Pauli-Lubanski vector

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,... 

In Cartesian notation, the Pauli-Lubanski vector of particle theory becomes a 4-vector of the classical electromagnetic field... 

The basic ansatz is that there is a field vector analogous to the Pauli-Lubanski vector of particle physics, a field vector defined by... 

In the rest frame of a photon with mass, the held and particle Pauli-Lubanski vectors are respectively... 

From this analysis, it is concluded that the B<3> component is identically nonzero, otherwise all the field components vanish in the B cyclic theorem (812) and Lie algebra (809). If we assume Eq. (803) and at the same time assume that B(3) is zero, then the Pauli-Lubanski pseudo-4-vector vanishes for all A0. Similarly, in the particle interpretation, if we assume the equivalent of Eq. (803) and assume that J(3) is zero, the Pauli-Lubanski vector vanishes. This is contrary to the definition of the helicity of the photon. Therefore, for finite field helicity, we need a finite B(3). 

As argued, infinitesimal field generators appear as a by-product of this novel quantization scheme, so that B° is rigorously nonzero from the symmetry of the Poincare group and the B cyclic theorem is an invariant of the classical field. The basics of infinitesimal field generators on the classical level are to be found in the theory of relativistic spin angular momentum [42,46] and relies on the Pauli-Lubanski pseudo-4-vector ... 

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