In most applications, one is interested in the helicity states. Therefore, one possible way of expressing the electron/positron spin is to store the information whether each macro-particle is in the helicity h=+1 state of -1 state. The unpolarized state is represented by an equal number of macro-particles with h=+1 and -1. The spin may flip at the interactions such as laser-Compton scattering and beamstrahlung.
However, this simple way cannot be applied to our case because, for example, a pure transverse polarization may become longitudinal during the precession in a magnetic field (beam-beam field or external field). In order to include such classical precession effects, the phase relation between the up and down components of the spinor is important.
This problem can be solved by using the density matrix. Let us express
an electron(positron) state by a two-component spinor .
The 22 density matrix is defined as
where denotes the Hermitian conjugate and is the average over a particle ensemble. Since is Hermitian and its trace is unity by normalization, can be written as
where is the Pauli matrices. The 3-vector is called polarization vector .
In the case of pure states, can be represented by a superposition
of spin up(down) states :
With the standard representation of the Pauli matrices
can be written as
and its length is unity: . CAIN allows so that each macro-particle is in a mixed state, representing an ensemble of particles having almost the same energy-momentum and space-time coordinate.
If one observes the particle spin with the quantization axis (), the probability to be found in the spin state is given by .
The polarization vector obeys the Thomas-BMT equation (33) in the absense of quantum phenomena.
A similar way is used for photon polarization, too. The polarization
vector (3-vector) (normalized as )
is orthogonal to the photon momentum .
It can be represented by the components along two unit vectors
and perpendicular to .
The three vectors (, ,
) form a right-handed orthonormal basis.
The density matrix is defined as
This is Hermitian with unit trace as in the case of electron density matrix so that it can be written as
The 3-vector is called the Stokes parameter. In the standard representation of the Pauli matrices, the three components of have the meaning
In any process involving polarizations, the transition rate (or crosssection)
is given by multiplying the density matrices and by taking the trace. Therefore,
the expressions for the rates are bilinear forms for each polarization vector,
initial/final electron/positron or photon. The final polarization needs
some comments. The transition rate is written in general as
where represents the final energy-momentum variables and w and are functions of . The vector itself is not the final polarization. Its direction is defined by the setup of the detectors. What the term means is that, if one observes the spin direction (), the probability to be found in the state is given by .
The final energy-momentum
distribution is determined by . For given , the final
polarization vector is (see , page 254)
Now, consider a process involving initial and final electrons, summing over
other possible particles. The transition rate is written as
where the subscripts i and f denote initial and final variables, represents transpose, and H is a 33 matrix. For given , the final energy-momentum distribution is determined by . In a Monte Carlo algorithm, is decided by using random numbers according to . Once is decided, the final polarization is definitely (without using random numbers) given by
This expression does not satisfy . If one does not allow a macro-particle in a mixed state, one has to choose a pure state by using random numbers.
The macro-particles which did not make transition must carefully be treated. One might say their final polarization is equal to but this is not correct because of the selection effect due to the term .
The probability that a transion does not occur in a time interval is , where the underlines indicates quantities integrated over the whole kinetic range of . Consider an ensemble (one macro-particle) of N (real) particles having the polarization vector (). Each of these is a unit vector and the average over the ensemble is .
Let us arbitrarily take the quatization axis . The probability
in the state is
and the non-transition probability is
Therefore, the sum of the final polarization along over the ensemble
The axis is arbitrary. Therefore, the sum of the final polarization vector is given by the above expression with taken away. The total number of particles without transition is . Thus, the final polarization vector is
The average final polarization over the whole ensemble, with and without transition, is then given by
If one can ignore the change of energy-momentum during the transition, the evolution of the polarization is described by the differential equation
3ex The present version of CAIN does not include all the polarization effects. The following table shows what effects are included. In any case, the correlation of polarization between final particles is not taken into account.