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We propose a new world-line Lagrangian model of the D=4 massless relativisticparticle with continuous spin and develop its twistorial formulation. Thedescription uses two Penrose twistors subjected to four first classconstraints. After the first quantization of the world-line twistorial model,the wave function is defined by an unconstrained function on thetwo-dimensional complex affine plane. We find the twistor transform thatdetermines the space-time field of the continuous spin particle through thecorresponding twistor one, which plays the role of a prepotential. It is shownthat this space-time field is an exact solution of the space-time constraintsdefining the irreducible massless representation of the Poincare group withcontinuous spin.

Model of massless relativistic particle with continuous spin and its twistorial description