^{1}

^{*}

^{2}

^{*}

The universal expression for the amplitude square for any matrix of interaction M is derived. It has obvious covariant form. It allows the avoidance of calculation of products of the Dirac’s matrices traces and allows easy calculation of cross-sections of any different processes with polarized and unpolarized particles.

Amplitude square

in quantum electrodynamics. The interaction matrix M is the combination of the Dirac matrices and their products. This circumstance causes very labor-intensive calculation even if the Feynman technique of trace of matrix products calculation is used [

into account. Usually for each particular process

However, all interaction matrices have the same structure and set of permissible matrices is restricted. Any

Here

and J―scalar and pseudoscalar,

In- and out-fermions are represented by Dirac’s bispinors of the same type:

Here

Thus the possible choices for

universal expression can be derived. This expression can be used for all possible interaction matrices. Similar problem was discussed in [

cussed in [

Let us write

Here

Which leads to:

Let us take into account that for the bispinor (2)

Here

in the fermion’s reference frame, where it is at rest.

Vector

Spin tensor

has coordinates in the fermion’s reference frame:

Here

For the

This product contains 400 terms. The trace of most of them is zero. Calculations with the rest of the 164 terms lead to:

Here

The usual expression for the dot-product is used

rentz’s covariant form. This expression helps to get rid of the time-consuming necessity of trace matrices products calculations for different processes. Results of such calculations are already included into (13)-(27). The only thing we need to do is to substitute specific coefficients I,

much simpler. As an example of such simplification let us use (13)-(27) for calculation of

The electron-muon system transaction probability per unit time from the initial state to the final state can be calculated in the usual way:

Here

(15) from (13)-(27). The following quantities are zeroes

Expression (29)-(31) determines the transaction probability per unit time for the scattering of polarized electrons and muons. For the unpolarized particles one must average (31) by the initial polarizations of the particles and summing by the final polarizations of electrons and muons. It can be easily done in expression (31): all terms with

In the particular case when muon is at rest:

Also expressions for the final states density of the system and quantity

Thus, the scattering probability per unit time becomes:

Here

Here

The covariant form of the expression for the amplitude square for any interaction matrix is derived. The expression allows calculating cross-sections of any processes with polarized particles. In particular, the universal expression was used in order to calculate the transaction probability per unit time for the scattering of polarized electrons and muons. Then this result was averaged by the initial polarizations of the particles and summed by the final polarizations of electrons and muons. The final expression coincides with well-know expression for unpolarized particles (Mott cross-section).

The authors would like to thank Prof. Lukyanets S.P., Prof. Lev B.I., Prof. Tomchuk P.M. and Prof. Cooney for stimulating discussions. We also thank the Editor and the referee for their comments.

KonstantinKarplyuk,OleksandrZhmudskyy, (2015) The Universal Expression for the Amplitude Square in Quantum Electrodynamics. Journal of Modern Physics,06,2219-2225. doi: 10.4236/jmp.2015.615226