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Description of Polarization

  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 tex2html_wrap_inline9274. The 2tex2html_wrap_inline91662 density matrix tex2html_wrap_inline9278 is defined as
where tex2html_wrap_inline7316 denotes the Hermitian conjugate and tex2html_wrap_inline9282 is the average over a particle ensemble. Since tex2html_wrap_inline9278 is Hermitian and its trace is unity by normalization, tex2html_wrap_inline9278 can be written as
where tex2html_wrap_inline9288 is the Pauli matrices. The 3-vector tex2html_wrap_inline9290 is called polarization vector .

In the case of pure states, tex2html_wrap_inline9274 can be represented by a superposition of spin up(down) states tex2html_wrap_inline9294:
With the standard representation of the Pauli matrices
tex2html_wrap_inline9290 can be written as
and its length is unity: tex2html_wrap_inline9298. CAIN allows tex2html_wrap_inline9300 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 tex2html_wrap_inline9302 (tex2html_wrap_inline9304), the probability to be found in the spin tex2html_wrap_inline9306 state is given by tex2html_wrap_inline9308.

The polarization vector tex2html_wrap_inline9290 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) tex2html_wrap_inline9312 (normalized as tex2html_wrap_inline9314) is orthogonal to the photon momentum tex2html_wrap_inline9316. It can be represented by the components along two unit vectors tex2html_wrap_inline7628 and tex2html_wrap_inline9320 perpendicular to tex2html_wrap_inline9316. The three vectors (tex2html_wrap_inline7628, tex2html_wrap_inline9320, tex2html_wrap_inline9328) 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 tex2html_wrap_inline7924 is called the Stokes parameter. In the standard representation of the Pauli matrices, the three components of tex2html_wrap_inline7924 have the meaning

Linear polarization along the direction tex2html_wrap_inline9336 (tex2html_wrap_inline9338) or tex2html_wrap_inline9340 (tex2html_wrap_inline9342)
Circular polarization
Linear polarization along the direction tex2html_wrap_inline7628 (tex2html_wrap_inline9350) or tex2html_wrap_inline9320 (tex2html_wrap_inline9354)

Completely polarized states have tex2html_wrap_inline9356. A single photon is always in a completely polarized state. Mixed states may have tex2html_wrap_inline9358.

1ex 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 tex2html_wrap_inline9360 represents the final energy-momentum variables and w and tex2html_wrap_inline9364 are functions of tex2html_wrap_inline9360. The vector tex2html_wrap_inline9368 itself is not the final polarization. Its direction is defined by the setup of the detectors. What the term tex2html_wrap_inline9370 means is that, if one observes the spin direction tex2html_wrap_inline9302 (tex2html_wrap_inline9304), the probability to be found in the state tex2html_wrap_inline9306 is given by tex2html_wrap_inline9378.

The final energy-momentum distribution is determined by tex2html_wrap_inline9380. For given tex2html_wrap_inline9360, 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, tex2html_wrap_inline9388 represents transpose, and H is a 3tex2html_wrap_inline91663 matrix. For given tex2html_wrap_inline9394, the final energy-momentum distribution is determined by tex2html_wrap_inline9396. In a Monte Carlo algorithm, tex2html_wrap_inline9360 is decided by using random numbers according to tex2html_wrap_inline9396. Once tex2html_wrap_inline9360 is decided, the final polarization is definitely (without using random numbers) given by
This expression does not satisfy tex2html_wrap_inline9298. 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 tex2html_wrap_inline9394 but this is not correct because of the selection effect due to the term tex2html_wrap_inline9408.

The probability that a transion does not occur in a time interval tex2html_wrap_inline8540 is tex2html_wrap_inline9412, where the underlines indicates quantities integrated over the whole kinetic range of tex2html_wrap_inline9360. Consider an ensemble (one macro-particle) of N (real) particles having the polarization vector tex2html_wrap_inline9418 (tex2html_wrap_inline9420). Each of these is a unit vector tex2html_wrap_inline9422 and the average over the ensemble is tex2html_wrap_inline9424.

Let us arbitrarily take the quatization axis tex2html_wrap_inline9302. The probability in the state tex2html_wrap_inline9306 is tex2html_wrap_inline9430 and the non-transition probability is tex2html_wrap_inline9432. Therefore, the sum of the final polarization along tex2html_wrap_inline9302 over the ensemble is
The axis tex2html_wrap_inline9302 is arbitrary. Therefore, the sum of the final polarization vector is given by the above expression with tex2html_wrap_inline9302 taken away. The total number of particles without transition is tex2html_wrap_inline9440. 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.


Longitudinal spin of electron/positron (or circular polarization of photon).
Transverse spin of electron/positron (or linear polarization of photon).
tex2html_wrap_inline9528100% circular polarization only.
Not computed. (No change for existing particles, zero for created particles)

next up previous contents index
Next: Beam Parameters Up: Particle Variables Previous: Arrays for Particles

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998