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.

1ex
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

- Linear polarization along the direction () or ()
- Circular polarization
- Linear polarization along the direction
() or ()

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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 [3], 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
is

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

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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.

- L
- Longitudinal spin of electron/positron (or circular polarization of photon).
- T
- Transverse spin of electron/positron (or linear polarization of photon).
- L
- 100% circular polarization only.
- N
- Not computed. (No change for existing particles, zero for created particles)
- -
- Irrelevant.

Thu Dec 3 17:27:26 JST 1998