitational waves and magnetic field We used the gravitational wave metric in the form

After the calculations we get the Hamiltonian

Let us rewrite the Hamiltionian in a more convenient form

where the new terms are

and

Gravitational waves and magnetic field

The operator H2 has the form

where

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 have but now there is the interaction

term because

Particle with anomalous magnetic moment in a magnetostatic field: This is the cases where and

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

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 .

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.

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.