The universal expression for the amplitude square in quantum electrodynamics

The universal expression for the amplitude square |u_f M u_i|^2 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.


INTRODUCTION
Amplitude square |ū f M u i | 2 calculations are necessary in order to find probability transactions for any processes 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 [1]. Especially labor-intensive calculations are when polarization of in-and out-particles is taken into account. That is why the such calculations often do not take particle polarization into account. Usually for each particular process |ū f M u i | 2 is calculated separately. There are very many papers devoted to calculation of |ū f M u i | 2 for a particular processes.
In-and out-fermions are represented by Dirac's bispinors of the same type: Here pς 1 = p x σ 01 +p y σ 02 +p z σ 03 , nς 2 = n x σ 23 +n y σ 31 + n z σ 12 , n three dimensional unit spin pseudo-vector in particle's own reference frame, n 0 = 0, n 1 = −n x , n 2 = −n y , n 3 = −n z . In particles's own reference frame it's linear momentum is zero. For the bispinor u the relativistically covariant normalizationūu = mc is used.
Thus the possible choices for |ū f M u i | 2 are restricted. So, for all of them |ū f M u i | 2 can be calculated and a universal expression can be derived. This expression can be used for all possible interaction matrices. Such an expression was derived in [2] but |ū f M u i | 2 is expressed through the three dimensional quantities in laboratory reference frame. In most cases it is preferable to have Lorentz's covariant expression which is derived below.

COVARIANT EXPRESSION FOR AMPLITUDE SQUARE
Let us write |ū f M u i | 2 as (ū f M u i )(ū f M u i ) * and use the equality: (4) Which leads to: Let us take into account that for the bispinor (2) Here s α spin pseudo-vector n α coordinates in the reference frame where a fermion has momentum p α . Vector s α has coordinates s 0 = 0, s 1 = n x , s 2 = n y , s 3 = n z , n · n = 1 (7) in the fermion's reference frame, where it is at rest.
(10) Here ε αβµν entirely anisymmertical tensor, ε 0123 = 1, the same tensor ε 0123 = −1. Note that For the |ū f M u i | 2 with a help of (5) and (6) we have: This product contains 400 terms. The trace of most of them is zero. Calculations with the rest of the 164 terms leads to: Hereς αβ dual to the ς αβ tensor Also usual designation for the dot product is used (a·b) = a α b α . Expression (13)-(27) determines the amplitude square |ū f M u i | 2 for any quantum electrodynamics process with polarized particles. It has obviously Lorentz'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, V α , W α , F αβ , J for the interaction matrix M into (13)-(27). It is essentially reducing and simplifing calculations especially for the polarized particles. Expression (13)- (27) is very cumbersome. This is our price for it's universality. Note that for the specific processes many of the quantities I, V α , W α , F αβ , J are zero so that only some fragments of the (13)-(27) are used. These fragments are marked by different numbers in (13)-(27). In each particular case expression (13)-(27) becomes much simpler. As an example of such simplification let us use (13)- (27) for calculation of |ū f M u i | 2 for an electron-muon collision.

ELECTRON-MUON COLLISION
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 α fine structure constant, V normalization volume, which contains one electron and one muon, ρ(E) final states density of the system with total energy E = cP 0 = c(p i0 + P i0 ) = c(p f 0 + P f 0 ) and 3D impulse P = p f + P f : dΩ solid angle, through the which electron is scattered.
In (29)-(30) for electron (muon) quantities lower-case (upper-case) letters are used. Expression |ū f γ α u iŪf γ α U i | 2 can be written as |ū f γ α u i V α | 2 or |v αŪ f γ α U i | 2 , where V α =Ū f γ α U i or v α =ū f γ α u i . Amplitude square |ū f γ α u iŪf γ α U i | 2 can be obtained using only fragment (15) from (13)-(27). The following quantities are zeroes I = 0, W α = 0, F αβ = 0, J = 0. Then we need to contract tensor coefficient in front of the V α V β , calculated for the electron, with the similar tensor coefficient calculated for the muon. Note that the real parts of these coefficients are symmetrical tensors and the imaginary parts are anti-symmetrical tensors. That is why we must contract them separately and add the contraction results: