Covariant spin vector

We consider the collision of longitudinally polarized electrons or muons (helicity A = ) with a polarized nucleon described by a covariant spin vector S/j,. 

Because the proton is polarized we need to slightly generalize the number densities nj(/c) used previously. Thus we introduce Uf K, A S) d K as the number of flavour / quarks with momentum in the range k —> K -l-d/c and with helicity A inside a proton of momentmn P and covariant spin vector [see eqn (15.6.2)]. After some algebra one finds for the forward hadronic matrix elements. 

For any octet of axial-vector currents Aj with matrix elements taken between a spin-half baryon octet whose polarization is specified by the covariant spin vector 5, one has an expression analogous to (A3.24) ... 

The last feature requires a new definition and formulation of SSP or FM in relativistic systems since spin is no more a good quantum number in relativistic theories spin couples with momentum and its direction changes during the motion. It is well known that the Pauli-Lubanski vector W1 is the four vector to represent the spin degree of freedom in a covariant form,... 

Equations (72) and (73) show that the spin-affine connection QM and vector potential behave similarly under a gauge transformation. The relation between covariant derivatives has been developed in Section III. 

The squared bracket in Eq. (36) denotes the behavior of the quaternion field with respect to its vector degrees of freedom alone. The covariant derivatives of the two-component spinor variables are as follows v(/ p = (0p + Op) ]/ and the spin-affine connection has two alternative (equivalent) forms [17] ... 

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