Introduction: One of the comprehensive developments of the idea of spinorization of the phase space has been achieved in the twistor theory. Using the condition of masslessness the twistor approach established a deep relation between phase space variables of massless spinning particles and spinor wave functions. In this formalism the equations of motion of massless particles have been transformed into algebraic equations over oscillator ladder operators of the helicity.
The success of the twistorial description of massless particles has inspired several authors to apply this tool to construct also a twistorial representation of the phase space of massive spinning particles. Dirac’s bispinors contain two independent Weyl’s spinors and in that sense keeps two times more information. This induces the idea to construct a dynamics of massive spinning systems using pairs of twistors corresponding to (two) massless particles. It turns out that a (double) phase space of classical massless object may serve as a building block for the construction of the irreducible phase space of a massive spinning particle. In some sense, the massive spinning particle obtained by the reduction procedure may be regarded as a bound (confined) system of two directly interacting massless spinning constituents.
The development of this approach in the quantum case requires the transformation of the Dirac equation into an algebraic system of equations over twistor variables. In the best approach to this program one has to exchange the mass parameter in the Dirac equation by two mutually complex conjugated values. This complex value arises as a simple consequence of the bispinor representation of momentum in the Dirac equation. The same result is obtained using the bispinor representation of the momentum in the basis of Dirac-gamma matrices, in that case starting from Pauli-Fierz identities, we obtain a Dirac like equation in six-dimensional momentum space, two components corresponding to the complex mass parameter. (Let us note, that these components accept another interpretation if bispinors are substituted by twistor coordinates.)
In this paper, we examine both representations: the spinorial- with well known van der Waerden symbols, and the vectorial expressed in terms of Dirac-gamma matrices. In this way the bridge between the twistorial description of massive particle and the concept of screws is established.
In Section 2, for the convenience of the reader, we simply summarize the assumptions and definitions used in the twistor’s theory of massless particles.
In Section 3, we build the twistor phase space for the massive particle. We show that the twistor representation of momentum demands a modification of the Dirac equation introducing a complex mass parameter.
In Section 4, the equation of motion of Lorentz-Bargmann-Michel-Telegdi in twistorial phase space is deduced in full detail.
The paper should be considered as an analysis and comparison of otherwise known formulations.