By Mike Swarbrick Jones
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Additional info for A Binomial Congruence
The transition from a certain state / to a certain state / w i l l be characterized by the quantity Sfi. A set of N2 quantities Sfi forms a scattering matrix, or S-matrix. If all interactions are switched off, then the S-matrix becomes the unit matrix / : each of the states goes over to itself. Hence physical processes take place if the Γ-matrix, which is determined by the relation S = I + iT, differs from zero. In what follows the quantity Mfi of Tfi = (2n)*d*(j>f-Pi)Mfi9 will be called the amplitude of the process.
The decay probability, expressed by means of the amplitude M, has the form (see Chapter 2) Λ4 \Μ\2 dk dq1 ™ , 2Εμ (2π)32Ε'(2π)32ω1 d\V=(2ny^— dq2 (2π) 3 2ω 2 xö4(k + q± + q2 - / ? ) , where k — (E, k) is the 4-momentum of the electron, tfi = ( ω ι? qi) is the 4-momentum of the antineutrino, qi = ( ω 2 5 qi) is the 4-momentum of the neutrino, p = (Εμ, p) is the 4-momentum of the muon. The four-dimensional delta-function 4 <5 (7c + #! + q2 - p) = δ(Ε + ωχ + ω2 - Εμ) d(k + qx + q2 - p) means that the total energy and each of the total momentum components are conserved.
It behaves as a vector not only under Lorentz transformations but also under reflection. e. it is an axial vector, like the magnetic field. So, the weak current jw is not invariant with respect to P-inversion: some of the terms consituting it change sign under P-inver sion, whereas the other ones do not. Since the weak interaction Lagrangian has the form Lw ~ jtjw, Lw is also not invariant with respect to P-inversion: products of the type V*VaandA*A^ involved in Lw do not change sign under P-inversion, whereas products of the type V£Aa and A* ¥Ά do.
A Binomial Congruence by Mike Swarbrick Jones