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.
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