According to one of many equivalent definitions of twistors a (null) twistor
is a null geodesic in Minkowski spacetime. Null geodesics can intersect at
points (events). The idea of Penrose was to think of a spacetime point as a
derived concept: points are obtained by considering the incidence of twistors.
One needs two twistors to obtain a point. Twistor is thus a ``square root'' of
a point. In the present paper we entertain the idea of quantizing the space of
twistors. Twistors, and thus also spacetime points become operators acting in a
certain Hilbert space. The algebra of functions on spacetime becomes an
operator algebra. We are therefore led to the realm of non-commutative
geometry. This non-commutative geometry turns out to be related to conformal
field theory and holography. Our construction sheds an interesting new light on
bulk/boundary dualities.Comment: 21 pages, figure
