Twistor Approach to Relativistic Dynamics and to the Dirac Equation — A Review

Abstract

Twistors, which can be regarded as spinors of the conformal group acting on compactified Minkowski space, form a relativistic phase space T of a massless particle with a non-vanishing helicity [1]. A relativistic extended phase space of a massive spinning charged particle [2, 3] may be regarded as “embedded” in a two twistor phase space [4, 5, 6] Tp(2) which is simply TxT with its diagonal deleted (in the sense of a symplectic reduction). We describe a procedure by which relativistic physical variables of a charged spinning and massive particle are identified as functions on Tp(2). Those representing spin of the particle and those representing its commuting and non-commuting Minkowski positions are given special attention. A non-holomorphic relativistic canonical quantization turns the so-identified relativistic physical variables on Tp(2) into the corresponding relativistic quantum mechanical Poincare covariant operators. To initiate a test of our idea, a function H, corresponding to the square of the rest mass (see (5.1)) operator arising from the Dirac equation describing a relativistic massive, charged spin ½ quantum particle in an external electromagnetic field, is used to generate a canonical flow on Tp(2) giving a reasonable Poincare covariant set of classical equations of motion. The non-holomorphic (re-)quantization turns H into a new Poincare scalar quantum operator. Its eigenvalue equation valid for any value of the quantized spin corresponds and mimics the second order formulation of the Dirac equation (with minimally coupled external electromagnetic field). The present paper is essentially a review of previous presentations on the subject [2, 3, 4, 5, 6, 7]. Some new work, at an undeveloped introductory level, appears in sections 5 and 6.